Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UF00090188/00001
## Material Information- Title:
- Nonlinear gap and Mindlin shell elements for the analysis of concrete structures
- Series Title:
- Nonlinear gap and Mindlin shell elements for the analysis of concrete structures
- Creator:
- Ahn, Kookjoon,
- Place of Publication:
- Gainesville FL
- Publisher:
- University of Florida
- Publication Date:
- 1990
## Subjects- Subjects / Keywords:
- Coordinate systems ( jstor )
Degrees of freedom ( jstor ) Eggshells ( jstor ) Engineering ( jstor ) Matrices ( jstor ) Shape functions ( jstor ) Shear stress ( jstor ) Stiffness ( jstor ) Stiffness matrix ( jstor ) Subroutines ( jstor )
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Ahn Kookjoon. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 023318537 ( alephbibnum )
23011869 ( oclc )
## UFDC Membership |

Downloads |

## This item has the following downloads:
UF00090188 ( .pdf )
UF00090188_00001.pdf 00006.txt 00199.txt UF00090188_00001_pdf.txt 00206.txt 00026.txt 00047.txt 00080.txt 00058.txt 00105.txt 00060.txt 00054.txt 00092.txt 00233.txt 00051.txt 00177.txt 00231.txt 00055.txt 00061.txt 00153.txt 00162.txt 00137.txt 00205.txt 00183.txt 00067.txt 00142.txt 00181.txt 00237.txt 00037.txt 00033.txt 00215.txt 00100.txt 00224.txt 00096.txt 00145.txt 00108.txt 00174.txt 00062.txt 00002.txt 00112.txt 00146.txt 00243.txt 00076.txt 00057.txt 00148.txt 00182.txt 00158.txt 00087.txt 00066.txt 00186.txt 00073.txt 00075.txt 00194.txt 00007.txt UF00090188_pdf.txt 00127.txt 00235.txt 00027.txt 00063.txt 00114.txt 00221.txt 00091.txt 00071.txt 00120.txt 00059.txt 00223.txt 00136.txt 00150.txt 00042.txt 00012.txt 00201.txt 00156.txt 00125.txt 00023.txt 00167.txt 00039.txt 00218.txt 00122.txt 00163.txt 00133.txt 00210.txt 00072.txt 00081.txt 00020.txt 00038.txt 00213.txt 00188.txt 00179.txt 00193.txt 00151.txt 00101.txt 00011.txt 00238.txt 00190.txt 00160.txt 00034.txt 00010.txt 00083.txt 00157.txt 00143.txt 00024.txt 00110.txt 00093.txt 00117.txt 00247.txt 00234.txt 00152.txt 00184.txt 00022.txt 00204.txt 00119.txt 00189.txt 00168.txt 00111.txt 00154.txt 00248.txt 00207.txt 00019.txt 00203.txt 00126.txt 00135.txt 00172.txt 00191.txt 00170.txt 00220.txt 00246.txt 00169.txt 00070.txt 00032.txt 00138.txt 00068.txt 00241.txt 00107.txt 00217.txt 00128.txt 00140.txt 00212.txt 00064.txt 00008.txt 00035.txt 00095.txt 00200.txt 00090.txt 00196.txt 00016.txt 00222.txt 00116.txt 00118.txt 00005.txt 00103.txt 00208.txt 00166.txt 00197.txt 00017.txt 00139.txt oai_xml.txt 00178.txt 00097.txt 00063a0001.txt 00050.txt 00121.txt 00085.txt 00195.txt 00018.txt 00227.txt 00098.txt Copyright.txt 00209.txt 00113.txt 00052.txt 00144.txt 00084.txt 00069.txt 00245.txt 00134.txt 00239.txt 00004.txt 00088.txt 00187.txt 00240.txt 00029.txt 00175.txt 00226.txt 00074.txt 00132.txt 00077.txt 00219.txt 00041.txt 00236.txt 00053.txt 00164.txt 00198.txt 00229.txt 00104.txt 00185.txt 00115.txt 00078.txt 00149.txt 00141.txt 00131.txt 00021.txt 00028.txt 00216.txt 00031.txt 00009.txt 00230.txt 00046.txt 00147.txt 00044.txt 00013.txt 00228.txt 00001.txt 00109.txt 00225.txt 00099.txt 00102.txt 00180.txt 00040.txt 00129.txt 00094.txt 00159.txt 00014.txt 00086.txt 00242.txt 00232.txt 00130.txt 00049.txt 00079.txt 00048.txt 00165.txt 00211.txt 00123.txt 00065.txt 00106.txt 00214.txt 00015.txt 00056.txt 00192.txt 00045.txt 00161.txt 00171.txt 00176.txt 00173.txt 00202.txt 00030.txt 00244.txt 00089.txt 00082.txt 00155.txt 00036.txt 00124.txt 00043.txt 00025.txt 00003.txt |

Full Text |

NONLINEAR GAP AND MINDLIN SHELL ELEMENTS FOR THE ANALYSIS OF CONCRETE STRUCTURES By KOOKJOON AHN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA U::i: L "At; 7 OF FLORIN LIfRPMR17 1990 ACKNOWLEDGEMENTS I would like to express my deep gratitude to professor Marc I. Hoit for his invaluable guidance and support. I also thank professors Clifford 0. Hays, Jr., Fernando E. Fagundo, John M. Lybas, and Paul W. Chun for being on my committee. I also express my gratitude to professor Duane S. Ellifritt for his help as my academic advisor at the start of my Ph.D. program. Thanks are also due to all the other professors not mentioned above and my fellow graduate students, Alfredo, Jose, Mohan, Prasan, Prasit, Shiv, Tom, Vinax, and Yuh-yi. Finally, I am thankful to every member of my family, especially my wife and son, for their patience and support in one way or another. The work presented in this dissertation was partially sponsored by the Florida Department of Transportation. TABLE OF CONTENTS page ACKNOWLEDGEMENTS ...................................... ii ABSTRACT ...................................... ........ v CHAPTERS 1 INTRODUCTION ....................... ............. 1 1.1 General Remarks ............................ 1 1.2 Link Element ............................... 2 1.3 Shell Element .............................. 5 1.4 Literature Review .......................... 5 2 GENERAL THEORIES OF NONLINEAR ANALYSIS .......... 13 2.1 Introduction ............................... 13 2.2 Motion of a Continuum ...................... 14 2.3 Principle of Virtual Work .................. 16 2.4 Updated Lagrangian Formulation ............. 18 2.5 Total Lagrangian Formulation ............... 22 2.6 Linearization of Equilibrium Equation ...... 26 2.7 Strain-Displacement relationship Using von Karman Assumptions ............... 28 3 THREE-DIMENSIONAL LINK ELEMENT .................. 34 3.1 Element Description ...................... 34 3.2 Formation of Element Stiffness ............. 43 3.3 Solution Strategy .......................... 51 3.4 Element Verification ....................... 52 4 LINEAR SHELL ELEMENT ............................ 59 4.1 Introduction ............................... 59 4.2 Formulation of Shape Functions ............. 59 4.3 The Inverse of Jacobian Matrix ............. 64 4.4 Membrane Element ........................... 66 4.5 Plate Bending Element .................... 73 5 NONLINEAR SHELL ELEMENT ........................ 92 5.1 Introduction ............. ................. 92 5.2 Element Formulation ........................ 93 iii 5.3 Finite Element Discretization .............. 100 5.4 Derivation of Element Stiffness ............ 113 5.5 Calculation of Element Stiffness Matrix .... 115 5.6 Element Stress Recovery .................... 119 5.7 Internal Resisting Force Recovery ......... 122 6 NONLINEAR SHELL ELEMENT PERFORMANCE ............ 126 6.1 Introduction ............................... 126 6.2 Large Rotation of a Cantilever ............ 126 6.3 Square Plates .............................. 133 7 CONCLUSIONS AND RECOMMENDATIONS.................. 143 APPENDICES A IMPLEMENTATION OF LINK ELEMENT .................. 146 B IMPLEMENTATION OF LINEAR SHELL ELEMENT ......... 170 C IMPLEMENTATION OF NONLINEAR SHELL ELEMENT ....... 219 REFERENCES ............................................ 230 SUPPLEMENTAL BIBLIOGRAPHY ............................ 238 BIOGRAPHICAL SKETCH ................................... 240 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR GAP AND MINDLIN SHELL ELEMENTS FOR THE ANALYSIS OF CONCRETE STRUCTURES By KOOKJOON AHN August, 1990 Chairman: Marc I. Hoit Major Department: Civil Engineering Segmental post-tensioned concrete box girders with shear keys have been used for medium to long span bridge structures due to ease of fabrication and shorter duration construction. Current design methods are predominantly based on linear elastic analysis with empirical constitutive laws which do not properly quantify the nonlinear effects, and are likely to provide a distorted view of the factor of safety. Two finite elements have been developed that render a rational analysis of a structural system. The link element is a two-dimensional friction gap element. It allows opening and closing between the faces of the element, controlled by the normal forces. The Mindlin flat shell element is a combination of membrane element and Mindlin plate element. This element considers the shear responses along the element thickness direction. The shell element is used to model the segment itself. The link element is used to model dry joints and has shown realistic element behavior. It opens under tension and closes under compression. The link element has shown some convergence problems and exhibited a cyclic behavior. The linear Mindlin shell element to model the concrete section of the hollow girder showed an excellent response within its small displacement assumption. The nonlinear Mindlin flat shell element has been developed from the linear element to predict large displacement and initial stress (geometric) nonlinearities. The total Lagrangian formulation was used for the description of motion. The incremental-iterative solution strategy was used. It showed satisfactory results within the limitation of moderate rotation. Three areas of further studies are recommended. The first is the special treatment of finite rotation which is not a tensorial quantity. The second is the displacement dependent loadings commonly used for shell elements. The third is the material nonlinearity of concrete which is essential to provide realistic structural response for safe and cost effective designs. CHAPTER 1 INTRODUCTION 1.1 General Remarks In the past few decades segmental post-tensioned concrete box girders have been used for medium to long span bridge structures. Highway aesthetics through long spans, economy due to ease of fabrication, shorter construction duration are some of the many advantages of precast segment bridge construction. The segments are hollow box sections, match cast with shear keys in a casting yard, then assembled in place, leaving the joints entirely dry. The shear keys are meant to transfer service level shears and to help in alignment during erection. Current design methods are heavily based on linear elastic analysis with empirically derived constitutive laws assuming homogeneous, isotropic materials. The behavior under load of the bridge system is very complex. Analyses which do not properly quantify the nonlinear effects including the opening of joints in flexure, are likely to provide a distorted view of the factor of safety existing in a structural system between service loads and failure. The potential sliding and separation at the joints due to shear, and by deformations generated by temperature gradients over the depth and width of the box further complicate the problem [1]. Two finite elements have been developed that render a rational analysis of the system. The link element is a two- dimensional friction gap element. It allows sliding between the faces of the element, controlled by a friction coefficient and the normal forces. It also accounts for zero stiffness in tension and a very high stiffness under compression. This link element was borrowed from rock mechanics and newly applied to this problem to model the dry joint between the segments. The Mindlin flat shell element is a combination of membrane element and Mindlin plate element. This element considers the shear responses along the element thickness direction. The shell element was used to model the segment itself. This element can handle large displacement and geometric nonlinearities. 1.2 Link Element A link element is a nonlinear friction gap element used to model discontinuous behavior in solid mechanics. Some examples are interfaces between dissimilar materials and joints, fractures in the material, and planes of weakness. These have been modeled using constraint equations, discrete springs and a quasi-continuum of small thickness [2]. The following characteristics of prototype joints were considered. 1. Joints can be represented as flat planes. 2. They offer high resistance to compression in the normal direction but may deform somewhat modeling compressible filling material or crushable irregularities. 3. They have essentially no resistance to a net tension force in the normal direction. 4. The shear strength of joints is frictional. Small shear displacements probably occur as shear stress builds up below the yield shear stress. A model for the mechanics of jointed rocks was developed by Goodman [3]. The finite element approximation was done as a decomposition of the total potential energy of a body into the sum of potential energies of all component bodies. Therefore, element stiffness is derived in terms of energy. The Goodman element was tested for several modeled cases. 1. Sliding of a joint with a tooth. 2. Intersection of joints. 3. Tunnel in a system of staggered blocks. A problem with the Goodman's two dimensional model is that adjacent elements can penetrate into each other. Zienkiewicz et al. [4] advocate the use of continuous isoparametric elements with a simple nonlinear material property for shear and normal stresses, assuming uniform strain in the thickness direction. Numerical difficulties may arise from ill conditioning of the stiffness matrix due to very large off-diagonal terms or very small diagonal terms which are generated by these elements in certain cases. A discrete finite element for joints was introduced which avoids such theoretical difficulties and yet is able to represent a wide range of joint properties, including positive and negative dilatency (expansion and compaction accompanying shear) [3]. The element uses relative displacements as the independent degrees of freedom. The displacement degrees of freedom of one side of the slip surface are transformed into the relative displacements between the two sides of the slip surface. This element has been incorporated into a general finite element computer program [5]. The use of relative displacement as an independent degree of freedom to avoid numerical sensitivity is discussed in detail [6]. An isoparametric formulation is given by Beer [2]. A four-node, two-dimensional link element and a eight-node plate bending element were used to model the dry jointed concrete box girder bridge with shear keys [7]. 1.3 Shell Element The shell element is formulated through the combination of two different elements, the membrane element and the Mindlin plate bending element. The Mindlin plate element is different from the Kirchhoff plate element in that the former allows transverse shear deformation while the latter does not. The nonlinearities included in the formulation of the flat shell element is for large displacement and geometric nonlinearity due to initial stress effects. The large displacement effects are caused by finite transverse displacements. These effects are taken into account by coupling transverse displacement and membrane displacements. The initial stress effects are caused by the actual stresses at the start of each iteration. These stresses change the element stiffness for the subsequent iteration. These effects are evaluated directly from the stresses at the start of each iteration and included in the element stiffness. 1.4 Literature Review The purpose of nonlinear analysis is to develop the capability for determining the nonlinear load-deflection behavior of the structures up to failure so that a proper evaluation of structural safety can be assured. There are two general approaches for nonlinear analysis. The first approach is a linearized incremental formulation by reducing the analysis to a sequence of linear solutions. The second approach is mathematical iterative techniques applied to the governing nonlinear equations [8]. The advantage of the incremental approach results from the simplicity and generality of the incremental equations written in matrix form. Such equations are readily programmed in general form for computer solutions [9]. A generalized incremental equilibrium equation for nonlinear analysis can be found in [10, 11, 12]. The formulation is valid for both geometrical and material nonlinearities, large displacements and rotations, conservative and displacement dependent (nonconservative) loads. There are two frames for the description of motion. The difference lies in the coordinate systems in which the motion is described. These are the total Lagrangian formulation which refers to the initial configuration [10, 11] and the updated Lagrangian formulation which refers to the deformed configuration [12]. There have evolved two types of notations in the description of motion. A correlation is given these two notations, the B-notations and the N-notations, currently used in the Lagrangian formulation of geometrically nonlinear analysis [13]. A short history of early theoretical development of nonlinear analysis can be found in [9, 14]. One form of updated Lagrangian formulation is the corotational stretch theory [15]. Shell elements are often derived from governing equations based on a classical shell theory. Starting from the field equations of the three-dimensional theory, various assumptions lead to a shell theory. This reduction from three to two dimensions is combined with an analytical integration over the thickness and is in many cases performed on arbitrary geometry. Static and kinematic resultants are used. These are referred to as classical shell elements. Alternatively, one can obtain shell elements by modifying a continuum element to comply with shell assumptions without resorting to a shell theory. These are known as degenerated shell elements. This approach was originally introduced by Ahmad, Irons, and Zienciewicz [16, 17]. Other applications can be found in [8, 18-25]. In large rotation analysis, the major problems arise from the verification of the kinematic assumptions. The displacement representation contains the unknown rotations of the normal in the arguments of trigonometric functions. Thus additional nonlinearity occurs. Further difficulties enter through the incremental procedure. Rotations are not tensorial variables, therefore, they cannot be summed up in an arbitrary manner [17]. One of the special treatment of finite rotation is that the rotation of the coordinate system is assumed to be accomplished by two successive rotations, an out-of-plane rotation followed by an in-plane rotation using updated Lagrangian formulation [26, 27]. Usually the loadings are assumed to be conservative, i.e., they are assumed not to change as the structure deforms. One of the well known exceptions is pressure loading which can be classified as conservative loading or a nonconservative loading [28]. Another is the concentrated loading that follows the deformed structure. For example, a tip loading on a cantilever beam will change its direction as the deformation gets larger. As loading is a vector quantity, the change in direction means that the loading is not conservative. Sometimes this is called a follower loading. The governing equation for large strain analysis can be used for small increments of strain and large increments of rotations [29]. This can be regarded as a generalization of nonlinearity of small strain with large displacement. If large strain nonlinearity is employed, an important question is which constitutive equation should be used [9]. The degree of continuity of finite element refers to the order of partial differential of displacements with respect to its coordinate system. Order zero means displacement itself must be continuous over the connected elements. Order one means that the first order differential of displacement must be continuous. Thus the higher order the continuity requirement, the higher the order of assumed displacement (shape, interpolation) function. Mindlin-Reissner elements require only Co continuity, so that much lower order shape functions can be used, whereas in Kirchhoff-Love type elements, high order shape functions must be used to satisfy the C1 continuity. Furthermore, since Mindlin-Reissner elements account for transverse shear, these elements can be used for a much larger range of shell thickness. The relaxed continuity requirements which permit the use of isoparametric mapping techniques gives good computational efficiency if formulated in the form of resultant stresses [30]. Unlike compressible continuum elements, which are quite insensitive to the order of the quadrature rule, curved Co shell elements require very precisely designed integration scheme. Too many integration points result in locking phenomena, while using an insufficient number of quadrature points results in rank deficiency or spurious modes [30]. While Gauss point stress results are very accurate for shallow and deep, regular and distorted meshes, the nodal stresses of the quadratic isoparametric Mindlin shell element are in great error because of the reduced integration scheme which is necessary to avoid locking [31]. The degenerate solid shell element based on the conventional assumed displacement method suffers from the locking effect as shell thickness becomes small due to the condition of zero inplane strain and zero transverse shear strain. Element free of locking for linear shell analysis using the formulation based on the Hellinger-Reissner principle with independent strain as variables in addition to displacement is presented in [32]. Shear locking is the locking phenomenon associated with the development of spurious transverse shear strain. Membrane locking is the locking phenomenon associated with the development of non-zero membrane strain under a state of constant curvature. Machine locking is the locking phenomenon associated with the different order of dependence of the flexural and real transverse shear strain energies on the element thickness ratio, and it is therefore strictly related to the machine finite word length [33]. Some of the solutions are as follows: 1. Assumed strain stabilization procedure using the Hu- Washizu or Hellinger-Reissner variational principles [33]. 2. The assumed strain or mixed interpolation approach [34, 35]. 3. Suppressing shear with assumed stress/strain field in a hybrid/mixed formulation [30]. Suppression of zero energy deformation mode using assumed stress finite element [36]. 4. Coupled use of reduced integration and nonconforming modes in quadratic Mindlin plate element [37]. 5. Higher order shallow shell element, with 17 to 25 nodes [38, 39]. 6. Global spurious mode filtering [40]. 7. Artificial stiffening of element to eliminating zero energy mode, special stabilizing element [41]. In the faceted elements, due to the faceted approximation of the shell surface, coupling between the membrane and the flexural actions is excluded within each individual element, the coupling is, however, achieved in the global model through the local to global coordinate transformation for the elements [39]. In geometrically nonlinear analysis with flat plate elements, it is common to use the von Karman assumptions when evaluating the strain-displacement relations. The assumption invoked is that the derivatives of the inplane displacements can be considered to be small and hence their quadratic variations neglected. However, this simplification of the nonlinear strain-displacement relationship of the plate, when used in conjunction with the total Lagrangian approach, implies that the resulting formulation is valid only when the rotation of the element from its initial configuration is moderate. Thus for the total Lagrangian approach to handle large rotations, simplifications of the kinematic relationship using the von Karman assumptions is not permitted [39]. Some of the special solution strategies to pass the limit point are given in references [25, 42-48]. A limit point is characterized by the magnitude of tangential stiffness. It is zero or infinite at a limit point. Thus conventional solution strategies fail at the limit point. Arc length method was introduced in reference [42], and applied in the case of cracking of concrete [43]. This was improved with line search and accelerations in references [44, 45]. Line search means the calculation of an optimum scalar step length parameter which scales the standard iterative vector. This can be applied to load and displacement control and arc length methods [44]. The traditional solution strategies are iterative solutions, for example, Newton-Raphson, constant stiffness, initial stiffness, constant displacement iteration, load increment [46] along with Cholesky algorithm with shifts for the eigensolution of symmetric matrices [47] for element testing for spurious displacement mode. The vector iteration method without forming tangent stiffness for the postbuckling analysis of spatial structures is also noted [48]. The linearized incremental formulation in total Lagrangian description has been used for this study of large displacement nonlinearity including initial stress effects. The special treatment of finite rotation is not included in the current study. Material nonlinearity is also excluded. CHAPTER 2 GENERAL THEORIES OF NONLINEAR ANALYSIS 2.1 Introduction The incremental formulations of motion in this chapter closely follow the paper by Bathe, Ramm, and Wilson [11]. Other references are also available [9, 10, 12, 14, 15, 49, 50, 51]. Using the principle of virtual work, the incremental finite element formulations for nonlinear analysis can be derived. Time steps are used as load steps for static nonlinear analysis. The general formulations include large displacements, large strains and material nonlinearities. Basically, two different approaches have been pursued in incremental nonlinear finite element analysis. In the first, Updated Lagrangian Formulation, static and kinematic variables, i.e., forces, stresses, displacements, and strains, are referred to an updated deformed configuration in each load step. In the second, Total Lagrangian Formulation, static and kinematic variables are referred to the initial undeformed configuration. It is noted that using either of two formulations should give the same results because they are based on the same continuum mechanics principles including all nonlinear effects. Therefore, the question of which formulation should be used merely depends on the relative numerical effectiveness of the methods. 2.2 Motion of a Continuum Consider the motion of a body in a Cartesian coordinate system as shown in Fig. 2-1. The body assumes the equilibrium positions at the discrete time points 0, dt, 2dt, ..., where dt is an increment in time. Assume that the solution for the static and kinematic variables for all time steps from time 0 to time t, inclusive, have been solved, and that the solution for time t+dt is required next. The superscript on left hand side of a variable shows the time at which the variable is measured, while the subscript on left hand side of a variable indicates the reference configuration to which the variable is measured. Thus the coordinates describing the configuration of the body using index notation are At time 0 = x At time t = txi At time t+dt = t+dtx P t+dt P( x ) t P( Xi) 0 P( Xi) Fig. 2-1 Motion of a Body The total displacements of the body are At time 0 = ui At time t = tui At time t+dt = t+dtu The configurations are denoted as At time 0 = C At time t = tc At time t+dt = t+dtc Thus, the updated coordinates at time t and time t+dt are txi = Oxi + tui t+dtx = Oxi + t+dtu The unknown incremental displacements from time t to time t+dt are denoted as (Note that there is no superscript at left hand side.) u = t+dtui tui (2.1) 2.3 Principle of Virtual Work Since the solution for the configuration at time t+dt is required, the principle of virtual work is applied to the equilibrium configuration at time t+dt. This means all the variables are those at time t+dt and are measured in the configuration at time t+dt and all the integration are performed over the area or volume in the configuration at time t+dt. Then the internal virtual work (IVW) by the corresponding virtual strain due to virtual displacement in t+dtC is S t+dt t+dt t+dt t+dt ij t+dt eij (+dt dV) (2.2) where, t+dt rj = Stresses at time t+dt measured in the t+dt j configuration at time t+dt. = Cauchy stresses. = True stresses. t+dt t+dt eij = Cauchy's infinitesimal(linear) strain tensor referred to the configuration at time t+dt. = Virtual strain tensor. 6 = Delta operator for variation. and the external virtual work (EVW) by surface tractions and body forces is EVW = t+dt t ] 6 [ t+dt u (t+dt dA) t+dt k t+dt k (t+dt St+dt t+dt b 6 t+dt u (t+dt dV) St+dt t+dt k t+dt uk (2.3) where, t+dt tk = Surface traction at time t+dt measured in t+dt the configuration at time t+dt. t+dt Uk = Total displacement at time t+dt measured in t+dt the configuration at time t+dt. 6 t+dt Uk = Variation in total displacement at time t+dt t+dt measured in configuration at time t+dt = Virtual displacement. t+dt p = Mass density per unit volume. t+dt t+dt bk = Body force per unit mass. t+dt k and all the integration is performed over the area and the volume at time t+dt. 2.4 Updated Lagrangian Formulation In this formulation all the variables in Eqs. (2.2) and (2.3) are referred to the updated configuration of the body, i.e, the configuration at time t. The equilibrium position at time t+dt is sought for the unknown incremental displacements from time t to t+dt. The internal virtual work, the volume integral in Eq. (2.2) measured in the configuration at time t+dt can be transformed to the volume integral measured in the configuration at time t in a similar manner that is given in reference [52] IVW t+dt t+dt e (t+dt dV) t+dt ij t+dt i (t dV) t Sij 6 Et ij (t dV) = EVW (2.4) I ~t t~d where, t+dt Sj = Second Piola-Kirchhoff (PK-II) stresses t measured in the configuration at time t. 6 t+dt ej = Variations in Green-Lagrange (GL) strain t tensor measured in the configuration at time t. The PK-II stress tensor at time t+dt, measured in the configuration at time t can be decomposed as t+dt t t Sij =t Sij + t Sij j + t Sij (2.5) because the second PK-II stress at time t measured in the configuration at time t is the Cauchy stress. From Eq. (2.1), the total displacements at time t+dt measured in the configuration at time t is t+dt t t Ui = t Ui + Ui = t ui (2.6) This is true because the displacement at time t measured in the configuration at time t is zero. In other words, the displacement at time t+dt with respect to the configuration at time t is the incremental displacement itself. And the GL strain is defined in terms of displacement as Eij = 1 (Ui,j + Uji + Uk,iUk,j) (2.7) E and U are used in the places of e and u to avoid confusion between general strain and incremental strain, and between general displacement and incremental displacement used in this formulation. It is noted that these finite strain components involve only linear and quadratic terms in the components of the displacement gradient. This is the complete finite strain tensor and not a second order approximation to it. Thus this is completely general for any three-dimensional continuum [52]. Then the GL strain tensor at time t+dt measured at time t can be calculated as +dt = i [(ui+ i)+ u) + ( j + tuj),i +{(uk + tuk),i)( tuk + uk),j)] = tuij + tuj,i+ tuki tuk,j] = teij + tij = tij (2.8) where, ttij = tei + t ij = Incremental GL strain in tC. tej = (ui,j + tuji ) = Linear portion of incremental GL strain in t. = This is linear in terms of unknown incremental displacement. = Linearized incremental GL strain in tc. ttij = t (uk,i tuk,j) = Nonlinear portion of incremental GL strain. The variations in Green-Lagrange strain tensor at time t+dt measured in the configuration at time t can be shown as using Eq. (2.8). t+dt t 6 t ij = 6 ( tij + Ei ) = 6 t (2.9) 6teij = 0 because teij is known. There is no variation in known quantity. Then using the Eqs. (2.5), (2.8) and (2.9), the integrand of Eq. (2.4) becomes t+dt t+dt t t Sij 6 qt =ij ( t i + tSij ) 6 tCij = trij + tSij)(6 teij + 6tij) =tij(6eij + 6t 1ij) + t~ij 6 te + rij 6 tij t t =tij 6 ti ij + ti 6eJ 6 (2.10) tj t t ttij 6 t tii j` t The constitutive relation between incremental PK-II stresses and GL strains are tSij = tCijkj t'kl (2.11) Finally the equilibrium Eq. (2.4) from the principle of virtual work using Eqs. (2.10) and (2.11) is I tCijkl tkl 6 tcij tdv + tri 6 t Vij tdV = EVW J ij 6 eij tdV t 1t (2.12) where, the external virtual work must be transformed from t+dtc to tC. This is not applicable to conservative loading, i.e., loading that is not changed during deformation. EVW= t+dt tk [ t+dt uk (tdA) + t+dt t+dt b t+dt td) (2 ) and this is the general nonlinear incremental equilibrium equation of updated Lagrangian formulation. 2.5 Total Lacrangian Formulation Total Lagrangian formulation is almost identical with the updated Lagrangian formulation. All the static and kinematic variables in Eqs. (2.2) and (2.3) are referred to the initial undeformed configuration of the body, i.e, the configuration at time 0. The terms in the linearized strain are also slightly different from those of updated Lagrangian formulation. The volume integral in Eq. (2.2) measured in the configuration at time t+dt can be transformed to the volume integral measured in the configuration at time 0 as [52] t+dt 'i 6 t+dt eij t+dt t+dt ]i t+dt o t Sij t+dt j (dV) (2.14) where, t+dtS = Second Piola-Kirchhoff stress tensor 0 measured in the configuration at time 0. 6 t+dtij = Variations in Green-Lagrange (GL) strain 0 tensor measured in the configuration at time 0. The PK-II stress tensor at time t+dt, measured in the configuration at time 0 can be decomposed as t+dtsij= tsij+ *ij (2.15) o1 ) o) 1 o01j (2.15) From Eq. (2.1), the total displacements at time t+dt measured in the configuration at time 0 is t+dtu = u + Ui (2.16) o 0i o0 o Then the GL strain tensor at time t+dt measured at time 0 can be calculated as t+dt = [(tui + oui)j + (tuj + ou i =tt+ +(( e*k+ oUk),i)(( Uk+ Uk) j)] = Eij + e + i = ij + Eij (2.17) where, tt tutu tu tu) oij = I (t u + + ) 0 o o rJ o k,i k,j = GL strain at time t in oC. ij = oeij + oij = Incremental GL strain in oC. oeij = (oi, + oUj1 + uk,i ouk,kj+ k,j ouk,i) = Linear portion of incremental GL strain in oC. =This is linear in terms of unknown incremental displacement. = Linearized incremental GL strain in oC. oij = i (oUk, ouk,j) = Nonlinear portion of incremental GL strain. The variations in Green-Lagrange strain tensor at time t+dt measured in the configuration at time t can be shown as using Eq. (2.17). Stdtij = 6 ( ij+ ij ) = 6 (2.18) o i= oec olu olt 6 te, = 0 because ej is known. There is no variation in o ij o-J known quantity. Then using the Eqs. (2.15), (2.17) and (2.18), the integrand of Eq. (2.14) becomes t+dt t+dt ts t+dt*i 6 t+dti ( tSi + S ) 6 e o o o o0J o'J = (ij oSij) oeij + 6 oiij) = Sij(6 eij + 6 ij) + oSij 6 e 6 + = Si 6 eij + tS i 6 ei + ij 6 o7ij (2.19) The constitutive relation between incremental PK-II stresses and GL strains are Sij = oCijkj ockl (2.20) Finally the equilibrium Eq. (2.14) from the principle of virtual work using Eqs. (2.19) and (2.20) is I oCijkl oekl 6 j dV + tsi 6 oij OdV = EVW tij 6 e dV o 1 oe ij (2.21) where, the external virtual work must be transformed from t+dtc to oC. This is not applicable to conservative loading, that is, loading that is not changed during deformation. EVW = t+dt t t+dt uk (dA) + t+dt p t+dt b t+dt uk (odV) (2.22) and this is the general nonlinear incremental equilibrium equation of total Lagrangian formulation. 2.6 Linearization of Equilibrium Equation The incremental strain from time t to t+dt is assumed to be linear, i.e., ekl = ekl in Eqs. (2.11), (2.12), (2.20), and (2,21). For the updated Lagrangian formulation, tSij = Cijkj tekl (2.23) and, ItCijkl tekl teij d tI t + tij 6 tij tdV = EVW rj 6 e tdV (2.24) For the total Lagrangian formulation, oSij = oCijkj oekl (2.25) and, IoCijkl oekl 6 oeij odV ddV + [sij ij dV = EVW Sj e odV o 0J o ij (2.26) It should be noted that the surface tractions and the body forces in the calculation of external virtual work may be treated configuration dependent when the structure undergoes large displacements or large strains. If this is the case, the external forces must be transformed to the current configuration at each iteration [10, 11, 12]. 2.7 Strain-Displacement Relationship Using the von Karman Assumptions The nonlinear strain terms can be simplified for the plate or shell type structures using von Karman assumption of large rotation. In the mechanics of continuum the measure of deformation is represented by the strain tensor Eij [52] and is given by using index notation. 2Eij = ( ui,j + uj,i + k,iuk,j ) (2.27) where, ui = Displacement in i-direction. uij = aui / axj xi = Rectangular Cartesian coordinate axes, i=1,2,3. uk,iuk,j = ul,iul,j + u2,iu2,j + u3,iu3,j The von Karman theory of plate is a nonlinear theory that allows for comparatively large rotations of line elements originally normal to the middle surface of plate. This plate theory assumes that the strains and rotations are both small compared to unity, so that we can ignore the changes in geometry in the definition of stress components and in the limits of integration needed for work and energy considerations [53]. It is also assumed that the order of the strains is much less than the order of rotations. If the linear strain eij and the linear rotation rij are defined as 2eij = uij + uji (2.28) 2rij = u ui (2.29) Then the sum of Eqs. (2.28) and (2.29) gives 2(eij + rij) = 2uij (2.30) and the subtraction of Eq. (2.29) from Eq. (2.28) gives 2(eij rij) = 2uj (2.31) From Eqs. (2.30) and (2.31), it is concluded that uk,j = ekj + rkj (2.32) uk,i = eik rik (2.33) Eq. (2.33) can be rewritten as uk,i = eki + rki (2.34) since eik = eki from the symmetry of linear strain terms and rik = -rki from the skew symmetry of the linear rotation terms. The strain-displacement Eq. (2.27) now becomes 2Eij = 2eij + (eki + rki)(ekj + rkj) (2.35) by substituting Eqs. (2.30) through (2.34) into Eq. (2.27). Thus the nonlinear strain terms have been decomposed into linear strain terms and linear rotation terms. From the assumption on the order of strains and rotations eki << rki and ekj << rkj (2.36) Thus Eq. (2.35) can be simplified as by ignoring eki and ekj. 2Eij = 2eij + rkirkj (2.37) The straight line remains normal to the middle surface and unextended in the Kirchhoff assumption, but it is not necessarily normal to the middle surface for the Mindlin assumption. For both assumptions the generic displacements u,v,w can be expressed by the displacements at middle surface. For the Kirchhoff plate [20], u(x,y,z) = uo(x,y) z[Wo(x,y),x] v(x,y,z) = vo(x,y) z[wo(x,y),y] (2.38) w(x,y,z) = Wo(x,y) where, uo, Vo, wo = Displacements of the middle surface in the direction of x, y, z. u, v, w = Displacements of an arbitrary point in the direction of x, y, z. Now the linear strain components eij and the linear rotation components rij can be calculated using Eqs. (2.28) and (2.29). ell = (ul,1 + Ul,1) = ul,1 = ,x el2 = (l,2 + u2,1) = 1(uly + Vx) el3 = (u1,3 + U3,1) = (-Wox + Wx) 22 = i(u2,2 + u2,2) = u2,2 = Vy (2.39) e23 = (u2,3 + u3,2) = (-woy + Wy) e33 = (u3,3 + u3,3) = u3,3 = The rotation terms r12, r13, r23 are the rotation quantities about the axes 3(z), 2(y) and l(x), respectively. For the plate located in the xy plane, the rotation about z axis rl2 is much smaller than rotation about x axis r23 and y axis r13 and therefore rl2 is assumed to be zero here. And it is noted further that wo(x,y) is the same as w(x,y) and is a function of only x and y so that w,3 = w,z = 0. lrl21 << lr231 or 1r13 (2.40) rl = (ul,1 ul,1) = 0 r12 = 1(ul,2 u2,1) = i(u, Vx) = 0 r13 = I(ul,3 u3,1) = (-Wox Wx) = -Wx r22 = 1(u2,2 u2,2) = 0 (2.41) 23 = 1(u2,3 u3,2) = (-Wo'y W'y) = -Wy r33 = i(u3,3 3,3) = 0 The linear strain component eij is symmetric and the linear rotation component rij is antisymmetric. eij = eji rij = -rji (2.42) The strain components from Eq. (2.37) can be rewritten using Eqs. (2.39) and (2.41). Ex = ell + (r112 + yy = e22 + (r122 + Ezz = e3 + 1(r132 + Exy = el2 + 2(rllrl2 Exz = e3 + (rllrl3 Eyz = e23 + U(rl2r13 r21 + r31 ) = ell + r31 r22 + r322) = e22 + r32 r232 + r332) = 1(r132 + r232) = 0 + r21r22 + r31r32) = el2 + r31r32 + r21r23 + r31r33) + r22r23 + r32r33) (2.43) Egz term is assumed to be zero because it does not have the linear term. Exz and Eyz terms are transverse shear terms which can be ignored for thin plate. Then Eq. (2.43) can be rearranged as follows using Eqs. (2.41) if all the zero terms are removed. Exx = ell + r312 = ell + (W,x)2 Eyy = e22 + r322 = 22 + )2 Ey = el2 + Ir31r32 = e2 + (W'x) (,y) Exz = e13 Eyz = e23 (2.44) 33 Thus the decomposition of exact strain components has been done using the Kirchhoff plate assumptions (2.38) and the von Karman assumption (2.40) on the magnitude of rotation. It is noted that all the inplane displacement gradients in nonlinear strain terms are ignored through von Karman assumptions [20]. This fact will be applied in chapter 5. CHAPTER 3 THREE DIMENSIONAL LINK ELEMENT 3.1 Element Description The link element used here is based on the two dimensional element developed by Cleary [54]. The link element is based on the following assumptions. Any normal compressive force is transferred to the other side of the link without any loss. To facilitate this, a very limited amount of loss through displacement should be allowed. Currently, this limited displacement is defaulted to .001 units, while it is a input parameter. The link separates in response to any net tension, losing its normal stiffness. To discuss the shear force transfer, some definitions for friction are needed. The force to start one body sliding along the other body is called the static friction force. The force to keep it moving is the kinetic friction force. There are two corresponding coefficients of friction, static friction coefficient and dynamic friction coefficient, where the static friction coefficient will generally be greater than the dynamic friction coefficient. Two laws of friction were used in the link element. The first law is that the frictional force is proportional to the normal force, with the constant of proportionality being the friction coefficient. The second law is that friction does not depend on the apparent area of the connecting solids, i.e., it is independent of the size of the bodies. The shear force is transferred through friction. The uncertainty in friction is the factor which limits the overall accuracy of the calculation. Therefore, it is assumed that the static friction coefficient is proportional to the dynamic friction coefficient. For nonmetallic materials, the ratio of dynamic coefficient to static coefficient is about 0.75. The link element is composed of two surfaces. If the shear force is less than or equal to the static friction force, i.e., coefficient of friction times the normal force, the shear force is balanced by the friction force and the total force is transferred. This is shown in Fig. 3-1. But if the shear force is greater than the static friction force, one surface of the link element will move along the other surface. In this case there will be a dynamic friction force which is less than the shear force. This dynamic friction force can only resist a portion of the shear and the system is not in static equilibrium. Therefore, if the shear force is greater than the static friction force, the link element will lose its shear stiffness. This can also be modeled with a body on roller and spring as shown in Fig. 3- 2. The spring model of the link element is shown in Fig. 3- 3. The link element here has four nodes and each node has three translational degrees of freedom in local u-, n-, and w-directions. The total number of element degrees of freedom is 12. The element degrees of freedom are shown in Fig. 3-4. The equivalent "strain" for the link element is defined as the average deformation at the center of the element. The average deformation corresponding to the translational degrees of freedom, i.e., uo, vo, and wo, can be directly calculated from the joint displacements by averaging the difference in nodal displacements at the ends of element in local u-, n-, and w-direction in turn. The relative rotation at the center of the element, ro, can be found using nodal displacements in local n-direction and the element length as shown in Fig. 3-5. This angle is not an "average" value but the "relative" change in angle of the center line due to rotation. The two joint parameters must be introduced. These are kn, the unit stiffness normal to the joint, and ks, the unit stiffness along the joint. The off-diagonal term kns to account for dilatation during shearing is ignored because this joint element will model the dry joint between concrete box girder bridge segments. No significant dilation is expected in this case. Some values of kn and ks were reported in geotechnique area [3]. As the values are those for natural joints, they do not directly apply to this case. From the test results [7], it can be seen that the shear stiffness of dry joint ranges from 70,000 to 286,000 psi per inch at ultimate. In case of frictional strength, this can be interpreted as linear behavior between the origin and the ultimate point. It seems reasonable that the normal stiffness of the element, kn, is assumed to be stiffer than the connected material by the order of 103 to transfer the normal force without any significant loss. The forces are either totally transferred in compression or totally lost in tension. The latter case has no problem related to the value of kn. The shear stiffness parameter is more difficult to define. The data available is so limited that even a statistical treatment cannot be done. But in the analysis of structural behavior up to the ultimate, these properties do not have great influence because the forces are transferred through friction. The shear stiffness becomes zero upon sliding. But there is some 'residual' shear force. This 'residual' force is equal to friction force. Therefore, if shear displacement is more than the displacement just before the sliding the shear stiffness is set to be zero. Ff P, N = External forces. F = Friction force. f m = Friction coefficient 1) P < or = mN then P = Friction Force. In Equilibrium. 2) P > mN then the body moves but the frictional force mN is acting against the other body. Fig. 3-1 Friction Force FRICTIONAL SPRING WITH SHEAR STIFFNESS Force in spring = mN N -%AAN- BEFORE SLIDING SFRICTIONAL SPRING WITH ZERO STIFFNESS N A F mN mN < F AFTER SLIDING Fig. 3-2 Spring Model of Friction Force __ __ Kn = Zero under tension. = Very large under compression. SPRING MODEL FOR NORMAL FORCES SPRING MODEL FOR SHEAR FORCES Fig. 3-3 Spring Model of Link Element 11 K Fig. 3-4 Element Degrees of Freedom of Link Element L 12 I 34 CENTER OF ELEMENT K UK NoLNo uo = [(UK + UL) (UI + UJ)] / 2 ro= p a S[(VK-VL) (VJ-VI)] / L VK O------ VI o o- Fig. 3-5 Element "Strain" uo-_ZL 3.2 Formation of Element Stiffness There are four nodes per element. Each node has three degrees of freedom corresponding the translational displacements in u-, n-, and w-direction resulting in 12 element degrees of freedom as shown in Fig. 3-4. The element stiffness is derived directly from the physical behavior of the element described in section 3.1. The mathematical symbol {} is used for a column vector and [] for a matrix. The nodal displacement column vector {q)(12) is composed of 12 translational nodal displacements corresponding to the 12 element degrees of freedom. (q} = { ui vi wi Uj Vj Wj uk vk wk ul vl wl )T The "strain" is defined as the average deformation at the center of the element as shown in Fig. 3-5. The "strain" column vector {e)(4) is {e} = { uo vo wo ro )T where, uo = ( uk + ul ) / 2 ( u + j ) / 2 o = vk + 1 ) / 2 ( + Vj ) / 2 w = (wk + 1 ) / 2 ( i + j ) / 2 ro = ( vk v ) / L ( vj vi ) / L where, L = The length of the element. uo, vo, wo = Average nodal displacements in local u-, n-, w-directions. ro = The relative angle change about local z axis. Therefore the relationship between "strain" and nodal displacements is (e}(4) = [B](4,12) (q)(12) The [B](4,12) matrix which gives strains due to unit values of nodal displacements is -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0 0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0 0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 1/L 0.0 0.0 -1/L 0.0 0.0 1/L 0.0 0.0 -1/L 0.0 The "stress" is defined as the normal and shear stress per unit of area. {s} is the average stress on the surface due to the two nodal forces exerted in the plane of the surface. This stress is in equilibrium with the stress on the other surface of the element as shown in Fig. 3-6. m is the moment of the nodal forces on one surface in local n- direction about the center of the element. This moment is also balanced by the moment of the nodal forces on the other surface of the element. This moment is used to define the distribution of the normal stress of the element as shown in Fig. 3-7. Pnl fl1 P...' uk -~- -~- -~- -~- -~- -~- -~- -~- -~- -~- uj -nj Pj J ~pnj -W- Stresses -- Nodal Forces Local Coordinate System Fig. 3-6 Nodal Forces and Stresses of Link Element / ni -W.- -.- -w -0--o 0--0 p FORCE TRANSFER THROUGH ONE EDGE OF THE LINK ELEMENT P L K Vo ro 2 Vo CENTER OF ELEMENT p J Fig. 3-7 Element "Strain", m The "stress" column vector (s}(4) is (s) = { Sx, sn, Sz, m ) The "stress-strain" relationship is (s)(4) = [E](4,4) {e)(4) where, kx 0 0 0 [E] = 0 kn 0 0 0 0 kz 0 0 0 0 km where km can be related to kn using the definition of the moment m, i.e., m = (Sn)(L)(t)(0.5)(L) = kmro = (km)(vo/(0.5)(L)) Thus, km = (knVo)(L)(t)(0.5)(L) / [Vo/(0.5)(L)] = (0.25)(t)(kn)(L3) where, t = Element thickness. This assumes that there is no coupling between the shear stress and normal stress. The element nodal force column matrix (P)(12) is composed of the 12 nodal forces shown in Fig. 3-6. (P) Pui Pni Pwi Puj Pnj Pwj Puk Pnk Pwk Pul nl Pwl T Stress can then be related to nodal forces using the definition of stress and force equilibrium between the two surfaces of the element. By the definition of stress, sn = (1/Lt)( Pnk + Pnl ) (3.1) sx = (1/Lt)( Puk + Pul ) (3.2) sz = (1/Lt)( Pwk + Pwl ) (3.3) m = Pnk(0.5)(L) Pnl(0.5)(L) (3.4) where, Lt = (L)(t) By force equilibrium of the two surfaces, Pi = -P and Pj = Pk (3.5) To express the element nodal forces in terms of the stress, we use Eqs. (3.1) through (3.5) to find the force recovery matrix [FR]. [FR] gives the nodal forces in equilibrium with the element stresses. From (Eq. (3.1) + Eq. (3.4)), 2Pnk = (L)(t) (sn) + 2(m)/L Pnk = 0.5(L) (t) (sn) + (l/L)(m) From Eq. (3.5), Pj = -Pk nj = -Pnk = -0.5(L)(t)(sn) (1/L)(m) From Eq. (3.1), Pn1 = (L)(t)(sn) Pnk = (L)(t)(sn) ( 0.5(L)(t)(sn) + (1/L)(m)) = 0.5(L) (t) (n) (1/L) (m) From Eq. (3.5), Pni = Pn1 = -0.5(L)(t)(sn) + (1/L)(m) From the assumption that Puk = Pul and Eq. (3.2), Puk = (L)(t)(sx)/2 Pul = (L)(t)(sx)/2 From Eq. (3.5), Pui = ul = -(L)(t)(sx)/2 Puj = uk = -(L)(t)(sx)/2 From the assumption that Pwk = Pwl and Eq. (3.3), Pwk = (L)(t)(sz)/2 Pwl = (L)(t)(sz)/2 From eqn 5, Pwi = wl = -(L)(t)(sz)/2 Pwj = Pwk = -(L)(t) (s)/2 Therefore, the force-stress relationship is {P)(12) = [FR](12,4) (s)(4) where the force recovery matrix [FR](12,4) is [FR] = -Lt/2 0 0 -Lt/2 0 0 Lt/2 0 0 Lt/2 0 0 0 -Lt/2 0 0 -Lt/2 0 0 Lt/2 0 0 Lt/2 0 0 0 -Lt/2 0 0 -Lt/2 0 0 Lt/2 0 0 Lt/2 0 1/L 0 0 -1/L 0 0 1/L 0 0 -1/L 0 And this relationship is further expanded using the stress-strain relationship and the strain-nodal displacement relationship as follows. (P}(12) = [FR](12,4) = [FR](12,4) = [Bt](12,4) [E](4,4) {e)(4) [E](4,4) [B](4,12) [E](4,4) [B](4,12) Then finally this can be symbolized as equilibrium equation. (q)(12) {q)(12) {P)(12)= [Ke](12,12) (q}(12) where [Ke] = [Bt][E][B] Here it is noted that [FR] = [Bt] and [Ke] = [Bt][E][B] just as in the case of common finite element method. The final element stiffness matrix [Ke] is (L)t 4 kx 0 0 kx 0 0 -kx 0 0 -kx 0 0 0 2kn 0 0 0 0 0 0 0 0 -2kn 0 0 0 kz 0 0 kz 0 0 -kz 0 0 -kz kx 0 0 kx 0 0 -kx 0 o -kx 0 0 0 0 0 0 2kn 0 0 -2kn 0 0 0 0 0 0 kz 0 0 kz 0 0 -kz 0 0 -kz -kx 0 0 -kx 0 0 kx 0 0 kx 0 0 0 0 0 0 -2kn 0 0 2kn 0 0 0 0 0 0 -kz 0 0 -kz 0 0 kz 0 0 kz -kx 0 0 -kx 0 0 kx 0 0 kx 0 0 0 -2kn 0 0 0 0 0 0 0 0 2kn 0 0 0 -kz 0 0 -kz 0 0 kz 0 0 kz This matrix can be standard rotation. rotated to any direction using the 3.3 Solution Strategy The structural stiffness changes because of the slip and debonding of the link. Therefore, the process of the resistance of the total structure physically becomes nonlinear. Correspondingly, special solution techniques for nonlinear behavior are needed. This can be done using the iterative solution technique with initial stiffness or tangent stiffness. The latter can be formed by assembling the structural stiffness at the beginning of each iteration and this converges faster than the initial stiffness. A third solution strategy for this case is event-to- event technique which is usually employed for the linear stiffnesses between any two "events," which are defined as the intersection point between two linear segments. This also provides means of controlling the equilibrium error. Any significant event occurring within any element determines a substep. The tangent stiffness is modified in each substep, and hence, the solution closely follows the exact response. 3.4 Element Verification 3.4.1 SIMPAL The finite element analysis program SIMPAL [55], is used to implement and verify the element formulation. SIMPAL was chosen for the initial implementation because that was the original implementation done by Cleary [54]. This way, the 3-D aspects could be implemented and verified using Cleary's original program. A table of the element verification is shown in Fig. 3-8 and Fig. 3-9. LOADING 10 10 Y 1 2 NODE THEORY SIMPAL ERROR DISP 2 -.1333 -.1333 .000 DISP 4 -.1333 -.1333 .000 STRESS N/A -80 -80 .000 NODE THEORY SIMPAL ERROR DISP 2 -.1333 -.1333 .000 DISP 4 -.1333 -.1333 .000 STRESS N/A -80 -80 .000 NODE THEORY SIMPAL ERROR DISP 2 -.1333 -.1337 .003 DISP 4 -.1333 -.1337 .003 STRESS N/A -80 -80 .000 * NODE 2 Y DISP =-0.1017-04 Z DISP = -0.8684-05 NODE 4 Z DISP =-0.8684-05 Y DISP =-0.1017-04 2 2 SQRT((.1017) + (.08684) )=0.1337 Fig. 3-8 Link Element Test Using SIMPAL 3 3 THICKNESS = .25 Ks = 3E6 Kn = 6E6 RESULTS LOADING 4 6 Aft Am .P -...... 3 5 1 _~~ RESULTS 8 Aft Ii MP, - Adh - 10 1 I 1 3 5 7 Y 1 3 5 7 Y Z 2 4 6 3 5 Fig. 3-9 Combined Test Model for SIMPAL Z 2 -~araooa a;a8oo~ uru - ia--m i-- i-o -n ---------- HtS~ ^~ k 10 8 10 3.4.2 ANSR The test examples used are the same as those used in the initial element verification using SIMPAL. The results from ANSR [56] are exactly the same as those from SIMPAL. The link element was tested further using a modeled membrane element composed of 22 truss elements as a membrane element was not available at the time of element verification in ANSR. The results are shown in Table 3-1 and the structures used are shown in Fig. 3-10 and Fig. 3-11. Table 3-1 Displacements of for ANSR Truss Model Node Truss Truss Diff. No. only w/ LINK (%) 10-x -.1027e-4 -.1049e-4 2.2 10-y -.1990e-5 -.2010e-5 0.9 ll-x -.9017e-5 -.9211e-5 2.2 11-y -.4906e-6 -.4973e-6 1.4 12-x -.9915e-5 -.1049e-4 2.2 12-y +.9742e-6 +.9703e-6 0.4 LINK ELEMENT 20 .2 20 .2 20 Fig. 3-10 Combined Test Model for ANSR Fig. 3-11 Truss Model for ANSR CHAPTER 4 LINEAR SHELL ELEMENT 4.1 Element Description The shell element is formulated through a combination of two different elements, the membrane element and the Mindlin plate bending element [57]. The Mindlin plate element is different from the Kirchhoff plate element in that the former allows transverse shear deformation while the latter does not. The common portions of the formulation of two elements are 1. Formation of the shape functions. 2. Formation of the inverse of Jacobian matrix. These processes can be done at the same time. The four- to nine-node shape functions and their derivatives in rs- space can be formed and then transformed into xy-space through the inverse of Jacobian matrix. 4.2 Formulation of Shape Functions The formulation of shape functions starts with three basic sets of shape functions shown in Fig. 4-1. 1. The bilinear shape functions for four-node element. 2. The linear-quadratic shape functions for nodes five to eight of the eight-node element. 3. The bubble shape function for node nine of nine-node element. These shape functions can be formulated directly from the local coordinates of the element nodes through the multiplication of the equations of the lines which have zero values in the assumed displacement shapes and the scale factor to force the shape function value to one at the node for which the shape function is formed. The derivative of each shape function with respect to r and s is then evaluated from the shape function expressed in terms of r and s. If node nine exists, the value at node nine of shape functions one to eight must be set to zero. The value of the bilinear shape functions for a four-node element at node nine is one fourth and the value of the linear-quadratic shape functions for the five- to eight-node element at the node nine is one half. This can be forced to zero using the bubble shape function of the nine-node element because this shape function has the value of one at node nine and zero at all other nodes. Therefore the modification is the subtraction of one fourth of the value the bubble shape function has at node nine from the each shape function for the corner nodes and the subtraction of one half of the value of the bubble shape function of node nine for the nodes five to eight, whichever exists. If any of the center nodes on the edge of the element (any one of nodes five to eight) exists, the bilinear shape functions of four-node element must be modified further because the value at center of the edge is one half in those bilinear shape functions. This can be done by subtracting one half of the linear-quadratic shape function for the newly defined center node on the edge of the element from the bilinear shape functions of the two adjacent corner nodes. The value of any five node shape functions at the corner node is zero. Therefore, no further consideration is needed except for the shifting of the shape functions in the computer implementation. These processes are shown in Fig. 4-2. If any of the linear-quadratic shape functions of nodes five to eight is missing, all the linear-quadratic shape functions thereafter and the bubble shape function must be shifted to the proper shape function number. For example, if linear-quadratic shape function five is missing, then the shape functions six to eight must be shifted to five through seven and the bubble shape function must be shifted to the node eight because all of the linear-quadratic shape functions have been defined and numbered as shape functions for the nodes five through eight and the bubble shape function for the node nine. Four Node Element Shape Function for Corner Node Five Node Element Shape Function for Edge Center Node Nine Node Element Shape Function for Element Center Node Fig. 4-1 Three Basic Shape Functions SF 1 SF 2 SF 3 SF 3 SF 4 = (SF 1) (1/4) (SF 3) SF 5 = (SF 2) (1/2) (SF 3) Fig. 4-2 Formation of Shape Functions 4.3 The Inverse of Jacobian Matrix While the generic displacements are expressed in terms of rs-coordinate, the partial differential with respect to the xy-coordinate is needed for the calculation of strain components. Thus the inverse of the Jacobian matrix must be calculated. This can directly be found from the chain rule using the notation (a,b) defined as the partial differential of function a with respect to the variable b for simplicity. f,x = (f,r)(r,x) + (f,s)(s,x) f,y = (f,r)(r,y) + (f,s)(s,y) In matrix form, f,x r,x s,x f,r Jll-1 J12-1 f,r f,y r,y s,y f,s J21-1 J22-1 f,s The inverse of Jacobian matrix But the terms in the inverse of the Jacobian matrix are not readily available because the rs-coordinate cannot be solved explicitly in terms of xy-coordinate. On the other hand, for the isoparametric formulation, the geometry is interpolated using the nodal coordinate values(constants) and the displacement shape functions in terms of r and s. Thus the generic coordinate x and y can be expressed in r and s easily and explicit partial differentials of x and y with respect to r and s can be performed. Therefore the Jacobian matrix is computed and then inverted. The Jacobian matrix is derived by the chain rule. f,r = (f,x)( x,r) + (f,y)( y,r) f,s = (f,x)( x,s) + (f,y)( y,s) In matrix form, f,r x,r y,r f,x Jll J12 f,x f,s x,s y,s f,y J21 J22 f, Jacobian matrix nn Let s be . i=1 where nn = number of nodes (4 to 9). From geometric interpolation equations, x = Z fi*xi y = Z fi*Yi The terms in the Jacobian matrix are J11 = x,r = (z fi*xi),r = Z ((fi,r) xi) J12 = y,r = (Z fi*yi),r = Z ((fi,r) yi) J21 = x,s = (Z fi*xi),s = Z ((fi,s) xi) J22 = y,s = (Z fi*yi),s = Z ((fi,s) yi) xi, yi are coordinate values of the element and are constants and therefore can be taken out of the partial differentiation. The inverse of two-by-two Jacobian matrix can be found -i Jl-1 = r,x = J22 / det(J) -i J12 = s,x = -J12 / det(J) -I J21- = r,y = -J21 / det(J) -i J22- = s,y = J11 / det(J) where det(J) = J11J22 12J21 4.4 Membrane Element The formulation of the membrane element used for the implementation follows the procedure shown on pages 115 through 118 in reference [57]. The ( ) symbol will be used for the column vectors. Nodal displacements are the nodal values of two in- plane translations and denoted as {ui vi}T. The generic displacements are defined as two translational displacements at a point and denoted as { u v )T. By the word generic it is meant that the displacement is measured at an arbitrary point within an element. The generic displacements u and v can be calculated using shape functions. The shape function is a continuous, smooth function defined over the closed element domain and is differentiable over the open domain of the element. The shape function is also the contribution of displacement of a node for which the shape function has been defined to the generic displacement. Thus the generic displacement at an arbitrary point can be found by summing up all the contributions of all the nodes of the element. The displacement interpolation equations are u = Z fi Ui v = Z fi Vi In the isoparametric formulation the geometry is interpolated using the same shape function assumed for the displacement interpolation. Therefore, the geometry interpolation is x = Z fi xi y = Z fi Yi where, fi = Shape function for node i. xi, Yi = Coordinates of node i. ui, vi = Displacements at the node i. u, v = Displacements at an arbitrary point within an element. The three in-plane strain components for a membrane element are { E ) = ( ex Cy 7xy ) These strain components can be found through the partial differentials of the generic displacements with respect to xy-coordinates. ex = ux Ey = v,y 7xy = uy + VX Using the inverse of the Jacobian matrix, the strain components can be evaluated. eX = u,x = (u,r)(r,x) + (u,s)(s,x) = (u,r)(J11-1) + (u,s)(J12-1) = ((Efiui),r)(J11-1) + ((EfiUi),s)(J12-1) = Z[(fi,r)(r,x) + (fi,s)(s,x)] ui y = v,y = (v,r)(r,y) + (v,s)(s,y) = (v,r)(J21-1) + (v,s)(J22-1 = ((.fivi),r)(J21-1) + ((Zfivi),s)(J22-1) = [(fi,r)(r,y) + (fi,s)(s,y)] vi 7xy = u,y + v,x = [(u,r)(r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)] = [((Zfiui),r)(J21-1) + ((Zfiui),s)(J22-1)] + [((Zfivi),r)(J11-) + ((Zfivi),s)(J12-1)] = Z[(fi,r)(r,y) + (fi,s)(s,y)] ui + Z[(fi,r)(r,x) + (fi,s)(s,x)] vi New notations are introduced here to simplify the equations. These are ai and bi and defined as follows: ai = (r,x)(fi,r) + (s,x)(fi,s) = fi,x bi = (r,y)(fi,r) + (s,y)(fi,s) = fiY Then the strain terms above become ex = Zaiui = Zf,x ui Ey = Zbivi = Zfiy vi 7xy = Zbiui + Zaivi = Zfi,y ui + Zfi,x vi In matrix form, ex ai 0 ui yE = 0 bi vi 7xy bi ai In symbolic form, [E] = E[Bi]Cqi] where, ai 0 fix 0 [Bi] = 0 bi 0 fi', bi ai fi'y fix and, ui [q] = Vi vi Therefore the strain at an arbitrary point within an element is [e] = [Bl][q1] + [B2][q2] + ... + [B9][q99 = [ B1 B2 B3 B4 B5 B6 B7 B8 B9 ] ql q2 q3 q4 95 q6 q7 q8 q9 The size of the vectors and matrix are [e(3,1)] = [B(3,18)][q(18,1)] In the actual calculation, this can be done by summing up the [Bij[qi] over all the nodes for the given coordinates of the point under consideration, i.e., the coordinates of one of the integration points. The stresses corresponding to the strains are { a } = { ax ay rxy )T The stress-strain relationship of an isotropic material is Ell E12 0 [E] = E21 E22 0 0 0 E33 where, E11 = E22 = E / ( 1 2 ) E12 = E21 = pE / ( 1 p2) E33 = G where, E = Young's modulus p = Poisson's ratio G = shear mQdulus = E / ( 2*(1+p)) The integration over volume must be introduced for the calculation of the element stiffness and equivalent nodal loads of distributed loads or temperature effects. As the thickness of the element is constant, the integration over volume can be changed to the integration over area. The element stiffness related to the degrees of freedom of the node i can be calculated through the volume integration of Bi(2,3)E(3,3)Bi(3,2). [Ki] = BT E Bi dV As [B] and [E] are constant about z, the integration through the thickness from -t/2 to t/2 can be performed on z only and yields T [Ki] = [ Bi(t)EBi] dA A B T - = [ Bi E Bi] dA A where, E = tE The size of membrane element stiffness is 18 by 18. [K] = [ E ] [ B1 B2 B3 ... B8 B9 ] dV (3,3) (3,18) B8 B9 (18,3) Equivalent nodal loads due to body forces on the membrane element are calculated as Pb Jv fTbdV =11 1-1 fTbIJI dr ds in which {b} = ( 0 0 bz T or { 0 b 0 }T or { bx 0 0 } in accordance with the direction of gravity in the coordinate system used. The nonzero quantities bx, by, or bz represent the body force per unit area in the direction of application. Equivalent loads caused by initial(temperature) strains are PO = BTEEO dV JV 1 1 BTEcIJI dr ds S-1 J-i where, (C0)= exxO yyO 0 0 0 )T = {aAT aAT 0 0 0 ) 4.5 Plate Bending Element The formulation of the plate bending element used for the implementation has followed the procedures shown on pages 217 through 221 in reference [57]. The { } symbol will be used for the column vectors. Many plate bending elements have been proposed. The most commonly used are Kirchhoff plate elements and Mindlin plate elements. Kirchhoff theory is applicable to thin plates, in which transverse shear deformation is neglected. The assumptions made on the displacement field are 1. All the points on the midplane(z = 0) deform only in the thickness direction as the plate deforms in bending. Thus there is no stretching of midplane. 2. A material line that is straight and normal to the midplane before loading is to remain straight and normal to the midplane after loading. Thus there is no transverse shear deformation (change in angle from the normal angle). 3. All the points not on the midplane have displacement components u and v only in the x and y direction, respectively. Thus there is no thickness change through the deformation. Strain energy in the Kirchhoff plate is determined entirely by in-plane strains ex, Cy, and 7xy which can be determined by the displacement field w(x,y) in the thickness direction. The interelement continuity of boundary-normal slopes is not preserved through any form of constraint. Mindlin theory considers bending deformation and transverse shear deformation. Therefore, this theory can be used to analyze thick plates as well as thin plates. When this theory is used for thin plates, however, they may be less accurate than Kirchhoff theory because of transverse shear deformation. The assumptions made on the displacement field are 1. A material line that is straight and normal to the midplane before loading is to remain straight but not necessarily normal to the midplane after loading. Thus transverse shear deformation (change in angle from normal angle) is allowed. 2. The motion of a point on the midplane is not governed by the slopes (w,x) and (w,y) as in Kirchhoff theory. Rather its motion depends on rotations Ox and 0 of the lines that were normal to the midplane of the undeformed plate. Thus 0, and 0 are independent of the lateral displacement w, i.e., they are not equal to (w,x) or (w,y). It is noted that if the thin plate limit is approached, -xz = 7yz = 0 because there is no transverse shear deformation. In this case the angles 0x and 0y can be equated to the (w,x) and (w,y) numerically but the second assumption still holds. The stiffness matrix of a Mindlin plate element is composed of a bending stiffness [kb] and a transverse shear stiffness [ks]. [kb] is associated with in-plane strains ex, cy, and 7xy. [ks] is associated with transverse shear strains 7xz and 7yz. As these two groups of strains are uncoupled, i.e., one group of the strains do not produce the other group of strains, the element stiffness can be shown as [82] [k] = ( BbEBb ) dA + (BsEBs) dA because BbEBs = BsEBb = 0 from uncoupling (corresponding E = 0). Each integration point used for the calculation of [kg] places two constraints to a Mindlin plate element, associated with two transverse shear strains lyz and yzx. If too many integration points are used, there will be too many constraints in transverse shear terms, resulting in locking. Therefore, a reduced or selective integration can prevent shear locking. Or, the transverse shear deformation can be redefined to avoid such locking. For example, a bilinear Mindlin plate element responds properly to pure bending with either reduced or selective integration. But full two-by-two integration is used for pure bending, shear strains appear at the Gauss points as shown in Fig 4-3. As the element becomes thin, its stiffness is due almost entirely to transverse shear. Thus, if fully integrated, a bilinear Mindlin plate element exhibits almost no bending deformations, i.e., the mesh "locks" against bending deformations. Nodal displacements for the plate bending consist of one out-of-plane translation and two out-of-plane rotations and are denoted as { wi 0xi 8yi )T. The rotations are chosen independently of the transverse displacement and are not related to it by differentiation. Thus the transverse shear strains jxz and 7yz are considered in the formulation resulting in five strain components. The generic displacements are defined as three translational displacements and denoted as { u v w }T. By the word generic it is meant that the displacement is measured at an arbitrary point within an element. These generic displacements are different quantities from the nodal displacements and therefore must be related to the nodal displacements. The generic displacements u and v can be calculated as functions of the generic out-of-plane rotations using the small strain(rotation) assumption. The relationship between generic displacements and rotation is shown in Fig 4-4. u = zBy v = -ZBx The generic displacements Ox and By can be found using the assumed displacement shape functions and the corresponding nodal displacements Oxi and 0yi. The generic displacement w does not need any conversion because it corresponds to the nodal displacement wi. In the isoparametric formulation the geometry is interpolated using the same shape function assumed for the displacement interpolation. The displacement interpolation is 8x = Z fi 0xi 6y = Z fi 0yi w = Z fi Wi Similarly, the geometric interpolation is x = Z fi xi y = Z fi Yi Zero Shear Strain One Point Gauss Integration I Non-zero Shear Strain Two Point Gauss Integration Fig. 4-3 Shear Strains at Gauss Point(s) i~Amlh Z + u y X Positive small rotational angle about y-axis gives positive generic displacement in x-direction ( u ). Shown is xz-plane. Positive small rotational angle about x-axis gives negative generic displacement in y-direction ( v Shown is yz-plane. Fig. 4-4 Displacements due to Rotations where, fi = shape function for node i xi, Yi = coordinates of node i Therefore, u = Zoy = z z fi 6yi v = -zOx = -z Z fi Oxi W = E fi Wi The five strain components for plate bending element are { ex ey 7xy 7xz Iyz )T. These strain components can be found through the partial differentials of the generic displacements with respect to xy-coordinates. e = uI, Ey = v,y 7xy = uy + V,X 7XZ = u,z + W,X 7yz = vz + w,y Using the inverse of the Jacobian matrix found, the strain components can be evaluated. Ex = u,X = (ZBy),X = (u,r)(r,x) + (u,s)(s,x) = (u,r)(J11-1) + (u,s)(J12-1) = ((ZzfiOyi),r)(J~l-1) + ((zzfiyi) ,) (J12-1) = z Z[(fi,r)(r,x) + (fi,s)(s,x)] Oyi = v,y = (-zOx),y = (v,r)(r,y) + (v,s)(s,y) = (v,r)(J21-1) + (v,s)(J22-1) = ((-zzfioxi),r)(J21-1) + ((-zZfi0xi),s)(J22-1) = -z Z[(fi,r)(r,y) + (fi,s)(s,y)] Oxi 7xy = u,y + v,x = (zoy),y + (-zox),X = [(u,r)(r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)] = [((zzfioyi),r)(J21-1) + ((zZfiyi),s)(J22-1)] + [((-zZfisxi),r)(Jll-1) + ((-zzfixi),s) (J12-1)] = z Z[(fi,r)(r,y) + (fi,s)(s,y)] Oyi -z Z[(fi,r)(r,x) + (fi,s)(s,x)] exi 7xz = (u,z) + (w,x) = (ZOy),Z + w,x = 9y + w,x = Zfisyi + [(w,r)(r,x) + (w,s)(s,x)] = Zfioyi + [((Zfiwi),r)(J11-1) + ((Zfiwi),s) (J12-1) = Zfieyi + [((Zfi,r)wi) (J11-1) + ((Zfi,s)wi)(J12-1)] = ZfiOyi + z[(fi,r)(r,x) + (fi,s)(s,x)] wi lyz = (v,z) + (w,y) = (-z0x),z + w,y = (-OX) + w,y = (-Zfioxi) + [(w,r)(r,y) + (w,s)(s,y)] = (-Zfioxi) + [((Zfiwi),r)(J21-1) +((Zfiwi),s)(J22-1)] = (-Zfioxi) + [((Zfi,r)wi)(J21-1) +((Zfi,s)wi)(J22-1)] = (-Zfi0xi) + Z[(fi,r)(r,y) +(fi,s)(s,y)] Wi New notations are introduced here to simplify the equations. These are ai and bi and defined as follows: ai = (r,x)(fi,r) + (s,x)(fi,s) = bi = (r,y)(fi,r) + (s,y)(fi,s) = Then the strain terms above become f.l i1,x ex = z aioyi Cy = -z Zbixi 7xy = z Zbiyi z Zaixi 7xz = fi0yi + Zaiwi Tyz = fioxi + Zbiwi In matrix form, eX fy 7xy = 7XZ Lyz In symbolic form, [e] = E[Bi][qi] 0 -zbi -zai 0 -fi i zai 0 zbi fi 0 wi exi Oyi where, [Bi] = or, [Bi] = 0 - zbi - zai 0 fi i 0 0 0 fi,x fi'y zai 0 zbi fi 0 0 - zfi,y - zfi,x 0 fi zfi,x 0 zfi,Y fi 0 and, [qi] wi ixi Oyi Therefore the strain at an arbitrary point within an element is [e] = [B][ql] ] + ... + [B2[ + + 9][q9] = [ B1 B2 B3 B4 B5 B6 B7 Bg B9 ] q91 92 93 94 95 96 97 98 99 The size of the vectors and matrix are [E(5,1)] = [B(5,27)][q(27,1)] In the actual calculation, this can be done by summing up the [Bi][qi] over all the nodes for the given coordinates of the point under consideration, i.e., the coordinates of one integration point. The stresses corresponding to the strains are { o ) = ( ax ay rxy rxz yz )TT The stress-strain relationship of an isotropic material is E11 E12 0 0 0 E21 E22 0 0 0 [E] = 0 E33 0 0 0 0 0 E44 0 0 0 0 0 E55 where, E11 = E22 = E / ( 1 p2 ) E12 = E21 = pE / ( 1 p2) E33 = G E44 = E55 = G / 1.2 where, p = Poisson's ratio G = shear modulus = E / ( 2*(1+p)) The form factor 1.2 for the E44 and E55 terms is provided to account for the parabolic distribution of the transverse shear stress rzx over a rectangular section. This form factor 1.2 can be shown from the difference in deflections of a cantilever beam at its free end [58]. Let a beam have a rectangular cross section of dimensions b by t with a length of L. If P is the transverse shear force, then the parabolic distribution of the transverse shear stress rz, is r'X = (3P/2bt3)(t2 4z2) where z = 0 at the neutral axis. Then the transverse shear strain energy from the parabolic distribution can be calculated by Us = (1/2)V (zx 2/G) dV = (1/2) [((3P/2bt3)(t2 4z2))2 / G] dV = [(1/2)(3P/2bt3)2]/G (t2 4z22 dAdz = (area)[(1(/)(3P/2bt3)2]/G (t2 4z2)2 dz = bL[(1/2)(3P/2bt3)2]/G (t2 4z22 dz = 1.2(P2L/btG)/2 While the transverse shear strain energy from the constant distribution is Us = (1/2) I(rx2/G) dV = (1/2) ((P/bt)2 /G) dV = [(1/2)(P/bt)2/G] (btL) = (p2L/btG)/2 This result suggests the view that a uniform stress P/bt acts over a modified area bt/1.2, so that the same Us results. Therefore the deflection at the free end of a cantilever beam with parabolically distributed transverse shear stress will be 1.2 times that with constantly distributed transverse shear stress. The Mindlin plate is generalized from the Mindlin beam. Thus the higher transverse shear stiffness from the assumption of constant transverse shear stress has been reduced by dividing the corresponding elastic constants by the factor 1.2 for flat plate element. The reduced stiffness will produce the more flexible response in shear that is expected from the actual parabolic distribution. The integration over volume must be introduced for the calculation of the element stiffness and equivalent nodal loads of distributed loads or temperature effects. As the thickness of the element is constant, the integration over volume can be changed to the integration over area. This can be accomplished through the partition of the strain-nodal displacement matrix [Bi] as follows: 0 0 zfj,x 0 zfi,y 0 [Bi] = 0 zfi,x zfi,y fi,x 0 fi fi,Y fi 0 The submatrices are named as follows: BiA ZBiA [Bi] = BiB BiB The element stiffness can be calculated through the T volume integration of Bi (3,5)E(5,5)Bi(5,3). Thus the [E] matrix is to be partitioned as follows: Ell E12 0 0 0 E21 E22 0 0 0 [E] = 0 0 E33 0 0 0 0 0 E44 0 0 0 0 0 E55 The submatrices are named as follows: E EA 0 E = 0 EB Then the stiffness of the element is FT B -T T [EA 0 ziA 1 [Ki] =I B E Bi dV = zBiA [ A B dV 0 EB BiB 2 -T T = [ BA EA BA] + [BB EB BB] As [B] and [E] are constant about z, the integration through the thickness from -t/2 to t/2 can be performed on z only and yields = T 3 [Ki] = [ BA(t3/12)EABA + BB(t)EBBB] dA I -T- - = [ BAEABA + BBEBBB] dA where, EA = (t3/12)EA and EB = tEB Then [Ki] can be rewritten as matrix equation as follows: -T T EA 0 BiA [Ki] = [ BiA BiB A dA 0 E B B iB d -T- - = BiE Bi dA The size of plate element stiffness will be 27 by 27. [K] = [ E ] (5,5) [ B1 B2 B3 ... Bg Bg ] dV (5,27) B8 B9J (27,5) The strain-nodal displacement matrix from which the constant thickness is taken out is defined as [Bi]. [Bi] = 0 0 0 fi,x fi,Y 0 - fi,y - fi,x 0 fi fi,x 0 fiy fi 0 BiA] BiB Equivalent nodal loads due to body forces on the plate element are calculated as Pb = fTb dV = 1 V -1 -i fTblJI dr ds in which {b} = { 0 0 bz )T or ( 0 by 0 }T or { bx 0 0 T in accordance with the direction of the gravity in the coordinate system used. The nonzero quantities bx, by, or bz represent the body force per unit area in the direction of application. Equivalent loads caused by initial strains are PO = BTEeo dV JV 1 1 = BTEO#IJI dr ds where, {40)= ( xx0 Oyy0 xy 0 0 )T = { aAT/2 aAT/2 0 0 0 T The stresses can be calculated from the equation [a] = [E][E] The corresponding generalized stresses, if desired, may be computed from M = ( Mxy My y QM Q y T = E ( B q 40 ) It is noted that the generalized stresses are actually moment and shear forces applied per unit length of the edge of the plate element. Therefore these can also be turned into common stresses using the formulation for the bending stress calculation. The moment of inertia for the unit length of the plate is t3 / 12. Then the in-plane stress at a point along the thickness can be calculated as a = Mz / I = M(t/2) / (t3/12) = 6M / t2 91 The transverse shear stresses can be found as r =Q/ t But this may be multiplied by a factor of 1.5 to get the maximum shear stress at a point on a neutral surface because the transverse shear stresses show parabolic distribution while the calculated stresses are average stresses coming from the assumption of a constant transverse strain along the element z axis. CHAPTER 5 NONLINEAR SHELL ELEMENT 5.1 Introduction The nonlinearities included in the formulation of the Mindlin flat shell element are those due to large displacements and those due to initial stress effects(geometric nonlinearity). The large displacement effects are caused by finite transverse displacements. These effects are taken into account by coupling transverse displacement and membrane displacements. The initial stress effects are caused by the stresses at the start of each iteration. These stresses change the element stiffness for the current iteration. These effects are evaluated directly from the stresses at the start of each iteration and are included in the element stiffness formation. The total Lagrangian formulation is used. If the updated Lagrangian formulation is used, the element coordinate system cannot be easily formed for the next iteration because the deformed shell is not usually planar [26]. The symbol {} is used for a column matrix (a vector) and the symbol [] is used for a matrix of multiple columns and rows throughout the chapter. 5.2 Element Formulation The generic displacements of Mindlin type shell element are translational displacements {u v w)T and denoted as {U}. The displacements and rotations at a point on the midplane are (uo vo wo 9x 0y)T and denoted as (Uo}. The generic displacements can be expressed in terms of the midplane displacements and z as u = Uo(x,y) + zOy(x,y) v = Vo(x,y) zOx(x,y) (5.1) w = Wo(x,y) The linearized incremental strain from Eq. (2.17) is e = ( ui,j + uj,i + uk,i uk,j + k,j k,i) (5.2) This equation can be written out for the strain terms to be used for shell element using the generic displacements {u, v, w)T exx = eyy= exy = u, U,y i ( 2' ez = i ( eyz = ( + tu u, + tv, v, + tw,x , + t,y U,y + t,y V,y + tw,y Wy U,y + V,x + tU, U,y + t,x V,y + tx w,y + u, tuy + V,x tvy + W,x ty ) ,z + w,x + tU u,z + tVx ,z + tW,x w,Z + u,x tu,z + V,x tv, + ,x tw,z ) V,z + w,y + tu,y u,z + t,y v,z + tWy w,z + u,y tU, + V,y tV, + w,y tW, ) (5.3) The derivatives of inplane displacements u and v with respect to x, y, and z are assumed to be small and thus the second order terms of these quantities can be ignored through von Karman assumption from Eqs. (2.44) [20, 21]. Furthermore the transverse displacement w is independent of z for the shell element which means that w,z is zero. Then Eqs. (5.3) can be reduced to xx = u, + tw,x w,x eyy = Uy + ty Wy exy = ( Uy + Vx + tw,x Wy + ,x wy ) exz = ( u,z + wx ) eyz = 1 ( v,z + W,y ) (5.4) The incremental Green's strains, sometimes called engineering strains, can then be shown as x = exx = Ux + tw,x w,x 4e = eyy = Uy + tw,y Wy Ixy = 2exy = Uy + v,x + tw,x Wy + Wx ty 7xz = 2exz = u,z + wx yz e = e = + Wy (5.5) It is noted that the linearized nonlinear strains are left only for inplane strain terms. By substituting Eqs. (5.1) into Eqs. (5.5), the Green's strain can be expressed in terms of midplane displacements. |

Full Text |

NONLINEAR GAP AND MINDLIN SHELL ELEMENTS
FOR THE ANALYSIS OF CONCRETE STRUCTURES By KOOKJOON AHN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA UNIVERSITY OF FLORIDA LIBRARIES 1990 ACKNOWLEDGEMENTS I would like to express my deep gratitude to professor Marc I. Hoit for his invaluable guidance and support. I also thank professors Clifford 0. Hays, Jr., Fernando E. Fagundo, John M. Lybas, and Paul W. Chun for being on my committee. I also express my gratitude to professor Duane S. Ellifritt for his help as my academic advisor at the start of my Ph.D. program. Thanks are also due to all the other professors not mentioned above and my fellow graduate students, Alfredo, Jose, Mohan, Prasan, Prasit, Shiv, Tom, Vinax, and Yuh-yi. Finally, I am thankful to every member of my family, especially my wife and son, for their patience and support in one way or another. The work presented in this dissertation was partially sponsored by the Florida Department of Transportation. ii TABLE OF CONTENTS gage, ACKNOWLEDGEMENTS Ãœ ABSTRACT v CHAPTERS 1 INTRODUCTION 1 1.1 General Remarks 1 1.2 Link Element 2 1.3 Shell Element 5 1.4 Literature Review 5 2 GENERAL THEORIES OF NONLINEAR ANALYSIS 13 2.1 Introduction 13 2.2 Motion of a Continuum 14 2.3 Principle of Virtual Work 16 2.4 Updated Lagrangian Formulation 18 2.5 Total Lagrangian Formulation 22 2.6 Linearization of Equilibrium Equation 26 2.7 Strain-Displacement relationship Using von Karman Assumptions 28 3 THREE-DIMENSIONAL LINK ELEMENT 34 3.1 Element Description 34 3.2 Formation of Element Stiffness 43 3.3 Solution Strategy 51 3.4 Element Verification 52 4 LINEAR SHELL ELEMENT 59 4.1 Introduction 59 4.2 Formulation of Shape Functions 59 4.3 The Inverse of Jacobian Matrix 64 4.4 Membrane Element 66 4.5 Plate Bending Element 73 5 NONLINEAR SHELL ELEMENT 92 5.1 Introduction 92 5.2 Element Formulation 93 iii 5.3 Finite Element Discretization 100 5.4 Derivation of Element Stiffness 113 5.5 Calculation of Element Stiffness Matrix .... 115 5.6 Element Stress Recovery 119 5.7 Internal Resisting Force Recovery 122 6 NONLINEAR SHELL ELEMENT PERFORMANCE 126 6.1 Introduction 126 6.2 Large Rotation of a Cantilever 126 6.3 Square Plates 133 7 CONCLUSIONS AND RECOMMENDATIONS 143 APPENDICES A IMPLEMENTATION OF LINK ELEMENT 146 B IMPLEMENTATION OF LINEAR SHELL ELEMENT 170 C IMPLEMENTATION OF NONLINEAR SHELL ELEMENT 219 REFERENCES 230 SUPPLEMENTAL BIBLIOGRAPHY 238 BIOGRAPHICAL SKETCH 240 ÃV Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy NONLINEAR GAP AND MINDLIN SHELL ELEMENTS FOR THE ANALYSIS OF CONCRETE STRUCTURES By KOOKJOON AHN August, 1990 Chairman: Marc I. Hoit Major Department: Civil Engineering Segmental post-tensioned concrete box girders with shear keys have been used for medium to long span bridge structures due to ease of fabrication and shorter duration construction. Current design methods are predominantly based on linear elastic analysis with empirical constitutive laws which do not properly quantify the nonlinear effects, and are likely to provide a distorted view of the factor of safety. Two finite elements have been developed that render a rational analysis of a structural system. The link element is a two-dimensional friction gap element. It allows opening and closing between the faces of the element, controlled by the normal forces. The Mindlin flat shell element is a combination of membrane element and Mindlin plate element. This element considers the shear responses along the element v thickness direction. The shell element is used to model the segment itself. The link element is used to model dry joints and has shown realistic element behavior. It opens under tension and closes under compression. The link element has shown some convergence problems and exhibited a cyclic behavior. The linear Mindlin shell element to model the concrete section of the hollow girder showed an excellent response within its small displacement assumption. The nonlinear Mindlin flat shell element has been developed from the linear element to predict large displacement and initial stress (geometric) nonlinearities. The total Lagrangian formulation was used for the description of motion. The incremental-iterative solution strategy was used. It showed satisfactory results within the limitation of moderate rotation. Three areas of further studies are recommended. The first is the special treatment of finite rotation which is not a tensorial quantity. The second is the displacement dependent loadings commonly used for shell elements. The third is the material nonlinearity of concrete which is essential to provide realistic structural response for safe and cost effective designs. vi CHAPTER 1 INTRODUCTION 1.1 General Remarks In the past few decades segmental post-tensioned concrete box girders have been used for medium to long span bridge structures. Highway aesthetics through long spans, economy due to ease of fabrication, shorter construction duration are some of the many advantages of precast segment bridge construction. The segments are hollow box sections, match cast with shear keys in a casting yard, then assembled in place, leaving the joints entirely dry. The shear keys are meant to transfer service level shears and to help in alignment during erection. Current design methods are heavily based on linear elastic analysis with empirically derived constitutive laws assuming homogeneous, isotropic materials. The behavior under load of the bridge system is very complex. Analyses which do not properly quantify the nonlinear effects including the opening of joints in flexure, are likely to provide a distorted view of the factor of safety existing in a structural system between service loads and failure. The 1 potential sliding and separation at the joints due to shear, and by deformations generated by temperature gradients over the depth and width of the box further complicate the problem [1]. Two finite elements have been developed that render a rational analysis of the system. The link element is a two- dimensional friction gap element. It allows sliding between the faces of the element, controlled by a friction coefficient and the normal forces. It also accounts for zero stiffness in tension and a very high stiffness under compression. This link element was borrowed from rock mechanics and newly applied to this problem to model the dry joint between the segments. The Mindlin flat shell element is a combination of membrane element and Mindlin plate element. This element considers the shear responses along the element thickness direction. The shell element was used to model the segment itself. This element can handle large displacement and geometric nonlinearities. 1.2 Link Element A link element is a nonlinear friction gap element used to model discontinuous behavior in solid mechanics. Some examples are interfaces between dissimilar materials and joints, fractures in the material, and planes of weakness. These have been modeled using constraint equations, discrete 3 springs and a quasi-continuum of small thickness [2]. The following characteristics of prototype joints were considered. 1. Joints can be represented as flat planes. 2. They offer high resistance to compression in the normal direction but may deform somewhat modeling compressible filling material or crushable irregularities. 3. They have essentially no resistance to a net tension force in the normal direction. 4. The shear strength of joints is frictional. Small shear displacements probably occur as shear stress builds up below the yield shear stress. A model for the mechanics of jointed rocks was developed by Goodman [3], The finite element approximation was done as a decomposition of the total potential energy of a body into the sum of potential energies of all component bodies. Therefore, element stiffness is derived in terms of energy. The Goodman element was tested for several modeled cases. 1. Sliding of a joint with a tooth. 2. Intersection of joints. 3. Tunnel in a system of staggered blocks. A problem with the Goodman's two dimensional model is that adjacent elements can penetrate into each other. Zienkiewicz et al. [4] advocate the use of continuous 4 isoparametric elements with a simple nonlinear material property for shear and normal stresses, assuming uniform strain in the thickness direction. Numerical difficulties may arise from ill conditioning of the stiffness matrix due to very large off-diagonal terms or very small diagonal terms which are generated by these elements in certain cases. A discrete finite element for joints was introduced which avoids such theoretical difficulties and yet is able to represent a wide range of joint properties, including positive and negative dilatency (expansion and compaction accompanying shear) [3]. The element uses relative displacements as the independent degrees of freedom. The displacement degrees of freedom of one side of the slip surface are transformed into the relative displacements between the two sides of the slip surface. This element has been incorporated into a general finite element computer program [5]. The use of relative displacement as an independent degree of freedom to avoid numerical sensitivity is discussed in detail [6]. An isoparametric formulation is given by Beer [2]. A four-node, two-dimensional link element and a eight-node plate bending element were used to model the dry jointed concrete box girder bridge with shear keys [7]. 5 1.3 Shell Element The shell element is formulated through the combination of two different elements, the membrane element and the Mindlin plate bending element. The Mindlin plate element is different from the Kirchhoff plate element in that the former allows transverse shear deformation while the latter does not. The nonlinearities included in the formulation of the flat shell element is for large displacement and geometric nonlinearity due to initial stress effects. The large displacement effects are caused by finite transverse displacements. These effects are taken into account by coupling transverse displacement and membrane displacements. The initial stress effects are caused by the actual stresses at the start of each iteration. These stresses change the element stiffness for the subsequent iteration. These effects are evaluated directly from the stresses at the start of each iteration and included in the element stiffness. 1â™¦4 Literature Review The purpose of nonlinear analysis is to develop the capability for determining the nonlinear load-deflection behavior of the structures up to failure so that a proper 6 evaluation of structural safety can be assured. There are two general approaches for nonlinear analysis. The first approach is a linearized incremental formulation by reducing the analysis to a sequence of linear solutions. The second approach is mathematical iterative techniques applied to the governing nonlinear equations [8]. The advantage of the incremental approach results from the simplicity and generality of the incremental equations written in matrix form. Such equations are readily programmed in general form for computer solutions [9]. A generalized incremental equilibrium equation for nonlinear analysis can be found in [10, 11, 12]. The formulation is valid for both geometrical and material nonlinearities, large displacements and rotations, conservative and displacement dependent (nonconservative) loads. There are two frames for the description of motion. The difference lies in the coordinate systems in which the motion is described. These are the total Lagrangian formulation which refers to the initial configuration [10, 11] and the updated Lagrangian formulation which refers to the deformed configuration [12]. There have evolved two types of notations in the description of motion. A correlation is given these two notations, the B-notations and the N-notations, currently used in the Lagrangian formulation of geometrically nonlinear analysis [13]. A short history of early theoretical development of nonlinear analysis can be found in [9, 14]. One form of updated Lagrangian formulation is the corotational stretch theory [15]. Shell elements are often derived from governing equations based on a classical shell theory. Starting from the field equations of the three-dimensional theory, various assumptions lead to a shell theory. This reduction from three to two dimensions is combined with an analytical integration over the thickness and is in many cases performed on arbitrary geometry. Static and kinematic resultants are used. These are referred to as classical shell elements. Alternatively, one can obtain shell elements by modifying a continuum element to comply with shell assumptions without resorting to a shell theory. These are known as degenerated shell elements. This approach was originally introduced by Ahmad, Irons, and Zienciewicz [16, 17]. Other applications can be found in [8, 18-25]. In large rotation analysis, the major problems arise from the verification of the kinematic assumptions. The displacement representation contains the unknown rotations of the normal in the arguments of trigonometric functions. Thus additional nonlinearity occurs. Further difficulties enter through the incremental procedure. Rotations are not tensorial variables, therefore, they cannot be summed up in an arbitrary manner [17]. One of the special treatment of finite rotation is that the rotation of the coordinate system is assumed to be accomplished by two successive rotations, an out-of-plane rotation followed by an in-plane rotation using updated Lagrangian formulation [26, 27]. Usually the loadings are assumed to be conservative, i.e., they are assumed not to change as the structure deforms. One of the well known exceptions is pressure loading which can be classified as conservative loading or a nonconservative loading [28]. Another is the concentrated loading that follows the deformed structure. For example, a tip loading on a cantilever beam will change its direction as the deformation gets larger. As loading is a vector quantity, the change in direction means that the loading is not conservative. Sometimes this is called a follower loading. The governing equation for large strain analysis can be used for small increments of strain and large increments of rotations [29]. This can be regarded as a generalization of nonlinearity of small strain with large displacement. If large strain nonlinearity is employed, an important question is which constitutive equation should be used [9]. The degree of continuity of finite element refers to the order of partial differential of displacements with respect to its coordinate system. Order zero means displacement itself must be continuous over the connected elements. Order one means that the first order differential of displacement must be continuous. Thus the higher order the continuity requirement, the higher the order of assumed displacement (shape, interpolation) function. Mindlin-Reissner elements require only CÂ° continuity, so that much lower order shape functions can be used, whereas in Kirchhoff-Love type elements, high order shape functions must be used to satisfy the C1 continuity. Furthermore, since Mindlin-Reissner elements account for transverse shear, these elements can be used for a much larger range of shell thickness. The relaxed continuity requirements which permit the use of isoparametric mapping techniques gives good computational efficiency if formulated in the form of resultant stresses [30]. Unlike compressible continuum elements, which are quite insensitive to the order of the quadrature rule, curved CÂ° shell elements require very precisely designed integration scheme. Too many integration points result in locking phenomena, while using an insufficient number of quadrature points results in rank deficiency or spurious modes [30]. While Gauss point stress results are very accurate for shallow and deep, regular and distorted meshes, the nodal stresses of the quadratic isoparametric Mindlin shell element are in great error because of the reduced integration scheme which is necessary to avoid locking [31]. The degenerate solid shell element based on the conventional assumed displacement method suffers from the locking effect as shell thickness becomes small due to the condition of zero inplane strain and zero transverse shear strain. Element free of locking for linear shell analysis using the formulation based on the Hellinger-Reissner principle with independent strain as variables in addition to displacement is presented in [32]. Shear locking is the locking phenomenon associated with the development of spurious transverse shear strain. Membrane locking is the locking phenomenon associated with the development of non-zero membrane strain under a state of constant curvature. Machine locking is the locking phenomenon associated with the different order of dependence of the flexural and real transverse shear strain energies on the element thickness ratio, and it is therefore strictly related to the machine finite word length [33]. Some of the solutions are as follows: 1. Assumed strain stabilization procedure using the Hu- Washizu or Hellinger-Reissner variational principles [33]. 2. The assumed strain or mixed interpolation approach [34, 35] . 3. Suppressing shear with assumed stress/strain field in a hybrid/mixed formulation [30]. Suppression of zero energy deformation mode using assumed stress finite element [36]. 4. Coupled use of reduced integration and nonconforming modes in quadratic Mindlin plate element [37]. 5. Higher order shallow shell element, with 17 to 25 nodes [38, 39]. 6. Global spurious mode filtering [40]. 7. Artificial stiffening of element to eliminating zero energy mode, special stabilizing element [41]. In the faceted elements, due to the faceted approximation of the shell surface, coupling between the membrane and the flexural actions is excluded within each individual element, the coupling is, however, achieved in the global model through the local to global coordinate transformation for the elements [39]. In geometrically nonlinear analysis with flat plate elements, it is common to use the von Karman assumptions when evaluating the strain-displacement relations. The assumption invoked is that the derivatives of the inplane displacements can be considered to be small and hence their quadratic variations neglected. However, this simplification of the nonlinear strain-displacement relationship of the plate, when used in conjunction with the total Lagrangian approach, implies that the resulting formulation is valid only when the rotation of the element from its initial configuration is moderate. Thus for the total Lagrangian approach to handle large rotations, simplifications of the kinematic relationship using the von Karman assumptions is not permitted [39]. Some of the special solution strategies to pass the limit point are given in references [25, 42-48]. A limit point is characterized by the magnitude of tangential stiffness. It is zero or infinite at a limit point. Thus conventional solution strategies fail at the limit point. Arc length method was introduced in reference [42], and applied in the case of cracking of concrete [43], This was improved with line search and accelerations in references [44, 45]. Line search means the calculation of an optimum scalar step length parameter which scales the standard iterative vector. This can be applied to load and displacement control and arc length methods [44]. The traditional solution strategies are iterative solutions, for example, Newton-Raphson, constant stiffness, initial stiffness, constant displacement iteration, load increment [46] along with Cholesky algorithm with shifts for the eigensolution of symmetric matrices [47] for element testing for spurious displacement mode. The vector iteration method without forming tangent stiffness for the postbuckling analysis of spatial structures is also noted [48]. The linearized incremental formulation in total Lagrangian description has been used for this study of large displacement nonlinearity including initial stress effects. The special treatment of finite rotation is not included in the current study. Material nonlinearity is also excluded. CHAPTER 2 GENERAL THEORIES OF NONLINEAR ANALYSIS 2.1 Introduction The incremental formulations of motion in this chapter closely follow the paper by Bathe, Ramm, and Wilson [11]. Other references are also available [9, 10, 12, 14, 15, 49, 50, 51]. Using the principle of virtual work, the incremental finite element formulations for nonlinear analysis can be derived. Time steps are used as load steps for static nonlinear analysis. The general formulations include large displacements, large strains and material nonlinearities. Basically, two different approaches have been pursued in incremental nonlinear finite element analysis. In the first, Updated Lagrangian Formulation, static and kinematic variables, i.e., forces, stresses, displacements, and strains, are referred to an updated deformed configuration in each load step. In the second, Total Lagrangian Formulation, static and kinematic variables are referred to the initial undeformed configuration. It is noted that using either of two formulations should give the same results because they are based on the 13 14 same continuum mechanics principles including all nonlinear effects. Therefore, the question of which formulation should be used merely depends on the relative numerical effectiveness of the methods. 2.2 Motion of a Continuum Consider the motion of a body in a Cartesian coordinate system as shown in Fig. 2-1. The body assumes the equilibrium positions at the discrete time points 0, dt, 2dt, ..., where dt is an increment in time. Assume that the solution for the static and kinematic variables for all time steps from time 0 to time t, inclusive, have been solved, and that the solution for time t+dt is required next. The superscript on left hand side of a variable shows the time at which the variable is measured, while the subscript on left hand side of a variable indicates the reference configuration to which the variable is measured. Thus the coordinates describing the configuration of the body using index notation are At time 0 = Â°xÂ¿ At time t = ^xÂ¿ At time t+dt = t+dtxi 15 Fig. 2-1 Motion of a Body The total displacements of the body are At time 0 At time t u j 'Uj At time t+dt = _ t+dt Uj The configurations are denoted as At time 0 = Â°C At time t = UC = At time t+dt = t+dt( Thus, the updated coordinates at time t and time t+dt are ^ = Â°Xi + ^ t+dtx. = Ox. + t+dtu. The unknown incremental displacements from time t to time t+dt are denoted as (Note that there is no superscript at left hand side.) u. = t+dtu. 'U: (2.1) 2.3 Principle of Virtual Work Since the solution for the configuration at time t+dt is required, the principle of virtual work is applied to the equilibrium configuration at time t+dt. This means all the variables are those at time t+dt and are measured in the configuration at time t+dt and all the integrations are performed over the area or volume in the configuration at time t+dt. Then the internal virtual work (IVW) by the corresponding virtual strain due to virtual displacement in t+dtC is IVW = t+dt .. t+dt s ' t+dt -.. t+dt 13 â€¢ - - (t+dt dv) (2.2) where, t+dt . , t+dt x3 Stresses at time t+dt measured in the configuration at time t+dt. = Cauchy stresses. = True stresses. *;+d^ eji = Cauchy's infinitesimal(linear) strain tensor t+dt referred to the configuration at time t+dt. = Virtual strain tensor. S Delta operator for variation. and the external virtual work (EVW) by surface tractions and body forces is EVW = t+dt x. Â£ t+dt t+dt k t+dt k 0 _ (t+dt dAj where, t+dt a. t+dt k t+dt t+dt uk t+dt t+dt b X t+dt u t+dt k t+dt p t+dt k 0 _ (t+dt dV) (2.3) Surface traction at time t+dt measured in the configuration at time t+dt. Total displacement at time t+dt measured in the configuration at time t+dt. 6 *"+u^ = Variation in total displacement at time + t+dt measured in configuration at time t+dt = Virtual displacement. = Mass density per unit volume. = Body force per unit mass. and all the integration is performed over the area and the volume at time t+dt. 2.4 Updated Laqranqian Formulation t+dt t+dt 9 t+dt H t+dt bk In this formulation all the variables in Eqs. (2.2) and (2.3) are referred to the updated configuration of the body, i.e, the configuration at time t. The equilibrium position at time t+dt is sought for the unknown incremental displacements from time t to t+dt. The internal virtual work, the volume integral in Eq. (2.2) measured in the configuration at time t+dt can be transformed to the volume integral measured in the configuration at time t in a similar manner that is given in reference [52] IVW t+dt t+dt ij t+dt t+dt (t+dt dv) t+dt c t Sij t+dt t e ij (fc dV) = EVW (2.4) where, t+dt s _ second Piola-Kirchhoff (PK-II) stresses t J measured in the configuration at time t. S eji = Variations in Green-Lagrange (GL) strain tensor measured in the configuration at time t. The PK-II stress tensor at time t+dt, measured in the configuration at time t can be decomposed as t+dt e t Sij ID tsij (2.5) because the second PK-II stress at time t measured in the configuration at time t is the Cauchy stress. From Eq. (2.1), the total displacements at time t+dt measured in the configuration at time t is t+dt Ui = t u. + ^ u. â€ž t u. (2.6) This is true because the displacement at time t measured in the configuration at time t is zero. In other words, the displacement at time t+dt with respect to the configuration at time t is the incremental displacement itself. And the GL strain is defined in terms of displacement as E ij = * (2.7) E and U are used in the places of e and u to avoid confusion between general strain and incremental strain, and between 20 general displacement and incremental displacement used in this formulation. It is noted that these finite strain components involve only linear and quadratic terms in the components of the displacement gradient. This is the complete finite strain tensor and not a second order approximation to it. Thus this is completely general for any three-dimensional continuum [52]. Then the GL strain tensor at time t+dt measured at time t can be calculated as t+dt t + {(-tuk + ^tUk + (2.8) where teij teij + t^ij Incremental GL strain in **C Linear portion of incremental GL strain in **C This is linear in terms of unknown incremental displacement. Linearized incremental GL strain in ^C. 21 tÂ»?ij i tUjcOJ = Nonlinear portion of incremental GL strain. The variations in Green-Lagrange strain tensor at time t+dt measured in the configuration at time t can be shown as using Eg. (2.8) . 8 t+dt Â£ij 8 ( t^ij +tei3 J 8 teij (2.9) 4Â» 4â€œ t t S^elj = 0 because is known. There is no variation m known quantity. Then using the Eqs. (2.5), (2.8) and (2.9), the integrand of Eq. (2.4) becomes t+dt s. s t+dt e. . _ ^ t 'i: t ID + Si j ) 6 . e. t 'ID t ID ' " t ID " (trij + tSiD)(5 teiD + 5t'?ij) =tSiD(5teiD + 5t'?iD) + trij 5 teiD + triD 6 t^D =tSiD 5 teiD + tfij 8 teiD + t rij 8 t^ij (2.10) The constitutive relation between incremental PK-II stresses and GL strains are tSij tCiDkj tCkl (2.11) 22 Finally the equilibrium Eq. (2.4) from the principle of virtual work using Eqs. (2.10) and (2.11) is tcijkl tekl 8 tâ‚¬ij fcdV + trij 'dV = EVW (2.12) where, the external virtual work must be transformed from t+dtc to tC. This is not applicable to conservative loading, i.e., loading that is not changed during deformation. EVW t+dt t t+dt t uk (fcdA) t+dt â€ž t p t+dt y. t bk s t+dt t Uk - (fcdV) (2.13) and this is the general nonlinear incremental equilibrium equation of updated Lagrangian formulation. 2.5 Total Lagrangian Formulation Total Lagrangian formulation is almost identical with the updated Lagrangian formulation. All the static and kinematic variables in Eqs. (2.2) and (2.3) are referred to the initial undeforxned configuration of the body, i.e, the configuration at time 0. The terms in the linearized strain are also slightly different from those of updated Lagrangian formulation. The volume integral in Eq. (2.2) measured in the configuration at time t+dt can be transformed to the volume integral measured in the configuration at time 0 as [52] t+dt , , t+dt XD t+dt e.. t+dt iD (t+dtdv) t+dt t+dt ID (Â°dV) (2.14) where, t+dtg _ second Piola-Kirchhoff stress tensor Â° measured in the configuration at time 0. 5 t+dtÂ£ _ variations in Green-Lagrange (GL) strain Â° tensor measured in the configuration at time 0. The PK-II stress tensor at time t+dt, measured in the configuration at time 0 can be decomposed as t+dt o (2.15) From Eq. (2.1), the total displacements at time t+dt measured in the configuration at time 0 is 24 t+dtu, = tu, u. (2.16) Then the GL strain tensor at time t+dt measured at time 0 can be calculated as t+dt. 3 â€˜ v qâ€œ3 ' 0U j ^ ' i O 'ij - i [(^ + 0Ui)(j + (V +< o XJ o XJ o +3 Oeij + oeij where, t oâ‚¬ij oeij eii o XJ oâ€™ij = 1 (tui -A + tu 4 + tuk 1 tuk -Â¡) = GL strain at time t in Â°C. ei-i + 'ii-i o +J o XJ (2.17) = Incremental GL strain in Â°C. = | ( \li -s + Uj i + tUl_ â€¢ u> + tu . u> i ) = Linear portion of incremental GL strain in Â°C. = This is linear in terms of unknown incremental displacement. = Linearized incremental GL strain in Â°C. " i = Nonlinear portion of incremental GL strain. The variations in Green-Lagrange strain tensor at time t+dt measured in the configuration at time t can be shown as using Eq. (2.17). c t+dt 0 â‚¬ ID 5 < oâ€˜ij + oij > = 5 Sij (2.18) S = 0 because te-!-i is known. There is no variation in o 1D o known quantity. Then using the Eqs. (2.15), (2.17) and (2.18), the integrand of Eq. (2.14) becomes t+dt o sii 6 t+dtfii = ( tsii + sii ) Â« J-D o 1D o 1D o 1D o 1D â– (oSij + (5 oeij + 5 oâ€™ii1 =oSij(i oeiJ + * O "ij â€™ +otsij Soeij +oSij S = S e-s-t + 5 S rii-: (2.19) o XD o 1D o -LD o 1D o ^-J o 1D The constitutive relation between incremental PK-II stresses and GL strains are *ij oCijkj 0ekl (2.20) Finally the equilibrium Eq. (2.14) from the principle of virtual work using Eqs. (2.19) and (2.20) is 26 en 5 e-s-Ã Â°dV 'ijkl 0ekl 0 0eij tsi-i 5 Â«Ãi-i Â°dv o -LJ O XJ = EVW - tsii S e^ Â°dV o 1J o +J (2.21) where, the external virtual work must be transformed from t+dtC to Â°C. This is not applicable to conservative loading, that is, loading that is not changed during deformation. EVW = 1 X -p -p + â€¢P 0 1 6 ' t+dt u 1 o K â€¢ L J - (Â°dA) t+dt o p t+dt bk 1 o K 6 t+dt .. uk O K â€¢ - (Â°dV) (2.22) and this is the general nonlinear incremental equilibrium equation of total Lagrangian formulation. 2.6 Linearization of Equilibrium Equation The incremental strain from time t to t+dt is assumed to be linear, i.e., ekl = ekl in Eqs. (2.11), (2.12), (2.20), and (2,21). 27 For the updated Lagrangian formulation, tSij ' tCijkj tekl (2.23) and, tcijkl tekl 5 teij dv t'U s tâ€™ij tdV = EVW - trij 5 teij tdV (2.24) For the total Lagrangian formulation, oSij 0Cijkj o6kl (2.25) and, Cijkl ekl ^ ein O Jvx o J-J 3dV + S o^ij Â°dV = EVW (2.26) It should be noted that the surface tractions and the body forces in the calculation of external virtual work may be treated configuration dependent when the structure undergoes large displacements or large strains. If this is the case, the external forces must be transformed to the current configuration at each iteration [10, 11, 12]. 2.7 Strain-Displacement Relationship Using the von Karman Assumptions The nonlinear strain terms can be simplified for the plate or shell type structures using von Karman assumption of large rotation. In the mechanics of continuum the measure of deformation is represented by the strain tensor E^j [52] and is given by using index notation. 2Eij = ( ui,j + uj,i + uk,iuk,j ) (2.2*7) where, uÂ¿ = Displacement in i-direction. ui,j = aui / axj xÂ¿ = Rectangular Cartesian coordinate axes, i=l,2,3. uk,iuk,j = ul,iul,j + u2,iu2,j + u3,iu3,j The von Karman theory of plate is a nonlinear theory that allows for comparatively large rotations of line elements originally normal to the middle surface of plate. This plate theory assumes that the strains and rotations are both small compared to unity, so that we can ignore the changes in geometry in the definition of stress components and in the limits of integration needed for work and energy considerations [53]. It is also assumed that the order of the strains is much less than the order of rotations. 29 If the linear strain e^j and the linear rotation r^j are defined as (2.28) 2eij = ui,j + uj,i (2.29) Then the sum of Eqs. (2.28) and (2.29) gives (2.30) and the subtraction of Eq. (2.29) from Eq. (2.28) gives (2.31) From Eqs. (2.30) and (2.31), it is concluded that (2.32) uk,j â€œ ekj + rkj uk,i = eik " rik (2.33) Eq. (2.33) can be rewritten as (2.34) uk,i â€œ eki + rki since e^^ = e^ from the symmetry of linear strain terms and r^ = -r^ from the skew symmetry of the linear rotation terms. The strain-displacement Eq. (2.27) now becomes 2Eij = 2eij + (eki + rki)(ekj + rkj) (2.35) by substituting Eqs. (2.30) through (2.34) into Eq. (2.27). Thus the nonlinear strain terms have been decomposed into linear strain terms and linear rotation terms. From the assumption on the order of strains and rotations eki << rki and ekj Â« rkj (2.36) Thus Eq. (2.35) can be simplified as by ignoring ekÂ¿ and ekj' 2Eij = 2eÂ¿j + rkirkj (2.37) The straight line remains normal to the middle surface and unextended in the Kirchhoff assumption, but it is not necessarily normal to the middle surface for the Mindlin assumption. For both assumptions the generic displacements u,v,w can be expressed by the displacements at middle surface. For the Kirchhoff plate [20], u(x,y,z) = uQ(x,y) - z[wQ(x,y),x] v(x,y,z) = vQ(x,y) - z[wQ(x,y),y] (2.38) w(x,y,z) = wQ(x,y) where, uQ, Vq, W0 = Displacements of the middle surface in the direction of x, y, z. u, v, w = Displacements of an arbitrary point in the direction of x, y, z. 31 Now the linear strain components e^j and the linear rotation components r^j can be calculated using Eqs. (2.28) and (2.29). ell = 1 2 (ul,l + ul,l) = ul,l = U'X e12 = 1 2 (ul,2 + u2,l) = I(u,y + v,x e13 â€” 1 2 (ul,3 + u3,l) = *(~wo'x + w e22 = 1 2 (u2,2 + u2,2> = u2,2 = v'y 0) to to = 1 2 (u2,3 + u3,2) = H-w0,y + w e33 = 1 2 (u3,3 + u3,3) = u3,3 = 0 (2.39) The rotation terms r12, ri3' r23 are the rotation quantities about the axes 3(z), 2(y) and l(x), respectively. For the plate located in the xy plane, the rotation about z axis r12 is much smaller than rotation about x axis r23 and y axis r13 and therefore r12 is assumed to be zero here. And it is noted further that wQ(x,y) is the same as w(x,y) and is a function of only x and y so that w,3 = w,z = 0. r12I Â« Ir23I or Ir13I (2.40) rll - 1(ulfl - u3 x) = 0 r12 = *(ul,2 â€œ u2,l) = Hu,y - V,x) = 0 r13 = Hu1/3 â€œ u3,l) = H-wD,x - w,x) = -w,x r22 = Hu2/2 â€œ u2, 2) = 0 r23 = = iu2,3 â€œ u3,2) = H-w0,y - w/y) = -w,y r33 = *(u3,3 " u3, 3) = 0 (2.41) 32 The linear strain component e^j is symmetric and the linear rotation component r^j is antisymmetric. eij = eji rij â€” -rj^ (2.42) The strain components from Eq. (2.37) can be rewritten using Eqs. (2.39) and (2.41). E E E E E E XX = en + 1 2 = e22 + 1 2 = e13 + 1 2 = e12 + 1 2 (rllr12 xz = e13 + 1 2 (rllr13 yz = e23 + A 2 (r12r13 2 21 + r312) = e1]L + |r31 2 22 + r322) = e22 + ^r32 2 23 + r332) = 3(r132 + r + r21r22 + r31r32> = e12 + "r31r32 + r21r23 + r31r33) + r22r23 + r32r33^ (2.43) Ezz term is assumed to be zero because it does not have the linear term. Exz and EyZ terms are transverse shear terms which can be ignored for thin plate. Then Eq. (2.43) can be rearranged as follows using Eqs. (2.41) if all the zero terms are removed. Exx = ell + iâ€” 2 2 r31 = ell + 1 2 (w,x)2 II >i >i w e22 + 2 r322 = e2 2 + 1 2 CM 'lx Exy e12 + 2r31r32 = e12 + 1 2 (W,x) (w/y) (2.44) Exz = e13 Eyz = e23 Thus the decomposition of exact strain components has been done using the Kirchhoff plate assumptions (2.38) and the von Karman assumption (2.40) on the magnitude of rotation. It is noted that all the inplane displacement gradients in nonlinear strain terms are ignored through von Karman assumptions [20], This fact will be applied in chapter 5. CHAPTER 3 THREE DIMENSIONAL LINK ELEMENT 3.1 Element Description The link element used here is based on the two dimensional element developed by Cleary [54]. The link element is based on the following assumptions. Any normal compressive force is transferred to the other side of the link without any loss. To facilitate this, a very limited amount of loss through displacement should be allowed. Currently, this limited displacement is defaulted to .001 units, while it is a input parameter. The link separates in response to any net tension, losing its normal stiffness. To discuss the shear force transfer, some definitions for friction are needed. The force to start one body sliding along the other body is called the static friction force. The force to keep it moving is the kinetic friction force. There are two corresponding coefficients of friction, static friction coefficient and dynamic friction coefficient, where the static friction coefficient will generally be greater than the dynamic friction coefficient. 34 Two laws of friction were used in the link element. The first law is that the frictional force is proportional to the normal force, with the constant of proportionality being the friction coefficient. The second law is that friction does not depend on the apparent area of the connecting solids, i.e., it is independent of the size of the bodies. The shear force is transferred through friction. The uncertainty in friction is the factor which limits the overall accuracy of the calculation. Therefore, it is assumed that the static friction coefficient is proportional to the dynamic friction coefficient. For nonmetallic materials, the ratio of dynamic coefficient to static coefficient is about 0.75. The link element is composed of two surfaces. If the shear force is less than or equal to the static friction force, i.e., coefficient of friction times the normal force, the shear force is balanced by the friction force and the total force is transferred. This is shown in Fig. 3-1. But if the shear force is greater than the static friction force, one surface of the link element will move along the other surface. In this case there will be a dynamic friction force which is less than the shear force. This dynamic friction force can only resist a portion of the shear and the system is not in static equilibrium. Therefore, if the shear force is greater than the static friction force, the link element will lose its shear stiffness. This can also be modeled with a body on roller and spring as shown in Fig. 3- 2. The spring model of the link element is shown in Fig. 3- 3. The link element here has four nodes and each node has three translational degrees of freedom in local u-, n-, and w-directions. The total number of element degrees of freedom is 12. The element degrees of freedom are shown in Fig. 3-4. The equivalent "strain" for the link element is defined as the average deformation at the center of the element. The average deformation corresponding to the translational degrees of freedom, i.e., uQ, vQ, and wQ, can be directly calculated from the joint displacements by averaging the difference in nodal displacements at the ends of element in local u-, n-, and w-direction in turn. The relative rotation at the center of the element, rQ, can be found using nodal displacements in local n-direction and the element length as shown in Fig. 3-5. This angle is not an "average" value but the "relative" change in angle of the center line due to rotation. The two joint parameters must be introduced. These are kn, the unit stiffness normal to the joint, and ks, the unit stiffness along the joint. The off-diagonal term kns to account for dilatation during shearing is ignored because this joint element will model the dry joint between concrete box girder bridge segments. No significant dilation is expected in this case. Some values of kn and ks were reported in geotechnique area [3]. As the values are those for natural joints, they do not directly apply to this case. From the test results [7], it can be seen that the shear stiffness of dry joint ranges from 70,000 to 286,000 psi per inch at ultimate. In case of frictional strength, this can be interpreted as linear behavior between the origin and the ultimate point. It seems reasonable that the normal stiffness of the element, kn, is assumed to be stiffer than the connected material by the order of 103 to transfer the normal force without any significant loss. The forces are either totally transferred in compression or totally lost in tension. The latter case has no problem related to the value of kn. The shear stiffness parameter is more difficult to define. The data available is so limited that even a statistical treatment cannot be done. But in the analysis of structural behavior up to the ultimate, these properties do not have great influence because the forces are transferred through friction. The shear stiffness becomes zero upon sliding. But there is some 'residual' shear force. This 'residual' force is equal to friction force. Therefore, if shear displacement is more than the displacement just before the sliding the shear stiffness is set to be zero. 38 N F = Friction force, f m = Friction coefficient 1) P < or = mN then P = Friction Force. In Equilibrium. 2) P > mN then the body moves but the frictional force mN is acting against the other body. Fig. 3-1 Friction Force FRICTIONAL SPRING WITH SHEAR STIFFNESS BEFORE SLIDING F FRICTIONAL SPRING WITH ZERO STIFFNESS mN < F AFTER SLIDING Fig. 3-2 Spring Model of Friction Force 40 SPRING MODEL FOR SHEAR FORCES Fig. 3-3 Spring Model of Link Element n A11 Ik Ã‰. i ^8 â€™ u w Fig. 3-4 Element Degrees of Freedom of Link Element Fig. 3-5 Element "Strain" 3.2 Formation of Element Stiffness There are four nodes per element. Each node has three degrees of freedom corresponding the translational displacements in u-, n-, and w-direction resulting in 12 element degrees of freedom as shown in Fig. 3-4. The element stiffness is derived directly from the physical behavior of the element described in section 3.1. The mathematical symbol {} is used for a column vector and [] for a matrix. The nodal displacement column vector {q}(12) is composed of 12 translational nodal displacements corresponding to the 12 element degrees of freedom. {q} = { uÂ± V;L Uj Vj Wj uk vk wk Ul wx }T The "strainâ€ is defined as the average deformation at the center of the element as shown in Fig. 3-5. The "strain" column vector (e}(4) is {e} = { uQ vQ wQ rQ }T where, uo = ( uk + U1 ) / 2 - ( Uj. + uj ) / 2 vo = ( vk + V1 ) / 2 - (vi + vj ) / 2 wo = ( wk + W1 ) / 2 - ( wA + Wj ) / 2 ro = ( vk - V1 ) / L - ( Vj - vi ) / L where, L = The length of the element. 44 uQ/ v0, wq = Average nodal displacements in local u-, n-, w-directions. rQ = The relative angle change about local z axis. Therefore the relationship between "strain" and nodal displacements is {e>(4) = [B](4,12) {q}(12) The [B](4,12) matrix which gives strains due to unit values of nodal displacements is -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0 0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 0.0 0.0 -0.5 0.0 0.0 -0.5 0.0 0.0 0.5 0.0 0.0 0.5 0.0 1/L 0.0 0.0 -1/L 0.0 0.0 1/L 0.0 0.0 -1/L 0.0 The "stress" is defined as the normal and shear stress per unit of area. {s> is the average stress on the surface due to the two nodal forces exerted in the plane of the surface. This stress is in equilibrium with the stress on the other surface of the element as shown in Fig. 3-6. m is the moment of the nodal forces on one surface in local n- direction about the center of the element. This moment is also balanced by the moment of the nodal forces on the other surface of the element. This moment is used to define the distribution of the normal stress of the element as shown in Fig. 3-7. 45 Local Coordinate System Fig. 3-6 Nodal Forces and Stresses of Link Element FORCE TRANSFER THROUGH ONE EDGE OF THE LINK ELEMENT Fig. 3-7 Element "Strain", m The "stress" column vector (s}(4) is {s} = { sx, sn, sz, m } The "stress-strain" relationship is {s}(4) = [E](4,4) {e}(4) where, 0 0 0 V where can be related to kn using the definition of the moment m, i.e., m = (sn)(L)(t)(0.5)(L) = *m*0 = ^(Vf0-5)^)) Thus, kjn = (knVD) (L) (t) (0.5) (L) / [V0/ (0.5) (L) ] = (0.25) (t) (kn) (L3) where, t = Element thickness. This assumes that there is no coupling between the shear stress and normal stress. The element nodal force column matrix (P}(12) is composed of the 12 nodal forces shown in Fig. 3-6. 48 {P} { Pui Pni pwi puj pnj pwj puk pnk pwk pul pnl Pwl >T Stress can then be related to nodal forces using the definition of stress and force equilibrium between the two surfaces of the element. By the definition of stress, sn = (1/Lt)( Pnk + Pnl ) (3.1) sx = (1/Lt)( Puk + Pul ) (3.2) sz = (1/Lt)( Pwk + Pwl ) (3.3) m =Pnk(0.5)(L) -Pnl(0.5)(L) (3.4) where, Lt = (L)(t) By force equilibrium of the two surfaces, PÂ¿ = -P1 and Pj = -Pk (3.5) To express the element nodal forces in terms of the stress, we use Eqs. (3.1) through (3.5) to find the force recovery matrix [FR]. [FR] gives the nodal forces in equilibrium with the element stresses. From (Eq. (3.1) + Eq. (3.4)), 2Pnk = (L)(t)(sn) + 2(m)/L Pnk = 0.5(L)(t)(sn) + (1/L)(m) 49 From Eq. (3.5), pnj = ~pnk = -0.5(L)(t)(sn) - (1/L)(m) From Eq. (3.1), Pnl = (L)(t)(sn) - Pnk = (L)(t)(sn) - ( 0.5(L)(t)(sn) + (1/L)(m)) = 0.5(L)(t)(sn) - (1/L)(m) From Eq. (3.5), pni = "pnl = -0.5(L)(t)(sn) + (1/L)(m) From the assumption that Puk = Pu^ and Eq. (3.2), Puk = (L)(t)(sx)/2 Pul = (L)(t)(sx)/2 From Eq. (3.5), Pui = - pul = "(L)(t)(sx)/2 Puj = - Puk = -(L)(t)(sx)/2 From the assumption that Pwk = Pw^ and Eq. (3.3), Pwk = (L)(t)(sz)/2 PW1 = (L)(t)(sz)/2 50 From eqn 5, pwi = - pwl = "(L> (tHSz)/2 pwj = " pwk = "(L)(t)(sz)/2 Therefore, the force-stress relationship is {P}(12) = [FR](12,4) {s}(4) where the force recovery matrix [FR](12,4) is -Lt/2 0 0 0 0 -Lt/2 0 1/L 0 0 -Lt/2 0 -Lt/2 0 0 0 0 -Lt/2 0 -1/L 0 0 -Lt/2 0 Lt/2 0 0 0 0 Lt/2 0 1/L 0 0 Lt/2 0 Lt/2 0 0 0 0 Lt/2 0 -1/L 0 0 Lt/2 0 And this relationship is further expanded using the stress-strain relationship and the strain-nodal displacement relationship as follows. {P >(12) = [FR](12,4) [E](4,4) (e)(4) = [FR](12,4) [E](4,4) [B](4,12) (q}(12) = [Bt](12,4) [E](4,4) [B](4,12) (q)(12) Then finally this can be symbolized as equilibrium equation. 51 {P>(12)â€” [Ke](12,12) {q}(12) where [Ke] = [Bt][E][B] Here it is noted that [FR] = [Bt] and [Ke] = [Bt][E][B] just as in the case of common finite element method. The final element stiffness matrix [Ke] is kx 0 0 kx 0 0 -kx 0 0 -kx 0 0 0 2kn 0 0 0 0 0 0 0 0 -2kn 0 0 0 kz 0 0 kz 0 0 -kz 0 0 -kz kx 0 0 kx 0 0 -kx 0 0 -kx 0 0 0 0 0 0 2kn 0 0 -2kn 0 0 0 0 0 0 kz 0 0 kz 0 0 -kz 0 0 -kz -kx 0 0 -kx 0 0 kx 0 0 kx 0 0 0 0 0 0 -2kn 0 0 2kn 0 0 0 0 0 0 -kz 0 0 -kz 0 0 kz 0 0 kz -kx 0 0 -kx 0 0 kx 0 0 kx 0 0 0 -2kn 0 0 0 0 0 0 0 0 2kn 0 0 0 -kz 0 0 -kz 0 0 kz 0 0 kz This matrix can be rotated to any direction using the standard rotation. 3.3 Solution Strategy The structural stiffness changes because of the slip and debonding of the link. Therefore, the process of the resistance of the total structure physically becomes nonlinear. Correspondingly, special solution techniques for nonlinear behavior are needed. This can be done using the iterative solution technique with initial stiffness or tangent stiffness. The latter can be formed by assembling the structural stiffness at the beginning of each iteration and this converges faster than the initial stiffness. A third solution strategy for this case is event-to- event technique which is usually employed for the linear stiffnesses between any two "events,â€ which are defined as the intersection point between two linear segments. This also provides means of controlling the equilibrium error. Any significant event occurring within any element determines a substep. The tangent stiffness is modified in each substep, and hence, the solution closely follows the exact response. 3.4 Element Verification 3.4.1 SIMPAL The finite element analysis program SIMPAL [55], is used to implement and verify the element formulation. SIMPAL was chosen for the initial implementation because that was the original implementation done by Cleary [54]. This way, the 3-D aspects could be implemented and verified using Cleary's original program. A table of the element verification is shown in Fig. 3-8 and Fig. 3-9. LOADING RESULTS NODE THEORY SIMPAL ERROR DISP 2 -.1333 -.1333 .000 DISP 4 -.1333 -.1333 .000 STRESS N/A -80 -80 .000 NODE THEORY SIMPAL ERROR DISP 2 -.1333 -.1333 .000 DISP 4 -.1333 -.1333 .000 STRESS N/A -80 -80 .000 NODE THEORY SIMPAL ERROR DISP 2 -.1333 * -.1337 .003 DISP 4 -.1333 â˜… -.1337 .003 STRESS N/A -80 -80 .000 THICKNESS = .25 Ks = 3E6 Kn = 6E6 NODE 2 NODE 4 YDISP =-0.1017-04 Z DISP =-0.8684-05 Z DISP = -0.8684-05 Y DISP = -0.1017-04 2 2 SQRT((.1017) + (.08684) ) = 0.1337 Fig. 3-8 Link Element Test Using SIMPAL 54 LOADING RESULTS Z Fig. 3-9 Combined Test Model for SIMPAL 55 3.4.2 ANSR The test examples used are the same as those used in the initial element verification using SIMPAL. The results from ANSR [56] are exactly the same as those from SIMPAL. The link element was tested further using a modeled membrane element composed of 22 truss elements as a membrane element was not available at the time of element verification in ANSR. The results are shown in Table 3-1 and the structures used are shown in Fig. 3-10 and Fig. 3-11. Table 3-1 Displacements of Truss Model for ANSR Node No. Truss only Truss w/ LINK Diff. (%) 10-x -.1027e-4 -.1049e-4 2.2 10-y -.1990e-5 -.2010e-5 0.9 11-x -.9017e-5 -.9211e-5 2.2 11-y -,4906e-6 -.4973e-6 1.4 12-x -.9915e-5 -.1049e-4 2.2 12-y +.9742e-6 +.9703e-6 0.4 57 Fig. 3-10 Combined Test Model for ANSR 58 Fig. 3-11 Truss Model for ANSR CHAPTER 4 LINEAR SHELL ELEMENT 4.1 Element Description The shell element is formulated through a combination of two different elements, the membrane element and the Mindlin plate bending element [57]. The Mindlin plate element is different from the Kirchhoff plate element in that the former allows transverse shear deformation while the latter does not. The common portions of the formulation of two elements are 1. Formation of the shape functions. 2. Formation of the inverse of Jacobian matrix. These processes can be done at the same time. The four- to nine-node shape functions and their derivatives in rs- space can be formed and then transformed into xy-space through the inverse of Jacobian matrix. 4.2 Formulation of Shape Functions The formulation of shape functions starts with three basic sets of shape functions shown in Fig. 4-1. 59 1. The bilinear shape functions for four-node element. 2. The linear-quadratic shape functions for nodes five to eight of the eight-node element. 3. The bubble shape function for node nine of nine-node element. These shape functions can be formulated directly from the local coordinates of the element nodes through the multiplication of the equations of the lines which have zero values in the assumed displacement shapes and the scale factor to force the shape function value to one at the node for which the shape function is formed. The derivative of each shape function with respect to r and s is then evaluated from the shape function expressed in terms of r and s. If node nine exists, the value at node nine of shape functions one to eight must be set to zero. The value of the bilinear shape functions for a four-node element at node nine is one fourth and the value of the linear-quadratic shape functions for the five- to eight-node element at the node nine is one half. This can be forced to zero using the bubble shape function of the nine-node element because this shape function has the value of one at node nine and zero at all other nodes. Therefore the modification is the subtraction of one fourth of the value the bubble shape function has at node nine from the each shape function for the corner nodes and the subtraction of one half of the value of the bubble shape function of node nine for the nodes five to eight, whichever exists. If any of the center nodes on the edge of the element (any one of nodes five to eight) exists, the bilinear shape functions of four-node element must be modified further because the value at center of the edge is one half in those bilinear shape functions. This can be done by subtracting one half of the linear-quadratic shape function for the newly defined center node on the edge of the element from the bilinear shape functions of the two adjacent corner nodes. The value of any five node shape functions at the corner node is zero. Therefore, no further consideration is needed except for the shifting of the shape functions in the computer implementation. These processes are shown in Fig. 4-2. If any of the linear-quadratic shape functions of nodes five to eight is missing, all the linear-quadratic shape functions thereafter and the bubble shape function must be shifted to the proper shape function number. For example, if linear-quadratic shape function five is missing, then the shape functions six to eight must be shifted to five through seven and the bubble shape function must be shifted to the node eight because all of the linear-quadratic shape functions have been defined and numbered as shape functions for the nodes five through eight and the bubble shape function for the node nine. Four Node Element Shape Function for Corner Node Shape Function for Edge Center Node Shape Function for Element Center Node Fig. 4-1 Three Basic Shape Functions 63 SF 4 = (SF 1) - (1/4) (SF 3) SF 5 = (SF 2) - (1/2) (SF 3) Fig. 4-2 Formation of Shape Functions 4.3 The Inverse of Jacobian Matrix While the generic displacements are expressed in terms of rs-coordinate, the partial differential with respect to the xy-coordinate is needed for the calculation of strain components. Thus the inverse of the Jacobian matrix must be calculated. This can directly be found from the chain rule using the notation (a,b) defined as the partial differential of function a with respect to the variable b for simplicity. f,x = (f,r)(r,x) + (f/S) (s,x) f,y = (f,r)(r,y) + (f,s)(s,y) In matrix form, f ,x r,x s,x f,r T "I J11 j -1 ' J12 f,r f,Y r,y s,y f,s T â€” 1 J21 T "I J22 f ,s The inverse of Jacobian matrix But the terms in the inverse of the Jacobian matrix are not readily available because the rs-coordinate cannot be solved explicitly in terms of xy-coordinate. On the other hand, for the isoparametric formulation, the geometry is interpolated using the nodal coordinate values(constants) and the displacement shape functions in terms of r and s. Thus the generic coordinate x and y can be expressed in r and s 65 easily and explicit partial differentials of x and y with respect to r and s can be performed. Therefore the Jacobian matrix is computed and then inverted. The Jacobian matrix is derived by the chain rule. f,r = (f/x)( x,r) + (f,y)( y,r) f/s = (f,x)( x,s) + (f,y)( y, s) In matrix form, f,r x, r y,r f,x 1 l-> J12 f ,X f ,s x, s y,s f,y J21 J22 f,y Jacobian matrix nn Let E be E . i=l where nn = number of nodes (4 to 9). From geometric interpolation equations, x = E fi*xÂ¿ y = E fi*yi The terms in the Jacobian matrix are li = x, r = (s fi*xi),r = E ( (f1/r) * xi) 12 = y,r = (S fi*yi) = E ((fi,r) * Yi) 21 = X, S = (E fi*Xi),s = E ((fi/S) * xi) 22 II *< CO li M fi*yi)/S = s * Yi) 66 xi' Yi are coordinate values of the element and are constants and therefore can be taken out of the partial differentiation. The inverse of two-by-two Jacobian matrix can be found r,x = J22 / det(J) s,x = -J12 / det(J) r,y = -J2i / det(J) s,y = Jji / det(J) where deh(J) â€” ^ 11*^22 ^ 12^"21 4.4 Membrane Element The formulation of the membrane element used for the implementation follows the procedure shown on pages 115 through 118 in reference [57]. The { } symbol will be used for the column vectors. Nodal displacements are the nodal values of two inÂ¬ plane translations and denoted as {u^ v^}T. The generic displacements are defined as two translational displacements at a point and denoted as { u v }T. By the word generic it is meant that the displacement is measured at an arbitrary point within an element. The generic displacements u and v can be calculated using shape functions. The shape function is a continuous, smooth function defined over the closed 67 element domain and is differentiable over the open domain of the element. The shape function is also the contribution of displacement of a node for which the shape function has been defined to the generic displacement. Thus the generic displacement at an arbitrary point can be found by summing up all the contributions of all the nodes of the element. The displacement interpolation equations are u = 2 fÂ¿ uÂ¿ V = 2 Â£Ã vÂ¡ In the isoparametric formulation the geometry is interpolated using the same shape function assumed for the displacement interpolation. Therefore, the geometry interpolation is x = E fÂ¿ y = z fi Yi where, f^ = Shape function for node i. x^, yÂ¿ = Coordinates of node i. uÂ¿, Vj_ = Displacements at the node i. u, v = Displacements at an arbitrary point within an element 68 The three in-plane strain components for a membrane element are < c > = < Â£x ey ^xy >T These strain components can be found through the partial differentials of the generic displacements with respect to xy-coordinates. ex = u,x ey = v,y 7xy = U/Y + v,x Using the inverse of the Jacobian matrix, the strain components can be evaluated. ex = U, X = (u,r)(r,x) + (u,s)(s,x) = (u,r)(J11_1) + (u,s)(J12_1) = ((SfiUi),r) (Jn-1) + ((SfiUi)^)^-1) = S[(fifr)(r,x) + (fi#s)(s,x)] uÂ¿ ey = v, y = (v,r)(r,y) + (v,s)(s,y) = (v,r)(J21-1) + (v,s)(J22â€œ1) = ((sfiVÂ±),r)(Js!-1) + ((sfiVi),s)(J22_1) = S[ (fifr) (r,y) + (f^sj^y)] ^xy = u,y + v,x = [ (u,r) (r,y) + (u,s)(s,y)] + [(v,r)(r,x) + (v,s)(s,x)] = [((sf^) ,r) + ((Sf^i) ,s) (J22_1)] + [ ((ZfjVi) ,r) (Jii-1) + ((Sf^i) ,s) (J12-1) ] = E[(fifr)(r,y) + (fi,s)(s,y)] uÂ¿ '+ S[(firr)(r,x) + (fi,s)(s,x)] vÂ± New notations are introduced here to simplify the equations. These are a^ and and defined as follows: = (r,x)(fi,r) + (sfx)(firs) = fÂ¿,x *>i = (r,y)(fi#r) + (s,y)(fÂ±,s) = fi#y Then the strain terms above become ex = Sa^u^ = Sf^,x u^ â‚¬y = EbiVi = sfi#y Vi 7xy = EbjUi + Sa^i = Sfi,y ui + Ef^x vÂ± In matrix form, ex ai 0 ui Â£y â– ^xy = s 0 bi bi ai vi In symbolic form, [ej = S[Bi][qi] where, ai 0 fi'X 0 0 bi s 0 bi ai i H) H- â–º< fi'x and, [qj.] V i Therefore the strain at an arbitrary point within an element is [e] â€” [B-jJ [q-jJ + [B2][q2] + â€¢â€¢â€¢"*" [Bg] [qg] ~ [ B1 B2 B3 B4 B5 Bg B? Bg Bg *1 *2 *3 q4 *5 <*6 q7 ^8 q9 The size of the vectors and matrix are [Â«(3,1)] = [B(3,18)][q(18,l)] In the actual calculation, this can be done by summing up the [BjJ [qjJ over all the nodes for the given coordinates of the point under consideration, i.e., the coordinates of one of the integration points. The stresses corresponding to the strains are T { <7 > = { (Tx CTy TXy } The stress-strain relationship of an isotropic material [E] E 11 Â¡21 0 0 E33 where, E11 = E22 = E / ( 1 - p2 ) E12 = E21 = ^E / ( i â€ E33 = G where, E = Young's modulus n = Poisson's ratio G = shear mpdulus = E / ( 2*(l+/Â¿)) The integration over volume must be introduced for the calculation of the element stiffness and equivalent nodal loads of distributed loads or temperature effects. As the thickness of the element is constant, the integration over volume can be changed to the integration over area. The element stiffness related to the degrees of freedom of the node i can be calculated through the volume T integration of (2,3)E(3,3)(3,2). 72 [Ki] B? E Bi dV V As [B] and [E] are constant about z, the integration through the thickness from -t/2 to t/2 can be performed on z only and yields [Ki] [ Bi(t)EBi] dA A T - = [ Bi E Bi] dA Ja where, E = tE The size of membrane element stiffness is 18 by 18. [K] [ E ] [ B2 B3 ... B8 B9 ] dV (3,3) (3,18) (18,3) Equivalent nodal loads due to body forces on the membrane element are calculated as pb fTb dV = â€¢1 r Jv -1 . fTb|j| dr ds in which (b> = { 0 0 b2 }T or ( 0 by 0 )T or { bx 0 0 )T in 73 accordance with the direction of gravity in the coordinate system used. The nonzero quantities bx, by, or bz represent the body force per unit area in the direction of application. Equivalent loads caused by initial(temperature) strains are P 0 BTEe0 dV V r1 r1 BTE e o IJI dr ds .-1 J-i where, {e0} { â‚¬xxO eyyO 0 { aAT Â«AT 0 0 0 }T 0 0 }T 4.5 Plate Bending Element The formulation of the plate bending element used for the implementation has followed the procedures shown on pages 217 through 221 in reference [57]. The { } symbol will be used for the column vectors. Many plate bending elements have been proposed. The most commonly used are Kirchhoff plate elements and Mindlin plate elements. Kirchhoff theory is applicable to thin plates, in which transverse shear deformation is neglected. The assumptions made on the displacement field are 1. All the points on the midplane(z = 0) deform only in the thickness direction as the plate deforms in bending. Thus there is no stretching of midplane. 2. A material line that is straight and normal to the midplane before loading is to remain straight and normal to the midplane after loading. Thus there is no transverse shear deformation (change in angle from the normal angle). 3. All the points not on the midplane have displacement components u and v only in the x and y direction, respectively. Thus there is no thickness change through the deformation. Strain energy in the Kirchhoff plate is determined entirely by in-plane strains ex, ey, and 7Xy which can be determined by the displacement field w(x,y) in the thickness direction. The interelement continuity of boundary-normal slopes is not preserved through any form of constraint. Mindlin theory considers bending deformation and transverse shear deformation. Therefore, this theory can be used to analyze thick plates as well as thin plates. When this theory is used for thin plates, however, they may be less accurate than Kirchhoff theory because of transverse shear deformation. The assumptions made on the displacement field are 1. A material line that is straight and normal to the midplane before loading is to remain straight but not necessarily normal to the midplane after loading. Thus transverse shear deformation (change in angle from normal angle) is allowed. 2. The motion of a point on the midplane is not governed by the slopes (w,x) and (w,y) as in Kirchhoff theory. Rather its motion depends on rotations 0X and 0y of the lines that were normal to the midplane of tire undeformed plate. Thus 0X and 0y are independent of the lateral displacement w, i.e., they are not equal to (w,x) or (w,y). It is noted that if the thin plate limit is approached, 7XZ = 7yZ = 0 because there is no transverse shear deformation. In this case the angles 0X and 0y can be equated to the (w,x) and (w,y) numerically but the second assumption still holds. The stiffness matrix of a Mindlin plate element is composed of a bending stiffness [kb] and a transverse shear stiffness [ks]. [kb] is associated with in-plane strains â‚¬x, Cy, and 7Xy. [ks] is associated with transverse shear strains 7XZ and 7yZ. As these two groups of strains are uncoupled, i.e., one group of the strains do not produce the other group of strains, the element stiffness can be shown as [82] [k] T ( BbEBb ) dA + (BgEBg) dA because BbEBs = BgEBb = 0 from uncoupling (corresponding E = 0). Each integration point used for the calculation of [kg] places two constraints to a Mindlin plate element, associated with two transverse shear strains 7yZ and 7ZX. If too many integration points are used, there will be too many constraints in transverse shear terms, resulting in locking. Therefore, a reduced or selective integration can prevent shear locking. Or, the transverse shear deformation can be redefined to avoid such locking. For example, a bilinear Mindlin plate element responds properly to pure bending with either reduced or selective integration. But full two-by-two integration is used for pure bending, shear strains appear at the Gauss points as shown in Fig 4-3. As the element becomes thin, its stiffness is due almost entirely to transverse shear. Thus, if fully integrated, a bilinear Mindlin plate element exhibits almost no bending deformations, i.e., the mesh "locks" against bending deformations. Nodal displacements for the plate bending consist of one out-of-plane translation and two out-of-plane rotations m , and are denoted as { w^ 6xÂ¿ 0yÂ¿ } . The rotations are chosen independently of the transverse displacement and are not related to it by differentiation. Thus the transverse shear strains 7XZ and 7yZ are considered in the formulation resulting in five strain components. The generic displacements are defined as three translational displacements and denoted as { u v w }T. By the word generic it is meant that the displacement is measured at an arbitrary point within an element. These generic displacements are different quantities from the nodal 77 displacements and therefore must be related to the nodal displacements. The generic displacements u and v can be calculated as functions of the generic out-of-plane rotations using the small strain(rotation) assumption. The relationship between generic displacements and rotation is shown in Fig 4-4. u = z0y v = â€”z*x The generic displacements 6X and 6y can be found using the assumed displacement shape functions and the corresponding nodal displacements 6X^ and 0yÂ¿. The generic displacement w does not need any conversion because it corresponds to the nodal displacement wÂ¿. In the isoparametric formulation the geometry is interpolated using the same shape function assumed for the displacement interpolation. The displacement interpolation is *x = 2 fi *xi 6y â€” Z fÂ£ ^yi W = Z f^ WÂ¿ Similarly, the geometric interpolation is x = Z f^ x^ y = s fi yi Zero Shear Strain One Point Gauss Integration Two Point Gauss Integration Fig. 4-3 Shear Strains at Gauss Point (s) Positive small rotational angle about y-axis gives positive generic displacement in x-direction ( u ). Shown is xz-plane. Positive small rotational angle about x-axis gives negative generic displacement in y-direction ( v ). Shown is yz-plane. Fig. 4-4 Displacements due to Rotations 80 where, = shape function for node i x^, = coordinates of node i Therefore, c II N II Z 2 fi *yi v = -zex = -Z 2 fi *xi w = 2 fi wi The five strain components for plate bending element are { ex ey 7Xy yxz 7yz }T. These strain components can be found through the partial differentials of the generic displacements with respect to xy-coordinates. ex = U,X ey = v,y 7Xy = u/Y + V,X 7XZ = u/z + W,X 7y 2 = v,z + w,y Using the inverse of the Jacobian matrix found, the strain components can be evaluated. ex = U , X = (Z0y) ,X = (u,r)(r,x) + (u,s)(s,x) = (u,r)(Jii-1) + (u,s)(J12_1) Tm [(x's) (s'Tj) + (x'a)(a'Tj)]2 + = t (x_ZTr) (^(s'Tj^)) + (x_I1:r) (tav(j/T5s) ) ] + = [(T.ZIr)(s'(Wjs)) + (T_TIr) U' (Wjs)) ] + = [(x's)(s'm) + (x'j)(a'w)] + = x'm + = x'n. + z'(^0z) = (x'w) + (z'n) = zxÂ¿ xx0 [(x's)(s'Tj) + (x'a) ]s z- [(A's)(s'Tj) + (A'j) (jc'Tj) ]s z = nT_ZTr) (s' (T^Tjsz-)) + (T_XIr) U' (fVjsz-)) ] + [ (x_Z2r) (S/(T^^TjSz)) + (x_TZr) U' ) ] = [ (x's) (s'a) + (x'j) (j'a) ] + [(A's)(s'n) + (A'j) (a'n)] = xâ€˜(x0z~) + A1(^0z) = x'a + A'n = Axl Tx0 [(A's)(s'Tj) + (A'a)(j'Tj)]z z- = (x_32r)(S/(TX^Tj^z-)) + (x_Tzf) (â– *'(TX0TJSZ-)) = (x_zzr)(s'A) + (x_xZr)(a'A) = (A's) (s'a) + (A'j) (j'a) = A' (X0Z~) = A'a = [(x's)(s'Tj) + (x'j)(a/Xj)3s z = (x_ZIr) (s# (TAeT5sz)) + (x_TIr) U' (TAffT5sz)) = 18 82 7yz = (V/Z) + (W,y) = (-Z6X),Z + w,y = (-0X) + w,y = (-Ef^i) + [(w,r)(r,y) + (w,s)(s,y)] = + [((sf^) ,r) (Jsi"1) +((Sfiwi) ,s) (J22-1)] = (-sfi^xi) + [((Sfi^rjwi) (J2-L-1) +((sfi#s)wi) (J22_1) ] = (-Sf^xi) + S[ (fÂ±/r) (r,y) +(fi,s)(s,y)] Wi New notations are introduced here to simplify the equations. These are a^ and and defined as follows: ai = (rrx)(fi#r) + (s,x)(fifs) = fi#x *>i = (rjyHf^r) + (s^Hf^s) = fi/y Then the strain terms above become Â£x = Z 2ai^yi Â£y = -z 2bi*xi 7xy = z Sbf^yi - z Sa^^xi 7xz = Sff^yi + EajW^ N II - Eff^xi + SbjW^ In matrix form, ex ey 7xy = E 7xz lyz 0 0 0 0 0 -f i w i ?xi Vi In symbolic form, [e] = E[BjJ [q-jj 83 where, 0 0 za^ 0 - zb^ 0 0 - za^ zbi ai 0 fi bi - fi 0 or, 0 0 zfÂ¿,x 0 â€œ zfA,y 0 0 - zfjL,x zfi,Y fi,x 0 fi fi'Y " fi 0 and, [qjj = wi *xi Vi Therefore the strain at an arbitrary point within element is [Â«] = [B-LHq-L] + [B2][q2] + ... + [Bg] [q9] â€œ [ B1 B2 B3 B4 B5 Bg B7 Bg Bg 51 52 ^3 q4 55 56 57 58 59 an The size of the vectors and matrix are [e(5,l)] = [B(5,27)][q(27,l)] In the actual calculation, this can be done by summing up the [BjJ [qÂ¿] over all the nodes for the given coordinates of the point under consideration, i.e., the coordinates of one integration point. The stresses corresponding to the strains are { CT } = { CTX (7y Tx y Txz TyZ } ^ The stress-strain relationship of an isotropic material is 0 0 0 0 0 0 E44 0 0 E55 where, E11 = E22 = E / ( 1 - n2 ) E^2 = ^21 = ^E / ( 1 - Â¡P") E33 - G E44 = E55 = G / 1.2 where, /x = Poisson's ratio G = shear modulus = E / ( 2*(l+/x)) [E] = 0 0 0 J11 J21 J12 322 0 0 0 0 E- 0' 0 J33 The form factor 1.2 for the E44 and E55 terms is provided to account for the parabolic distribution of the transverse shear stress rzx over a rectangular section. This form factor 1.2 can be shown from the difference in deflections of a cantilever beam at its free end [58]. Let a beam have a rectangular cross section of dimensions b by t with a length of L. If P is the transverse shear force, then the parabolic distribution of the transverse shear stress rzx is tzx = (3P/2bt3)(t2 - 4z2) where z = 0 at the neutral axis. Then the transverse shear strain energy from the parabolic distribution can be calculated by U s (1/2) V (rzxVG) dV (1/2) [((3P/2bt3)(t2 [(1/2)(3P/2bt3)2]/G - 4z2))2 / G] dV (t2 - 4z2)2 dAdz = (area)[(1/2)(3P/2bt3)2 = bL[(1/2)(3P/2bt3)2]/G = 1.2(P2L/btG)/2 ]/G (t2 - 4z2)2 (t2 - 4z2)2 dz dz While the transverse shear strain energy from the constant distribution is 86 Us = (1/2) (rzxVG) dV V = (1/2) ((P/bt)2 /G) dV = [(1/2)(P/bt)2/G](btL) (P2L/btG)/2 This result suggests the view that a uniform stress P/bt acts over a modified area bt/1.2, so that the same Us results. Therefore the deflection at the free end of a cantilever beam with parabolically distributed transverse shear stress will be 1.2 times that with constantly distributed transverse shear stress. The Mindlin plate is generalized from the Mindlin beam. Thus the higher transverse shear stiffness from the assumption of constant transverse shear stress has been reduced by dividing the corresponding elastic constants by the factor 1.2 for flat plate element. The reduced stiffness will produce the more flexible response in shear that is expected from the actual parabolic distribution. The integration over volume must be introduced for the calculation of the element stiffness and equivalent nodal loads of distributed loads or temperature effects. As the thickness of the element is constant, the integration over volume can be changed to the integration over area. This can be accomplished through the partition of the strain-nodal displacement matrix [BÂ¿] as follows: 87 0 0 zfi,x 0 - zfÂ±fy 0 0 - zfitx zfj.,y firX 0 fi fi'Y - fi 0 The submatrices are named as follows: BiA 1 < â€¢H 1 PQ N BiB BiB The element stiffness can be calculated T volume integration of (3,5)E(5,5)B^(5,3). matrix is to be partitioned as follows: [E] r- E11 E12 0 0 0 E21 E22 0 0 0 o o e33 0 0 0 0 0 E44 0 0 0 0 0 E55 The submatrices are named as follows: E 0 E B through the Thus the [E] 0 88 Then the stiffness of the element is [KÂ±] T Bi E Bjl dV zBiA I BiB * dV [z2 Ba Ea Ba] + [Bg Eg Bg] As [B] and [E] are constant about z, the integration through the thickness from -t/2 to t/2 can be performed on z only and yields [KÂ±] where, [ SA(t3/12)EA5A + Bg(t)EgBg] dA C baeaba + bbebbb] dA EA _ (t3/12)EA and Eg = tEg Then [KjJ can be rewritten as matrix equation as follows: -T T ea 0 r Â®iA i [Kj.] = [ BiA BiB ^ _ dA â€¢ 0 eb L BiB J -T , B^E BÂ¿ dA The size of plate element stiffness will be 27 by 27. [K] = [ E ] (5,5) [ B, B2 B3 B8 B9 ] dV (5,27) (27,5) The strain-nodal displacement matrix from which the constant thickness is taken out is defined as [B^] [BÂ±] = 0 0 0 fi,x fi 'Y - fi,y - fifX - f; fi,x :i,y fi 0 = [BiA] L BiB j Equivalent nodal loads due to body forces on the plate calculated as â€¢ *1 |* fTb dV V -1 â€¢ fTb|j| dr ds -1 in which (b) = { 0 0 bz }T or { 0 by 0 }T or { bx 0 0 T â€¢ . . } m accordance with the direction of the gravity m the coordinate system used. The nonzero quantities bx, by, or b represent the body force per unit area in the direction of application. Equivalent loads caused by initial strains are P 0 BTEe0 dV V r1 r1 - 5tEÂ¿0|j| dr ds -1 J-l where, T = { Â¿xxO ^yyO ^xyO 0 0 } = { a AT/2 aAT/2 0 0 0 }T The stresses can be calculated from the equation M = [E][6] The corresponding generalized stresses, if desired, may be computed from M = { Mxx Myy Mxy Qx Qy }T = Ã ( 5 q - It is noted that the generalized stresses are actually moment and shear forces applied per unit length of the edge of the plate element. Therefore these can also be turned into common stresses using the formulation for the bending stress calculation. The moment of inertia for the unit length of the plate is t3 / 12. Then the in-plane stress at a point along the thickness can be calculated as a = Mz / I = M(t/2) / (t3/12) = 6M / t2 91 The transverse shear stresses can be found as r = Q / t But this may be multiplied by a factor of 1.5 to get the maximum shear stress at a point on a neutral surface because the transverse shear stresses show parabolic distribution while the calculated stresses are average stresses coming from the assumption of a constant transverse strain along the element z axis. CHAPTER 5 NONLINEAR SHELL ELEMENT 5.1 Introduction The nonlinearities included in the formulation of the Mindlin flat shell element are those due to large displacements and those due to initial stress effects(geometric nonlinearity). The large displacement effects are caused by finite transverse displacements. These effects are taken into account by coupling transverse displacement and membrane displacements. The initial stress effects are caused by the stresses at the start of each iteration. These stresses change the element stiffness for the current iteration. These effects are evaluated directly from the stresses at the start of each iteration and are included in the element stiffness formation. The total Lagrangian formulation is used. If the updated Lagrangian formulation is used, the element coordinate system cannot be easily formed for the next iteration because the deformed shell is not usually planar [26], The symbol {} is used for a column matrix (a vector) and the symbol [] is used for a matrix of multiple columns and rows throughout the chapter. 92 93 5.2 Element Formulation The generic displacements of Mindlin type shell element are translational displacements {u v w)T and denoted as {U}. The displacements and rotations at a point on the midplane are (uQ vQ wQ 0X 0y)T and denoted as {UQ}. The generic displacements can be expressed in terms of the midplane displacements and z as u = u0(x,y) + z0y(x,y) v = vQ(x,y) - z0x(x,y) (5.1) w = wQ(x,y) The linearized incremental strain from Eq. (2.17) is eij = * ( ui,j + uj,i + Sc,i uk,j + tuk,j uk,i> (5.2) This equation can be written out for the strain terms to be used for shell element using the generic displacements (u, v, w)T. exx X d ii + u,x U'X + tv'X V'X + tw'X W,x eyy = u,y + ^/y U,y + tv'y v,y + tw w / y W,y exy 1 2 ( U,y + V'X + U,y + tv'X V' y + W,y + u/x tu'y + v'x tv' y + W,x ) exz = 1 2 ( uÂ»z + W'X + Vjc u,z + tv'X V' z + tw'x W'Z + UÂ»x tu/z + V'X tv/ z + w'x Vz ) eyz = 1 2 ( V'Z + W,y + Vy U/Z + ^ty vf z + t^,y W,z + u'y tu/Z + V,y fcV, z + w/y tw/Z ) (5. 3) The derivatives of inplane displacements u and v with respect to x, y, and z are assumed to be small and thus the second order terms of these quantities can be ignored through von Karman assumption from Eqs. (2.44) [20, 21]. Furthermore the transverse displacement w is independent of z for the shell element which means that w,z is zero. Then Eqs. (5.3) can be reduced to exx = u'x + tw'x W'X d !i >i 0) y + fcw,, >i Â¡s p â€” Ã Â«Xy 2 ( >i d + V,x + W ,y + W,X ) P s 1 exz 2 ( u,z + W,x ) p â€” 1 eyz 2 ( V'Z + >1 Â¡5 (5.4) The incremental Green's strains, sometimes called engineering strains, can then be shown as Â£xx exx = U'X + W'X W'X >1 >1 II (D *< = u,y + W,y 7xy = 2exy >i d II + V,x + W,y + W,X 7xz = 2exz = U/Z + W,x 7yz = 2eyz = V'Z + w,y It is noted that the linearized nonlinear strains are left only for inplane strain terms. By substituting Eqs. (5.1) into Eqs. (5.5), the Green's strain can be expressed in terms of midplane displacements. 95 Â£xx = U0'X + Ziy/X + W0'X W0'X â‚¬yy = c 0 *< " WQ / y Wq f y 7xy = uo'y + ZÃy fy + VQ,X - Z0X + tw wo' x wo'y + wo'x wo'y 7xz = 6y + wG / x fyz = ~9x + wo'y This can be simplified as {ep} Z{eb} {*1} {6} = {0} + {es} { f2 } where, {e} = {exx â‚¬yy 7Xy ^xz ^yz) = Incremental strains for shell element. {el} = {Â£XX ^yy 7Xy } = Incremental inplane strains. {Â£2} â€” Ã7xz 7yz) = Incremental transverse shear strains. uo'x + wo'x wo'x vo'y + Wqfy Wq/y uo'y + vo'x + two'x wo'y + wo'x two'y = Linearized incremental inplane strains. (0) = {0 0}T = Zero vector. â€” {^y/x ~8X'y (^y/y "" 8X'x) ^ = Linear bending strains. (5.6) (5.7) (5.8) (5.9) (5.10) (5.11) (5.12) 96 {es} = {(WQ/X + 0y) (Wqty ~ 6x)}T (5.13) = Linear shear strains. If Piola-Kirchhoff II (PK-II) stresses are denoted as {S}, the internal virtual work due to the virtual strain 5{e} corresponding to the virtual displacements 5{q} can be calculated using the conjugate Green's strains as SVIa S{e)T{S)dV ' t/2 exsx+5 eysy+57Xysxy+5'irxzsxz+iTryzsyz)dzdA (5.14) where, {S) = (Sx sy Sxy sX2 Syz)T = PK-II stresses. The integrand of Eg. (5.14) can be further expanded as Integrand = [ S( ep(l) + zeb(l) )SX + Ã( ep(2) + Zeb(2) )Sy + S( ep(3) + Zeb(3) )Sxy + 5eS(l)Sxz + 5eS(2)Syz ]dzdA If the integration over thickness is performed analytically, this can be rearranged as Integrand = [ 5{ep)T{SSp) + 5{eb)T{SSb) + S(es}T{SSs} ]dA [5{ee}T{SS} ]dA where, {ce} = { into inplane, bending, and shear strains. {SS} = { (SSp}T {SSb}T {SSs}T }T = Generalized stresses arranged for {ee}. t/2 {SSp} = r -] * t/2 Nx SX dz = Nâ€ž ss sâ€ž Nxy -t/2 sy bxy Â« â€¢ {SSb} = {SI} = {Sx Sy Sxy} T -t/2 {Sl}dz (5.15) CM M llX. SX M y M 1Axy -t/2 sy sxy zdz = â€¢ t/2 -t/2 {Sl}zdz (5.16) * t/2 Qx sxz (SSs) = [ qy J â€¢ -t/2 syz dz = t/2 -t/2 {S2}dz (5.17) = Inplane Piola-Kirchhoff II stresses. These generalized stresses (stress resultants along the element thickness per unit length) can be further expanded incorporating the stress-strain relationship between PK-II stress and the conjugate Green's strain. 98 {S} = [E]{Â£> where the constitutive matrix [E] can be subdivided as [E (5,5) ] = [El] = [E2 ] (SSp) l-/xâ€˜ E [El](3,3) [0 ] (3,2) [0 ] (2,3) [E2](2,2) 1 n 0 H 1 0 0 0 (1-^/2 (1-J0/2 0 0 (l-n)/2 t/2 (Sl)dz = t/2 [El]{eljdz -t/2 t/2 -t/2 = Integl + Integ2 -t/2 [El]{Â£p + Zfbjdz where, Integl = t/2 [El](ep)dz Integ2 = [El]{ ZÂ£b }dz (i -t/2 't/2 -t/2 For a single layer element, integ2 is always zero. (SSb) = ' t/2 -t/2 t/2 -t/2 z{Sl}dz = ' t/2 -t/2 z[El]{Â£l}dz [El](ZÂ£p + z^fbjdz .18) (5.19) 99 {SSs} t/2 -t/2 ' t/2 {S2}dz = t/2 [E2] { e2}dz -t/2 [E2]{es}dz (5.20) â€œt/2 Eqs. (5.18),(5.19) and (5.20) can be put together to form the generalized "stress-strain" relationship of a shell element as (SS)(8,1) = [D](8,8){Â£Â£>(8,1) where, [D] CDiiia^S)][D12(3# 3)3 CO (3,2)] [D21(3,3)][D22(3,3)][0 (3,2)] [0 (2,3)][0 (2,3)][D33(2,2)] (5.21) [Du] = 't/2 -t/2 [El]dz tD123 â€œ [Â°21^ â€™t/2 z[El]dz -t/2 [d22] [D33] t/2 -t/2 't/2 -t/2 z2[El]dz [E2]dz It is noted that [D12] and [D21] are zero for a single layer element because it is the integration of an odd function over the open domain (-t/2,t/2). For a multi-layer system, these must be kept as is because material properties of layers change as the element deforms. o o o o o 100 5.3 Finite Element Discretization 5.3.1 Linearized Incremental Strain-Displacement Relationship By rearranging Eqs. (5.11) through (5.13), the strain- inidplane displacement relationship is (ee) eb eS U0'X + (twO'X >1 0 > + (two'y uo'y + vo' X + ^y'x â€œflx'y 9Y'Y - 6X'X X 0 > + eY woÂ»y - 0* (wo'x) (wo'y> (two'x)(wo'y) + (^o^) (wQ,x) d/dx 0 d/dy 0 (hr d/dy (hr d/dx (rw O'X o'y O'X 0 0 0 0 0 0 0 d/dx 0 d/dy )(d/dx) )(d/dy) ) (d/dyi + fS^y 0 0 )(d/dx) 0 0 -d/dy -d/dx 0 -1 0 0 0 d/dx 0 d/dy 1 0 u. w [d](8,5) [U0](5,l) (5.22) 101 5.3.2 Generic Displacement-Nodal Displacement Relationship The generic displacements at midsurface {U0} can be interpolated using the shape (interpolation, displacement) functions and the nodal displacements at midsurface (q). (U0> uo !iui vo n fivi wo = s fiwi Â«X i=l !i*xi [ *Y J f i^yi flul + f2u2 + â€¢ â€¢ + Â£nun !lvl + !2v2 + â€¢ â€¢ + ?nvn !iwi + f 2W2 + * â€¢ + fnwn Â£l*xl + 1l29x2 + â€¢ â€¢ 4* ^n^xn f l^yl + t2ey2 + â€¢ â€¢ + fn*yn fx 0 0 0 0 f2 0 0 0 fx 0 0 0 0 f2 o 0 0 fx 0 0 0 0 f2 000 fÂ± 0000 0000 fx 000 fn 0 0 0 0 0 fn 0 0 0 0 fn 0 0 0 fâ€ž 0 0 0 0 0 0 0n f n Ui W, 'xl 'yi u w- 'x2 ?y2 u w. n n n xn yn 102 = [ f![15] f2[I5] ... fn[I5] qi <*2 q n = [[%] [N2] ... [Nn3 3 {q} {UQ} = [N]{q} where, n = Number of element nodes. [15] 1 0 0 0 0 0 10 0 0 0 0 10 0 0 0 0 1 0 0 0 0 0 1 [N] = [ ^[15] f2[I5] ... fn[I5] ] = C [% ] [N2 ] ... [Nn] ] {gi} = {uL Vi Wi exi eyi}T (q) = { {qi)T (q2}T â€¢â€¢â€¢ {qn>T >T 5.3.3 Strain-Nodal Displacement Relationship If Eq. (5.24) is substituted into Eq. (5.22), the incremental strain is {ee) = [d]{U0) = [d] [N](q) = [Blz] (q> (5.23) (5.24) (5.25) (5.26) 103 where, [Blz] = [d][N] (5.27) = Linearized strain-nodal displacement matrix. 5.3.4 Evaluation of Linearized Strain-Nodal Displacement Matrix Each term of the [BLZ] matrix is calculated using Eq. (5.27) . [Blz] = [d] (8,5) [N] (5, (5n)) = [d] [ [N-l] (5,5) [N2](5,5) ... [Nn](5,5) ] Let [B-jJ = [dHNjJ, then Cblz^ â€œ tBi b2 â€¢â€¢â€¢ Bn^ where, [BÂ±] = [dJCNi] d/dx 0 (Ãw0,x)(d/dx) 0 d/dy (rw0,y)(d/dy) d/dy d/dx (S^,*) (d/dy) 0 0 0 0 0 0 0 0 0 0 0 d/dx 0 0 d/dy 0 0 0 0 w0,v)(d/dx) 0 0 2 0 d/dx -d/dy 0 -d/dx d/dy 0 1 -1 0 fi 0 0 0 0 0 0 0 0 0 0 fj, 0 0 0 0 0 fÂ¿ 0 0 0 0 0 X 104 (fi' x) 0 0 0 0 (fi' y> (*â€¢ i'y) (fi'x> twÂ°fy (fi'y) t (Si'y) (fi/^ + i^o^x) (fi'y) 0 0 0 0 0 0 0 0 ( f i / x) 0 0 0 0 0 0 0 (fi/y) 0 0 (!i'x> 0 fi 0 0 (fi/y) -fi 0 being denoted as [BlPi](3,2) (0] (3,2) [0] (2,2) [Bn, )(3,3) [BlÃ‰il(3,3) [BlS-jJ (2,3) (5.28) and decomposed into two parts for later use in element stiffness calculation. [BlPi ] (3,2) [0] (3,3) [Bl] = [0] (3,2) [Blbi ] (3,3) (5 .29) [0] (2,2) [BlSi 3(2,3) [0] (3,2) [BnÂ± ] (3,3) [Bn] = [0] (3,2) [0] (3,3) (5 30) [0] (2,2) [0] (2,3) 5.3.5 Evaluation of Nonlinear Strain-Nodal Displacement Matrix From Eq. (2.17), the nonlinear incremental strain components ij * (uk,i uk,j) are 105 If Eq. (5.1) is substituted into Eq. (5.30), then ^XX â€œ * (U/X u, X + V'X V'X + W,x W,x r,yY - 1 (u,y U,y + V/y v,y + W,y W, y Â»7xy = | (u f X U/y + V'X v,y + w,x W,y ^xz = *
u,z+ V'X V'Z + W,x W,z Â»?yZ = 5 (U,y U,z + v,y V'Z + W'Z W, z This can be simplified using the same assumptions for linearized strains. â€™'xx = 1 2 w,x w,x -3 II 1 2 W,y W,y ^xy = 1 2 W,x W,y (5.32) ixz = 0 riyZ = 0 And only non-zero terms are contained for corresponding Green's strain and denoted as (n). nxx ^xx * W,x w,x ^yy Â»?yy â€” 2 W,y W,y (5.33) nxy = 2r?xy = W,x W,y If Eq. (5.1) is substituted into Eq. (5.33), then [49] (n) I(wQ,x i(wO'y (wo'x ) (wo 'Y ) wO' X 0 W0'X _ 1 â€” 1 0 Wq / y wo'y wo'x Wq / y = HA]{i) (5.34) 106 As [A] and {0} are linear functions of {q>, i.e., linear functions of (w,x) and (w,y), the strain is quadratic in nodal displacements. But the (w,x), (w,y) values from the previous iteration are used in [A] as approximations to the true values. The strain is linearized in this manner and used in the evaluation of element stiffness matrix for the iterative solution of nonlinear equilibrium equation. To apply the principle of virtual work, it is desirable to express nonlinear strain in terms of nodal displacements {q}- . {n} = \ [A]{*> = f({q}) and the displacement gradient {0} can be written in terms of {q}- n [2 i=l (fjWi)] 'X n [2 (f^) ] ,y i=l (?l'x>wl + if2'x)w2 (fl,y)wl + (f2,y)W2 + + + J?n'x!wn + (fn,y)wn 107 0 0 0 0 0 0 f2Â»jj 0 0 flfy 00 00 f2,y 0 0 fn,x 0 0 0 0 fn,y 0 0 Ui w u 1 xl yi w- *x2 9y2 u w. n rn n xn yn = [ [Gi] [G2] [Gn] ] *1 *2 q n = [G]{q} where, (5.35) [GiJ 0 0 (fi#x) 0 0 0 0 (fi/y) 0 0 [G] = ( [Gx] [G2] tGn] ] 108 5.3.6 Discretization The word "discretization" means that the continuous displacement field is approximated using displacements at discretized nodal points. For the total Lagrangian formulation, incremental iterative equilibrium equation from Eq. (2.25) is oCijkl Qekl eii o 3dV 4 Â°dV = EVW Â°dV This can be shown in a matrix form as follows. The term "linear" means the first order differential of displacements with respect to coordinate variables while the term "nonlinear" means the second order differential of displacements with respect to coordinate variables. For example, if (u,x) is defined as au / ax and (u,xx) is defined as 32u / ax2, (u,x) is "linear" while (u,xx) is "nonlinear" in reference to strain terms. The linearized incremental strain (e) is related to incremental nodal displacements {q} through linearized strain-displacement matrix [BLZ] from Eq. (5.27). (e> = [BLZ]{q) 109 The variation of linearized incremental strain is then 5{e} = i[BLZ]{q} + [BLZ]i{q} = [BLZ]5{q> (5.36) because [BLZ] is constant about the unknown incremental displacements and the variation thereof is zero, i.e., S[Bl2] is zero. The nonlinear incremental strain {n} can be shown as the multiplication of two matrices [A] and {Â£}, which contains only linear terms. From Eq. (5.34), (n) = | [A]{0> The virtual variation in nonlinear strain terms is s(n> = sl[A]{0} = l(fi[A)){i) + I[A](5{0 } ) = HA](S{0}) + i[A](i{i>) = (A] (5{^ >) (5.37) because (5[A]){0) is equal to [A](5{^}) as shown below. 5W0'X 0 wo'x 0 Ãw0,y wo'y *wo'y 5W0'X [A]6(0} = w, 0 w, O'X o'y w, o'y O'X (*w0,x) (5wo'y Thus, (Â£ [A]){8} = [A](8{8}) *(w0,x)(wo'x) 5(wo'y^ (wo'y) *(wo'x> (wo'y) + iwo'x)5(wo'y) The gradient of displacements {0} is related to incremental nodal displacements through [G] matrix. From Eq. (5.35), {0} = [G] {q) and the variation thereof is 5(0} = 5[G]{q) + [G]5{q) = (G]5{q) (5.38) because [G] is constant about the unknown incremental displacements and the variation thereof is zero, i.e., 5[G] is zero. Now the incremental iterative equilibrium equation can be put into a matrix equation. It is noted that the engineering strains, (e) and (n), are used in the places of ekl' eij and ^ij corresponding constitutive matrices. cijkl ekl 5 eij = 5ie}T[cHe} = ([BLZ]5{q))T[C](CBlz](q) ) = Â«(q}T[BLZ]T[C][BLZ]{q} = 6(q}T[K1]{q} sij 5 *lij = S{n}T{S) = ([A]5{0})T{S) = 5<0}T[A]T{S} = ([G]5{q})T[A]T(S) = 5{q)T[G]T(S]{0) = *{q>T[G]T[S][G]{q} = 5{q}T[K2]{q) The relationship, [A]T{S) = [S]{0> [49], is simple mathematical equivalence by rearranging the elements of the matrices in different format to relate the nonlinear strain Ill {n} to incremental displacement {q>. It is noted that [S] is a multi-column and multi-row matrix and {S} is a column matrix (a vector). This will be discussed in 5.4. The matrices [K-jJ and [K2] are newly defined as [*l] = [BLZ]T[C][BLZ] [K2] = [G]T[S][G] Sij 8 e^j = 5{e}^{S} = Â«{q}T[BLZ]T{S} External virtual work due only to nodal forces is EVW = 5{q}T{P} Then the incremental equilibrium equation becomes S{ q)T [KiHq) dV + 5{q)T [K2]{q) dV = S{q}T{P) - Ã{q}T [Blz]T{S) dV Let Cklz] = [%] dV [Kg] = [K2] dV 112 {RI} [Blz]T{S) dV If the volume integration is changed to area integration using analytical integration through thickness, [rlz] [Kj] dA (5.39) [KG] [Kn] dA (5.40) {RI} [Blz]T{SS} dA (5.41) where, [Kil â€œ [Blz]T[D][Blz] [Kj-j-] = [G]T[SS][G] It is noted that stresses are in a resultant form [SS] with the corresponding constitutive matrix [D]. Then the equilibrium equation becomes i(q)T (( [Klz] + [Kg] ) (q) - (P) + (RI) ) = {0} and this must be satisfied for any virtual displacements, 5{q}, meaning that ${q} cannot always be {0}, thus, (( [Klz] + [Kg] ) {q} - {P} + {RI} ) = {0} And finally the usual form, [K] {q} = {R}, can be obtained. ( [Klz] + [KG] ) {q} = {P} - {RI} (5.42) 113 5.4 Derivation of Element Stiffness Matrix 5.4.1 Linearized Element Stiffness The linearized incremental element stiffness due to linear and large displacement effects is evaluated using Eq. (5.39). For the efficiency in calculation, [BLZ] is divided into [Bl] and [Bn] in Eqs. (5.29) and (5.30), then [Blz]T[D][Blz] = [[Bl]+[Bn]]T [D] [[Bl]+[Bn]] = [B1JT[D][B1] + [Bl]T[D][Bn] + [Bn]T[D][Bl] + [Bn]T[D][Bn] [klz3 [B1]T[D][B1] dA + ([B1]T[D][Bn]+[Bn]T[D][Bl]+[Bn]T[D][Bn]) dA [Kl] + [Kid] (5.43) where, [Kl] = [B1]a[D][B1] dA (5.44) = Linear element stiffness. [Kid] = ( [B1]T[D] [Bn] + [Bn]T[D] [B;L] + [Bn]T[D] [Bâ€ž] ) dA = [Kldl] + [Kld2] + [Kld3] (5.45) = Large displacement element stiffness. 114 5.4.2 Geometric Element Stiffness The element stiffness due to initial stress effects is calculated using Eq. (5.40). The [SS] matrix must be found using [A] and {SS}. From Eq. (5.34), [A] = wo'x 0 0 w w o'y w o'y o'x {*} = WG'X wo' y The {SS} corresponding to nonlinear strain {n> is {SS} = {Nx Ny Nxy} The relationship between [SS] and {SS} is by simple rearrangement of matrix elements. [A]t{S} = [SS]{0} wo'x Â® wo'y 0 w0#y wQ,x x xy Nx N Nxy Ny* w0 r X WQf y Thus matrix form of generalized stresses [SS] is defined as [SS] = Nx N NXy N/ (5.46) Thus the geometric element stiffness becomes [Kg] [G]T[SS][G] dA (5.47 = Geometric element stiffness. 5.5 Calculation of Element Stiffness Matrix 5.5.1 Calculation of TK11 Each term of linear element stiffness is calculated from Eq. (5.42) block by block. From Eq. (29), [Bli]T = [BlPi] T(2,3) [0] t(2,3) [Â°] m (2,2) [0] (3,3) [BlbjJ T(3,3) [BlSi]T (3,2) [Blp-i ] [0] (3,2) [0] (3,2) [0] (2,2) [Blj] = (3,3) [Blbj] (3,3) [BlSj] (2,3) From Eq. (5.21), [D] (D11(3,3)][D12(3,3)][0 (3,2)] [D21(3,3)][D22(3,3)][0 (3,2)] [0 (2,3)][0 (2,3)][D33(2,2)] Thus, [B1]T[D][B1](5,5) [Kipp](2,2) [Klpb] (2,3) [Klbp](3,2) [Klbs] (3,3) 116 where, [Kipp] = [Blpi]T[D11][Blpj] [Klpb] = [Blpi]T[D12][Blbj] [Klbp] = [Blbi]T[D21][Blpj] [Klbs] = [Blbi]T[D22][Blbj] + [Blsi]T[D33][BlSjJ Note that for a single layer element, both [Klpb] and [Klbp] are zero matrices. 5.5.2 Calculation of TKldl Each term of nonlinear element stiffness from large displacement effects is calculated from Eq. (5.45) block by block. / 5.5.2.1 Calculation of TB11â€”TD1TBnl for TKldll [B1]t[D][Bn] [0] (2,2) [Kin] (2,3) [0] (3,2) [0] (3,3) where, [Kin] = [Blpi]T[D11][Bnj] Actual calculation gives Kln(l,1) 0 0 Kln(2,1) 0 0 [Kin] = 117 where, Kin(1, 1) D11 (If 1) (fi- X^ Bnj (1. â– 1) + D11 (I, 2) (fi< 'x) Bnj (2, 1) + D11 (3, 3) (fi< -y) Bnj (3, -1) Kin(2, 1) = D11 (2, 1) (fi< ry) Bnj (1- -1) + D11 (2, 2) (fÂ±, ry) Bnj (2, -1) + D11 (3, 3) (fi- rX^ Bnj (3, rl) [Bnj] = [Bn] evaluated for node j 5.5.2.2 Calculation of TBnlâ€”TD1TB11 for TKld21 [Bn]T[D][Bl] [0] (2,2) [0](2,3) [Knp](3,2) [Knb](3,3) where, [Knp] = [Bn^tD^nBlpj] [Knb] = [Bni^tD^Hblbj] Note that for a single layer element, [Knb] is zero matrix. Actual calculation gives [Knp]= Knp(1,1) Knp(1,2) 0 0 0 0 where, Knp(1,1) = ( D11(l,l)Bni(l,l)+D11(2,l)Bni(2,l) )(fj,x) + Dj^ÃS^Bnip,!) (fj,y) 118 Knp(l,2) = ( D11(l/2)Bni(l,l)+D11(2,2)Bni(2/l) )(fj,y) + D11(3,3)Bni(3/l)(fj/x) [Bnj] = [Bn] evaluated for node j [Knb] 0 Knb(1,2) Knb(1,3) 0 0 0 0 0 0 where, Knb(1,2) = (D12(l/2)Bni(l,l)+D12(2,2)Bni(2,l))(-fj/y) + D12(3#3)Bni(3,l)("fj,x) Knb(1,3) = (D12(l,l)Bni(l,l)+D12(2fl)Bni(2fl))(fj#x) + D12(3,3)Bni(3,l)(fj/y) 5.5.2.3 Calculation of fBnlâ€”TD1TBnl for TKld31 [Bn]T[D][Bn] [0](2,2) [0] (2,3) [0](3,2) [Knn] (3,3) where, [Knn] = [Bni]T[D11][Bnj] Actual calculation gives [Knn] Knn(1,1) 0 0 0 0 0 0 0 0 where, Knn(l,1) = (D11(l/l)Bni(l,l)+D11(2/l)Bni(2,l))Bnj(l,l) + (D11(l,2)Bni(l,l)+D11(2,2)Bni(2,l))Bnj(2,l) + D11(3,3)Bni(3,l)Bnj(3,1) 5.5.3 Calculation of TKC1 Each term of nonlinear element stiffness from stress effects is calculated from Eq. (5.47) block by block. [Gi^tSSHGj] 0 0 0 0 0 0 0 0 0 0 = 00 Kg(3,3) 0 0 0 0 0 0 0 0 0 0 0 0 where, Kg(3,3) = Nx(fi/x) (fjfx) + Nxy[(fi/x)(fj,y)+(fj,x)(fi/y)] + Ny(fi/y) (fj,y) 5.6. Element Stress Recovery Stresses can be calculated from the Eqs. (5.15), (5.16) and (5.17). {SSp} t/2 (Sl)dz = -t/2 *t/2 [El]{el)dz -t/2 *t/2 [El]{ep + Zeb}dz -t/2 By explicit integration along thickness nlyr {SSp} = E [El][(ep*thk(k) +cb*thk2(k)] k=l (5.48) where, nlyr = number of layers thk(k) = the thickness of k-th layer thk2(k) = | (h(k+l)2 - h(k)2) h(k) = the dimension from bottom of element to the bottom of k-th layer Similarly, {SSb} and {SSs} can be found as {SSb} = z{Sl}dz = z[El]{el}dz [El](Zep + z2eb)dz nlyr E [El][ep*thk2(k) +eb*thk3(k)] k=l (5.49) where thk3(k) = (1/3)(h(k+l)3 - h(k)3) {SSs} = {S2}dz = [E2]{e 2}dz [E2](eS)dz nlyr E [E2][thk(k)eS] k=l (5.50) These stresses are generalized stresses. Thus the common stresses for the determination of layer or element state must be calculated using the definition of generalized stresses from Eqs. (5.48), (5.49) and (5.50). From {SSp}, i.e., (Nx Ny NXy}T, which are the resultants of the inplane stresses for unit length of element edges, PK-II stresses Sx, Sy, and SXy can be calculated as 'x = Nx / thk Sâ€ž = Nv / thk Svâ€ž = Nvv / thk y y 'xy â€ iTxy From the (SSb), i.e., (Mx My MXy), which are the moment resultants of the inplane stresses for the unit length of element edges, PK-II stresses Sx, Sy and SXy can be found as 'x Mx(thk/2) / II = Mx * S6 Sy = My(thk/2) / II = My * S6 SXy = This is a torsional moment and very complicated in nature but can be approximated as (1/3)(1-0.63thk)thk3. where, thk = Element or layer thickness. II = Moment of inertia of the unit length of element section. = (1)(tkh3)/12 S6 = The inverse of section modulus of the unit length of the element section. = 11/(thk/2) = thk2/6 And these two components from {SSp} and {SS^} must be summed up for total stresses. Similarly PK-II stresses Sxz, and SyZ can be calculated from {SSs}. sxz = Qx / thk Syz = Qy / thk 5.7. Internal Resisting Force Recovery Once the general stresses are obtained, internal resisting forces at node i can be evaluated as ,T [Bi]x{SS} dA (5.51) where, (Pi) = {PXi, Py^ Pzi# Rxit Ry-jJ T {Pmi}(2,l) {PPj_} (3,1) PxÂ¿, PyÂ¿, PzÂ¿ = Concentrated nodal forces in x, y, z directions. RxÂ¿ = Nodal moment about x-axis. RyÂ¿ = Nodal moment about y-axis. {Pmi}= {PXi, Pyi}T = Concentrated nodal forces from membrane behavior. 123 (Ppi}= {Pzif RxÂ±l RYi}1 = Concentrated nodal forces from plate behavior. CBÂ±] [BlPi](3,2) [Bni] (3,3) [0] (3,2) [BlbjJ(3,3) [0] (2,2) [Blsi](2,3) (SS) = { (SSp}T (SSb}T {SSs}T }T â€” { N, Ny , Njjy , , My , Mj^y , QjÂ£ , Qy } Note: These generalized stresses are evaluated at the current integration point for the numerical integration of the internal resisting forces. The integrand of Eq. (5.51) can be evaluated using submatrices as (Pmi) = [BlPi]T{SSp} {PPi> = [Bni]T{SSp) + [Blbi]T{SSb) + [Bls^tSSs} Actual calculation will be (Pmi) = [Blpi]T{SSp} = (fi'x) 0 (fi'y) Nx 0 (fi'y) (fi/X) Ny PXi - (fi,X)*Nx + (fi,y)*NXy PYi = (fify)*Ny + (fi,X)*NXy The concentrated nodal forces from plate behavior have three components. 124 The first component is [Bni]T{SSp> = Bnll Bn21 Bn31 Nx 0 0 0 Nv 0 0 0 xy Thus, Pzp^ = Bni;L*Nx + Bn21*Ny + Bn31*Nxy RxpÂ¿ = 0 RyPi = 0 where, Bni;L = Bnp^ (1,1) Bn21 = Bnp^(2,l) Bn31 = BnpÂ¿(3,l) Note : These are all the nonzero terms in [BlpjJ . PzpÂ¿ = Component of PzÂ¿ from nonlinear strain terms, RxpÂ¿ = Component of Rx^ from nonlinear strain terms, RyPj. = Component of Ry^ from nonlinear strain terms, The second component is [Blib]T{SSb} = 0 0 0 MX 0 -(fi,y) M y (fi'x) 0 J (fi/y) M"* X1xy Thus, PzbÂ¿ = 0 RxbÂ¿ = -(fi,y)*My - Rybi = (fi,x)*Mx + (fi,y)*Mxy 125 where, PzbÂ¿ = Component of PzÂ¿ from plate behavior. Rxbj^ = Component of RxÂ¿ from plate behavior. Rybi = Component of Ry^ from plate behavior. The third component is [Blis]T{SSs) = Qx QY Thus, Pzsi = (fi#x)*Qx + (fi/y)*Qy Rxsi = -(fi)*Qy RysÂ± = (fi)*Qx where, PzsÂ¿ = Component of PzÂ¿ from shear stresses. Rxs^ = Component of RxÂ¿ from shear stresses. RysÂ¿ = Component of Ry^ from shear stresses. Therefore the concentrated nodal forces are Pzi = PzpÂ¿ + PzbÂ¿ + PzsÂ¿ Rx^ = RxpÂ¿ + RxbÂ¿ + Rxs^ (fi) 0 RYi = RyPi + Ryt>i + RysÂ¿ CHAPTER 6 NONLINEAR SHELL ELEMENT PERFORMANCE 6.1 Introduction The nonlinear Mindlin shell element directly derived from the linearized incremental equilibrium equation presented in chapter 5 has been implemented in the general nonlinear analysis program ANSR developed at the University of California, Berkeley [56], Linear material property is assumed for all the test runs. Three commonly used examples are tested. These are a cantilever beam with free end moment, a clamped square plate with distributed load, and a simply supported square plate with distributed load. 6.2 Large Rotation of a Cantilever For the cantilever with free end moment shown in Fig. 6-1, the analytical solution can be found as follows. From the geometry, the length does not change as the beam deforms. 2jtR * { Thus, R = L / 126 127 From the moment-curvature relationship, From Eq. (6.2), M = El / L (6.3) and if Eq. (6.2) is substituted into Eq. (6.1), R = El / M (6.4) The free end displacements u, v can be found using geometry. u = L - R sin^ = L - (EI/M) sin (ML/EI) (6.5) v = R - R cos<Â¿ = R ( 1 - cos = (EI/M)( 1 - COS (ML/EI)) (6.6) The data used are E = 30000 ksi I = (1)(0.1)3/12 in4 El = 2.5 Kips-in2 L = 10 in The analytical solution and numerical solution from ANSR are given in Table 6-1 and plotted in Fig. 6-2 and Fig. 6-3 for the vertical displacement and horizontal displacements. The figures show the excellent response within moderate rotation limits, in this case a total change in angle of n/8 radians. They also show the deviation in horizontal and vertical displacements as the rotation becomes large. Width = 1 in. Fig. 6-1 Cantilever under Free End Moment Table 6-1 Displacements of Cantilever Beam under Free End Moment Load Level Moment (K-in) Z-Disp. ANSR Z-Disp. ANAL X-Disp. ANSR X-Disp. ANAL 0 0.0000 0.0000 0.0000 0.0000 0.0000 1 0.0196 0.3926 0.3925 0.0205 0.0103 2 0.0393 0.7819 0.7838 0.0611 0.0411 3 0.0589 1.1660 1.1727 0.1208 0.0923 4 0.0785 1.5430 1.5579 0.1982 0.1637 5 0.0982 1.9100 1.9384 0.2919 0.2550 6 0.1178 2.2690 2.3129 0.4000 0.3660 7 0.1374 2.6160 2.6803 0.5211 0.4962 8 0.1571 2.9530 3.0396 0.6533 0.6451 9 0.1767 3.2780 3.3896 0.7952 0.8122 10 0.1963 3.5920 3.7297 0.9454 0.9968 11 0.2160 3.8960 4.0576 1.1030 1.1984 12 0.2356 4.1890 4.3737 1.2660 1.4161 13 0.2553 4.4720 4.6767 1.4340 1.6491 14 0.2749 4.7450 4.9657 1.6060 1.8967 15 0.2945 5.0100 5.2399 1.7820 2.1579 16 0.3142 5.2650 5.4987 1.9610 2.4317 17 0.3338 5.5120 5.7412 2.1410 2.7173 18 0.3534 5.7510 5.9670 2.3240 3.0135 19 0.3731 5.9330 6.1755 2.5080 3.3194 20 0.3927 6.2080 6.3662 2.6930 3.6338 Displacement (in) 4 Fig. 6-2 Vertical Free End Displacement of Cantilever Beam under Free End Moment 131 Displacement (in) Fig. 6-3 Horizontal Free End Displacement of Cantilever Beam under Free End Moment 132 133 6.3 Square Plate The second test model used is shown in Fig. 6-4. This is a square plate under distributed loads. The boundary conditions can either be fixed or simply supported. The data used are n = 0.3 = Poisson's ratio a = 300 in = Side length t = 3 in = Thickness E = 30000 ksi q = Distributed load The analytical linear solutions for the displacement at center of plate [59] are w = 0.00126qa4/D for clamped square plate w = 0.00406qa4/D for the simply supported square plate where, D = Et3 / 12(l-/Â¿2) = plate stiffness The linear analytical solution and numerical solution from ANSR are given in Table 6-2, Table 6-3 and plotted in Fig. 6-5 and Fig. 6-6 for clamped plate and simply supported plate, respectively. The comparison of nonlinear responses is given in Tables 6-4 and 6-5. The relative effects of large displacements and initial stresses with respect to total nonlinear effects are given in Table 6-6 and Fig. 6-7. 134 Size = 300 in. x 300 in. Thickness = 3 in. E = 30000 ksi Poisson's ratio = 0.3 Fig. 6-4 Square Plate under Distributed Loads Table 6-2 Displacements of Square Plate with Fixed Support under Distributed Loads Step No. Wt (pcf) q (psi) Linear (in) ANSR (in) 0 0 0.0000 0.0000 0.0000 1 1000 1.7361 0.2389 0.2389 2 2000 3.4722 0.4778 0.4716 3 3000 5.2083 0.7166 0.6949 4 4000 6.9444 0.9555 0.9065 5 5000 8.6806 1.1944 1.1060 6 6000 10.4167 1.4333 1.2930 7 7000 12.1528 1.6721 1.4680 8 8000 13.8889 1.9110 1.6310 9 9000 15.6250 2.1499 1.7850 10 10000 17.3611 2.3888 1.9300 11 11000 19.0972 2.6276 2.0660 12 12000 20.8333 2.8665 2.1950 13 13000 22.5694 3.1054 2.3180 14 14000 24.3056 3.3443 2.4340 15 15000 26.0417 3.5831 2.5440 16 16000 27.7778 3.8220 2.6500 17 17000 29.5139 4.0609 2.7510 18 18000 31.2500 4.2998 2.8470 19 19000 32.9861 4.5386 2.9400 20 20000 34.7222 4.7775 3.0290 Displacement (in) 5 Fig. 6-5 Center Displacement of Clamped Square Plate under Distributed Load 136 Table 6-3 Displacements of Square Plate with Simple Support under Distributed Loads Step No. Wt (pcf) q (psi) Linear (in) ANSR (in) 0 0 0.0000 0.0000 0.0000 1 1000 1.7361 0.7697 0.6618 2 2000 3.4722 1.5394 1.1340 3 3000 5.2083 2.3091 1.4940 4 4000 6.9444 3.0788 1.7850 5 5000 8.6806 3.8485 2.0290 6 6000 10.4167 4.6183 2.2400 7 7000 12.1528 5.3880 2.4260 8 8000 13.8889 6.1577 2.5930 9 9000 15.6250 6.9274 2.7450 10 10000 17.3611 7.6971 2.8850 11 11000 19.0972 8.4668 3.0140 12 12000 20.8333 9.2365 3.1340 13 13000 22.5694 10.0062 3.2470 14 14000 24.3056 10.7759 3.3540 15 15000 26.0417 11.5456 3.4540 16 16000 27.7778 12.3153 3.5500 17 17000 29.5139 13.0850 3.6410 18 18000 31.2500 13.8548 3.7290 19 19000 32.9861 14.6245 3.8120 20 20000 34.7222 15.3942 3.8930 Displacement (in) 16 Fig. 6-6 Center Displacement of Simply Supported Square Plate under Distributed Load 138 Table 6-4 Comparison of Displacements of Square Plate with Simple Support Load Load Exact ANSR ERROR (psi) steps (in) (in) (%) 0.000 0 0.0000 0.0000 0.0000 2.748 10 1.005 1.019 1.39 10.980 10 2.454 2.345 4.44 43.950 10 4.41 4.212 4.49 175.800 10 7.2 6.258 13.08 100 10.99 4.35 703.200 200 11.49 11.24 2.17 400 11.31 1.57 2813.100 200 18.21 17.80 2.25 The exact values are quoted from reference [20] Table 6-5 Comparison of Displacements of Square Plate with Clamped Support Load Load Exact ANSR ERROR (psi) Steps (in) (in) (%) 0.00 0 0.0000 0.0000 0.0000 5.337 10 0.711 0.7082 0.39 11.490 10 1.413 1.3960 1.20 20 1.4020 0.78 19.020 10 2.085 2.041 2.11 20 2.057 1.34 28.500 10 2.736 2.649 3.18 20 2.679 2.08 40.470 20 3.363 3.274 2.65 55.200 20 3.969 3.841 3.22 73.500 20 4.563 4.396 3.66 95.400 20 5.142 4.931 4.10 120.600 20 5.706 5.436 4.73 The exact values are quoted from reference [20] Table 6-6 Displacements of Square Plate with Fixed Support under Distributed Loads Using Different Nonlinear Stiffnesses without Iterations step no. wt (pcf) GEO (psi) LD (in) ALL (in) 0 0 0.0000 0.0000 0.0000 1 1000 0.2524 0.2524 0.2524 2 2000 0.5023 0.5030 0.5005 3 3000 0.7474 0.7483 0.7396 4 4000 0.9859 0.9854 0.9664 5 5000 1.2160 1.2120 1.1790 6 6000 1.4830 1.4280 1.3780 7 7000 1.6500 1.6320 1.5640 8 8000 1.8540 1.8250 1.7360 9 9000 2.0480 2.0060 1.8970 10 10000 2.2330 2.1780 2.0480 11 11000 2.4110 2.3400 2.1900 12 12000 2.5810 2.4930 2.3240 13 13000 2.7430 2.6390 2.4500 14 14000 2.8990 2.7770 2.5690 15 15000 3.0490 2.9080 2.6830 16 16000 3.1930 3.0330 2.7910 17 17000 3.3320 3.1530 2.8940 18 18000 3.4660 3.2680 2.9930 19 19000 3.5950 3.3780 3.0880 20 20000 3.7200 3.4840 3.1790 Displacement (in) Fig. 6-7 Effects of Different Nonlinear Stiffnesses on Clamped Square Plate under Distributed Load 142 CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS Two finite elements have been developed and implemented in ANSR [56] for the analysis of a hollow box girder for post-tensioned construction. The three-dimensional link element used to model a dry joint has shown realistic element behavior. It opens under tension and closes under compression. The link element has exhibited a cyclic convergence behavior. The linear Mindlin shell element used to model the concrete section of the hollow girder has shown an excellent response within its small displacement assumption. The nonlinear Mindlin shell element has been developed to predict large displacement and initial stress (geometric) nonlinearities. It has been derived directly from the linearized incremental equilibrium equation. This is basically a linear equilibrium equation within each increment. Thus the formulation is similar to that of linear element. The total Lagrangian formulation was used for the description of motion. The disadvantage of this formulation is that it needs special treatment for large rotations because finite rotation is not a tensorial (vector) quantity. One of the solutions to this limitation is co- 143 144 rotational formulation which is basically an updated Lagrangian formulation for rotations only [26, 27]. The displacement dependent loads, which are usual for shell structures, is also recommended for further study. Another area of further research is the material nonlinearity of concrete. This nonlinearity is essential to provide realistic structural response for safe and cost effective designs. Some of the complex concrete properties are nonlinear material properties, cracking in concrete, shear transfer across cracked interfaces, time dependent effects such as creep, shrinkage, and transient temperature distribution [60]. The nonlinear material properties of concrete have long been the subject of research. The first reliable test results on the two dimensional constitutive relationship were reported by Kupfer and Hilsdorf [61, 62] and were used to develop the equivalent uniaxial strain concept [63]. Various constitutive models of concrete can be found in [60, 64]. An entirely different approach, endochronic theory, to materials in which the inelastic strain accumulates gradually was also suggested. It consists of characterizing the inelastic strain accumulation by a certain scalar parameter, called intrinsic time, whose increment is a function of strain increment [65]. The cracks developing in concrete have been studied for a long time. Theories and applications have been developed for the mechanical behavior of individual crack [66, 67], discrete parallel continuous crack [68], distributed (smeared) in a fixed directions cracks [60, 69, 70, 71] , distributed rotating cracks[72]. The smeared crack theory and fracture localization in concrete is well documented in [73]. In a cracked reinforced concrete flexural member, the intact concrete between each pair of adjacent tensile cracks assists the tensile steel in carrying the internal tensile force, and therefore contributes to the overall bending stiffness of the member. This is called tension stiffening [74], The shear transfer through aggregate interlock was described in [75, 76, 77, 78]. The combination of all the effects from cracks, aggregate interlock, dowel action, and tension stiffening in concrete response can be found in [79]. It is well known that there can be numerical instability, and sensitivity on finite element mesh size, in the solution process for the strain softening material [80]. This has been overcome through the shear band concept [80] or a specific element formulation, for example, the four node isoparametric element suitable for modelling cracks in [81]. All these complex nonlinear material properties of concrete must be incorporated for the realistic analysis of any concrete structure including the hollow box girder for bridge structures. APPENDIX A IMPLEMENTATION OF LINK ELEMENT The three dimensional link element was implemented to the ANSR-III program. ANSR requires that ten subroutines be written for an element to be used by the program [56], The following is a description of the subroutines written for the link element; INEL06, STIF06, STAT06, RINT06, EVNT06, OUTS06. The subroutines RDYN06, INIT06, MDSE06, EOUT06, and CRLD06 are not used by the current version of the link element. As a result, dummy subroutines that consist of return statements are written for these subroutines. The element group number is set to six for the link element. A.1 General Implementation Details Implicit double precision is used throughout the interface subroutines. IMPLICIT DOUBLE PRECISION(A-H,0-Z) The labeled common block /INFGR/ for element group information is defined as follows. COMMON /INFGR / NGR,NELS,MFST,IGRHED(10),NINFC,LSTAT, LSTF,LSTC,NDOF,DKO,DKT,EPROP(3,200) 146 where, NGR = Element group number. NELS = Number of elements within the group. MFST = Element number of the first element in this group. IGRHED(IO) = Element group heading. LSTAT = Length of words of state information variables in /INFEL/. LSTF = Length of words of stiffness information variables. LSTC = Length of words of stiffness control variables. NINFC = Length of words of variables in /INFEL/. = LSTAT + LSTF + LSTC NDOF = Number of element degrees of freedom. DKO = Initial stiffness damping factor. DKT = Current tangent stiffness damping factor/ DKO, DKT are not used in the link element but are included in /INFGR/ because these are used by the base program. The labeled common block /INFEL/ for element information is defined as COMMON /INFEL / IMEM,KST,LM(12),NODE(5),MAT,CDEL,XL, XX(5),YY(5),ZZ(5),T(3,3),XKS,XKN,XKSO, XKNO,XNU,THK,DISU,DIST,DISV,DISW,STRU, STRV,STRT,STRM,IJTS,UTO,FKP(78) where, state information variables are IMEM through STRM. The length of the state information variables in terms of integer words, LSTAT, is 100 for this case because there are twenty integer variables (IMEM through MAT) and forty real variables (CDEL through STEM). A real word has a length of two integer words. IMEM, KST, and LM(12) must be at the beginning of the labeled common block /INFEL/ in the given sequence. LM is 12 long since there are 12 element degrees of freedom for the link element. IMEM = Element number. KST = Stiffness update code. LM(12) = Location matrix. NODE(5) = Node numbers, no. of 4-el nodes & K node. MAT = Material property number. CDEL = Allowable compressive displacement for the passing of two joint nodes. XL = Element length. XKS = Current element shear stiffness. XKSO = Original element shear stiffness. XKN = Current element normal stiffness. XKNO = Original element normal stiffness. XNU = Friction coefficient. THK = Element thickness. DISU = Element deformation in u(x) direciton. DIST = Element deformation in t(y) direction. DISV = Element deformation in v(z) direction. DISW = Element rotation. STRU = Element stress in u(x) direction. STRT Element stress in t(y) direction. 149 STRV = Element stress in v(z) direction. STRM = Element moment which shows the distribution of element stress. XX (5) = X coordinates of four nodes and the K node. YY(5) = Y coordinates of four nodes and the K node. ZZ (5) = Z coordinates of four nodes and the K node. T(3,3) = Transformation matrix. (3,3) submatrix. The stiffness control information variables are UTO,IJTS. The word length of these variables, LSTC, is two. IJTS = INDEX FOR JOINT SLIDING. (1 = SLID, 0 = NOT SLID) IJTO = INDEX FOR JOINT OPENING. (1 = OPENED, 0 = CLOSED) The stiffness information variables are FK(78). The word length of this group, LSTF, is 156. The actual size of the element stiffness matrix is twelve by twelve which contains 144 real numbers. Because of the symmetry of the element stiffness, only the lower half of the element stiffness matrix, including the diagonal is to be saved. The number of terms in that portion of the element stiffness matrix is 78. FK(78) = Element stiffness matrix. A.2 Subroutine INEL06 Subroutine INEL06 is the input subroutine. Its purpose is to read and print the input data for link elements and to initialize the variables in the labeled common blocks /INFGR/ and /INFEL/. This subroutine has the form of SUBROUTINE INEL06 (NJT,NDKOD,X,Y,Z,KEXEC) In the inel06 subroutine, the element group control variables are to be set. The first one is MFST, the element number of the first element in this group. MFST is input data read by the base program. It is defaulted to be one at the beginning of the inel06 subroutine if it is not given in input file. The index for the stiffness change, KST, is set to one. This indicates to form the stiffness since this is the first pass. KST can be updated in accordance with the actual status of element stiffness, i.e., if element stiffness is not changed KST is assigned the value of zero. The length of the element information and the number of element degrees of freedom are also to be set in inel06 subroutine. LSTAT = 100 LSTF =156 LSTC = 2 NINFC = 258 NDOF 12 Then element group information is printed. NGR, IGRHED, NELS, MFST are read in base program. All others are set in inel06 subroutine. The number of material properties and the material properties are read and then echo-printed. CDEL is the compressive deformation limit to define joint overlap. This is the amount of displacement overlap allowed before the stiffness is increased to prevent the loss in forces during transfer through displacement. NMAT is the number of material properties. For the link element, three material properties are required, i.e., joint normal stiffness, joint shear stiffness, and coefficient of friction. An index for the error, INERR, is set up to detect an error during the data input in inel06 subroutine. It is initially set to zero. If an error is detected it is set to one. The control information for element data is calculated. IMEM is the element number, which will be increased by one after the end of each element data line is read or generated. NLAST, the the last element number in this group, calculated by taking the first element number plus the number of elements. An index to check the number of the lines of element data, ICNTR is set up. Element data are read element by element. The required element data are NEL = Element number. NODE(5) MAT = Numbers of four nodes and the third node. = Material property of the element. THK = Thickness of the element. NGEN = Number of elements to be generated including the element specified. (NGEN-1) elements are to be generated. KINCR = Increment of the number of the third node for element generation. If no generation of element data is specified, i.e., NGEN is zero. The numbers of node three and four are interchanged for ease of input and then element data will be processed by the subroutine EVEL06. The node numbering for the formulation and data input is 4 * * 3 1 * * 2 Node numbering for formulation. 3 * * 4 1 * * 2 Node numbering for data input. If generation of element data are requested, i.e., positive NGEN, (NGEN-1) sets of element data will be generated followed by the element data processing. Before element data are processed by the base program, all element data is printed. Element data processing is performed by calling the subroutine EVEL06 which is part of the element input subroutine. All these processes are done for each link element. For the last element, the element number is checked against the 153 last element number NLAST. If there is a discrepancy, KEXEC is set to one for data check mode and the subroutine EXIT is called to terminate the program. Element data processing is composed of 1. Continuous element numbering from the first MFST. 2. Fill LM array. 3. Form transformation matrix. 4. Initializing variables in /INFEL /. 5. Compute stiffness matrix profile. 6. Transfer element data to tape. Continuous element numbering is done simply by increasing the element number IMEM by one at the end of each element processing. The LM array contains the global degree of freedom numbers corresponding to the element degree of freedom numbers. The subroutine NCODLM must be called as many times as the number of the terms in LM array. This subroutine is called 12 times for the link element, since there are 12 element degrees of freedom. ANSR numbers all the structural degrees of freedom in the x-direction first and then those in the y-direction and in the z-direction, respectively. The transformation matrix is formed as follows: 1. Dimension a (3,3) matrix, i.e., T. 2. Find the element direction vector({i*}, local u-direction vector) from the coordinates of the start and end nodes and then normalize it by 154 calling VECTOR. Put the three components into the first row of the [T] matrix. 3. Form the third direction vector({kk), local third direction vector) from input by calling VECTOR. This is the local n direction vector for the link element. Therefore, normalize it({k'}) and put the three components into the second row of the [T] matrix. 4. Perform the vector cross {i* > x {kâ€¢> to find local w-direction vector and normalize it({j *)) by calling CROSS. Put the three components into the third row of [T] matrix. 5. The resulting matrix is the transformation matrix [T]. The length of the element is also calculated during the formation of the transformation matrix. The profile of the stiffness matrix is updated by calling the subroutine BAND. The element data are transferred to tape by calling the subroutine COMPAC. A.3 Subroutine STIF06 Subroutine STIF06 is the element stiffness formation subroutine. This subroutine is called whenever the structural stiffness is to be formed or modified. If the total element stiffness matrix is to be formed, the subroutine is called once for each element. If a change in stiffness is being formed, the subroutine is called only for those elements which have undergone a stiffness change. The total stiffness or change in stiffness must be returned in array FK for assembly into the structure stiffness. This subroutine has the form of SUBROUTINE STIFO6(ISTEP,NDF,CDKO,CDKT,FK,INDFK,ISTFC) The variables used in this subroutine are FK = Element stiffness matrix(NDOF,NDOF). FKG = Global element stiffness matrix. FKL = Local element stiffness matrix. FKP = Previous global element stiffness. ISTFC = Stiffness matrix content index. 1 = Total element stiffness matrix. 0 = Change in element stiffness matrix. INDFK = FK storage index. 1 = Lower half, compacted columnwise. 0 = Square(all). ISTEP = Current step no. in step-by-step integration. NDOF = Number of element degrees of freedom. The index for the stiffness storage scheme, INDFK, is set to zero in this routine since square storage compacted column-wise is used. The index for stiffness matrix content , ISTFC, indicates whether the total stiffness matrix or the change in stiffness is needed. This is sent in by the base program and not set in the subroutine. The linear part of element stiffness is formed and rotated to global coordinate system. There are two options for element stiffness formation. If total element stiffness is requested, newly formed element stiffness FKG will be transferred to FK. Otherwise, the change in element stiffness is calculated by FKG minus FKP and then this change will be transferred to FK. The element stiffness in the local coordinate system is formed in the subroutine LSTF06. The element stiffness matrix is initialized with zeros and then the element properties are updated. Each term of the upper triangular and diagonal of the element stiffness is evaluated and then lower triangular of the element stiffness is filled using symmetry. The transformation of the element stiffness from local coordinates to global coordinates is done in the subroutine TRAN06 using the three by three submatrix. This is done for the efficiency and storage savings. The 12 by 12 element stiffness matrix is divided into three by three submatrices and the rotation is performed for each submatrix. This will eliminate the unnecessary multiplication of zeros while saving storage slightly. The process is FKG = TT*FKL*T using 3x3 submatrices 1. Zero FKG(K,L). 2. Divide FKL(12x12) into 3x3 submatrix. 3. Zero TEMP3(3,3) for each manipulation. 4. TEMP3=TT*FKL 5. FKG=(TT*FKL)*T A.4 Subroutine STAT06 Subroutine STAT06 is the state determination subroutine. Its purpose is to update the element state 157 information in /INFEL/, given the current state and the increment of nodal displacements in the global axes(array Q) set up by the base program. The subroutine has the form of SUBROUTINE STAT06 (NDF,Q,TIME) The variables used in this subroutine are = INCREMENTAL global nodal displacements. Therefore, if total stresses are needed, the element displacements are to be added up at the end of each iteration. = INCREMENTAL local nodal displacements = Incremental element deformation. These are to be saved in infel for use in rint06 to find FE for the equilibrium check of incremental external loads. = total elemental deformation, where * = U,T,V, or W If geometric (large displacement) nonlinear analysis is requested, i.e., KGEOM is one, Coordinates of all the nodes are to be updated and the new lengths of the elements with new direction cosines are to be calculated and then the transformation matrix is to be updated. Otherwise skip to the small displacement analysis step. For the analysis with the small displacement assumption, the incremental global nodal displacements (Q)(NDOF) are rotated to local coordinates (QDL)(NDOF). Then the average incremental element displacements, DISUI, DISTI, DISVI, DISWI are calculated. These incremental element Q(NDF) QDL(12) DIS*I DIS* 158 displacements are then added up to form total average element displacements, DISU, DIST, DISV, DISW. The total average element stresses, STRU, STRT, STRV, STRM are then calculated from the total average element displacements. These total stress are used for the state determination along with the total average element displacements. XNU is the static friction coefficient and XNUK the kinetic friction coefficient. XKNU is assumed to be 0.75 times XNU. If the joint has slid previously (xks=0.), kinetic friction stress has developed. This stress should be incorporated in the internal resisting force recovery for the equilibrium check. Sliding is defined in three categories: 1. Sliding in u-direction. 2. Sliding in v-direction. 3. Sliding in both u- and v-direction. If any one of these happens, the shear stiffness is set to zero. The frictional force is treated as the unbalanced force in the corresponding direction for the next iteration. Joint sliding is defined as the state where the vectorial sum of the element stresses in u- and v-directions is greater than the magnitude of the frictional stress if the normal stress is compressive. If the normal stiffness, XKN, is zero, then no shear stress will develop, which was considered in average stress calculation above. The normal stress is normal stiffness, XKN, times the corresponding displacement, DIST. The shear stiffness, XKS, is set to zero whenever the normal stiffness is set to zero. Before checking deformations, the stresses and determining the state of the element, the stiffness change index, KST is set to zero. If any change in element state occurs, this index will be set to one so that the element stiffness can be updated in the next iteration. The change in element state is checked against four possible cases. The element state change modes are 1. Closed to open. 2. Open to closed. 3. Stopped to sliding. 4. Sliding to stopped. If none of these four changes in state occurs, there is no change in element state and the index KST remains zero. The state 'open' is defined is defined as 1. Avg. normal deformation dist> or = zero. 2. or strt > or = zero. If a joint is opened, then there will be no shear and no normal stiffness. This is indicated by setting the joint opening index, IJTO, to one. The normal and shear stiffness are also set to zero for the next iteration in the subroutine STIF06. If these are to be set to zero here this will affect the following decision statements in the rest of the STAT06 routine. As a result, SKX is used as an argument 160 in the decision if-statement. XKS is the shear stiffness at the beginning of the STAT06 routine. 'Overlap' is defined as the state where normal compressive displacement is algebraically less than the negative value of the given limit of compressive displacement. This value, CDEL, is a positive number in the input data. If joint nodes have overlapped beyond the limit specified by CDEL, reactivate the shear stiffness XKS and increase the normal stiffness XKN to prevent overlapping in the next iteration. If (abs(dist).ge.cdel) is used, a large tension disp will be taken as joint overlap, which is not true. If a joint has been closed without overlapping, the normal stiffness and the shear stiffness will be reset to the original values saved in the labeled common block /INFEL/. The element state 'slid' is defined as sqrt(stru**2+strv**2) > -xnu*strt if strt < 0. The case where the normal stress STRT is greater than or equal to zero was covered in the joint opening decision. The decision of whether the displacement, sqrt(disu**2+disv**2), is greater than zero cannot be used as a definition of sliding. This is because there is a slight displacement before sliding occurs. This displacement is not precisely known for every case to be analyzed. If the joint is sliding, then no shear stiffness is maintained. If the joint is sliding in one direction, it is assumed that the joint is sliding in both directions. The joint sliding index, IJTS, is set to one. The shear stiffness, XKS may be set to zero because XKS is not used in anymore if-statement. For the consistency in the program structure, this is done in the subroutine STIF06, the same as XKN. If the joint is in the state of 'stop' which has occurred from the 'sliding' state, shear stiffness is recovered and IJTS is set to one. The state determination for creep strain and large displacement is to be done here. These are not included for the current version of the link element. A.5 Subroutine RINT06 Subroutine RINT06 is the element force recovery subroutine. This subroutine is called for each element at the beginning of the analysis and after each state determination. Its purpose is to compute the element forces, i.e., the nodal loads which are in equilibrium with the current state of stress. These may be stated as equivalent nodal loads which can cause the current state of element stresses. Therefore, if these equivalent nodal loads are subtracted from the actual nodal loads applied, current unbalanced nodal loads will be obtained. This subroutine has the form of SUBROUTINE RINT06 (NDF,Q,VEL,FE,FD,TIME) The meaning of the variables used in this subroutine are Q = Nodal displacements. VEL = Nodal velocity. FD = Dynamic nodal force when TIME > 0. FE = Elasto-plastic nodal force when TIME = 0. = Nodal loads which is in equilibrium with current state of stresses. = Imaginary forces which act on element to introduce current element stresses. The diagram for the element stress, element force, internal resisting force and the external load is <â€”Â° > O ELTMT 0 o m * <â€”0 0 * EL STRESS FE RI NODE element force internal external resisting force force 163 The sign convention for the element shear stresses is A< I I I I >v positive shear stress The recovered element forces which are in equilibrium with the element stresses can be turned into internal resisting forces by changing the direction of the forces. These internal resisting forces are not usually in equilibrium with external forces. The difference is the unbalanced forces. In ANSR, the norm(sum of squares of each difference) is used as decision variables for convergence. For the equilibrium check, ANSR uses the total loads. Therefore, the element forces are to be recovered using the total element displacement. The element forces recovered from the total element displacements are in the local coordinates. These must be rotated to global coordinates so that these forces can be compared with the global loads for the equilibrium check. The rotation from the local to global is done by (local) = [T](global) (global) = [T transpose](local) The recovered element forces are transferred to the base program through the FE array after the rotation from local to global coordinates. The nodal forces due to damping are supposed to be calculated in this subroutine. The current version does not include damping. If damping is considered, the variable TIME will be greater than zero. This is compared with the constant zero to see if damping is included. A.6 Subroutine EVNT06 Subroutine EVNT** is called for each element at frequent intervals during the analysis. Its purpose is to calculate the proportion of the displacement increment, Q, which can be applied to an element before a significant nonlinear event occurs. Typical events are yielding, gap closure, and unloading, i.e., the intersection of two linear portions in structural stiffness. This subroutine has the form of SUBROUTINE EVNT06 (NDF,Q,VEL,ACC,EVFAC,IEV) The variables used in this subroutine are IEV = Event type index. 0 = No event. 1 = Link opened. 2 = Link slid. 3 = Link closed. EVFAC = Event factor = Force used to cause the event / Force applied. If no event happens, EVFAC = 1. The large displacements nonlinear analysis is not included for the current version of the link element. 165 The number 0.999999 is defined as PTNINE to avoid the numerical difficulty in comparing the calculated ratio to one to predict an event. Small displacements are assumed for the current version of the link element. The nodal displacements are more useful than the average displacements at the center of the element because nodal displacements can be used to define the joint closing to avoid passing of the nodes by providing higher stiffness at the point of joint closing. The element nodal displacements are recovered from the given global element nodal displacements through rotation from global to local coordinates. If several events happened, the smallest event factor governs. The events are identified in sequence and the corresponding event type assigned. Then the event factor is calculated and compared with the current smallest event factor. At the beginning of the decision process, the event type is set to zero and the event factor to one. Two additional variables are required as input data to define events. These are the element normal and shear forces at the opening event and sliding event, respectively. The first event is a link opening event. This is defined by the normal element displacement and the normal stiffness. The previous contact state is identified by the nonzero normal stiffness. The opening is identified by the 166 ratio of the normal displacement at the event over actual normal displacement. The second event is link sliding. The absolute shear displacement, DIS, is calculated by the square root of the sum of the squares of the displacements in local x and y direction. The absolute displacement is then compared with the shear displacement at the event. The third event is link closing. If this happens, the event factor is set to a very small number because there is a sudden change in element behavior. The analysis process needs to set back to the closing point and the normal stiffness set to its input value. A fourth event is node overlap. In this event the event factor is given a very small number and the normal stiffness is set to a higher value by a trial factor of ten to avoid overlap in the next iteration. A.7 Subroutine QUTS06 Subroutine OUTS06 is the output subroutine. Its purpose is to print the current element stresses, strains, and status information from the information in /INFEL/. The subroutine has the form SUBROUTINE OUTS06 (KPR,TIME) If no print is requested, KPR is set to zero, return to the base program. For the first element, KHED, is set to zero to write heading, print index, and print request type. For all subsequent elements, only the element information is written. This is identified by the KHED. If KHED is not zero, this element is not the first element. The current element information printed in this subroutine is element number, element node numbers, and the element stresses in local u, n, and w directions. A.8 Link Element Data Input Guide (1) Control information (1.1) First control line COLUMNS NOTE NAME DATA 4- 5(1) NGR Element group number = 6 6-10(1) NELS Number of link elements 11-15(1) MFST Element number of first link element 16-25(F) DKO Initial stiffness damping factor 26-35(F) DKT Tangent stiffness damping factor 41-80(A) Optional heading for link element (1.2) Second control line COLUMNS NOTE NAME DATA 1 -5(1) NMAT Number of material properties 6-15(F) CDEL Allowable compressible deformation (2) Material property data NMAT lines COLUMNS NOTE NAME DATA 1- 5(1) MAT Material property number 6-15(E) XKS Element shear stiffness 16-25(E) XKN Element normal stiffness 26-35(F) XNU Friction coefficient (3) Element data NELS lines COLUMNS NOTE NAME DATA 1- 5(1) NEL Element number 6-10(1) NODE1 Node 1 11-15(1) NODE 2 Node 2 16-20(1) NODE 3 Node 3 21-25(1) NODE4 Node 4 26-30(1) NODEK Node K for transformation 31-35(1) MAT Material property number 36-45(F) THK Element thickness 46-50(1) 51-55(1) NGENX KINCR Number of elements generated Node K increment for el. generation NOTES (1) Local coordinate system N0DE1,NODE2,NODE3,NODE4,NODEK n K o A I I 3 I | 4 oâ€” - - 1 â€” o O""' / / 2 w Note: Local w axis is decided by right-hand rule coming out of the page. APPENDIX B IMPLEMENTATION OF LINEAR SHELL ELEMENT The implementation is based on the linear version of three to nine node shell element from the simple analysis (SIMPAL) program written by Dr. Marc I. Hoit. Isotropic materials are assumed and arbitrary orientation of element in global coordinate system is considered. The shell element was implemented in the ANSR-III program. The subroutines written are; INEL13, EVEL13, VECT13, TLIST1, TLIST2, SELFWT, FRMST1, FRMST2, STIF13, SHSTF1, SHSTF2, ELAW1, ELAW2, FORMH, GD, TRANS, TRIPL, STAT13, RINT13, 0UTS13. The dummy subroutines for current shell element are; RDYN13, CRLD13, E0UT13, EVNT13, MDSE13. These are required by ANSR-III but not used for shell element. Thus, all the dummy subroutines have only return statements. Implicit double precision was used for all the subroutines. B.l Subroutine INEL13(NJT.NDKOD.X.Y.Z.KEXECn This subroutine reads element data for two dimensional shell element. The element is formulated through the 170 combination of membrane element and plate bending element. The assigned element group number is 13. Named common blocks and the variables are as follows. COMMON/INFGR/NGR,NELS,MFST,IGRHED(10),NINFC,LSTAT,LSTF,LSTC NDOF,DKO,DKT,PROP(4,225),RS(2,8),RS4(2,4),WG(8),WG4(4),NMAT ITMPLD,ALPHA,REFTEM,ITMOFF,IDUM(121) NGR = Group number for current elements = 13 for shell NELS = Number of current elements. MFST = Element number of the first element in current group. IGRHED(10) = Element group heading. NINFC = Length of common block /INFGR / in terms of integer words. LSTAT = Length of state information. LSTF = Length of stiffness control information. LSTC = Length of stiffness information. NDOF = Number of element degrees of freedom. DKO = Initial stiffness damping factor. DKT = Current tangent stiffness damping factor. DKO, DKT are not used for shell element but are included in /INFGR / for the compatibility with base program. The element property array EPROP(IOOO) in the labelled common block /infgr/ of base program has been divided into small blocks of group information. PROP(4,225) = Element property array which carries Young's modulus, Poisson's ratio, shear modulus and self weight. There is a limit of 225 on the total number of element property sets. RS (2,8) and RS(2,4) = The coordinates of Gauss points in local rs-systexn for the eight point and four point numerical integration, respectively. The numerical integration is needed in the evaluation of element stiffness, initial loads, and stresses. Selective numerical integration has been used. WG(8) and WG4(4) = Numerical Integration weights. NMAT = Total number of material properties. ITMPLD = Temperature load index for input. 1 = Input temperature of top and bottom sides. The same temperature over the top or bottom sides. 2 = Input different temperatures at each node. ALPHA = Coefficient of thermal expansion. REFTEM = Reference temperature for temperature change. This is needed for the calculation of in-plane thermal strains. ITMOFF = Index to ignore the effect of temperature in local r- or s- direction to model beam-type structures. 1 = Ignore temp loading in r-direction(local x). 2 = Ignore temp loading in s-direction(local y). IDUM(121) = Dummy integer array to make the length of the total element group property exactly 1000. COMMON/INFEL/IMEM,KST,LM(54),N0DE(9),NN,MAT,NRIP,THK,XX(9),Y Y(9),ZZ(9),TT(3,3),XY(2,9),XM(54),DISPT(54),SIG(24),TEMPLD(5 4),SELFLD(54),SIGI(24),ST(24,54),FKP(1485) This named common block carries the following element information. State information is IMEM through ST(24,54). The length is 3298 integer words consisting of 68 integer words and 1615 real words. There is no stiffness control information for shell element. Stiffness information is 173 FKP(1485) with the length of 2970 integer words from 1485 real words. IMEM = Current element number. KST = Control variable for stiffness change. 1 = Stiffness has been changed. 0 = Stiffness has not been changed. LM(54) = Location matrix that contains global degree of freedom numbers corresponding to element degree of freedom numbers. NODE(9) = Node numbers rearranged for formulation. NN Number of nodes (4 to 9) of the element. NRIP Number of numerical integration points. MAT Material property number. THK Element thickness. XX (9) Nodal coordinates in global x. YY (9) Nodal coordinates in global y. ZZ (9) Nodal coordinates in global z. TT(3,3) = Transpose of transformation matrix for the local X-, y-, z- vectors. XY(2,9) = Local dimensions dx, dy to be used for the calculation of Jacobian matrix. XM (54) = Nodal mass matrix (to be used for dynamics). DISPT(54) = Total nodal displacements. SIG(24) = Six stress components at four integration points to be extrapolated to nodal stresses. TEMPLD(54) = Equivalent nodal loads for temperature. SELFLD(54) = Equivalent nodal loads for self weight. SIGI(24) = Initial stresses to be subtracted from total stresses to yield actual stresses. 174 ST(24,54) = Element stress recovery array for linear version. Stresses can be recovered from local element nodal displacements as follows. [stress] = [E][strain] = [E][d][u] = [E][d][f][q] = [E][ B ][q] = [ST ][q] where, [E] = Stress-strain matrix = [C] in the program. [d] = Differential operator relating strain and [u]. [u] = Generic displacement at a point within an element. [f] = Displacement function. [q] = Nodal displacement in local coordinate system. = Displacement [u] at element nodes. FKP(1485) = Upper triangular portion of element stiffness or change in element stiffness(54,54). COMMON /WORK / NNODE(9),IDUM,IJG(2),TEM(9) NNODE(9) = Element node numbers for read-in. IDUM = Integer dummy variable to make the length even. IJG(2) = Number of elements to be generated in local i- and j- direction including the one specified. TEM(9) = Temporary array for temperature information. DIMENSION NDKOD(NJT,6),X(NJT),Y(NJT),Z(NJT),I0RD(9), NODGEN(9),TNODE(2,9) NDKOD = Index array of global degree of freedom numbers corresponding to element degrees of freedom numbers. NJT = Number of joints(nodes) in structure. X(NJT) = X-coordinates of all nodes. Y(NJT) = Y-coordinates of all nodes. Z(NJT) = Z-coordinates of all nodes. IORD(9)= Order of node numbers for formulation. NODGEN(9) = Node numbers for generation. TNODE(2,9) = Top and bottom temperatures at nodes. The array iord(9) has the order of element node numbers used for the formulation which is different from the order used for data read-in. This array will be used for the rearrangement of element node numbers for formulation at the time of data input. This array must be dimensioned because the array is filled by data statement and thus cannot be included in the common block. Other variables used in this subroutine are NODE(9) = ordered node numbers for formulation WG = integration weights IEL = number of elements to be generated in i-direction JEL = number of elements to be generated in j-direction INC = node number increment in i-direction for element generation JNC = node number increment in j-direction for element generation IJG(2) = number of elements to be generated including the element specified in i (IJG(l)) and j (IJG(2)) direction. Element generation works only for rectangular elements. The element number of the first shell element MFST is set to one unless otherwise specified. The element stiffness is currently constant for the linear shell element and therefore the stiffness change index KST is set to zero. The word lengths of element information groups are figured out. Length of element state information variables LSTAT, length of stiffness control variables LSTC, length of 176 stiffness information variables LSTF and total length of common block /infel /, NINFC are calculated. The number of element degrees of freedom in 3-D global coordinate system NDOF is set to 54. There are six dofs per node, i.e., three translations and rotations. Thus the total number of element dofs is 54 dofs for nine nodes. Default integration weight is set to 0.999 if optional integration weight is not provided through input. The shell element uses 8 point Gauss Quadrature. Integration constants are calculated based on optional or default integration weight and saved in WG(8) for numerical integration. AW is the primary integration point calculated and is to be used in the stress calculation. The array of integration point coordinates in the local system is built and saved in RS(2,8). Four point integration parameters are calculated in local coordinates and saved in RS4(2,4) with the corresponding weights WG(4) for four point integration on the plate element. Element property set is read and written. This includes material property number, Young's modulus, Poisson's ratio, shear modulus and self weight. If shear modulus is not given, assume isotropic material, so that G = (1/2)*( E / (1 + 2*P0I)) The self weight was assumed to be given in lb/cf and this is changed to kips/cubic inch because the shell element is used basically to model the concrete box girder itself. The element number will be added up at the end of each element data line including the one to be generated. Thus, let it be one less than the element number of the first shell element. The element number of the last shell element is calculated. This will be compared with the element number of the last element data to check if all the element data lines have been read. Element information is then read line by line. Each element input line has an element number, material property number, nine node numbers, thickness, number of elements to be generated. The element thickness is defaulted to one if not specified. Temperature information is read using the temperature input control variable ITMPLD. Temperature can be given in two ways. If temperatures at top and bottom faces are constant, then ITMPLD is one and temperature at top face and temperature at bottom face is given in one line. If temperatures at top and bottom faces are varying, ITMPLD is two and the temperatures at each node for top and bottom faces are given in two lines. The order of nodal temperatures is then re-organized to match the one used for formulation. If element generation is specified, new elements are generated as necessary. IJG(l) elements are generated in i- direction, ijg(2) elements are generated in j-direction including the element specified. Thus (IJG(1 or 2) - 1) elements will actually be generated. The order of element nodes for data input and formulation is as follows: 7 8 9 4 7 3 4 5 6 -> 8 9 6 12 3 15 2 Node Numbering for Data Input Node Numbering for Formulation Once element data are read then they are processed for each element by calling the EVEL13 subroutine. If there remain more element data lines after the processing of the previous element data, the next input line is read. Otherwise, the last element number is checked for termination of data input. In case of any error in element data input, an error message will be written and then the program will be stopped. B.2 Subroutine EVEL13(NJT.NDKOD.X.Y.Z.NODGEN.TNODE. ICNTR,INERR) Newly introduced variables are: AA(4) = Direction cosines of r-axis( local x). 179 BB(4) = Direction cosines of s-axis( local y). CC(4) = Direction cosines of t-axis( local z). This subroutine processes element data element by element. Element number is updated and the number of elements processed is counted for input control. Dispt(54), fkp(1485), selfld(54), lm(54) arrays are initialized. The coordinates of four corner nodes are recovered from the global coordinate arrays. Four corner nodes one, two, three and four used for formulation and local element coordinate system are shown in Fig. B-l. The node number three is the origin for the local coordinate system for the calculation of local i- and j- and k-direction vectors. Local x-axis goes from node three to node four. Local y-axis goes from node three to two. The element i-vector, a unit direction vector in local x- direction and the element j-vector, a unit vector in local y-direction are formed directly from the coordinates of the nodes by calling VECT13. The element k-vector, a unit vector in local z-direction can be found by a vector cross. k = i*j for right-handed coordinate system. Local j vector must then be modified using the second vector cross. j = k*i The transpose of transformation matrix can then be assembled from the unit local coordinate vectors as follows. 180 3 dy(l) Fig. B-l Local Coordinate System and Dimensions 181 i' = aa(l)i + j ' = bb(l) i + k' = cc(l) i + i' aa(l) j' bb(l) k' cc(1) aa(2)j + aa(3)k bb(2)j + bb(3)k cc(2)j + cc(3)k aa(2) aa(3) bb(2) bb (3) cc(2) cc(3) i j k [ local ] = [ T ] [ global ] Local dimensions dx, dy for the current element are calculated using the coordinates of start and ending points and local coordinate vectors through vector dot product to get the projection of the element dimension onto the element coordinate system. This is shown in Fig. B-l. The number of nodes in the current element is counted to collapse local dimensions. The array XY(2,9) has been dimensioned for a nine-node element. If any node is missing, the above process is skipped. Thus the values of the next node are shifted to the space for the current node, which is missing. These dimensions will be used in the calculation of the elements of Jacobian matrix. Refer to the subroutine FORMH. Element location matrix is set up and the number of integration points set. One point integration is used for 4 or less nodes and four point integration is used if the number of nodes is greater than four. The stress recovery array ST is formed for linear stress recovery. [Stresses] equals [ST][q], where [q] is 182 element nodal displacements in local coordinate system. The ST array relates the local nodal displacements to the element stresses at the integration points. After initialization of the ST array, it is formed by calling the subroutines FRMST1 and FRMST2 for membrane and plate portion, respectively. Equivalent nodal loads for temperature loads and initial stresses due to the temperature loads are calculated. These initial stresses will be subtracted from the stresses calculated using the total element displacements because the temperature strain does not introduce stresses. Sometimes the total strain is divided into two components, i.e., mechanical strain and thermal strain. Only the former produces stresses. The equivalent loads must be subtracted from the internal resisting forces formed by the total displacements or total stresses for equilibrium check in RINT13 subroutine. These calculations are done in the subroutines TLIST1 and TLIST2 for membrane and plate elements, respectively. This calculation is skipped if there is no temperature loading using the temperature load index ITMPLD. Equivalent nodal loads for self weight are calculated by calling the subroutine SELFWT. These equivalent nodal loadings are not temperature-type loadings even though these are treated similarly in this subroutine. Thus the equivalent nodal loads due to self weight will not be 183 subtracted from the internal resisting forces recovered from the total displacements. Element stiffness matrix profile is computed by calling the subroutine BAND. The element data processing is over after the data are transferred to tape through the subroutine COMPAC. B.3 Subroutine VECT13fV.XI.YI.ZI,XJ.YJ.ZJ) This subroutine finds the unit vector of a given vector specified by the coordinates of starting point and ending point. V(4) is dimensioned and Xi, Yi, Zi are the coordinates of the starting point of a given vector and Xj, Yj, Zj are the coordinates of the ending point of a given vector. The magnitude of the vector can be found by the square root of the sum of the squares of the coordinate differences in three global coordinate directions. The magnitude is stored in V(4). The three components of the unit vector can be found simply by dividing the coordinate difference in corresponding direction by the magnitude and are stored in V(l), V(2) and V(3). B.4 Subroutine TLIST1(TNODE) This subroutine forms equivalent element nodal loading due to temperature differential in local coordinate system for a membrane element and then rotates them into global coordinate system. COMMON /WORK / H(3,9),EB(3),B(3,18),PHI01(3),tem(54) + PHIOl(3) = Initial strain due to temperature loading for membrane portion of shell element. TEM(54) = Temporary array for the rotation of temperature load to global coordinate system. DIMENSION ini(18), in2(12), in3(18),tnode(2,9),cm(3,3) DATA in2 71,2,4,7,8,10,13,14,16,19,20,22/ in3 /l,2,7,8,13,14,19,20,25,26,31,32,37,38,43,44,49,50/ CM(3,3) = Constitutive matrix for membrane element. The data in2 contain the numbers corresponding to inÂ¬ plane stresses out of six stresses for four integration points. This will be used in locating the stresses from membrane portion in the 24 stresses possible for shell element in global coordinate system. The data in3 contain the numbers of in-plane membrane element dofs out of 54 global dofs of shell element in global coordinate system. This will be used the transformation of membrane portion of equivalent nodal loads into global loading which has a size of 54. The number of columns of local membrane element strain- nodal displacement matrix is set to the number of nodes times two. For nine node element, this will be 18. The arrays b(3,18), templd(54), tem(54) are initialized. The element properties needed in this subroutine are recovered. These are Young's modulus and 185 Poisson's ratio. The isotropic stress-strain law for membrane elements is evaluate by a call to ELAW1. The equivalent nodal loadings due to temperature effects are calculated through the loop over all integration points using numerical integration. The following procedures are performed for each integration point and the results are summed up. The shape functions,their derivatives and Jacobian matrix at current integration point are formed through FORMH. The weighting factor for the current integration point including the determinant of Jacobian matrix and thickness is calculated. Then the strain-displacement matrix, b(3,18) is calculated. Refer to frmstl for details. The initial strain phiol(3) for the membrane portion of the shell is computed using given temperatures. The membrane has only the three in-plane strains out of the five strain components of a shell element. If uniform temperature differential is given, the temperature differential to calculate initial strains for the membrane element is the difference between average temperature at neutral surface and the given reference temperature. The temperature difference for membrane strains at the current integration point, DELTI, is sum of the difference in each contribution of the temperature difference at each node. This contribution can be found by multiplying the numerical value of the shape function at the current integration point and the corresponding temperature difference at the node considered. Each difference is the average temperature minus reference temperature for membrane strains. Otherwise, the difference is calculated and multiplied by the corresponding shape function for the contribution of the temperature differential for the current node. These contributions are then summed up for all the nodes. The in-plane initial strain array due to temperature change PHI01(3) is initialized and calculated. The initial strain can be calculated as the temperature difference times the thermal expansion coefficient for the material specified. The thermal strain component for in-plane shear, Ã“xy, is zero because the temperature differential is the same in local x- and y- directions. This means that no inÂ¬ plane shear stresses will be introduced by the uniform temperature differential. Once the initial in-plane strains are calculated, equivalent nodal loading and initial stresses can be found through numerical integration. Temperature loads are the negative values of those recovered from the initial strain due to temperature loads because equivalent load(54) is set to the negative values of templd(54) in the subroutines ansr/static/templd.f and ansr/load/elfrc.f. Equivalent nodal loading is calculated through numerical integration. The thickness term is included in the integration weight. The numerical calculations for each integration point are as follows. The strain displacement matrix [B] is formed by choosing proper terms from H(3,9) evaluated by a call to FORMH subroutine. The equivalent nodal loads due to temperature change are then obtained through the numerical integration of [B]transpose*[E][phiol] over the volume of the element. Here this integration is performed over the area as the thickness is constant and has been included in the integration weight. Local temperature loads are then rotated to global coordinates for later assembly into global loads. The initial stresses due to thermal loadings can be evaluated through the loop over the integration points. One point or four point integration scheme is used depending on the number of element nodes. Once the formation of membrane element initial stresses in the local coordinate system has been done, these will be subtracted from the element stresses recovered from the total displacements in subroutine STAT13. As the element stresses are calculated in the local coordinate system, these initial stresses of membrane element are placed at the corresponding locations 188 of the shell element initial stresses for possible combination with plate element initial stresses. The following processes are done for each integration point. The shape functions, their derivatives, Jacobian, and its inverse are numerically evaluated for the current integration point. The initial strain PHI01(3) for the membrane portion of the shell is formed. The initial stress matrix for current Gauss point is calculated through the numerical integration of [E][phiol] over the area. B.5 Subroutine TLIST2(TNODE) This subroutine forms the equivalent nodal loads and initial stresses of plate element due to temperature loads in local coordinate system and fills these into the shell element temperature loads and initial stresses for the combination with plane membrane temperature loads and initial stresses. COMMON /WORK / b(5,27),eb(5),h(3,9),phio2(5),tem(54) PHI02(5) = Initial strain of plate bending element due to temperature loading. DIMENSION in2(20),in3(27),tnode(2,9),cp(5,5),c2(5,5) CP(5,5) = Constitutive matrix of plate bending element. C2(5,5) = Constitutive matrix with thickness terms taken out for stress calculation. DATA in2 /l,2,4,5,6,7,8,10,11,12,13,14,16,17,18,19,20,22,23, 24/ 189 in3 73,4,5,9,10,11,15,16,17,21,22,23,27,28,29,33,34,35, 39,40,41,45,46,47,51,52,53/ The data in2 contain the numbers corresponding to inÂ¬ plane stresses and shear stresses out of six stresses for four integration points. This will be used in locating the stresses from plate portion in the 24 stresses possible for shell element in global coordinate system. The data in3 contain the numbers of plate element dofs out of 54 global dofs of shell element in global coordinate system. This will be used the transformation of plate portion of equivalent nodal loads into global loading which has a size of 54. The number of columns of strain-nodal displacement matrix for plate is set to the number of nodes times three. In case of nine node element, this will be 27. The isotropic constitutive matrix for plate element CP(5,5) is formed by a call to subroutine ELAW2. Element properties needed in this subroutine are recovered. These are Young's modulus, Poisson's ratio and shear modulus. The equivalent nodal loads are formed through the loop over integration points. The calculations for each integration point are as follows. The shape functions, their derivatives, Jacobian, and its inverse are evaluated for the current integration point. The strain-displacement matrix B(5,27) is then formed through the choice of proper terms from H(3,9) matrix. The initial thermal strain phio2(5) due to the temperature loads is calculated. Temperature loads are the negative values of those recovered from the initial strain due to temperature loads because equivalent load(54) is set to the negative values of templd(54) in the subroutines of base program ansr/static/templd.f and ansr/load/elfrc.f. Temperature difference at current integration point, DELTI, is calculated in a similar manner with the membrane portion of the shell element. But the difference between top and bottom temperatures is used for the initial strains of plate element. The initial strains for plate bending element due to temperature loading can be found by temperature difference times thermal expansion coefficient divided by the element thickness. There are only two non-zero terms out of five terms. If temperature loading in x- or y- direction needs to be ignored to model beam type structure, the corresponding thermal strain is set to zero. This is for the comparison of the results with those from beam theory for verification. Once initial strains are calculated, equivalent nodal loads due to thermal loading are evaluated through numerical integration of [B]transpose[E][phio2] over the area. The local temperature loads are then rotated to global coordinates for later assembly into global loads. The initial stresses due to thermal loadings can be found through the loop over the integration points by numerical integration as follows. The initial stresses of the plate element due to temperature loads in the local coordinate system are computed, which will be subtracted from the recovered total element stresses. These stresses are then placed at the proper locations of the shell element initial stresses for the combination with plane membrane stresses using data in2. The initial stresses are calculated through the loop over stress output points (integration points). The calculation procedures are as follows: General stresses are defined as moment resultant over the thickness per unit length and thus have the unit of moment per unit length while common stresses have the unit of forces per unit area. The conversion can be done by removing thickness terms from constitutive matrix. The resulting matrix has been named C2(5,5). The factor 1.5 is divided for maximum shear at center line because the formulation gives only average transverse shear stresses while the actual distribution is a parabola. The remaining processes are identical with those for membrane element. B.6 Subroutine SELFWT This subroutine forms equivalent nodal loads for the self weight applied in the direction of gravity in global coordinate system. WGHT is self weight in local thickness direction per unit thickness which was stored in prop(4,mat). Self weight is divided into local rst-components using the transformation matrix. The direction of gravity in global coordinates is defined as follows. application direction X Y Z -X -Y -Z IGRAVD 12 3-1 -2 -3 1 = T g Let 1 be local components of self weight (br,bs,bt) transpose, T be usual transformation matrix and g be global self weight which has only one component in the direction of gravity, for example, {0,by,0> transpose, where by is self weight(wght). Therefore the relationship can be shown as br = t(l,2)*wght (local x component) bs = t(2,2)*wght (local y component) bt = t(3,2)*wght (local z component) As the transpose of transformation matrix has been formed, the terms for transformation matrix are expressed by the gravity direction index. The negative signs of local gravity components are due to the assumption that the gravity in global negative direction is defined as being positive. The equivalent nodal loads for self weight are evaluated through the loop over integration points. The procedures are as follows: Determine the shape functions, their derivatives, Jacobian, and its inverse numerically for the current integration point and store them in h(3,9), the array of shape functions h(l,i) = fÂ¿ h(2,i) = fÂ¿,x h(3,i) = (f^,y). Integration weight is retrieved for the current integration point. The equivalent local nodal loads due to self weight are calculated by integrating local components of gravity using shape functions and corresponding integration weight. These are then transformed into a global coordinate system using the transformation matrix. B.7 Subroutine FRMST1 This subroutine forms the stress-nodal displacement array [ST] for membrane element. COMMON /WORK / H(3,9),B(3,18),CP(3,3) H(3,9) are shape functions and their derivatives with respect to global x and y for each node up to nine nodes. node # 123456789 row #1 f^ ... fÂ¿ ... fg row #2 f1#x â€” fi,x â€” fg,x row # 3 f1#y ... fify ... fg,y B(3,18) is strain-nodal displacement matrix for membrane element and cp(3,3) is constitutive law for membrane element. The data in2(12) give the locations of in-plane streses from the membrane portion of the shell element out of 24 stress terms at four integration points. Each integration point has six stress terms. There are three non-zero stress terms from membrane portion of shell element for each integration point. This shows the row number of [ST] array for membrane stresses. Stresses are calculated at each of the four integration point. There exist three non-zero stresses for membrane portion of shell element. For integration point number one, the numbers for these non-zero stresses are existing(membrane) 1 2 4 non-zero stresses XX yy xy possible stresses XX yy ZZ xy yz zx 1 2 3 4 5 6 The data in3(18) has the numbers corresponding to two in-plane translational degrees of freedom at nine nodes out of 54 degrees of freedom for nine node element. This shows the column number in [ST] array for membrane stresses. The numbers for node number one are existing(membrane) 1 2 dofs x y possible dofs x y z xx yy zz in global system 123456 195 The number of columns in stress-nodal displacement matrix is two times the number of element nodes for membrane element. There are two translational degrees of freedom per node. The element properties, Young's modulus and Poisson's ratio, are recovered to evaluate isotropic constitutive matrix for membrane. The stress-nodal displacement array [ST] can be calculated through the loop over stress output points (integration points). [stress] = [E][strain ] (6) = [E] [d] [u ] (6.6)(6,3)(3,1) = [E] [d] [f] [q] (6.6)(6,3)(3,54)(54,1) = [E][ B ][q] (6,6)(6,54)(54,1) = [st ][q] (6,54) (54,1) (numbers for one integration point) For four integration points, the size is [ST ][q ] [24,54 ][54] Six stress components at four integration points makes a total of 24 stresses. In general, the relationship between strain and generic displacements can be shown as follows [strain]=[differential operator][generic displacements] e(ij)= (1/2)(u(i,j)+u(j,i)+u(k,i)u(k,j)) 196 i.e., e(ll)= e(22)= e(33)= e(12)= e(23)= e(13)= /2) (u1#1+ulf /2)(U2 2+u2,2+u1,22+U2,22+U: ./2) (Uo o+Uo ,+Ui 3^+Uo o -1-" ;+U, 2 i â€¢ ^2 2'U3'2; ) ) ) +n3,lU3,2\ +U3,2U3,3Â¡ +U3,lu3,3) This can be re-arranged as follows: 12 3 2e(ll) 2dll 0 0 2e(22) 0 2d22 0 2e(33) 0 0 2d33 2e(12) = d22 dll 0 2e(23) 0 d33 d22 2e(13) d33 0 dll ul U2 U3 where d** is a differential operator. Thus, the size of [d] is six by three in general. The general relationship between generic displacements and nodal displacements can be shown as follows: 12 3 ul fll u2 = fll u3 fll qll q22 q33 q44 q55 q66 (3,1) (3,54) (54,1) where, fll = {fl f2 f3 f4 f5 f6 f7 f8 f9} 197 and, qll = {ql q7 ql3 ql9 q25 q31 q37 q43 q49} transpose q22 = (q2 q8 ql4 q20 q26 q32 q38 q44 q50) transpose q33 = {q3 q9 ql5 q21 q27 q33 q39 q45 q51) transpose q44 = {q4 qlO ql6 q22 q28 q34 q40 q46 q52} transpose q55 = {q5 qll ql7 q23 q29 q35 q41 q47 q53) transpose q66 = {q6 ql2 ql8 q24 q30 q36 q42 q48 q54) transpose This shows the size of [f] which is six by fifty-four. The shape functions and their derivatives with respect to to global x and y at the current integration point are calculated by calling FORMH subroutine. For details, refer to the subroutine FORMH. The strain-nodal displacement matrix [B] is then formed by selecting proper terms from H(3,9). In subroutine FORMH, all the components of [B] matrix have been calculated. Here the correct components from H(3,9) are simply placed at the proper locations in [B]. b(3,18) = ^(3,2) , i=l,9 node # 1 ... i row#l | fx,x 0 | row#2 I 0 f1#y I row#3 I flry flfx I I 0 I I o fify I I fiÂ»Y ffrX | 9 The stress-displacement matrix [ST] is equal to [E][B] for membrane element and is placed at the correct places in the shell [ST] using data in2 and in3. The base index ii shows the location of row number for current stress in data in2. B.8 Subroutine FRMST2 This subroutine forms the stress-nodal displacement matrix [ST] for the plate portion of the shell element. COMMON /WORK / CP(5,5),C2(5,5),H(3,9),B(5,27),EB(5) CP(5,5) is constitutive law for the plate bending element for stiffness formulation and c2(5,5) is constitutive law for plate bending element for stress recovery. Thickness terms have been stripped. B(5,27) is strain-nodal displacement matrix for the plate bending element and eb(5) is an temporary array for [E][B] calculation. The data in2(12) give the locations of stresses from the plate portion of shell element out of 24 stress terms at four integration points. Each integration point has six stress terms. There are five non-zero stress terms from plate portion of shell element for each integration point. This shows the row number of [ST] array for plate stresses. Stresses are calculated at each of the four integration points. There exist five non-zero stresses for the plate portion of the shell element. For the integration point number one, the numbers for these non-zero stresses are existing(plate) 1 2 4 5 6 non-zero stresses XX yy xy yz zx possible stresses XX yy zz xy yz zx 1 2 3 4 5 6 199 The data in3(18) has the numbers corresponding to one out-of-plane translational degree of freedom and two out-ofÂ¬ plane rotational degrees of freedom of plate element at nine nodes out of 54 degrees of freedom available for nine node element. This shows the column number in [ST] array for plate stresses. The numbers for node number one is existing(plate) 345 element dofs x y z xx yy zz in global system 123456 The number of columns in stress-nodal displacement matrix is three times number of element nodes for plate element. There are three degrees of freedom per node. After the arrays b(5,27), h(3,9),c2(5,5) have been initialized, the element properties needed for constitutive law are recovered. These are Young's modulus em, Poisson's ratio poi and shear modulus G. The isotropic stress-strain law for plate elements is evaluated by a call to subroutine ELAW2. The thickness term is removed from the constitutive matrix for stress calculation of plate element. The stress-nodal displacement array [ST] can be found through the loop over stress output points in a similar way with the procedures used for membrane portion of the shell element by adding up of stress contribution for all the integration points. 200 The procedures are as follows: the shape functions, their derivatives, Jacobian, and its inverse are formed first. The strain-displacement matrix b(5,27) is formed using the proper terms from h(3,9). b(5,27) = bi(5,3), i= =1,9 node # 1 â€¢ â€¢ â€¢ i * â€¢ â€¢ row#l 1 0 0 fl,x 1 â€¢ â€¢ â€¢ 1 0 0 fi,x row# 2 1 o -Â£lfy 0 1 â€¢ â€¢ â€¢ 1 o -fj_,y 0 row# 3 1 o -f1#x fi,y 1 * * * 1 o -fi,x fi'Y row# 4 1 f,,X 0 fi 1 * * * 1 fifx 0 fi row# 5 1 fl'Y "fl 0 1 â€¢ t 1 1 fi'Y -fi 0 The stress-displacement matrix [ST] for current Gauss point is calculated, added up and then placed into proper places of stress recovery array [ST] of shell element using data in2(row) and in3(column). B.9 Subroutine STIF13(ISTEP.NDF,CDKO,CDKT.FK.INDFK.ISTFC^ This subroutine calculates the shell element stiffness by combining the stiffnesses of both the membrane element and plate bending element. The shell element stiffness formed in the local coordinate system is then rotated to the global coordinate system. It is noted that stiffness is a second order tensor and therefore it follows a different transformation law from the one used for forces or displacements which are a first order tensor (a vector). [global stiffness] = [T transpose][local stiffness][T] COMMON /WORK / S(45,45) The dimension statement defines global shell element stiffness FK, constitutive matrices of membrane and plate elements, CM and CP. DIMENSION FK(54,54),cm(3,3),cp(5,5) FK = current global element stiffness matrix FKL = S = current local element stiffness matrix FKP = previous global element stiffness FKC = current global element stiffness ISTFC = stiffness matrix content index (supplied by the base program) 1 = total element stiffness matrix 0 = change in element stiffness matrix If the change in element stiffness is local, the unchanged element stiffness can be stored and only the changed portion may be calculated and updated. INDFK = FK storage index 1 = lower half, compacted column-wise 0 = square (all) ISTEP = current step no. in step-by-step integration >1 = form dynamic stiffness 1 = form dynamic stiffness at start of new phase < 0 = form static stiffness 0 = form static (and geometric) stfns at start NDOF = number of element degrees of freedom The stiffness matrix storage scheme, INDFK is set to zero. If the lower half compacted column-wise can be used, it is better for the symmetric element stiffness matrix but this causes some problem. Therefore INDFK is set to 0 for square matrix storage scheme. All the element properties are recovered for the calculation of constitutive matrices. These are Young's modulus, Poisson's ratio and shear modulus. The calculation of MEMBRANE stiffness has been done as follows. The isotropic stress-strain law for the membrane element is evaluated by calling the subroutine ELAW1. Then the lower triangular portion of membrane element stiffness is formed in the local coordinate system and placed at the proper places of local shell element stiffness S(45,45). This is done in the subroutine SHSTF1. The calculation of plate contribution has been done in a similar manner. The isotropic stress-strain law for plate element is calculated in the subroutine ELAW2. Then the lower triangular portion of plate element stiffness in the local coordinate system is formed and placed at the proper places of local shell element stiffness S(45,45) in the subroutine SHSTF2. The upper part of local shell element stiffness matrix is filled. The local shell element stiffness is transformed from local to global coordinate system by calling the subroutine TRANS. Transfer element stiffness information to base program in accordance with the control variable ISTFC. This variable is set to one for the first iteration for full element stiffness. From the subsequent iteration, it is set to zero only for the change in element stiffness by base program. B.10 Subroutine SHSTFlfcn This subroutine forms lower triangular portion of element stiffness in local coordinate system for membrane element and puts them into full local shell element stiffness S(45,45). Five degrees of freedom per node times nine nodes gives a total degrees of freedom of 45. And hence the size of element stiffness matrix is 45 by 45. COMMON /WORK / S(45,45) C(3,3) = stress-strain law for membrane element S(45,45) = full element stiffness matrix for shell element. Element stiffness is the integration of ([B]transpose[E][B]) over element volume. Therefore the procedure will be as follows. The shape functions, their derivatives and Jacobian matrix are formed to evaluate strain-nodal displacement matrix [B]. The element stiffness is calculated by numerical integration of [B]transpose[E][B] over element volume using the constitutive matrix brought in through C. The data ini contain the numbers of degrees of freedom of shell element which correspond to the degrees of freedom of membrane element. These are two in-plane translational degrees of freedom. The loop over integration points for stiffness calculation is as follows: 1) Form shape functions. 2) Calculate current integration weight factor. 3) Calculate strain-displacement matrix [B]. 4) Perform Gauss quadrature on point r,s to form stiffness matrix [B]transpose[E][B], where [E]=[cm(3,3)] 5) Put them into the corresponding locations of full shell element stiffness matrix using data /ini/. B.11 Subroutine SHSTF2fC^ This forms the lower triangular portion of the plate element stiffness in the local coordinate system and fills these terms into the shell element stiffness. The data ini is the numbers of degrees of freedom of plate bending element in the shell element. These are one out-of-plane translational degree of freedom and two out-of- plane rotational degrees of freedom. The procedures are the same as those of the membrane element except for the size of the [B] and [E] matrices. 205 B.12 Subroutine ELAWlfEM.POI.CM This subroutine evaluates isotropic stress-strain law for membrane elements. cm(3,3) = constitutive matrix em = Young's modulus poi = Poisson's ratio Temperature effects on em and/or poi ignored. Initial em, poi values used. Em,poi may vary with temperature. The constants are set to the values required. cm(3,3) cl c2 0 c2 cl 0 0 0 c3 where, cl = em / (1.0 - poi * poi) c2 = cl * poi c3 = cl * (1.0 - poi) * 0.5 B.13 Subroutine ELAW2 fEM.POI,G.THK.CP^ This subroutine forms Isotropic stress-strain law ignoring temperature dependence of elastic constants, cp(5,5) = constitutive matrix for plate element em = Young's modulus poi = Poisson's ratio g shear modulus 206 thk = plate thickness The plate thickness is assumed to be constant and is taken out of the integration and entered here. The constants for plate stress-strain matrix are. cp(5,5) cp2 0 0 0 cpl 000 0 cp3 0 0 0 0 cp4 0 0 00 cp4 where, ea = thk * thk * thk / 12.0 dem = 1.0 - (poi * poi) eed=em*ea/dem cpl = eed cp2 = poi * eed cp3 = g * ea cp4 = g * thk / 1.2 B.14 Subroutine FORMH(R.S.NODE.XY.H.DJ.NN) This subroutine forms four to nine node shape functions and their derivatives in rs-space then transforms them into xy-space through the inverse of Jacobian matrix. DIMENSION NODE(9),XY(2,9),H(3,9),D(2,2), N ( 2,9) N(2,9) = Coordinates of nine nodes in local system. r,s = Natural coordinates in local system. NODE(9) = Node numbers defining element. XY(2,9) = local x,y coordinates of nodes H(3,9) = Array of shape functions. h(l,i)=f(i) h (2, i) =f (i),x h(3,i) =f (i) ,y dj = Determinant of Jacobian, nn = Number of nodes defining element. D(2,2)=Jacobian matrix for 2D, replaced by Inverse of J The initial functions are evaluated at the current integration point by calling the subroutine GD two times with the coordinate r and s of the current integration point, respectively. The subroutine GD provides the numerical initial functions and their derivatives corresponding to the r and s coordinates provided. The evaluation of shape functions starts with four bilinear shape functions for the four corner nodes. If node nine exists, the shape functions for the four corner nodes need to be modified because the values of these corner node shape functions at node nine is one fourth. This must be forced to zero and can be done using the shape function of the node nine because this shape function has the value of one at node nine and zero at all other nodes. Therefore the modification is simple subtraction of one fourth of the shape function at node nine from the each shape function for the corner node. If any of the center nodes on the edge of the element(nodes five to eight) exists, the shape functions of corner nodes need to be modified further because the values at center of the edge is one half. This can be done by subtracting one half of the shape functions for the adjacent center nodes on the edge of the element from the each of the shape function for the four corner nodes. The results are stored in the array H(3,9). H(l,i) are the numerical value of the shape function evaluated at the current integration point. H(2,i) and H(3,i) is the numerical values of the derivatives of current shape function with respect to to local variable r and s, respectively. Once shape functions are evaluated, Jacobian matrix can be calculated as follows. x = sum(fjL*xÂ¿) x,r = sum(f^,r*xÂ¿) y = sum(fi*yi) y,r = sumif^^yj.) x = sum(f^*xÂ¿) x,s = sum(f^,s*xÂ¿) y = sum(fi*yi) y,s = sum(fi,s*yi) where sum(qty(i)) = sum of qty(i) over i=l,nn fÂ¿ = the first row of h(3,9) matrix f^,r = the second row of h(3,9) matrix f^,s = the third row of h(3,9) matrix xÂ¿ = the first row of xy(2,9) matrix yÂ¿ = the second row of xy(2,9) matrix It is noted that xÂ¿, y^ are not the actual global coordinates of node i of the element but they are local element geometry coordinates. These values can be found by the dot product of local axis unit direction vector before being mapped into the natural coordinates and the vector from the local origin and the node under consideration. [J] x,r y,r x,s y,s The inversion of two by two Jacobian matrix can be done as follows. The determinant of Jacobian matrix is found. D(1,1) and D(2,2) are interchanged and the signs are changed to the opposite for the terms D(l,2) and D(2,l). All these values are then divided by the determinant of Jacobian matrix. The element connectivity can be checked using the determinant of Jacobian matrix. The determinant must be greater than zero for the properly connected elements. If the determinant is less than or equal to zero, an error message is written. The partial differential of shape functions with respect to global coordinates x and y can be calculated as follows. fi,x = (f-^rH^x) + (fi,s)(s,x) where, fÂ¿,r = h(2,i), r,x = invJ(l,l), fÂ¿,s = h(3,i), s,x = invJ(1,2) 210 fiÂ»Y = (f-^rH^y) + (fifs)(s,y) where f^,r = h(2,i), r,y = invJ(2,1), f^,s = h(3,i), s,y = invJ(2,2) where invJ is the inverse of the Jacobian matrix. B.15 Subroutine GD(B.IB.G.D) b = Coordinate of local r or s of the current integration point. ib = Coordinate of local r or s of the current node. g = Contribution to shape function in the current r or s direction. d = Derivative of g with respect to r or s direction. For bottom corner nodes, G=(1.0-B)*0.5 D=-. 5 For center nodes along the edge of element, G=1.0-B*B D=-2.0*B For top corner nodes, G=(1.0+B)*0.5 D=. 5 B.16 Subroutine TRANS (NN.TT.FKL.FKG} This subroutine performs local-global transformations of element stiffness. nn = Number of nodes for the current shell element. tt(3,3) = Transpose of transformation matrix. Sometimes called as [A] matrix for the second order tensor. fkl(45,45) = Local shell element stiffness. This has a size of 45 by 45. There are five dofs at each node, three translational dofs and two out-of-plane rotational dofs. The maximum number of element nodes is nine. And hence 45 local dofs. fkg(54,54) = Global shell element stiffness. This has a size of 54 by 54. In global coordinate system, even one local rotational dof may have three components in global coordinate system and thus there are six global dofs at each node. The maximum number of element nodes is nine. And hence 54 local dofs. Is(18) = Location matrix to put local 45 by 45 into global 54 by 54 stiffness matrix. 54 dofs has been divided into 18 groups of three orthogonal dofs and will be transformed in blocks of the three dofs. It(18) = Location matrix for transformation matrix for the stiffness transformation in blocks of three. For the three translational dofs we need all the three rows of transformation matrix. For the two out-of-plane rotational dofs we need two rows of three by three transformation matrix. Stiffness transformation in blocks of 3 is done through the subroutine TRIPL, expanding 45x45 local stiffness to 54x54 global stiffness. As the above calculation is done only for lower half of the matrix, the upper part of stiffness is filled up using symmetry. B.17 Subroutine TRIPLE(LI.L2.M.N.K1.K2.TT.A.AA.NR1.NCI. NR2,NC2^ This subroutine calculates the matrix triple product, aa = t(transpose) * a * t in blocked form (3x3). As the transformation matrix has been formed in its transpose, the product becomes aa=tt*a*tt(transpose). where, a(nrl,ncl) = Local element stiffness matrix. aa(nr2,nc2)= Global element stiffness matrix. tt(3,3) = Transpose of transformation matrix. tk(3,3) = Temporary array to carry the results of the first multiplication tt(3,3)*corresponding a(3,3) . 11, 12 = Indices of element stiffness matrix in local system. m,n = Indices of transpose of transformation matrix. kl, k2 = Indices of element stiffness matrix in global system. nrl = Number of rows in "aâ€, ncl = Number of columns in "a". nr2 = Number of rows in "aa". nc2 = Number of columns in "aa". The first multiplication, tt*a is performed followed by the second multiplication [tt*a]*[tt(transpose)]. 213 B.18 Subroutine INIT13 (NJTS.NDF.RF.FACILE This subroutine forms initial loadings for the shell element. Self weight is also included in this subroutine. The application factor of temperature loading FACT and/or the factor of self weight FACS are recovered from the array FACIL transferred from base program. The applied loads are then calculated using the equivalent nodal loads for temperature differential and self weight computed in subroutine EVEL13 and placed in the initial force array RF. These equivalent nodal loads will be added to the global load vector in the base program. B.19 Subroutine STAT13 (NDF.0.TIME.FACAL.FACIL.ALFA) This subroutine is for state determination calculations. COMMON /STLDPT/ This common block will be used to find load application factor in RINT13. Q(NDF) = INCREMENTAL global nodal displacements Therefore, if total stresses are needed, the incremental displacements are to be added up at the end of each iteration. QDL(54) = INCREMENTAL element nodal displacements FACAL(1) = total load application factor FACIL(l) = incremental load application factor 214 ALFA = factor used in FACAL tt = transpose of element transformation matrix nd = number of global displacements at nodes The Gauss point parameters for linear stress interpolation are set up. The global incremental displacements are then added up and transformed to form total local displacements using 3x3 submatrix to remove multiplications with zeros. Stresses at integration points are then calculated as follows. There are six stress components at each integration point. [stress(24)] = [ST(24,54)][g(54)] The initial stresses due to temperature loading are subtracted from the calculated total stresses. The stresses at four corner nodes are calculated linear extrapolation from the stresses evaluated at the integration points and put back into stress array sig(24). B.20 Subroutine RINT13 fNDF.O.VEL.FE.FD,TIME,FACAL. FACIL,ALFA) This subroutine calculates element forces in the global coordinate system. COMMON /STLDPT/NPNF,NPFF,NPTP,NPP4,NPDP,NETP,NESWNITSR, NSPATT npnf = Number of nodal force patterns. 215 npff = Number of follower force patterns. nptp = Number of nodal temperature patterns. npdp = Number of nodal displacement patterns. netp = Number of temperature loadings through element data. news = Number of self weight loadings through element data. COMMON /WORK / RFTEM(54),fktem(54,54) rftem(54) = Temporary array for element force rotation. fktem(54,54) = Temporary array for element stiffness transformation. Q(NDF) = NODAL DISPLACEMENTS. VEL(NDF)= NODAL VELOCITY FD(NDF) = DYNAMIC NODAL FORCE WHEN TIME > 0 FE(NDF) = ELASTO-PLASTIC NODAL FORCE WHEN TIME = 0 TT = transpose of transformation matrix. The element forces in the global coordinate system can be calculated as [FE global]=[Ke global][Q global] As the element stiffness is in global coordinate system and stored in the array FKP(1485). The element forces can then be recovered directly from the global nodal displacements. The load application factor for temperature and/or self weight loading is calculated. FACT is a factor for temperature loads. FACS is a factor for self weight. These 216 are calculated whichever loading is applied. The load application can be identified by the numbers of load patterns, i.e., if NETP or NESP is positive, then there is temperature loading or self weight, respectively. IF temperature load is applied, subtract initial temperature loadings from element forces. Templd(54) is formed in initl3. Negative value of actual templd(54) has been calculated because the base program uses -templd(54) as applied loads. The + sign for subtraction is due to this fact. The index for gravity can be used to identify if self weight is included. B.21 Subroutine OUTS13 (KPR.TIME^ This subroutine is to print the time history of the current state from /INFEL/ including stresses, strains, status information, etc. If the current element is the first element, write the heading for the element information data and element information. Otherwise print element information directly. If no element information is requested, skip this subroutine. From the second element, write the element information only. This includes element number, node number and stresses at the four corner nodes. These stresses have been linearly extrapolated from the integration points. B.22 Linear Shell Element Data Input Guide (1) Control information (1- 1) First control line COLUMNS NAME DATA 1 - 5(1) NGR Element group number = 13 6 - io(i) NELS Number of shell elements 11 ~ 15(1) MFST Element # of first shell element 16 - 25(F) DKO Initial stiffness damping factor 26 - 35(F) DKT Tangent stiffness damping factor 41 - 80(A) Optional heading for shell element (1. 2) Second control line COLUMNS NAME DATA 1 - 5(1) NMAT Number of shell material properties 6 - 10(1) IGRAVD Direction of gravity 11 - 15(1) ITMPLD Type of temperature load 16 - 30(E) ALPHA Thermal expansion coefficient 31 - 40(F) REFTEM Reference temperature 41 - 45(1) ITMOFF Temperature load turn-off index 46 - 55(F) WGT Integration weight (default = 0.999) Notes: IGRAVD : direction of gravity global X Y Z -X -Y -Z igravd 1 2 3 -1 -2 -3 ITMPLD : index for temperature load 1 = input top and bottom temperatures only 2 = input temperatures for all the nodes else = no temperature effects no input for the AHPHA,REFTEM,ITMOFF ITMOFF : index to turn off temperature effect 1 = in local x direction (^xx = 0.0) 2 = in local y direction (<6yy = 0.0) (2) Material property data NMAT sets of material properties COLUMNS NAME DATA 1 - 5(1) MAT 6 - 15(F) E 16 - 25(F) POI 26 - 35(F) G Shell material property number Young's modulus Poisson's ratio Shear modulus [default=E/(2.0*(1+POI))] 218 36 - 45(F) WGHT Self weight per cubic foot (3) Element data NELS sets of element data COLUMNS NAME DATA 1 - 5(1) NEL Element number 6 - 10(1) MAT Material property number 11 - 55(1) NODE(9) Node numbers (915, 0 for missing node) 56 - 65(F) THK Element thickness (default = 1.0) 66 - 75(1) IJG(2) Number of element to be generated If ITMPLD is 1, add a second line to element data. 1 - 10(F) TMPTOP Temperature at top surface 11 - 20(F) TMPBOT Temterature at bottom surface If ITMPLD is 2, add two lines to element data. 1 - 64(F) Temperature at top surface for nine nodes (9F8.2) 1 - 64(F) Temperature at bottom surface for nine nodes (9F8.2) Notes: NODE(9): element node numbers 7* 8* 9* 4* 5* 6* 1* 2* 3* IJG(2) : number of elements to be generated in i- and j- direction including the element specified, (works only for nine node elements) If no temperature effects are desired for specific element, input the same values for TMPTOP and TMPBOT, i.e., (REFTEM,REFTEM) APPENDIX C IMPLEMENTATION NONLINEAR SHELL ELEMENT This appendix describes the implementation of a incremental nonlinear finite shell element into ANSR-III program. Only those features that are different from those of a linear shell element in subroutines INEL13, STIF13, NONSTF, STAT13, STRESS, RINT13 are dealt with. For detailed equations and elements of matrices, refer to chapters four and five. C.l Subroutine INEL13fNJT.NDKOD.X.Y.Z.KEXEC) The local coordinates rs9(2,9) and the weights wg9(9) for the standard 3X3 Gauss integration were added to /INFGR/ for the calculation of stiffnesses, stresses, and internal resisting forces. For the solution of the incremental equilibrium equation in total Lagrangian description, the total displacements and the total displacement gradients of previous iteration are required and thus stored in /INFEL/. These are ut9(9) through ayyt9(9). Some arrays for the explicit integration through the thickness are also added. These are thklyr(lO) through dhc(10). Accordingly, the 219 length of element information block LSTAT was changed. The control variables for nonlinear analysis and layered element are IL = Number of layers. KLD = Large displacement nonlinearity index. 1 = on, 0 = off KGM = Geometric nonlinearity index. 1 = on, 0 = off KMAT = Material nonlinearity index (not used). 1 = on, 0 = off KUL = Motion description index. 1 = Total Lagrangian formulation. 2 = Updated Lagrangian formulation(not used). The coordinate for 3x3 integration is aw = 0.774596669241483 bw = aw The layer information that can be handled currently is 10 layers with different thicknesses and materials, nlyr = Number of layers. thklyr(lO) = Layer thickness. matlyr(lO) = Layer material. C.2 Subroutine EVEL13(NJT.NDKOD. X. Y, Z .NODGEN. TNODE.ICNTR.INERR} The element is divided into layers for explicit integration over element thickness (-t/2 to t/2). Integrations needed are htop htop dz , z dz , and hbot hbot w ht z hbot Â§P z^ dz, where, htop = The coordinate of top of the layer. hbot = The coordinate of bottom of the layer. Thus the coordinates of top and bottom of each layer are required and these are calculated and stored in hh(ll). Currently the coordinate of the center line is set to zero. But this can be changed for arbitrary location along the thickness direction if an input parameter is given. The explicit integration over the thickness becomes [htop - hbot], [0.5(htop2 - hbot2)], and [(1/3)(htop3 - hbotJ)]. These are calculated and stored in dhs(10), dhc(10). It is noted that [htop - hbot] is equal to the layer thickness itself. The stress-nodal displacement matrix [ST] cannot be used as it changes continuously in incremental nonlinear analysis. [ST] is defined as {stress} = [ST]{nodal displacement} 222 C.3 SUBROUTINE STIF13(ISTEP.NDF.CDKO.CDKT. FK. INDFK. ISTFC) The plate stiffness matrix [D] is calculated by performing integration of constitutive matrix [E] through thickness layer by layer using the subroutines ELAW1 and ELAW2. The incremental nonliner element stiffness is calculated by calling the subroutine NONSTF. The rotation of the element stiffness and the storage scheme are identical to those for linear shell element. C.4 Subroutine NONSTF (dm.dp.ds.lvr) The standard 3x3 Gauss quadrature is used for the numerical integration of nonlinear element stiffness. The incremental nonlinear element stiffness requires the total displacement gradient (w,x) and (w,y) from the previous iteration. These were saved in wxt9(9) and wyt9(9) for each integration point in subroutine STRESS and are recovered for use. Only the lower triangular portion of element stiffness is calculated in this subroutine. The linear element stiffness is formed using subroutines SHSTF1 and SHSTF2. The nonlinear element stiffness is calculated in two groups; the large displacement stiffness [KLD] and the geometric stiffness [KGM]. The large displacement matrix is evaluated using the three component matrices [KLD1], [KLD2], and [KLD3]. These are [KLD1] = [Bli^tDHBn-j] [KLD2] = [Bni]T[D][Blj] [KLD3] = [Bni]T[D][Bnj] The geometric stiffness [KGM] is calculated using the element stresses at current integration points as [G]T[STRM][G]. The inplane element stresses Nx, Ny, and NXy for the current integration point are recovered from STRSL array to form [STRM]. [G] consists of only (fi,x) and (fi,y)â€¢ C.5 Subroutine STAT13 (NDF.O.TIME.FACAL.FACIL.ALFA) The element stresses are recovered through subroutine STRESS at integration points using the incremental displacements and then extrapolated to element nodes for analysis and design purposes. The locations of integration points for stress evaluation are 4 X I I * 4 8 * 8 X I I * 1 I 1 X + (y) -7 X A * 7 i (0,0) * 9 X * 5 3 X I * 3 | I > *66 X I * 2 | 5 x 2 x + (x) where, * = Integration points, x = Node number for formulation. This can be done using the shape functions [82] or bilinear extrapolation for the inplane and shear stresses and biquadratic extrapolation for the bending stresses. Another approach is the least squares fit. C. 6 Subroutine STRESS (ndf. cr) Element stress evaluation is done at integration points but is not a numerical integration. Thus no weighting factor is used. The arrays used are strsl(9,8,10) = Layer stresses at 9 integration points, for 8 stress components, Nx, Ny, Nxy, Mx, My,Mxy, Qx, Qy, and for 10 layers. 225 elstri(8,9) = Incremental element stresses of 8 components at 9 integration points. strnip(8;9) = Total element strains of 8 components at 9 integ points. The global displacements are transformed to local coordinates for stress recovery. The material properties are recovered to form the constitutive law or the stress-strain relationship through subroutines ELAW11 and ELAW22. For each layer the constants for explicit integration are recovered and stresses are calculated at integration points. Stresses are usually discontinuous and less accurate if they are recovered directly at nodes. 3x3 integrations are used for inplane and bending stresses and 2x2 integration points are used for shear stress evaluation. Incremental strains are calculated as {strain} = [Linearized B matrix](nodal displacements} = [BLZ]{q} As [Blz] is the function of (w,x) and (w,y), these incremental quantities, wx9(9) and wy9(9), must be calculated first for the current integration point. (w,x)= [sum of (fi)(wi)],x = [sum of (fi,x)(wi)] (w,y)= [sum of (fi)(wi)],y = [sum of (fi,y)(wi)] 226 (fi,x) and (fi,y) are element of h(3,9) and wi is the local nodal displacements transformed from the global nodal displacements. And these are added up to yield total displacement gradients, wxt9(9) and wyt9(9). Total strains, strnip(8.9), are calculated because some failure criterion requires principal strains. These are not needed currently but will be used for material nonlinearity. Once incremental strain is evaluated, then incremental stresses can be calculated through explicit integration across thickness and then added up for the total stresses. C.7 Subroutine RINT13 fNDF.0.VEL.FE.FD.TIME.FACAL. FACIL,ALFA) This subroutine calculates element forces in global coordinates, FE, the numerical integration of [BLZ]T{stress} over the area. The {stress} are generalized, i.e., the integration along thickness has been performed. [BLZ] is evaluated in the same procedures used for stress recovery. The integration scheme is compatible with that of stress recovery, too. A 3x3 integration for the internal resisting forces from inplane and bending stresses and a 2x2 for those from shear stresses were used. 227 C. 8 Nonlinear Shell Element Data Input Guide (1) Control information (1. 1) First control line COLUMNS NAME DATA 1 â€” 5(1) NGR Element group number = 13 6 - 10(1) NELS Number of shell elements 11 - 15(1) MFST Element # of first shell element 16 - 25(F) DKO Initial stiffness damping factor 26 41 - 35(F) 80(A) DKT Tangent stiffness damping factor Optional heading for shell element (1. 2) Second control line COLUMNS NAME DATA 1 â€” 5(1) NMAT Number of shell material properties 6 - 10(1) IGRAVD Direction of gravity 11 - 15(1) ITMPLD Type of temperature load 16 â€” 30(E) ALPHA Thermal expansion coefficient 31 - 40(F) REFTEM Reference temperature 41 - 45(1) ITMOFF Temperature load turn-off index 46 - 55(F) WGT Integration weight (default = 0.999999) 56 60(1) ILYR Index for layer analysis 1 = layers of same thickness and material. Use for no layer analysis. 2 = 10 layers of different thicknesses and materials. 61 mm 65(1) KLD Index for large displacement analysis. 0 = no 1 = yes 66 70(1) KGM Index for geometric nonlinear analysis. 0 = no 1 = yes Note: KLD and KGM must be unity(1) for 'geometric' nonlinear analysis if it includes the effects of large displacements and initial stresses as used by some authors. 71 - 75(1) KMAT Index for material nonlinear analysis. (Currently not used.) Notes: IGRAVD : direction of gravity global X Y Z -X -Y -Z igravd 1 2 3 -1 -2 -3 228 ITMPLD : index for temperature load 1 = input top and bottom temperatures only 2 = input temperatures for all the nodes else = no temperature effects no input for the ALPHA,REFTEM,ITMOFF ITMOFF : index to turn off temperature effect 1 = in local x direction (Â¿xx = 0.0) 2 = in local y direction ( (2) Material property data NMAT sets of material properties COLUMNS NAME DATA 1 - 5(1) MAT Shell material property number 6 - 15(F) E Young's modulus 16 - 25(F) POI Poisson's ratio 26 - 35(F) G Shear modulus [default=E/(2.0*(1+POI))] 36 - 45(F) WGHT Self weight per cubic foot (3) Layer data if ILYR = 1 1 - 5(1) NLYR Number of layers(Current max. = 10) 6 - 15(F) THKLAY Thickness of layer 16 - 20(1) MATLAY Material Index of layer (4) Layer data if ILYR = 2 1 " 5(1) NLYR Number of layers(Current max. = 10) 1 - 80(F) THKLYR Thicknesses of layers( one line) 1 - 80(F) MATLYR Material Properties of layers(one line) (5) Element data NELS sets of element data COLUMNS NAME DATA 1 - 5(1) NEL Element number 6 - 10(1) MAT Material property number 11 - 55(1) NODE(9) Node numbers (915, 0 for missing node) 56 - 65(F) THK Element thickness (default = 1.0) 66 - 75(1) IJG(2) Number of element to be generated If ITMPLD is 1, add a second line to element data. 1 - 10(F) TMPTOP Temperature at top surface 11 - 20(F) TMPBOT Temperature at bottom surface 229 If ITMPLD is 2, 1 - 64(F) 1 - 64(F) add two lines to element data. Temperature at top surface for nine nodes (9F8.2) Temperature at bottom surface for nine nodes (9F8.2) Notes: NODE(9): element node numbers 7* 8* 9* 4* 5* 6* 1* 2* 3* IJG(2) : number of elements to be generated in i- and j- direction including the element specified, (works only for nine node elements) If no temperature effects are desired for specific element, input the same values for TMPTOP and TMPBOT, i.e., (REFTEM,REFTEM) REFERENCES 1. Hoit, M. I., "Development of Analysis Capabilities for Nonlinear Post-Tensioned Concrete Box Sections," Project Proposal submitted to Florida Department of Transportation. 1987. 2. Beer, G., "An Isoparametric Joint/Interface Element for Finite Element Analysis," I.J. for Numerical Methods in Engineering, Vol. 21. 1985, pp. 585-600. 3. Goodman, R. E., Taylor, R. L., and Brekke, T. L., "A Model for the Mechanics of Jointed Rock," Journal of the Soil Mechanics and Foundations Division. Proceedings of the ASCE, Vol. 94, No. SM 3. May, 1968, p. 637. 4. Zienkiewicz, O. C., "Analysis of Nonlinear Problems in Rock Mechanics with Particular Reference to Jointed Rock Systems," Proceedings of the 2nd Congress of the International Society for Rock Mechanics. Belgrade, Yugoslavia, 1970. 5. Isenberg, J., "Analytic Modeling of Rock-Structure Interaction," Agbabian Associates Final Technical Report to U. S. Bureau of Mines. Contract No. H0220035. April, 1973. 6. G. Gudehus, Finite Elements in Geomechanics. John Wiley & Sons, New York, N.Y., 1977. 7. Soongswang, K., "Ultimate Strength of Segmental Box Girder Bridges with Unbonded Tendons," Doctoral Dissertation, University of Florida, 1987. 8. Brebbia, C., Connor, J., "Geometrically Nonlinear Finite Element Analysis," Journal of the Engineering Mechanics Division. ASCE. EM2. April, 1969, pp. 463- 483. 9. Hibbit, H. D., Marcal, P. V. and Rice, J. R., "A Finite Element Formulation for Problems of Large Strain and Large Displacement," I.J. of Solids and Structures. 1969, pp. 1069-1096. 230 10. Frey, F., Cescotto, S., "Some New Aspects of Incremental Total Lagrangian Description in Nonlinear Analysis," Finite Elements in Nonlinear Mechanics. Vol. I, Tapir, Norweign Institute of Technology, Trondheim, 1978, pp. 323-343. 11. Bathe, K. J., Ramm, E., Wilson, E. L.,"Finite Element Formulation for Large Deformation Dynamic Analysis," I.J. for Numerical Methods in Engineering, Vol. 9. 1975, pp. 353-386. 12. Yaghmai, S., Popov, E. P., "Incremental Analysis of Large Deflections of Shells of Revolution," I.J. of Solids and Structures. Vol. 7. 1971, pp. 1375-1393. 13. Wood, R. D., Schrefler, B., "Geometrically Nonlinear Analysis-A Correlation of Finite Element Notations," I.J. for Numerical Methods in Engineering, Vol. 12. 1978, pp. 635-642. 14. Powell, G. H., "Theory of Nonlinear Elastic Structures," Journal of the Structural Division. Proceedings of the ASCE, ST 12.. December 1969, pp. 2687-2701 15. Belytschko, T., Glaum, L. W., "Applications of Higher Order Corotational Stretch Theories to Nonlinear Finite Element Analysis," Computers & Structures. Vol. 10. 1978, pp. 175-182. 16. Kanok-Nukulchgai, W., Taylor, R. L., Hughes, T. J. R., "A Large Deformation Formulation for Shell Analysis by the Finite Element Method," Computers & Structures. Vol. 13. 1981, pp. 19-27. 17. Ramm, E., "A Plate/Shell Element for Large Deflection and Rotations," Formulations and Computational Algorithms in Finite Element Analysis: US-Germanv Symposium. 1977, pp. 264-293. 18. Chang, T. Y., Sawamiphakdi, K., "Large Deformation Analysis of Laminated Shells by Finite Element Method," Computers & Structures, Vol. 13. 1981, pp. 331-340. 19. Parish, H., "Large Displacements of Shells Including Material Nonlinearities," Computer Methods in Applied Mechanics and Engineering. Vol. 27. 1981, pp. 183-214. 232 20. Pica, A., Wood, R. D. and Hinton, E., "Finite Element Analysis of Geometrically Nonlinear Plate Behavior Using a Mindlin Fornulation," Computers and Structures. Vol. 11. 1980, pp. 203-215. 21. Pica. A., Wood., R. D., "Post Buckling Behavior of Plates and Shells Using A Mindlin Shallow Shell Formulation," Computers and Structures. Vol. 12. 1980, pp. 759-768. 22. Bathe, K. J., Bolourchi, S., "A Geometric and Material Nonlinear Plate and Shell Element," Computers & Structures. Vol. 11.. 1980, pp. 23-48. 23. Horrigmoe, G., Bergan, P., "Nonlinear Analysis of Free- Formed Shells by Flat Finite Element," Computer Methods in Applied Mechanics and Engineering, Vol. 16. 1978, pp. 11-35. 24. Bergan, P. G., Clough, R. W., "Large Deflection Analysis of Plates and Shellow Shells Using the Finite Element Method," I.J. for the Numerical methods in Engineering. Vol. 5. 1973, pp. 543-556. 25. Murray, D. W., Wilson, E. L., "Finite Element Large Deflection Analysis of Plates," ASCE Journal of Engineering Mechanics. Vol. 95. No. EMI. Feb. 1969. 26. Hsiao, K., Hung, H., "Large Deflection Analysis of Shell Structures by Using Corotational Total Lagrangian Formulation," Computer Methods in Applied Mechanics and Engineering. 1989, pp. 209-225. 27. Hsiao, K., "Nonlinear Analysis of General Shell Structures by Flat Triangular Shell Element," Computers and Structures. Vol. 25. 1987, pp. 665-675. 28. Schweizerhof, K., Ramm, E., "Displacement Dependent Pressure Loads in Nonlinear Finite Element Analysis," Computers & Structures. Vol. 18., No. 6.. 1984, pp. 1099-1114. 29. Nagtegaal, J. C., De Jong, J. E., "Some Computational Aspects of Elastic-Plastic Large Strain Analysis," I.j. for Numerical Methods in Engineering, Vol. 17. 1981, pp. 15-41. 30. Belytscheko, T., Wong, B. L., "Assumed Strain Stabilization Procedure for the Nine Node Lagrangian Shell Element," I.J. for Numerical Methods in Engineering. Vol. 28. 1989, pp. 385-414. 233 31. Kamoulakos, A., "Understanding and Improving the Reduced Integration of Mindlin Shell Elements," I.J. for Numerical Methods in Engineering, Vol. 26. 1988, pp. 2009-2029. 32. Rhiu, J. J., Lee, S. W., "A Nine Node Finite Element for Analysis of Geometrically Nonlinear Shells," I.J. for Numerical Methods in Engineering. Vol. 26. 1988, pp. 1945-1962. 33. Briassoulis, D., "Machine Locking of Degenerated Thin Shell Elements," I.J. for Numerical Methods in Engineering, Vol. 26. 1988, 1749-1768. 34. Yeom, C. H., Lee, W., "An Assumed Strain Finite Element Model for Large Deflection Composite Shells," I.J. for Numerical Methods in Engineering, Vol. 28. 1989, pp. 1749-1768. 35. Pinsky, P. M., Jasti, R. V., "A mixed Finite Element Formulation for Reissner-Mindlin Plates Based on the Use of Bubble Bubble Functions," I.J. for Numerical Methods in Engineering. Vol. 28. 1989, pp. 1677-1702. 36. Pian, T. H. H., Chen, D., "On the Suppression of Zero Energy Deformation Modes," I.J. for Numerical Methods in Engineering, Vol. 19. 1983, pp. 1741-1752. 37. Choi, C. K., Kim, S. H., "Coupled Use of Reduced Integration and Non-conforming Modes in Quadratic Mindlin Plate Element," I.J. for Numerical Methods in Engineering, Vol. 28. 1989, pp. 1909-1928. 38. Chan, H. C., Chung, W. C., "Geometrically Nonlinear Analysis of Shallow Shells Using Higher Order Finite Elements," Computers & Structures. Vol. 31, No. 3. 1989, pp. 329-338. 39. Meek, J. L., Tan, H. S., "Instability Analysis of Thin Plates and Arbitrary Shells Using a Faceted Shell Element with Loof Nodes," Computer Methods in Applied Mechanics and Engineering. Vol. 57. 1986, pp. 143-170. 40. Vu-Quoc, L., Mora, J. A., "A Class of Simple and Efficient Degenerated Shell Elements-Analysis of Global Spurious-Mode Filtering," Computer Methods in Applied Mechanics and Engineering, Vol. 74.. 1989, pp. 117-175. 234 41. Briassoulis, D., "The Zero Energy Modes Problem of the Nine-Node Lagrangian Degenerated Shell Element," I.J. for Numerical Methods in Engineering, Vol. 30. No. 6. 1988, pp. 1389-1402. 42. Forde, B. W., Steimer, S. F., "Improved Arc Length Orthogonality Methods for Nonlinear Finite Element Analysis," Computers & Structures. Vol. 27, No. 5. 1987, pp. 625-630. 43. Crisfield, M. A., "Snap-Through and Snap-Back response in Concrete Structures and the Dangers of Under- Integration, " I.J. for Numerical Methods in Engineering, Vol. 22. 1986, pp. 751-767. 44. Crisfield, M. A., "An Arc Length Method including Line Searchs and Accelerations," I.J. for Numerical Methods in Engineering, Vol. 19. 1983, pp. 1269-1289. 45. Crisfield, M. A., "A Fast Incremental/Iterative Solution Procedure that Handles 'Snap-Through'," Computers & Structures, Vol. 13. 1981, pp. 55-62. 46. Powell, G., Simons, J., "Improved Iteration Strategy for Nonlinear Structures," I.J for Numerical Methods in Engineering. Vol. 17. 1981, pp. 1455-1467. 47. Meirovitch, L., Baruh, H., "On the Chilesky Algorithm with Shifts for the Eigensolution of Real Symmetric Matrices," I.J for Numerical Methods in Engineering. Vol. 17. 1981, pp. 923-930. 48. Papadrakakis, M., "Post-Buckling Analysis of Spatial Structures by Vector Iteration Methods," Computers & Structures. Vol. 14, No. 5-6. 1981, pp. 393-402. 49. Zienkiewicz, O. C., The Finite Element Method. McGrow- Hill Book Company (UK) Limited, London, U.K., 1977. 50. Desai, C. S., Abel, J. F., Introduction to the Finite Element Method. Van Nostrand Reinold Company, New York, N.Y., 1972. 51. Biot, M. A., Mechanics of Incremental Deformations. John Wiley & Sons, Inc., New York, N.Y., 1965. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, N.J., 1969, p. 161. 52. 53. Shames, I. H. and Dym, C. L., Energy and Finite Element Methods in Structural Mechanics. Hemisphere Publishing Corporation, New York, N.Y., 1985, pp. 401-403. 54 Cleary, P. J., "An analysis of the behavior of key joints in concrete box girder bridges," Master Thesis, University of Florida, 1986. 55. Hoit, M. I.,"New Computer Programming Technique for Structural Engineering," Doctoral Dissertation. University of California, Berkeley, California, 1983. 56. Oughourlian, C. V., and Powell, G. H., ANSR - III. General Purpose Computer Program for Nonlinear Structural Analysis. University of California, Berkeley, California, 1982. 57. Weaver, Jr. W. and Johnston, P. R., Finite Elements for Structural Analysis. Prentice-Hall, Englewood Cliffs, N.J., 1984. 58. McGuire, W. and Gallagher, R. H., Matrix Structural Analysis. John Wiley & Sons, New York, 1979. 59. Timoshenko, S., Woinowsky-Krieger, S., Theory of Plates and Shells. Van Nostrand Reinold Company, New York, N.Y., 1968. 60. Buyukozturk, O., Shareef, S. S., "Constitutive Modeling of Concrete in Finite Element Analysis," Computers & Structures. Vol. 21, No. 3. 1985, pp. 581-610. 61. Kupfer, H. B., Gerstle, K. H., "Behavior of Concrete under Biaxial Stresses," ASCE Journal of Engineering Mechanics. Aug. 1973, pp. 853-866. 62. Kupfer, H. , Hilsdorf, H. K., Rusch, H. "Behavior of Concrete Biaxial Stresses," ACT Journal. Aug. 1969, pp. 656-666. 63. Darwin, D., Pecknold, D. A., "Nonlinear Biaxial Stress- Strain Law for Concrete," ASCE Journal of Engineering Mechnics, Apr. 1977, pp. 229-241. 64. Chen, W. F., Suzuki, H., "Constitutive Model for Concrete," Computers & Structures. Vol. 12. 1980 65. Bazant, Z. P., Bhat, P. D., "Endochronic Theory of Inelasticity and Failure of Concrete," ASCE Journal of Engineering Mechnics. Aug. 1976, pp. 701-722. 236 66. Mehlhorn, G., Kollegger, J., Keuser, M., Kolmar, W., "Nonlinear Contact Problems-a Finite Element Approach Implemented in ADINA," Computers & Structures. Vol. 21, No. 1/2. 1985, pp. 69-80. 67. Walraven, J. C., Reinhardt, H. W., "Theory and Experiments on the Mechanical Behavior of Cracks in Plain and Reinforced Concrete Subjected to Shear Loading," Heron. Vol. 26, NO. 1A. 1981, pp. 1-68. 68. Bazant, Z. P., Oh, B. H., "Deformation of Cracked Net- Reinforced Concrete Walls," ASCE Journal of Structural Engineering. Jan. 1983, pp. 93-108. 69. Suzuki, H., Chen, W. F., "Elastic-Plastic Fracture Analysis of Concrete Structures," Computers & Structures. Vol. 16. No. 6. 1983, pp. 697-705. 70. Owen, D. R. J., Figueiras, J. A., Damjanic, F., "Finite Element Analysis of Reinforced and Prestressed Concrete Structures including Thermal Loading," Computer Methods in Applied Mechnics and Engineering, Vol. 41. 1983, pp. 323-366. 71. Chen, W. F., Suzuki, H., Chang, T. Y., "Nonlinear Analysis of Concrete Cylinder Structures under Hydrostatic Loading," Computers & Structures, Vol. 12. 1980, pp. 559-570. 72. Milford, R. V., Schnobrich, W. C., "The Application of the Rotating Crack Model to the Analysis of Reinforced Concrete Shells," Computers & Structures. Vol. 20. No. 1-3. 1985, pp. 225-234. 73. Rots, J. G., Nauta, P., Kusters, G. M. A., Blaauwendraad, J., "Smeared Crack Approach and Fracture Localization in Concrete," Heron. Vol. 30, No. 1. 1985, pp. 1-48. 74. Gilbert, R. I., Waner, A., "Tension Stiffening in Reinforced Concrete Slabs," ASCE Journal of Structural Division. Dec. 1978, pp. 1885-1900. 75. Bazant, Z. P., Gambarova, P., "Rough Cracks in Reinforced Concrete," ASCE Journal of Structural Division. Apr. 1980, pp. 819-843. 76. Fardis, M. N., Buyukozturk, O., "Shear Stiffness of Concrete by Finite Elements," ASCE Journal of Structural Division. Jun. 1980, pp. 1311-1327. 237 77. Bazant, Z. P., Tsubaki, T., "Slip-Dilatation for Cracked Reinforced Concrete," ASCE Journal of Structural Division. Sep. 1980, pp. 1947-1966. 78. Fardis, M. N., Buyukozturk, O., "Shear Transfer Model for Reinforced Concrete," ASCE Journal of Engineering Mechanics Division. Apr. 1979, pp. 255-275. 79. Frantzeskakis, C., Theillout, J. N., "Nonlinear Finite Element Analysis of Reinforced Concrete Structures with a Particular Strategy Following the Cracking Process," Computers & Structures, Vol. 31. No. 3. 1989, pp. 395- 412. 80. St. Pietruszczak, Mroz., Z., "Finite Element Analysis of Deformation of Strain-Hardening Materials," I.J. for Numerical Methods in Engineering, Vol. 17. 1981, pp. 327-334. 81. Gupta, A. K., Akbar, H., "A Finite Element for the Analysis of Reinforced Concrete Structures," I.J. for Numerical Methods in Engineering, Vol. 19. 1983, pp. 1705-1712. 82. Cook, R. D., Malkus, D. S., Plesha, M. E., Concepts and Applications of Finite Element Analysis. John Wiley & Sons, New York, N.Y., 1989. SUPPLEMENTAL BIBLIOGRAPHY Ghaboussi, J., and Wilson, E. L., and Isenberg, J., "Finite Element for Rock Joints and Interface," Journal of the Soil Mechanics and Foundations Division. Proceedings of ASCE, Vol. 99, No. SM 10. October, 1973, p. 833. Wunderlich, Stein, Bathe, Nonlinear Finite Element Analysis in Structural Mechnics. Springer-Verlag, New York, N.Y.,1981. Fafard, M., Dhatt, G., Batoz, J. L., "A New Discrete Kirchhoff Plate/Shell Element with Updated Procedures," Computers & Structures. Vol. 31, No. 4. 1989, pp. 591- 606. Yuan, F. G., Miller, R. E., A New Element for Laminated Composite Beams, Computers & Structures. Vol. 31. No. 5, 1989, pp. 737-745. Liao, C. L., Reddy, J. N., Engelstad, S. P., "A solid- shell Transition Element for Geometrically Non-linear Analysis of Laminated Composite Structures," I.J. for Numerical Methods in Engineering, Vol. 26. 1988, pp. 1843-1854. Ortiz, M., Morris, G. R., "CÂ° Finite Element Discretization of Kirchhoff's Equations of Thin Plate Bending," I.J. for Numerical Methods in Engineering. Vol. 26. 1988, pp. 1551-1566. Kamoulakos, A., "Understanding and Improving the Reduced Integration of Mindlin Shell Elements," I.J. for Numerical Methods in Engineering, Vol. 26. 1988, pp. 2009-2029. Dhatt, G., Marcotte, L., Matte, Y, "A New Triangular Discrete Kirchhoff Plate/Shell Element," I.J. for Numerical Methods in Engineering, Vol. 23. 1986, pp. 453-470. Hughes, T. Jr., Hinton, E., Finite Element Methods for Plate and Shell Structures. Vol. l. Pineridge Press International, Swansea, U.K., 1986. Hughes, T. Jr., Hinton, E., Finite Element Methods for Plate and Shell Structures. Vol. 2. Pineridge Press International, Swansea, U.K., 1986. 238 239 Yang, T. Y., Saigal, S., "A curved Quadrilateral Element for Static Analysis of Shells with Geometric and Material Nonlinearities," I.J. for Numerical Methods in Engineering, Vol. 21. 1985, pp. 617-635. Chao, W. C., Reddy, J. N., "Analysis of Laminated Composite Shells Using a Degenerated 3-D Element," I.J. for Numerical Methods in Engineering, Vol. 20. 1984, pp. 1991-2007. Oliver, J., Onate, E., "A Toatal Lagrangian Formulation for the Geometrically Nonlinear Analysis of Structures Using Finite Elements. Part I. Two-Dimensional Problems: Shell and Plate Structures," I.J. for Numerical Methods in Engineering, Vol. 20. 1984, pp. 2253-2281. Spilker, R. L., "Hybrid Stress Eight Node Elements for Thin and Thick Multilayer Laminated Plates," I.J. for Numerical Methods in Engineering, Vol. 18. 1982, pp. 801-828. Noor, A. K., "Mixed Models and Reduced/Selective Integration Displacement Models for Nonlinear Shell Analysis," I.J. for Numerical Methods in Engineering. Vol. 18. 1982, pp. 1429-1454. Sander, G., Idelsohn, S., "A Family of Conforming Finite Elements for Deep Shell Analysis," I.J. for Numerical Methods in Engineering, Vol. 18. 1982, pp. 363-380. Hughes, T. J. R., Liu, W. K., "Nonlinear Finite Analysis of Shells-Part II. Two-Dimensional Shells," Computer Methods in Applied Mechanics and Engineering. 1981, pp. 167-181. Hinton, E., Owen, R., Computational Modelling of Reinforced Concrete Structures. Pineridge Press International, Swansea, U.K., 1986. Arnesen, A., Sorensen, S. I., Bergan, P. G., "Nonlinear analysis of Reinforced Concrete," Computers & Structures. Vol. 12. 1980, pp. 571-579. Nilson, A. H., Design of Prestressed Concrete. John Wiley & Sons, New YOrk, 1987. BIOGRAPHICAL SKETCH The author was born in Ham-ahn, Korea, in 1956. He graduated from Busan Senior High School in 1974 and from Seoul National University in 1978 with a B.S. degree in architectural engineering. He then joined Korea Electric Power Corporation and was involved in the construction of nuclear power plants for 7 years. The author received his M.E. degree in construction engineering and management from the University of Florida in 1986. He expects to obtain his Ph.D. degree in structural engineering in 1990. 240 I certify that I have read tluis study and that in my opinion it conforms to /acceptably standards of scholarly presentation and is fulw adequaty in scope and quality, as a dissertation for the/^egree ofyDoctor of Philosophy. Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. liffÃ¼Fd 0. Hays/ Jr./ Cliffy Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Fernando E. Fagundo^ Associate Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jphn M. Lybas Associate Professor of ivil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Pagi W. Chun jfessor of Biochemistry and Molecular Biology This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1990 Dean, lips of Engineering Madelyn M. Lockhart Dean, Graduate School 07/02/2008 05:12 AM "UFDissertations" To cc bcc Subject UF Libraries:Digital Dissertation Project Dear Dr. Kookjoon Ahn, The George A. Smathers Libraries at the University of Florida has initiated a project to retrospectively digitize and make available on the Internet any dissertation written by a University of Florida doctoral candidate and accepted by the University of Florida. It is our goal to make the documents fully text searchable and easily harvested by Internet search engines, allowing the full breadth and scope of scholarship produced at the University of Florida to be made available across the world quickly and easily. The Library is bearing the full cost of the project. We would like to add your dissertation, Nonlinear gap and Mindlin shell elements for the analysis of concrete structures /, published in 1990, to the project. In order to do so we need a signed, legal original Internet Distribution Consent Agreement for our files. If you want your dissertation included in the project, please print the Consent Agreement on the page below, sign it and mail it back to the Libraries at the address listed. We will keep you informed of the progress of your dissertation as it works its way through the project. If you have any questions, please reply to UFdissertations @ uflib.ufl.edu. Thank you, Cathy Martyniak, Project Coordinator Christy Shorey, Project Technician The Foundation for The Gator Nation Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: TITLE: Ahn, Kookjoon Nonlinear gap and Mindlin shell elements for the analysis of concrete PUBLICATION DATE: structures / (record number: 1583924) 1990 [ I, M A H } as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-based versions as appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Signature of Copyright Holder \fC*>c>\,Csroot+l AH si Printed or Typed Name of Copyright Holdei/Licensee Personal information blurred 7/ Zl ( Zoe?3 Date of Signature Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Preservation Department University of Florida Libraries P.O. Box 117007 Gainesville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Â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f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n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f >@ 7KH HOHPHQW XVHV UHODWLYH GLVSODFHPHQWV DV WKH LQGHSHQGHQW GHJUHHV RI IUHHGRP 7KH GLVSODFHPHQW GHJUHHV RI IUHHGRP RI RQH VLGH RI WKH VOLS VXUIDFH DUH WUDQVIRUPHG LQWR WKH UHODWLYH GLVSODFHPHQWV EHWZHHQ WKH WZR VLGHV RI WKH VOLS VXUIDFH 7KLV HOHPHQW KDV EHHQ LQFRUSRUDWHG LQWR D JHQHUDO ILQLWH HOHPHQW FRPSXWHU SURJUDP >@ 7KH XVH RI UHODWLYH GLVSODFHPHQW DV DQ LQGHSHQGHQW GHJUHH RI IUHHGRP WR DYRLG QXPHULFDO VHQVLWLYLW\ LV GLVFXVVHG LQ GHWDLO >@ $Q LVRSDUDPHWULF IRUPXODWLRQ LV JLYHQ E\ %HHU >@ $ IRXUQRGH WZRGLPHQVLRQDO OLQN HOHPHQW DQG D HLJKWQRGH SODWH EHQGLQJ HOHPHQW ZHUH XVHG WR PRGHO WKH GU\ MRLQWHG FRQFUHWH ER[ JLUGHU EULGJH ZLWK VKHDU NH\V >@ PAGE 11 6KHOO (OHPHQW 7KH VKHOO HOHPHQW LV IRUPXODWHG WKURXJK WKH FRPELQDWLRQ RI WZR GLIIHUHQW HOHPHQWV WKH PHPEUDQH HOHPHQW DQG WKH 0LQGOLQ SODWH EHQGLQJ HOHPHQW 7KH 0LQGOLQ SODWH HOHPHQW LV GLIIHUHQW IURP WKH .LUFKKRII SODWH HOHPHQW LQ WKDW WKH IRUPHU DOORZV WUDQVYHUVH VKHDU GHIRUPDWLRQ ZKLOH WKH ODWWHU GRHV QRW 7KH QRQOLQHDULWLHV LQFOXGHG LQ WKH IRUPXODWLRQ RI WKH IODW VKHOO HOHPHQW LV IRU ODUJH GLVSODFHPHQW DQG JHRPHWULF QRQOLQHDULW\ GXH WR LQLWLDO VWUHVV HIIHFWV 7KH ODUJH GLVSODFHPHQW HIIHFWV DUH FDXVHG E\ ILQLWH WUDQVYHUVH GLVSODFHPHQWV 7KHVH HIIHFWV DUH WDNHQ LQWR DFFRXQW E\ FRXSOLQJ WUDQVYHUVH GLVSODFHPHQW DQG PHPEUDQH GLVSODFHPHQWV 7KH LQLWLDO VWUHVV HIIHFWV DUH FDXVHG E\ WKH DFWXDO VWUHVVHV DW WKH VWDUW RI HDFK LWHUDWLRQ 7KHVH VWUHVVHV FKDQJH WKH HOHPHQW VWLIIQHVV IRU WKH VXEVHTXHQW LWHUDWLRQ 7KHVH HIIHFWV DUH HYDOXDWHG GLUHFWO\ IURP WKH VWUHVVHV DW WKH VWDUW RI HDFK LWHUDWLRQ DQG LQFOXGHG LQ WKH HOHPHQW VWLIIQHVV A /LWHUDWXUH 5HYLHZ 7KH SXUSRVH RI QRQOLQHDU DQDO\VLV LV WR GHYHORS WKH FDSDELOLW\ IRU GHWHUPLQLQJ WKH QRQOLQHDU ORDGGHIOHFWLRQ EHKDYLRU RI WKH VWUXFWXUHV XS WR IDLOXUH VR WKDW D SURSHU PAGE 12 HYDOXDWLRQ RI VWUXFWXUDO VDIHW\ FDQ EH DVVXUHG 7KHUH DUH WZR JHQHUDO DSSURDFKHV IRU QRQOLQHDU DQDO\VLV 7KH ILUVW DSSURDFK LV D OLQHDUL]HG LQFUHPHQWDO IRUPXODWLRQ E\ UHGXFLQJ WKH DQDO\VLV WR D VHTXHQFH RI OLQHDU VROXWLRQV 7KH VHFRQG DSSURDFK LV PDWKHPDWLFDO LWHUDWLYH WHFKQLTXHV DSSOLHG WR WKH JRYHUQLQJ QRQOLQHDU HTXDWLRQV >@ 7KH DGYDQWDJH RI WKH LQFUHPHQWDO DSSURDFK UHVXOWV IURP WKH VLPSOLFLW\ DQG JHQHUDOLW\ RI WKH LQFUHPHQWDO HTXDWLRQV ZULWWHQ LQ PDWUL[ IRUP 6XFK HTXDWLRQV DUH UHDGLO\ SURJUDPPHG LQ JHQHUDO IRUP IRU FRPSXWHU VROXWLRQV >@ $ JHQHUDOL]HG LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ IRU QRQOLQHDU DQDO\VLV FDQ EH IRXQG LQ > @ 7KH IRUPXODWLRQ LV YDOLG IRU ERWK JHRPHWULFDO DQG PDWHULDO QRQOLQHDULWLHV ODUJH GLVSODFHPHQWV DQG URWDWLRQV FRQVHUYDWLYH DQG GLVSODFHPHQW GHSHQGHQW QRQFRQVHUYDWLYHf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f IXQFWLRQ 0LQGOLQ5HLVVQHU HOHPHQWV UHTXLUH RQO\ &r FRQWLQXLW\ VR WKDW PXFK ORZHU RUGHU VKDSH IXQFWLRQV FDQ EH XVHG ZKHUHDV LQ .LUFKKRII/RYH W\SH HOHPHQWV KLJK RUGHU VKDSH IXQFWLRQV PXVW EH XVHG WR VDWLVI\ WKH & FRQWLQXLW\ )XUWKHUPRUH VLQFH 0LQGOLQ5HLVVQHU HOHPHQWV DFFRXQW IRU WUDQVYHUVH VKHDU WKHVH HOHPHQWV FDQ EH XVHG IRU D PXFK ODUJHU UDQJH RI VKHOO WKLFNQHVV 7KH UHOD[HG FRQWLQXLW\ UHTXLUHPHQWV ZKLFK SHUPLW WKH XVH RI LVRSDUDPHWULF PDSSLQJ WHFKQLTXHV JLYHV JRRG FRPSXWDWLRQDO HIILFLHQF\ LI IRUPXODWHG LQ WKH IRUP RI UHVXOWDQW VWUHVVHV >@ 8QOLNH FRPSUHVVLEOH FRQWLQXXP HOHPHQWV ZKLFK DUH TXLWH LQVHQVLWLYH WR WKH RUGHU RI WKH TXDGUDWXUH UXOH FXUYHG &r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r[Â $W WLPH W A[Â $W WLPH WGW WGW[L PAGE 21 )LJ 0RWLRQ RI D %RG\ PAGE 22 7KH WRWDO GLVSODFHPHQWV RI WKH ERG\ DUH $W WLPH $W WLPH W X M n8M $W WLPH WGW B WGW 8M 7KH FRQILJXUDWLRQV DUH GHQRWHG DV $W WLPH r& $W WLPH W 8& $W WLPH WGW WGW 7KXV WKH XSGDWHG FRRUGLQDWHV DW WLPH W DQG WLPH WGW DUH A r;L A WGW[ 2[ WGWX 7KH XQNQRZQ LQFUHPHQWDO GLVSODFHPHQWV IURP WLPH W WR WLPH WGW DUH GHQRWHG DV 1RWH WKDW WKHUH LV QR VXSHUVFULSW DW OHIW KDQG VLGHf X WGWX n8 f 3ULQFLSOH RI 9LUWXDO :RUN 6LQFH WKH VROXWLRQ IRU WKH FRQILJXUDWLRQ DW WLPH WGW LV UHTXLUHG WKH SULQFLSOH RI YLUWXDO ZRUN LV DSSOLHG WR WKH HTXLOLEULXP FRQILJXUDWLRQ DW WLPH WGW 7KLV PHDQV DOO WKH YDULDEOHV DUH WKRVH DW WLPH WGW DQG DUH PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW DQG DOO WKH LQWHJUDWLRQV DUH SHUIRUPHG RYHU WKH DUHD RU YROXPH LQ WKH FRQILJXUDWLRQ DW WLPH WGW 7KHQ WKH LQWHUQDO YLUWXDO ZRUN ,9:f E\ WKH PAGE 23 FRUUHVSRQGLQJ YLUWXDO VWUDLQ GXH WR YLUWXDO GLVSODFHPHQW LQ WGW& LV ,9: WGW WGW V n WGW WGW f WGW GYf f ZKHUH WGW WGW [ 6WUHVVHV DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW &DXFK\ VWUHVVHV 7UXH VWUHVVHV rGA HML &DXFK\nV LQILQLWHVLPDOOLQHDUf VWUDLQ WHQVRU WGW UHIHUUHG WR WKH FRQILJXUDWLRQ DW WLPH WGW 9LUWXDO VWUDLQ WHQVRU 6 'HOWD RSHUDWRU IRU YDULDWLRQ DQG WKH H[WHUQDO YLUWXDO ZRUN (9:f E\ VXUIDFH WUDFWLRQV DQG ERG\ IRUFHV LV (9: WGW [ e WGW WGW N WGW N B WGW G$M ZKHUH WGW D WGW N WGW WGW XN WGW WGW E ; WGW X WGW N WGW S WGW N B WGW G9f f 6XUIDFH WUDFWLRQ DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW 7RWDO GLVSODFHPHQW DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW PAGE 24 rXA 9DULDWLRQ LQ WRWDO GLVSODFHPHQW DW WLPH WGW PHDVXUHG LQ FRQILJXUDWLRQ DW WLPH WGW 9LUWXDO GLVSODFHPHQW 0DVV GHQVLW\ SHU XQLW YROXPH %RG\ IRUFH SHU XQLW PDVV DQG DOO WKH LQWHJUDWLRQ LV SHUIRUPHG RYHU WKH DUHD DQG WKH YROXPH DW WLPH WGW 8SGDWHG /DTUDQTLDQ )RUPXODWLRQ WGW WGW WGW + WGW EN ,Q WKLV IRUPXODWLRQ DOO WKH YDULDEOHV LQ (TV f DQG f DUH UHIHUUHG WR WKH XSGDWHG FRQILJXUDWLRQ RI WKH ERG\ LH WKH FRQILJXUDWLRQ DW WLPH W 7KH HTXLOLEULXP SRVLWLRQ DW WLPH WGW LV VRXJKW IRU WKH XQNQRZQ LQFUHPHQWDO GLVSODFHPHQWV IURP WLPH W WR WGW 7KH LQWHUQDO YLUWXDO ZRUN WKH YROXPH LQWHJUDO LQ (T f PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW FDQ EH WUDQVIRUPHG WR WKH YROXPH LQWHJUDO PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W LQ D VLPLODU PDQQHU WKDW LV JLYHQ LQ UHIHUHQFH >@ ,9: WGW WGW LM WGW WGW WGW GYf WGW F W 6LM WGW W H LM IF G9f (9: f PAGE 25 ZKHUH WGW V B VHFRQG 3LROD.LUFKKRII 3.,,f VWUHVVHV W PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W 6 HML 9DULDWLRQV LQ *UHHQ/DJUDQJH */f VWUDLQ WHQVRU PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W 7KH 3.,, VWUHVV WHQVRU DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W FDQ EH GHFRPSRVHG DV WGW H W 6LM ,' WVLM f EHFDXVH WKH VHFRQG 3.,, VWUHVV DW WLPH W PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W LV WKH &DXFK\ VWUHVV )URP (T f WKH WRWDO GLVSODFHPHQWV DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W LV WGW 8L W X A X f W X f 7KLV LV WUXH EHFDXVH WKH GLVSODFHPHQW DW WLPH W PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W LV ]HUR ,Q RWKHU ZRUGV WKH GLVSODFHPHQW DW WLPH WGW ZLWK UHVSHFW WR WKH FRQILJXUDWLRQ DW WLPH W LV WKH LQFUHPHQWDO GLVSODFHPHQW LWVHOI $QG WKH */ VWUDLQ LV GHILQHG LQ WHUPV RI GLVSODFHPHQW DV ( LM r XLM 8ML XNLXNM! f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f ZKHUH WHLM WHLM WALM ,QFUHPHQWDO */ VWUDLQ LQ rr& /LQHDU SRUWLRQ RI LQFUHPHQWDO */ VWUDLQ LQ rr& 7KLV LV OLQHDU LQ WHUPV RI XQNQRZQ LQFUHPHQWDO GLVSODFHPHQW /LQHDUL]HG LQFUHPHQWDO */ VWUDLQ LQ A& PAGE 27 W}"LM L W8MF21RQOLQHDU SRUWLRQ RI LQFUHPHQWDO */ VWUDLQ 7KH YDULDWLRQV LQ *UHHQ/DJUDQJH VWUDLQ WHQVRU DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W FDQ EH VKRZQ DV XVLQJ (J f WGW eLM WALM WHL WHLM f } f W W 6AHOM EHFDXVH LV NQRZQ 7KHUH LV QR YDULDWLRQ P NQRZQ TXDQWLW\ 7KHQ XVLQJ WKH (TV f f DQG f WKH LQWHJUDQG RI (T f EHFRPHV WGW V V WGW H B A W nL W ,' 6L M f H W n,' W ,' n W ,' WULM W6L'f WHL' Wn"LMf W6L'WHL' Wn"L'f WULM WHL' WUL' WA' W6L' WHL' WILM WHL' W ULM WALM f 7KH FRQVWLWXWLYH UHODWLRQ EHWZHHQ LQFUHPHQWDO 3.,, VWUHVVHV DQG */ VWUDLQV DUH W6LM W&L'NM W&NO f PAGE 28 )LQDOO\ WKH HTXLOLEULXP (T f IURP WKH SULQFLSOH RI YLUWXDO ZRUN XVLQJ (TV f DQG f LV WFLMNO WHNO WfLM IFG9 WULM nG9 (9: f ZKHUH WKH H[WHUQDO YLUWXDO ZRUN PXVW EH WUDQVIRUPHG IURP WGWF WR W& 7KLV LV QRW DSSOLFDEOH WR FRQVHUYDWLYH ORDGLQJ LH ORDGLQJ WKDW LV QRW FKDQJHG GXULQJ GHIRUPDWLRQ (9: WGW W WGW W XN IFG$f WGW f W S WGW \ W EN V WGW W 8N IFG9f f DQG WKLV LV WKH JHQHUDO QRQOLQHDU LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ RI XSGDWHG /DJUDQJLDQ IRUPXODWLRQ 7RWDO /DJUDQJLDQ )RUPXODWLRQ 7RWDO /DJUDQJLDQ IRUPXODWLRQ LV DOPRVW LGHQWLFDO ZLWK WKH XSGDWHG /DJUDQJLDQ IRUPXODWLRQ $OO WKH VWDWLF DQG NLQHPDWLF YDULDEOHV LQ (TV f DQG f DUH UHIHUUHG WR PAGE 29 WKH LQLWLDO XQGHIRU[QHG FRQILJXUDWLRQ RI WKH ERG\ LH WKH FRQILJXUDWLRQ DW WLPH 7KH WHUPV LQ WKH OLQHDUL]HG VWUDLQ DUH DOVR VOLJKWO\ GLIIHUHQW IURP WKRVH RI XSGDWHG /DJUDQJLDQ IRUPXODWLRQ 7KH YROXPH LQWHJUDO LQ (T f PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGW FDQ EH WUDQVIRUPHG WR WKH YROXPH LQWHJUDO PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH DV >@ WGW WGW ;' WGW H WGW L' WGWGYf WGW WGW ,' rG9f f ZKHUH WGWJ B VHFRQG 3LROD.LUFKKRII VWUHVV WHQVRU r PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH WGWe B YDULDWLRQV LQ *UHHQ/DJUDQJH */f VWUDLQ r WHQVRU PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH 7KH 3.,, VWUHVV WHQVRU DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH FDQ EH GHFRPSRVHG DV WGW R f )URP (T f WKH WRWDO GLVSODFHPHQWV DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH LV PAGE 30 WGWX WX X f 7KHQ WKH */ VWUDLQ WHQVRU DW WLPH WGW PHDVXUHG DW WLPH FDQ EH FDOFXODWHG DV WGW f Y Tf n 8 M A n L 2 nLM L >A 8LfM 9 R8fF RX.fnLfRX. RXNnMf Ar W ÂÂ r R ;R ;R 2HLM RHLM ZKHUH W RfLM RHLM HLL R ;RfLM WXL $ WX WXN WXN cf */ VWUDLQ DW WLPH W LQ r& HLL nLLL R R ;f ,QFUHPHQWDO */ VWUDLQ LQ r& ?OL V 8M L W8OB f X! WX X! L f /LQHDU SRUWLRQ RI LQFUHPHQWDO */ VWUDLQ LQ r& 7KLV LV OLQHDU LQ WHUPV RI XQNQRZQ LQFUHPHQWDO GLVSODFHPHQW /LQHDUL]HG LQFUHPHQWDO */ VWUDLQ LQ r& L RfNL XNM! 1RQOLQHDU SRUWLRQ RI LQFUHPHQWDO */ VWUDLQ 7KH YDULDWLRQV LQ *UHHQ/DJUDQJH VWUDLQ WHQVRU DW WLPH WGW PHDVXUHG LQ WKH FRQILJXUDWLRQ DW WLPH W FDQ EH VKRZQ DV PAGE 31 XVLQJ (T f F WGW f ,' RfLM RLM 6LM f 6 EHFDXVH WHL LV NQRZQ 7KHUH LV QR YDULDWLRQ LQ R R NQRZQ TXDQWLW\ 7KHQ XVLQJ WKH (TV f f DQG f WKH LQWHJUDQG RI (T f EHFRPHV WGW R VLL WGWILL WVLL VLL f m -' R R R R Â‘ R6LM RHLM RfLL R6LML RHLr 2 LM f RWVLM 6RHLM R6LM 6 6 HVW 6 ULL f R ;' R R /' R R AR 7KH FRQVWLWXWLYH UHODWLRQ EHWZHHQ LQFUHPHQWDO 3.,, VWUHVVHV DQG */ VWUDLQV DUH rLM R&LMNM HNO f )LQDOO\ WKH HTXLOLEULXP (T f IURP WKH SULQFLSOH RI YLUWXDO ZRUN XVLQJ (TV f DQG f LV PAGE 32 HQ HV rG9 nLMNO HNO HLM WVLL mLL rGY R /2 ;(9: WVLL 6 HA rG9 R R f ZKHUH WKH H[WHUQDO YLUWXDO ZRUN PXVW EH WUDQVIRUPHG IURP WGW& WR r& 7KLV LV QRW DSSOLFDEOH WR FRQVHUYDWLYH ORDGLQJ WKDW LV ORDGLQJ WKDW LV QRW FKDQJHG GXULQJ GHIRUPDWLRQ (9: ; S S f3 n WGW X R f / rG$f WGW R S WGW EN R WGW XN 2 f rG9f f DQG WKLV LV WKH JHQHUDO QRQOLQHDU LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ RI WRWDO /DJUDQJLDQ IRUPXODWLRQ /LQHDUL]DWLRQ RI (TXLOLEULXP (TXDWLRQ 7KH LQFUHPHQWDO VWUDLQ IURP WLPH W WR WGW LV DVVXPHG WR EH OLQHDU LH HNO HNO LQ (TV f f f DQG f PAGE 33 )RU WKH XSGDWHG /DJUDQJLDQ IRUPXODWLRQ W6LM n W&LMNM WHNO f DQG WFLMNO WHNO WHLM GY Wn8 V WfLM WG9 (9: WULM WHLM WG9 f )RU WKH WRWDO /DJUDQJLDQ IRUPXODWLRQ R6LM &LMNM RNO f DQG &LMNO HNO A HLQ 2 -Y[ R -G9 6 RALM rG9 (9: f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f rf ZKHUH XÂ 'LVSODFHPHQW LQ LGLUHFWLRQ XLM DXL D[M [Â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f HLM XLM XML f 7KHQ WKH VXP RI (TV f DQG f JLYHV f DQG WKH VXEWUDFWLRQ RI (T f IURP (T f JLYHV f )URP (TV f DQG f LW LV FRQFOXGHG WKDW f XNM f HNM UNM XNL HLN ULN f (T f FDQ EH UHZULWWHQ DV f XNL f HNL UNL VLQFH HAA HA IURP WKH V\PPHWU\ RI OLQHDU VWUDLQ WHUPV DQG UA UA IURP WKH VNHZ V\PPHWU\ RI WKH OLQHDU URWDWLRQ WHUPV 7KH VWUDLQGLVSODFHPHQW (T f QRZ EHFRPHV (LM HLM HNL UNLfHNM UNMf f PAGE 36 E\ VXEVWLWXWLQJ (TV f WKURXJK f LQWR (T f 7KXV WKH QRQOLQHDU VWUDLQ WHUPV KDYH EHHQ GHFRPSRVHG LQWR OLQHDU VWUDLQ WHUPV DQG OLQHDU URWDWLRQ WHUPV )URP WKH DVVXPSWLRQ RQ WKH RUGHU RI VWUDLQV DQG URWDWLRQV HNL UNL DQG HNM m UNM f 7KXV (T f FDQ EH VLPSOLILHG DV E\ LJQRULQJ HNÂ DQG HNMn (LM HÂM UNLUNM f 7KH VWUDLJKW OLQH UHPDLQV QRUPDO WR WKH PLGGOH VXUIDFH DQG XQH[WHQGHG LQ WKH .LUFKKRII DVVXPSWLRQ EXW LW LV QRW QHFHVVDULO\ QRUPDO WR WKH PLGGOH VXUIDFH IRU WKH 0LQGOLQ DVVXPSWLRQ )RU ERWK DVVXPSWLRQV WKH JHQHULF GLVSODFHPHQWV XYZ FDQ EH H[SUHVVHG E\ WKH GLVSODFHPHQWV DW PLGGOH VXUIDFH )RU WKH .LUFKKRII SODWH >@ X[\]f X4[\f ]>Z4[\f[@ Y[\]f Y4[\f ]>Z4[\f\@ f Z[\]f Z4[\f ZKHUH X4 9T : 'LVSODFHPHQWV RI WKH PLGGOH VXUIDFH LQ WKH GLUHFWLRQ RI [ \ ] X Y Z 'LVSODFHPHQWV RI DQ DUELWUDU\ SRLQW LQ WKH GLUHFWLRQ RI [ \ ] PAGE 37 1RZ WKH OLQHDU VWUDLQ FRPSRQHQWV HAM DQG WKH OLQHDU URWDWLRQ FRPSRQHQWV UAM FDQ EH FDOFXODWHG XVLQJ (TV f DQG f HOO XOO XOOf XOO 8n; H XO XOf ,X\ Y[ H fÂ§ XO XOf raZRn[ Z H X X! X Yn\ f WR WR X Xf +Z\ Z H X Xf X f 7KH URWDWLRQ WHUPV U ULn U DUH WKH URWDWLRQ TXDQWLWLHV DERXW WKH D[HV ]f \f DQG O[f UHVSHFWLYHO\ )RU WKH SODWH ORFDWHG LQ WKH [\ SODQH WKH URWDWLRQ DERXW ] D[LV U LV PXFK VPDOOHU WKDQ URWDWLRQ DERXW [ D[LV U DQG \ D[LV U DQG WKHUHIRUH U LV DVVXPHG WR EH ]HUR KHUH $QG LW LV QRWHG IXUWKHU WKDW Z4[\f LV WKH VDPH DV Z[\f DQG LV D IXQFWLRQ RI RQO\ [ DQG \ VR WKDW Z Z] U, m ,U, RU ,U, f UOO XOIO X [f U rXO f XOf +X\ 9[f U +X f XOf +Z'[ Z[f Z[ U +X f X f U LX f Xf +Z\ Z\f Z\ U rX X f f PAGE 38 7KH OLQHDU VWUDLQ FRPSRQHQW HAM LV V\PPHWULF DQG WKH OLQHDU URWDWLRQ FRPSRQHQW UAM LV DQWLV\PPHWULF HLM HML ULM fÂ§ UMA f 7KH VWUDLQ FRPSRQHQWV IURP (T f FDQ EH UHZULWWHQ XVLQJ (TV f DQG f ( ( ( ( ( ( ;; HQ UQ \\ H U ]] H U [\ H UOOU [] H UOOU \] H $ UU Uf H@/ _U Uf H AU Uf U U UU UU! H UU UU UUf UU UUA f (]] WHUP LV DVVXPHG WR EH ]HUR EHFDXVH LW GRHV QRW KDYH WKH OLQHDU WHUP ([] DQG (\= WHUPV DUH WUDQVYHUVH VKHDU WHUPV ZKLFK FDQ EH LJQRUHG IRU WKLQ SODWH 7KHQ (T f FDQ EH UHDUUDQJHG DV IROORZV XVLQJ (TV f LI DOO WKH ]HUR WHUPV DUH UHPRYHG ([[ HOO LfÂ§ U HOO Z[f ,, !L !L Z H U H &0 nO[ ([\ H UU H :[f Z\f f ([] H (\] H PAGE 39 7KXV WKH GHFRPSRVLWLRQ RI H[DFW VWUDLQ FRPSRQHQWV KDV EHHQ GRQH XVLQJ WKH .LUFKKRII SODWH DVVXPSWLRQV f DQG WKH YRQ .DUPDQ DVVXPSWLRQ f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nUHVLGXDOn VKHDU IRUFH 7KLV nUHVLGXDOn IRUFH LV HTXDO WR IULFWLRQ IRUFH 7KHUHIRUH LI VKHDU GLVSODFHPHQW LV PRUH WKDQ WKH GLVSODFHPHQW MXVW EHIRUH WKH VOLGLQJ WKH VKHDU VWLIIQHVV LV VHW WR EH ]HUR PAGE 44 1 ) )ULFWLRQ IRUFH I P )ULFWLRQ FRHIILFLHQW f 3 RU P1 WKHQ 3 )ULFWLRQ )RUFH ,Q (TXLOLEULXP f 3 P1 WKHQ WKH ERG\ PRYHV EXW WKH IULFWLRQDO IRUFH P1 LV DFWLQJ DJDLQVW WKH RWKHU ERG\ )LJ )ULFWLRQ )RUFH PAGE 45 )5,&7,21$/ 635,1* :,7+ 6+($5 67,))1(66 %()25( 6/,',1* ) )5,&7,21$/ 635,1* :,7+ =(52 67,))1(66 P1 ) $)7(5 6/,',1* )LJ 6SULQJ 0RGHO RI )ULFWLRQ )RUFH PAGE 46 635,1* 02'(/ )25 6+($5 )25&(6 )LJ 6SULQJ 0RGHO RI /LQN (OHPHQW PAGE 47 Q $ ,N Â‹ L A f X Z )LJ (OHPHQW 'HJUHHV RI )UHHGRP RI /LQN (OHPHQW PAGE 48 )LJ (OHPHQW 6WUDLQ PAGE 49 )RUPDWLRQ RI (OHPHQW 6WLIIQHVV 7KHUH DUH IRXU QRGHV SHU HOHPHQW (DFK QRGH KDV WKUHH GHJUHHV RI IUHHGRP FRUUHVSRQGLQJ WKH WUDQVODWLRQDO GLVSODFHPHQWV LQ X Q DQG ZGLUHFWLRQ UHVXOWLQJ LQ HOHPHQW GHJUHHV RI IUHHGRP DV VKRZQ LQ )LJ 7KH HOHPHQW VWLIIQHVV LV GHULYHG GLUHFWO\ IURP WKH SK\VLFDO EHKDYLRU RI WKH HOHPHQW GHVFULEHG LQ VHFWLRQ 7KH PDWKHPDWLFDO V\PERO ^` LV XVHG IRU D FROXPQ YHFWRU DQG >@ IRU D PDWUL[ 7KH QRGDO GLVSODFHPHQW FROXPQ YHFWRU ^T`f LV FRPSRVHG RI WUDQVODWLRQDO QRGDO GLVSODFHPHQWV FRUUHVSRQGLQJ WR WKH HOHPHQW GHJUHHV RI IUHHGRP ^T` ^ Xs 9/ 8M 9M :M XN YN ZN 8O Z[ `7 7KH VWUDLQf LV GHILQHG DV WKH DYHUDJH GHIRUPDWLRQ DW WKH FHQWHU RI WKH HOHPHQW DV VKRZQ LQ )LJ 7KH VWUDLQ FROXPQ YHFWRU H`f LV ^H` ^ X4 Y4 Z4 U4 `7 ZKHUH XR XN 8 f 8M XM f YR YN 9 f YL YM f ZR ZN : f Z$ :M f UR YN 9 f / 9M YL f / ZKHUH / 7KH OHQJWK RI WKH HOHPHQW PAGE 50 X4 Y ZT $YHUDJH QRGDO GLVSODFHPHQWV LQ ORFDO X Q ZGLUHFWLRQV U4 7KH UHODWLYH DQJOH FKDQJH DERXW ORFDO ] D[LV 7KHUHIRUH WKH UHODWLRQVKLS EHWZHHQ VWUDLQ DQG QRGDO GLVSODFHPHQWV LV ^H!f >%@f ^T`f 7KH >%@f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`f LV ^V` ^ V[ VQ V] P ` 7KH VWUHVVVWUDLQ UHODWLRQVKLS LV ^V`f >(@f ^H`f ZKHUH 9 ZKHUH FDQ EH UHODWHG WR NQ XVLQJ WKH GHILQLWLRQ RI WKH PRPHQW P LH P VQf/fWff/f rPr A9IfAff 7KXV NMQ NQ9'f /f Wf f /f >9 f /f @ f Wf NQf /f ZKHUH W (OHPHQW WKLFNQHVV 7KLV DVVXPHV WKDW WKHUH LV QR FRXSOLQJ EHWZHHQ WKH VKHDU VWUHVV DQG QRUPDO VWUHVV 7KH HOHPHQW QRGDO IRUFH FROXPQ PDWUL[ 3`f LV FRPSRVHG RI WKH QRGDO IRUFHV VKRZQ LQ )LJ PAGE 54 ^3` ^ 3XL 3QL SZL SXM SQM SZM SXN SQN SZN SXO SQO 3ZO !7 6WUHVV FDQ WKHQ EH UHODWHG WR QRGDO IRUFHV XVLQJ WKH GHILQLWLRQ RI VWUHVV DQG IRUFH HTXLOLEULXP EHWZHHQ WKH WZR VXUIDFHV RI WKH HOHPHQW %\ WKH GHILQLWLRQ RI VWUHVV VQ /Wf 3QN 3QO f f V[ /Wf 3XN 3XO f f V] /Wf 3ZN 3ZO f f P 3QNf/f 3QOf/f f ZKHUH /W /fWf %\ IRUFH HTXLOLEULXP RI WKH WZR VXUIDFHV 3Â 3 DQG 3M 3N f 7R H[SUHVV WKH HOHPHQW QRGDO IRUFHV LQ WHUPV RI WKH VWUHVV ZH XVH (TV f WKURXJK f WR ILQG WKH IRUFH UHFRYHU\ PDWUL[ >)5@ >)5@ JLYHV WKH QRGDO IRUFHV LQ HTXLOLEULXP ZLWK WKH HOHPHQW VWUHVVHV )URP (T f (T ff 3QN /fWfVQf Pf/ 3QN /fWfVQf /fPf PAGE 55 )URP (T f SQM aSQN /fWfVQf /fPf )URP (T f 3QO /fWfVQf 3QN /fWfVQf /fWfVQf /fPff /fWfVQf /fPf )URP (T f SQL SQO /fWfVQf /fPf )URP WKH DVVXPSWLRQ WKDW 3XN 3XA DQG (T f 3XN /fWfV[f 3XO /fWfV[f )URP (T f 3XL SXO /fWfV[f 3XM 3XN /fWfV[f )URP WKH DVVXPSWLRQ WKDW 3ZN 3ZA DQG (T f 3ZN /fWfV]f 3: /fWfV]f PAGE 56 )URP HTQ SZL SZO /! W+6]f SZM SZN /fWfV]f 7KHUHIRUH WKH IRUFHVWUHVV UHODWLRQVKLS LV ^3`f >)5@f ^V`f ZKHUH WKH IRUFH UHFRYHU\ PDWUL[ >)5@f LV /W /W / /W /W /W / /W /W /W / /W /W /W / /W $QG WKLV UHODWLRQVKLS LV IXUWKHU H[SDQGHG XVLQJ WKH VWUHVVVWUDLQ UHODWLRQVKLS DQG WKH VWUDLQQRGDO GLVSODFHPHQW UHODWLRQVKLS DV IROORZV ^3 !f >)5@f >(@f Hff >)5@f >(@f >%@f T`f >%W@f >(@f >%@f Tff 7KHQ ILQDOO\ WKLV FDQ EH V\PEROL]HG DV HTXLOLEULXP HTXDWLRQ PAGE 57 ^3!ffÂ§ >.H@f ^T`f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f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nV RULJLQDO SURJUDP $ WDEOH RI WKH HOHPHQW YHULILFDWLRQ LV VKRZQ LQ )LJ DQG )LJ PAGE 59 /2$',1* 5(68/76 12'( 7+(25< 6,03$/ (5525 ',63 ',63 675(66 1$ 12'( 7+(25< 6,03$/ (5525 ',63 ',63 675(66 1$ 12'( 7+(25< 6,03$/ (5525 ',63 r ',63 675(66 1$ 7+,&.1(66 .V ( .Q ( 12'( 12'( <',63 = ',63 = ',63 < ',63 6457f f f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bf [ H H \ H H [ H H \ H H [ H H \ H H PAGE 63 )LJ &RPELQHG 7HVW 0RGHO IRU $165 PAGE 64 )LJ 7UXVV 0RGHO IRU $165 PAGE 65 &+$37(5 /,1($5 6+(// (/(0(17 (OHPHQW 'HVFULSWLRQ 7KH VKHOO HOHPHQW LV IRUPXODWHG WKURXJK D FRPELQDWLRQ RI WZR GLIIHUHQW HOHPHQWV WKH PHPEUDQH HOHPHQW DQG WKH 0LQGOLQ SODWH EHQGLQJ HOHPHQW >@ 7KH 0LQGOLQ SODWH HOHPHQW LV GLIIHUHQW IURP WKH .LUFKKRII SODWH HOHPHQW LQ WKDW WKH IRUPHU DOORZV WUDQVYHUVH VKHDU GHIRUPDWLRQ ZKLOH WKH ODWWHU GRHV QRW 7KH FRPPRQ SRUWLRQV RI WKH IRUPXODWLRQ RI WZR HOHPHQWV DUH )RUPDWLRQ RI WKH VKDSH IXQFWLRQV )RUPDWLRQ RI WKH LQYHUVH RI -DFRELDQ PDWUL[ 7KHVH SURFHVVHV FDQ EH GRQH DW WKH VDPH WLPH 7KH IRXU WR QLQHQRGH VKDSH IXQFWLRQV DQG WKHLU GHULYDWLYHV LQ UV VSDFH FDQ EH IRUPHG DQG WKHQ WUDQVIRUPHG LQWR [\VSDFH WKURXJK WKH LQYHUVH RI -DFRELDQ PDWUL[ )RUPXODWLRQ RI 6KDSH )XQFWLRQV 7KH IRUPXODWLRQ RI VKDSH IXQFWLRQV VWDUWV ZLWK WKUHH EDVLF VHWV RI VKDSH IXQFWLRQV VKRZQ LQ )LJ PAGE 66 7KH ELOLQHDU VKDSH IXQFWLRQV IRU IRXUQRGH HOHPHQW 7KH OLQHDUTXDGUDWLF VKDSH IXQFWLRQV IRU QRGHV ILYH WR HLJKW RI WKH HLJKWQRGH HOHPHQW 7KH EXEEOH VKDSH IXQFWLRQ IRU QRGH QLQH RI QLQHQRGH HOHPHQW 7KHVH VKDSH IXQFWLRQV FDQ EH IRUPXODWHG GLUHFWO\ IURP WKH ORFDO FRRUGLQDWHV RI WKH HOHPHQW QRGHV WKURXJK WKH PXOWLSOLFDWLRQ RI WKH HTXDWLRQV RI WKH OLQHV ZKLFK KDYH ]HUR YDOXHV LQ WKH DVVXPHG GLVSODFHPHQW VKDSHV DQG WKH VFDOH IDFWRU WR IRUFH WKH VKDSH IXQFWLRQ YDOXH WR RQH DW WKH QRGH IRU ZKLFK WKH VKDSH IXQFWLRQ LV IRUPHG 7KH GHULYDWLYH RI HDFK VKDSH IXQFWLRQ ZLWK UHVSHFW WR U DQG V LV WKHQ HYDOXDWHG IURP WKH VKDSH IXQFWLRQ H[SUHVVHG LQ WHUPV RI U DQG V ,I QRGH QLQH H[LVWV WKH YDOXH DW QRGH QLQH RI VKDSH IXQFWLRQV RQH WR HLJKW PXVW EH VHW WR ]HUR 7KH YDOXH RI WKH ELOLQHDU VKDSH IXQFWLRQV IRU D IRXUQRGH HOHPHQW DW QRGH QLQH LV RQH IRXUWK DQG WKH YDOXH RI WKH OLQHDUTXDGUDWLF VKDSH IXQFWLRQV IRU WKH ILYH WR HLJKWQRGH HOHPHQW DW WKH QRGH QLQH LV RQH KDOI 7KLV FDQ EH IRUFHG WR ]HUR XVLQJ WKH EXEEOH VKDSH IXQFWLRQ RI WKH QLQHQRGH HOHPHQW EHFDXVH WKLV VKDSH IXQFWLRQ KDV WKH YDOXH RI RQH DW QRGH QLQH DQG ]HUR DW DOO RWKHU QRGHV 7KHUHIRUH WKH PRGLILFDWLRQ LV WKH VXEWUDFWLRQ RI RQH IRXUWK RI WKH YDOXH WKH EXEEOH VKDSH IXQFWLRQ KDV DW QRGH QLQH IURP WKH HDFK VKDSH IXQFWLRQ IRU WKH FRUQHU QRGHV DQG WKH VXEWUDFWLRQ RI RQH KDOI RI WKH PAGE 67 YDOXH RI WKH EXEEOH VKDSH IXQFWLRQ RI QRGH QLQH IRU WKH QRGHV ILYH WR HLJKW ZKLFKHYHU H[LVWV ,I DQ\ RI WKH FHQWHU QRGHV RQ WKH HGJH RI WKH HOHPHQW DQ\ RQH RI QRGHV ILYH WR HLJKWf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f f 6) f 6) 6) f f 6) f )LJ )RUPDWLRQ RI 6KDSH )XQFWLRQV PAGE 70 7KH ,QYHUVH RI -DFRELDQ 0DWUL[ :KLOH WKH JHQHULF GLVSODFHPHQWV DUH H[SUHVVHG LQ WHUPV RI UVFRRUGLQDWH WKH SDUWLDO GLIIHUHQWLDO ZLWK UHVSHFW WR WKH [\FRRUGLQDWH LV QHHGHG IRU WKH FDOFXODWLRQ RI VWUDLQ FRPSRQHQWV 7KXV WKH LQYHUVH RI WKH -DFRELDQ PDWUL[ PXVW EH FDOFXODWHG 7KLV FDQ GLUHFWO\ EH IRXQG IURP WKH FKDLQ UXOH XVLQJ WKH QRWDWLRQ DEf GHILQHG DV WKH SDUWLDO GLIIHUHQWLDO RI IXQFWLRQ D ZLWK UHVSHFW WR WKH YDULDEOH E IRU VLPSOLFLW\ I[ IUfU[f I6f V[f I\ IUfU\f IVfV\f ,Q PDWUL[ IRUP I [ U[ V[ IU 7 M n IU I< U\ V\ IV 7 fÂ§ 7 I V 7KH LQYHUVH RI -DFRELDQ PDWUL[ %XW WKH WHUPV LQ WKH LQYHUVH RI WKH -DFRELDQ PDWUL[ DUH QRW UHDGLO\ DYDLODEOH EHFDXVH WKH UVFRRUGLQDWH FDQQRW EH VROYHG H[SOLFLWO\ LQ WHUPV RI [\FRRUGLQDWH 2Q WKH RWKHU KDQG IRU WKH LVRSDUDPHWULF IRUPXODWLRQ WKH JHRPHWU\ LV LQWHUSRODWHG XVLQJ WKH QRGDO FRRUGLQDWH YDOXHVFRQVWDQWVf DQG WKH GLVSODFHPHQW VKDSH IXQFWLRQV LQ WHUPV RI U DQG V 7KXV WKH JHQHULF FRRUGLQDWH [ DQG \ FDQ EH H[SUHVVHG LQ U DQG V PAGE 71 HDVLO\ DQG H[SOLFLW SDUWLDO GLIIHUHQWLDOV RI [ DQG \ ZLWK UHVSHFW WR U DQG V FDQ EH SHUIRUPHG 7KHUHIRUH WKH -DFRELDQ PDWUL[ LV FRPSXWHG DQG WKHQ LQYHUWHG 7KH -DFRELDQ PDWUL[ LV GHULYHG E\ WKH FKDLQ UXOH IU I[f [Uf I\f \Uf IV I[f [Vf I\f \ Vf ,Q PDWUL[ IRUP IU [ U \U I[ O! I ; I V [ V \V I\ I\ -DFRELDQ PDWUL[ QQ /HW ( EH ( L O ZKHUH QQ QXPEHU RI QRGHV WR f )URP JHRPHWULF LQWHUSRODWLRQ HTXDWLRQV [ ( ILr[Â \ ( ILr\L 7KH WHUPV LQ WKH -DFRELDQ PDWUL[ DUH OL [ U V ILr[LfU ( IUf r [Lf \U 6 ILr\Lf ( ILUf r PAGE 72 [Ln PAGE 73 HOHPHQW GRPDLQ DQG LV GLIIHUHQWLDEOH RYHU WKH RSHQ GRPDLQ RI WKH HOHPHQW 7KH VKDSH IXQFWLRQ LV DOVR WKH FRQWULEXWLRQ RI GLVSODFHPHQW RI D QRGH IRU ZKLFK WKH VKDSH IXQFWLRQ KDV EHHQ GHILQHG WR WKH JHQHULF GLVSODFHPHQW 7KXV WKH JHQHULF GLVSODFHPHQW DW DQ DUELWUDU\ SRLQW FDQ EH IRXQG E\ VXPPLQJ XS DOO WKH FRQWULEXWLRQV RI DOO WKH QRGHV RI WKH HOHPHQW 7KH GLVSODFHPHQW LQWHUSRODWLRQ HTXDWLRQV DUH X IÂ XÂ 9 e Yc ,Q WKH LVRSDUDPHWULF IRUPXODWLRQ WKH JHRPHWU\ LV LQWHUSRODWHG XVLQJ WKH VDPH VKDSH IXQFWLRQ DVVXPHG IRU WKH GLVSODFHPHQW LQWHUSRODWLRQ 7KHUHIRUH WKH JHRPHWU\ LQWHUSRODWLRQ LV [ ( IÂ \ ] IL PAGE 74 7KH WKUHH LQSODQH VWUDLQ FRPSRQHQWV IRU D PHPEUDQH HOHPHQW DUH F e[ H\ A[\ !7 7KHVH VWUDLQ FRPSRQHQWV FDQ EH IRXQG WKURXJK WKH SDUWLDO GLIIHUHQWLDOV RI WKH JHQHULF GLVSODFHPHQWV ZLWK UHVSHFW WR [\FRRUGLQDWHV H[ X[ H\ Y\ [\ 8< Y[ 8VLQJ WKH LQYHUVH RI WKH -DFRELDQ PDWUL[ WKH VWUDLQ FRPSRQHQWV FDQ EH HYDOXDWHG H[ 8 ; XUfU[f XVfV[f XUf-Bf XVf-Bf 6IL8LfUf -Qf 6IL8LfAfAf 6>ILIUfU[f ILVfV[f@ XÂ H\ Y \ YUfU\f YVfV\f YUf-f YVf-ff VIL9sfUf-Vf VIL9LfVf-Bf 6> ILIUf U\f IAVMA\f@ PAGE 75 A[\ X\ Y[ > XUf U\f XVfV\f@ >YUfU[f YVfV[f@ >VIAf Uf 6IALf Vf -Bf@ > =IM9Lf Uf -LLf 6IALf Vf -f @ (>ILIUfU\f ILVfV\f@ XÂ n 6>ILUUfU[f ILVfV[f@ Ys 1HZ QRWDWLRQV DUH LQWURGXFHG KHUH WR VLPSOLI\ WKH HTXDWLRQV 7KHVH DUH DA DQG DQG GHILQHG DV IROORZV U[fILUf VI[fILUVf IÂ[ r!L U\fILUf V\fIsVf IL\ 7KHQ WKH VWUDLQ WHUPV DERYH EHFRPH H[ 6DAXA 6IA[ XA f\ (EL9L VIL\ 9L [\ (EM8L 6DAL 6IL\ XL (IA[ Ys ,Q PDWUL[ IRUP H[ DL XL e\ Â‘A[\ V EL EL DL YL ,Q V\PEROLF IRUP >HM 6>%L@>TL@ PAGE 76 ZKHUH DL ILn; EL V EL DL L +f + ILn[ DQG >TM@ 9 L 7KHUHIRUH WKH VWUDLQ DW DQ DUELWUDU\ SRLQW ZLWKLQ DQ HOHPHQW LV >H@ fÂ§ >%M>TM>%@>T@ fffr >%J@ >TJ@ a > % % % % % %J %" %J %J r r r T r r T A T 7KH VL]H RI WKH YHFWRUV DQG PDWUL[ DUH >mf@ >%f@>TOf@ ,Q WKH DFWXDO FDOFXODWLRQ WKLV FDQ EH GRQH E\ VXPPLQJ XS WKH >%M>TMRYHU DOO WKH QRGHV IRU WKH JLYHQ FRRUGLQDWHV RI WKH SRLQW XQGHU FRQVLGHUDWLRQ LH WKH FRRUGLQDWHV RI RQH RI WKH LQWHJUDWLRQ SRLQWV PAGE 77 7KH VWUHVVHV FRUUHVSRQGLQJ WR WKH VWUDLQV DUH 7 ^ ^ 7[ &7\ 7;\ ` 7KH VWUHVVVWUDLQ UHODWLRQVKLS RI DQ LVRWURSLF PDWHULDO >(@ ( c ( ZKHUH ( ( ( S f ( ( A( L f ( ZKHUH ( PAGE 78 >.L@ %" ( %L G9 9 $V >%@ DQG >(@ DUH FRQVWDQW DERXW ] WKH LQWHJUDWLRQ WKURXJK WKH WKLFNQHVV IURP W WR W FDQ EH SHUIRUPHG RQ ] RQO\ DQG \LHOGV >.L@ > %LWf(%L@ G$ $ 7 > %L ( %L@ G$ -D ZKHUH ( W( 7KH VL]H RI PHPEUDQH HOHPHQW VWLIIQHVV LV E\ >.@ > ( @ > % % % % @ G9 f f f (TXLYDOHQW QRGDO ORDGV GXH WR ERG\ IRUFHV RQ WKH PHPEUDQH HOHPHQW DUH FDOFXODWHG DV SE I7E G9 f U -Y I7E_M_ GU GV LQ ZKLFK E! ^ E `7 RU E\ f7 RU ^ E[ f7 LQ PAGE 79 DFFRUGDQFH ZLWK WKH GLUHFWLRQ RI JUDYLW\ LQ WKH FRRUGLQDWH V\VWHP XVHG 7KH QRQ]HUR TXDQWLWLHV E[ E\ RU E] UHSUHVHQW WKH ERG\ IRUFH SHU XQLW DUHD LQ WKH GLUHFWLRQ RI DSSOLFDWLRQ (TXLYDOHQW ORDGV FDXVHG E\ LQLWLDOWHPSHUDWXUHf VWUDLQV DUH 3 %7(H G9 9 U U %7( H R ,-, GU GV -L ZKHUH ^H` ^ f[[2 H\\2 ^ D$7 m$7 `7 `7 3ODWH %HQGLQJ (OHPHQW 7KH IRUPXODWLRQ RI WKH SODWH EHQGLQJ HOHPHQW XVHG IRU WKH LPSOHPHQWDWLRQ KDV IROORZHG WKH SURFHGXUHV VKRZQ RQ SDJHV WKURXJK LQ UHIHUHQFH >@ 7KH ^ ` V\PERO ZLOO EH XVHG IRU WKH FROXPQ YHFWRUV 0DQ\ SODWH EHQGLQJ HOHPHQWV KDYH EHHQ SURSRVHG 7KH PRVW FRPPRQO\ XVHG DUH .LUFKKRII SODWH HOHPHQWV DQG 0LQGOLQ SODWH HOHPHQWV PAGE 80 .LUFKKRII WKHRU\ LV DSSOLFDEOH WR WKLQ SODWHV LQ ZKLFK WUDQVYHUVH VKHDU GHIRUPDWLRQ LV QHJOHFWHG 7KH DVVXPSWLRQV PDGH RQ WKH GLVSODFHPHQW ILHOG DUH $OO WKH SRLQWV RQ WKH PLGSODQH] f GHIRUP RQO\ LQ WKH WKLFNQHVV GLUHFWLRQ DV WKH SODWH GHIRUPV LQ EHQGLQJ 7KXV WKHUH LV QR VWUHWFKLQJ RI PLGSODQH $ PDWHULDO OLQH WKDW LV VWUDLJKW DQG QRUPDO WR WKH PLGSODQH EHIRUH ORDGLQJ LV WR UHPDLQ VWUDLJKW DQG QRUPDO WR WKH PLGSODQH DIWHU ORDGLQJ 7KXV WKHUH LV QR WUDQVYHUVH VKHDU GHIRUPDWLRQ FKDQJH LQ DQJOH IURP WKH QRUPDO DQJOHf $OO WKH SRLQWV QRW RQ WKH PLGSODQH KDYH GLVSODFHPHQW FRPSRQHQWV X DQG Y RQO\ LQ WKH [ DQG \ GLUHFWLRQ UHVSHFWLYHO\ 7KXV WKHUH LV QR WKLFNQHVV FKDQJH WKURXJK WKH GHIRUPDWLRQ 6WUDLQ HQHUJ\ LQ WKH .LUFKKRII SODWH LV GHWHUPLQHG HQWLUHO\ E\ LQSODQH VWUDLQV H[ H\ DQG ;\ ZKLFK FDQ EH GHWHUPLQHG E\ WKH GLVSODFHPHQW ILHOG Z[\f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f LV DOORZHG 7KH PRWLRQ RI D SRLQW RQ WKH PLGSODQH LV QRW JRYHUQHG E\ WKH VORSHV Z[f DQG Z\f DV LQ .LUFKKRII WKHRU\ 5DWKHU LWV PRWLRQ GHSHQGV RQ URWDWLRQV ; DQG \ RI WKH OLQHV WKDW ZHUH QRUPDO WR WKH PLGSODQH RI WLUH XQGHIRUPHG SODWH 7KXV ; DQG \ DUH LQGHSHQGHQW RI WKH ODWHUDO GLVSODFHPHQW Z LH WKH\ DUH QRW HTXDO WR Z[f RU Z\f ,W LV QRWHG WKDW LI WKH WKLQ SODWH OLPLW LV DSSURDFKHG ;= \= EHFDXVH WKHUH LV QR WUDQVYHUVH VKHDU GHIRUPDWLRQ ,Q WKLV FDVH WKH DQJOHV ; DQG \ FDQ EH HTXDWHG WR WKH Z[f DQG Z\f QXPHULFDOO\ EXW WKH VHFRQG DVVXPSWLRQ VWLOO KROGV 7KH VWLIIQHVV PDWUL[ RI D 0LQGOLQ SODWH HOHPHQW LV FRPSRVHG RI D EHQGLQJ VWLIIQHVV >NE@ DQG D WUDQVYHUVH VKHDU VWLIIQHVV >NV@ >NE@ LV DVVRFLDWHG ZLWK LQSODQH VWUDLQV f[ &\ DQG ;\ >NV@ LV DVVRFLDWHG ZLWK WUDQVYHUVH VKHDU VWUDLQV ;= DQG \= $V WKHVH WZR JURXSV RI VWUDLQV DUH XQFRXSOHG LH RQH JURXS RI WKH VWUDLQV GR QRW SURGXFH WKH RWKHU JURXS RI VWUDLQV WKH HOHPHQW VWLIIQHVV FDQ EH VKRZQ DV >@ >N@ 7 %E(%E f G$ %J(%Jf G$ EHFDXVH %E(%V %J(%E IURP XQFRXSOLQJ FRUUHVSRQGLQJ ( f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Â \Â ` 7KH URWDWLRQV DUH FKRVHQ LQGHSHQGHQWO\ RI WKH WUDQVYHUVH GLVSODFHPHQW DQG DUH QRW UHODWHG WR LW E\ GLIIHUHQWLDWLRQ 7KXV WKH WUDQVYHUVH VKHDU VWUDLQV ;= DQG \= DUH FRQVLGHUHG LQ WKH IRUPXODWLRQ UHVXOWLQJ LQ ILYH VWUDLQ FRPSRQHQWV 7KH JHQHULF GLVSODFHPHQWV DUH GHILQHG DV WKUHH WUDQVODWLRQDO GLVSODFHPHQWV DQG GHQRWHG DV ^ X Y Z `7 %\ WKH ZRUG JHQHULF LW LV PHDQW WKDW WKH GLVSODFHPHQW LV PHDVXUHG DW DQ DUELWUDU\ SRLQW ZLWKLQ DQ HOHPHQW 7KHVH JHQHULF GLVSODFHPHQWV DUH GLIIHUHQW TXDQWLWLHV IURP WKH QRGDO PAGE 83 GLVSODFHPHQWV DQG WKHUHIRUH PXVW EH UHODWHG WR WKH QRGDO GLVSODFHPHQWV 7KH JHQHULF GLVSODFHPHQWV X DQG Y FDQ EH FDOFXODWHG DV IXQFWLRQV RI WKH JHQHULF RXWRISODQH URWDWLRQV XVLQJ WKH VPDOO VWUDLQURWDWLRQf DVVXPSWLRQ 7KH UHODWLRQVKLS EHWZHHQ JHQHULF GLVSODFHPHQWV DQG URWDWLRQ LV VKRZQ LQ )LJ X ]\ Y fÂ§]r[ 7KH JHQHULF GLVSODFHPHQWV ; DQG \ FDQ EH IRXQG XVLQJ WKH DVVXPHG GLVSODFHPHQW VKDSH IXQFWLRQV DQG WKH FRUUHVSRQGLQJ QRGDO GLVSODFHPHQWV ;A DQG \Â 7KH JHQHULF GLVSODFHPHQW Z GRHV QRW QHHG DQ\ FRQYHUVLRQ EHFDXVH LW FRUUHVSRQGV WR WKH QRGDO GLVSODFHPHQW ZÂ ,Q WKH LVRSDUDPHWULF IRUPXODWLRQ WKH JHRPHWU\ LV LQWHUSRODWHG XVLQJ WKH VDPH VKDSH IXQFWLRQ DVVXPHG IRU WKH GLVSODFHPHQW LQWHUSRODWLRQ 7KH GLVSODFHPHQW LQWHUSRODWLRQ LV r[ IL r[L \ fÂ§ = Ie A\L : = IA :Â 6LPLODUO\ WKH JHRPHWULF LQWHUSRODWLRQ LV [ = IA [A \ V IL \L PAGE 84 =HUR 6KHDU 6WUDLQ 2QH 3RLQW *DXVV ,QWHJUDWLRQ 7ZR 3RLQW *DXVV ,QWHJUDWLRQ )LJ 6KHDU 6WUDLQV DW *DXVV 3RLQW Vf PAGE 85 3RVLWLYH VPDOO URWDWLRQDO DQJOH DERXW \D[LV JLYHV SRVLWLYH JHQHULF GLVSODFHPHQW LQ [GLUHFWLRQ X f 6KRZQ LV []SODQH 3RVLWLYH VPDOO URWDWLRQDO DQJOH DERXW [D[LV JLYHV QHJDWLYH JHQHULF GLVSODFHPHQW LQ \GLUHFWLRQ Y f 6KRZQ LV \]SODQH )LJ 'LVSODFHPHQWV GXH WR 5RWDWLRQV PAGE 86 ZKHUH VKDSH IXQFWLRQ IRU QRGH L [A FRRUGLQDWHV RI QRGH L 7KHUHIRUH F ,, 1 ,, = IL r\L Y ]H[ = IL r[L Z IL ZL 7KH ILYH VWUDLQ FRPSRQHQWV IRU SODWH EHQGLQJ HOHPHQW DUH ^ H[ H\ ;\ \[] \] `7 7KHVH VWUDLQ FRPSRQHQWV FDQ EH IRXQG WKURXJK WKH SDUWLDO GLIIHUHQWLDOV RI WKH JHQHULF GLVSODFHPHQWV ZLWK UHVSHFW WR [\FRRUGLQDWHV H[ 8; H\ Y\ ;\ X< 9; ;= X] :; \ Y] Z\ 8VLQJ WKH LQYHUVH RI WKH -DFRELDQ PDWUL[ IRXQG WKH VWUDLQ FRPSRQHQWV FDQ EH HYDOXDWHG H[ 8 ; =\f ; XUfU[f XVfV[f XUf-LLf XVf-Bf PAGE 87 7P >[nVf Vn7Mf [nDfDn7Mf@ W [B=7Uf AVn7MAff [B,Uf WDYM7Vf f @ >7=,UfVn:MVff 7B7,Uf 8n :MVff @ >[nVfVnPf [nMfDnZf@ [nP [nQ ]nA]f [nZf ]nQf ][Â [[ >[nVfVn7Mf [nDf @V ] >$nVfVn7Mf $nMf MFn7Mf @V ] Q7B=7Uf Vn 7A7MV]ff 7B;,Uf 8n I9MV]ff @ > [B=Uf 67AA7M6]ff [B7=Uf 8n f @ > [nVf VnDf [nMf MnDf @ >$nVfVnQf $nMf DnQf@ [f[]af $A]f [nD $nQ $[O 7[ >$nVfVn7Mf $nDfMn7Mf@] ] [BUf67;A7MA]ff [B7]If Â‘rn7;7-6=ff [B]]UfVn$f [B[=UfDn$f $nVf VnDf $nMf MnDf $n ;=af $nD >[nVfVn7Mf [nMfD;MfV ] [B=,Uf V 7$H7V]ff [B7,Uf 8n 7$II7V]ff PAGE 88 \] 9=f :\f =;f= Z\ ;f Z\ (IALf >ZUfU\f ZVfV\f@ >VIAf Uf -VLf 6ILZLf Vf -f@ VILA[Lf >6ILAUMZLf -/f VILVfZLf -Bf @ 6IA[Lf 6> IsUf U\f ILVfV\f@ :L 1HZ QRWDWLRQV DUH LQWURGXFHG KHUH WR VLPSOLI\ WKH HTXDWLRQV 7KHVH DUH DA DQG DQG GHILQHG DV IROORZV DL UU[fILUf V[fILIVf IL[ r!L UM\+IAUf VA+IAVf IL\ 7KHQ WKH VWUDLQ WHUPV DERYH EHFRPH e[ = DLA\L e\ ] ELr[L [\ ] 6EIA\L ] 6DAA[L [] 6IIA\L (DM:A 1 ,, (IIA[L 6EM:A ,Q PDWUL[ IRUP H[ H\ [\ ( [] O\] I L Z L "[L 9L ,Q V\PEROLF IRUP >H@ (>%M>TMM PAGE 89 ZKHUH ]DA ]EA ]DA ]EL DL IL EL IL RU ]IÂ[ f ]I$\ ]IM/[ ]IL< IL[ IL ILn< IL DQG >TMM ZL r[L 9L 7KHUHIRUH WKH VWUDLQ DW DQ DUELWUDU\ SRLQW ZLWKLQ HOHPHQW LV >m@ >%/+T/@ >%@>T@ >%J@ >T@ f > % % % % % %J % %J %J A T DQ PAGE 90 7KH VL]H RI WKH YHFWRUV DQG PDWUL[ DUH >HOf@ >%f@>TOf@ ,Q WKH DFWXDO FDOFXODWLRQ WKLV FDQ EH GRQH E\ VXPPLQJ XS WKH >%M>TÂ@ RYHU DOO WKH QRGHV IRU WKH JLYHQ FRRUGLQDWHV RI WKH SRLQW XQGHU FRQVLGHUDWLRQ LH WKH FRRUGLQDWHV RI RQH LQWHJUDWLRQ SRLQW 7KH VWUHVVHV FRUUHVSRQGLQJ WR WKH VWUDLQV DUH ^ &7 ` ^ &7; \ 7[ \ 7[] 7\= ` A 7KH VWUHVVVWUDLQ UHODWLRQVKLS RI DQ LVRWURSLF PDWHULDO LV ( ( ZKHUH ( ( ( Q f (A A A( c3f ( ( ( ZKHUH [ 3RLVVRQnV UDWLR VKHDU PRGXOXV ( rO[ff >(@ ( n - PAGE 91 7KH IRUP IDFWRU IRU WKH ( DQG ( WHUPV LV SURYLGHG WR DFFRXQW IRU WKH SDUDEROLF GLVWULEXWLRQ RI WKH WUDQVYHUVH VKHDU VWUHVV U][ RYHU D UHFWDQJXODU VHFWLRQ 7KLV IRUP IDFWRU FDQ EH VKRZQ IURP WKH GLIIHUHQFH LQ GHIOHFWLRQV RI D FDQWLOHYHU EHDP DW LWV IUHH HQG >@ /HW D EHDP KDYH D UHFWDQJXODU FURVV VHFWLRQ RI GLPHQVLRQV E E\ W ZLWK D OHQJWK RI / ,I 3 LV WKH WUDQVYHUVH VKHDU IRUFH WKHQ WKH SDUDEROLF GLVWULEXWLRQ RI WKH WUDQVYHUVH VKHDU VWUHVV U][ LV W][ 3EWfW ]f ZKHUH ] DW WKH QHXWUDO D[LV 7KHQ WKH WUDQVYHUVH VKHDU VWUDLQ HQHUJ\ IURP WKH SDUDEROLF GLVWULEXWLRQ FDQ EH FDOFXODWHG E\ 8 V f 9 U][9*f G9 f >3EWfW >f3EWf@* ]ff *@ G9 W ]f G$G] DUHDf>f3EWf E/>f3EWf@* 3/EW*f @* W ]f W ]f G] G] PAGE 92 :KLOH WKH WUDQVYHUVH VKHDU VWUDLQ HQHUJ\ IURP WKH FRQVWDQW GLVWULEXWLRQ LV 8V f U][9*f G9 9 f 3EWf *f G9 >f3EWf*@EW/f 3/EW*f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Â@ DV IROORZV ]IL[ ]IsI\ ]ILW[ ]IM\ ILU; IL ILn< IL 7KH VXEPDWULFHV DUH QDPHG DV IROORZV %L$ f+ 34 1 %L% %L% 7KH HOHPHQW VWLIIQHVV FDQ EH FDOFXODWHG 7 YROXPH LQWHJUDWLRQ RI f(f%Af PDWUL[ LV WR EH SDUWLWLRQHG DV IROORZV >(@ U ( ( ( ( R R H ( ( 7KH VXEPDWULFHV DUH QDPHG DV IROORZV ( ( % WKURXJK WKH 7KXV WKH >(@ PAGE 94 7KHQ WKH VWLIIQHVV RI WKH HOHPHQW LV >.s@ 7 %L ( %MO G9 ]%L$ %L% r G9 >] %D (D %D@ >%J (J %J@ $V >%@ DQG >(@ DUH FRQVWDQW DERXW ] WKH LQWHJUDWLRQ WKURXJK WKH WKLFNQHVV IURP W WR W FDQ EH SHUIRUPHG RQ ] RQO\ DQG \LHOGV >.s@ ZKHUH > 6$Wf($$ %JWf(J%J@ G$ & EDHDED EEHEEE@ G$ ($ B Wf($ DQG (J W(J 7KHQ >.MFDQ EH UHZULWWHQ DV PDWUL[ HTXDWLRQ DV IROORZV 7 7 HD U pL$ L >.M@ > %L$ %L% A B G$ f HE / %L% 7 %A( %Â G$ PAGE 95 7KH VL]H RI SODWH HOHPHQW VWLIIQHVV ZLOO EH E\ >.@ > ( @ f > % % % % % @ G9 f f 7KH VWUDLQQRGDO GLVSODFHPHQW PDWUL[ IURP ZKLFK WKH FRQVWDQW WKLFNQHVV LV WDNHQ RXW LV GHILQHG DV >%A@ >%s@ IL[ IL n< IL\ ILI; I IL[ L\ IL >%L$@ / %L% M (TXLYDOHQW QRGDO ORDGV GXH WR ERG\ IRUFHV RQ WKH SODWH FDOFXODWHG DV f r _r I7E G9 9 f I7E_M_ GU GV LQ ZKLFK Ef ^ E] `7 RU ^ E\ `7 RU ^ E[ 7 f ` P DFFRUGDQFH ZLWK WKH GLUHFWLRQ RI WKH JUDYLW\ P WKH FRRUGLQDWH V\VWHP XVHG 7KH QRQ]HUR TXDQWLWLHV E[ E\ RU E UHSUHVHQW WKH ERG\ IRUFH SHU XQLW DUHD LQ WKH GLUHFWLRQ RI DSSOLFDWLRQ PAGE 96 (TXLYDOHQW ORDGV FDXVHG E\ LQLWLDO VWUDLQV DUH 3 %7(H G9 9 U U W(Â_M_ GU GV -O ZKHUH 7 ^ Â[[2 A\\2 A[\2 ` ^ D $7 D$7 `7 7KH VWUHVVHV FDQ EH FDOFXODWHG IURP WKH HTXDWLRQ 0 >(@>@ 7KH FRUUHVSRQGLQJ JHQHUDOL]HG VWUHVVHV LI GHVLUHG PD\ EH FRPSXWHG IURP 0 ^ 0[[ 0\\ 0[\ 4[ 4\ `7 Â T W! f ,W LV QRWHG WKDW WKH JHQHUDOL]HG VWUHVVHV DUH DFWXDOO\ PRPHQW DQG VKHDU IRUFHV DSSOLHG SHU XQLW OHQJWK RI WKH HGJH RI WKH SODWH HOHPHQW 7KHUHIRUH WKHVH FDQ DOVR EH WXUQHG LQWR FRPPRQ VWUHVVHV XVLQJ WKH IRUPXODWLRQ IRU WKH EHQGLQJ VWUHVV FDOFXODWLRQ 7KH PRPHQW RI LQHUWLD IRU WKH XQLW OHQJWK RI WKH SODWH LV W 7KHQ WKH LQSODQH VWUHVV DW D SRLQW DORQJ WKH WKLFNQHVV FDQ EH FDOFXODWHG DV D 0] 0Wf Wf 0 W PAGE 97 7KH WUDQVYHUVH VKHDU VWUHVVHV FDQ EH IRXQG DV U 4 W %XW WKLV PD\ EH PXOWLSOLHG E\ D IDFWRU RI WR JHW WKH PD[LPXP VKHDU VWUHVV DW D SRLQW RQ D QHXWUDO VXUIDFH EHFDXVH WKH WUDQVYHUVH VKHDU VWUHVVHV VKRZ SDUDEROLF GLVWULEXWLRQ ZKLOH WKH FDOFXODWHG VWUHVVHV DUH DYHUDJH VWUHVVHV FRPLQJ IURP WKH DVVXPSWLRQ RI D FRQVWDQW WUDQVYHUVH VWUDLQ DORQJ WKH HOHPHQW ] D[LV PAGE 98 &+$37(5 121/,1($5 6+(// (/(0(17 ,QWURGXFWLRQ 7KH QRQOLQHDULWLHV LQFOXGHG LQ WKH IRUPXODWLRQ RI WKH 0LQGOLQ IODW VKHOO HOHPHQW DUH WKRVH GXH WR ODUJH GLVSODFHPHQWV DQG WKRVH GXH WR LQLWLDO VWUHVV HIIHFWVJHRPHWULF QRQOLQHDULW\f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` LV XVHG IRU D FROXPQ PDWUL[ D YHFWRUf DQG WKH V\PERO >@ LV XVHG IRU D PDWUL[ RI PXOWLSOH FROXPQV DQG URZV WKURXJKRXW WKH FKDSWHU PAGE 99 (OHPHQW )RUPXODWLRQ 7KH JHQHULF GLVSODFHPHQWV RI 0LQGOLQ W\SH VKHOO HOHPHQW DUH WUDQVODWLRQDO GLVSODFHPHQWV ^X Y Zf7 DQG GHQRWHG DV ^8` 7KH GLVSODFHPHQWV DQG URWDWLRQV DW D SRLQW RQ WKH PLGSODQH DUH X4 Y4 Z4 ; \f7 DQG GHQRWHG DV ^84` 7KH JHQHULF GLVSODFHPHQWV FDQ EH H[SUHVVHG LQ WHUPV RI WKH PLGSODQH GLVSODFHPHQWV DQG ] DV X X[\f ]\[\f Y Y4[\f ][[\f f Z Z4[\f 7KH OLQHDUL]HG LQFUHPHQWDO VWUDLQ IURP (T f LV HLM r XLM XML 6FL XNM WXNM XNL! f 7KLV HTXDWLRQ FDQ EH ZULWWHQ RXW IRU WKH VWUDLQ WHUPV WR EH XVHG IRU VKHOO HOHPHQW XVLQJ WKH JHQHULF GLVSODFHPHQWV X Y Zf7 H[[ ; G LL X[ 8n; WYn; 9n; WZn; :[ H\\ X\ A\ 8\ WYn\ Y\ WZ Z \ :\ H[\ 8\ 9n; 8\ WYn; 9n \ :\ X[ WXn\ Yn[ WYn \ :[ f H[] X}] :n; 9MF X] WYn; 9n ] WZn[ :n= 8}[ WX] 9n; WY ] Zn[ 9] f H\] 9n= :\ 9\ 8= AW\ YI ] WA\ :] Xn\ WX= 9\ IF9 ] Z\ WZ= f f PAGE 100 7KH GHULYDWLYHV RI LQSODQH GLVSODFHPHQWV X DQG Y ZLWK UHVSHFW WR [ \ DQG ] DUH DVVXPHG WR EH VPDOO DQG WKXV WKH VHFRQG RUGHU WHUPV RI WKHVH TXDQWLWLHV FDQ EH LJQRUHG WKURXJK YRQ .DUPDQ DVVXPSWLRQ IURP (TV f > @ )XUWKHUPRUH WKH WUDQVYHUVH GLVSODFHPHQW Z LV LQGHSHQGHQW RI ] IRU WKH VKHOO HOHPHQW ZKLFK PHDQV WKDW Z] LV ]HUR 7KHQ (TV f FDQ EH UHGXFHG WR H[[ Xn[ WZn[ :n; G L !L f \ IFZ !L cV S fÂ§ m;\ !L G 9[ : \ :; f 3 V H[] X] :[ f S fÂ§ H\] 9n= c f 7KH LQFUHPHQWDO *UHHQnV VWUDLQV VRPHWLPHV FDOOHG HQJLQHHULQJ VWUDLQV FDQ WKHQ EH VKRZQ DV e[[ H[[ 8n; :n; :n; ! ,, r X\ :\ [\ H[\ !L G ,, 9[ :\ :; [] H[] 8= :[ \] H\] 9n= Z\ ,W LV QRWHG WKDW WKH OLQHDUL]HG QRQOLQHDU VWUDLQV DUH OHIW RQO\ IRU LQSODQH VWUDLQ WHUPV %\ VXEVWLWXWLQJ (TV f LQWR (TV f WKH *UHHQnV VWUDLQ FDQ EH H[SUHVVHG LQ WHUPV RI PLGSODQH GLVSODFHPHQWV PAGE 101 e[[ 8n; =L\; :n; :n; f\\ F r :4 \ :T I \ [\ XRn\ =\ I\ 94; =; WZ ZRn [ ZRn\ ZRn[ ZRn\ [] \ Z* [ I\] a[ ZRn\ 7KLV FDQ EH VLPSOLILHG DV ^HS` =^HE` ^r` ^` ^` ^HV` ^ I ` ZKHUH ^H` ^H[[ f\\ ;\ A[] A\]f ,QFUHPHQWDO VWUDLQV IRU VKHOO HOHPHQW ^HO` ^e;; A\\ ;\ ` ,QFUHPHQWDO LQSODQH VWUDLQV ^e` fÂ§ Â[] \]f ,QFUHPHQWDO WUDQVYHUVH VKHDU VWUDLQV XRn[ ZRn[ ZRn[ YRn\ :TI\ :T\ XRn\ YRn[ WZRn[ ZRn\ ZRn[ WZRn\ /LQHDUL]HG LQFUHPHQWDO LQSODQH VWUDLQV f ^ `7 =HUR YHFWRU fÂ§ ^A\[ a;n\ A\\ ;n[f A /LQHDU EHQGLQJ VWUDLQV f f f f f f f PAGE 102 ^HV` ^:4; \f :TW\ a [f`7 f /LQHDU VKHDU VWUDLQV ,I 3LROD.LUFKKRII ,, 3.,,f VWUHVVHV DUH GHQRWHG DV ^6` WKH LQWHUQDO YLUWXDO ZRUN GXH WR WKH YLUWXDO VWUDLQ ^H` FRUUHVSRQGLQJ WR WKH YLUWXDO GLVSODFHPHQWV ^T` FDQ EH FDOFXODWHG XVLQJ WKH FRQMXJDWH *UHHQnV VWUDLQV DV 69,D 6^Hf7^6fG9 n W H[V[ H\V\;\V[\nLU[]V[]L7U\]V\]fG]G$ f ZKHUH ^6f 6[ V\ 6[\ V; 6\]f7 3.,, VWUHVVHV 7KH LQWHJUDQG RI (J f FDQ EH IXUWKHU H[SDQGHG DV ,QWHJUDQG > 6 HSOf ]HEOf f6; HSf =HEf f6\ 6 HSf =HEf f6[\ H6Of6[] H6f6\] @G]G$ ,I WKH LQWHJUDWLRQ RYHU WKLFNQHVV LV SHUIRUPHG DQDO\WLFDOO\ WKLV FDQ EH UHDUUDQJHG DV ,QWHJUDQG > ^HSf7^66Sf ^HEf7^66Ef 6HV`7^66V` @G$ >^HH`7^66` @G$ PAGE 103 ZKHUH ^FH` ^ HS`7 ^FE`7 H6`7 `7 ,QFUHPHQWDO VKHOO HOHPHQW VWUDLQV JURXSHG LQWR LQSODQH EHQGLQJ DQG VKHDU VWUDLQV ^66` ^ 66S`7 ^66E`7 ^66V`7 `7 *HQHUDOL]HG VWUHVVHV DUUDQJHG IRU ^HH` W ^66S` U @ r W 1[ 6; G] 1f VV Vf 1[\ W V\ E[\ m f ^66E` ^6,` ^6[ 6\ 6[\` 7 W ^6O`G] f &0 0 OO; 6; 0 \ 0 $[\ W V\ V[\ ]G] f W W ^6O`]G] f r W 4[ V[] 66Vf > T\ f W V\] G] W W ^6`G] f ,QSODQH 3LROD.LUFKKRII ,, VWUHVVHV 6! f V[] V\]f7 7UDQVYHUVH VKHDU 3.,, VWUHVVHV 7KHVH JHQHUDOL]HG VWUHVVHV VWUHVV UHVXOWDQWV DORQJ WKH HOHPHQW WKLFNQHVV SHU XQLW OHQJWKf FDQ EH IXUWKHU H[SDQGHG LQFRUSRUDWLQJ WKH VWUHVVVWUDLQ UHODWLRQVKLS EHWZHHQ 3.,, VWUHVV DQG WKH FRQMXJDWH *UHHQnV VWUDLQ PAGE 104 ^6` >(@^e! ZKHUH WKH FRQVWLWXWLYH PDWUL[ >(@ FDQ EH VXEGLYLGHG DV >( f @ >(O@ >( @ 66Sf O[f ( >(O@f > @ f > @ f >(@f Q + A OQf W 6OfG] W >(O@^HOMG] W W W ,QWHJO ,QWHJ W >(O@^eS =IEMG] ZKHUH ,QWHJO W >(O@HSfG] ,QWHJ >(O@^ =eE `G] L W nW W )RU D VLQJOH OD\HU HOHPHQW LQWHJ LV DOZD\V ]HUR 66Ef n W W W W ]^6O`G] n W W ]>(O@^eO`G] >(O@=eS ]AIEMG] f f PAGE 105 ^66V` W W n W ^6`G] W >(@ ^ H`G] W >(@^HV`G] f fW (TV ff DQG f FDQ EH SXW WRJHWKHU WR IRUP WKH JHQHUDOL]HG VWUHVVVWUDLQ UHODWLRQVKLS RI D VKHOO HOHPHQW DV 66ff >'@f^ee!f ZKHUH >'@ &'LLLDA6f@>' f &2 f@ >'f@>'f@> f@ > f@> f@>'f@ f >'X@ nW W >(O@G] W' f >rA fW ]>(O@G] W >G@ >'@ W W nW W ]>(O@G] >(@G] ,W LV QRWHG WKDW >'@ DQG >'@ DUH ]HUR IRU D VLQJOH OD\HU HOHPHQW EHFDXVH LW LV WKH LQWHJUDWLRQ RI DQ RGG IXQFWLRQ RYHU WKH RSHQ GRPDLQ WWf )RU D PXOWLOD\HU V\VWHP WKHVH PXVW EH NHSW DV LV EHFDXVH PDWHULDO SURSHUWLHV RI OD\HUV FKDQJH DV WKH HOHPHQW GHIRUPV PAGE 106 R R R R R )LQLWH (OHPHQW 'LVFUHWL]DWLRQ /LQHDUL]HG ,QFUHPHQWDO 6WUDLQ'LVSODFHPHQW 5HODWLRQVKLS %\ UHDUUDQJLQJ (TV f WKURXJK f WKH VWUDLQ LQLGSODQH GLVSODFHPHQW UHODWLRQVKLS LV HHf HE H6 8n; WZ2n; ! WZRn\ XRn\ YRn ; A\n[ fIO[n\ PAGE 107 *HQHULF 'LVSODFHPHQW1RGDO 'LVSODFHPHQW 5HODWLRQVKLS 7KH JHQHULF GLVSODFHPHQWV DW PLGVXUIDFH ^8` FDQ EH LQWHUSRODWHG XVLQJ WKH VKDSH LQWHUSRODWLRQ GLVSODFHPHQWf IXQFWLRQV DQG WKH QRGDO GLVSODFHPHQWV DW PLGVXUIDFH Tf 8! XR LXL YR Q ILYL ZR V ILZL m; L O Lr[L > r< I LA\L IOXO IX f f eQXQ OYO Y f f "QYQ LZL I : r f IQZQ eOr[O O[ f f r AQA[Q I OA\O WH\ f f IQr\Q I[ I I[ I R I[ I Is I[ IQ IQ IQ If Q I Q 8L : n[O n\L X Z n[ "\ X Z Q Q Q [Q \Q PAGE 108 > I>@ I>,@ IQ>,@ TL r T Q >>b@ >1@ >1Q ^T` ^84` >1@^T` ZKHUH Q 1XPEHU RI HOHPHQW QRGHV >@ >1@ > A>@ I>,@ IQ>,@ @ & >b @ >1 @ >1Q@ @ ^JL` ^X/ 9L :L H[L H\L`7 Tf ^ ^TLf7 T`7 fff ^TQ!7 !7 6WUDLQ1RGDO 'LVSODFHPHQW 5HODWLRQVKLS ,I (T f LV VXEVWLWXWHG LQWR (T f WKH LQFUHPHQWDO VWUDLQ LV ^HHf >G@^8f >G@ >1@Tf >%O]@ T! f f f f PAGE 109 ZKHUH >%O]@ >G@>1@ f /LQHDUL]HG VWUDLQQRGDO GLVSODFHPHQW PDWUL[ (YDOXDWLRQ RI /LQHDUL]HG 6WUDLQ1RGDO 'LVSODFHPHQW 0DWUL[ (DFK WHUP RI WKH >%/=@ PDWUL[ LV FDOFXODWHG XVLQJ (T f >%O]@ >G@ f >1@ Qff >G@ > >1O@ f >1@f >1Q@f @ /HW >%M>G+1MWKHQ &EO]A f W%L E fff %QA ZKHUH >%s@ >G-&1L@ GG[ Z[fGG[f GG\ UZ\fGG\f GG\ GG[ 6Arf GG\f GG[ GG\ ZYfGG[f GG[ GG\ GG[ GG\ IL IM IÂ ; PAGE 110 ILn [f ILn \! rf Ln\f ILn[! WZrI\ ILn\f W 6Ln\f ILA LARA[f ILn\f I L [f IL\f Ln[! IL IL\f IL EHLQJ GHQRWHG DV >%O3L@f @ f >@ f >%Q ff >%OÂ‹LOf >%O6Mf f DQG GHFRPSRVHG LQWR WZR SDUWV IRU ODWHU XVH LQ HOHPHQW VWLIIQHVV FDOFXODWLRQ >%O3L @ f >@ f >%O@ >@ f >%OEL @ f f >@ f >%O6L f >@ f >%Qs @ f >%Q@ >@ f >@ f f >@ f >@ f (YDOXDWLRQ RI 1RQOLQHDU 6WUDLQ1RGDO 'LVSODFHPHQW 0DWUL[ )URP (T f WKH QRQOLQHDU LQFUHPHQWDO VWUDLQ FRPSRQHQWV LM r XNL XNMf DUH PAGE 111 ,I (T f LV VXEVWLWXWHG LQWR (T f WKHQ A;; f r 8; X ; 9n; 9n; :[ :[ U\< X\ 8\ 9\ Y\ :\ : \ }[\ X I ; 8\ 9n; Y\ Z[ :\ A[] r 8n; X] 9n; 9n= :[ :] }"\= 8\ 8] Y\ 9n= :n= : ] 7KLV FDQ EH VLPSOLILHG XVLQJ WKH VDPH DVVXPSWLRQV IRU OLQHDUL]HG VWUDLQV fn[[ Z[ Z[ ,, :\ :\ A[\ :[ :\ f L[] UL\= $QG RQO\ QRQ]HUR WHUPV DUH FRQWDLQHG IRU FRUUHVSRQGLQJ *UHHQnV VWUDLQ DQG GHQRWHG DV Qf Q[[ A[[ r :[ Z[ A\\ }"\\ fÂ§ :\ :\ f Q[\ U"[\ :[ :\ ,I (T f LV VXEVWLWXWHG LQWR (T f WKHQ >@ Qf ,Z4[ LZ2n\ ZRn[ f ZR n< f Z2n ; :n; B fÂ§ :T \ ZRn\ ZRn[ :T \ +$@^Lf f PAGE 112 $V >$@ DQG ^` DUH OLQHDU IXQFWLRQV RI ^T! LH OLQHDU IXQFWLRQV RI Z[f DQG Z\f WKH VWUDLQ LV TXDGUDWLF LQ QRGDO GLVSODFHPHQWV %XW WKH Z[f Z\f YDOXHV IURP WKH SUHYLRXV LWHUDWLRQ DUH XVHG LQ >$@ DV DSSUR[LPDWLRQV WR WKH WUXH YDOXHV 7KH VWUDLQ LV OLQHDUL]HG LQ WKLV PDQQHU DQG XVHG LQ WKH HYDOXDWLRQ RI HOHPHQW VWLIIQHVV PDWUL[ IRU WKH LWHUDWLYH VROXWLRQ RI QRQOLQHDU HTXLOLEULXP HTXDWLRQ 7R DSSO\ WKH SULQFLSOH RI YLUWXDO ZRUN LW LV GHVLUDEOH WR H[SUHVV QRQOLQHDU VWUDLQ LQ WHUPV RI QRGDO GLVSODFHPHQWV ^T` ^Q` ? >$@^r! I^T`f DQG WKH GLVSODFHPHQW JUDGLHQW ^` FDQ EH ZULWWHQ LQ WHUPV RI ^T` Q > L O IM:Lf@ n; Q > IAf @ \ L O "On[!ZO LIn[fZ IO\fZO I\f: -"Qn[ZQ IQ\fZQ PAGE 113 I}MM IOI\ I\ IQ[ IQ\ 8L Z X [O \L Z r[ \ X Z Q UQ Q [Q \Q > >*L@ >*@ >*Q@ @ r r T Q >*@^T` ZKHUH f >*LIL[f IL\f >*@ >*[@ >*@ W*Q@ @ PAGE 114 'LVFUHWL]DWLRQ 7KH ZRUG GLVFUHWL]DWLRQ PHDQV WKDW WKH FRQWLQXRXV GLVSODFHPHQW ILHOG LV DSSUR[LPDWHG XVLQJ GLVSODFHPHQWV DW GLVFUHWL]HG QRGDO SRLQWV )RU WKH WRWDO /DJUDQJLDQ IRUPXODWLRQ LQFUHPHQWDO LWHUDWLYH HTXLOLEULXP HTXDWLRQ IURP (T f LV R&LMNO 4HNO HLL R G9 rG9 (9: rG9 7KLV FDQ EH VKRZQ LQ D PDWUL[ IRUP DV IROORZV 7KH WHUP OLQHDU PHDQV WKH ILUVW RUGHU GLIIHUHQWLDO RI GLVSODFHPHQWV ZLWK UHVSHFW WR FRRUGLQDWH YDULDEOHV ZKLOH WKH WHUP QRQOLQHDU PHDQV WKH VHFRQG RUGHU GLIIHUHQWLDO RI GLVSODFHPHQWV ZLWK UHVSHFW WR FRRUGLQDWH YDULDEOHV )RU H[DPSOH LI X[f LV GHILQHG DV DX D[ DQG X[[f LV GHILQHG DV X D[ X[f LV OLQHDU ZKLOH X[[f LV QRQOLQHDU LQ UHIHUHQFH WR VWUDLQ WHUPV 7KH OLQHDUL]HG LQFUHPHQWDO VWUDLQ Hf LV UHODWHG WR LQFUHPHQWDO QRGDO GLVSODFHPHQWV ^T` WKURXJK OLQHDUL]HG VWUDLQGLVSODFHPHQW PDWUL[ >%/=@ IURP (T f H! >%/=@^Tf PAGE 115 7KH YDULDWLRQ RI OLQHDUL]HG LQFUHPHQWDO VWUDLQ LV WKHQ ^H` L>%/=@^T` >%/=@L^T` >%/=@^T! f EHFDXVH >%/=@ LV FRQVWDQW DERXW WKH XQNQRZQ LQFUHPHQWDO GLVSODFHPHQWV DQG WKH YDULDWLRQ WKHUHRI LV ]HUR LH 6>%O@ LV ]HUR 7KH QRQOLQHDU LQFUHPHQWDO VWUDLQ ^Q` FDQ EH VKRZQ DV WKH PXOWLSOLFDWLRQ RI WZR PDWULFHV >$@ DQG ^e` ZKLFK FRQWDLQV RQO\ OLQHDU WHUPV )URP (T f Qf >$@^! 7KH YLUWXDO YDULDWLRQ LQ QRQOLQHDU VWUDLQ WHUPV LV VQ! VO>$@^` OIL>$ff^Lf ,>$@^ ` f +$@6^`f L>$@L^L!f $@ ^A !f f EHFDXVH >$@f^f LV HTXDO WR >$@^A`f DV VKRZQ EHORZ :n; ZRn[ Z\ ZRn\ rZRn\ :n; >$@` Z Z 2n; Rn\ Z Rn\ 2n; rZ[f ZRn\ 7KXV e >$@f^` >$@^`f rZ[fZRn[f ZRn\A ZRn\f rZRn[! ZRn\f LZRn[fZRn\f PAGE 116 7KH JUDGLHQW RI GLVSODFHPHQWV ^` LV UHODWHG WR LQFUHPHQWDO QRGDO GLVSODFHPHQWV WKURXJK >*@ PDWUL[ )URP (T f ^` >*@ ^Tf DQG WKH YDULDWLRQ WKHUHRI LV ` >*@^Tf >*@^Tf *@^Tf f EHFDXVH >*@ LV FRQVWDQW DERXW WKH XQNQRZQ LQFUHPHQWDO GLVSODFHPHQWV DQG WKH YDULDWLRQ WKHUHRI LV ]HUR LH >*@ LV ]HUR 1RZ WKH LQFUHPHQWDO LWHUDWLYH HTXLOLEULXP HTXDWLRQ FDQ EH SXW LQWR D PDWUL[ HTXDWLRQ ,W LV QRWHG WKDW WKH HQJLQHHULQJ VWUDLQV Hf DQG Qf DUH XVHG LQ WKH SODFHV RI HNOn HLM DQG ALM FRUUHVSRQGLQJ FRQVWLWXWLYH PDWULFHV FLMNO HNO HLM LH`7>F+H` >%/=@^Tff7>&@&%O]@Tf f mT`7>%/=@7>&@>%/=@^T` T`7>.@^T` VLM rOLM 6^Q`7^6f >$@^`f7^6f `7>$@7^6` >*@^T`f7>$@76f ^Tf7>*@76@^f r^T!7>*@7>6@>*@^T` ^T`7>.@^Tf 7KH UHODWLRQVKLS >$@7^6f >6@^! >@ LV VLPSOH PDWKHPDWLFDO HTXLYDOHQFH E\ UHDUUDQJLQJ WKH HOHPHQWV RI WKH PDWULFHV LQ GLIIHUHQW IRUPDW WR UHODWH WKH QRQOLQHDU VWUDLQ PAGE 117 ,OO ^Q` WR LQFUHPHQWDO GLVSODFHPHQW ^T! ,W LV QRWHG WKDW >6@ LV D PXOWLFROXPQ DQG PXOWLURZ PDWUL[ DQG ^6` LV D FROXPQ PDWUL[ D YHFWRUf 7KLV ZLOO EH GLVFXVVHG LQ 7KH PDWULFHV >.MDQG >.@ DUH QHZO\ GHILQHG DV >rO@ >%/=@7>&@>%/=@ >.@ >*@7>6@>*@ 6LM HAM ^H`A^6` m^T`7>%/=@7^6` ([WHUQDO YLUWXDO ZRUN GXH RQO\ WR QRGDO IRUFHV LV (9: ^T`7^3` 7KHQ WKH LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ EHFRPHV 6^ Tf7 >.L+Tf G9 ^Tf7 >.@^Tf G9 6^T`7^3f ^T`7 >%O]@7^6f G9 /HW &NO]@ >b@ G9 >.J@ >.@ G9 PAGE 118 ^5,` >%O]@7^6f G9 ,I WKH YROXPH LQWHJUDWLRQ LV FKDQJHG WR DUHD LQWHJUDWLRQ XVLQJ DQDO\WLFDO LQWHJUDWLRQ WKURXJK WKLFNQHVV >UO]@ >.M@ G$ f >.*@ >.Q@ G$ f ^5,` >%O]@7^66` G$ f ZKHUH >.LO f >%O]@7>'@>%O]@ >.MM@ >*@7>66@>*@ ,W LV QRWHG WKDW VWUHVVHV DUH LQ D UHVXOWDQW IRUP >66@ ZLWK WKH FRUUHVSRQGLQJ FRQVWLWXWLYH PDWUL[ >'@ 7KHQ WKH HTXLOLEULXP HTXDWLRQ EHFRPHV LTf7 >.O]@ >.J@ f Tf 3f 5,f f ^` DQG WKLV PXVW EH VDWLVILHG IRU DQ\ YLUWXDO GLVSODFHPHQWV ^T` PHDQLQJ WKDW ^T` FDQQRW DOZD\V EH ^` WKXV >.O]@ >.J@ f ^T` ^3` ^5,` f ^` $QG ILQDOO\ WKH XVXDO IRUP >.@ ^T` ^5` FDQ EH REWDLQHG >.O]@ >.*@ f ^T` ^3` ^5,` f PAGE 119 'HULYDWLRQ RI (OHPHQW 6WLIIQHVV 0DWUL[ /LQHDUL]HG (OHPHQW 6WLIIQHVV 7KH OLQHDUL]HG LQFUHPHQWDO HOHPHQW VWLIIQHVV GXH WR OLQHDU DQG ODUJH GLVSODFHPHQW HIIHFWV LV HYDOXDWHG XVLQJ (T f )RU WKH HIILFLHQF\ LQ FDOFXODWLRQ >%/=@ LV GLYLGHG LQWR >%O@ DQG >%Q@ LQ (TV f DQG f WKHQ >%O]@7>'@>%O]@ >>%O@>%Q@@7 >'@ >>%O@>%Q@@ >%-7>'@>%@ >%O@7>'@>%Q@ >%Q@7>'@>%O@ >%Q@7>'@>%Q@ >NO] >%@7>'@>%@ G$ >%@7>'@>%Q@>%Q@7>'@>%O@>%Q@7>'@>%Q@f G$ >.O@ >.LG@ f ZKHUH >.O@ >%@D>'@>%@ G$ f /LQHDU HOHPHQW VWLIIQHVV >.LG@ >%@7>'@ >%Q@ >%Q@7>'@ >%/@ >%Q@7>'@ >%f@ f G$ >.OGO@ >.OG@ >.OG@ f /DUJH GLVSODFHPHQW HOHPHQW VWLIIQHVV PAGE 120 *HRPHWULF (OHPHQW 6WLIIQHVV 7KH HOHPHQW VWLIIQHVV GXH WR LQLWLDO VWUHVV HIIHFWV LV FDOFXODWHG XVLQJ (T f 7KH >66@ PDWUL[ PXVW EH IRXQG XVLQJ >$@ DQG ^66` )URP (T f >$@ ZRn[ Z Z Rn\ Z Rn\ Rn[ ^r` :*n; ZRn \ 7KH ^66` FRUUHVSRQGLQJ WR QRQOLQHDU VWUDLQ ^Q! LV ^66` ^1[ 1\ 1[\` 7KH UHODWLRQVKLS EHWZHHQ >66@ DQG ^66` LV E\ VLPSOH UHDUUDQJHPHQW RI PDWUL[ HOHPHQWV >$@W^6` >66@^` ZRn[ p ZRn\ Z\ Z4[ [ [\ 1[ 1 1[\ 1\r Z U ; :4I \ 7KXV PDWUL[ IRUP RI JHQHUDOL]HG VWUHVVHV >66@ LV GHILQHG DV >66@ 1[ 1 1;\ 1 f PAGE 121 7KXV WKH JHRPHWULF HOHPHQW VWLIIQHVV EHFRPHV >.J@ >*@7>66@>*@ G$ *HRPHWULF HOHPHQW VWLIIQHVV &DOFXODWLRQ RI (OHPHQW 6WLIIQHVV 0DWUL[ &DOFXODWLRQ RI 7. (DFK WHUP RI OLQHDU HOHPHQW VWLIIQHVV LV FDOFXODWHG IURP (T f EORFN E\ EORFN )URP (T f >%OL@7 >%O3L@ 7f >@ Wf >r@ P f >@ f >%OEM7f >%O6L@7 f >%OSL @ >@ f >@ f >@ f >%OM@ f >%OEM@ f >%O6M@ f )URP (T f >'@ 'f@>'f@> f@ >'f@>'f@> f@ > f@> f@>'f@ 7KXV >%@7>'@>%@f >.LSS@f >.OSE@ f >.OES@f >.OEV@ f PAGE 122 ZKHUH >.LSS@ >%OSL@7>'@>%OSM@ >.OSE@ >%OSL@7>'@>%OEM@ >.OES@ >%OEL@7>'@>%OSM@ >.OEV@ >%OEL@7>'@>%OEM@ >%OVL@7>'@>%O6M1RWH WKDW IRU D VLQJOH OD\HU HOHPHQW ERWK >.OSE@ DQG >.OES@ DUH ]HUR PDWULFHV &DOFXODWLRQ RI 7.OGO (DFK WHUP RI QRQOLQHDU HOHPHQW VWLIIQHVV IURP ODUJH GLVSODFHPHQW HIIHFWV LV FDOFXODWHG IURP (T f EORFN E\ EORFN &DOFXODWLRQ RI 7%fÂ§7'7%QO IRU 7.OGOO >%@W>'@>%Q@ >@ f >.LQ@ f >@ f >@ f ZKHUH >.LQ@ >%OSL@7>'@>%QM@ $FWXDO FDOFXODWLRQ JLYHV .OQOf .OQf >.LQ@ PAGE 123 ZKHUH .LQ f ,I f IL ;A %QM Â‘f f IL n[f %QM f f IL \f %QM f .LQ f f IL U\f %QM f f Is U\f %QM f f IL U;A %QM UOf >%QM@ >%Q@ HYDOXDWHG IRU QRGH M &DOFXODWLRQ RI 7%QOfÂ§7'7% IRU 7.OG >%Q@7>'@>%O@ >@ f >@f >.QS@f >.QE@f ZKHUH >.QS@ >%QAW'AQ%OSM@ >.QE@ >%QLAW'A+EOEM@ 1RWH WKDW IRU D VLQJOH OD\HU HOHPHQW >.QE@ LV ]HUR PDWUL[ $FWXDO FDOFXODWLRQ JLYHV >.QS@ .QSf .QSf ZKHUH .QSf 'OOf%QLOOf'Of%QLOf fIM[f 'MA6A%QLSf IM\f PAGE 124 .QSOf 'Of%QLOOf'f%QLOf fIM\f 'f%QLOfIM[f >%QM@ >%Q@ HYDOXDWHG IRU QRGH M >.QE@ .QEf .QEf ZKHUH .QEf 'Of%QLOOf'f%QLOffIM\f 'f%QLOfIM[f .QEf 'OOf%QLOOf'IOf%QLIOffIM[f 'f%QLOfIM\f &DOFXODWLRQ RI I%QOfÂ§7'7%QO IRU 7.OG >%Q@7>'@>%Q@ >@f >@ f >@f >.QQ@ f ZKHUH >.QQ@ >%QL@7>'@>%QM@ $FWXDO FDOFXODWLRQ JLYHV >.QQ@ .QQf PAGE 125 ZKHUH .QQOf 'OOf%QLOOf'Of%QLOff%QMOOf 'Of%QLOOf'f%QLOff%QMOf 'f%QLOf%QMf &DOFXODWLRQ RI 7.& (DFK WHUP RI QRQOLQHDU HOHPHQW VWLIIQHVV IURP VWUHVV HIIHFWV LV FDOFXODWHG IURP (T f EORFN E\ EORFN >*LAW66+*M@ .Jf ZKHUH .Jf 1[IL[f IMI[f 1[\>IL[fIM\fIM[fIL\f@ 1\IL\f IM\f (OHPHQW 6WUHVV 5HFRYHU\ 6WUHVVHV FDQ EH FDOFXODWHG IURP WKH (TV f f DQG f ^66S` W 6OfG] W rW >(O@^HOfG] W PAGE 126 rW >(O@^HS =HE`G] W %\ H[SOLFLW LQWHJUDWLRQ DORQJ WKLFNQHVV QO\U ^66S` ( >(O@>HSrWKNNf FErWKNNf@ N O f ZKHUH QO\U QXPEHU RI OD\HUV WKNNf WKH WKLFNQHVV RI NWK OD\HU WKNNf KNOf KNff KNf WKH GLPHQVLRQ IURP ERWWRP RI HOHPHQW WR WKH ERWWRP RI NWK OD\HU 6LPLODUO\ ^66E` DQG ^66V` FDQ EH IRXQG DV ^66E` ]^6O`G] ]>(O@^HO`G] >(O@=HS ]HEfG] QO\U ( >(O@>HSrWKNNf HErWKNNf@ N O f ZKHUH WKNNf fKNOf KNff ^66V` ^6`G] >(@^H `G] >(@H6fG] QO\U ( >(@>WKNNfH6@ N O f PAGE 127 7KHVH VWUHVVHV DUH JHQHUDOL]HG VWUHVVHV 7KXV WKH FRPPRQ VWUHVVHV IRU WKH GHWHUPLQDWLRQ RI OD\HU RU HOHPHQW VWDWH PXVW EH FDOFXODWHG XVLQJ WKH GHILQLWLRQ RI JHQHUDOL]HG VWUHVVHV IURP (TV f f DQG f )URP ^66S` LH 1[ 1\ 1;\`7 ZKLFK DUH WKH UHVXOWDQWV RI WKH LQSODQH VWUHVVHV IRU XQLW OHQJWK RI HOHPHQW HGJHV 3.,, VWUHVVHV 6[ 6\ DQG 6;\ FDQ EH FDOFXODWHG DV n[ 1[ WKN 6f 1Y WKN 6Yf 1YY WKN \ \ n[\ f L7[\ )URP WKH 66Ef LH 0[ 0\ 0;\f ZKLFK DUH WKH PRPHQW UHVXOWDQWV RI WKH LQSODQH VWUHVVHV IRU WKH XQLW OHQJWK RI HOHPHQW HGJHV 3.,, VWUHVVHV 6[ 6\ DQG 6;\ FDQ EH IRXQG DV n[ 0[WKNf ,, 0[ r 6 6\ 0\WKNf ,, 0\ r 6 6;\ 7KLV LV D WRUVLRQDO PRPHQW DQG YHU\ FRPSOLFDWHG LQ QDWXUH EXW FDQ EH DSSUR[LPDWHG DV fWKNfWKN ZKHUH WKN (OHPHQW RU OD\HU WKLFNQHVV ,, 0RPHQW RI LQHUWLD RI WKH XQLW OHQJWK RI HOHPHQW VHFWLRQ fWNKf 6 7KH LQYHUVH RI VHFWLRQ PRGXOXV RI WKH XQLW OHQJWK RI WKH HOHPHQW VHFWLRQ WKNf WKN PAGE 128 $QG WKHVH WZR FRPSRQHQWV IURP ^66S` DQG ^66A` PXVW EH VXPPHG XS IRU WRWDO VWUHVVHV 6LPLODUO\ 3.,, VWUHVVHV 6[] DQG 6\= FDQ EH FDOFXODWHG IURP ^66V` V[] 4[ WKN 6\] 4\ WKN ,QWHUQDO 5HVLVWLQJ )RUFH 5HFRYHU\ 2QFH WKH JHQHUDO VWUHVVHV DUH REWDLQHG LQWHUQDO UHVLVWLQJ IRUFHV DW QRGH L FDQ EH HYDOXDWHG DV SL! 7 >%L@[^66` G$ f ZKHUH 3Lf ^3;L 3\A 3]L 5[LW 5\M7 ^3PL`Of ^33MB` f 3[Â 3\Â 3]Â &RQFHQWUDWHG QRGDO IRUFHV LQ [ \ ] GLUHFWLRQV 5[Â 1RGDO PRPHQW DERXW [D[LV 5\Â 1RGDO PRPHQW DERXW \D[LV ^3PL` ^3;L 3\L`7 &RQFHQWUDWHG QRGDO IRUFHV IURP PHPEUDQH EHKDYLRU PAGE 129 3SL` ^3]LI 5[sO 5 PAGE 130 7KH ILUVW FRPSRQHQW LV >%QL@7^66S! %QOO %Q %Q 1[ 1Y [\ 7KXV 3]SA %QL/r1[ %Qr1\ %Qr1[\ 5[SÂ 5\3L ZKHUH %QL/ %QSA f %Q %QSAOf %Q %QSÂOf 1RWH 7KHVH DUH DOO WKH QRQ]HUR WHUPV LQ >%OSM3]SÂ &RPSRQHQW RI 3]Â IURP QRQOLQHDU VWUDLQ WHUPV 5[SÂ &RPSRQHQW RI 5[A IURP QRQOLQHDU VWUDLQ WHUPV 5\3M &RPSRQHQW RI 5\A IURP QRQOLQHDU VWUDLQ WHUPV 7KH VHFRQG FRPSRQHQW LV >%OLE@7^66E` 0; IL\f 0 \ ILn[f IL\f 0r ;[\ 7KXV 3]EÂ 5[EÂ IL\fr0\ 5\EL IL[fr0[ IL\fr0[\ PAGE 131 ZKHUH 3]EÂ &RPSRQHQW RI 3]Â IURP SODWH EHKDYLRU 5[EMA &RPSRQHQW RI 5[Â IURP SODWH EHKDYLRU 5\EL &RPSRQHQW RI 5\A IURP SODWH EHKDYLRU 7KH WKLUG FRPSRQHQW LV >%OLV@7^66Vf 4[ 4< 7KXV 3]VL IL[fr4[ IL\fr4\ 5[VL ILfr4\ 5\Vs ILfr4[ ZKHUH 3]VÂ &RPSRQHQW RI 3]Â IURP VKHDU VWUHVVHV 5[VA &RPSRQHQW RI 5[Â IURP VKHDU VWUHVVHV 5\VÂ &RPSRQHQW RI 5\A IURP VKHDU VWUHVVHV 7KHUHIRUH WKH FRQFHQWUDWHG QRGDO IRUFHV DUH 3]L 3]SÂ 3]EÂ 3]VÂ 5[A 5[SÂ 5[EÂ 5[VA ILf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r ^M! Qf / 7KXV 5 / M! f PAGE 133 )URP WKH PRPHQWFXUYDWXUH UHODWLRQVKLS M! 0/ (O f )URP (T f 0 (O! / f DQG LI (T f LV VXEVWLWXWHG LQWR (T f 5 (O 0 f 7KH IUHH HQG GLVSODFHPHQWV X Y FDQ EH IRXQG XVLQJ JHRPHWU\ X / 5 VLQA / (,0f VLQ 0/(,f f Y 5 5 FRVÂ 5 FRVM! f (,0f &26 0/(,ff f 7KH GDWD XVHG DUH ( NVL ff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f ='LVS $165 ='LVS $1$/ ;'LVS $165 ;'LVS $1$/ PAGE 137 'LVSODFHPHQW LQf )LJ 9HUWLFDO )UHH (QG 'LVSODFHPHQW RI &DQWLOHYHU %HDP XQGHU )UHH (QG 0RPHQW PAGE 138 'LVSODFHPHQW LQf )LJ +RUL]RQWDO )UHH (QG 'LVSODFHPHQW RI &DQWLOHYHU %HDP XQGHU )UHH (QG 0RPHQW PAGE 139 6TXDUH 3ODWH 7KH VHFRQG WHVW PRGHO XVHG LV VKRZQ LQ )LJ 7KLV LV D VTXDUH SODWH XQGHU GLVWULEXWHG ORDGV 7KH ERXQGDU\ FRQGLWLRQV FDQ HLWKHU EH IL[HG RU VLPSO\ VXSSRUWHG 7KH GDWD XVHG DUH Q 3RLVVRQnV UDWLR D LQ 6LGH OHQJWK W LQ 7KLFNQHVV ( NVL T 'LVWULEXWHG ORDG 7KH DQDO\WLFDO OLQHDU VROXWLRQV IRU WKH GLVSODFHPHQW DW FHQWHU RI SODWH >@ DUH Z TD' IRU FODPSHG VTXDUH SODWH Z TD' IRU WKH VLPSO\ VXSSRUWHG VTXDUH SODWH ZKHUH (W OÂf SODWH VWLIIQHVV 7KH OLQHDU DQDO\WLFDO VROXWLRQ DQG QXPHULFDO VROXWLRQ IURP $165 DUH JLYHQ LQ 7DEOH 7DEOH DQG SORWWHG LQ )LJ DQG )LJ IRU FODPSHG SODWH DQG VLPSO\ VXSSRUWHG SODWH UHVSHFWLYHO\ 7KH FRPSDULVRQ RI QRQOLQHDU UHVSRQVHV LV JLYHQ LQ 7DEOHV DQG 7KH UHODWLYH HIIHFWV RI ODUJH GLVSODFHPHQWV DQG LQLWLDO VWUHVVHV ZLWK UHVSHFW WR WRWDO QRQOLQHDU HIIHFWV DUH JLYHQ LQ 7DEOH DQG )LJ PAGE 140 6L]H LQ [ LQ 7KLFNQHVV LQ ( NVL 3RLVVRQnV UDWLR )LJ 6TXDUH 3ODWH XQGHU 'LVWULEXWHG /RDGV PAGE 141 7DEOH 'LVSODFHPHQWV RI 6TXDUH 3ODWH ZLWK )L[HG 6XSSRUW XQGHU 'LVWULEXWHG /RDGV 6WHS 1R :W SFIf T SVLf /LQHDU LQf $165 LQf PAGE 142 'LVSODFHPHQW LQf )LJ &HQWHU 'LVSODFHPHQW RI &ODPSHG 6TXDUH 3ODWH XQGHU 'LVWULEXWHG /RDG PAGE 143 7DEOH 'LVSODFHPHQWV RI 6TXDUH 3ODWH ZLWK 6LPSOH 6XSSRUW XQGHU 'LVWULEXWHG /RDGV 6WHS 1R :W SFIf T SVLf /LQHDU LQf $165 LQf PAGE 144 'LVSODFHPHQW LQf )LJ &HQWHU 'LVSODFHPHQW RI 6LPSO\ 6XSSRUWHG 6TXDUH 3ODWH XQGHU 'LVWULEXWHG /RDG PAGE 145 7DEOH &RPSDULVRQ RI 'LVSODFHPHQWV RI 6TXDUH 3ODWH ZLWK 6LPSOH 6XSSRUW /RDG /RDG ([DFW $165 (5525 SVLf VWHSV LQf LQf bf 7KH H[DFW YDOXHV DUH TXRWHG IURP UHIHUHQFH >@ PAGE 146 7DEOH &RPSDULVRQ RI 'LVSODFHPHQWV RI 6TXDUH 3ODWH ZLWK &ODPSHG 6XSSRUW /RDG /RDG ([DFW $165 (5525 SVLf 6WHSV LQf LQf bf 7KH H[DFW YDOXHV DUH TXRWHG IURP UHIHUHQFH >@ PAGE 147 7DEOH 'LVSODFHPHQWV RI 6TXDUH 3ODWH ZLWK )L[HG 6XSSRUW XQGHU 'LVWULEXWHG /RDGV 8VLQJ 'LIIHUHQW 1RQOLQHDU 6WLIIQHVVHV ZLWKRXW ,WHUDWLRQV VWHS QR ZW SFIf *(2 SVLf /' LQf $// LQf PAGE 148 'LVSODFHPHQW LQf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f QRQOLQHDULWLHV ,W KDV EHHQ GHULYHG GLUHFWO\ IURP WKH OLQHDUL]HG LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ 7KLV LV EDVLFDOO\ D OLQHDU HTXLOLEULXP HTXDWLRQ ZLWKLQ HDFK LQFUHPHQW 7KXV WKH IRUPXODWLRQ LV VLPLODU WR WKDW RI OLQHDU HOHPHQW 7KH WRWDO /DJUDQJLDQ IRUPXODWLRQ ZDV XVHG IRU WKH GHVFULSWLRQ RI PRWLRQ 7KH GLVDGYDQWDJH RI WKLV IRUPXODWLRQ LV WKDW LW QHHGV VSHFLDO WUHDWPHQW IRU ODUJH URWDWLRQV EHFDXVH ILQLWH URWDWLRQ LV QRW D WHQVRULDO YHFWRUf TXDQWLW\ 2QH RI WKH VROXWLRQV WR WKLV OLPLWDWLRQ LV FR PAGE 150 URWDWLRQDO IRUPXODWLRQ ZKLFK LV EDVLFDOO\ DQ XSGDWHG /DJUDQJLDQ IRUPXODWLRQ IRU URWDWLRQV RQO\ > @ 7KH GLVSODFHPHQW GHSHQGHQW ORDGV ZKLFK DUH XVXDO IRU VKHOO VWUXFWXUHV LV DOVR UHFRPPHQGHG IRU IXUWKHU VWXG\ $QRWKHU DUHD RI IXUWKHU UHVHDUFK LV WKH PDWHULDO QRQOLQHDULW\ RI FRQFUHWH 7KLV QRQOLQHDULW\ LV HVVHQWLDO WR SURYLGH UHDOLVWLF VWUXFWXUDO UHVSRQVH IRU VDIH DQG FRVW HIIHFWLYH GHVLJQV 6RPH RI WKH FRPSOH[ FRQFUHWH SURSHUWLHV DUH QRQOLQHDU PDWHULDO SURSHUWLHV FUDFNLQJ LQ FRQFUHWH VKHDU WUDQVIHU DFURVV FUDFNHG LQWHUIDFHV WLPH GHSHQGHQW HIIHFWV VXFK DV FUHHS VKULQNDJH DQG WUDQVLHQW WHPSHUDWXUH GLVWULEXWLRQ >@ 7KH QRQOLQHDU PDWHULDO SURSHUWLHV RI FRQFUHWH KDYH ORQJ EHHQ WKH VXEMHFW RI UHVHDUFK 7KH ILUVW UHOLDEOH WHVW UHVXOWV RQ WKH WZR GLPHQVLRQDO FRQVWLWXWLYH UHODWLRQVKLS ZHUH UHSRUWHG E\ .XSIHU DQG +LOVGRUI > @ DQG ZHUH XVHG WR GHYHORS WKH HTXLYDOHQW XQLD[LDO VWUDLQ FRQFHSW >@ 9DULRXV FRQVWLWXWLYH PRGHOV RI FRQFUHWH FDQ EH IRXQG LQ > @ $Q HQWLUHO\ GLIIHUHQW DSSURDFK HQGRFKURQLF WKHRU\ WR PDWHULDOV LQ ZKLFK WKH LQHODVWLF VWUDLQ DFFXPXODWHV JUDGXDOO\ ZDV DOVR VXJJHVWHG ,W FRQVLVWV RI FKDUDFWHUL]LQJ WKH LQHODVWLF VWUDLQ DFFXPXODWLRQ E\ D FHUWDLQ VFDODU SDUDPHWHU FDOOHG LQWULQVLF WLPH ZKRVH LQFUHPHQW LV D IXQFWLRQ RI VWUDLQ LQFUHPHQW >@ 7KH FUDFNV GHYHORSLQJ LQ FRQFUHWH KDYH EHHQ VWXGLHG IRU D ORQJ WLPH 7KHRULHV DQG PAGE 151 DSSOLFDWLRQV KDYH EHHQ GHYHORSHG IRU WKH PHFKDQLFDO EHKDYLRU RI LQGLYLGXDO FUDFN > @ GLVFUHWH SDUDOOHO FRQWLQXRXV FUDFN >@ GLVWULEXWHG VPHDUHGf LQ D IL[HG GLUHFWLRQV FUDFNV > @ GLVWULEXWHG URWDWLQJ FUDFNV>@ 7KH VPHDUHG FUDFN WKHRU\ DQG IUDFWXUH ORFDOL]DWLRQ LQ FRQFUHWH LV ZHOO GRFXPHQWHG LQ >@ ,Q D FUDFNHG UHLQIRUFHG FRQFUHWH IOH[XUDO PHPEHU WKH LQWDFW FRQFUHWH EHWZHHQ HDFK SDLU RI DGMDFHQW WHQVLOH FUDFNV DVVLVWV WKH WHQVLOH VWHHO LQ FDUU\LQJ WKH LQWHUQDO WHQVLOH IRUFH DQG WKHUHIRUH FRQWULEXWHV WR WKH RYHUDOO EHQGLQJ VWLIIQHVV RI WKH PHPEHU 7KLV LV FDOOHG WHQVLRQ VWLIIHQLQJ >@ 7KH VKHDU WUDQVIHU WKURXJK DJJUHJDWH LQWHUORFN ZDV GHVFULEHG LQ > @ 7KH FRPELQDWLRQ RI DOO WKH HIIHFWV IURP FUDFNV DJJUHJDWH LQWHUORFN GRZHO DFWLRQ DQG WHQVLRQ VWLIIHQLQJ LQ FRQFUHWH UHVSRQVH FDQ EH IRXQG LQ >@ ,W LV ZHOO NQRZQ WKDW WKHUH FDQ EH QXPHULFDO LQVWDELOLW\ DQG VHQVLWLYLW\ RQ ILQLWH HOHPHQW PHVK VL]H LQ WKH VROXWLRQ SURFHVV IRU WKH VWUDLQ VRIWHQLQJ PDWHULDO >@ 7KLV KDV EHHQ RYHUFRPH WKURXJK WKH VKHDU EDQG FRQFHSW >@ RU D VSHFLILF HOHPHQW IRUPXODWLRQ IRU H[DPSOH WKH IRXU QRGH LVRSDUDPHWULF HOHPHQW VXLWDEOH IRU PRGHOOLQJ FUDFNV LQ >@ $OO WKHVH FRPSOH[ QRQOLQHDU PDWHULDO SURSHUWLHV RI FRQFUHWH PXVW EH LQFRUSRUDWHG IRU WKH UHDOLVWLF DQDO\VLV RI DQ\ FRQFUHWH VWUXFWXUH LQFOXGLQJ WKH KROORZ ER[ JLUGHU IRU EULGJH VWUXFWXUHV PAGE 152 $33(1',; $ ,03/(0(17$7,21 2) /,1. (/(0(17 7KH WKUHH GLPHQVLRQDO OLQN HOHPHQW ZDV LPSOHPHQWHG WR WKH $165,,, SURJUDP $165 UHTXLUHV WKDW WHQ VXEURXWLQHV EH ZULWWHQ IRU DQ HOHPHQW WR EH XVHG E\ WKH SURJUDP >@ 7KH IROORZLQJ LV D GHVFULSWLRQ RI WKH VXEURXWLQHV ZULWWHQ IRU WKH OLQN HOHPHQW ,1(/ 67,) 67$7 5,17 (917 2876 7KH VXEURXWLQHV 5'<1 ,1,7 0'6( (287 DQG &5/' DUH QRW XVHG E\ WKH FXUUHQW YHUVLRQ RI WKH OLQN HOHPHQW $V D UHVXOW GXPP\ VXEURXWLQHV WKDW FRQVLVW RI UHWXUQ VWDWHPHQWV DUH ZULWWHQ IRU WKHVH VXEURXWLQHV 7KH HOHPHQW JURXS QXPEHU LV VHW WR VL[ IRU WKH OLQN HOHPHQW $ *HQHUDO ,PSOHPHQWDWLRQ 'HWDLOV ,PSOLFLW GRXEOH SUHFLVLRQ LV XVHG WKURXJKRXW WKH LQWHUIDFH VXEURXWLQHV ,03/,&,7 '28%/( 35(&,6,21$+=f 7KH ODEHOHG FRPPRQ EORFN ,1)*5 IRU HOHPHQW JURXS LQIRUPDWLRQ LV GHILQHG DV IROORZV &20021 ,1)*5 1*51(/60)67,*5+('f1,1)&/67$7 /67)/67&1'2)'.2'.7(3523f PAGE 153 ZKHUH 1*5 (OHPHQW JURXS QXPEHU 1(/6 1XPEHU RI HOHPHQWV ZLWKLQ WKH JURXS 0)67 (OHPHQW QXPEHU RI WKH ILUVW HOHPHQW LQ WKLV JURXS ,*5+(',2f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f12'(f0$7&'(/;/ ;;f< PAGE 154 WZHQW\ LQWHJHU YDULDEOHV ,0(0 WKURXJK 0$7f DQG IRUW\ UHDO YDULDEOHV &'(/ WKURXJK 67(0f $ UHDO ZRUG KDV D OHQJWK RI WZR LQWHJHU ZRUGV ,0(0 .67 DQG /0f PXVW EH DW WKH EHJLQQLQJ RI WKH ODEHOHG FRPPRQ EORFN ,1)(/ LQ WKH JLYHQ VHTXHQFH /0 LV ORQJ VLQFH WKHUH DUH HOHPHQW GHJUHHV RI IUHHGRP IRU WKH OLQN HOHPHQW ,0(0 (OHPHQW QXPEHU .67 6WLIIQHVV XSGDWH FRGH /0f /RFDWLRQ PDWUL[ 12'(f 1RGH QXPEHUV QR RI HO QRGHV t QRGH 0$7 0DWHULDO SURSHUW\ QXPEHU &'(/ $OORZDEOH FRPSUHVVLYH GLVSODFHPHQW IRU WKH SDVVLQJ RI WZR MRLQW QRGHV ;/ (OHPHQW OHQJWK ;.6 &XUUHQW HOHPHQW VKHDU VWLIIQHVV ;.62 2ULJLQDO HOHPHQW VKHDU VWLIIQHVV ;.1 &XUUHQW HOHPHQW QRUPDO VWLIIQHVV ;.12 2ULJLQDO HOHPHQW QRUPDO VWLIIQHVV ;18 )ULFWLRQ FRHIILFLHQW 7+. (OHPHQW WKLFNQHVV ',68 (OHPHQW GHIRUPDWLRQ LQ X[f GLUHFLWRQ ',67 (OHPHQW GHIRUPDWLRQ LQ W\f GLUHFWLRQ ',69 (OHPHQW GHIRUPDWLRQ LQ Y]f GLUHFWLRQ ',6: (OHPHQW URWDWLRQ 6758 (OHPHQW VWUHVV LQ X[f GLUHFWLRQ 6757 (OHPHQW VWUHVV LQ W\f GLUHFWLRQ PAGE 155 6759 (OHPHQW VWUHVV LQ Y]f GLUHFWLRQ 6750 (OHPHQW PRPHQW ZKLFK VKRZV WKH GLVWULEXWLRQ RI HOHPHQW VWUHVV ;; f ; FRRUGLQDWHV RI IRXU QRGHV DQG WKH QRGH < PAGE 156 $ 6XEURXWLQH ,1(/ 6XEURXWLQH ,1(/ LV WKH LQSXW VXEURXWLQH ,WV SXUSRVH LV WR UHDG DQG SULQW WKH LQSXW GDWD IRU OLQN HOHPHQWV DQG WR LQLWLDOL]H WKH YDULDEOHV LQ WKH ODEHOHG FRPPRQ EORFNV ,1)*5 DQG ,1)(/ 7KLV VXEURXWLQH KDV WKH IRUP RI 68%5287,1( ,1(/ 1-71'.2';<=.(;(&f ,Q WKH LQHO VXEURXWLQH WKH HOHPHQW JURXS FRQWURO YDULDEOHV DUH WR EH VHW 7KH ILUVW RQH LV 0)67 WKH HOHPHQW QXPEHU RI WKH ILUVW HOHPHQW LQ WKLV JURXS 0)67 LV LQSXW GDWD UHDG E\ WKH EDVH SURJUDP ,W LV GHIDXOWHG WR EH RQH DW WKH EHJLQQLQJ RI WKH LQHO VXEURXWLQH LI LW LV QRW JLYHQ LQ LQSXW ILOH 7KH LQGH[ IRU WKH VWLIIQHVV FKDQJH .67 LV VHW WR RQH 7KLV LQGLFDWHV WR IRUP WKH VWLIIQHVV VLQFH WKLV LV WKH ILUVW SDVV .67 FDQ EH XSGDWHG LQ DFFRUGDQFH ZLWK WKH DFWXDO VWDWXV RI HOHPHQW VWLIIQHVV LH LI HOHPHQW VWLIIQHVV LV QRW FKDQJHG .67 LV DVVLJQHG WKH YDOXH RI ]HUR 7KH OHQJWK RI WKH HOHPHQW LQIRUPDWLRQ DQG WKH QXPEHU RI HOHPHQW GHJUHHV RI IUHHGRP DUH DOVR WR EH VHW LQ LQHO VXEURXWLQH /67$7 /67) /67& 1,1)& 1'2) PAGE 157 7KHQ HOHPHQW JURXS LQIRUPDWLRQ LV SULQWHG 1*5 ,*5+(' 1(/6 0)67 DUH UHDG LQ EDVH SURJUDP $OO RWKHUV DUH VHW LQ LQHO VXEURXWLQH 7KH QXPEHU RI PDWHULDO SURSHUWLHV DQG WKH PDWHULDO SURSHUWLHV DUH UHDG DQG WKHQ HFKRSULQWHG &'(/ LV WKH FRPSUHVVLYH GHIRUPDWLRQ OLPLW WR GHILQH MRLQW RYHUODS 7KLV LV WKH DPRXQW RI GLVSODFHPHQW RYHUODS DOORZHG EHIRUH WKH VWLIIQHVV LV LQFUHDVHG WR SUHYHQW WKH ORVV LQ IRUFHV GXULQJ WUDQVIHU WKURXJK GLVSODFHPHQW 10$7 LV WKH QXPEHU RI PDWHULDO SURSHUWLHV )RU WKH OLQN HOHPHQW WKUHH PDWHULDO SURSHUWLHV DUH UHTXLUHG LH MRLQW QRUPDO VWLIIQHVV MRLQW VKHDU VWLIIQHVV DQG FRHIILFLHQW RI IULFWLRQ $Q LQGH[ IRU WKH HUURU ,1(55 LV VHW XS WR GHWHFW DQ HUURU GXULQJ WKH GDWD LQSXW LQ LQHO VXEURXWLQH ,W LV LQLWLDOO\ VHW WR ]HUR ,I DQ HUURU LV GHWHFWHG LW LV VHW WR RQH 7KH FRQWURO LQIRUPDWLRQ IRU HOHPHQW GDWD LV FDOFXODWHG ,0(0 LV WKH HOHPHQW QXPEHU ZKLFK ZLOO EH LQFUHDVHG E\ RQH DIWHU WKH HQG RI HDFK HOHPHQW GDWD OLQH LV UHDG RU JHQHUDWHG 1/$67 WKH WKH ODVW HOHPHQW QXPEHU LQ WKLV JURXS FDOFXODWHG E\ WDNLQJ WKH ILUVW HOHPHQW QXPEHU SOXV WKH QXPEHU RI HOHPHQWV $Q LQGH[ WR FKHFN WKH QXPEHU RI WKH OLQHV RI HOHPHQW GDWD ,&175 LV VHW XS (OHPHQW GDWD DUH UHDG HOHPHQW E\ HOHPHQW 7KH UHTXLUHG HOHPHQW GDWD DUH 1(/ (OHPHQW QXPEHU PAGE 158 12'(f 0$7 1XPEHUV RI IRXU QRGHV DQG WKH WKLUG QRGH 0DWHULDO SURSHUW\ RI WKH HOHPHQW 7+. 7KLFNQHVV RI WKH HOHPHQW 1*(1 1XPEHU RI HOHPHQWV WR EH JHQHUDWHG LQFOXGLQJ WKH HOHPHQW VSHFLILHG 1*(1f HOHPHQWV DUH WR EH JHQHUDWHG .,1&5 ,QFUHPHQW RI WKH QXPEHU RI WKH WKLUG QRGH IRU HOHPHQW JHQHUDWLRQ ,I QR JHQHUDWLRQ RI HOHPHQW GDWD LV VSHFLILHG LH 1*(1 LV ]HUR 7KH QXPEHUV RI QRGH WKUHH DQG IRXU DUH LQWHUFKDQJHG IRU HDVH RI LQSXW DQG WKHQ HOHPHQW GDWD ZLOO EH SURFHVVHG E\ WKH VXEURXWLQH (9(/ 7KH QRGH QXPEHULQJ IRU WKH IRUPXODWLRQ DQG GDWD LQSXW LV r r r r 1RGH QXPEHULQJ IRU IRUPXODWLRQ r r r r 1RGH QXPEHULQJ IRU GDWD LQSXW ,I JHQHUDWLRQ RI HOHPHQW GDWD DUH UHTXHVWHG LH SRVLWLYH 1*(1 1*(1f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f PDWUL[ LH 7 )LQG WKH HOHPHQW GLUHFWLRQ YHFWRU^Lr` ORFDO XGLUHFWLRQ YHFWRUf IURP WKH FRRUGLQDWHV RI WKH VWDUW DQG HQG QRGHV DQG WKHQ QRUPDOL]H LW E\ PAGE 160 FDOOLQJ 9(&725 3XW WKH WKUHH FRPSRQHQWV LQWR WKH ILUVW URZ RI WKH >7@ PDWUL[ )RUP WKH WKLUG GLUHFWLRQ YHFWRU^NNf ORFDO WKLUG GLUHFWLRQ YHFWRUf IURP LQSXW E\ FDOOLQJ 9(&725 7KLV LV WKH ORFDO Q GLUHFWLRQ YHFWRU IRU WKH OLQN HOHPHQW 7KHUHIRUH QRUPDOL]H LW^Nn`f DQG SXW WKH WKUHH FRPSRQHQWV LQWR WKH VHFRQG URZ RI WKH >7@ PDWUL[ 3HUIRUP WKH YHFWRU FURVV ^Lr [ ^Nf! WR ILQG ORFDO ZGLUHFWLRQ YHFWRU DQG QRUPDOL]H LW^M rff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f 7KH YDULDEOHV XVHG LQ WKLV VXEURXWLQH DUH ). (OHPHQW VWLIIQHVV PDWUL[1'2)1'2)f ).* *OREDO HOHPHQW VWLIIQHVV PDWUL[ )./ /RFDO HOHPHQW VWLIIQHVV PDWUL[ ).3 3UHYLRXV JOREDO HOHPHQW VWLIIQHVV ,67)& 6WLIIQHVV PDWUL[ FRQWHQW LQGH[ 7RWDO HOHPHQW VWLIIQHVV PDWUL[ &KDQJH LQ HOHPHQW VWLIIQHVV PDWUL[ ,1'). ). VWRUDJH LQGH[ /RZHU KDOI FRPSDFWHG FROXPQZLVH 6TXDUHDOOf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r)./r7 XVLQJ [ VXEPDWULFHV =HUR ).*./f 'LYLGH )./[f LQWR [ VXEPDWUL[ =HUR 7(03f IRU HDFK PDQLSXODWLRQ 7(03 77r)./ ).* 77r)./fr7 $ 6XEURXWLQH 67$7 6XEURXWLQH 67$7 LV WKH VWDWH GHWHUPLQDWLRQ VXEURXWLQH ,WV SXUSRVH LV WR XSGDWH WKH HOHPHQW VWDWH PAGE 163 LQIRUPDWLRQ LQ ,1)(/ JLYHQ WKH FXUUHQW VWDWH DQG WKH LQFUHPHQW RI QRGDO GLVSODFHPHQWV LQ WKH JOREDO D[HVDUUD\ 4f VHW XS E\ WKH EDVH SURJUDP 7KH VXEURXWLQH KDV WKH IRUP RI 68%5287,1( 67$7 1')47,0(f 7KH YDULDEOHV XVHG LQ WKLV VXEURXWLQH DUH ,1&5(0(17$/ JOREDO QRGDO GLVSODFHPHQWV 7KHUHIRUH LI WRWDO VWUHVVHV DUH QHHGHG WKH HOHPHQW GLVSODFHPHQWV DUH WR EH DGGHG XS DW WKH HQG RI HDFK LWHUDWLRQ ,1&5(0(17$/ ORFDO QRGDO GLVSODFHPHQWV ,QFUHPHQWDO HOHPHQW GHIRUPDWLRQ 7KHVH DUH WR EH VDYHG LQ LQIHO IRU XVH LQ ULQW WR ILQG )( IRU WKH HTXLOLEULXP FKHFN RI LQFUHPHQWDO H[WHUQDO ORDGV WRWDO HOHPHQWDO GHIRUPDWLRQ ZKHUH r 879 RU : ,I JHRPHWULF ODUJH GLVSODFHPHQWf QRQOLQHDU DQDO\VLV LV UHTXHVWHG LH .*(20 LV RQH &RRUGLQDWHV RI DOO WKH QRGHV DUH WR EH XSGDWHG DQG WKH QHZ OHQJWKV RI WKH HOHPHQWV ZLWK QHZ GLUHFWLRQ FRVLQHV DUH WR EH FDOFXODWHG DQG WKHQ WKH WUDQVIRUPDWLRQ PDWUL[ LV WR EH XSGDWHG 2WKHUZLVH VNLS WR WKH VPDOO GLVSODFHPHQW DQDO\VLV VWHS )RU WKH DQDO\VLV ZLWK WKH VPDOO GLVSODFHPHQW DVVXPSWLRQ WKH LQFUHPHQWDO JOREDO QRGDO GLVSODFHPHQWV 4f1'2)f DUH URWDWHG WR ORFDO FRRUGLQDWHV 4'/f1'2)f 7KHQ WKH DYHUDJH LQFUHPHQWDO HOHPHQW GLVSODFHPHQWV ',68, ',67, ',69, ',6:, DUH FDOFXODWHG 7KHVH LQFUHPHQWDO HOHPHQW 41')f 4'/f ',6r, ',6r PAGE 164 GLVSODFHPHQWV DUH WKHQ DGGHG XS WR IRUP WRWDO DYHUDJH HOHPHQW GLVSODFHPHQWV ',68 ',67 ',69 ',6: 7KH WRWDO DYHUDJH HOHPHQW VWUHVVHV 6758 6757 6759 6750 DUH WKHQ FDOFXODWHG IURP WKH WRWDO DYHUDJH HOHPHQW GLVSODFHPHQWV 7KHVH WRWDO VWUHVV DUH XVHG IRU WKH VWDWH GHWHUPLQDWLRQ DORQJ ZLWK WKH WRWDO DYHUDJH HOHPHQW GLVSODFHPHQWV ;18 LV WKH VWDWLF IULFWLRQ FRHIILFLHQW DQG ;18. WKH NLQHWLF IULFWLRQ FRHIILFLHQW ;.18 LV DVVXPHG WR EH WLPHV ;18 ,I WKH MRLQW KDV VOLG SUHYLRXVO\ [NV f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nRSHQn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n2YHUODSn LV GHILQHG DV WKH VWDWH ZKHUH QRUPDO FRPSUHVVLYH GLVSODFHPHQW LV DOJHEUDLFDOO\ OHVV WKDQ WKH QHJDWLYH YDOXH RI WKH JLYHQ OLPLW RI FRPSUHVVLYH GLVSODFHPHQW 7KLV YDOXH &'(/ LV D SRVLWLYH QXPEHU LQ WKH LQSXW GDWD ,I MRLQW QRGHV KDYH RYHUODSSHG EH\RQG WKH OLPLW VSHFLILHG E\ &'(/ UHDFWLYDWH WKH VKHDU VWLIIQHVV ;.6 DQG LQFUHDVH WKH QRUPDO VWLIIQHVV ;.1 WR SUHYHQW RYHUODSSLQJ LQ WKH QH[W LWHUDWLRQ ,I DEVGLVWfJHFGHOf LV XVHG D ODUJH WHQVLRQ GLVS ZLOO EH WDNHQ DV MRLQW RYHUODS ZKLFK LV QRW WUXH ,I D MRLQW KDV EHHQ FORVHG ZLWKRXW RYHUODSSLQJ WKH QRUPDO VWLIIQHVV DQG WKH VKHDU VWLIIQHVV ZLOO EH UHVHW WR WKH RULJLQDO YDOXHV VDYHG LQ WKH ODEHOHG FRPPRQ EORFN ,1)(/ 7KH HOHPHQW VWDWH nVOLGn LV GHILQHG DV VTUWVWUXrrVWUYrrf [QXrVWUW LI VWUW 7KH FDVH ZKHUH WKH QRUPDO VWUHVV 6757 LV JUHDWHU WKDQ RU HTXDO WR ]HUR ZDV FRYHUHG LQ WKH MRLQW RSHQLQJ GHFLVLRQ 7KH GHFLVLRQ RI ZKHWKHU WKH GLVSODFHPHQW VTUWGLVXrrGLVYrrf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nVWRSn ZKLFK KDV RFFXUUHG IURP WKH nVOLGLQJn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f 7KH PHDQLQJ RI WKH YDULDEOHV XVHG LQ WKLV VXEURXWLQH DUH 4 1RGDO GLVSODFHPHQWV 9(/ 1RGDO YHORFLW\ )' '\QDPLF QRGDO IRUFH ZKHQ 7,0( )( (ODVWRSODVWLF QRGDO IRUFH ZKHQ 7,0( 1RGDO ORDGV ZKLFK LV LQ HTXLOLEULXP ZLWK FXUUHQW VWDWH RI VWUHVVHV ,PDJLQDU\ IRUFHV ZKLFK DFW RQ HOHPHQW WR LQWURGXFH FXUUHQW HOHPHQW VWUHVVHV 7KH GLDJUDP IRU WKH HOHPHQW VWUHVV HOHPHQW IRUFH LQWHUQDO UHVLVWLQJ IRUFH DQG WKH H[WHUQDO ORDG LV fÂ§r 2 (/707 R P r fÂ§ r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f LV XVHG DV GHFLVLRQ YDULDEOHV IRU FRQYHUJHQFH )RU WKH HTXLOLEULXP FKHFN $165 XVHV WKH WRWDO ORDGV 7KHUHIRUH WKH HOHPHQW IRUFHV DUH WR EH UHFRYHUHG XVLQJ WKH WRWDO HOHPHQW GLVSODFHPHQW 7KH HOHPHQW IRUFHV UHFRYHUHG IURP WKH WRWDO HOHPHQW GLVSODFHPHQWV DUH LQ WKH ORFDO FRRUGLQDWHV 7KHVH PXVW EH URWDWHG WR JOREDO FRRUGLQDWHV VR WKDW WKHVH IRUFHV FDQ EH FRPSDUHG ZLWK WKH JOREDO ORDGV IRU WKH HTXLOLEULXP FKHFN 7KH URWDWLRQ IURP WKH ORFDO WR JOREDO LV GRQH E\ ORFDOf >7@JOREDOf JOREDOf >7 WUDQVSRVH@ORFDOf 7KH UHFRYHUHG HOHPHQW IRUFHV DUH WUDQVIHUUHG WR WKH EDVH SURJUDP WKURXJK WKH )( DUUD\ DIWHU WKH URWDWLRQ IURP ORFDO WR JOREDO FRRUGLQDWHV PAGE 170 7KH QRGDO IRUFHV GXH WR GDPSLQJ DUH VXSSRVHG WR EH FDOFXODWHG LQ WKLV VXEURXWLQH 7KH FXUUHQW YHUVLRQ GRHV QRW LQFOXGH GDPSLQJ ,I GDPSLQJ LV FRQVLGHUHG WKH YDULDEOH 7,0( ZLOO EH JUHDWHU WKDQ ]HUR 7KLV LV FRPSDUHG ZLWK WKH FRQVWDQW ]HUR WR VHH LI GDPSLQJ LV LQFOXGHG $ 6XEURXWLQH (917 6XEURXWLQH (917rr LV FDOOHG IRU HDFK HOHPHQW DW IUHTXHQW LQWHUYDOV GXULQJ WKH DQDO\VLV ,WV SXUSRVH LV WR FDOFXODWH WKH SURSRUWLRQ RI WKH GLVSODFHPHQW LQFUHPHQW 4 ZKLFK FDQ EH DSSOLHG WR DQ HOHPHQW EHIRUH D VLJQLILFDQW QRQOLQHDU HYHQW RFFXUV 7\SLFDO HYHQWV DUH \LHOGLQJ JDS FORVXUH DQG XQORDGLQJ LH WKH LQWHUVHFWLRQ RI WZR OLQHDU SRUWLRQV LQ VWUXFWXUDO VWLIIQHVV 7KLV VXEURXWLQH KDV WKH IRUP RI 68%5287,1( (917 1')49(/$&&(9)$&,(9f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f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f &RQWURO LQIRUPDWLRQ f )LUVW FRQWURO OLQH &2/8016 127( 1$0( '$7$ f 1*5 (OHPHQW JURXS QXPEHU f 1(/6 1XPEHU RI OLQN HOHPHQWV f 0)67 (OHPHQW QXPEHU RI ILUVW OLQN HOHPHQW )f '.2 ,QLWLDO VWLIIQHVV GDPSLQJ IDFWRU )f '.7 7DQJHQW VWLIIQHVV GDPSLQJ IDFWRU $f 2SWLRQDO KHDGLQJ IRU OLQN HOHPHQW f 6HFRQG FRQWURO OLQH &2/8016 127( 1$0( '$7$ f 10$7 1XPEHU RI PDWHULDO SURSHUWLHV )f &'(/ $OORZDEOH FRPSUHVVLEOH GHIRUPDWLRQ f 0DWHULDO SURSHUW\ GDWD 10$7 OLQHV &2/8016 127( 1$0( '$7$ f 0$7 0DWHULDO SURSHUW\ QXPEHU (f ;.6 (OHPHQW VKHDU VWLIIQHVV (f ;.1 (OHPHQW QRUPDO VWLIIQHVV )f ;18 )ULFWLRQ FRHIILFLHQW f (OHPHQW GDWD 1(/6 OLQHV &2/8016 127( 1$0( '$7$ f 1(/ (OHPHQW QXPEHU f 12'( 1RGH f 12'( 1RGH f 12'( 1RGH f 12'( 1RGH f 12'(. 1RGH IRU WUDQVIRUPDWLRQ f 0$7 0DWHULDO SURSHUW\ QXPEHU )f 7+. (OHPHQW WKLFNQHVV PAGE 175 f f 1*(1; .,1&5 1XPEHU RI HOHPHQWV JHQHUDWHG 1RGH LQFUHPHQW IRU HO JHQHUDWLRQ 127(6 f /RFDO FRRUGLQDWH V\VWHP 1'(12'(12'(12'(12'(. Q R $ , RfÂ§ fÂ§ R 2n Z 1RWH /RFDO Z D[LV LV GHFLGHG E\ ULJKWKDQG UXOH FRPLQJ RXW RI WKH SDJH PAGE 176 $33(1',; % ,03/(0(17$7,21 2) /,1($5 6+(// (/(0(17 7KH LPSOHPHQWDWLRQ LV EDVHG RQ WKH OLQHDU YHUVLRQ RI WKUHH WR QLQH QRGH VKHOO HOHPHQW IURP WKH VLPSOH DQDO\VLV 6,03$/f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f1,1)&/67$7/67)/67& 1'2)'.2'.73523f56f56f:*f:*f10$7 ,703/'$/3+$5()7(0,702)),'80f 1*5 *URXS QXPEHU IRU FXUUHQW HOHPHQWV IRU VKHOO 1(/6 1XPEHU RI FXUUHQW HOHPHQWV 0)67 (OHPHQW QXPEHU RI WKH ILUVW HOHPHQW LQ FXUUHQW JURXS ,*5+('f (OHPHQW JURXS KHDGLQJ 1,1)& /HQJWK RI FRPPRQ EORFN ,1)*5 LQ WHUPV RI LQWHJHU ZRUGV /67$7 /HQJWK RI VWDWH LQIRUPDWLRQ /67) /HQJWK RI VWLIIQHVV FRQWURO LQIRUPDWLRQ /67& /HQJWK RI VWLIIQHVV LQIRUPDWLRQ 1'2) 1XPEHU RI HOHPHQW GHJUHHV RI IUHHGRP '.2 ,QLWLDO VWLIIQHVV GDPSLQJ IDFWRU '.7 &XUUHQW WDQJHQW VWLIIQHVV GDPSLQJ IDFWRU '.2 '.7 DUH QRW XVHG IRU VKHOO HOHPHQW EXW DUH LQFOXGHG LQ ,1)*5 IRU WKH FRPSDWLELOLW\ ZLWK EDVH SURJUDP 7KH HOHPHQW SURSHUW\ DUUD\ (3523,222f LQ WKH ODEHOOHG FRPPRQ EORFN LQIJU RI EDVH SURJUDP KDV EHHQ GLYLGHG LQWR VPDOO EORFNV RI JURXS LQIRUPDWLRQ 3523f (OHPHQW SURSHUW\ DUUD\ ZKLFK FDUULHV PAGE 178 56 f DQG 56f 7KH FRRUGLQDWHV RI *DXVV SRLQWV LQ ORFDO UVV\VWH[Q IRU WKH HLJKW SRLQW DQG IRXU SRLQW QXPHULFDO LQWHJUDWLRQ UHVSHFWLYHO\ 7KH QXPHULFDO LQWHJUDWLRQ LV QHHGHG LQ WKH HYDOXDWLRQ RI HOHPHQW VWLIIQHVV LQLWLDO ORDGV DQG VWUHVVHV 6HOHFWLYH QXPHULFDO LQWHJUDWLRQ KDV EHHQ XVHG :*f DQG :*f 1XPHULFDO ,QWHJUDWLRQ ZHLJKWV 10$7 7RWDO QXPEHU RI PDWHULDO SURSHUWLHV ,703/' 7HPSHUDWXUH ORDG LQGH[ IRU LQSXW ,QSXW WHPSHUDWXUH RI WRS DQG ERWWRP VLGHV 7KH VDPH WHPSHUDWXUH RYHU WKH WRS RU ERWWRP VLGHV ,QSXW GLIIHUHQW WHPSHUDWXUHV DW HDFK QRGH $/3+$ &RHIILFLHQW RI WKHUPDO H[SDQVLRQ 5()7(0 5HIHUHQFH WHPSHUDWXUH IRU WHPSHUDWXUH FKDQJH 7KLV LV QHHGHG IRU WKH FDOFXODWLRQ RI LQSODQH WKHUPDO VWUDLQV ,702)) ,QGH[ WR LJQRUH WKH HIIHFW RI WHPSHUDWXUH LQ ORFDO U RU V GLUHFWLRQ WR PRGHO EHDPW\SH VWUXFWXUHV ,JQRUH WHPS ORDGLQJ LQ UGLUHFWLRQORFDO [f ,JQRUH WHPS ORDGLQJ LQ VGLUHFWLRQORFDO \f ,'80f 'XPP\ LQWHJHU DUUD\ WR PDNH WKH OHQJWK RI WKH WRWDO HOHPHQW JURXS SURSHUW\ H[DFWO\ &20021,1)(/,0(0.67/0f1'(f110$715,37+.;;f< PAGE 179 ).3f ZLWK WKH OHQJWK RI LQWHJHU ZRUGV IURP UHDO ZRUGV ,0(0 &XUUHQW HOHPHQW QXPEHU .67 &RQWURO YDULDEOH IRU VWLIIQHVV FKDQJH 6WLIIQHVV KDV EHHQ FKDQJHG 6WLIIQHVV KDV QRW EHHQ FKDQJHG /0f /RFDWLRQ PDWUL[ WKDW FRQWDLQV JOREDO GHJUHH RI IUHHGRP QXPEHUV FRUUHVSRQGLQJ WR HOHPHQW GHJUHH RI IUHHGRP QXPEHUV 12'(f 1RGH QXPEHUV UHDUUDQJHG IRU IRUPXODWLRQ 11 1XPEHU RI QRGHV WR f RI WKH HOHPHQW 15,3 1XPEHU RI QXPHULFDO LQWHJUDWLRQ SRLQWV 0$7 0DWHULDO SURSHUW\ QXPEHU 7+. (OHPHQW WKLFNQHVV ;; f 1RGDO FRRUGLQDWHV LQ JOREDO [ << f 1RGDO FRRUGLQDWHV LQ JOREDO \ == f 1RGDO FRRUGLQDWHV LQ JOREDO ] 77f 7UDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ IRU WKH ORFDO ; \ ] YHFWRUV ; PAGE 180 67f (OHPHQW VWUHVV UHFRYHU\ DUUD\ IRU OLQHDU YHUVLRQ 6WUHVVHV FDQ EH UHFRYHUHG IURP ORFDO HOHPHQW QRGDO GLVSODFHPHQWV DV IROORZV >VWUHVV@ >(@>VWUDLQ@ >(@>G@>X@ >(@>G@>I@>T@ >(@> % @>T@ >67 @>T@ ZKHUH >(@ 6WUHVVVWUDLQ PDWUL[ >&@ LQ WKH SURJUDP >G@ 'LIIHUHQWLDO RSHUDWRU UHODWLQJ VWUDLQ DQG >X@ >X@ *HQHULF GLVSODFHPHQW DW D SRLQW ZLWKLQ DQ HOHPHQW >I@ 'LVSODFHPHQW IXQFWLRQ >T@ 1RGDO GLVSODFHPHQW LQ ORFDO FRRUGLQDWH V\VWHP 'LVSODFHPHQW >X@ DW HOHPHQW QRGHV ).3f 8SSHU WULDQJXODU SRUWLRQ RI HOHPHQW VWLIIQHVV RU FKDQJH LQ HOHPHQW VWLIIQHVVf &20021 :25. 112'(f,'80,-*f7(0f 112'(f (OHPHQW QRGH QXPEHUV IRU UHDGLQ ,'80 ,QWHJHU GXPP\ YDULDEOH WR PDNH WKH OHQJWK HYHQ ,-*f 1XPEHU RI HOHPHQWV WR EH JHQHUDWHG LQ ORFDO L DQG M GLUHFWLRQ LQFOXGLQJ WKH RQH VSHFLILHG 7(0f 7HPSRUDU\ DUUD\ IRU WHPSHUDWXUH LQIRUPDWLRQ ',0(16,21 1'.2'1-7f;1-7f<1-7f=1-7f,5'f 12'*(1f712'(f 1'.2' ,QGH[ DUUD\ RI JOREDO GHJUHH RI IUHHGRP QXPEHUV FRUUHVSRQGLQJ WR HOHPHQW GHJUHHV RI IUHHGRP QXPEHUV 1-7 1XPEHU RI MRLQWVQRGHVf LQ VWUXFWXUH ;1-7f ;FRRUGLQDWHV RI DOO QRGHV <1-7f PAGE 181 712'(f 7RS DQG ERWWRP WHPSHUDWXUHV DW QRGHV 7KH DUUD\ LRUGf KDV WKH RUGHU RI HOHPHQW QRGH QXPEHUV XVHG IRU WKH IRUPXODWLRQ ZKLFK LV GLIIHUHQW IURP WKH RUGHU XVHG IRU GDWD UHDGLQ 7KLV DUUD\ ZLOO EH XVHG IRU WKH UHDUUDQJHPHQW RI HOHPHQW QRGH QXPEHUV IRU IRUPXODWLRQ DW WKH WLPH RI GDWD LQSXW 7KLV DUUD\ PXVW EH GLPHQVLRQHG EHFDXVH WKH DUUD\ LV ILOOHG E\ GDWD VWDWHPHQW DQG WKXV FDQQRW EH LQFOXGHG LQ WKH FRPPRQ EORFN 2WKHU YDULDEOHV XVHG LQ WKLV VXEURXWLQH DUH 12'(f RUGHUHG QRGH QXPEHUV IRU IRUPXODWLRQ :* LQWHJUDWLRQ ZHLJKWV ,(/ QXPEHU RI HOHPHQWV WR EH JHQHUDWHG LQ LGLUHFWLRQ -(/ QXPEHU RI HOHPHQWV WR EH JHQHUDWHG LQ MGLUHFWLRQ ,1& QRGH QXPEHU LQFUHPHQW LQ LGLUHFWLRQ IRU HOHPHQW JHQHUDWLRQ -1& QRGH QXPEHU LQFUHPHQW LQ MGLUHFWLRQ IRU HOHPHQW JHQHUDWLRQ ,-*f QXPEHU RI HOHPHQWV WR EH JHQHUDWHG LQFOXGLQJ WKH HOHPHQW VSHFLILHG LQ L ,-*Off DQG M ,-*ff GLUHFWLRQ (OHPHQW JHQHUDWLRQ ZRUNV RQO\ IRU UHFWDQJXODU HOHPHQWV 7KH HOHPHQW QXPEHU RI WKH ILUVW VKHOO HOHPHQW 0)67 LV VHW WR RQH XQOHVV RWKHUZLVH VSHFLILHG 7KH HOHPHQW VWLIIQHVV LV FXUUHQWO\ FRQVWDQW IRU WKH OLQHDU VKHOO HOHPHQW DQG WKHUHIRUH WKH VWLIIQHVV FKDQJH LQGH[ .67 LV VHW WR ]HUR 7KH ZRUG OHQJWKV RI HOHPHQW LQIRUPDWLRQ JURXSV DUH ILJXUHG RXW /HQJWK RI HOHPHQW VWDWH LQIRUPDWLRQ YDULDEOHV /67$7 OHQJWK RI VWLIIQHVV FRQWURO YDULDEOHV /67& OHQJWK RI PAGE 182 VWLIIQHVV LQIRUPDWLRQ YDULDEOHV /67) DQG WRWDO OHQJWK RI FRPPRQ EORFN LQIHO 1,1)& DUH FDOFXODWHG 7KH QXPEHU RI HOHPHQW GHJUHHV RI IUHHGRP LQ JOREDO FRRUGLQDWH V\VWHP 1'2) LV VHW WR 7KHUH DUH VL[ GRIV SHU QRGH LH WKUHH WUDQVODWLRQV DQG URWDWLRQV 7KXV WKH WRWDO QXPEHU RI HOHPHQW GRIV LV GRIV IRU QLQH QRGHV 'HIDXOW LQWHJUDWLRQ ZHLJKW LV VHW WR LI RSWLRQDO LQWHJUDWLRQ ZHLJKW LV QRW SURYLGHG WKURXJK LQSXW 7KH VKHOO HOHPHQW XVHV SRLQW *DXVV 4XDGUDWXUH ,QWHJUDWLRQ FRQVWDQWV DUH FDOFXODWHG EDVHG RQ RSWLRQDO RU GHIDXOW LQWHJUDWLRQ ZHLJKW DQG VDYHG LQ :*f IRU QXPHULFDO LQWHJUDWLRQ $: LV WKH SULPDU\ LQWHJUDWLRQ SRLQW FDOFXODWHG DQG LV WR EH XVHG LQ WKH VWUHVV FDOFXODWLRQ 7KH DUUD\ RI LQWHJUDWLRQ SRLQW FRRUGLQDWHV LQ WKH ORFDO V\VWHP LV EXLOW DQG VDYHG LQ 56f )RXU SRLQW LQWHJUDWLRQ SDUDPHWHUV DUH FDOFXODWHG LQ ORFDO FRRUGLQDWHV DQG VDYHG LQ 56f ZLWK WKH FRUUHVSRQGLQJ ZHLJKWV :*f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f HOHPHQWV DUH JHQHUDWHG LQ L GLUHFWLRQ LMJf HOHPHQWV DUH JHQHUDWHG LQ MGLUHFWLRQ LQFOXGLQJ WKH HOHPHQW VSHFLILHG 7KXV ,-* RU f f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f 1HZO\ LQWURGXFHG YDULDEOHV DUH $$f 'LUHFWLRQ FRVLQHV RI UD[LV ORFDO [f PAGE 185 %%f 'LUHFWLRQ FRVLQHV RI VD[LV ORFDO \f &&f 'LUHFWLRQ FRVLQHV RI WD[LV ORFDO ]f 7KLV VXEURXWLQH SURFHVVHV HOHPHQW GDWD HOHPHQW E\ HOHPHQW (OHPHQW QXPEHU LV XSGDWHG DQG WKH QXPEHU RI HOHPHQWV SURFHVVHG LV FRXQWHG IRU LQSXW FRQWURO 'LVSWf INSf VHOIOGf OPf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rM IRU ULJKWKDQGHG FRRUGLQDWH V\VWHP /RFDO M YHFWRU PXVW WKHQ EH PRGLILHG XVLQJ WKH VHFRQG YHFWRU FURVV M NrL 7KH WUDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ FDQ WKHQ EH DVVHPEOHG IURP WKH XQLW ORFDO FRRUGLQDWH YHFWRUV DV IROORZV PAGE 186 G\Of )LJ %O /RFDO &RRUGLQDWH 6\VWHP DQG 'LPHQVLRQV PAGE 187 Ln DDOfL M n EEOf L Nn FFOf L Ln DDOf Mn EEOf Nn FFf DDfM DDfN EEfM EEfN FFfM FFfN DDf DDf EEf EE f FFf FFf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f 7KLV VXEURXWLQH ILQGV WKH XQLW YHFWRU RI D JLYHQ YHFWRU VSHFLILHG E\ WKH FRRUGLQDWHV RI VWDUWLQJ SRLQW DQG HQGLQJ SRLQW 9f LV GLPHQVLRQHG DQG ;L PAGE 190 IRU D PHPEUDQH HOHPHQW DQG WKHQ URWDWHV WKHP LQWR JOREDO FRRUGLQDWH V\VWHP &20021 :25. +f(%f%f3+,fWHPf 3+,2Of ,QLWLDO VWUDLQ GXH WR WHPSHUDWXUH ORDGLQJ IRU PHPEUDQH SRUWLRQ RI VKHOO HOHPHQW 7(0f 7HPSRUDU\ DUUD\ IRU WKH URWDWLRQ RI WHPSHUDWXUH ORDG WR JOREDO FRRUGLQDWH V\VWHP ',0(16,21 LQLf LQf LQfWQRGHfFPf '$7$ LQ LQ O &0f &RQVWLWXWLYH PDWUL[ IRU PHPEUDQH HOHPHQW 7KH GDWD LQ FRQWDLQ WKH QXPEHUV FRUUHVSRQGLQJ WR LQn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f WHPSOGf WHPf DUH LQLWLDOL]HG 7KH HOHPHQW SURSHUWLHV QHHGHG LQ WKLV VXEURXWLQH DUH UHFRYHUHG 7KHVH DUH PAGE 191 3RLVVRQn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f LV FDOFXODWHG 5HIHU WR IUPVWO IRU GHWDLOV 7KH LQLWLDO VWUDLQ SKLROf IRU WKH PHPEUDQH SRUWLRQ RI WKH VKHOO LV FRPSXWHG XVLQJ JLYHQ WHPSHUDWXUHV 7KH PHPEUDQH KDV RQO\ WKH WKUHH LQSODQH VWUDLQV RXW RI WKH ILYH VWUDLQ FRPSRQHQWV RI D VKHOO HOHPHQW ,I XQLIRUP WHPSHUDWXUH GLIIHUHQWLDO LV JLYHQ WKH WHPSHUDWXUH GLIIHUHQWLDO WR FDOFXODWH LQLWLDO VWUDLQV IRU WKH PHPEUDQH HOHPHQW LV WKH GLIIHUHQFH EHWZHHQ DYHUDJH WHPSHUDWXUH DW QHXWUDO VXUIDFH DQG WKH JLYHQ UHIHUHQFH WHPSHUDWXUH 7KH WHPSHUDWXUH GLIIHUHQFH IRU PHPEUDQH VWUDLQV DW WKH FXUUHQW LQWHJUDWLRQ SRLQW '(/7, LV VXP RI WKH GLIIHUHQFH PAGE 192 LQ HDFK FRQWULEXWLRQ RI WKH WHPSHUDWXUH GLIIHUHQFH DW HDFK QRGH 7KLV FRQWULEXWLRQ FDQ EH IRXQG E\ PXOWLSO\LQJ WKH QXPHULFDO YDOXH RI WKH VKDSH IXQFWLRQ DW WKH FXUUHQW LQWHJUDWLRQ SRLQW DQG WKH FRUUHVSRQGLQJ WHPSHUDWXUH GLIIHUHQFH DW WKH QRGH FRQVLGHUHG (DFK GLIIHUHQFH LV WKH DYHUDJH WHPSHUDWXUH PLQXV UHIHUHQFH WHPSHUDWXUH IRU PHPEUDQH VWUDLQV 2WKHUZLVH WKH GLIIHUHQFH LV FDOFXODWHG DQG PXOWLSOLHG E\ WKH FRUUHVSRQGLQJ VKDSH IXQFWLRQ IRU WKH FRQWULEXWLRQ RI WKH WHPSHUDWXUH GLIIHUHQWLDO IRU WKH FXUUHQW QRGH 7KHVH FRQWULEXWLRQV DUH WKHQ VXPPHG XS IRU DOO WKH QRGHV 7KH LQSODQH LQLWLDO VWUDLQ DUUD\ GXH WR WHPSHUDWXUH FKDQJH 3+,f LV LQLWLDOL]HG DQG FDOFXODWHG 7KH LQLWLDO VWUDLQ FDQ EH FDOFXODWHG DV WKH WHPSHUDWXUH GLIIHUHQFH WLPHV WKH WKHUPDO H[SDQVLRQ FRHIILFLHQW IRU WKH PDWHULDO VSHFLILHG 7KH WKHUPDO VWUDLQ FRPSRQHQW IRU LQSODQH VKHDU Â•[\ LV ]HUR EHFDXVH WKH WHPSHUDWXUH GLIIHUHQWLDO LV WKH VDPH LQ ORFDO [ DQG \ GLUHFWLRQV 7KLV PHDQV WKDW QR LQn SODQH VKHDU VWUHVVHV ZLOO EH LQWURGXFHG E\ WKH XQLIRUP WHPSHUDWXUH GLIIHUHQWLDO 2QFH WKH LQLWLDO LQSODQH VWUDLQV DUH FDOFXODWHG HTXLYDOHQW QRGDO ORDGLQJ DQG LQLWLDO VWUHVVHV FDQ EH IRXQG WKURXJK QXPHULFDO LQWHJUDWLRQ 7HPSHUDWXUH ORDGV DUH WKH QHJDWLYH YDOXHV RI WKRVH UHFRYHUHG IURP WKH LQLWLDO VWUDLQ GXH WR WHPSHUDWXUH ORDGV EHFDXVH HTXLYDOHQW ORDGf LV VHW WR WKH QHJDWLYH YDOXHV RI PAGE 193 WHPSOGf LQ WKH VXEURXWLQHV DQVUVWDWLFWHPSOGI DQG DQVUORDGHOIUFI (TXLYDOHQW QRGDO ORDGLQJ LV FDOFXODWHG WKURXJK QXPHULFDO LQWHJUDWLRQ 7KH WKLFNQHVV WHUP LV LQFOXGHG LQ WKH LQWHJUDWLRQ ZHLJKW 7KH QXPHULFDO FDOFXODWLRQV IRU HDFK LQWHJUDWLRQ SRLQW DUH DV IROORZV 7KH VWUDLQ GLVSODFHPHQW PDWUL[ >%@ LV IRUPHG E\ FKRRVLQJ SURSHU WHUPV IURP +f HYDOXDWHG E\ D FDOO WR )250+ VXEURXWLQH 7KH HTXLYDOHQW QRGDO ORDGV GXH WR WHPSHUDWXUH FKDQJH DUH WKHQ REWDLQHG WKURXJK WKH QXPHULFDO LQWHJUDWLRQ RI >%@WUDQVSRVHr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f IRU WKH PHPEUDQH SRUWLRQ RI WKH VKHOO LV IRUPHG 7KH LQLWLDO VWUHVV PDWUL[ IRU FXUUHQW *DXVV SRLQW LV FDOFXODWHG WKURXJK WKH QXPHULFDO LQWHJUDWLRQ RI >(@>SKLRO@ RYHU WKH DUHD % 6XEURXWLQH 7/,67712'(f 7KLV VXEURXWLQH IRUPV WKH HTXLYDOHQW QRGDO ORDGV DQG LQLWLDO VWUHVVHV RI SODWH HOHPHQW GXH WR WHPSHUDWXUH ORDGV LQ ORFDO FRRUGLQDWH V\VWHP DQG ILOOV WKHVH LQWR WKH VKHOO HOHPHQW WHPSHUDWXUH ORDGV DQG LQLWLDO VWUHVVHV IRU WKH FRPELQDWLRQ ZLWK SODQH PHPEUDQH WHPSHUDWXUH ORDGV DQG LQLWLDO VWUHVVHV &20021 :25. EfHEfKfSKLRfWHPf 3+,f ,QLWLDO VWUDLQ RI SODWH EHQGLQJ HOHPHQW GXH WR WHPSHUDWXUH ORDGLQJ ',0(16,21 LQfLQfWQRGHfFSfFf &3f &RQVWLWXWLYH PDWUL[ RI SODWH EHQGLQJ HOHPHQW &f &RQVWLWXWLYH PDWUL[ ZLWK WKLFNQHVV WHUPV WDNHQ RXW IRU VWUHVV FDOFXODWLRQ '$7$ LQ O PAGE 195 LQ 7KH GDWD LQ FRQWDLQ WKH QXPEHUV FRUUHVSRQGLQJ WR LQn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f LV IRUPHG E\ D FDOO WR VXEURXWLQH (/$: (OHPHQW SURSHUWLHV QHHGHG LQ WKLV VXEURXWLQH DUH UHFRYHUHG 7KHVH DUH PAGE 196 WKURXJK WKH FKRLFH RI SURSHU WHUPV IURP +f PDWUL[ 7KH LQLWLDO WKHUPDO VWUDLQ SKLRf GXH WR WKH WHPSHUDWXUH ORDGV LV FDOFXODWHG 7HPSHUDWXUH ORDGV DUH WKH QHJDWLYH YDOXHV RI WKRVH UHFRYHUHG IURP WKH LQLWLDO VWUDLQ GXH WR WHPSHUDWXUH ORDGV EHFDXVH HTXLYDOHQW ORDGf LV VHW WR WKH QHJDWLYH YDOXHV RI WHPSOGf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f 7KH FDOFXODWLRQ SURFHGXUHV DUH DV IROORZV *HQHUDO VWUHVVHV DUH GHILQHG DV PRPHQW UHVXOWDQW RYHU WKH WKLFNQHVV SHU XQLW OHQJWK DQG WKXV KDYH WKH XQLW RI PRPHQW SHU XQLW OHQJWK ZKLOH FRPPRQ VWUHVVHV KDYH WKH XQLW RI IRUFHV SHU XQLW DUHD 7KH FRQYHUVLRQ FDQ EH GRQH E\ UHPRYLQJ WKLFNQHVV WHUPV IURP FRQVWLWXWLYH PDWUL[ 7KH UHVXOWLQJ PDWUL[ KDV EHHQ QDPHG &f 7KH IDFWRU LV GLYLGHG IRU PD[LPXP VKHDU DW FHQWHU OLQH EHFDXVH WKH IRUPXODWLRQ JLYHV RQO\ DYHUDJH WUDQVYHUVH VKHDU VWUHVVHV ZKLOH WKH DFWXDO GLVWULEXWLRQ LV D SDUDEROD 7KH UHPDLQLQJ SURFHVVHV DUH LGHQWLFDO ZLWK WKRVH IRU PHPEUDQH HOHPHQW PAGE 198 % 6XEURXWLQH 6(/):7 7KLV VXEURXWLQH IRUPV HTXLYDOHQW QRGDO ORDGV IRU WKH VHOI ZHLJKW DSSOLHG LQ WKH GLUHFWLRQ RI JUDYLW\ LQ JOREDO FRRUGLQDWH V\VWHP :*+7 LV VHOI ZHLJKW LQ ORFDO WKLFNQHVV GLUHFWLRQ SHU XQLW WKLFNQHVV ZKLFK ZDV VWRUHG LQ SURSPDWf 6HOI ZHLJKW LV GLYLGHG LQWR ORFDO UVWFRPSRQHQWV XVLQJ WKH WUDQVIRUPDWLRQ PDWUL[ 7KH GLUHFWLRQ RI JUDYLW\ LQ JOREDO FRRUGLQDWHV LV GHILQHG DV IROORZV DSSOLFDWLRQ GLUHFWLRQ ; < = ; < = ,*5$9' 7 J /HW EH ORFDO FRPSRQHQWV RI VHOI ZHLJKW EUEVEWf WUDQVSRVH 7 EH XVXDO WUDQVIRUPDWLRQ PDWUL[ DQG J EH JOREDO VHOI ZHLJKW ZKLFK KDV RQO\ RQH FRPSRQHQW LQ WKH GLUHFWLRQ RI JUDYLW\ IRU H[DPSOH ^E\! WUDQVSRVH ZKHUH E\ LV VHOI ZHLJKWZJKWf 7KHUHIRUH WKH UHODWLRQVKLS FDQ EH VKRZQ DV EU WOfrZJKW ORFDO [ FRPSRQHQWf EV WfrZJKW ORFDO \ FRPSRQHQWf EW WfrZJKW ORFDO ] FRPSRQHQWf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f WKH DUUD\ RI VKDSH IXQFWLRQV KOLf IÂ KLf IÂ[ KLf IA\f ,QWHJUDWLRQ ZHLJKW LV UHWULHYHG IRU WKH FXUUHQW LQWHJUDWLRQ SRLQW 7KH HTXLYDOHQW ORFDO QRGDO ORDGV GXH WR VHOI ZHLJKW DUH FDOFXODWHG E\ LQWHJUDWLQJ ORFDO FRPSRQHQWV RI JUDYLW\ XVLQJ VKDSH IXQFWLRQV DQG FRUUHVSRQGLQJ LQWHJUDWLRQ ZHLJKW 7KHVH DUH WKHQ WUDQVIRUPHG LQWR D JOREDO FRRUGLQDWH V\VWHP XVLQJ WKH WUDQVIRUPDWLRQ PDWUL[ % 6XEURXWLQH )5067 7KLV VXEURXWLQH IRUPV WKH VWUHVVQRGDO GLVSODFHPHQW DUUD\ >67@ IRU PHPEUDQH HOHPHQW &20021 :25. +f%f&3f +f DUH VKDSH IXQFWLRQV DQG WKHLU GHULYDWLYHV ZLWK UHVSHFW WR JOREDO [ DQG \ IRU HDFK QRGH XS WR QLQH QRGHV QRGH URZ IA IÂ IJ URZ I[ fÂ§ IL[ fÂ§ IJ[ URZ I\ ILI\ IJ\ PAGE 200 %f LV VWUDLQQRGDO GLVSODFHPHQW PDWUL[ IRU PHPEUDQH HOHPHQW DQG FSf LV FRQVWLWXWLYH ODZ IRU PHPEUDQH HOHPHQW 7KH GDWD LQf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f QRQ]HUR VWUHVVHV ;; \\ [\ SRVVLEOH VWUHVVHV ;; \\ == [\ \] ][ 7KH GDWD LQf KDV WKH QXPEHUV FRUUHVSRQGLQJ WR WZR LQSODQH WUDQVODWLRQDO GHJUHHV RI IUHHGRP DW QLQH QRGHV RXW RI GHJUHHV RI IUHHGRP IRU QLQH QRGH HOHPHQW 7KLV VKRZV WKH FROXPQ QXPEHU LQ >67@ DUUD\ IRU PHPEUDQH VWUHVVHV 7KH QXPEHUV IRU QRGH QXPEHU RQH DUH H[LVWLQJPHPEUDQHf GRIV [ \ SRVVLEOH GRIV [ \ ] [[ \\ ]] LQ JOREDO V\VWHP PAGE 201 7KH QXPEHU RI FROXPQV LQ VWUHVVQRGDO GLVSODFHPHQW PDWUL[ LV WZR WLPHV WKH QXPEHU RI HOHPHQW QRGHV IRU PHPEUDQH HOHPHQW 7KHUH DUH WZR WUDQVODWLRQDO GHJUHHV RI IUHHGRP SHU QRGH 7KH HOHPHQW SURSHUWLHV PAGE 202 LH HOOf Hf Hf Hf Hf Hf f XXOI f8 XX88 f 8R R8R 8L A8R R 8 L f A n8n f f f QO8? 88c 8OXf 7KLV FDQ EH UHDUUDQJHG DV IROORZV HOOf GOO Hf G Hf G Hf G GOO Hf G G Hf G GOO XO 8 8 ZKHUH Grr LV D GLIIHUHQWLDO RSHUDWRU 7KXV WKH VL]H RI >G@ LV VL[ E\ WKUHH LQ JHQHUDO 7KH JHQHUDO UHODWLRQVKLS EHWZHHQ JHQHULF GLVSODFHPHQWV DQG QRGDO GLVSODFHPHQWV FDQ EH VKRZQ DV IROORZV XO IOO X IOO X IOO TOO T T T T T f f f ZKHUH IOO ^IO I I I I I I I I` PAGE 203 DQG TOO ^TO T TO TO T T T T T` WUDQVSRVH T T T TO T T T T T Tf WUDQVSRVH T ^T T TO T T T T T Tf WUDQVSRVH T ^T TO2 TO T T T T T T` WUDQVSRVH T ^T TOO TO T T T T T Tf WUDQVSRVH T ^T TO TO T T T T T Tf WUDQVSRVH 7KLV VKRZV WKH VL]H RI >I@ ZKLFK LV VL[ E\ ILIW\IRXU 7KH VKDSH IXQFWLRQV DQG WKHLU GHULYDWLYHV ZLWK UHVSHFW WR WR JOREDO [ DQG \ DW WKH FXUUHQW LQWHJUDWLRQ SRLQW DUH FDOFXODWHG E\ FDOOLQJ )250+ VXEURXWLQH )RU GHWDLOV UHIHU WR WKH VXEURXWLQH )250+ 7KH VWUDLQQRGDO GLVSODFHPHQW PDWUL[ >%@ LV WKHQ IRUPHG E\ VHOHFWLQJ SURSHU WHUPV IURP +f ,Q VXEURXWLQH )250+ DOO WKH FRPSRQHQWV RI >%@ PDWUL[ KDYH EHHQ FDOFXODWHG +HUH WKH FRUUHFW FRPSRQHQWV IURP +f DUH VLPSO\ SODFHG DW WKH SURSHU ORFDWLRQV LQ >%@ Ef Af L O QRGH L URZO I[[ URZ I\ URZ IOU\ IOI[ , , R ILI\ , IL}< IIU; 7KH VWUHVVGLVSODFHPHQW PDWUL[ >67@ LV HTXDO WR >(@>%@ IRU PHPEUDQH HOHPHQW DQG LV SODFHG DW WKH FRUUHFW SODFHV LQ WKH VKHOO >67@ XVLQJ GDWD LQ DQG LQ 7KH EDVH LQGH[ LL VKRZV WKH ORFDWLRQ RI URZ QXPEHU IRU FXUUHQW VWUHVV LQ GDWD LQ PAGE 204 % 6XEURXWLQH )5067 7KLV VXEURXWLQH IRUPV WKH VWUHVVQRGDO GLVSODFHPHQW PDWUL[ >67@ IRU WKH SODWH SRUWLRQ RI WKH VKHOO HOHPHQW &20021 :25. &3f&f+f%f(%f &3f LV FRQVWLWXWLYH ODZ IRU WKH SODWH EHQGLQJ HOHPHQW IRU VWLIIQHVV IRUPXODWLRQ DQG Ff LV FRQVWLWXWLYH ODZ IRU SODWH EHQGLQJ HOHPHQW IRU VWUHVV UHFRYHU\ 7KLFNQHVV WHUPV KDYH EHHQ VWULSSHG %f LV VWUDLQQRGDO GLVSODFHPHQW PDWUL[ IRU WKH SODWH EHQGLQJ HOHPHQW DQG HEf LV DQ WHPSRUDU\ DUUD\ IRU >(@>%@ FDOFXODWLRQ 7KH GDWD LQf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f QRQ]HUR VWUHVVHV ;; \\ [\ \] ][ SRVVLEOH VWUHVVHV ;; \\ ]] [\ \] ][ PAGE 205 7KH GDWD LQf KDV WKH QXPEHUV FRUUHVSRQGLQJ WR RQH RXWRISODQH WUDQVODWLRQDO GHJUHH RI IUHHGRP DQG WZR RXWRIn SODQH URWDWLRQDO GHJUHHV RI IUHHGRP RI SODWH HOHPHQW DW QLQH QRGHV RXW RI GHJUHHV RI IUHHGRP DYDLODEOH IRU QLQH QRGH HOHPHQW 7KLV VKRZV WKH FROXPQ QXPEHU LQ >67@ DUUD\ IRU SODWH VWUHVVHV 7KH QXPEHUV IRU QRGH QXPEHU RQH LV H[LVWLQJSODWHf HOHPHQW GRIV [ \ ] [[ \\ ]] LQ JOREDO V\VWHP 7KH QXPEHU RI FROXPQV LQ VWUHVVQRGDO GLVSODFHPHQW PDWUL[ LV WKUHH WLPHV QXPEHU RI HOHPHQW QRGHV IRU SODWH HOHPHQW 7KHUH DUH WKUHH GHJUHHV RI IUHHGRP SHU QRGH $IWHU WKH DUUD\V Ef KfFf KDYH EHHQ LQLWLDOL]HG WKH HOHPHQW SURSHUWLHV QHHGHG IRU FRQVWLWXWLYH ODZ DUH UHFRYHUHG 7KHVH DUH PAGE 206 7KH SURFHGXUHV DUH DV IROORZV WKH VKDSH IXQFWLRQV WKHLU GHULYDWLYHV -DFRELDQ DQG LWV LQYHUVH DUH IRUPHG ILUVW 7KH VWUDLQGLVSODFHPHQW PDWUL[ Ef LV IRUPHG XVLQJ WKH SURSHU WHUPV IURP Kf Ef ELf L QRGH f f f L r f f URZO IO[ f f f IL[ URZ R eOI\ f f f R IMB\ URZ R I[ IL\ r r r R IL[ ILn< URZ I; IL r r r ILI[ IL URZ IOn< IO f W ILn< IL 7KH VWUHVVGLVSODFHPHQW PDWUL[ >67@ IRU FXUUHQW *DXVV SRLQW LV FDOFXODWHG DGGHG XS DQG WKHQ SODFHG LQWR SURSHU SODFHV RI VWUHVV UHFRYHU\ DUUD\ >67@ RI VKHOO HOHPHQW XVLQJ GDWD LQURZf DQG LQFROXPQf % 6XEURXWLQH 67,),67(31')&'.2&'.7).,1').,67)&A 7KLV VXEURXWLQH FDOFXODWHV WKH VKHOO HOHPHQW VWLIIQHVV E\ FRPELQLQJ WKH VWLIIQHVVHV RI ERWK WKH PHPEUDQH HOHPHQW DQG SODWH EHQGLQJ HOHPHQW 7KH VKHOO HOHPHQW VWLIIQHVV IRUPHG LQ WKH ORFDO FRRUGLQDWH V\VWHP LV WKHQ URWDWHG WR WKH JOREDO FRRUGLQDWH V\VWHP ,W LV QRWHG WKDW VWLIIQHVV LV D VHFRQG RUGHU WHQVRU DQG WKHUHIRUH LW IROORZV D GLIIHUHQW WUDQVIRUPDWLRQ ODZ IURP WKH RQH XVHG IRU IRUFHV RU GLVSODFHPHQWV ZKLFK DUH D ILUVW RUGHU WHQVRU D YHFWRUf PAGE 207 >JOREDO VWLIIQHVV@ >7 WUDQVSRVH@>ORFDO VWLIIQHVV@>7@ &20021 :25. 6f 7KH GLPHQVLRQ VWDWHPHQW GHILQHV JOREDO VKHOO HOHPHQW VWLIIQHVV ). FRQVWLWXWLYH PDWULFHV RI PHPEUDQH DQG SODWH HOHPHQWV &0 DQG &3 ',0(16,21 ).fFPfFSf ). FXUUHQW JOREDO HOHPHQW VWLIIQHVV PDWUL[ )./ 6 FXUUHQW ORFDO HOHPHQW VWLIIQHVV PDWUL[ ).3 SUHYLRXV JOREDO HOHPHQW VWLIIQHVV ).& FXUUHQW JOREDO HOHPHQW VWLIIQHVV ,67)& VWLIIQHVV PDWUL[ FRQWHQW LQGH[ VXSSOLHG E\ WKH EDVH SURJUDPf WRWDO HOHPHQW VWLIIQHVV PDWUL[ FKDQJH LQ HOHPHQW VWLIIQHVV PDWUL[ ,I WKH FKDQJH LQ HOHPHQW VWLIIQHVV LV ORFDO WKH XQFKDQJHG HOHPHQW VWLIIQHVV FDQ EH VWRUHG DQG RQO\ WKH FKDQJHG SRUWLRQ PD\ EH FDOFXODWHG DQG XSGDWHG ,1'). ). VWRUDJH LQGH[ ORZHU KDOI FRPSDFWHG FROXPQZLVH VTXDUH DOOf ,67(3 FXUUHQW VWHS QR LQ VWHSE\VWHS LQWHJUDWLRQ IRUP G\QDPLF VWLIIQHVV IRUP G\QDPLF VWLIIQHVV DW VWDUW RI QHZ SKDVH IRUP VWDWLF VWLIIQHVV IRUP VWDWLF DQG JHRPHWULFf VWIQV DW VWDUW 1'2) QXPEHU RI HOHPHQW GHJUHHV RI IUHHGRP 7KH VWLIIQHVV PDWUL[ VWRUDJH VFKHPH ,1'). LV VHW WR ]HUR ,I WKH ORZHU KDOI FRPSDFWHG FROXPQZLVH FDQ EH XVHG LW LV EHWWHU IRU WKH V\PPHWULF HOHPHQW VWLIIQHVV PDWUL[ EXW WKLV FDXVHV VRPH SUREOHP 7KHUHIRUH ,1'). LV VHW WR IRU VTXDUH PDWUL[ VWRUDJH VFKHPH PAGE 208 $OO WKH HOHPHQW SURSHUWLHV DUH UHFRYHUHG IRU WKH FDOFXODWLRQ RI FRQVWLWXWLYH PDWULFHV 7KHVH DUH PAGE 209 LV VHW WR RQH IRU WKH ILUVW LWHUDWLRQ IRU IXOO HOHPHQW VWLIIQHVV )URP WKH VXEVHTXHQW LWHUDWLRQ LW LV VHW WR ]HUR RQO\ IRU WKH FKDQJH LQ HOHPHQW VWLIIQHVV E\ EDVH SURJUDP % 6XEURXWLQH 6+67)OIFQ 7KLV VXEURXWLQH IRUPV ORZHU WULDQJXODU SRUWLRQ RI HOHPHQW VWLIIQHVV LQ ORFDO FRRUGLQDWH V\VWHP IRU PHPEUDQH HOHPHQW DQG SXWV WKHP LQWR IXOO ORFDO VKHOO HOHPHQW VWLIIQHVV 6f )LYH GHJUHHV RI IUHHGRP SHU QRGH WLPHV QLQH QRGHV JLYHV D WRWDO GHJUHHV RI IUHHGRP RI $QG KHQFH WKH VL]H RI HOHPHQW VWLIIQHVV PDWUL[ LV E\ &20021 :25. 6f &f VWUHVVVWUDLQ ODZ IRU PHPEUDQH HOHPHQW 6f IXOO HOHPHQW VWLIIQHVV PDWUL[ IRU VKHOO HOHPHQW (OHPHQW VWLIIQHVV LV WKH LQWHJUDWLRQ RI >%@WUDQVSRVH>(@>%@f RYHU HOHPHQW YROXPH 7KHUHIRUH WKH SURFHGXUH ZLOO EH DV IROORZV 7KH VKDSH IXQFWLRQV WKHLU GHULYDWLYHV DQG -DFRELDQ PDWUL[ DUH IRUPHG WR HYDOXDWH VWUDLQQRGDO GLVSODFHPHQW PDWUL[ >%@ 7KH HOHPHQW VWLIIQHVV LV FDOFXODWHG E\ QXPHULFDO LQWHJUDWLRQ RI >%@WUDQVSRVH>(@>%@ RYHU HOHPHQW YROXPH XVLQJ WKH FRQVWLWXWLYH PDWUL[ EURXJKW LQ WKURXJK & 7KH GDWD LQL FRQWDLQ WKH QXPEHUV RI GHJUHHV RI IUHHGRP RI VKHOO HOHPHQW ZKLFK FRUUHVSRQG WR WKH GHJUHHV RI IUHHGRP PAGE 210 RI PHPEUDQH HOHPHQW 7KHVH DUH WZR LQSODQH WUDQVODWLRQDO GHJUHHV RI IUHHGRP 7KH ORRS RYHU LQWHJUDWLRQ SRLQWV IRU VWLIIQHVV FDOFXODWLRQ LV DV IROORZV f )RUP VKDSH IXQFWLRQV f &DOFXODWH FXUUHQW LQWHJUDWLRQ ZHLJKW IDFWRU f &DOFXODWH VWUDLQGLVSODFHPHQW PDWUL[ >%@ f 3HUIRUP *DXVV TXDGUDWXUH RQ SRLQW UV WR IRUP VWLIIQHVV PDWUL[ >%@WUDQVSRVH>(@>%@ ZKHUH >(@ >FPf@ f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f FRQVWLWXWLYH PDWUL[ HP PAGE 212 WKN SODWH WKLFNQHVV 7KH SODWH WKLFNQHVV LV DVVXPHG WR EH FRQVWDQW DQG LV WDNHQ RXW RI WKH LQWHJUDWLRQ DQG HQWHUHG KHUH 7KH FRQVWDQWV IRU SODWH VWUHVVVWUDLQ PDWUL[ DUH FSf FS FSO FS FS FS ZKHUH HD WKN r WKN r WKN GHP SRL r SRLf HHG HPrHDGHP FSO HHG FS SRL r HHG FS J r HD FS J r WKN % 6XEURXWLQH )250+5612'(;<+'-11f 7KLV VXEURXWLQH IRUPV IRXU WR QLQH QRGH VKDSH IXQFWLRQV DQG WKHLU GHULYDWLYHV LQ UVVSDFH WKHQ WUDQVIRUPV WKHP LQWR [\VSDFH WKURXJK WKH LQYHUVH RI -DFRELDQ PDWUL[ ',0(16,21 12'(f; PAGE 213 ; PAGE 214 DW FHQWHU RI WKH HGJH LV RQH KDOI 7KLV FDQ EH GRQH E\ VXEWUDFWLQJ RQH KDOI RI WKH VKDSH IXQFWLRQV IRU WKH DGMDFHQW FHQWHU QRGHV RQ WKH HGJH RI WKH HOHPHQW IURP WKH HDFK RI WKH VKDSH IXQFWLRQ IRU WKH IRXU FRUQHU QRGHV 7KH UHVXOWV DUH VWRUHG LQ WKH DUUD\ +f +OLf DUH WKH QXPHULFDO YDOXH RI WKH VKDSH IXQFWLRQ HYDOXDWHG DW WKH FXUUHQW LQWHJUDWLRQ SRLQW +Lf DQG +Lf LV WKH QXPHULFDO YDOXHV RI WKH GHULYDWLYHV RI FXUUHQW VKDSH IXQFWLRQ ZLWK UHVSHFW WR WR ORFDO YDULDEOH U DQG V UHVSHFWLYHO\ 2QFH VKDSH IXQFWLRQV DUH HYDOXDWHG -DFRELDQ PDWUL[ FDQ EH FDOFXODWHG DV IROORZV [ VXPIM/r[Âf [U VXPIAUr[Âf \ VXPILr\Lf \U VXPLIAA\Mf [ VXPIAr[Âf [V VXPIAVr[Âf \ VXPILr\Lf \V VXPILVr\Lf ZKHUH VXPTW\Lff VXP RI TW\Lf RYHU L OQQ IÂ WKH ILUVW URZ RI Kf PDWUL[ IAU WKH VHFRQG URZ RI Kf PDWUL[ IAV WKH WKLUG URZ RI Kf PDWUL[ [Â WKH ILUVW URZ RI [\f PDWUL[ \Â WKH VHFRQG URZ RI [\f PDWUL[ PAGE 215 ,W LV QRWHG WKDW [Â \A DUH QRW WKH DFWXDO JOREDO FRRUGLQDWHV RI QRGH L RI WKH HOHPHQW EXW WKH\ DUH ORFDO HOHPHQW JHRPHWU\ FRRUGLQDWHV 7KHVH YDOXHV FDQ EH IRXQG E\ WKH GRW SURGXFW RI ORFDO D[LV XQLW GLUHFWLRQ YHFWRU EHIRUH EHLQJ PDSSHG LQWR WKH QDWXUDO FRRUGLQDWHV DQG WKH YHFWRU IURP WKH ORFDO RULJLQ DQG WKH QRGH XQGHU FRQVLGHUDWLRQ >-@ [U \U [V \V 7KH LQYHUVLRQ RI WZR E\ WZR -DFRELDQ PDWUL[ FDQ EH GRQH DV IROORZV 7KH GHWHUPLQDQW RI -DFRELDQ PDWUL[ LV IRXQG 'f DQG 'f DUH LQWHUFKDQJHG DQG WKH VLJQV DUH FKDQJHG WR WKH RSSRVLWH IRU WKH WHUPV 'Of DQG 'Of $OO WKHVH YDOXHV DUH WKHQ GLYLGHG E\ WKH GHWHUPLQDQW RI -DFRELDQ PDWUL[ 7KH HOHPHQW FRQQHFWLYLW\ FDQ EH FKHFNHG XVLQJ WKH GHWHUPLQDQW RI -DFRELDQ PDWUL[ 7KH GHWHUPLQDQW PXVW EH JUHDWHU WKDQ ]HUR IRU WKH SURSHUO\ FRQQHFWHG HOHPHQWV ,I WKH GHWHUPLQDQW LV OHVV WKDQ RU HTXDO WR ]HUR DQ HUURU PHVVDJH LV ZULWWHQ 7KH SDUWLDO GLIIHUHQWLDO RI VKDSH IXQFWLRQV ZLWK UHVSHFW WR JOREDO FRRUGLQDWHV [ DQG \ FDQ EH FDOFXODWHG DV IROORZV IL[ IAU+A[f ILVfV[f ZKHUH IÂU KLf U[ LQY-OOf IÂV KLf V[ LQY-f PAGE 216 IL}< IAU+A\f ILIVfV\f ZKHUH IAU KLf U\ LQY-f IAV KLf V\ LQY-f ZKHUH LQYLV WKH LQYHUVH RI WKH -DFRELDQ PDWUL[ % 6XEURXWLQH *'%,%*'f E &RRUGLQDWH RI ORFDO U RU V RI WKH FXUUHQW LQWHJUDWLRQ SRLQW LE &RRUGLQDWH RI ORFDO U RU V RI WKH FXUUHQW QRGH J &RQWULEXWLRQ WR VKDSH IXQFWLRQ LQ WKH FXUUHQW U RU V GLUHFWLRQ G 'HULYDWLYH RI J ZLWK UHVSHFW WR U RU V GLUHFWLRQ )RU ERWWRP FRUQHU QRGHV %fr )RU FHQWHU QRGHV DORQJ WKH HGJH RI HOHPHQW %r% r% )RU WRS FRUQHU QRGHV %fr % 6XEURXWLQH 75$16 1177)./).*` 7KLV VXEURXWLQH SHUIRUPV ORFDOJOREDO WUDQVIRUPDWLRQV RI HOHPHQW VWLIIQHVV QQ 1XPEHU RI QRGHV IRU WKH FXUUHQW VKHOO HOHPHQW PAGE 217 WWf 7UDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ 6RPHWLPHV FDOOHG DV >$@ PDWUL[ IRU WKH VHFRQG RUGHU WHQVRU INOf /RFDO VKHOO HOHPHQW VWLIIQHVV 7KLV KDV D VL]H RI E\ 7KHUH DUH ILYH GRIV DW HDFK QRGH WKUHH WUDQVODWLRQDO GRIV DQG WZR RXWRISODQH URWDWLRQDO GRIV 7KH PD[LPXP QXPEHU RI HOHPHQW QRGHV LV QLQH $QG KHQFH ORFDO GRIV INJf *OREDO VKHOO HOHPHQW VWLIIQHVV 7KLV KDV D VL]H RI E\ ,Q JOREDO FRRUGLQDWH V\VWHP HYHQ RQH ORFDO URWDWLRQDO GRI PD\ KDYH WKUHH FRPSRQHQWV LQ JOREDO FRRUGLQDWH V\VWHP DQG WKXV WKHUH DUH VL[ JOREDO GRIV DW HDFK QRGH 7KH PD[LPXP QXPEHU RI HOHPHQW QRGHV LV QLQH $QG KHQFH ORFDO GRIV ,Vf /RFDWLRQ PDWUL[ WR SXW ORFDO E\ LQWR JOREDO E\ VWLIIQHVV PDWUL[ GRIV KDV EHHQ GLYLGHG LQWR JURXSV RI WKUHH RUWKRJRQDO GRIV DQG ZLOO EH WUDQVIRUPHG LQ EORFNV RI WKH WKUHH GRIV ,Wf /RFDWLRQ PDWUL[ IRU WUDQVIRUPDWLRQ PDWUL[ IRU WKH VWLIIQHVV WUDQVIRUPDWLRQ LQ EORFNV RI WKUHH )RU WKH WKUHH WUDQVODWLRQDO GRIV ZH QHHG DOO WKH WKUHH URZV RI WUDQVIRUPDWLRQ PDWUL[ )RU WKH WZR RXWRISODQH URWDWLRQDO GRIV ZH QHHG WZR URZV RI WKUHH E\ WKUHH WUDQVIRUPDWLRQ PDWUL[ 6WLIIQHVV WUDQVIRUPDWLRQ LQ EORFNV RI LV GRQH WKURXJK WKH VXEURXWLQH 75,3/ H[SDQGLQJ [ ORFDO VWLIIQHVV WR [ JOREDO VWLIIQHVV $V WKH DERYH FDOFXODWLRQ LV GRQH RQO\ IRU ORZHU KDOI RI WKH PDWUL[ WKH XSSHU SDUW RI VWLIIQHVV LV ILOOHG XS XVLQJ V\PPHWU\ PAGE 218 % 6XEURXWLQH 75,3/(/,/01..77$$$151&, 151&A 7KLV VXEURXWLQH FDOFXODWHV WKH PDWUL[ WULSOH SURGXFW DD WWUDQVSRVHf r D r W LQ EORFNHG IRUP [f $V WKH WUDQVIRUPDWLRQ PDWUL[ KDV EHHQ IRUPHG LQ LWV WUDQVSRVH WKH SURGXFW EHFRPHV DD WWrDrWWWUDQVSRVHf ZKHUH DQUOQFOf /RFDO HOHPHQW VWLIIQHVV PDWUL[ DDQUQFf *OREDO HOHPHQW VWLIIQHVV PDWUL[ WWf 7UDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ WNf 7HPSRUDU\ DUUD\ WR FDUU\ WKH UHVXOWV RI WKH ILUVW PXOWLSOLFDWLRQ WWfrFRUUHVSRQGLQJ Df ,QGLFHV RI HOHPHQW VWLIIQHVV PDWUL[ LQ ORFDO V\VWHP PQ ,QGLFHV RI WUDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ NO N ,QGLFHV RI HOHPHQW VWLIIQHVV PDWUL[ LQ JOREDO V\VWHP QUO 1XPEHU RI URZV LQ Df QFO 1XPEHU RI FROXPQV LQ D QU 1XPEHU RI URZV LQ DD QF 1XPEHU RI FROXPQV LQ DD 7KH ILUVW PXOWLSOLFDWLRQ WWrD LV SHUIRUPHG IROORZHG E\ WKH VHFRQG PXOWLSOLFDWLRQ >WWrD@r>WWWUDQVSRVHf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f 7KLV VXEURXWLQH LV IRU VWDWH GHWHUPLQDWLRQ FDOFXODWLRQV &20021 67/'37 7KLV FRPPRQ EORFN ZLOO EH XVHG WR ILQG ORDG DSSOLFDWLRQ IDFWRU LQ 5,17 41')f ,1&5(0(17$/ JOREDO QRGDO GLVSODFHPHQWV 7KHUHIRUH LI WRWDO VWUHVVHV DUH QHHGHG WKH LQFUHPHQWDO GLVSODFHPHQWV DUH WR EH DGGHG XS DW WKH HQG RI HDFK LWHUDWLRQ 4'/f ,1&5(0(17$/ HOHPHQW QRGDO GLVSODFHPHQWV )$&$/f WRWDO ORDG DSSOLFDWLRQ IDFWRU )$&,/Of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f@ >67f@>Jf@ 7KH LQLWLDO VWUHVVHV GXH WR WHPSHUDWXUH ORDGLQJ DUH VXEWUDFWHG IURP WKH FDOFXODWHG WRWDO VWUHVVHV 7KH VWUHVVHV DW IRXU FRUQHU QRGHV DUH FDOFXODWHG OLQHDU H[WUDSRODWLRQ IURP WKH VWUHVVHV HYDOXDWHG DW WKH LQWHJUDWLRQ SRLQWV DQG SXW EDFN LQWR VWUHVV DUUD\ VLJf % 6XEURXWLQH 5,17 I1')29(/)()'7,0()$&$/ )$&,/$/)$f 7KLV VXEURXWLQH FDOFXODWHV HOHPHQW IRUFHV LQ WKH JOREDO FRRUGLQDWH V\VWHP &20021 67/'37131)13))137313313'31(731(6:1,765 163$77 QSQI 1XPEHU RI QRGDO IRUFH SDWWHUQV PAGE 221 QSII 1XPEHU RI IROORZHU IRUFH SDWWHUQV QSWS 1XPEHU RI QRGDO WHPSHUDWXUH SDWWHUQV QSGS 1XPEHU RI QRGDO GLVSODFHPHQW SDWWHUQV QHWS 1XPEHU RI WHPSHUDWXUH ORDGLQJV WKURXJK HOHPHQW GDWD QHZV 1XPEHU RI VHOI ZHLJKW ORDGLQJV WKURXJK HOHPHQW GDWD &20021 :25. 5)7(0fINWHPf UIWHPf 7HPSRUDU\ DUUD\ IRU HOHPHQW IRUFH URWDWLRQ INWHPf 7HPSRUDU\ DUUD\ IRU HOHPHQW VWLIIQHVV WUDQVIRUPDWLRQ 41')f 12'$/ ',63/$&(0(176 9(/1')f 12'$/ 9(/2&,7< )'1')f '<1$0,& 12'$/ )25&( :+(1 7,0( )(1')f (/$6723/$67,& 12'$/ )25&( :+(1 7,0( 77 WUDQVSRVH RI WUDQVIRUPDWLRQ PDWUL[ 7KH HOHPHQW IRUFHV LQ WKH JOREDO FRRUGLQDWH V\VWHP FDQ EH FDOFXODWHG DV >)( JOREDO@ >.H JOREDO@>4 JOREDO@ $V WKH HOHPHQW VWLIIQHVV LV LQ JOREDO FRRUGLQDWH V\VWHP DQG VWRUHG LQ WKH DUUD\ ).3f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f LV IRUPHG LQ LQLWO 1HJDWLYH YDOXH RI DFWXDO WHPSOGf KDV EHHQ FDOFXODWHG EHFDXVH WKH EDVH SURJUDP XVHV WHPSOGf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f &RQWURO LQIRUPDWLRQ f )LUVW FRQWURO OLQH &2/8016 1$0( '$7$ f 1*5 (OHPHQW JURXS QXPEHU LRLf 1(/6 1XPEHU RI VKHOO HOHPHQWV a f 0)67 (OHPHQW RI ILUVW VKHOO HOHPHQW )f '.2 ,QLWLDO VWLIIQHVV GDPSLQJ IDFWRU )f '.7 7DQJHQW VWLIIQHVV GDPSLQJ IDFWRU $f 2SWLRQDO KHDGLQJ IRU VKHOO HOHPHQW f 6HFRQG FRQWURO OLQH &2/8016 1$0( '$7$ f 10$7 1XPEHU RI VKHOO PDWHULDO SURSHUWLHV f ,*5$9' 'LUHFWLRQ RI JUDYLW\ f ,703/' 7\SH RI WHPSHUDWXUH ORDG (f $/3+$ 7KHUPDO H[SDQVLRQ FRHIILFLHQW )f 5()7(0 5HIHUHQFH WHPSHUDWXUH f ,702)) 7HPSHUDWXUH ORDG WXUQRII LQGH[ )f :*7 ,QWHJUDWLRQ ZHLJKW GHIDXOW f 1RWHV ,*5$9' GLUHFWLRQ RI JUDYLW\ JOREDO ; < = ; < = LJUDYG ,703/' LQGH[ IRU WHPSHUDWXUH ORDG LQSXW WRS DQG ERWWRP WHPSHUDWXUHV RQO\ LQSXW WHPSHUDWXUHV IRU DOO WKH QRGHV HOVH QR WHPSHUDWXUH HIIHFWV QR LQSXW IRU WKH $+3+$5()7(0,702)) ,702)) LQGH[ WR WXUQ RII WHPSHUDWXUH HIIHFW LQ ORFDO [ GLUHFWLRQ A[[ f LQ ORFDO \ GLUHFWLRQ \\ f f 0DWHULDO SURSHUW\ GDWD 10$7 VHWV RI PDWHULDO SURSHUWLHV &2/8016 1$0( '$7$ f 0$7 )f ( )f 32, )f 6KHOO PDWHULDO SURSHUW\ QXPEHU PAGE 224 )f :*+7 6HOI ZHLJKW SHU FXELF IRRW f (OHPHQW GDWD 1(/6 VHWV RI HOHPHQW GDWD &2/8016 1$0( '$7$ f 1(/ (OHPHQW QXPEHU f 0$7 0DWHULDO SURSHUW\ QXPEHU f 12'(f 1RGH QXPEHUV IRU PLVVLQJ QRGHf )f 7+. (OHPHQW WKLFNQHVV GHIDXOW f f ,-*f 1XPEHU RI HOHPHQW WR EH JHQHUDWHG ,I ,703/' LV DGG D VHFRQG OLQH WR HOHPHQW GDWD )f 703723 7HPSHUDWXUH DW WRS VXUIDFH )f 703%27 7HPWHUDWXUH DW ERWWRP VXUIDFH ,I ,703/' LV DGG WZR OLQHV WR HOHPHQW GDWD )f 7HPSHUDWXUH DW WRS VXUIDFH IRU QLQH QRGHV )f )f 7HPSHUDWXUH DW ERWWRP VXUIDFH IRU QLQH QRGHV )f 1RWHV 12'(f HOHPHQW QRGH QXPEHUV r r r r r r r r r ,-*f QXPEHU RI HOHPHQWV WR EH JHQHUDWHG LQ L DQG M GLUHFWLRQ LQFOXGLQJ WKH HOHPHQW VSHFLILHG ZRUNV RQO\ IRU QLQH QRGH HOHPHQWVf ,I QR WHPSHUDWXUH HIIHFWV DUH GHVLUHG IRU VSHFLILF HOHPHQW LQSXW WKH VDPH YDOXHV IRU 703723 DQG 703%27 LH 5()7(05()7(0f PAGE 225 $33(1',; & ,03/(0(17$7,21 121/,1($5 6+(// (/(0(17 7KLV DSSHQGL[ GHVFULEHV WKH LPSOHPHQWDWLRQ RI D LQFUHPHQWDO QRQOLQHDU ILQLWH VKHOO HOHPHQW LQWR $165,,, SURJUDP 2QO\ WKRVH IHDWXUHV WKDW DUH GLIIHUHQW IURP WKRVH RI D OLQHDU VKHOO HOHPHQW LQ VXEURXWLQHV ,1(/ 67,) 12167) 67$7 675(66 5,17 DUH GHDOW ZLWK )RU GHWDLOHG HTXDWLRQV DQG HOHPHQWV RI PDWULFHV UHIHU WR FKDSWHUV IRXU DQG ILYH &O 6XEURXWLQH ,1(/I1-71'.2';<=.(;(&f 7KH ORFDO FRRUGLQDWHV UVf DQG WKH ZHLJKWV ZJf IRU WKH VWDQGDUG ; *DXVV LQWHJUDWLRQ ZHUH DGGHG WR ,1)*5 IRU WKH FDOFXODWLRQ RI VWLIIQHVVHV VWUHVVHV DQG LQWHUQDO UHVLVWLQJ IRUFHV )RU WKH VROXWLRQ RI WKH LQFUHPHQWDO HTXLOLEULXP HTXDWLRQ LQ WRWDO /DJUDQJLDQ GHVFULSWLRQ WKH WRWDO GLVSODFHPHQWV DQG WKH WRWDO GLVSODFHPHQW JUDGLHQWV RI SUHYLRXV LWHUDWLRQ DUH UHTXLUHG DQG WKXV VWRUHG LQ ,1)(/ 7KHVH DUH XWf WKURXJK D\\Wf 6RPH DUUD\V IRU WKH H[SOLFLW LQWHJUDWLRQ WKURXJK WKH WKLFNQHVV DUH DOVR DGGHG 7KHVH DUH WKNO\UO2f WKURXJK GKFf $FFRUGLQJO\ WKH PAGE 226 OHQJWK RI HOHPHQW LQIRUPDWLRQ EORFN /67$7 ZDV FKDQJHG 7KH FRQWURO YDULDEOHV IRU QRQOLQHDU DQDO\VLV DQG OD\HUHG HOHPHQW DUH ,/ 1XPEHU RI OD\HUV ./' /DUJH GLVSODFHPHQW QRQOLQHDULW\ LQGH[ RQ RII .*0 *HRPHWULF QRQOLQHDULW\ LQGH[ RQ RII .0$7 0DWHULDO QRQOLQHDULW\ LQGH[ QRW XVHGf RQ RII .8/ 0RWLRQ GHVFULSWLRQ LQGH[ 7RWDO /DJUDQJLDQ IRUPXODWLRQ 8SGDWHG /DJUDQJLDQ IRUPXODWLRQQRW XVHGf 7KH FRRUGLQDWH IRU [ LQWHJUDWLRQ LV DZ EZ DZ 7KH OD\HU LQIRUPDWLRQ WKDW FDQ EH KDQGOHG FXUUHQWO\ LV OD\HUV ZLWK GLIIHUHQW WKLFNQHVVHV DQG PDWHULDOV QO\U 1XPEHU RI OD\HUV WKNO\UO2f /D\HU WKLFNQHVV PDWO\UO2f /D\HU PDWHULDO PAGE 227 & 6XEURXWLQH (9(/1-71'.2' ; < = 12'*(1 712'(,&175,1(55` 7KH HOHPHQW LV GLYLGHG LQWR OD\HUV IRU H[SOLFLW LQWHJUDWLRQ RYHU HOHPHQW WKLFNQHVV W WR Wf ,QWHJUDWLRQV QHHGHG DUH KWRS KWRS G] ] G] DQG KERW KERW Z KW ] KERW i3 ]A G] ZKHUH KWRS 7KH FRRUGLQDWH RI WRS RI WKH OD\HU KERW 7KH FRRUGLQDWH RI ERWWRP RI WKH OD\HU 7KXV WKH FRRUGLQDWHV RI WRS DQG ERWWRP RI HDFK OD\HU DUH UHTXLUHG DQG WKHVH DUH FDOFXODWHG DQG VWRUHG LQ KKOOf &XUUHQWO\ WKH FRRUGLQDWH RI WKH FHQWHU OLQH LV VHW WR ]HUR %XW WKLV FDQ EH FKDQJHG IRU DUELWUDU\ ORFDWLRQ DORQJ WKH WKLFNQHVV GLUHFWLRQ LI DQ LQSXW SDUDPHWHU LV JLYHQ 7KH H[SOLFLW LQWHJUDWLRQ RYHU WKH WKLFNQHVV EHFRPHV >KWRS KERW@ >KWRS KERWf@ DQG >fKWRS KERW-f@ 7KHVH DUH FDOFXODWHG DQG VWRUHG LQ GKVf GKFf ,W LV QRWHG WKDW >KWRS KERW@ LV HTXDO WR WKH OD\HU WKLFNQHVV LWVHOI 7KH VWUHVVQRGDO GLVSODFHPHQW PDWUL[ >67@ FDQQRW EH XVHG DV LW FKDQJHV FRQWLQXRXVO\ LQ LQFUHPHQWDO QRQOLQHDU DQDO\VLV >67@ LV GHILQHG DV ^VWUHVV` >67@^QRGDO GLVSODFHPHQW` PAGE 228 & 68%5287,1( 67,),67(31')&'.2&'.7 ). ,1'). ,67)&f 7KH SODWH VWLIIQHVV PDWUL[ >'@ LV FDOFXODWHG E\ SHUIRUPLQJ LQWHJUDWLRQ RI FRQVWLWXWLYH PDWUL[ >(@ WKURXJK WKLFNQHVV OD\HU E\ OD\HU XVLQJ WKH VXEURXWLQHV (/$: DQG (/$: 7KH LQFUHPHQWDO QRQOLQHU HOHPHQW VWLIIQHVV LV FDOFXODWHG E\ FDOOLQJ WKH VXEURXWLQH 12167) 7KH URWDWLRQ RI WKH HOHPHQW VWLIIQHVV DQG WKH VWRUDJH VFKHPH DUH LGHQWLFDO WR WKRVH IRU OLQHDU VKHOO HOHPHQW & 6XEURXWLQH 12167) GPGSGVOYUf 7KH VWDQGDUG [ *DXVV TXDGUDWXUH LV XVHG IRU WKH QXPHULFDO LQWHJUDWLRQ RI QRQOLQHDU HOHPHQW VWLIIQHVV 7KH LQFUHPHQWDO QRQOLQHDU HOHPHQW VWLIIQHVV UHTXLUHV WKH WRWDO GLVSODFHPHQW JUDGLHQW Z[f DQG Z\f IURP WKH SUHYLRXV LWHUDWLRQ 7KHVH ZHUH VDYHG LQ Z[Wf DQG Z\Wf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f DQG IL\ff & 6XEURXWLQH 67$7 1')27,0()$&$/)$&,/$/)$f 7KH HOHPHQW VWUHVVHV DUH UHFRYHUHG WKURXJK VXEURXWLQH 675(66 DW LQWHJUDWLRQ SRLQWV XVLQJ WKH LQFUHPHQWDO GLVSODFHPHQWV DQG WKHQ H[WUDSRODWHG WR HOHPHQW QRGHV IRU DQDO\VLV DQG GHVLJQ SXUSRVHV PAGE 230 7KH ORFDWLRQV RI LQWHJUDWLRQ SRLQWV IRU VWUHVV HYDOXDWLRQ DUH ; , r r ; , r ; \f ; $ r L f r ; r ; r r ; r [ [ [f ZKHUH r ,QWHJUDWLRQ SRLQWV [ 1RGH QXPEHU IRU IRUPXODWLRQ 7KLV FDQ EH GRQH XVLQJ WKH VKDSH IXQFWLRQV >@ RU ELOLQHDU H[WUDSRODWLRQ IRU WKH LQSODQH DQG VKHDU VWUHVVHV DQG ELTXDGUDWLF H[WUDSRODWLRQ IRU WKH EHQGLQJ VWUHVVHV $QRWKHU DSSURDFK LV WKH OHDVW VTXDUHV ILW & 6XEURXWLQH 675(66 QGI FUf (OHPHQW VWUHVV HYDOXDWLRQ LV GRQH DW LQWHJUDWLRQ SRLQWV EXW LV QRW D QXPHULFDO LQWHJUDWLRQ 7KXV QR ZHLJKWLQJ IDFWRU LV XVHG 7KH DUUD\V XVHG DUH VWUVOf /D\HU VWUHVVHV DW LQWHJUDWLRQ SRLQWV IRU VWUHVV FRPSRQHQWV 1[ 1\ 1[\ 0[ 0\0[\ 4[ 4\ DQG IRU OD\HUV PAGE 231 HOVWULf ,QFUHPHQWDO HOHPHQW VWUHVVHV RI FRPSRQHQWV DW LQWHJUDWLRQ SRLQWV VWUQLSf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` >/LQHDUL]HG % PDWUL[@QRGDO GLVSODFHPHQWV` >%/=@^T` $V >%O]@ LV WKH IXQFWLRQ RI Z[f DQG Z\f WKHVH LQFUHPHQWDO TXDQWLWLHV Z[f DQG Z\f PXVW EH FDOFXODWHG ILUVW IRU WKH FXUUHQW LQWHJUDWLRQ SRLQW Z[f >VXP RI ILfZLf@[ >VXP RI IL[fZLf@ Z\f >VXP RI ILfZLf@\ >VXP RI IL\fZLf@ PAGE 232 IL[f DQG IL\f DUH HOHPHQW RI Kf DQG ZL LV WKH ORFDO QRGDO GLVSODFHPHQWV WUDQVIRUPHG IURP WKH JOREDO QRGDO GLVSODFHPHQWV $QG WKHVH DUH DGGHG XS WR \LHOG WRWDO GLVSODFHPHQW JUDGLHQWV Z[Wf DQG Z\Wf 7RWDO VWUDLQV VWUQLSf DUH FDOFXODWHG EHFDXVH VRPH IDLOXUH FULWHULRQ UHTXLUHV SULQFLSDO VWUDLQV 7KHVH DUH QRW QHHGHG FXUUHQWO\ EXW ZLOO EH XVHG IRU PDWHULDO QRQOLQHDULW\ 2QFH LQFUHPHQWDO VWUDLQ LV HYDOXDWHG WKHQ LQFUHPHQWDO VWUHVVHV FDQ EH FDOFXODWHG WKURXJK H[SOLFLW LQWHJUDWLRQ DFURVV WKLFNQHVV DQG WKHQ DGGHG XS IRU WKH WRWDO VWUHVVHV & 6XEURXWLQH 5,17 I1')9(/)()'7,0()$&$/ )$&,/$/)$f 7KLV VXEURXWLQH FDOFXODWHV HOHPHQW IRUFHV LQ JOREDO FRRUGLQDWHV )( WKH QXPHULFDO LQWHJUDWLRQ RI >%/=@7^VWUHVV` RYHU WKH DUHD 7KH ^VWUHVV` DUH JHQHUDOL]HG LH WKH LQWHJUDWLRQ DORQJ WKLFNQHVV KDV EHHQ SHUIRUPHG >%/=@ LV HYDOXDWHG LQ WKH VDPH SURFHGXUHV XVHG IRU VWUHVV UHFRYHU\ 7KH LQWHJUDWLRQ VFKHPH LV FRPSDWLEOH ZLWK WKDW RI VWUHVV UHFRYHU\ WRR $ [ LQWHJUDWLRQ IRU WKH LQWHUQDO UHVLVWLQJ IRUFHV IURP LQSODQH DQG EHQGLQJ VWUHVVHV DQG D [ IRU WKRVH IURP VKHDU VWUHVVHV ZHUH XVHG PAGE 233 & 1RQOLQHDU 6KHOO (OHPHQW 'DWD ,QSXW *XLGH f &RQWURO LQIRUPDWLRQ f )LUVW FRQWURO OLQH &2/8016 1$0( '$7$ fÂ§ f 1*5 (OHPHQW JURXS QXPEHU f 1(/6 1XPEHU RI VKHOO HOHPHQWV f 0)67 (OHPHQW RI ILUVW VKHOO HOHPHQW )f '.2 ,QLWLDO VWLIIQHVV GDPSLQJ IDFWRU )f $f '.7 7DQJHQW VWLIIQHVV GDPSLQJ IDFWRU 2SWLRQDO KHDGLQJ IRU VKHOO HOHPHQW f 6HFRQG FRQWURO OLQH &2/8016 1$0( '$7$ fÂ§ f 10$7 1XPEHU RI VKHOO PDWHULDO SURSHUWLHV f ,*5$9' 'LUHFWLRQ RI JUDYLW\ f ,703/' 7\SH RI WHPSHUDWXUH ORDG fÂ§ (f $/3+$ 7KHUPDO H[SDQVLRQ FRHIILFLHQW )f 5()7(0 5HIHUHQFH WHPSHUDWXUH f ,702)) 7HPSHUDWXUH ORDG WXUQRII LQGH[ )f :*7 ,QWHJUDWLRQ ZHLJKW GHIDXOW f f ,/<5 ,QGH[ IRU OD\HU DQDO\VLV OD\HUV RI VDPH WKLFNQHVV DQG PDWHULDO 8VH IRU QR OD\HU DQDO\VLV OD\HUV RI GLIIHUHQW WKLFNQHVVHV DQG PDWHULDOV PP f ./' ,QGH[ IRU ODUJH GLVSODFHPHQW DQDO\VLV QR \HV f .*0 ,QGH[ IRU JHRPHWULF QRQOLQHDU DQDO\VLV QR \HV 1RWH ./' DQG .*0 PXVW EH XQLW\f IRU nJHRPHWULFn QRQOLQHDU DQDO\VLV LI LW LQFOXGHV WKH HIIHFWV RI ODUJH GLVSODFHPHQWV DQG LQLWLDO VWUHVVHV DV XVHG E\ VRPH DXWKRUV f .0$7 ,QGH[ IRU PDWHULDO QRQOLQHDU DQDO\VLV &XUUHQWO\ QRW XVHGf 1RWHV ,*5$9' GLUHFWLRQ RI JUDYLW\ JOREDO ; < = ; < = LJUDYG PAGE 234 ,703/' LQGH[ IRU WHPSHUDWXUH ORDG LQSXW WRS DQG ERWWRP WHPSHUDWXUHV RQO\ LQSXW WHPSHUDWXUHV IRU DOO WKH QRGHV HOVH QR WHPSHUDWXUH HIIHFWV QR LQSXW IRU WKH $/3+$5()7(0,702)) ,702)) LQGH[ WR WXUQ RII WHPSHUDWXUH HIIHFW LQ ORFDO [ GLUHFWLRQ Â[[ f LQ ORFDO \ GLUHFWLRQ M!\\ f f 0DWHULDO SURSHUW\ GDWD 10$7 VHWV RI PDWHULDO SURSHUWLHV &2/8016 1$0( '$7$ f 0$7 6KHOO PDWHULDO SURSHUW\ QXPEHU )f ( PAGE 235 ,I ,703/' LV )f )f DGG WZR OLQHV WR HOHPHQW GDWD 7HPSHUDWXUH DW WRS VXUIDFH IRU QLQH QRGHV )f 7HPSHUDWXUH DW ERWWRP VXUIDFH IRU QLQH QRGHV )f 1RWHV 12'(f HOHPHQW QRGH QXPEHUV r r r r r r r r r ,-*f QXPEHU RI HOHPHQWV WR EH JHQHUDWHG LQ L DQG M GLUHFWLRQ LQFOXGLQJ WKH HOHPHQW VSHFLILHG ZRUNV RQO\ IRU QLQH QRGH HOHPHQWVf ,I QR WHPSHUDWXUH HIIHFWV DUH GHVLUHG IRU VSHFLILF HOHPHQW LQSXW WKH VDPH YDOXHV IRU 703723 DQG 703%27 LH 5()7(05()7(0f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t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t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t 6WUXFWXUHV 9RO 1R SS &ULVILHOG 0 $ 6QDS7KURXJK DQG 6QDS%DFN UHVSRQVH LQ &RQFUHWH 6WUXFWXUHV DQG WKH 'DQJHUV RI 8QGHU ,QWHJUDWLRQ ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS &ULVILHOG 0 $ $Q $UF /HQJWK 0HWKRG LQFOXGLQJ /LQH 6HDUFKV DQG $FFHOHUDWLRQV ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS &ULVILHOG 0 $ $ )DVW ,QFUHPHQWDO,WHUDWLYH 6ROXWLRQ 3URFHGXUH WKDW +DQGOHV n6QDS7KURXJKn &RPSXWHUV t 6WUXFWXUHV 9RO SS 3RZHOO 6LPRQV ,PSURYHG ,WHUDWLRQ 6WUDWHJ\ IRU 1RQOLQHDU 6WUXFWXUHV ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS 0HLURYLWFK / %DUXK + 2Q WKH &KLOHVN\ $OJRULWKP ZLWK 6KLIWV IRU WKH (LJHQVROXWLRQ RI 5HDO 6\PPHWULF 0DWULFHV ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS 3DSDGUDNDNLV 0 3RVW%XFNOLQJ $QDO\VLV RI 6SDWLDO 6WUXFWXUHV E\ 9HFWRU ,WHUDWLRQ 0HWKRGV &RPSXWHUV t 6WUXFWXUHV 9RO 1R SS =LHQNLHZLF] 2 & 7KH )LQLWH (OHPHQW 0HWKRG 0F*URZ +LOO %RRN &RPSDQ\ 8.f /LPLWHG /RQGRQ 8. 'HVDL & 6 $EHO ) ,QWURGXFWLRQ WR WKH )LQLWH (OHPHQW 0HWKRG 9DQ 1RVWUDQG 5HLQROG &RPSDQ\ 1HZ PAGE 241 6KDPHV + DQG '\P & / (QHUJ\ DQG )LQLWH (OHPHQW 0HWKRGV LQ 6WUXFWXUDO 0HFKDQLFV +HPLVSKHUH 3XEOLVKLQJ &RUSRUDWLRQ 1HZ PAGE 242 0HKOKRUQ .ROOHJJHU .HXVHU 0 .ROPDU : 1RQOLQHDU &RQWDFW 3UREOHPVD )LQLWH (OHPHQW $SSURDFK ,PSOHPHQWHG LQ $',1$ &RPSXWHUV t 6WUXFWXUHV 9RO 1R SS :DOUDYHQ & 5HLQKDUGW + : 7KHRU\ DQG ([SHULPHQWV RQ WKH 0HFKDQLFDO %HKDYLRU RI &UDFNV LQ 3ODLQ DQG 5HLQIRUFHG &RQFUHWH 6XEMHFWHG WR 6KHDU /RDGLQJ +HURQ 9RO 12 $ SS %D]DQW = 3 2K % + 'HIRUPDWLRQ RI &UDFNHG 1HW 5HLQIRUFHG &RQFUHWH :DOOV $6&( -RXUQDO RI 6WUXFWXUDO (QJLQHHULQJ -DQ SS 6X]XNL + &KHQ : ) (ODVWLF3ODVWLF )UDFWXUH $QDO\VLV RI &RQFUHWH 6WUXFWXUHV &RPSXWHUV t 6WUXFWXUHV 9RO 1R SS 2ZHQ 5 )LJXHLUDV $ 'DPMDQLF ) )LQLWH (OHPHQW $QDO\VLV RI 5HLQIRUFHG DQG 3UHVWUHVVHG &RQFUHWH 6WUXFWXUHV LQFOXGLQJ 7KHUPDO /RDGLQJ &RPSXWHU 0HWKRGV LQ $SSOLHG 0HFKQLFV DQG (QJLQHHULQJ 9RO SS &KHQ : ) 6X]XNL + &KDQJ 7 < 1RQOLQHDU $QDO\VLV RI &RQFUHWH &\OLQGHU 6WUXFWXUHV XQGHU +\GURVWDWLF /RDGLQJ &RPSXWHUV t 6WUXFWXUHV 9RO SS 0LOIRUG 5 9 6FKQREULFK : & 7KH $SSOLFDWLRQ RI WKH 5RWDWLQJ &UDFN 0RGHO WR WKH $QDO\VLV RI 5HLQIRUFHG &RQFUHWH 6KHOOV &RPSXWHUV t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t 6WUXFWXUHV 9RO 1R SS 6W 3LHWUXV]F]DN 0UR] = )LQLWH (OHPHQW $QDO\VLV RI 'HIRUPDWLRQ RI 6WUDLQ+DUGHQLQJ 0DWHULDOV ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS *XSWD $ $NEDU + $ )LQLWH (OHPHQW IRU WKH $QDO\VLV RI 5HLQIRUFHG &RQFUHWH 6WUXFWXUHV ,IRU 1XPHULFDO 0HWKRGV LQ (QJLQHHULQJ 9RO SS &RRN 5 0DONXV 6 3OHVKD 0 ( &RQFHSWV DQG $SSOLFDWLRQV RI )LQLWH (OHPHQW $QDO\VLV -RKQ :LOH\ t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f > 0 $ + `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f DV ZHOO DV WR WKH PDLQWHQDQFH DQG SUHVHUYDWLRQ RI D GLJLWDO DUFKLYH FRS\ 'LJLWL]DWLRQ DOORZV WKH 8QLYHUVLW\ RI )ORULGD WR JHQHUDWH LPDJH DQG WH[WEDVHG YHUVLRQV DV DSSURSULDWH DQG WR SURYLGH DQG HQKDQFH DFFHVV XVLQJ VHDUFK VRIWZDUH 7KLV JUDQW RI SHUPLVVLRQV SURKLELWV XVH RI WKH GLJLWL]HG YHUVLRQV IRU FRPPHUFLDO XVH RU SURILW 6LJQDWXUH RI &RS\ULJKW +ROGHU ?I&r!F!?&VURRWO $+ VL 3ULQWHG RU 7\SHG 1DPH RI &RS\ULJKW +ROGHL/LFHQVHH 3HUVRQDO LQIRUPDWLRQ EOXUUHG =O =RH" 'DWH RI 6LJQDWXUH 3OHDVH SULQW VLJQ DQG UHWXUQ WR &DWKOHHQ 0DUW\QLDN 8) 'LVVHUWDWLRQ 3URMHFW 3UHVHUYDWLRQ 'HSDUWPHQW 8QLYHUVLW\ RI )ORULGD /LEUDULHV 32 %R[ *DLQHVYLOOH )/ xml record header identifier oai:www.uflib.ufl.edu.ufdc:UF0009018800001datestamp 2009-03-16setSpec [UFDC_OAI_SET]metadata oai_dc:dc xmlns:oai_dc http:www.openarchives.orgOAI2.0oai_dc xmlns:dc http:purl.orgdcelements1.1 xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.openarchives.orgOAI2.0oai_dc.xsd dc:title Nonlinear gap and Mindlin shell elements for the analysis of concrete structures dc:creator Ahn, Kookjoon,dc:type Bookdc:identifier http://www.uflib.ufl.edu/ufdc/?b=UF00090188&v=00001001583924 (alephbibnum)23011869 (oclc)dc:source University of Florida |