CRITICAL INVESTIGATION ABOUT BOND-SLIP IN BEAM-COLUMN JOINT MACRO-MODEL

Transcription

1 Istituto Universitario di Studi Superiori Università degli Studi di Pavia EUROPEAN SCHOOL FOR ADVANCED STUDIES IN REDUCTION OF SEISMIC RISK ROSE SCHOOL CRITICAL INVESTIGATION ABOUT BOND-SLIP IN BEAM-COLUMN JOINT MACRO-MODEL A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in EARTHQUAKE ENGINEERING by SARA BROGLIO Supervisor: Dr. Rui Pinho Co-Supervisor: Prof. Haluk Sucuoglu May, 2009

2 The dissertation entitled Critical Investigation About Beam-Column Joint Macro-Model, by Sara Broglio, has been approved in partial fulfilment of the requirements for the Master Degree in Earthquake Engineering. Dr. Rui Pinho Prof.Haluk Sucuoglu

3 Abstract ABSTRACT Beam-column joint is considered a critical region in the structures under severe earthquake attacks. The joint region is subjected to horizontal and vertical actions: the magnitude of these forces is higher than in the adjacent beams and columns. Experimental researches indicate that elongation and slip of tensile reinforcements in a joint could result in significant rotation that increases total lateral displacement of the structure. The purpose of this work is to study the effect of bond slip and interface interaction in a T beam-column joint. In traditional analytical models this aspect is not considered and the intersection between the elements is supposed to be rigid. Starting from the joint model proposed [Mitra and Lowes, 2004], a critical approach is followed: some points of their theory are reviewed and some approximations are introduced. A flexible joint is developed and inserted in the structural model. The pseudo-static analyses are performed using a fibre-based Finite Element program. The results in terms of strains, internal actions, vertical and lateral displacements are presented and discussed. Keywords: beam-column joint; bond-slip; interface interaction; flexible joint; fibre-based Finite Element; i

4 Aknowledgements AKNOWLEDGEMENTS I would like to sincerely thank my supervisor Dr. Rui Pinho for being willing and available to help me. Thanks also to Prof. Haluk Sucuoglu for his effort to facilitate my research during his period in Italy. I would like to thank my family for its continuous support throughout this work. ii

5 Index TABLE OF CONTENTS Page ABSTRACT... i AKNOWLEDGEMENTS... ii TABLE OF CONTENTS... iii LIST OF FIGURES...v LIST OF TABLES... viii 1. INTRODUCTION Scope Objectives Outline SHEAR MECHANISM IN BEAM-COLUMN JOINTS Features of joint behaviour Equilibrium Criteria Shear Strength DESCRIPTION OF THE JOINT MODELS PROPOSED BY LOWES Lowes [1999] Lowes and Altoontash [2003] L.N.Lowes and Nilanjan Mitra [2004] EXPERIMENTAL TEST Design of As-Built Test Model Loading Material and Construction Concrete Properties...26 iii

6 Index Concrete Elastic Modulus Reinforcing Steel Properties Test Procedure Instrumentation Experimental Response of the As-Built Connection Load-Displacements Relationship Anchorage and Reinforcements ANALYSIS PROCEDURE AND RESULTS Description of Structural Model Traditional Model: Rigid Internal Joint Model with Flexible Joint:Bond Slip Springs Model with Flexible Joint: Slip Springs and Interface Springs INTRODUCTION OF A SIMPLIFIED RELATIONSHIP FOR BOND-SLIP SPRINGS CONCLUSIONS...67 iv

7 Index LIST OF FIGURES Page Figure 2.1. Features of column and joint behaviour [Paulay and Priestley,1992]...5 Figure 2.2. Mechanism of shear transfer at an interior joint [Paulay and Priestley, 1992]...6 Figure 2.3. External actions and internal stress resultants at an interior joint [Paulay and Priestley, 1992]...6 Figure 2.4. Compression strut model [Lowes and Mitra, 2004]...7 Figure 2.5. Bond stress and bar stress distributed for a bar anchored in a beam column joint [Lowes and Altoontash, 2003]...8 Figure 3.1. Finite Element mesh in the vicinity of a bond zone [Lowes, 1999]...10 Figure 3.2. Finite Element mesh in the vicinity of a bond zone with cover concrete element [Lowes, 1999]...10 Figure 3.3. Beam column connection core subjected to uniform shear loading for the case of uniform bond stress in anchorage zones [Lowes, 1999]...11 Figure 3.4. Beam column bridge connection core subjected to diagonal compression strut shear loading for the case of significant bond stress only in compression zone Figure 3.5. Finite element model of panels tested by Vecchio and Nieto [1991]...12 Figure 3.6. Discrete concrete cracking as computed using the model with flexible load frame and no initial concrete damage [Lowes, 1999]...14 Figure 3.7. Discrete concrete cracking as computed using model with initial damage[lowes, 1999]...14 v

8 Index Figure 3.8. Discrete concrete cracking as compute using model without initial damage [Lowes, 1999]...14 Figure 3.9. Computed and observed shear stress versus shear strain history for different panel models [Lowes. 1999]...15 Figure Reinforced concrete beam-column joint model [Lowes and Altoontash, 2003].16 Figure Definition of component deformation and generalized forces (a) Component deformation and forces, (b) Shear forces acting on shear panel component [Lowes and Altoontash, 2003]...16 Figure One-dimensional material model: (a) Material states, (b) Impact of hysteretic damage on load-deformation response [Lowes and Altoontash, 2003]...17 Figure Behaviour of reinforced concrete panel SE8 tested by Stevens et al.[1990] under cyclic shear loading: (a) Simulated response, (b) Observed response...18 Figure Bond stress and bar stress distribution for a bar anchored in a beam column joint...19 Figure Observed (a, b) and simulated response (c, d) of the Park and Ruitong [1988] building sub-assemblages Figure 4.1. Prototype multi-column frame [Lowes and Moehle, 1999]...22 Figure 4.2. Reinforced concrete bridge frame sub-assemblages tested by Lowes and Moehle [1999]...24 Figure 4.3 Loading scheme. Gravity and Earthquake loads [Lowes and Moehle, 1999]...25 Figure 4.4. Instrumentation used to measure global displacements and applied loads using in Lowes and Moehle test...26 Figure 4.5. Instrumentation used to measured joint deformation in Lowes and Moehle Test [1999]...29 Figure 4.6. Steel reinforcement strain gages. Lowes and Moehle Test [1999]...30 Figure 4.7. Instrumentation to measured column bar slip. Lowes and Moehle Test [1999]...30 Figure 4.8. Column longitudinal reinforcement strain [Lowes and Moehle, 1999]...31 Figure 4.9. Zone of disconnected concrete [Lowes and Moehle, 1999]...31 Figure Load versus beam longitudinal reinforcement [Lowes and Moehle, 1999]...32 Figure Beam longitudinal reinforcement strain history [Lowes and Moehle, 1999]...33 Figure Load versus reinforcement strain history [Lowes and Moehle, 1999]...34 Figure Load versus displacements [Lowes and Moehle, 1999]...35 vi

9 Index Figure Nominal joint shear strain measured with internal and external gages [Lowes and Moehle, 1999]...37 Figure Nominal joint shear stress versus shear strain measured with internal gages [Lowes and Moehle, 1999]...37 Figure Nominal joint shear stress versus shear strain measured with external gages [Lowes and Moehle, 1999]...37 Figure Nominal joint shear strain from external gages [Lowes and Moehle, 1999]...38 Figure Load versus column internal bar strain measured with gage sg4 [Lowes and Moehle, 1999]...39 Figure Load versus column internal bar strain measured with gage sg5 [Lowes and Moehle, 1999]...39 Figure Displacement versus slip of column longitudinal reinforcement (sl2) [Lowes and Moehle, 1999]...40 Figure Displacement versus slip of column longitudinal reinforcement (sl1) [Lowes and Moehle, 1999]...40 Figure 5.1. Discretization of RC cross-section in a fire based model [SeismoSoft]...41 Figure 5.2. Traditional model implemented with SeismoStruct [SeismoSoft]...42 Figure 5.3. Geometrical dimensions...42 Figure 5.4. Assumed displacement history applied at the column base for pseudo-static analysis...43 Figure 5.5. Beam section in which steel strain is reached for first...43 Figure 5.6. Column longitudinal reinforcements strains developed under experimental cyclic loading [Lowes and Moehle, 1999]...44 Figure 5.7. Strains in column: comparison between traditional model and experimental test 45 Figure 5.8. Shear at the column base in traditional model and in experimental test...45 Figure 5.9. Macro-model with slip springs...46 Figure Geometrical dimensions of the model with joint...47 Figure Force versus slip relationship for spring in the upper side of the beam...47 Figure Tensile stress versuss slip constitutive relationship [Lowes and Altoontash, 2003] and a possible approximation...48 Figure System stiffness definition: steel reinforcement as springs in parallel (beam section)...49 Figure Beam section in which steel strain is reached for first...49 vii

10 Index Figure Comparison of traditional model, model with slip springs and experimental test results in column...50 Figure Shear at the column base in traditional model, in model with slip springs and experimental test...50 Figure Traditional model and model with springs beam shear force comparison...51 Figure Subdivision of the beam...52 Figure Column lateral displacements...52 Figure Beam A, lateral displacement...52 Figure Beam A, vertical displacements...53 Figure Macro-model with slip and interface spring...53 Figure Idealized shear deformations behaviour...54 Figure Shear spring law for beams and column...56 Figure Beam Section in which steel strain is reached for first...56 Figure Strain comparison obtained with experimental test, analytical model with slip springs only and model with interface springs...57 Figure Shear at the column base in traditional model, in model with slip springs and experimental test...58 Figure Comparison of shear forces in the beam...59 Figure Displacement in x direction in BEAM element...59 Figure Displacements in x direction for COLUMN element...59 Figure Displacements in z direction for BEAM element...60 Figure 6.1. Bilinear approximation for bond-slip relationship...61 Figure 6.2. Beam section in which steel strain is reached for first...62 Figure 6.3. Strain results for asymmetrical tri-linear and bilinear approximated model Figure 6.4. Time history of shear at the column base. Comparison between values of shear obtained with asymmetrical tri-linear constitutive relationship and approximated bilinear model...64 Figure 6.5. Displacements in x direction for BEAM element...65 Figure 6.6. Displacement in x direction for COLUMN element...66 viii

11 Index LIST OF TABLES Page Table 4.1. Concrete mix used for sub assemblages model...26 Table 4.2. Quantities used in concrete mix...27 Table 5.1. Steel strain limit. Numerical output for traditional model Table 5.2. Column strains comparison between traditional model results and experimental output...44 Table 5.3. Shear Forces at the column base: comparison between experimental outputs and traditional model outputs...45 Table 5.4. Average bond strength as a function of steel stress state [Lowes and Altoontash, 2003]...48 Table 5.5. Steel strain limit. Numerical output for model with slip springs only...49 Table 5.6. Beam shear results for traditional model and for model with springs...51 Table 5.7. Steel strain limit. Numerical output for model with slip springs only...56 Table 5.8. Column strains comparison between model with interface springs results and experimental output...57 Table 5.9. Shear Forces: comparison of results for all the models analyzed...57 Table Shear forces in the beam for the model with interface springs...58 Table 6.1. Steel strain limit. Numerical output for Model with slip springs only...62 Table 6.2. Comparison between results obtain with tri-linear model and approximated bilinear model...62 Table 6.3. Steel strains in the column. Comparison of results ix

12 Index Table 6.4. Shear forces in the beam for the model with approximated bilinear constitutive relationship...64 Table 6.5. Beam displacements in x direction. Numerical Values...65 Table 6.6. Column displacements in x direction. Numerical Values...66 x

13 Chapter 1. introduction 1. INTRODUCTION 1.1 Scope Beam-column joints are critical regions. In reinforced concrete structures, during severe earthquake attacks, brittle shear failure can occur in the joints, as well as cracking and frictional sliding under reverse cyclic loading. The ultimate resistance capacity depends directly on their different material behaviour (concrete damage, steel plasticity) but it must deal with crack opening and degradation of bonding between concrete and steel. The integration of all these items could make possible a realistic approach of global structures response. Bond behaviour is important from this point of view because its degradation increases the period of vibration (more flexible structures), decreases energy dissipation capacity and produces a global redistribution of internal actions. A traditional fibre-based Finite Element model, where the beam-column joint is supposed to remain elastic during the earthquake, cannot catch the real response of reinforced concrete structures. Laboratory testing of built sub-assemblages, with design details typical of pre constructions shows that joints with little or no transverse reinforcements and relatively high shear and bond stress demands exhibit severe stiffness and strength loss [Meinheit and Jirsa, 1977; Leon, 1990; Walker, 2001]. For this reason many models have been developed. One of these models [Lowes and Altoontash, 2003] represents the response of reinforced concrete beam-column joint under reverse cyclic loading. This model provides a representation of the primary inelastic mechanisms that determine joint behaviour: failure of joint core under shear loading and anchorage failure of beam-column longitudinal reinforcement embedded in the joint. Constitutive relationship are developed to define the load deformation response. Based on previous work, modifying it to improve the prediction of response and to extend the range of applicability, a new model formulation is proposed by Lowes and Altoontash [2003] to simulate the shear-strain response of joint core. This represents well stiffness and strength response parameters for joint with a wide range of design parameters. Both previous models show some limitations: the first, in terms of constitutive relationship, seems to be too much detailed to described the bond slip behaviour. In the second one, the main limit is represented by the location of springs that are placed at the centroid of the compressive and tensile zone respectively. 1

14 Chapter 1. introduction 1.2 Objectives Lowes and Altoontash [2004] model is mainly detailed in the definition of bond-slip relationship. The introduction of some approximations could improve its applicability without a great loss of efficiency. By making an extensive review of the previous models the main aims of this research are the following: the first one is to discuss the different theories about the joint modelling (models from 1999 to 2004 are considered [Lowes et al]) to understand which are the common points and the limits. The second purpose is to explore the possibility of using a simplified model, taking into account which could be the limitations of its own approximations. The checks on these simplified models are made using experimental data [Lowes and Moehle, 1999] when possible; Otherwise results are compared with the output of a traditional fibre-based Finite Element model. 1.3 Outline This study is organized into seven chapters. It covers several aspects ranging from the motivation behind the selection of the research topic, the literature review, the development of structural models with flexible joints. Chapter 1 is devoted to the presentation of the research topic and the identification of the general scope and specific objective. Theoretical aspects and description of the internal mechanism acting in the joint with particular attention for bond-slip mechanism, are described in Chapter 2, in order to understand some choices taken during the development of the flexible joint properties. The description of the evolution of the joint model is an important point in this research. Starting from the model proposed by Lowes [1999] to the last model implemented in 2004, common points and limitations are underlined to understand if such detailed models are useful and if some simplification are possible or not. This excursus is illustrated in Chapter 3. In Chapter 4, an experimental test is shown [Lowes and Moehle, 1999]. The goal of this test is to identify issues critical to the evaluation and retrofit of beam-column T-joint in older reinforced concrete bridge frame. The data of this work are used to define the material properties, the geometry of the structures and loading distribution useful for a correct computer model. Where possible, the analytical results are compared to the experimental output. Fibre-based Finite Element models are developed to check the behaviour of a structure with flexible beam-column joint. The output results are compared with the output of traditional model with elastic rigid joint. Dispersion diagrams are developed and discussed in Chapter 5. In Chapter 6 the effect on the global structural behaviour of a simplified model for bond slip springs is discussed. 2

15 Chapter 1. introduction In Chapter 7 the conclusion of this research are discussed. 3

16 Chapter 2. shear mechanism in beam-column joints 2. SHEAR MECHANISM IN BEAM-COLUMN JOINTS Beam-column joints are considered critical regions in the structures designed for inelastic response, under a severe earthquake attack. The joint regions are subjected to horizontal and vertical actions. The magnitude of these forces is typically higher than in the adjacent beams and columns. If not designed for, joint shear failure can result. The reversal moment across the joint means that the beam reinforcement is required to be in compression on one side of the joint. The bond stress needed to sustain this force gradient is high and this may cause bond failure and corresponding degradation of moment capacity accompanied by excessive drift. Detailed studies of joint for buildings in seismic regions have been undertaken only in the past 20 years. Between 1984 and 1989 significant effort, including coordinated experimental works, by researcher from United States, New Zealand, Japan and China were made. 2.1 Features of joint behaviour Under seismic action large shear forces could be generated into beam-column joints. These forces may cause failure in the joints core due to breakdown of shear or bond mechanism or both Equilibrium Criteria The joints are the elements generated by the connection between beams and columns. The forces acting on the joints are the actions carried by the framing elements: the joint regions are subjected to horizontal and vertical shear forces higher than in the adjacent beams and columns. It is possible to consider the joint as a free body on which forces generated by framing beams and columns are applied. Actions introduced by beam reinforcements (symmetrical ones in this case) to the joint are shown in Figure 2.1 to be internal tension T b and compression forces C b and vertical beam shear forces V b. 4

17 Chapter 2. shear mechanism in beam-column joints Figure 2.1. Features of column and joint behaviour [Paulay and Priestley, 1992] The joint as free body must be in equilibrium. Making the approximation: C b = (2.1) and considering that the beams shears on the opposite interfaces on the joint are equal, the equilibrium of the joint requires an horizontal column shear force of: (2.2) Writing the horizontal equilibrium equation, the large horizontal shear force across the joint region is given by: (2.3) Shear Strength Internal forces transmitted from adjacent members to the joints result in shear forces in horizontal and vertical direction. These shear forces lead to diagonally compression and tension stresses in the joint core and often diagonal cracks occur. The mechanism of shear resistance at this stage changes drastically: some of the internal forces, particularly those generated in the concrete, will combine to develop a diagonal strut. To prevent shear failure by diagonal tension, both horizontal and vertical shear reinforcements will be required. The amount of horizontal joint shear reinforcements may be significantly more than what would normally be provided in columns in the form of ties or hoops, in particular when the axial load in the column is small. 5

18 Chapter 2. shear mechanism in beam-column joints Figure 2.2. Mechanism of shear transfer at an interior joint [Paulay and Priestley, 1992] Figure 2.3. External actions and internal stress resultants at an interior joint [Paulay and Priestley, 1992] Typical moments, shear and axial forces introduced in the joint are represented in Figure 2.3. (2.4) As just defined in equation 2.4, in seismic design, V c is computed considering the beam flexural overstrength. Taking into account the equilibrium of vertical forces at the joint, it would lead to expression for vertical joint shear forces. Compression forces induced to the joint by the beam and the column at diagonal opposite corners of the joint are combined into a single diagonal compression force carried by a diagonal concrete truss, as shown in Figure 2.4. The shear forces developed in beams and 6

19 Chapter 2. shear mechanism in beam-column joints column are transmitted to the joint via the respective flexural compressive zones. Considering the position of neutral axis in each section, it is possible to define the width of the concrete truss. The total horizontal force is transmitted by the beam top flexural reinforcements to the joint by means of bond. A fraction of this force is transferred to the diagonal strut. When the axial load is not applied to the column, the inclination in the strut is similar to the potential linear plane, otherwise the inclination is steeper. Bond strength begins to deteriorate when the yield strain in steel is exceeded. In elastic joints higher average bond stress can be maintained. Bond deterioration due to plastic strain in a bar embedded in a joint core can contribute up to 50% of the overall deflection in beam-column sub-assemblages. Bond is not constant along the reinforcement bars; it is rapidly reduced outside the joint concrete core. Within the column joint core, high bond can be developed Figure 2.4 Figure 2.4. Compression strut model [Lowes and Mitra, 2004] because some confinement of concrete perpendicular to the beam longitudinal bars is always present. The bond slip relationship varies along the bar in function of the region of embedment considered. For this reason it is difficult to develop a simple model for the global reinforcement bond slip response. Bond slip behaviour is investigated by experimental test similar to traditional pull-out test. In such test the bond stresses are uniformly distributed around the periphery of the bar. Their effect is a uniform distribution of tangential and radial stresses in surrounding concrete. However, this is not the real situation because a bar at the top of a beam is in a more unfavourable condition respect to that within the joint. Because of the very large bond force, a splitting crack along the bar will usually form. The total bond force from a top beam bar needs to be transfer predominantly downward into the diagonal compression field of joint core. Therefore the distribution of bond stress is not uniform as in the standard case. Much larger bond stress will need to be generated in the side of the bar facing the joint core. Any bond in excess of about 15% of the total, which might be transferred toward the column, will have to enter the joint core: this involves shear transfer by shear friction across the horizontal splitting cracks. 7

20 Chapter 2. shear mechanism in beam-column joints Figure 2.5. Bond stress and bar stress distributed for a bar anchored in a beam-column joint [Lowes and Altoontash, 2003] 8

21 Chapter 3. description of the joint models proposed by Lowes 3. DESCRIPTION OF THE JOINT MODELS PROPOSED BY LOWES Many models are developed to represent the real behaviour of beam-column joints. Early works to simulate the response of reinforced concrete frames, relied on the calibration of plastic hinges introduced at the end of beam-column line element to represent the inelastic behaviour of the joints. The next generation of models considers separately the inelastic response of beams, columns and joints, introducing zero length rotational spring elements [El- Metwally and Chen,1989; Alath and Kunnath, 1995]. Calibration of these models is based on experimental data characterizing the response of joint sub-assemblages. In this case the development of an objective, transparent model is difficult because the effects of multiple inelastic response mechanism are combined into a single moment rotation relationship. More recently, researchers have begun using continuum-type elements to represent the response of reinforced concrete joints in combination with transition elements, used to maintain compatibility with beam-column line elements. 3.1 Lowes [1999] The proposed Finite Element model incorporates non standard element formulations, solution algorithms and material models. Two aspects of the proposed model, requiring the most significant modification of the base Finite Element code, include meshing of the model to accommodate the bond element for appropriate representation of bond-zone behaviour and introduction of a solution algorithm appropriate for systems including material models with degrading strength and stiffness. Bond elements are introduced to represent the microscopic behaviour of the concrete and steel in the vicinity of the interface. A bond element has zero width, in order to represent radial forces developed in association with tangential bond forces; the model includes definition of radial bond response mode. Figure 3.1 shows how the bond element is idealized. 9

22 Chapter 3. description of the joint models proposed by Lowes Figure 3.1. Finite Element mesh in the vicinity of a bond zone [Lowes, 1999] Two nodes with identical coordinates and different element connectivity may not be possible within the structure of many codes and this requires the introduction of an independent meshing algorithm. The concept of cover concrete is introduced to represent the three-dimensional bond zone response. The concrete element in line with the bond elements presents the thickness of concrete equal to the thickness of the bond zone. If the radial forces associated with bond response are significant, the concrete cove could carry tension in direction perpendicular to the axis of reinforcing bars and develop splitting type crack. The deformation of the cover concrete element is modelled as not compatible with deformation of concrete element in plane with the bond and reinforcing steel element as shown in Figure 3.2. Figure 3.2. Finite Element mesh in the vicinity of a bond zone with cover concrete element [Lowes, 1999] 10

23 Chapter 3. description of the joint models proposed by Lowes The simplest representation of concrete-steel bond is achieved by introducing a one dimensional bond element: its introduction requires use of only one additional, repeated node while two additional elements are needed for the introduction of the two-dimensional bond elements. The response of reinforced concrete bond zone is defined by a number of parameters including the distribution of concrete damage, the embedded reinforcing steel stress-strain distribution, local bond slip response, average bond strength, global and local load displacement history. The analytical model results are compared with those given by laboratory tests. Correlation between flexural-bond behaviour computed by the Finite Element model and that observed in laboratory specimens is good. The Finite Element model represents well the fundamental characteristics of flexural bond zone in reinforced concrete elements. The model predicts crack-spacing and average bond strength typical of that observed in laboratory. It seems to evaluate well the orientation of localized cracks, while it is less accurate in representing localized crack orientation along discrete crack surface. The model predicts temporary losses in system strength, associated with initiation of significant discrete concrete cracks, typically not observed in laboratory. The next phase of model verification proposed by Lowes [1999] is focused on evaluating the response of reinforced concrete sub-system subjected to shear type loading. In these systems, anchorage zone defines the perimeter of the beam-column connection core. If the distribution of bond stress is quite uniform, the connection core is subjected to an approximately uniform shear loading, as illustrated in Figure 3.3. If the bond stress is significant only near the compression end of the bar, in the flexural compression zone, the connection core is subjected to shear in form of diagonal compression loading. Connection strength may be determined by the capacity of the core concrete connections to carry uniform shear and compression-strut shear. Figure 3.3. Beam-column connection core subjected to uniform shear loading for the case of uniform bond stress in anchorage zones [Lowes, 1999] 11

24 Chapter 3. description of the joint models proposed by Lowes Figure 3.4. Beam-column bridge connection core subjected to diagonal compression strut shear loading for the case of significant bond stress only in compression zone. While laboratory loading of these panels consists of applying uniform tension followed by unloading and reloading under uniform shear in combination with tension or compression, such loading is not possible in computer simulation. For the computer model, initial preloading and the resulting concrete damage is introduced by pre-damaging concrete element in the vicinity of the crack observed in the laboratory specimen. The goal of pre-cracking is simply to define the orientation of the fictitious concrete crack planes. It should be noted that if the concrete elements are initially cracked under tensile loading, the crack surfaces are established at 0 and 90 degree rotation from the horizontal. As a result, another crack surface cannot develop at a 45 degree angle from the horizontal, as observed in laboratory. Vecchio and Nieto[1991] tested two different frames, shown in Figure 3.5: Figure 3.5. Finite Element model of panels tested by Vecchio and Nieto [1991] 12

25 Chapter 3. description of the joint models proposed by Lowes In these analytical model the concrete mesh is 18 by 18 elements. In the most heavily reinforced areas, reinforcement is 2-6 mm diameter bars spaced at 50 mm. All reinforcing bars are connected to concrete element via bond element and the grey elements shown in Figure 3.5 are damaged prior to application of shear load. Vecchio and Nieto [1991] indicate that the loading is achieved through the application of approximately uniform loads to the five shear keys embedded along each of panel edges. Load control is not possible in computer simulation nor is the application of shear load in combination with level of tension or compression loading. In the computer simulation, only the case of pure shear load is considered and this loading is applied under displacement control. As declared before, two different load frames are used. Load Frame A is a stiff frame composed by truss elements, connected to every node of the panel edge and subjected to point loading through displacement control at the two extreme nodes. The stiff frame provides the redistribution along the edges of the panel as a function of the panel damage. Given the stiffness of the exterior frame and the concrete damage pattern significant redistribution is observed and an absolutely uniform loading is not achieved. Load Frame B consists of a relatively flexible frame composed of axial elements and connected to the panel in five points along each edge. The exterior frame is flexible; only a little redistribution is possible and an approximately constant shear is applied to each edge of the panel. For Finite Element models, average and shear strain are defined in different way for the two different load frames. For Load Frame A, shear stress is defined as the total load applied to the extreme nodes of the load frame divided by the shear surface of the panel. For the flexible frame, shear stress is defined on the basis of the load transferred from the flexible frame to the concrete panel at each node on the panel edge. For both model shear deformation is the relative deformation of the model nodes at the measurement point used in laboratory experiments. The computed and observed behaviours of the panel are compared on the basis of concrete damage pattern. The Figures 3.6, 3.7 and 3.8 describe the computed concrete pattern under moderate levels of shear loading for three different models. These data show the orientation of discrete cracking in the concrete panel prior to the development of highly localized failure mechanism. In general, data present discrete cracking oriented perpendicular to the direction of maximum principal tensile stress at an angle of 45 degrees, as observed in laboratory. 13

26 Chapter 3. description of the joint models proposed by Lowes Figure 3.6. Discrete concrete cracking as computed using model with initial damage[lowes, 1999] Figure 3.7. Figure 3.7. Discrete concrete cracking as compute using model without initial damage [Lowes, 1999] Figure 3.8. Discrete concrete cracking as computed using the initial concretee damage [Lowes, 1999] model with flexible load frame and no 14

27 Chapter 3. description of the joint models proposed by Lowes Within the most heavily damage region of the panel, the concrete elements, that are initially damaged, accumulate less damage respect to adjacent elements. In laboratory test, the initial concrete cracking does not appear to affect the orientation of concrete cracking developed under shear loading. The computed and observed panel behaviour is compared also in terms of shear stress versus shear strain histories. The different behaviours are shown and compared in Figure 3.9. Figure 3.9. Computed and observed shear stress versus shear strain history for different panel models [Lowes, 1999] The computed response is initially linear elastic. Once cracking has occurred, shear strength decreases with increasing shear strain as damage distributes throughout the panel. One or more discrete crack zones dominate the response and shear strength begins to increase with increasing shear strain. Failure of the system is reached when all the reinforcing steel, crossing the centre of the panel, yields and the model becomes instable. Since the laboratory testing is conducted under load control, it is not possible to observe a region in which shear strength deteriorates. The definition of an initial shear-friction strength equal to the concrete compressive strength appears to produce quite accurate computed response. The introduction of two orthogonal cracks is not sufficient to represent the behaviour defined by damage on a single crack surface direction between the two cracks. Finally, data from this study underline the importance of accurate representation of system boundary conditions. While the investigators suggest that the applied loading of the panel is uniform, evaluation of reinforcement and shear keys detailing in the panel, as well as the computed response, indicate that either is not the case or that uniform loading from the actuators is distributed along the panel edge by reinforcing steel, shear keys and shear keys anchorage dowel. 15

28 Chapter 3. description of the joint models proposed by Lowes 3.2 Lowes and Altoontash [2003] Figure 3.10 shows the joint model proposed by Lowes and Altoontash [2003]. At the perimeter of the joint the element displacement field is defined by two translation and one rotation at each of four external nodes. The element formulation is compatible with line elements used in two dimensional structural analysis. The joint element is a classical superelement, composed of four internal nodal translations and 13 one-dimensional components. Determination of the element material state requires the solution of a non linear system to determine the internal translations that satisfy equilibrium. Figure Reinforced concrete beam-column joint model [Lowes and Altoontash, 2003] Figure Definition of component deformation and generalized forces (a) Component deformation and forces, (b) Shear forces acting on shear panel component [Lowes and Altoontash, 2003] The generalized displacement history of four external and four internal element nodes defines the material state of the joint element and the deformation history of the eight bar-slip springs, four interface shear springs and one shear panel component that form the joint model. One-dimensional material constitutive relationship defines component forces presented in Figure 3.11 as f i, function of the component deformation histories. For the shear panel, the complementary component force f 13 is the nominal shear stress developed in the joint core multiplied by the volume of joint core. Complementary to the 16 internal and external nodal displacements, there is a set of 16 internal and external nodal resultants. Nodal resultants can be computed from component forces imposing equilibrium at both internal and external degrees of freedom. 16

29 Chapter 3. description of the joint models proposed by Lowes Given a set of imposed external and internal nodal displacements, determination of the element material state requires a solution for four unknown internal displacements in order to satisfy the internal equilibrium of the element. The proposed joint model formulation enables the simulation of inelastic behaviour, due to any kind of possible failure that can occur in the joint (anchorage failure, failure of joint core under shear loading and failure of interface-transfer mechanism). Constitutive relationships are developed to define the load deformation behaviour of shear panel component and shear component as a function of material properties, joint geometry and joint reinforcement layout. To define the constitutive laws, fundamental material behaviours and experimental data are used. The joint element and constitutive material relationships are compared with the response of some laboratory tests. Few data are available for use in evaluating a constitutive relationship for the interface-shear component of the model. A general one-dimensional hysteretic model is used to represent the response of each of the components of the element. A response envelope, an unloading-reloading path and three damage rules, that control evolution of this path, define the one-dimensional material model. Hysteretic damage is simulated through deterioration in unloading stiffness (unloading stiffness degradation), deterioration in strength achieved at previously unachieved deformation demands (strength degradation) and determination in the strength development in the vicinity of the maximum and minimum deformation demands (reloading strength degradation). Figure One-dimensional material model: (a) Material states, (b) Impact of hysteretic damage on load-deformation response [Lowes and Altoontash, 2003] 17

30 Chapter 3. description of the joint models proposed by Lowes Figure 3.13 Behaviour of reinforced concrete panel SE8 tested by Stevens et al.[1990] under cyclic shear loading: (a) Simulated response, (b) Observed response As previously discussed, an earthquake attack results in shear loading at the joint core. The inelastic response is simulated by the shear panel (simulation of inelastic core behaviour). On the basis of previous research a constitutive relationship is developed to define deformation of shear panel on the basis of material properties and joint geometry. This constitutive model is developed employing Modified Compression Field Theory (MCFT) [Vecchio and Collins, 1986] to define the envelope of shear stress strain history of the joint core; experimental data provided by Stevens et al [1991] are used to described response under cyclic loading. This procedure allows the user to define behaviour using material properties, joint geometry, and joint reinforcing steel ratio. Only few experimental investigations consider the response of reinforced concrete element under pure shear. Figure 3.13 shows the linear interpolation of the monotonic envelope, as a result of the MCFT model, and the simulated response of reversed-cyclic loading. Discrepancies in the results are due to the relative simplicity of the model applied. Earthquake loading results in substantial bond demand for longitudinal reinforcements. A constitutive model is developed for the load-deformation history of bond slip springs that simulate the inelastic anchorage-zone response. This model is developed using data from experimental testing of anchorage-zone specimens and assumption about the bond-stress distribution within the joint. The envelope of the bar-stress versus slip relationship is based on several approximations and simplified assumptions: constant bond stress for reinforcements in elastic range and slip defined as the relative movement of reinforcing bar with respect to the perimeter of the joint and function of strain distribution along the bar. The bar is assumed to exhibit zero slip for zero stress. 18

31 Chapter 3. description of the joint models proposed by Lowes Figure 3.14 Figure Bond stress and bar stress distribution for a bar anchored in a beam column joint Average bond strength is based on experimental data and previous investigations. The cyclic response is calibrated on data provided by Eligehausen et al.[1983] and Hawkins et al.[1982], used for defining these model parameters. (a) Comparison of Simulated and Observed Response. The model is evaluated through the comparison of simulated and observed response for a series of building frame subassemblages tested in laboratory under pseudo-static reversed cyclic loading by Park and Ruiton [1988]. Numerical models were developed to simulate Park and Ruitong [1988] experiment using MATLAB; these models comprised lumped plasticity beam-column Figure Observed (a, b) and simulated response (c, d) of the Park and Ruitong [1988] building sub-assemblages. 19

32 Chapter 3. description of the joint models proposed by Lowes elements and the proposed joint element. Compared results are in Figure The comparison of simulated and observed response for a series of beam-column joints indicates that the proposed model represents well the fundamental characteristics of response for joints subjected to moderate shear demands. 3.3 Lowes and Mitra [2004] A study by Mitra and Lowes [2004] suggests a way to modify the joint model proposed by Altoontash and Lowes [2003]. The main modifications are: Bar-slip springs are located at the centroid of the beam and column flexural tension and compression zones rather than at the perimeter of the joint. A new model is used to calibrate joint-panel component, assuming a diagonal compression strut mechanism for load transfer within the joint [Paulay et a, 1978] rather than an uniform field. The new model simulates deterioration of strength. A new bond slip model is proposed, in order not to exhibit negative stiffness prior to reinforcing steel reaching ultimate strength to avoid numerical instability. The shear panel component of the joint element is used to simulate strength and stiffness degradation due to joint core damage under severe demands. This new model assumes that the shear is carried by a concrete compression strut and the transverse reinforcement acts to increase the strength and deformation capacity of this strut. The new shear model is calibrated using simulated and observed data. Assuming that joint shear load is transferred via concrete compression strut, it is possible to predict shear response of confined concrete. The strut orientation is supposed to be constant and defined by the geometry of the beams and columns flexural compression zone. Depth of the strut is defined as the maximum of out of plane depth of the beam and the column. The confined concrete model is the Mander [1988] one and it defines the stress and strain response of the strut. Column longitudinal and joint horizontal reinforcing steel confines the joint core concrete. Only the components of confining forces acting perpendicular to the strut direction are considered. Concrete compressive strength is reduced to take into account cracking parallel to the axis of the strut [Vecchio and Collins, 1986; Belarbi and Hsu, 1995]. The result of previous research indicates that concrete compressive strength is reduced by tensile cracking due to cyclic loading [Stevenson et al, 1991]. The joint carries shear only through compression strut. The bar-slip used for the joint model simulates stiffness and strength loss associated to the deterioration of beam and column reinforcement anchorage in the joint. A review of experimental data by Mitra and Lowes [2004] indicates that the 3.00 mm of slip proposed by Lowes and Altoontash [2003] was too conservative. (a) Comparison of Simulated and Observed Response. Failure Mechanism. The inelastic failure mechanism is correctly simulated. Initial and Unloading Stiffness. The proposed model represents well the observed initial stiffness and the unloading stiffness at the maximum load. 20

33 Chapter 3. description of the joint models proposed by Lowes Post-yield Tangent Stiffness. The post-yield stiffness is well predicted. The coefficient of variation between observed and simulated result is about 22%, but considering the high number of parameters influencing the results (post peak response of shear panel, the hardening response, flexural stiffness of beams and columns), it is possible to consider this value sufficiently low. Maximum Strength. The model represents well the observed maximum strength of the sub-assemblage, with the average ratio of simulated to observed response equal to Drift at Maximum Strength. Drift at maximum strength is simulated with less accuracy than strength. Strength Loss at Final Drift Level. On average the model predicts well the observed strength during the final load cycle. These results validate the proposed shear panel calibration model and the proposed strength reduction model for joints that exhibit yielding of beam longitudinal reinforcement steel prior to the joint failure. Pinching Ratio. On average the model predicts the observed pinching ratio closely. 21

34 Chapter 4. experimental test 4. EXPERIMENTAL TEST The experimental test considered in this study was performed by Mohele and Lowes [1999]. The goal of this research was to identify issues critical to the evaluation and retrofit of beam column T-joints in older reinforced concrete bridge frame. The whole experimental test considered three models: As-built model (Model One) Two retrofit models sub-assemblies from a single prototype frame (Model Two and Model Three) In this study the results given by the test on the as-built model (Model One) are considered. In order to assess the design details controlling the response of as-built connections to earthquake loading, a prototype multi-column frame was designed. Figure 4.1. Prototype multi-column frame [Lowes and Moehle,1999] This Model was tested in laboratory. The Model One displayed essentially no ductility under simulated earthquake loading. Cracking of concrete cover in the joint area was observed during the initial cycles of loading and, during the test, the damage accumulated in the joint. 22

35 Chapter 4. experimental test Throughout the test, instrumentation measured progressive slip of column reinforcement; measurement of beam longitudinal reinforcement strains appears to indicate that this reinforcement was also slipping through the joint. The relatively poor response of Model One was attributed to loss of anchorage of column longitudinal reinforcements terminating in the joint, degradation of joint concrete due to high joint stresses and reduced bond slip capacity along the length of the beam longitudinal reinforcement passing through the joint. 4.1 As-Built Model The as-build prototype was designed considering a review of engineering drawings of RC bridges designed and constructed between 1950s and 1960s. This frame included details considered as critical for the beam-column joint behaviour. The bridge review focused on the older RC bridges in California, characterized by a single level and multi-column frame perpendicular to the roadway. Details of interest for this study are the following: Gross member dimension Reinforcement details in the interior beam-column joint Flexural reinforcement ratio for beams and columns Location of cut-off for flexural reinforcements in beams Anchorage lengths for column flexural reinforcements terminating in the interior beam column joint Column axial load Foundation conditions During the 1950s and 1960s, common practice did not dictate any consideration about the relative flexural capacity of the beams and columns framing into a joint, about force transfer through the joint capacity or concerning member shear capacity as function of member flexural capacity. For this reason it is common to find existing bridges of this period in which undesiderable failure mechanism could occur during an earthquake. For the Model One the following target design concepts are considered: The sum of the cap beam nominal flexural strength exceeds the column nominal flexural strength. Yield of the beam is expected prior to flexural yielding of the column. Considering the method recommended by the ACI-ASCE Committee 352 [A1], the nominal horizontal joint shear developed when the column reaches its nominal flexural strength, is expected as follows: Where: A ACI is the effective joint area in in2 V j =6.2 A ACI f c, lbs 23

36 Chapter 4. experimental test fc is the concrete compressive strength in psi Transverse reinforcements were designed for the shear force corresponding to the ultimate flexural strength. This resulted in the use of significantly more transverse reinforcements than usually found in existing bridges of this time. Figure 4.2. Reinforced concrete bridge frame sub-assemblages tested by Lowes and Moehle [1999] The model tested in laboratory is not at real scale. The selected scale for the specimen is 1:3 with appropriate boundary conditions. During the test only a single interior column and a portion of the cap beam extending on both sides of the joint were considered. The idealized loading system is characterized by: Gravity load simulated by two points on the cap beam, at the external surface. Gravity load are reacted at the column base. Lateral load is applied at the base of the column through an idealized pin connection. This load is equilibrated by the horizontal reactions at the beam cap restraint. The relatively large size of the beam-column joints in actual elevated highway structures determined the choice of a scaled model test. To reduce the effect of scaling on the test 24

37 Chapter 4. experimental test results, a length scaling factor of 1/3 was selected. The areas were scaled of 1/9. Both reinforcement bar diameters and concrete cover are scaled approximately 1/3. Figure 4.3 Loading scheme. Gravity and Earthquake loads [Lowes and Moehle,1999] 4.2 Loading The subassembly is rotated of 90 degrees. In this configuration the column is horizontal and the beam vertical. Because the stress distribution within the structure due to the self weight was minimal, rotating the system of 90 degrees was not expected to affect the test result. Earthquake loading is applied using a 500 kip actuator to cycle the column base through a prescribed displacement path. A Flexural Hinge was used at the end of the actuator: in this way no moment is transferred to the column base. The applied loads were reacted by reaction frame. Gravity loads are simulated with two hydraulic 60 kip jacks. A load cell was placed in series with each jack and used during the testing to monitor the load applied through the jack. The applied load was kept constant during the test. 4.3 Material and Construction Using scaling factor of 1/3, the test specimens were sufficiently large to allow the use of the same construction material of actual structures. 25

38 Chapter 4. experimental test All reinforcement bars are Grade 40, with exception of #2 specially produced for this project. Five steel coupons were tested in tension each for #5 and #6, Grade 40 reinforcement. Three steel coupons were tested in tension for #3, Grade 40. The concrete mix for as built model was normal weight with a maximum aggregate of one inch. Figure 4.4. Instrumentation used to measure global displacements and applied loads using in Lowes and Moehle [1999] test Concrete Properties Table 4.1. Concrete mix used for sub-assemblages model Supplier Mix Number Description Expected Strength Expected Slump Concrete Mix One Sugar City Building Materials Pinole CA SC601220P 6.0 SK, 1 PM AD days 4 inches 26

39 Chapter 4. experimental test Table 4.2. Quantities used in concrete mix SSD Weights (lbs) % Used SP GR Abs. Val 308 Cement Type II Pozzolan 10% repl Water air 2% Gravel 1X4 Radum 55.1 % Top Sand Radum 36.1 % Blend Sand 8.8% Total % Concrete Elastic Modulus For each concrete batch the concrete elastic modulus was computed as the average of the modulus measured for the material testing, E CHORD, and the modulus predicted by ACI Committee 318 [A2] relationship between concrete elastic modulus and compressive strength, E ACI. Concrete elastic modulus was not measured in accordance with ASTM. Compression tests were performed in accordance with ASTM C39-88 Standard Test Method for Compressive Strength of Cylindrical Concrete Strength. The elastic modulus was measured for a monotonically increasing load path rather than for cyclic loading as it is recommended by ASTM. E CHORD was computed as follows: Where: f SU =0.4 f C, psi, ε SU is the concrete strain corresponding to f SU measured in/in f 50 is the concrete stress corresponding to ε 50 measured in psi ε 50 is the measured concrete compressive strain of approximately 50 micro-strain measured in/in E ACI is computed as recommended y ACI Committee 318 [A2]: 27

40 Chapter 4. experimental test E CI f, in psi Where f c is the concrete compression strength measured in psi. This equation is based on ASTM C469 prescriptions. The concrete cylinders were tested in a dry surface state which usually results in a slightly higher compressive strength and a slightly lower elastic modulus being measured. Additionally concrete cylinders, in measuring Elastic Modulus, were loaded to failure at uniform rate, which could be expected to result in a slightly higher measurement Reinforcing Steel Properties Tension testing of steel coupons was done to determine the material properties of Grade 40 reinforcements used in as-built portion of each model. For each of the bar size used in the model, five steel coupons were tested in accordance with ASTM A Engineering stress was computed as the applied load divided by the smallest cross section area of the coupon prior to the testing. Engineering strain was computed as the measure of the elongation of specified gage length divided by initial gage length. The yield stress is defined as the lowest point of the yield plateau. The ultimate stress is the maximum stress carried by the steel coupon. E s is computed with a linear regression on the elastic portion of the stress strain path. 4.4 Test Procedure The test followed these steps: Initialization of instrumentation Application of simulated gravity load Application of simulated earthquake load Test finished when the model strength deteriorated significantly or when the limit of the test set up were reached For the interpretation of the results DAS85 (Data Acquisition System 1985) was used. This system manages the signals provided by the instrumentation used to measure deformations and displacements. With this code, measurement from any instrument could be read at any time. Additionally this code allowed the user to store the measurement from all instruments to a data file at any time. When the instrumentation was zeros, the model was loaded with gravity load using gravity jack cylinders. Gravity loads were measured using the load cells in series with the jacks. With gravity load applied, the simulated earthquake load was applied as a shear force at the column base. The shear force followed a target displacement path. Displacement control is used. The simulated earthquake load is applied pseudo-statically, in this way cracks and damage are allowed to propagate in an accentuated manner. The result of the pseudo-static test is a softer 28

41 Chapter 4. experimental test weaker system response respect to the one expected under dynamic loading. The displacement path consisted in three cycles at each of a series of increasing maximum displacement. A cycle consist of travelling from zero displacement in one direction to the target maximum displacement in the opposite direction and back to zero. Three cycles where chosen so that effects of degradation would be apparent but so that unrealistic fatigue would be avoided. 4.5 Instrumentation 70 channels: Data were stored to computer data files at intervals along displacement path. DCDT (Direct Current Differential Transformers): they measured global displacements in small displacements range. A series of ¼ of inch diameter threaded rods were embedded in the model surface. DCDT connected to the protruding ends of the bars measured the displacements between discrete points. DCDT located at the beam-joint at the column-joint interface was expected to provide information about slip of longitudinal reinforcement bars anchored in the joint. Strain Gages: They were used to measured concrete and steel strain. Anchorage of column flexural reinforcement in the joint was of particular interest, in particular the slip of the two column bars with respect to the top of the joint. Figure 4.5. Instrumentation used to measured joint deformation in Lowes and Moehle Test [1999] 29

42 Chapter 4. experimental test Figure 4.6. Steel reinforcement strain gages. Lowes and Moehle Test [1999] Figure 4.7. Instrumentation to measured column bar slip. Lowes and Moehle Test [1999] 30

43 Chapter 4. experimental test 4.6 Experimental Response of the As-Built Connection The results of this experimental test are mainly in terms of strains, element displacements, reinforcement elongation and relative movement between steel and concrete. For the as built model we have informations about: Deterioration of joint core area and localization of disconnection zone; Force versus displacement relationship at the column base; Global behaviour of the joint in terms of shear strains measured with external and internal gages; Hysteretic relationship between nominal joint shear stress and strain measured with external and internal gages; Beam and column longitudinal reinforcement stains; Beam and column bar strains; Column bar slip versus displacements. Many of these outputs are too complex to be read and used as references for the developed numerical model developed. This is mainly due to the poor description of experimental procedure applied to perform the experimental test. For instance, the applied loads (static and pseudo-static) are not specified in the report and we do not have detailed indication about the characteristics of the cyclic load applied to the column base. Only spot informations are available about thatonly few of this data will be used to check the analytical model Figure 4.9. Zone of disconnected concrete Figure 4.8. Column longitudinal reinforcement strain [Lowes and Moehle,1999] 31

44 Chapter 4. experimental test Figure 4.9. Zone of disconnected concrete [Lowes and Moehle,1999] Figure Load versus beam longitudinal reinforcement [Lowes and Moehle,1999] 32

45 Chapter 4. experimental test Figure Beam longitudinal reinforcement strain history [Lowes and Moehle,1999] 33

46 Chapter 4. experimental test Figure Load versus reinforcement strain history [Lowes and Moehle,1999] 34

47 Chapter 4. experimental test Load-Displacements Relationship When subjected to simulated earthquake loading, the as-built test model did not perform in ductile manner. The model carried the simulated gravity forces throughout the test and the peak simulated earthquake load was only 66% of the load corresponding to the column developing nominal flexural strength. The model exhibited a large reduction in stiffness and load capacity. At peak load, significant cracking occurred and continued to accumulate in the joint. The model began to lose earthquake load capacity. Testing was concluded when the model exhibited only minimal earthquake load carrying capacity. Figure Load versus displacements [Lowes and Moehle,1999] Figure 4.13 shows measured and calculated load versus displacement relationship. Because this model began to lose load capacity following the application of maximum load, it is unreasonable to define a yield displacement for the model and discuss about ductility. As testing of as-built model proceeded, the first cycle to an increased maximum displacement was often marked by change in model response.in Figure 4.13 many points are identified: 35

48 Chapter 4. experimental test A. Measured slip at the end of the column reinforcement with respect to the top of the joint was initially observed during the first cycle to a maximum displacement of one half inch: Δ=0.5 inch (12.5 mm) P=23 kip ( kn) Slip equal to 0.01 (0.25 mm) was measured for the column reinforced bar in tension. Initial diagonal cracking in joint region appeared in the concrete over the anchored column reinforcement and in the flexural compressive zone. B. Measurable slip of column flexural reinforcement in tension was observed also at a negative displacement of 0.5 inches: Δ=0.5 inch (12.5 mm) P=20 kip (88.96 kn) Slip equal to 0.03 ( C. During the first cycle to a maximum displacement of one inch, the maximum load was reached: Δ=1 inch (25 mm) P=26.6 kip ( kn) This load is 82% of the load expected to develop nominal flexural strength of the beams and 66% of the load expected to develop nominal flexural strength. Slip equal to 0.06 inches (1.5 mm) was measured for the column longitudinal reinforcement. Significant diagonal cracking, with maximum crack width of 0.1 inches (2.5 mm) was observed in joint region. D. Displacement cycles to a maximum displacement of 1.5 inches (37.5 mm) resulted in a minor loss of strength and a significant loss of stiffness: Δ=1.5 inches (40 mm) P=25.7 kip (114.4 kn) Slip equal to 0.13 inches (3.25 mm)was measured for the tension column reinforcement. Additional cracking with maximum crack widths of 0.2 inches (5 mm) was observed in the joint Tapping with a hammer on the surface of the model indicated a zone of cover concrete that had lost connection with the core concrete and that could be expected to spall during subsequent displacement cycles. E. At the maximum displacement of three inches, a large quantity of concrete had spalled from the joint. This was accompanied by a noticeable loss of strength and stiffness: Δ=3.1 inches (78 mm) P=19.0 kip (84.55 kn) Cover concrete spalled in the region over the end of the column reinforcement. Very wide cracks opened in the top of the cap beam of the beam joint interface. F. At the maximum displacement of four inches, the joint was heavily damaged: Δ=3.9 inches (97.5 mm) P=19.0 kip (84.55 kn) The accumulated damage in the joint resulted in almost the entire length of the column flexural reinforcements. At the end of the test, the model had minimal earthquake load capacity and the joint showed extensive damage; gravity load resistance was maintained. 36

49 Chapter 4. experimental test Figure Nominal joint shear stress versus shear strain measured with external gages Figure Nominal joint shear stress versus shear strain measured with internal gages [Lowes and Moehle,1999] Figure Nominal joint shear strain measured with internal and external gages [Lowes and Moehle,1999] 37

50 Chapter 4. experimental test Figure Nominal joint shear strain from external gages [Lowes and Moehle,1999] Figures 4.16 and 4.17 show nominal joint shear strain as computed from the external DCDT gages and from embedded concrete gages. At small nominal shear strain there is a good correlation between the two computed values, while at larger strains, peak nominal joint shear strain as computed from the embedded concrete gages were significantly less than those computed from the external gage data. This discrepancy is due to the loss of abrasive coating on the concrete gages allowing them to slip. As show in Figure 4.16 data from embedded concrete gages were available only during the first cycles Anchorages and Reinforcements Data provided from strain gages placed on the column and beam flexural reinforcements provided additional informations about anchorage and reinforcements. Figure 4.18 and 4.19 shown the axial strain in column reinforcements at the column joint interfaces as measured by the attached strain gages. 38

51 Chapter 4. experimental test Figure Load versus column internal bar strain measured with gage sg5 [Lowes and Moehle,1999] Figure Load versus column internal bar strain measured with gage sg4 [Lowes and Moehle,1999] 39

52 Chapter 4. experimental test Figure Displacement versus slip of column longitudinal reinforcement (sl1) [Lowes and Moehle,1999] Figure Displacement versus slip of column longitudinal reinforcement (sl2) [Lowes and Moehle,1999] 40

53 Chapter 5. analysis procedure and results 5. ANALYSIS PROCEDURE AND RESULTS Experimental researches indicate that elongation and slip of tensile reinforcements at beam column interface could result in significant fixed end rotation. These additional rotations at beam column fixed ends can increase the member total lateral displacements. This aspect is not considered in a beam column traditional modelling. In this study a joint model that consider the real features of joint behaviour is proposed. 5.1 Description of Structural Model The models studied in this research are implemented by SeismoStruct. SeismoStruct is an internet downloadable fibre-based package, capable of predicting the large displacement behaviour of space frames under static and dynamic loading, considering both geometric non linearity and material inelasticity. The stress-strain state of a beam-column element is obtained through the integration of nonlinear uniaxial material response of the individual fibre, in which the section has been subdivided. This fully accounts for the spread of inelasticity along the member length and across the section depth. Discretization of typical reinforced cross section is defined in Figure 5.1. Figure 5.1. Discretization of RC cross-section in a fire based model [SeismoStruct] If a sufficient number of fibres ( fibres in special analysis) is employed the distribution of material non linearity across the section is accurately modelled. A critical step by step procedure is applied to this study to understand the different behaviour of a structure modelled with a rigid beam-column joint (that remains elastic during the reverse cyclic loading) and a structure with flexible joint. 41

54 Chapter 5. analysis procedure and results Traditional Model: Rigid Internal Joint A simple model is created with SeismoStruct using the element geometry and material properties proposed in the experimental test [Lowes and Moehle, 1999]. Considering some outputs proposed in the experimental test report, the model is checked. The joint is supposed to be elastic and the total horizontal displacements could be smaller than the lateral experimental displacements. Figure 5.2. Traditional model implemented with SeismoStruct (a) Geometry of the structure. The geometrical dimensions of the structure are shown in Figure 5.3. To model the elements, dimensions taken from the centreline are considered. Dimensions in Figure 5.3 are in centimeters. Figure 5.3. Geometrical dimensions (b) Restraints. The column base is pinned. Both rotations and horizontal displacements are allowed as prescribed in the experimental test report [Lowes and Moehle, 1999]. The right 42

55 Chapter 5. analysis procedure and results extreme of the beam is pinned and the other one is fully fixed to ensure the horizontal balance of the structure under horizontal loading. (c) Loading. Gravity and pseudo-static loads are applied to the structure. In the experimental test report [Lowes and Moehle, 1999] the magnitude of the gravity loads applied on the beam is not specified. So it is assumed that the hydraulic jacks used for the gravity loads simulation, work at maximum capacity (60 kips are applied on each beam). For the pseudo-static cyclic load, a displacement history is applied following some informations given by the report which is not sufficiently clear about this point. A displacement history is assumed and applied at the column base. In the experimental test, forces are applied at the column base. Because the nonlinearity of the system, the results could be quite different from the experimental ones. Displacement TH_Lowes Moehle Data 150 displacements [mm] Figure 5.4. Assumed displacement history applied time [sec] at the column base for pseudo-static analysis First of all Performance Criteria are checked. As prescribed in the Experimental test report, Figure 5.5. Beam section in which steel strain is reached for first beams crack before the column that seems to remain elastic during the cyclic test. In the beams strain steel limit (0.06) is reached. For the column no warning about that are shown. Table 5.1. Steel strain limit. Numerical output for traditional model. Element Time [sec] Section Steel Strain Beam beam 2A beam 1D Column ELASTIC 43

56 Chapter 5. analysis procedure and results Figure 5.6. Column longitudinal reinforcements strains developed under experimental cyclic loading [Lowes and Moehle, 1999] Results in terms of strains, internal forces and lateral displacements are considered and commented. In Table 5.2 strain comparison in longitudinal column reinforcements is shown. The results given by experimental test and by the analytical model are quite close: the traditional model seems to give a good approximation in term of maximum and minimum strain values. In Table 5.3 the relationship between applied displacements and generated forces is shown. In the model displacements are applied at the column base. Otherwise in the experimental test shear forces are applied. The relation is nonlinear so the expected results are not the same but close one to each other. It is possible to observe that the results are comparable. The differences between the results could be due to the effect of slippage between concrete and reinforcements, since this aspect is not considered in the traditional model. Table 5.2. Column strains comparison between traditional model results and experimental output strain values Min Max traditional model Min Max experimental test 44

57 Chapter 5. analysis procedure and results 2.00E-03 strains comparison 1.50E E-03 strains 5.00E E+00 traditional model result experimental test result -5.00E E Figure 5.7. Strains in column: comparison between traditional model and experimental test Table 5.3. Shear Forces at the column base: comparison between experimental outputs and traditional model outputs Displacements [m] Experim. Forces [kn] Traditional Model output [kn] shear at the column base shear force at the column base [kn] traditional model resut experimental test result displacements [m] Figure 5.8. Shear at the column base in traditional model and in experimental test 45

58 Chapter 5. analysis procedure and results Model with Flexible Joint: Bond Slip Springs A crude joint model is inserted in the structure. The model is composed by two diagonal elastic rigid elements transmitting shear into the joint and by six springs considering the effect of slippage of reinforcements on global behaviour of the structure. Figure 5.9. Macro-model with slip springs (a) Geometry of the structure. To obtain realistic values in terms of strains and internal forces, the geometry of the structure has been modified. The introduction of the flexible joint as 2D element, reduces the length of the elements. If column and beam are not increased in longitudinal dimensions they give not comparable output values. Dimensions in Figure 5.10 are in centimetres. (b) Joint Shear Mechanism. Shear is transmitted in the joint by a diagonal concrete truss. Two trusses are modeled to take into account the two possible earthquake directions. The trusses are elastic and infinitely rigid. They don t take into account the degradation due to the cyclic load applied to the structure. 46

59 Chapter 5. analysis procedure and results Figure Geometrical dimensions of the model with joint [cm] (c) Bond Slip Springs. Six tri-linear asymmetric springs are introduced to consider the slippage between concrete and reinforcements in function of the state of stress in the steel as proposed by Lowes and Altoontash [2003]. These springs are at the perimeter of the joint. In the model proposed by Mitra and Lowes [2004], the slip springs are located at the centroid of beam and column flexural tension and compression zones rather than at the perimeter of the joint. This means to perform a Moment Curvature Analysis to define the position of neutral axis. The position of neutral axis changes with axial load, varying at any time during a cyclic loading, either in column or beams. The position of neutral axis changes also in function of the state of stress of steel reinforcements (elastic range or post yielding range). This behaviour is impossible to model with SeismoStruct and details about that are not available in Lowes studies, so bond slip spring located at the perimeter in the joint could be a good solution. slip vs force relationship Force [kn] bars, 16 mm diameter 2 bars, 16 mm diameter 8 bars, 20 mm diameter slip [m] Figure Force versus slip relationship for spring in the upper side of the beam The constitutive laws proposed in Figure 5.11 are an approximation of the non linear relationships presented in previous work [Lowes and Altoontash, 2003]. In that model [Lowes and Altoontash, 2003] slip is defined in function of the state of stress in the longitudinal reinforcements. So when the stress in steel is lower than the yield stress, slip 47

60 Chapter 5. analysis procedure and results is defined as: slip = (5.1) E When the stress in steel is higher than the yield stress, it is defined as: with: (5.2) = = = (5.3) Table 5.4. Average bond strength as a function of steel stress state [Lowes and Altoontash, 2003] tension steel stress vs slip_qualitative Behaviour fs Slip [mm] Figure Tensile stress versus slip constitutive relationship [Lowes and Altoontash, 2003] and a possible approximation 48

61 Chapter 5. analysis procedure and results Figure System stiffness definition: steel reinforcement as springs in parallel (beam section) Performance Criteria are checked. As prescribed in the Experimental test report, beams crack before the column that seems to remain elastic during the cyclic test. In the beams, steel strain limit (0.06) is reached. For the column no warnings about that are shown. Figure Beam section in which steel strain is reached for first Table 5.5. Steel strain limit. Numerical output for model with slip springs only Element Time [sec] Section Steel Strain Beam beam 2A beam 1C Column ELASTIC The results in terms of strain reinforcements are compared with the outputs of both the traditional model and the experimental test, shown in Figure As expected the values of strains are smaller than the values obtained with the traditional model and they are closer to those of the experimental test. The structure considered is more flexible than the previous one with rigid internal node. To define the initial elastic stiffness and the post-yield stiffness, the reinforcements are considered to work in parallel. The stiffness of the system is the sum of each single reinforcement stiffness considered in tension and in compression, assuming neutral axis at the half of the height of the section. The same is done for the system yield force starting from the yielding stress value. These assumptions [Lowes, 1999] depend on the fact that the failure in reinforcements system occurs when all the reinforcements crossing the center of the panel yield. 49

62 Chapter 5. analysis procedure and results strains comparison 2.00E E E-03 strains 5.00E E+00 experimental test traditional model model with slip springs -5.00E E Figure Comparison of traditional model, model with slip springs and experimental test results in column In Figure 5.16 shear forces at the column base are shown. As in the previous model (Traditional Model) displacements are applied at the column base. As expected the model with slip springs gives lower values of shear. The differences between the model and the experimental output values increase during the cycles. The same phenomenon is observed for the traditional model. 200 shear at the column base 150 traditional model shear force at the column base[kn] experimental test model with slip springs time [sec] Figure Shear at the column base in traditional model, in model with slip springs and experimental test The shear actions in the element are compared: as expected the actions in the beam of the model with slip springs are lower than those obtained with the traditional model. In Table 5.6 the values of beam shear force for both models have been recorded. 50

63 Chapter 5. analysis procedure and results 200 comparison: shear force in beam 150 Shear Force [kn] traditional model model with slip spring beam segment Figure Traditional model and model with springs beam shear force comparison Table 5.6. Beam shear results for traditional model and for model with springs beam 1 Max Min Max Min beam 2 Max Min Max Min beam 3 Max Min Max Min kn kn kn kn kn kn kn kn kn kn kn kn Traditional Model Model with Slip Spring Traditional Model Model with Slip Spring Traditional Model Model with Slip Spring 51

64 Chapter 5. analysis procedure and results Figure Subdivision of the beam Lateral displacements in traditional model and in the model with flexible joint are compared. Larger displacements for the second model than for the traditional one are observed. This is due to the effect of slippage of reinforcements in the concrete modelled with springs. This phenomenon increases the total lateral displacement of the structure. displacements in x direction [COLUMN] displacements in x direction [m] traditional model model with slip springs column section Figure Column lateral displacements displacements in x direction [BEAM A] displacements in x direction [m] traditional model model with slip spring beam section Figure Beam A lateral displacements 52

65 Chapter 5. analysis procedure and results displacements in z directions [BEAM A] traditional model displacement in z direction[m] model with slip springs beam section Figure Beam a vertical displacements To check the higher flexibility of the system in all directions, vertical displacements are shown in Figure Model with Flexible Joint: Bond Slip Springs and Interface Springs Figure Macro-model with slip and interface spring In this model three interface springs are added to simulate the interaction between the joint and the elements. Calvi et al [2005] introduced a method to construct a force-deformation response curve for reinforced concrete that takes into account the effects of shear cracking. Their theoretical formulation is based on the mechanical equations for stiffness after shear cracking, proposed originally by Dilger [1966] and reported afterwards by Park and Paulay [1975]. (a) Construction of the force-deformation response curve. The force-deformation response curve used in Calvi et al [2005] procedure can be visualized as the combination of an idealized flexural response curve and an idealized shear response curve. The idealization 53

66 Chapter 5. analysis procedure and results consists in representing the response by three straight lines, which span across three predetermined member limit states. These member limit states are 1) flexural cracking, 2) shear cracking, 3) maximum shear capacity. The slope of each straight line is the stiffness that characterized the corresponding response phase. For the elastic response the stiffness is computed as: K se = GA v H (5.4) G is the elastic shear modulus and it is computed as: G= E c 2(1+ν) (5.5) The shear area A v is given by the elastic theory of mechanics and it depends on the shape of the cross section under consideration. For rectangular shape is: A v = 5 6 bd (5.6) If more details are not available, the evaluation of shear stiffness after flexural cracking is proportional to GA ve. In this case, the shear area A ve is based on effective section properties to reflect the influence of flexural cracking. Under the assumption that the reduction in shear stiffness is proportional to the reduction of flexural stiffness, the effective shear area can be expressed as: So the shear stiffness in Phase II is given by: A ve =A v I e I g (5.7) = GA ve (5.8) L In Phase III the shear stiffness is due to the behaviour of diagonal truss. This mechanism is not considered in this study. Figure Idealized shear deformations behaviour (b) Definition of limit states. The flexural limit state represents the transition from a fully elastic section to a cracked section during the element response. Flexural cracking is determined assuming an elastic behaviour of the concrete section. The applied moment that 54

67 Chapter 5. analysis procedure and results causes the tension capacity in the extreme concrete fibres to be exceeded, is known as cracking moment and it is defined as: Where: (5.9) S g = I g y t :Concrete section modulus to tension fibres f t =C t f c : Concrete tension strength f pc = P A g :Average axial stress in concrete C t is assumed constant and equal to The lateral force is defined as: F cr = M cr (5.10) L The shear cracking limit state represents the point at which inclined shear cracks appear in the member. The limit state is characterized by a reduction of the member stiffness which normally leads into shear failure. The equation used to define the limit force is the following: The effective shear sectional area is given by: V cr =0.215 s d ν cr A w (5.11) The shear cracking stress is determined as: A w =b w d (5.12) P ν cr =0.5 f c ' 1+ (5.13) 0.5 f c 'A g The three interface springs have the properties defined with previous formula. Other six rigid elements are added to the joint model to define the terminal nodes of the interface springs. So the joint is idealized as a rigid panel with nine springs, six for bond-slip and three for interface interaction between the joint and the elements. The lateral displacements are expected to be smaller than those found in the model with slip springs only. This phenomenon is due to the rigid panel at the beam column intersection. 55

68 Chapter 5. analysis procedure and results shear springs law shear spring beahavior [COLUMN] force [kn] shear spring behavior [BEAM] displacement [m] Figure Shear spring law for beams and column As shown in Figure 5.25 the shear behaviour for the beams is supposed to be bilinear instead of being tri-linear. The limit for phase I is too low and it is not considered in the analysis. For the column shear behaviour a tri-linear law is used. The limit values for column are higher than the beams ones: this is due to the effect of axial load acting on the element. For the beams the axial load is supposed to be zero. Performance Criteria are checked. As prescribed in the Experimental test report, beams crack before the column that seems to remain elastic during the cyclic test. In the beams steel strain limit (0.06) is reached. For the column no warnings about that are shown. Figure Beam Section in which steel strain is reached for first Table 5.7. Steel strain limit. Numerical output for model with slip springs only Element Time [sec] Section Steel Strain Beam beam 2A beam 1D Column ELASTIC 56

69 Chapter 5. analysis procedure and results Table 5.8. Column strains comparison between model with interface springs results and experimental output strain values Min Max traditional model Min Max experimental test The model with interface springs gives higher values for maximum strain than the model with slip springs only, because of the introduction of the rigid panel in the joint: this induces higher deformation in column steel reinforcements strains comparison experimental test model with interface springs strain model with slip spring only Figure Strain comparison obtained with experimental test, analytical model with slip springs only and model with interface springs Table 5.9. Shear Forces: comparison of results for all the models analyzed Displacements [m] Experimental Test [kn] Traditional Model output [kn] Model with slip springs only [kn] Model with interface springs [kn]

70 Chapter 5. analysis procedure and results The shear forces found using the model with interface spring are quite close to the values obtained with the model with slip springs only. In both cases the difference with experimental model increases during the cycles. The differences between experimental results and analytical results could depend on the nonlinearity of the members. In the experimental test, forces are applied at the column base, while displacements are applied in the computer model. Another reason could be a degradation law not able to catch the real behaviour of the structure. In Table 5.10 and in Figure 5.27 values of shear in the beams are presented. The shear action in the beam of the model with interface springs seems to be lower than in the other computer model, because the shear interface spring absorbs a certain quantity of shear force. 200 shear at the column base shear at the column base [kn] experimental test model with interface springs model with slip springs only time [sec] Figure Shear at the column base in traditional model, in model with slip springs and experimental test Table Shear forces in the beam for the model with interface springs beam 1 Max kn Min kn beam 2 Max kn Min -39 kn beam 3 Max kn Min -41 kn 58

71 Chapter 5. analysis procedure and results 200 comparison: shear force in beam 150 Shear Force [kn] traditional model model with slip spring model with interface springs beam segment Figure Comparison of shear forces in the beam displacements in x direction [BEAM A] 0.00 displacements in x direction [m] tra ditional model model with slip spring model with interface springs beam section Figure Displacement in x direction in BEAM element displacements in x direction [COLUMN] displacements in x direction [m] traditional model model with slip springs model with interface springs column section Figure Displacements in x direction for COLUMN element 59

72 Chapter 5. analysis procedure and results confronto disp beam in z displacement in z direction[m] model with interface springs traditional joint beam section Figure Displacements in z direction for BEAM element As shown in Figure 5.30 and in Figure 5.31 displacements in x-direction (horizontal displacements) are close to the displacements in x-direction obtained with the traditional model. This is caused by the geometry of the panel of the joint: it is rigid and a bidimensional model instead that a one-dimensional element as the joint, as considered in traditional model. This could influence the behaviour of the structure in terms of horizontal displacements. 60

73 Chapter 6. introduction of a simplified relationship for bond-slip springs 6. INTRODUCTION OF A SIMPLIFIED RELATIONSHIP FOR BOND-SLIP SPRINGS Models proposed by Lowes and Altoontash [2003] and by Lowes and Mitra [2004] are complex. Appling directly the model, a nonlinear curve is obtained. The first constitutive relationship proposed is characterized by degradation with negative slope: this could cause instability in numerical models used in the analysis code. The second one developed in 2004 solved this problem but it is anyway complex: many codes do not allow the definition of a non linear constitutive relationship like the one proposed. In SeismoStruct there is not a material constitutive law useful to define the bond-slip behaviour as proposed by Lowes. So a first approximation is done: a tri-linear asymmetric relationship without degradation is used. An horizontal plateau is considered instead of softening with negative slope (introduction of negative slope is not allowed in this SeismoStruct model) Tri-linear asymmetrical model bilinear approximation 400 Force [kn] Slip [m] Figure 6.1. Bilinear approximation for bond-slip relationship This model seems to represent quite well the real behaviour of the structure as discuss in Chapter 5. The differences are mainly due to the definition of the central rigid panel as shear transmission element. This element does not consider the effect of the damage in the joint 61

74 Chapter 6. introduction of a simplified relationship for bond-slip springs core (that means that reduction of total stiffness of the structure is not considered) and the core deformation that could be not relevant. In this chapter the effect of a second simplification on structural behaviour is discussed. Instead of an asymmetrical tri-linear relationship, a symmetrical bilinear law is introduced. The used elastic stiffness is equal to the elastic tension stiffness computed for the tri-linear model and the yield force is fixed. The post yield hardening is not considered: this is a rough approximation but the results obtained are not so different from the output obtained with trilinear model. As expected the structure shows a more flexible behaviour than the model with tri-linear springs. Performance Criteria are checked. As prescribed in the Experimental test report, beams cracks before the column that seems to remain elastic during the cyclic test. In the beams steel strain limit (0.06) is reached. For the column no warnings about that are shown. Figure 6.2. Beam section in which steel strain is reached for first Table 6.1. Steel strain limit. Numerical output for model with slip springs only Element Time [sec] Section Steel Strain beam 2A Beam beam 1D Column ELASTIC Strain results (Table 6.1) are compared with previous models results. Lower values of steel strains are achieved. Anyway differences in terms of steel are not relevant. Table 6.2. Steel strains in the Beam. Sequence of yielding sections. Element Δ time [sec] Δ strain Beam 2A Beam 1D The strains obtained are similar to those of the model with the tri-linear constitutive relationship. 62

75 Chapter 6. introduction of a simplified relationship for bond-slip springs Table 6.3. Steel strains in the column. Comparison of results. strain values Difference [%] Min Max A Min Max tri-linear model - - B Min Max bilinear model C Min Max experimental test In Table 6.3 differences in strains are underlined. In row B asymmetrical tri-linear and symmetrical bilinear relationship results are compared. The difference between the two models is not relevant ( % for minima and 0.00% for maxima). The bilinear model seems to be a good approximation for the experimental data. 2.00E-03 strains comparison 1.50E E-03 strains 5.00E E+00 experimental test tri-linear constitutive relationship approximated bilinear constitutive relationship -5.00E E Figure 6.3. Strain results for asymmetrical tri-linear and bilinear approximated model. 63

76 Chapter 6. introduction of a simplified relationship for bond-slip springs Shear at the column for both models is shown in Figure 6.4. It is possible to observe that the values obtain using the model with bilinear approximation are close to the values of the other model. For selected displacements the results are quite the same Shear at the column base [kn] 0-50 model with tri-linear constitutive relationship model with approximated bilinear constitutive relationship tri-linear model shear values for selected displacements -100 aproximated bilinear model shear values for selected displacements time Figure 6.4. Time history of shear at the column base. Comparison between values of shear obtained with asymmetrical tri-linear constitutive relationship and approximated bilinear model In Table 6.4 and in Figure 6.4 values of shear in the beams are shown. Table 6.4. Shear forces in the beam for the model with approximated bilinear constitutive relationship beam 1 Max kn Min kn beam 2 Max kn Min kn beam 3 Max kn Min kn 64

77 Chapter 6. introduction of a simplified relationship for bond-slip springs As expected the shear values obtained are smaller than the output shear forces of the tri-lineal model (Table 6.5). The symmetrical bilinear bond slip model does not consider hardening. Two shear values are higher than in the previous model. The differences between the two models are of the order of 2 kn and it could be due to numerical precision. The increment of total lateral displacement is the result of the deterioration of bond in the joint region. The introduction of this simplified model could overestimate this parameter. In this study only the comparison with the previous model developed is possible because the data given by the experimental test are not exhaustive displacements in x direction [BEAM A] traditional model displacements in x direction [m] model with bilinear constitutive relationship model withtri-linear constitutive relationship beam section Figure 6.5. Displacements in x direction for BEAM element As expected the displacements obtained with the model with approximated bilinear relationship are larger than the displacements obtained with the other constitutive relationship. There is not a sensible difference between the results of the two model. Table 6.5. Beam displacements in x direction. Numerical values BEAM section Displacement in x direction [m] Bilinear Tri-linear constitutive constitutive relationship relationship Difference in % % % % For the column model with bilinear constitutive relationship the results are the same as the previous one in terms of displacements. 65

78 Chapter 6. introduction of a simplified relationship for bond-slip springs displacements in x direction [COLUMN] displacements in x direction [m] model with bilinear constitutive relationship model with trilinear constitutive relationship column section Figure 6.6. Displacement in x direction for COLUMN element Table 6.6. Column displacements in x direction. Numerical values Displacement in x direction [m] COLUMN Bilinear Tri-linear constitutive constitutive Difference in % section relationship relationship % % % 66

79 Chapter 7. CONCLUSIONS 7. CONCLUSIONS Joint behaviour is tested in this study with particular attention for the bond slip phenomenon and its effect on the global response of the structure. Some results are compared with experimental outputs. This check has been not possible for all parameters because the report proposed by Lowes and Moehle [1999] does not always provide all the necessary informations. For this reason a great number of data are compared with other simpler structural models. It is important to underline that the traditional model with nonlinear behaviour gives results not so different from those obtained with experimental test. Using a flexible joint the response of the structure is closer to the real one. The flexible joint is composed by six springs, taking into account the slippage between concrete and reinforcements, and three springs used to model the interface response. Some results are influenced by the introduction of a rigid elastic shear transferring rigid element (rigid panel). For example, the horizontal displacements obtained using the model with rigid panel and interface springs are smaller than the displacements computed with the analytical model developed with slip springs only. During the analysis some approximations are made: the position of the slip springs is not the same as the one proposed by Lowes et al [2004] and the constitutive relationship used varies because of some incompatibilities in SeismoStruct code. So an asymmetrical tri-linear constitutive law is used: to define it with the analysis code ten parameters are needed; some of these data are available from material properties indicated in the experimental test report, other are simply assumed. For the interface behaviour nothing was clarified in the papers. Lowes only specified that the sprigs used to model the shear response have to be defined as stiff and elastic. So we use a model proposed by Miranda et al [1994]. The results given by the model with this characteristics gives coherent results as explained in Chapter 5. The model proposed by Lowes is a complex and detailed one. As said before, ten parameter are needed to define an approximation for that relationship. Using this simpler model the main response characteristics are not changed. A symmetric bilinear model is used to understand if such a detailed model is useful in the definition of the global response of the structure. The results of the analysis are close (in certain case the same) to the those of the tri-linear model. It is possible to affirm that the model proposed by Lowes is very rigorous and not convenient for practical implementation. In this particular case a simplified model yields to reasonable results. 67

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