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## 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,
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- Gainesville FL
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- 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 )
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- University of Florida
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- University of Florida
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- 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.
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- 001583924 ( alephbibnum )
23011869 ( oclc )
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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. |

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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. 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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. 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