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PAGE 1 1 CHARACTERIZING A REINFORCED CONCRETE CONNECTION FOR PROGRESSIVE COLLAPSE ASSESSMENT By LOO YONG TAN A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2010 PAGE 2 2 20 10 Loo Yong Tan PAGE 3 3 To my lovely wife in Singapore PAGE 4 4 ACKNOWLEDGMENTS I would like to express my gratitude to my thesis advisor, Professor Theodor Krauthammer for h is decisive guidance and advice. I would like to thank Dr Sedar Astarlioglu for his patience with my questions and encouragement. I would like to thank Dr Yim Hy u n Chang for his advice on the finite element software. I am grateful for the Defence Science and Technology Agency, Singapore, for the postgraduate scholarship. I thank all my friends at the Center for Infrastructure Protection and Physical Security and my Singaporean friends in Gainesville for their support and making my stay here a wonderful experience. Lastl y, I would like to thank my wife for her sacrifices and support from my family in Singapore throughout my course. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ...................................................................................................... 4 LIST OF TABLES ................................................................................................................ 7 LIST OF FIGURES .............................................................................................................. 8 LIST OF ABBREVIATIONS .............................................................................................. 11 ABSTRACT ........................................................................................................................ 15 C H APT ER 1 PROBLEM STATEMENT ........................................................................................... 16 Objectives and Scope ................................................................................................. 17 Research Significance ................................................................................................ 17 2 BACKGROUND AND LITERATURE REVIEW .......................................................... 18 Introduction ................................................................................................................. 18 Beam Column Connections ....................................................................................... 18 Two Member Connections / Knee Connections ................................................. 19 Three Member Connections / T Connections ..................................................... 20 Four Member Connections / Interior Connections .............................................. 20 Connection Behavior .................................................................................................. 20 Strut And Tie Model ............................................................................................. 20 Compressive Strut and Truss Mechanism Model ............................................... 21 Equilibrium Criteria ............................................................................................... 21 Connection Design Parameters ................................................................................. 22 Column Longitudinal Reinf orcement ................................................................... 22 Connection Transverse Reinforcement ............................................................... 23 Shear Strength ..................................................................................................... 24 Flexure .................................................................................................................. 26 Development of Reinforcement ........................................................................... 27 Beam Transverse Reinforcement ........................................................................ 28 Plastic Hinge ............................................................................................................... 28 Modeling of Beam Column Connections ................................................................... 30 Experiments and T ests ............................................................................................... 34 Moment Rotation Resistance of Connections ........................................................... 35 Materials ...................................................................................................................... 36 Concrete Compressive and Tensile Strength ..................................................... 36 Reinforcement ...................................................................................................... 37 PAGE 6 6 Finite Element Analysis .............................................................................................. 37 Verification and Validation ................................................................................... 39 The FE Code Abaqus .......................................................................................... 39 Concrete damaged plasticity model ............................................................. 41 Defining parameters of CDP ......................................................................... 43 Classical metal plasticity ............................................................................... 44 Elements ........................................................................................................ 45 Connector elements ...................................................................................... 46 Progressive Collapse .................................................................................................. 47 Summary ..................................................................................................................... 49 3 RESEARCH APPROACH .......................................................................................... 67 Introduction ................................................................................................................. 67 Characterization of RC Beam Column Connection ................................................... 68 Simplified FE Model .................................................................................................... 69 Summary ..................................................................................................................... 70 4 RESULTS AND DISCUSSIONS ................................................................................ 75 Introduction ................................................................................................................. 75 Verification and Validation .......................................................................................... 75 Verification of Abaqus .......................................................................................... 75 Validation of Abaqus ............................................................................................ 76 RC beam subjected to dynamic loads .......................................................... 77 RC beam column connections subjected to monotonic loads ..................... 78 Parameters of CDP .............................................................................................. 80 RC Beam Column Connection in Study .................................................................... 81 Resistance Function of RC Beam Column Connection ............................................ 81 Moment Rotation Resistance Function ............................................................... 83 Replacing Moment Rotation Resistance by Connector ...................................... 84 Verification of Predominantly Structural Based FE Model ........................................ 85 5 CONCLUSION AND DISCUSSION ......................................................................... 100 Summary ................................................................................................................... 100 Conclusions .............................................................................................................. 101 Validations .......................................................................................................... 101 Characterization of Moment Resistance of Interior Connection ....................... 101 A Simplified Structural FE Model ....................................................................... 101 Limitations ................................................................................................................. 101 Recommendations .................................................................................................... 102 LIST OF REFERENCES ................................................................................................. 104 BIOGRAPHICAL SKETCH .............................................................................................. 108 PAGE 7 7 LIST OF TAB LES Table page 2 1 values for beam column connections ................................................................ 65 2 2 Tensile strength of concrete................................................................................... 65 2 3 Occupancy categories ............................................................................................ 65 2 4 Occupancy categories and design requirements .................................................. 66 4 1 Dynamic increase factor ......................................................................................... 99 4 2 Parameters of CDP ................................................................................................ 99 4 3 Ultimate moment rotation of simulations ............................................................... 99 4 4 Comparison of FE models ..................................................................................... 99 PAGE 8 8 LIST OF FIGURES Figure page 2 1 RC beam column c onnections ............................................................................... 49 2 2 Typical monolithic interior beam column connection ............................................ 50 2 3 Kneeconnection subjected to opening loads ....................................................... 50 2 4 Diagonal and radial bars configuration .................................................................. 51 2 5 Typical threemember connection details .............................................................. 51 2 6 Strut andtie model ................................................................................................. 52 2 7 Joint loads and forces on an interior connection due to (a) gravity loads and (b) lateral loads ....................................................................................................... 52 2 8 Strut andtie model ................................................................................................. 53 2 9 Confinement within connections ............................................................................ 53 2 10 Development of r einforcement ............................................................................... 54 2 11 Computation flow chart from finite difference scheme .......................................... 55 2 12 Experimental setup ................................................................................................. 56 2 13 T section testing ..................................................................................................... 56 2 14 T16 Load deflection curve ...................................................................................... 57 2 15 Corner sectio n testing ............................................................................................ 57 2 16 Load deflection curve for U25 ................................................................................ 58 2 17 Interior connection testing ...................................................................................... 58 2 18 Interior connection load deflection curve for U69 .................................................. 59 2 19 Modified Hognestad curve ..................................................................................... 59 2 20 Stress strain curve of reinforcement using bilinear model .................................... 60 2 21 Process of finite element analysis ......................................................................... 60 2 22 Verification pr ocess ................................................................................................ 61 PAGE 9 9 2 23 Validation process .................................................................................................. 61 2 24 Tensile stressstrain curve of CDP ........................................................................ 62 2 25 Compressive stress strain curve of CDP .............................................................. 62 2 26 C3D8R continuum elements .................................................................................. 63 2 27 Beam elements ...................................................................................................... 63 2 28 Simplified frame model ........................................................................................... 64 2 29 Elevation view of RC structure to be subjected to progressive collapse test ....... 64 3 1 Beam column connections of a RC structure ........................................................ 71 3 2 Freebody diagram of Abaqus model .................................................................... 71 3 3 Global rotation ........................................................................................................ 72 3 4 Formation of plastic hinge ...................................................................................... 72 3 5 Rotation points ........................................................................................................ 73 3 6 Simplified structural based FE model .................................................................... 73 3 7 Flowchart of research approach ............................................................................ 74 4 1 Elevation of a common RC structure ..................................................................... 85 4 2 Section properties and details for RC beam 1C .................................................. 86 4 3 Loading time history for beam 1C ........................................................................ 86 4 4 Abaqus model showing 1 inch mesh and reinforcement ...................................... 87 4 5 Stress strain curve of concrete for Beam 1C ....................................................... 87 4 6 Stress strain curve of steel for Beam 1C ............................................................. 88 4 7 Deflection time history for RC beams subjected to dynamic loads ...................... 88 4 8 Concrete stress strain curve for T16, U25 & U69 ................................................. 89 4 9 Reinforcement stress strain curve for T16, U25 & U69 ........................................ 89 4 10 Loaddeflection curve for T16 ................................................................................ 90 4 11 Loaddeflection curve for U25 ................................................................................ 90 PAGE 10 10 4 12 Loaddeflection curve for U69 ................................................................................ 91 4 13 Elevation view of RC structure and interior connection ........................................ 91 4 14 Details of RC beams .............................................................................................. 92 4 15 Details of columns .................................................................................................. 92 4 16 Reinforcement stress strain curve ......................................................................... 93 4 17 Concrete stress strain curve .................................................................................. 93 4 18 Continuum FE model of RC beam column assemblage ...................................... 94 4 19 Positions of me asurement for local rotation .......................................................... 94 4 20 Comparison of dense and simple model of beam column assemblage .............. 95 4 21 Loading of RC beam column assemblage in opposite direction .......................... 95 4 22 Moment rotation resistance functions .................................................................... 96 4 23 Moment rotation resistance functions subjected to gravity load ........................... 96 4 24 Moment rotation resistance functions subjected to opposite load ....................... 97 4 25 Isotropic hardening behavior for connectors ......................................................... 97 4 26 Output for connector for gravity loading ................................................................ 98 4 27 Output for connector for opposite l oading ............................................................. 98 PAGE 11 11 LIST OF ABBREVIATION S cA Current area sA Area of steel reinforcement 0A Initial area va Shear span bb Transvers e width of beam cb Transverse width of column jb Effective connection width B Strain displacement matrix C Constitutive matrix C Damping matrix bd Diameter of reinforcement d Nodal displacements D Nodal displacement vector DL Dead load cE Elastic modulus of concrete bsf Bond stress cf Co ncretes cylinder strength crf Split cylinder strength of concrete Fc Force due to the concrete sf Stress applied to reinforcement Fs Force due to the reinforcement PAGE 12 12 rf Modulus of rupture of concret e tf Direct tensile strength of concrete yf Yield stress of reinforcement yif Yield stress of radial hoops f External nodal load F Force matrix h Depth of column 0K ik Initial rotational stiffness K Ratio of second stress invariant on the tensile meridian to compressive meridian K Element stiffness matrix K Global stiffness matrix l Anchorage length dhl Minimum development length pl Length of plastic hinge LL Live load M Moment M Mass matrix n shape factor N Shape function u Displacement matrix .u Velocity matrix PAGE 13 13 ..u Acceleration matrix uV Design shear force nV Nominal shear strength cw Density of concrete z Distance of critical section to the point of contraflexure of a cantilever Diameter of reinforcement Stress multiplier for reinforcement Rotation of connection 0 Plastic rotation of connection Load factor c Strain of concrete s Strain of reinforcement 0 Strain of concrete corresponding to maximum stress nom Nominal strain pl ln Plastic strain Strain vector Stress vector Dilation angle of concrete Fri ction angle of concrete 0 / 0 fc fb Ratio of unibiaxial compressive strength to intial uniaxial strength of concrete in CDP Eccentricity of plastic potential surface of concrete in CDP PAGE 14 14 Viscosity para meter in CDP true True stress or Cauchy stress nom Nominal stress PAGE 15 15 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CHARACTERIZING A REINFORCED CONCRETE CONNECTION FOR PROGRESSIVE COLLAPSE ASSESSMENT By L oo Yong T an M ay 2010 Chair: Theodor Krauthammer Major: Civil Engineering Advanced numerical tools, such as finite element codes, are requir ed for accurate assessment of reinforced concrete frames for progressive collapse mitigation. Previous studies of moment resisting steel frames have shown that the accuracy in modeling the connections had a large effect on the accuracy of the entire simula tion. This study is focused on a particular type of reinforced concrete (RC) beam column connection subjected to monotonic loading. The main objective is to use a high fidelity continuum model to derive a simplified mechanical model of the connection unde r monot o nic loadings to be used for progressive collapse assessment The approach was validated by comparing simulations for RC elements with available test data. The derived resistance functions for the connections will be incorporated in numerical model of a full scale RC frame that is scheduled for progressive collapse tests. The numerical results will eventually be compared with the test data for assessing the effectiveness of the proposed numerical approach. PAGE 16 16 CHAPTER 1 PROBLEM STATEMENT Reinforced concrete (RC) structures are very common in many count ries due to the easily availability and relatively low cost of the raw materials. The convenience and advantage of insitu construction in which the wet concrete could be made to the contours of different shapes and sizes of the buildings and the boom in population and economic activities further drive up the need for new buildings, especially RC structures. RC beam column connections, however, have complex stress distributions due to load and geometric discontinuities and cannot be described by the classical Beam theory (Krauthammer, 2008) The behavior of beam column connections under abnormal loads is not well understood and only a few models exist in the literature for design of such connections. Wh en subjected to an abnormal loading, such as blast, the beam column connection may become the primary plane of weakness. In essence, the failure may be initiated at the connection even though the loads have not reached the design load levels of the primary structural members. This results in catastrophic consequences in which a portion or the entire structure may fail from progressive collapse of the main structural elements, causing injuries or even fatalities and destruction of properties. Watson and Inke ster (1982) clearly identified the importance of connections relative to the main structural elements here: A joint was studied because it has been seen from bomb damage that joints often sustain more damage than the point of impac t.(Watson and Inkester, 1982) Advanced numerical tools are seldom used for progressive collapse analysis of RC structures because it is computationally too expensive and time consuming to model PAGE 17 17 an entire RC structure for progressive collapse simulation. Currently, engineers desig n buildings against progressive collapse by following guidelines by authorities such as Unified Facilities Criteria (UFC) or General Services Administration (GSA). Objectives and Scope The main objectives of this study are to develop a better understandi ng of the behavior of RC beam column connections under monotonic loads by employing an predominantly continuum typed finite element (FE) model and extract from it a simplified connector model that can be used along structural type elements for fast progre ssive collapse analysis. The scope of this study is limited to the analysis of a particular type of reinforced concrete interior connection of a RC building that is scheduled to be tested for progressive collapse in late 2010 It will ignore the effects o n the connections from the transverse RC beams and slabs. It will also not look into other types of reinforced concrete connections. Research Significance By using the simplified connector model and structural elements to represent the beams and columns, i t is faster and computationally inexpensive to simulate the progressive collapse of a RC structure. This can act as a n additional check for buildings designed to current progressive collapse guidelines PAGE 18 18 CHAPTER 2 BACKGROUND AND LITERATURE REVIEW Introduction The primary structural elements of a reinforced concrete structure can only fully exhibit their design strength and ductility if the connections between these elements are adequately designed to transmit such forces. Due to constraints posed on the d imensions of the connection because of the sizes of the columns and beams framing into it and inadequate detailing of reinforcement, the connections often have reduced ductility compared to the structural members they are adjoining. To ensure properly deta iled joint reinforcements, it is critical to understand the complex forces within the connection that result from the forces in the connecting structural elements. This chapter will look into detail the forces within a connection and the design of the conn ection accordingly to these forces. It will also look at material models, various tests and models, and experiments on beam column connections being carried out by others. Beam Column Connections A beam column joint is defined as that portion of the colum n within the depth of the deepest beam that frames into the column. (ACI 352R 02, 2002) Beam column connections (slabs not shown) can be classified into many types shown in Figure 21. Nilson, Darwin and Dolan (2004) gave a good representation of internal and exterior joints as shown in Figure 2 2. A typical internal joint is made up of all four beams coming together into the column. An exterior joint is formed by either beam 1, 2 and 3 or only beam 1 and 3 coming together into the column. A corner joint i s formed by beam 1 and 2 coming together into the column. According to ACI 352 R02, type 1 connection is a moment resisting connection designed on the basis of the PAGE 19 19 strength in accordance to ACI 31802 for structural members without significant inelastic de formation. Type 2 connects members that are required to have sustained strength and dissipate energy through reversals of deformation into the inelastic range. Two Member Connections / Knee Connections A two member connection or kneeconnection is most co mmonly found at the corner of portal frames. The connection may be subjected to either opening or closing load. MacGregor and Wight (2005) portrayed the behavior for a knee connection under opening loads in Figure 23. T he strut andtie model can be used t o describe the behavior of a knee connection. Nilsson and Losberg (1976) and Balint and Taylor (1972) conducted several tests for a large number of connections with different bar details. The test shown that the efficiency of a joint the ratio between th e theoretical and the measured moment capacities is highest at lower reinforcement ratio. Nilsson (1973) showed that addition of diagonal bars in the connection increases the efficiency of a joint. The diagonal bar resisted the crack from developing at th e inner corner and allows the compressive stress to flow around the corner, thus preventing tensile cracks from occurring. It is shown that the area of reinforcement required for this diagonal bar is half of the area of the main tensile steel. Another effect of the diagonal stirrups is that it helps to increase confinement in the concrete core and thus increase the strength and ductility of the joint. Krauthammer and Ku (1996) developed a numerical model to determine the influence of diagonal and radial rei nforcements (Figure 24 ) on kneeconnections. Diagonal bars were determined to increase the strength of the connection by resisting the diagonal tensile forces at the inside corner. Radial bars act against the diagonal PAGE 20 20 tensile forces across the connection. However, radial bars were found to affect the tensile stress of the diagonal bars. ThreeMember Connections / T Connections A threemember connection or T connection is usually found at roof beam column joints, exterior column connections or at base of r etaining walls (Figure 25) Each T connection will be subjected to both an opening and closing load. Tests by (Balint and Taylor, 1972) and (Taylor, 1974) shown that connections are stronger in closing loads than opening loads. Figure 2 6 indicates the st rut andtie model for the T connection. Four Member Connections / Interior Connections An interior connection or a four member connection is designed to resist forces due to gravity or lateral loadings as shown in Figure 27 The joint must be designed to resist all the forces, including moments, shears and axial loads that the beam and column transfer through the joint. Connection Behavior To ensure connections are being adequately designed, it is critical to understand the mechanics and behavior of the c onnection. Different models of explanation and theories on the behavior of the connection lead to emphasis on different characteristics of the connection and correspondingly different design and detailing requirements. StrutAnd Tie Model Nilson Darwin an d Dolan (2004) illustrated the use of a strut andtie model to analyze the beam column connection as shown in Figure 28 It shows an interior connection subjected to a lateral load causing clockwise moments from the beams balanced by anti clockwise moment s from the columns. The compressive force from the moments develops compressive struts within the joints, balanced by the tension ties. PAGE 21 21 There must be adequate development length for the tension bars to achieve its yield strength. In the case of a corner co nnection, hook anchorages in the form of Lbar or U bar is used. The maximum force experienced in the connection is not the force determined by structural analysis, but it is the force calculated based on the nominal strengths of the connecting members. In this case, the maximum tensile force in the tensile steel is y sf A where sA is the area of the tensile steel while yf is the yield stress of the tensile steel. In a strut andtie model, column ties are required by conventional design procedures to prevent buckling of the column ties, provide containment to the concrete within the joint under compression and prevent cracking which might occur due to the tension force perpendicular to the compression strut. Compressive Strut and Truss Mechanism Model Park and Paulay (1975) idealized the behavior of a connection into two mechanisms, shear transfer by compression strut and shear transfer through truss mechanism, assuming the two mechanisms are additive. The compressive forces and shear forces combined to form the diagonal strut which is resisted by the compressive strength of the concrete. In the truss mechanism, the bond forces in the steel set up diagonal compressive forces and vertical or horizontal tensile forces within the entire panel zone. The diagonal compressive forces can be resisted by the concrete while vertical and horizontal reinforcement are required to resist the tensile forces. Equilibrium Criteria Paulay (1989) and Paulay and Priestly (1992) used the laws of statics or equilibrium criteria to show that connection shear forces due to lateral (seismic) loading PAGE 22 22 caused extensive diagon al cracks in the concrete core which in turn generated significant orthogonal tensile forces. The d iagonal cr acks in the concrete core due to the large lateral force causes a shift in the internal tensile forces along the column and lead to the development of significant tensile forces in all vertical column bars. Tensile concrete strength lost due to the crackin g of the concrete needs to be replaced by orthogonal tensile reinforcement to continually sustain the shear resistance. As a consequence of these effects, significant orthogonal tensile forces are generated. Paulay suggested that the connection shear reinf orcement is required to sustain the diagonal compression field rather than to provide confinement to the compressed concrete. Connection Design Parameters For the connection to perform its intended function to connect the different elements of a structure and fully exhibit the intended strength and ductility, it must be designed to meet the following requirements. Column Longitudinal Reinforcement Park and Paulay (1975), Paulay (1989) and Paulay and Priestly (1992) determined that the tensile forces developed within the concrete core must be resisted by the longitudinal reinforcement of the column. In normal design, the column longitudinal reinforcement that runs through the connection will be sufficiently strong enough to resist the tensile forces developed in the connection, hence the code requirements is more focused towards a uniform distribution of the longitudinal reinforcements within the joint. ACI 352 08 indicat e s that column longitudinal reinforcements need to satisfy CL 10.9.1 and 10.9.2 of ACI 3 180 5 which states : PAGE 23 23 Area for the longitudinal reinforcement, stA should be larger than greater than gA 01 0 (gross sectional area of column) but less than gA 08 0 The minimum numbers of longitudinal reinfor cements are three, four, and six for triangular ties, rectangular ties or circular ties and bars enclosed by spirals respectively. ACI 352R 02 also gives guidance on the detailing requirements for both Type 1 and 2 connections for the offsetting of the l ongitudinal reinforcements within a joint. For Type 2 connections, the distance between longitudinal bars should be less than the larger of 8 inch or 1/3 of the column crosssection dimension ( or diameter), and not larger than 12 inch. Connection Transverse Reinforcement Park and Paulay (1975) reported that the diagonal compressive stresses within the connection are responsible for the failure of the concrete core due to cyclic loading, especially if the transverse reinforcement is allowed to yield. Effecti ve confinement through transverse reinforcement is critical for improving the connection performance. Nilson Darwin and Dolan (2004) indicated that confinement strengthened the core concrete, improved its strain capacity and prevent ed the vertical column bar s from buckling outward. Panatazopoulou and Bonacci (199 3 ) reported that the shear resistance of joints increases with the amount of hoop reinforcements in the joint. T he primary functions of ties in a tied column are to restrain the outward buckling of the column longitudinal bars, to improve the bond capacity of column bars, and to provide some confinement to the joint core. Confinement of the joint core is intended to maintain the integrity of joint concrete toughness and to reduce the rate of stiff ness and strength deterioration (ACI 352R 02 2002) For Type 1 connections transverse reinforcement is not necessary as adequate confinement is provided as shown in Figure 29 (Nilson Darwin and Dolan, 2004). Confinement in the other direction is not considered in the presence of a third beam framing into the column. Transverse reinforcement is required for the direction which is not adequately confined by the beams. PAGE 24 24 Vertical joint transverse reinforcement is also required for connections with discontinu ous columns. In this case, vertical reinforcement must be provided throughout the depth of the connection and there must be at least 2 layers between the outermost longitudinal column bars. Type 2 connections have different detailing requirements specified under ACI 352R 02. The main difference is that Type 2 connections have transverse reinforcements that are closed with a seismic hook. Shear Strength Joint transverse reinforcement, combined with uniformly distributed longitudinal column reinforcement, h a s been shown to increase the shear resistance of connections by Hanson and Conner (1967). Panatazopoulou and Bonacci (1992) have shown that shear strength of a connection depends on the compressive strength of the concrete that can be mobilized. Stirrups not only increase the shear stiffness of the reinforcement but also the shear stiffness of the concrete core through confinement The maximum shear within a connection is limited by amount of shear reinforcement and the magnitude of the diagonal compress ive stresses (Paulay and Priestley, 1992). T o ensure constructability t he amount of shear reinforcement that can be placed within the connection, considering the amount of beam and column reinforcement passing through the connection, is limited. A large d iagonal compressive force may cause extensive diagonal cracking within the connection core Therefore, it is necessary to limit the horizontal shear force to prevent the connection core from cracking and causing the failure of the connection. The horizonta l shear force may be PAGE 25 25 limited by maintaining the angle of the diagonal compressive forces within acceptable limits. psi f Vc u1300 25 0' (2 1) ACI 352R 02 limits the shear force on a horizontal plane passing through a connection to values established by tests. n uV V (2 2) where uV refers to the design shear force and nV is the nominal shear strength of the connection and is taken as 0.85. The nominal shear strength is given by h b f Vj c n' (psi) (2 3) h b f Vj c n' 083 0 (MPa) (2 4) where jb is the effective connection width, h is the depth of the column in the direction of the load, cf is the concrete compressive strength of concrete at 28 days Strength of the concrete is limited to 6000 psi due to the lack of research on high strength concrete. value is a constant that varies according to the confinement of the connection provided by the beam framing into it as shown in Table 21. There are restrictions to be applied when determining values. For interior conn ections, the width of the beams must be at least the width of the columns and the depth of the shallowest beam must be at least the depth of the deepest beam. Connections that do not meet these criteria will be considered exterior connection. An exteri or connection must have at least 2 beams framing from opposite sides of the connection. The width of the beam of the 2 opposite face should be at least the width PAGE 26 26 of the column and the depth of the two beams should not be less than the total depth of th e deepest beam. Connections that do not meet these criteria will be considered corner connection. ACI 352 R02 states that the effective connection width, jb depends on both the average width of the beam s ( bb ) that f rame into the column and the width of the column ( cb ). In the case of bb smaller than cb jb is: 2c b jb b b and h b bb j (2 5) I f the beam is flush ed with one surface of the column, effective connection width jb is as given: 2c b jb b b and 2 h b bb j (2 6) And if bb exceeds cb effective connection wi dth jb is the same as cb Flexure For Type 2 connections that are a part of the primary resistance to seismic loading, it is important to maintain a weak beam strong column ratio. This is to ensure that plastic hinge s are formed in beams rather than columns. A structure needs to be sufficiently ductile to dissipate the seismic energy through inelastic rotations within the beams (Uma, 2006). Having inelastic rotations within the beams instead of the column ensures that the inelastic rotational demand is not excessively high such that detailing within the connection is not possible. ACI 352R 02 recommends that the minimum column flexural strength be at least 1.2 times the beam flexural strength at the connection. The va lue of 1.2 is PAGE 27 27 recommended as a compromise to prevent plastic hinging from occurring at the column and the need to keep column sizes and reinforcement within economic and constructability purposes, although studies have shown that that a higher factor than 1.2 has been required in cases where the structure is extremely flexible and higher modes of failure contributes to the response. Ehsani and Wight (1984) recommend a value of greater than 1.4 for bare connections and 1.2 for connections where slab and transverse beams are present, for plastic hinges to form in the beams for ductile frames. For a stiff frame where the value is 1.0, the plastic hinge will form in the column. Development of Reinforcement For connections, it is important that the anchorage le ngth of the reinforcement (Figure 210 ) is sufficient to fully develop the tensile and compressive forces caused by the design loads on the structural element (Holmes and Martin, 1983) to prevent premature failure of the bond between concrete and the rebar O ne can determine the length that is necessary to prevent bond failure equating the force in the bar to the bond stress l f fbs s 4 .2 (2 7) where sf is the stress applied to the bar bsf is the bond stre ss and the anchorage length l bs sf f l 4 (2 8) While there is sufficient length to provide anchorage for interior connections where the flexural reinforcement of the beam enters a connection and continues straight on to PAGE 28 28 b ecome the flexural reinforcement of the opposite beam there is not sufficient length to provide anchorage for the tensile reinforcement in exterior and corner connections. In such cases, 90 hooks extending towards and beyond the connection are used. The bottom reinforcements only require to make a ninety degree hook if it the beam is part of a lateral load resisting system. Beam Transverse Reinforcement For Type 2 connections where the beam is wider than the column with computed beam shear stresses less than ) ( 2 psi fc or ) (' 17 0 MPa fc the maximum spacing of reinforcement within the beams plastic hinging zone should be the least of the effective beams depth, eight times the diameter of the longitudinal bar or 24 times the diameter of the stirrup according to ACI 352R02. This is to address the lower shear stresses experienced in wide beam s compared to current stringent provisions for shear. However, ACI 352 02 does not provide guidance on the detailing for the widebeams longitudin al bars into the beam column connections. If the outer reinforcement of a wide beam passes outside the width of the column, the diagonal strut will form outside the column width. In such cases the vertical compressive strut needed to balance the diagonal strut cannot be developed. Nilson et al (2004) suggested placing all the beam outer bars within the outer bars of the column or if this is not possible, placing vertical stirrups throughout the connection to balance the diagonal strut. Plastic Hinge Reinf orced concrete beam column connections are designed with strong column weak beam philosophy Type 2 connections (ACI 352R 02 ) can undergo significant inelastic deformations and the sum of the column flexural strengths will be more than PAGE 29 29 the sum of the bea m flexural strengths. When subjected to large shear forces, the column above and below a joint will remain elastic while plastic hinges will develop in the beams (Paulay, 1989). Plastic hinges are allowed to develop in the beams because it is possible to d esign for the inelastic deformations through proper detailing (Uma and Prasad, 2006) but is difficult to cater to the inelastic demand through detailing in the columns if plastic hinges are allowed to form in the columns. With the rotation of the beam due to the formation of the plastic hinges, one can represent the behavior of a reinforced concrete beam column connection with a moment rotation curve. The rotation of the plastic hinges can be determined from actual tests or through simulation models while the moment is simply the product of the force/pressure onto the beam and the lever arm. Several empirical formulas are proposed for the effective plastic hinge length. Paulay and Priestley (1992) gave a good estimate of eff ective plastic hinge length of a cantilever beam with a tip load as shown below. In the case of typical beam and column sections, h lp5 0 ) ( 022 0 08 0 MPa f d l ly b P (2 9) ) ( 15 0 08 0 ksi f d l ly b P (2 10) where l is the length of the cantilever h is the se ction depth, bd is the diameter of the longitudinal reinforcement and yf is the strength of the reinforcement Corley (1966) proposed that the equivalent plastic hinge length to be: ) ( 2 0 5 0 d z d d lp (2 11) PAGE 30 30 wher e d is the effective depth of the beam in inches and z is the distance from the critical section to the point of contraflexure or tip of a cantilever Mattock (1967) suggested that the equivalent plastic hinge length to be: z d lp05 0 5 0 (2 12) Finally, Sawyer (1964) proposed the equivalent plastic hinge length to be: z d lp075 0 25 0 (2 13) ASCE (1999) stated that the plastic hinge size is about equal to the members depth for plastic hinge to be dev eloped in flexure. Modeling of Beam Column Connections Alath and Kunnath (1995) modeled inelastic shear deformation in reinforced concrete beam column connections by idealizing the connection region as a panel zone characterized by pure shear deformation. The column moments and rotation are assumed to be different for the beam moments and rotation. bM b cM and c are the moments and rotations of the joint region and p is the shear deformation of the joint region or the relative difference in the rotations. The prediction by the model was able to correlate well with the test results conducted at Cornell (Beres, El Borgi, White and Gergely, 1992). c b p (2 14) The relationship between the moments and the rotations is: b c p b cG V M M 1 1 1 1 (2 15) where pV = volume of the panel and G = shear modulus of the material PAGE 31 31 El Metwally and Chen (1988) used thermod ynamic field theory to develop the moment rotation relationships for reinforced concrete beam column connections idealizing the connections as a rotational spring. The rotational spring is characterized by three parameters: initial rotational stiffness of the connection, ultimate capacity of the connection and internal variable that reflects the energy dissipation of the connection. The model assumes the connection to be properly designed, detailed and has sufficient shear strength. The results generated fr om the model showed good agreement with the experimental data conducted by Viwathanatepa Popov and Bertero (1979). The analytical model slightly overestimates experimental data because the model was based on the assumption that the free and dissipated energies of the RC connections are unbounded at failure. In reality, the RC connections have a limited capacity to gain free energy and dissipate energy and thus both the energies are bounded at failure Panatazopoulou and Bonacci (1992) idealized connections as 2 dimensional (2 D) panels reinforced in both orthogonal directions, and acted upon by inplane shear and normal stresses. The total shear stiffness K of the panel consists of cK shear stiffness for concrete, and sK shear stiffness for the reinforcement and both inter dependent. In tests where the connection transverse reinforcements were replaced by longitudinal beam reinforcements uniformly distributed along the height of the connections, the connections were able to resist the shear demanded to develop beam hinging but a rapid deterioration of joint shear resistance was observed with cyclic loads. The tests showed the stirrups were able to increase the shear resistance of the connections dire ctly by increasing the shear stiffness of the steel and indirectly by confining the concrete core and thus increasing the shear stiffness of the concrete, thereby PAGE 32 32 increasing the overall shear stiffness. T he stress of concrete decreas ed after yielding of th e stirrup and loss of confinement within the concrete. Yielding of the stirrup will be followed by either the yielding of the longitudinal column reinforcement or the crushing of concrete in the direction of the principal compressive stress. Lowes and Alt oontash (2003) developed numerical models to simulate reinforced concrete beam column connections subjected to cyclic loading. A classical super element, composed of 4 internal nodal translations and thirteen onedimensional components was used to model th e connection element. Behavior of the connection is characterized by a combination of one dimensional shear panel, bar slip and interface shear components. Each external node consists of two translations and one rotation and each internal node consists of a translation. The sixteen internal and external nodal displacements are complemented by 16 internal and external nodal resultants and can be represented by the following component deformations, forces and equations. 4 1 9 1 4 1: : : : v v u u A (2 16) 13 1 4 1 12 1: : : f f A F FT (2 17) PAGE 33 33 where vector of component deformations, u internal nodal displacements, v external nodal displacements, f component f orces, F external nodal resultants, internal nodal resultants Constitutive models are developed for the load deformation response for the shear panel and bar slip based on material properties, joint geometry and the distribution of the reinforcements distribution. The model was able to represent well the fundamental characteristics of response for the connections based on the comparison with actual observed test data. Krauthammer and Ku (1996) used a hyb rid finite element (FE) finite difference (FD) approach to investigate the behavior, response and influence of reinforcements in structural concrete kneeconnections (twomember connections) under impulsive opening loads. The connection area was modeled using FD numerical method while the adjoining structural elements were modeled using the FE code DYNA3D. The FD approach was advantageous for simple geometry but the FE approach has the advantageous when the boundary and loading conditions are complicated. In the FD approach computing time and memory requirements are reduced since no stiffness and mass matrices need to be calculated I n the case of FE method, much time is required in the constant updating of the matrices is required in the inelastic domai n. Nonlinear material models are also developed for concrete and reinforcements. For the dynamic response, the FE code DYNA3D was used to calculate the stress and strains, due to impulsive loads, at the nodal points between the connection boundaries and t his output is passed to the FD code to determine the displacement fields of the connections. The displacement field is then used to compute PAGE 34 34 strains and lastly, the material constitutive model is used to compute the stresses. For both time steps, the steel and concrete were checked for possible failures. Figure 2 11 shows the computational flow chart of the FD portion. Experiments and Tests Hii (2007) carried out tests on four beam column sub assemblages to develop a better understanding of beam column connections subjected to monotonic loads. Three of the specimens have higher column strength than the beam and connections concrete strength, with two of them carrying different reinforcements within the connection and the last one did not have any additional reinforcements within the connection but had the same column and beam strength. Figure 2 12 shows the experimental setup. All the specimens failed because of diagonal cracking within the connection. The addition of reinforcement in the joints increase the failure loads and caused the diagonal cracks to be more organized and delayed its formation. The specimen which had a higher concrete strength only had a slight higher failure load which indicates that the strength of the connection does not vary much for small increase in concrete strength. However, small increase in concrete strength may result in significant increase in shear strength. Nilsson (1973) performed several tests on beam column connections such as the one shown in Figure 2 13. The connections are laid onto the floor with the T lying on the ground on roller bearings to eliminate the effects of self weight in the test. A point load was applied at the tip of the column and the load deflection history was observed. Loaddeflection curve of Test T16 is shown in Figure 214 The ultimate test moment of 2465kgfm is about 94% of that of the design moment of 2630kgfm. PAGE 35 35 Nilsson also performed tests on corner RC beam column connections or kneejoints shown in Figure 215 The conditions for the tests are similar to those of the T connections. Load deflection of Test U25 is shown in Figure 2 16 The ultimate failure load is 3600kgf. Another series of tests Nilsson did was on interior RC connections subjected to monotonic loads as shown in Figure 2 17 The testing conditions were similar to that of the previous 2 tests and the load deflection curve of U69 is shown in Figure 2 18 Moment Rotation Resistance of Connections Reinforced beam column connections are loaded statically or dynamically to determine the moment rotational resistance function that characterize s the behavior of the connection. FE modeling is one of the methods able to simulate the moment rotational resistance to represent the behavior of the connection. Mathematical expressions are al so used to quantify the moment rotational relationship in steel connections. The Power Model or the RichardAbbot equation (Richard and Abbott, 1975) is commonly used to represent moment rotational relationships of steel connections. Yim (2007) used the Power Model to describe the resistance function simulated in Abaqus reasonably well. n n ik M1 01 (2 18) where ik is the initial connection stiffness is the rotation, 0 is the plast ic rotation n is the shape factor and M is the moment PAGE 36 36 Material s Concrete Compressive and Tensile Strength In general, concrete compressive strength is measured in reference to the uniaxial compressive strength obtained by the compression of concrete in a standard test cylinder. Composed of different materials, it is not a homogenous compound and thus has variations in its strength. The modified Hognestad stress strain curve (Fig ure 2 19 ) is commonly used to repre sent the stress strain relationship for concrete with strengths up to 6000 psi. It consists of a quadratic function with a maximum stress of "cf where "cf = 9 0cf at a strain of c cE f / 8 1 a nd a straight line with a negative slope terminating at stress of 85 0cf and strain of 0.0038. Elastic modulus cE is given in ACI 31808 as: 335 1 c c cf w E (2 19) for concrete with density of 3 3/ 160 / 90 ft lb w ft lbc 57000c cf E (2 20) for normal weighted concrete. Although weak in tension, tensile stress in concrete is measured in terms of rf modulus of rupture through different tests (Table 25). ACI recommends that 5 7c rf f normal weight concrete. This value is multiplied by 0.85 for sand lightweight concrete and 0.75 for all lightweight concrete. PAGE 37 37 Reinforcement Concrete is strong in compression but weak in tension. In reinforced concrete structures, reinforcement i s added to develop the tensile force within the structure while the compressive forces will be resisted by the concrete. Hsu (1993) observed that reinforcement behave differently when they are embedded within the concrete. The stress strain curve of a rei nforcement (Figure 220 ) refers to the stress strain at a particular location while the stress strain curve for a reinforcement embedded in concrete refers to the average stress strain value of a large length of bar over many cracks. Hsu proposed the following bi linear average stress strain properties for reinforcement embedded within concrete. s s sE f. for 'y sf f (2 21) s s y c sE B f B f 25 0 02 0 2 91 0 1000 45 / 2 1 for 'y sf f (2 22) y c yf B f 2 93 0 ) 1000 45 / 2 1 ( (2 23) 5 11 y crf f B (2 24) Finite Element Analysis Finite element analysis (FEA) is commonly used nowadays to solve engineering analysis problems especially with the advances made in computers. In the field of civil engineering, FEA is commonly used to analyze a struct ure subjected to certain loadings. Bathe (1996) summarizes the procedure using FEA to solve a physical problem (Figure 221). PAGE 38 38 The initial step is to idealize the physical problem into a mathematical model. The mathematical model will be governed by numerous assumptions and differential equations. It must be noted that the solution obtained by FEA is characterized by the mathematical model defined, governed by the assumptions and differential equations. Even very accurate FEA may be at odds with physical reality if the mathematical model is inappropriate or inadequate. (Cooks, Malkus, Plesha and Witt, 2002). It is almost compulsory to validate FEA by employing it in certain classic problems with known solution in whichever field that the FEA will be employed in. Only then, can the FEA be used with confidence to obtain valid results for the problem to be investigated. In finite element, we are concerned with the numerical solution of field problems (Cooks et.al, 2002). The spatial displacement of an element w ithin a body is given by: d N u (2 25) w here u is the displacement of the element N is the shape function, also known as interpolation function which approximates the displacement of a point based on the nodal displacements d is the nodal displacements d B (2 26) w here is the strain vector and B is the strain displacement matrix From Hookes Law, we can determi ne the constitutive relationship: C (2 27) w here C is the constitutive matrix By applying the virtual work principle: Internal work done = External work done PAGE 39 39 f D k (2 28) w here k is the element stiffness matrix, d B C B kT is the volume of the element f is the external nodal load and D is the nodal displacement vector Verification and Validation Ve rification and validation of FE codes is necessary before the FE code is used for analysis and simulation for any topics of interest. This is to impart greater confidence in the FE codes used and the r esults obtained. It is even more critical to determine the reliability and robustness for complicated or highconsequence systems that could not ever be physically tested (Oberkampf, Trucano and Tirsch, 2002). The process of verification is (Fig ure 2 5 8 ) necessary to ensure that the accuracy of the solution t o computational model with a known solution It provides evidence that the computational solution corresponds to the conceptional model. It does not, however, attempt to relate the relationship of the conceptional model with the real world. Validation (Fi g ure 2 5 9 ) on the other hand attempts to assess the accuracy of the computational solution to the real world or experimental data. There is no reason to believe that the experimental data will be more accurate than the computational solution but only that the experimental data is the closest benchmarks for validation. The FE Code Abaqus Abaqus is a finite element analysis software that is able to accurately simulate the predictions of strength and deformations in structures in both linear and nonlinear re gime (H ibbitt, Karlsson & Sorensen 200 8 ). It has a wide range of capabilities for structural modeling with nonlinear implicit and explicit static and dynamic analysis capabilities Abaqus/Explicit can model brief transient dynamic events well, such as PAGE 40 40 bla st and impact problems while Abaqus/Standard is more suitable for static or quasi static events. Abacus/Explicit uses an explicit integration method for the analysis while Abaqus/Standard uses an implicit integration method. Explicit integration method sa tisfies the equation of motion of a structure at current time it and extrapolates to determine the solution at time 1 it Because it extrapolates instead of satisfying the equation at time 1 it the time steps need to be small enough to minimize the error due to the extrapolation. Since the time step used is already small, it is useful for brief transient event such as blast and impact problem. The equation of motion at the ith step is: t F u C u M t Fi ii ext int .. (2 29) For the implicit integration method in Abaqus Standard, the equation of motion is satisfied at time 1 it A full matrix inversion is required to determine the solution and the time step used can be larger. 1 1 1 .. 1 i i i iu K u C u M t F (2 30) The disadvantage of implicit integration is that matrix inversion is required to find the solution at each time step and this is computationally expensive as compared to explicit integration. Another advantage of explicit integration is that it is able to handle nonlinearity better while implicit integration might have problems converging to a solution. PAGE 41 41 In a study by Yim (2007) on steel moment connections for structures under blast and collapse loading rates, Abaqus/Standard was used to determine the resistance functions of the steel connectors while Abacus/Explicit is used to develop the threedimensional finite model (Yim, 2007). Abaqus Explicit was used by Tran (2009) to validate the effect of short durationhigh impulse variable axial and transverse loads on reinforced concrete columns. Concrete damaged plasticity m odel Abaqus has three concrete material models available for modeling plain or reinforced concrete. They are concrete smeared cracking model, cracking model for concrete and concrete damaged plasticity model (CDP). All three models can be used for plain concrete, even though they used primarily for reinforced concrete. However, Concrete smeared cracking model can only be used in Abaqus/Standard while cracking model for conc rete can only be used in Abaqus/Explicit. CDP was chosen as the material model of choice for concrete as it can be used in both Abaqus/Standard and Abaqus/Explicit. CDP is suitable for use in which the structure is subjected to monotonic, cyclic and/or dy namic loading under low confining pressures. The concrete behavior is modeled by the theory of isotropic damaged elasticity combined with isotropic compressive and tensile plasticity. The two main failure mechanisms are the compressive failure and the tens ile cracking of the concrete. Tension stiffening of the cracked concrete is a required input in Abaqus and this effect simulates the interaction between the reinforcement and the concrete. Tension stiffening can either be defined in Abaqus as a postfailur e stressstrain curve or by PAGE 42 42 applying a fracture energy criterion in terms of postfailure stress and cracking displacement. The unloading portion of the stress strain curve is characterized by two damage variables, td and cd These two damage variables govern the slope of the unloading curve. They can range from zero (no damage) to one, a total loss of strength. Tension stiffening or the unloading portion of the tensile behavior of the concrete is defined by its post f ailure stress and cracking strain, ck t ~ The cracking strain is equal to the total strain minus the elastic strain of the post failure stress. el t t ck t 0 ~ ~ where 0 0 ~/ Et el t (2 31) Abaqus will convert the cracking strain to plastic strain, pl t ~ using the relationship: 0 ) ~ ~1( E d dt t t ck t pl t (2 32) In CDP, the effects between the reinforcement and concrete interface such as dowel action and bond slip can be modeled by introducing tension stiffeni ng (Hibbitt, Karlsson and Sorensen, 2008) to simulate the transfer of load over the cracks along the reinforcement. Abaqus suggest that a good estimate of the strain softening is to reduce the failure stress over a ultimate failure strain of 10 times the initial failure strain. Too little tension stiffening might probably cause local failure of the concrete due to cracking and cause unstable behavior in the concrete affecting the response of the entire model. Generally, increasing the tension stiffening makes it easier to obtain numerical solutions. PAGE 43 4 3 Compressive behavior or the unloading portion of the compressive stress strain curve is defined by its post failure stress and inelastic (crushing) strain, in c ~ The inelastic strain is equal to the total strain minus the elastic strain of the post failure stress. el c t in c 0 ~ ~ where 0 0 ~/ Ec el c (2 33) Abaqus will convert the inelastic strain to plastic strain, pl c ~ using the relationship: 0 ) ~ ~1 ( E d dc c c ck c pl c (2 34) Defining p arameters of CDP is the dilation angle in the pq plane under high confining pressure (Jankowiak and Lodygowski, 2005). Laboratory tests were performed by Jankowiak and Lodygowski to identify the materials for the concrete damage plasticity model. The material parameters are obtained from the analysis of the threepoint bending singleedge notched concrete beam specimen and the four point bending singleedge notched concrete beam under static loadings. Nielsen (199 8 ) showed the friction angle of concrete against the strength of the concrete and correlated the friction angle of concrete with the dilation angle of concrete, ) tan() sin( However, Mirmi ran, Zagers and Yuan (2000) indicate that dilation angle established the flow rule in the plasticity of concrete. If the dilation angle is equal to the internal friction angle of concrete, plastic straining will occur perpendicular to the yield surface wit h some volumetric expansion. If the dilation angle is less than the internal angle of friction, less volumetric expansion developed and there will be no PAGE 44 44 volumetric expansion if dilation angle is zero. It was found that a dilation angle of zero could be used to predict the response of concrete columns wrapped with fiber reinforced concrete polymer. 0 /0 fc fb is the ratio of the unibiaxial compressive strength to the initial uniaxial strength and the default in Abaqus is 1.16. is the eccentricity of the plastic potential surface and the default value in Abaqus is 0.1. K is the ratio of the second stress invariant on the tensile meridian to that of the compressive meridian (Hi bbitt, Karlsson and Sorenson, 2008) The default value in Abaqus is 0.667. Viscosity parameter, is used to describe the visco plastic regularization of the constitutive law for concret e. The default values in Abaqus/ Standard is 0.0 but this parameter is ignored in Abaqus Expli cit. Classical metal p lasticity Isotropic hardening found under Classical Metal Plasticity material models in Abaqus (Hibbitt, Karlsson & Sorensen, 2008) may be used to describe the behavior of rei nforcement. It uses Mises yield surfaces associated with plastic flow and can be combine with linear elastic material model. In this model, the elasticity is required to define the recoverable portion of the strain. This is the gradient of the stress strain curve exhibiting linear elastic behavior. Most metals ex hibit ductile behavior where the yield strain is orders of magnitudes smaller than the elastic modulus of the material. As such, material data definition in Abaqus is in terms of true stress or Cauchy stress and logarithmic plastic strain. Nominal stress strain data can be converted to true stress and PAGE 45 45 logarithmic plastic strain by the following equations. In Abaqus, the first set of data must be the true initial yield stress corresponding to the plastic strain, which will be equal to zero. c nom nom trueA P ) 1 ( (2 35) 0A Pnom (2 36) c nom trueA A0 (2 37) Etrue nom pl ) 1 ln(ln (2 38) w here E is the Youngs modulus and P is the load onto the metal Elements Linear 8 nodes continuum elements, C3D8R (Figure 226 ), with reduced integration and hourglass control were used to model the concrete. Linear 2nodes beam elements, B31 (Figure 227 ), are used to model the reinforcement and embedded elements technique is used to embed the reinforcement into the concrete. In the embedded element technique, the host will be the solid concrete elements while the reinforcement will be the embedded elements. In Abaqus, the nodes of the embedded elements will lose its translational degrees of freedom in the sense that they are constrained to the translational degrees of freedom of the host elements. However, the embedded elements are allowed to retain their rotational degrees of freedoms which are not constrained. The number of rotational degrees of freedom allowed in a node of an embedded element is identical to the number of translational degrees of freedom of the host element. In a threedimensional model with beam elements in solid elements, PAGE 46 46 since each node at the solid element have three degrees of transl ational freedom, each beam elements are able to have three rotational degrees of freedom at each node. For the reinforcement, it will retain its own translational degrees of freedom while taking on the interpolated values of the translational degrees of fr eedom of the host elements. January and Krauthammer (2003) found that beam elements are more suitable than truss elements for modeling of the reinforcement because the former exhibit shear and bending behavior while the latter only resist axial forces. T he study determined that the shear and bending behavior of the beam elements is necessary when refined meshes are used to obtain convergence of results to test data. Connector e lements Connector elements are able to connect or join two different parts in A baqus (H ibbitt, Karlsson & Sorensen, 2008) with discrete geometry but having complex kinetic and kinematic relationships. These elements can have relative displacements and rotations local to the elements and are defined by individual attributes. Most sing le connector elements are able to describe a connection but connectors can also be used in parallel to describe complicated connection. Connectors available in Abaqus include: 1 Basic translational connectors with available translational degree of freedom at both nodes. The rotational degrees of the first of both nodes might also be available. 2 Basic rotational connectors with available rotational degree of freedom at both nodes. 3 Specialized rotational connectors with available degree of freedoms and also othe r degree of freedom at the nodes. 4 Assembled connectors which is a combination of the predefined basic translational and rotational connectors or translational and specialized rotational connectors. 5 Complex connectors with specialized degree of freedom at t he nodes which cannot be defined by any individual connectors. PAGE 47 47 Progressive Collapse During the lifetime of a structure, it might be exposed to abnormal loading that was not designed for and this might result in the local failure of one or more structural elements. As a result, the loading path changes and this might create extreme stress on other structural elements not designed for this sudden increment in load. Progressive collapse occurs when a structure has its loading pattern, or boundary conditions, changed such that structural elements are loaded beyond their capacity and fail (Krauthammer 2008) This catastrophic collapse results in destruction of buildings and loss of lives. In a RC structure, it is just as important to understand the behavior o f the beam column connections in addition to the main structural elements such as beams and columns in progressive collapse. Under UFC 4 01001, it is compulsory for all buildings more than three stories, new and existing, to be designed for progressive c ollapse. Under UFC4 023 03 (2009), the level of design (Table 2 3 ) for structures against progressive collapse is based on the Occupancy Category (OC) of the structure (Table 24 ). For t ie f orces, the abnormal load will be transferred from the damage po rtion of the building to the undamaged portion. In the Alternate Path method, the abnormal load must be able to bridged over the damage area. Finally, in Enhanced Local Resistance, the shear and flexural capacity of the walls and columns are increased to provide stronger resistance and reduced the probability and extend of initial damage. The GSA guidelines (200 5 ) state that all new buildings must be designed to withstand the loss of column one level above grade at the building perimeter without progressi ve collapse. This is to ensure adequate load paths redistribution due to a local damage. In the case of the loss of an internal column, the designer has to show that PAGE 48 48 load redistribution to alternate path is possible. Or the designer can simply ignore progr essive collapse calculation by protecting the column such that the column would not have a chance to be damaged critically. The analysis can be performed either statically or dynamically. The following vertical load would have to be applied to the structur e for static analysis: Load = 2 (DL + 0.25LL) (2 39) w here DL is dead load and LL is live load. Hayes Jr et al. (2005) suggested that seismic upgrading of the Murrah Federal building will improve the resistance to both blast loads and progressive collapse. Increasing the member sizes and the addition of transverse and longitudinal reinforcements increases the flexural and shear strength of the structure, counteracts the abnormal loadings and increase the structure resistance to the blast loads and progressi ve collapse. Detailed modeling of the entire RC structure against blast loadings for progressive collapse using Abacus/Explicit will be computationally expensive and timeconsuming because of the large amount of elements in the detail model. Yim (2007) used Abaqus/Explicit to develop a simplified building frame model (Figure 26 6 ) consisting of beam elements instead of beams and columns, and connector element for connection components, whil e ensuring acceptable accuracy to the detailed model. The connector element mechanical properties are the resistance function of the steel moment connections derived earlier by Abaqus/Standard. Figure 229 shows an elevation view of the RC structure which is scheduled for progressive collapse test by the removal of a colum n at the first storey. The structure PAGE 49 49 made up entirely of RC columns, beams and slabs will be attached to very strong steel reaction frame. More details of the RC structure will be shown in the later chapters. Summary Different types of RC beam column conne ctions respond differently to loadings depending on their geometry and detailing of the reinforcements. Different models and theories on the behavior of the concrete connections have also lead to the emphasis on different characteristics and parameters of the connections during analysis which is in turn used for the design and detailing of the connections. Due to the lack of understanding of the RC beam column connection, many researchers have us ed experiment al and numerical modes in an attempt to better understand and explain connections behavior. Finally, progressive collapse of a structure is introduced through the study of the effects of a structures response to an abnormal load or the sudden removal of a structural element. Figure 21. RC beam col umn connections [Adapted from ACI 352R 02. 2002. ACI 352R 02. (Page 352R 2, Figure 1.1).] PAGE 50 50 Figure 22. Typical monolithic interior beam column connection [Adapted from Nilson, Darwin and Dolan. 2004 Design of concrete structures. (Page 349, Figure 11.2).] Figure 23. Knee connection subjected to opening loads [Adapted from MacGregor and Wight 2005. Reinforced concrete mechanics and design. (Page 916, Figure 1845).] COLUMN BEAM 1 BEAM 2 BEAM 3 BEAM 4 PAGE 51 51 Figure 24 Diagonal and radial bars configuration [ Reprinted with permission from Krauthammer and Ku. 1996 A hybrid computational approach for the analysis of blast resistant connections. (Page 841, Figure 11).] Figure 25 Typical threemember connection details [Adapted from Nilson, Darwin and Dola n 2004 Design of concrete structures (Page 368, Figure 11.21).] PAGE 52 52 Figure 26 Strut andtie model [Adapted from Nilson, Darwin and Dolan 2004. Design of concrete structures (Page 368, Figure 11.22).] Figure 27 Joint loads and forces on an interi or connection due to (a) gravity loads and (b) lateral loads [Adapted from Nilson, Darwin and Dolan. 2004 Design of concrete structures (Page 350, Figure 11.3 & 11.4).] PAGE 53 53 Figure 28 Strutandtie model [Adapted from Nilson, Darwin and Dolan 2004. Des ign of concrete structures (Page 361, Figure 11.12).] Figure 29 Confinement within connections [Adapted from Nilson, Darwin and Dolan. 2004. Design of concrete structures (Page 353, Figure 11.7).] PAGE 54 54 Figure 210 Development of reinforcement [Adapted from Holmes and Martin. 1983. Analysis and design of structural connections: Reinforced concrete and steel. (Page 63, Figure 2.30).] PAGE 55 55 Figure 211 Computation flow chart from finite difference scheme [Adapted from Krauthammer and Ku. 1996. A hybrid computational approach for the analysis of blast resistant connection. ( Page 836, Figure 4 )] PAGE 56 56 Figure 212 Experimental setup [Adapted from Hii. 2007. Influence of concrete strength on the behavior of external RC beam column joint. (Page 41 Figure 3.7) ] Figure 213 T section testing [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. (Page 166, Figure 9.6)] PAGE 57 57 Figure 214 T16 Load deflection curve. [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. ( Page 170 Figure 9.13)] Figure 215 Corner section testing [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. (Page 209)] PAGE 58 58 Figure 216 Load deflection curve for U25 [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. ( Page 225 Figure A.5 )] Figure 217 Interior connection testing [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. ( Page 186, Figure 9.34)] PAGE 59 59 Figure 218. Interior connection load deflection curve for U69 [Adapted from Nilsson. 1973. Reinforced concrete corners and joints subjected to bending moment. (Page 186, Figure 9.35)] Figure 219 Modified Hognestad curve. [Adapted from MacGregor and Wight 2005. Reinforced concrete mechanics and design. (Page 71, Figure 319(a))] PAGE 60 60 Figure 220 Stress strain curve of reinforcement using bilinear model [Adapted from Hsu. 1993. U nified theory of reinforced concrete. ( Page 206, Figure 7.8)] Figure 221 Process of finite element analysis [Adapted from Bathe. 1996. Finite element procedures. (Page 3, Figure 1.1)] PAGE 61 61 Figure 222 Verification process [Adapted from Oberkampf, Trucano and Tirsch. 2002. Verification, validation, and predictive capability in computational engineering and physics. (Page 10, Figure 2)] Figure 223 Validation process [Adapted from Oberkampf, Trucano and Tirsch. 2002. Verification, validation, and predictive capability in computational engineering and physics. (Page 11, Figure 3)] PAGE 62 62 Figure 224 Tensile stress strain curve of CDP. [Adapted from Hi bbitt, Karlsson and Sorenson. 2008. Abaqus 6.8 Analysis Users Manual. (Figure 19.6.3 3)] Figur e 225 Compressive stressstrain curve of CDP. [Adapted from Hi bbitt, Karlsson and Sorenson. 2008. Abaqus 6.8 Analysis Users Manual. (Figure 19.6.36)] PAGE 63 63 Figure 226. C3D8R continuum elements. [Adapted from Hi bbitt, Karlsson and Sorenson. 2008. Abaqus 6.8 Analysis Users Manual. (Figure 22.1.1 2)] Figure 227. Beam elements. [Reprinted with permission from Yim. 2007. A study of steel moment connections for structures under blast and progressive collapse loading rates. Ph.D. dissertation. (Page 33, Figure 2 18). The Pennsylvania State University, Pennsylvania] PAGE 64 64 Figure 228 Simplified frame model [Reprinted with permission from Yim. 2007. A study of steel moment connections for structures under blast and progressive collapse loading rates. Ph.D. dissertation. (Page 68, Figure 413). The Pennsylvania State University, Pennsylvania] Figure 229. Elevation view of RC structure to be subjected to progressive collapse test PAGE 65 65 Table 2 1. values for beam column connections [Adapted from ACI 352R 02. 200 2. (Page 352R 9, Table 1).] Connection Type Classification 1 2 A. Joints with a continuous column A.1 Joints effectively confined on all four vertical faces 24 20 A.2 Joints effectively confined on three vertical faces or on two opposite vertical faces 20 15 A.3 Other faces 15 12 B. Joints with a discontinuous column B.1 Joints effectively confined on all four vertical faces 20 15 B.2 Joints effectively confined on three vertical faces or on two opposite vertical face s 15 12 B.3 Other faces 12 8 Table 2 2. Tensile strength of concrete (Nilson, Darwin and Dolan, 2005) Normal Weight Concrete, psi Lightweight Concrete, psi Direct tensile strength tf 3 to 5 cf 2 to 3 cf Split cylinder strength ctf 6 to 8 cf 4 to 6 cf Modulus of rupture rf 8 to 12 cf 6 to 8 cf Table 2 3. Occupancy categ ories. [Adapted from Unified Facilities Criteria. 2009. Design of buildings to resist progressive collapse. (Page 16, Table 21)] Nature of Occupancy Occupancy Category Buildings in Occupancy Category I in Table 1 of UFC 331001 Low Occupancy Buildings A I Buildings in Occupancy Category II in Table 1 of UFC 3 310 01 Inhabited buildings with less than 50 personnel, primary gathering buildings, billeting and high occupancy family housing A,B II Buildings in Occupancy Category III in Table 1 of UFC 3 310 01 III Buildings in Occupancy Category IV in Table 1 of UFC 331001 Buildings in Occupancy Category V in Table 1 of UFC 3 310 01 IV A As defined by UFC 401001 Minimum Antiterrorism Standards for Buildings B Occupancy Category II is the minimum occupan cy category for these buildings, as their population or function may require designation as Occupancy Catergory III, IV or V. PAGE 66 66 Table 2 4 Occupancy categories and design requirements [Adapted from Unified Facilities Criteria. 2009. Design of buildings t o resist progressive collapse. (Page 17, Table 2 2)] Occupancy Category Design Requirements I No specific requirements II Option 1: Tie Forces for the entire structure and Enhanced Local Resistance for the corner and penultimate columns or walls at the f irst story. OR Option 2: Alternate Path for specified column and wall removal locations. III Alternate Path for specified column and wall removal locations; Enhanced Local Resistance for all perimeter first story columns or walls. IV Tie Forces: Alternat e Path for specified column and wall removal locations; Enhanced Local Resistance for all perimeter first and second story columns or walls. PAGE 67 67 CHAPTER 3 RESEARCH APPROACH Introduction The focus of this study is to use a predominantly continuum typed FE model to simulate a n RC beam column connection and extract a simple mechanical model which can be used along structural typed elements for faster progressive collapse analysis to be used in comprehensive progressive collapse assess ment from the analysis Before using the FE code to model the actual RC beam column connection, it is critical to ensure that the FE code is being verified and validated to have confidence in its usage and the computational solution. Verification of Abaqu s is required to provide evidence that the FE code executes correctly the mathematical algorithms In the Verification Manual for Abaqus (Hibbitt, Karlsson & Sorensen, 200 8 ), there are extensive verification tests for elements, materials and other options in the code. These verification tests are usually done on problems with well known solutions. It does not, however, attempt to relate the relationship of the conceptional model with the real world. Several examples of Abaqus verification will be provided i n Chapter 4. For validation Abaqus/Explicit (Hibbitt, Karlsson & Sorensen, 200 8 ) will be validated against a RC beam exposed to a dynamic load by Feldman and Seiss (1956 ). This is to validate the dynamic behavior of Abaqus/Explicit for reinforced concrete elements against dynamic loadings. Abaqus/Standard (H ibbitt, Karlsson & Sorensen 2008 ) against three RC beam column connections exposed to monotonic loadings by Nilsson (1973). These three connections are chosen because they are the most represent ative b eam column connections in a RC structure, namely a corner PAGE 68 68 connection, T connection and interior connection (Figure 3 1). Although we are only focusing on the interior connection in this study, the validation of the corner connection and T connection facilitates the future study of progressive collapse for the entire RC structure. Once the FE code has being verified and v alidated, Abaqus/Standard can be used to extract the mechanical model (resistance function) for the RC beam column connection. Connector elements, with the properties of the mechanical model, will be used together with structural typed elements to replace the predominantly continuum based FE model. The approach will be covered in this chapter in more details. Characterization of RC Beam Colu mn Connection The interior RC beam column connection together with the attached beams and columns will be modeled and subjected to monotonic load. This is to ensure that the true representation of the actual experiment is being replicated in the FE analysi s. T he detailed model of the RC beam column connection (modeled as an interior joint), up to d distance from the column face, is being modeled as fine mesh of continuum elements for the concrete and beam elements for the reinforcement while coarse mesh of continuum elements are used to represent the RC beams and columns further than d distance away (Figure 32 ). This will enable us to capture in details the behavior of the RC beam column connection, which is the main focus of the study. Although using coars e continuum elements to model the RC beams and columns might not be very accurate, it will not affect our study as we only need the element s to transfer the force and moments throughout the RC beams and columns to obtain the resistance function of the conn ection. The main objective is to obtain the moment rotation re sistance function of the RC beam column connection. Modeling of a detailed continuum FE PAGE 69 69 model of the entire assemblage is not recommended because it is complicated, computationally expensive and time consuming. Rotation at connections The global rotation of the connection can be easily found using the deflection at the tip of the beam and the length of the beam as shown in Fig ure 3 3 Rotation, is simply L d1tan This however, does not represent the rotation at the plastic hinge as it does not take into account of the length of the plastic hinge formed. As such, Fig ure 34 shows a more accurate depiction of the behavior of the connection considering the fo rmation of a plastic hinge. The rotation at the connection can be found considering the plastic hinge and the vertical and horizontal displacement of the points on the plastic hi nge shown in Figure 35 (Yim, 2007) The rotation will be the average of the c alculations taken at the top, middle and bottom of the beam. The rotation, will be: L u u v v 1 2 1 2tan (3 1) where L = pl plastic hinge length Simplified FE Model After obtaining the moment rotation of the connection from a predominantly continuum based FE model, connector elements will be used to represent the resistance function and beam elements to represent the RC beams and columns to form a simplified structural based FE model (Figure 36) Both models must be equivalent and they will be tested by subjecting them to the same loading and boundary conditions T he RC beams and columns of the structural based FE model (Figure 36) PAGE 70 70 will be modeled as beam elements while the beam column connection will be repres ented by connector elements JOIN and CARDAN. Connector JOIN is a translational basic connector and it freezes the position of two nodes. The Cartesian coordinates of node b relative to node a will always be fixed, regardless of whether the two nodes are l ocated at the same position. Connector CARDAN is a basic rotational basic connector which provides a rotational connection and relationship between 2 nodes defined by Cardan angles. The three available degree of freedom s are the relative Cardan angles of node b relative to node a and the kinetic moment in a Cardan connection is directly proportional to the relative Cardan angles. A combination of these two basic translational and rotational connectors will be able to describe the behavior of the moment rot ation relationship at the beam column connection. The connector JOIN links the RC beam to the RC column while CARDAN allows us to describe the behavior of the RC beam column connection by a moment rotation resistance function found earlier. Power Model (Ri chard and Abbott, 1975) will be used to see if it is a good fit to the moment rotational resistance function obtained from the simulation. The simplified structural based FE model can be tested against progressive collapse by the sudden removal of a colum n using Abaqus/Explicit However, this will not be in the scope of this study. Summary The above describes the characterization of the resistance function for RC interior beam column connection from a predominantly based continuum FE model to be incorporat ed in a simplified predominantly structural based FE model for faster PAGE 71 71 progressive collapse assessment. The flowchart (Figure 39 ) below describes clearly the research approach discussed. Figure 31. Beam column connections of a RC structure Figure 32 Free body diagram of Abaqus model PAGE 72 72 Figure 33. Global rotation Figure 34. Formation of plastic hinge d LOAD L d LOAD L l p, plastic hinge PAGE 73 73 Figure 35. Rotation points [Reprinted with permission from Yim. 2007. A study of steel moment connections for structures under blast and progressive collapse loading rates. Ph.D. dissertation. (Page 41, Figure 31(c)) The Pennsylvania State University, Pennsylvania] Figure 36. Simplified structural based FE m odel PAGE 74 74 Figure 37 Flowchart of research approach PAGE 75 75 CHAPTER 4 RESULTS AND DISCUSSI ONS Introduction Firstly, the verification and validation of Abaqus will be described in this chapter. Following this, a brief description of the RC structure containing the beam column conne ction will be given. The validated FE code, Abaqus /Standard, will be used to extract the resistance function of the RC beam column connection by subjecting it to a monotonic load. A connector will be used to replace the resistance function of the RC beam c olumn connection and beam elements will be used to model the RC beams and columns in a simplified structural based FE model This simplified structural based FE model is subjected to the same monotonic load and boundary conditions to verify that the same m oment rotation relationship will be obtained as the predominantly continuum based FE model Finally, the simplified structural based FE model will be ready to be subjected to progressive collapse assessment. Verification and Validation Verification of Abaqus Verification of Abaqus is necessary to have confidence that it can be used to describe reinforced concrete elements and structures. The focus of this study will not be on the verification as the main objective is to characterize the resistance function of a RC beam column connection. As such, the verification done and reported in Abaqus Verification Manual Version 6.8 (Hibbitt, Karlsson & Sorensen, 2008 ) will be highlighted. CDP is v erifi ed in Abaqus/Standard and Abaqus/Explicit using continuum elements for inelastic behavior of concrete based upon the theory of isotropic damaged PAGE 76 76 elasticity of concrete which depends on the isotropic compressive and tensile plasticity behavior The CARDAN connector element is verified by testing in Abaqus/Standard and Ab aqus/Explicit for elastic behavior and performance when subjected to a damped, free vibration analysis or sinusoidal excitation of a damped springmass system. Continuum C38DR elements are tested for the features of hourglass control, kinematic formulation tied contact surfaces, multi point constraints and shear in both Abaqus/Standard and Abaqus/Explicit. Beam B31 elements are tested for the features of small displacements and large rotations when subjected to loadings in both Abaqus/Standard and Abaqus/E xplicit. Validation of Abaqus Validation of Abaqus is important to compare its accuracy against experiments of RC elements or structures subjected to dynamic and monotonic loadings. Abaqus/Explicit is validated against RC beams subjected to dynamic loads w hile Abaqus/Standard is validated against RC beam column connections subjected to monotonic. For dynamic behavior of concrete elements, validation is performed for a RC b eam subjected to dynamic loading in the Feldman and Seiss tests. For RC beam column be havior, validations are performed for corner connections, T connections and interior connection as these are the most commonly found beam column connections to be found in RC structures (Figure 41). Although this stud y will look into only a particular int erior connection, validating the two other types of connections is useful for future study of progressive collapse of the entire building. PAGE 77 77 RC b eam s ubjected to d ynamic l oads This validation is performed because for future progressive collapse assessment of RC structures, the external or abnormal load might be a blast or impact load which will cause the RC structure to behave dynamically. Abaqus / Explicit is used to simulate a simply supported RC beam (Feldman and Seiss, 195 6 ) subjected to a dynamic lo ad at its midspan. Beam 1C has a length of 10 feet simply supported between 106 inches. It has a column stub at midspan to transmit the dynamic load. The concrete beam is reinforced with 2 no. of # 7 tensile reinforcements and 2 no. #6 compressive reinforcements with #3 stirrups at 7 inch intervals. The reinforcement details of the RC beam are shown in Figure 42 Figure 43 shows the loading time history for beam 1C Solid C3D8R elements are used to model the concrete and beam B31 elements are used for the reinforcement. In this validation, i t is found that 1 inch mesh (Figure 4 4 ) is most appropriate for t his simulation as the results converge and it takes a relatively short time to run the simulation. Dynamic Increase Factor s (DIF) (Table 41 ) per UFC 3 340 02 w ere applied to the material properties of concrete and steel to account for the response of structural materials to high loading rates (ASCE, 1999). Hsus material model (1993) is used for the steel while CDP is used for the concrete. The stress strain curves for concrete and steel for static loading and high loading rates in Beam 1C are shown in Figure 45 and Figure 46 The default parameters of CDP in Abaqus as described earlier are used (Table 42 ). It was found that varying the value of these par ameters do not change the response by a significant amount. A parametric study is done to investigate the optimum value of PAGE 78 78 dilation angle and it was found to be 31 Figure 4 7 shows the deflection time history for far design range (DIF) for various dilat ion angles and the test data. Abaqus / Explicit obtain ed very close results with the test data for the peak displacement for far design range for dilation angle of 31 The peak deflection of the model was less than 1% different compared to the actual test d ata. The residual response of the beam as the load starts to decrease does not match closely the experimental results. Damage to a structural element can be classified in terms of peak displacement and in this test; the inelastic demand of the beam is represented by the peak displacement. Based on the closeness between the peak displacements of the test and simulation w e would conclude that Abaqus / Explicit is able to model the dynamic behavior of reinforced concrete beams. RC beam column connections subj ected to monotonic l oads Abaqus / Standard i s used to simulate the response of a RC T connection, T16 (Figure 2 13 ), corner connection U25 (Figure 2 15) and interior connection U69 (Figure 2 17 ) subjected to monotonic loads (Nilsson, 1973). The same Abaqus modeling process as described in the RC beams was carried out. The tests are conducted with the RC beam column assembly that was placed on the floor to eliminate the effects of gravity on the results Displacement control testing was carried out to obtain the load deflection data as recorded in the tests by increasing the displacement followed by measuring the load required. CDP was used for the concrete and Hsus model for embedded reinforcement in concrete was used for reinforcement. The stress strain curve for the concrete and reinforcement for the 3 tests are shown in Figure 48 and Figure 49 The parameters PAGE 79 79 for CDP are also set at Abaqus default with the dilation angle of the concrete being varied. It was found that the most appropriate value of the v iscosity parameter was 0.01 as it allows the results to converge faster. For T16 (Figure 410 ), t he peak failure load of both the simulation and the experiment occurs at nearly the same deflection, 40mm in both the simulation and experiment, at dilation angle of 10. The p eak failure load of 18.168 kN in the simulation is about 5.7% compare to that of 17.187kN in the experiment. However, the simulation did not exhibit the same strain softening behavior or ductility as the experiment after the peak failure load. The experiment recorded a maximum deflection of around 62 mm but the maximum deflection in the simulation is around 41 mm. Based on the closeness of the peak failure load and its occurrence, the author believe s that the simulation does validate the behavior of the T c onnection tested. For U25 (Figure 4 1 1 ) at a dilation angle of 20, the initial failure load of 35.629kN for the simulation and 35.316kN for the experiment is really close, with a difference of just 0.89% between them. Both the initial failure also occurs at around the same deflection of around 17 mm Similar to T 16, the simulation does not exhibit the same strain hardening or ductility as shown in the test. Based on the almost identical initial failure load and occurrence the author believes that the simu lation is able to model the behavior of the corner connection, U25. For U69 (Figure 4 12) a parametric test by varying the dilation angles was carried out. It was found that for dilation angle of 10 the simulations peak load of 44.052 k N was about 21.1 8 % from the test peak load of 36.355 kN Both the peak load of the test and simulation occur near the same deflection. The peak load of the simulation reduces PAGE 80 80 with reducing dilation angles. But this reduction flattens off after a dilation angle of 10 degrees where it is seen that a dilation of 5 degrees produced similar results. In the experiment, cracks will be formed in the concrete with increasing load. T he reinforcement might not be able to reach its full capacity as the distance in between the cracks might be smaller than the development length of the reinforcement. However, this phenomenal does not occur in the Abaqus simulation where the stress of the reinforcement is only a function of the structure at that particular locati on. The Abaqus model migh t be stiffer than the actual test and this might explained the reason why the simulation has a higher load resistance compared to the experiment. O nly one test was carried out by Nilsson (1973), and since only one experiment was done, there might be some v ariability in the results as the materials, testing procedures or equipment might not be functioning as it should during the experiment. The author believes that Abaqus is able to model the behavior of the internal connection reasonably well at a dilation angle of 10 degrees because the peak load of both the model and the experiment occurs at nearly the same deflection and the 21.18% variation in the peak load could possibly be due to the two reasons stated above. Parameters of CDP During the validation of Abaqus/Explicit and Abaqus/Standard, all parameters of the CDP material model w ere kept constant to the default values in Abaqus except for the viscosity parameter and dilation angle. In Abaqus/Explicit, viscosity parameter will be zero regardless of what values we re assigned to it. For Abaqus/Standard, it was found out that a viscosity value of 0, 0.01 PAGE 81 81 and 0.001 does not have adverse effect on the results of the simulation but having a viscosity parameter of 0.01 makes it faster for the simulation runs fa ster. RC Beam Column Connection in Study An interior RC beam column connection of a concrete structure will be modeled using Abaqus/ Standard to determine its moment rotation resistance function. Figure 41 3 shows the elevation view of the reinforced concr ete structure and the interior connection to be modeled. Figure 4 1 4 and 41 5 shows the details of the beams and columns respectively. The concrete has an fc of 4000psi while the reinforcement used is Grade 60 steel conforming to the latest ASTM specific ations A616 or A706. The stress strain curves for the concrete and reinforcement are shown in Figure 41 6 and Figure 417 respectively Hsus material model for reinforcement and CDP material model for concrete is used in Abaqus/Standard for the simulation. Resistance Function of RC Beam Column Connection The dashed region (Figure 41 3 ) shows the beam column assemblage that will be modeled using Abaqus / Standard. One end of the beams will be loaded by the exterior column transferring the weight from the abov e floors, when the column beneath is removed, while the other end will be pinned onto a very strong steel resistance frame. The support condition at the steel frame will be a pinned support. Figure 4 18 shows the predominantly continuum FE model of the RC beam column assemblage. The focus of the simulation is to determine the behavior of the connection subjected to monotonic loads. As a strong column weak beam design, the plastic hinge will develop within the RC beam. For the plastic hinge to form on the RC beam and rotate about the edge of the column, the length of the plastic hinge will be d (distance PAGE 82 82 from the outer most compressive fibers of the beam to the tensile steel) from the edge of the column. D etailed modeling is done up to d the distance from the face of the connection at the beams and columns to accurately capture the behavior of the beam column connection. The local rotation at the plastic hinge at each face of the connection is obtained by capturing the translations and i s obtained as shown in Figure 35. The local rotation is taken as the average at the top, middle and bottom faces of the RC beam (Figure 4 19 ). The moment is simply the product of the load applied and the lever arm. The moment rotation relationship for the connection for this face of the RC beam column is obtained. Further away at more than d distance, the actual behavior of the RC beams and columns are not of importance in this study. They can either be represented by using appropriate beam elements or continuum elements of coarser mesh whose main purpose is t o transfer the loading and boundary conditions to the connection to maintain compatibility. In this case, continuum elements (Figure 4 18) are used to represent these sections and they are meshed into coarse mesh to save computing time. Figure 4 20 compares the assemblage consisting of the entire assemblage modeled of continuum and beam elements of 1 inch mesh compared to another model that only models in detailed the beam column connection in 1 inch me sh The amount of elements in the former model is about 7.5 times more than that in the latter model. We can save tremendous resources and computing time by just modeling in detail up to d distance of the beam column connection as we are only interested in the connection behavior. PAGE 83 83 Since the RC beam is not symmetrical, the moment rotation resistance function of the RC beam column connection is not symmetrical either. Therefore, separate moment rotation resistance functions have to be determined for a load pushing towards the ground and another in the opposite direction (Figure 4 2 1 ). Moment Rotation Re sistance Function Figure 42 2 shows the moment rotation resistance function generated by subjecting the continuum FE model in both directions in Abaqus/Standar d Three simulations are run with dilation angles of 31, 20 and 10. For all the runs, it was found out that the local rotations at the top, middle and bottom face of the beam is similar and this further reinforced our assumption that the plastic hinge will developed at d distance away from the column face. The maximum moment rotation results from the simulations are shown in Table 43 For gravity loading, simulation with dilation angle of 10 has the highest moment of 7530 kips inches while simulation with dilation angle of 31 has the highest moment of 5258 kips inch for loading in the opposite direction. Dynamic Structure Analysis Suite (DSAS) Version 3 .0 ( Astarlioglu and Krauthammer, 2009 ) was used to determine the maximum moment resistance based on the s ection details of the RC beam at the face of the connection. The maximum moment resistance when subjected to gravity loading based on the section properties is about 4884 kips inch and 3919 kips inch when subjected to loading in the opposite direction. Ba sed on these figures, a dilation angle of 20 is a good estimate to use for the simulation as it is the closest to the moment resistance calculated from the section properties. PAGE 84 84 Figure 42 3 compared the Power Model (Richard and Abbott, 1975) against the sim ulation for gravity loading. It can be seen that the Power Model over estimates the moment and is not a good fit for the simulation results. However, using a 3rd degree polynomial can fit the simulation results perfectly. This behavior is consistent for the simulation when the load was applied in the opposite direction. A 3rd degree polynomial (Figure 42 4 ) can fit the simulation results perfectly. Replacing Moment Rotation Re sistance by Connector C onnector elements will replace the resistance function of t he RC beam column connection in a s tructural based FE model. The connectors to be used in Abaqus/Standard are the combination of the basic translational connector JOIN and rotational connector CARDAN The RC column and beams will be represented by beam elements and JOIN will connect the node of the beam and the column at the intersection point. At the same location, CARDAN will provide a rotational connection between the node at the beam and the node at the column, parameterized by CARDAN angles through a m oment rotation resistance constitutive relationship of the connection found earlier. In Abaqus, this constitutive moment rotation resistance function will be defined by both l inear elastic behavior and nonlinear elastic behavior. The linear elastic behav ior will be the moment over rotation at initial failure while the nonlinear elastic behavior will be the corresponding sets of yield moment and plastic rotation values after initial failure that described the isotropic hardening behavior. Figure 42 5 show s the isotropic hardening behavior of the connectors in both loading directions. PAGE 85 85 Verification of Predominantly Structural Based FE Model The simplified s tructure based FE model shown in Figure 3 6 consists of beam elements to represent the RC beams and col umns and connectors to represent the RC beam column connection. When subjected to the same loading and boundary conditions used in the characterization of the moment rotation resistance of the connection, the CARDAN connector must exhibit the same moment r otation output for verification that the simplified predominantly structure based FE model is performing its intended function. Figure 42 6 and 427 compares the output of the simplified structural based FE model against the moment rotation resistance func tion obtained earlier from the predominantly continuum based FE model The results show that the outputs of the two models are exactly the same This gives us confidence to replace the detailed RC beam column assemblage with the simplified predominantly st ructural based FE model. More importantly, for the same output, the simplified structural based FE model was able to perform the simulation around 10 minutes compared to about 150 minutes for the predominantly continuum FE model (Table 44 ) Figure 41 Elevation of a common RC structure PAGE 86 86 Figure 42 Section properties and details for RC beam 1C [Adapted from Feldman and Seiss 195 6. Investigation of resistance and behavior of reinforced concrete members subjected to dynamic loading. (Figure 20)] Figure 43 Loading time history for beam 1C PAGE 87 87 Figure 44 Abaqus model showing 1 inch mesh and reinforcement Figure 45 Stress strain curve of concrete for Beam 1C PAGE 88 88 Figure 46 Stress stra in curve of steel for Beam 1C Figure 47 Deflection time history for RC beams subjected to dynamic loads PAGE 89 89 Figure 48. Concrete stress strain curve for T16, U25 & U69 Figure 49. Reinforcement stress strain curve for T16, U25 & U69 PAGE 90 90 Figure 410 Loaddeflection curve for T16 Figure 411 Loaddeflection curve for U25 PAGE 91 91 Figure 41 2 Load deflection curve for U69 Figure 413 Elevation view of RC structure and interior connection PAGE 92 92 Figure 41 4 Details of RC beams Figure 41 5 Details of columns PAGE 93 93 Figure 41 6 Reinforcement stress strain curve Figure 417 Concrete stress strain curve PAGE 94 94 Figure 418 Continuum FE model of RC beam column assemblage Figure 419 Positions of measurement for local rotation PAGE 95 95 Figure 42 0. Comparison of dense and simple model of beam column assemblage Figure 421 Loading of RC beam column assemblage in opposite direction PAGE 96 96 Figure 422 Moment rotation resistance functions Figure 42 3 Moment rotation resistance functions subje cted to gravity load PAGE 97 97 Figure 42 4 Moment rotation resistance functions subjected to opposite load Figure 42 5 Isotropic hardening behavior for connectors PAGE 98 98 Figure 426 Output for connector for gravity loading Figure 427 Output for connector for opposite loading PAGE 99 99 Table 4 1. Dynamic increase factor [Adapted from Unified Facilities Criteria. UFC 334002: Structures to resist the effects of accidental explosions. (Page 1068, Table 4 1)] Type of Stress FAR DESIGN RANGE CLOSE IN DESIGN RANGE Re inforcing Bars Concrete Reinforcing Bars Concrete y dyf f / y duf f / '/c dcf f y dyf f / y duf f / '/c dcf f Bending 1.17 1.05 1.19 1.23 1.05 1.25 Diagonal Tension 1.0 0 1.00 1.10 1.00 1.00 Direct Shear 1.10 1.00 1.10 1.10 1.00 1.10 Bond 1.17 1.05 1.00 1.23 1.05 1.00 Compression 1.10 1.12 1.13 1.10 Table 4 2 Parameters of CDP Description Symbol Value Flow Potential Eccentricity 0.1 R atio of Initial Equibiaxial Compressive Yield Stress to Initial Uniaxial Compressive Yield Stress 0 0 cb 1.16 Ratio of Second Stress Invariant on Tensile Meridian to Compressive Meridian cK 0.666 Viscosity Parameter 0 Table 4 3 Ultimate moment rotation of simulations Gravity Loading Opposite Loading Dilation Angle Moment, kips inch Rotation, rad Moment, kips inch Rotation, rad 31 6223 0.008547 5258 0.007224 20 5479 0.006883 4925 0.006687 10 7530 0.008458 5091 0.007637 PAGE 100 100 CHAPTER 5 CONCLUSION AND DISCUSSION Summary A numerical method employing an advanced nonlinear finite element (FE) code to extract the moment rotation curve from an internal RC beam column connection was presented. C hapt er 2 introduced the different types of RC beam column connections and the different structural behavior and connection design parameters associated with it. Material models for concrete and reinforcement wer e being reviewed. Finite element analysis was int roduced as a tool commonly used to solve engineering analysis problems and Abaqus was discussed in more details. Finally, progressive collapse was briefly examined through the effects of a structures response to an abnormal load or from the sudden removal of a structural element. C hapter 3 describe d the research methodology to extract the mechanical resistance function of an internal RC beam column connection subjected to monotonic loads. The actual process and the results of validation were described in C hapter 4. Extraction of the mechanical resistance of the inter ior RC beam column connection was done and a simplified structural based FE model with connector elements representing the RC beam column connection, was developed to replace the RC beam column assemblage. PAGE 101 101 Conclusions Validations The behavior of a RC beam under dynamic loadings was validated using Abaqus/Explicit. This is to validate the behavior of concrete elements under dynamic loadings. The behavior of three different types of RC beam colum n connections, namely corner connection, T connection and interior connection, under monotonic loadings wer e being validated with existing test data using Abaqus/ Standard. This ensured confidence in the usage of Abaqus/Standard to derive the mechanical re sistance of these three different types of RC beam column connections for further study. Characterization of M oment R esistance of I nterior C onnection The RC beam column connection, interior connection, in our study was modeled using Abaqus/Standard and a m oment rotation resistance function was successfully derived. A Simplified Structural FE M odel A simplified structural FE model consisting of beam elements to represent the RC beams and columns and connectors to replace the moment rotation resistance funct ion of the RC beam column connections was developed and shown to exhibit the same behavior as the detailed model. Most importantly, the new model was able to save much computing time and resources. This simplified structural FE model can be used for progre ssive collapse analysis on an entire structure in future studies Limitations The simulation of the RC beam column assemblage was done with fine mesh of continuum elements (concrete) and embedded beam elements (reinforcement) in the PAGE 102 102 interior connection up to d distance but further away, continuum elements of coarse mesh was used to represent the RC beams and columns. This is because the behavior of the connection is of interest while the role of the RC beams and columns is to transfer the loads to the conn ection. Due to the large dimensions of the RC beams and columns, simulating the RC beams and columns in detail similar to the interior connection will take a very long time. This limitation can be resolved with faster computers or computers that are conne cted in parallels to run the simulation s With that, the entire assemblage could be simulated in details with in just a few days and will provide a better simulation of load transfer from the beams and columns to the connections Currently, the material mod els for concrete and reinforcement in Abaqus could not couple together and adjust the stress in the reinforcement because of the reduction in the bonding length between the reinforcement and the concrete due to cracks This might result in a stiffer or str onger Abaqus model than the actual behavior of RC elements. This limitation can be solved by introducing a material model or code that could take into considerations the effect of the development length of the reinforcement. Recommendations Progressive col lapse of an entire structure subjected to abnormal loadings or the sudden removal of a structural element can be carried out using the simplified structural based FE model and Abaqus/Explicit in future study The RC beams and columns was represented by con tinuum elements having very coarse mesh and a linear elastic material property with yield stress ten times of normal concrete. A detailed study to represent the RC beams and columns by single resistance functions or material property can be carried out suc h that simpler continuum or beam PAGE 103 103 elements can replace the RC beams and columns to save time and resource during the analysis. Material tests to be carried out for concrete and reinforcement to better describe the parameters required in Abaqus for a more a ccurate simulation. Future study can investigate the influence of transverse RC beams and adjoining RC slabs on the moment resistance function of the RC beam column connection. 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Impact loading of a rei nforced concrete beam to column joint, Special Publications, ACI J .,73, 143164. Yim, H.C. (2007). A study of steel moment connections for structures under blast and progressive collapse loading rates. Ph.D thesis, The Pennsylvania State University, P ennsylvania, U.S. PAGE 108 108 BIOGRAPHICAL SKETCH Loo Yong Tan was born in Singapore in 1976. He attended secondary school and junior college in Singapore. He was drafted into the army for his national service from 1995 to 1997. He began his undergraduate studi es in civil and structures engineering at Nanyang Technological University, Singapore, in July 1997 and obtained his B achelor of Engineering degree in July 2001 In September 2001, he joined the Defence Science and Technology Agency, Singapore, as a projec t engineer. He continued with his part time graduate studies in industrial and systems engineering at the National University of Singapore, Singapore, in January 2004 and obtained his Master of Science degree in December 2006. In 2008, he was awarded a postgraduate scholarship to pursue a masters degree in civil engineering at the University of Florida, specializing in the field o f protective structures. 