Analytical Modeling of the Contribution of Transverse Reinforcement in Prestressed Concrete Bridge Girders

Transcription

1 Analytical Modeling of the Contribution of Transverse Reinforcement in Prestressed Concrete Bridge Girders Katie Boris, REU Student Padmanabha Rao Tadepalli, Graduate Mentor Y. L. Mo, Faculty Advisor Final Report Department of Civil and Environmental Engineering University of Houston Houston, TX Sponsored by the National Science Foundation July 2009

2 Abstract The role of transverse steel in prestressed reinforced concrete beams was modeled by varying the reinforcement ratio under different loading schemes. The program Simulation of Concrete Structures developed within the framework of OpenSees (Open System for Earthquake Engineering Simulation) at the University of Houston was used in the finite element analysis. A TxDOT Type-A beam model was loaded to simulate failure in shear and flexure. The results indicate that transverse steel greatly contributes to both the strength and ductility of the beam. These results show good agreement with observations from previous studies. Introduction Transverse reinforcement in concrete structural elements, such as beams or columns, plays an important role in the performance of that structure. Such reinforcement increases the strength, strain at maximum stress, and overall ductility of concrete structures by exerting a confining pressure on a core of concrete. The amount of transverse steel is important as all elements of a reinforcement cage work together to strengthen the element. This research investigates the contribution of transverse reinforcement in web shear and flexure shear failure through the use of finite element methods. Literature Review Over the past twenty-five years, many studies have investigated the effect of transverse steel in confining concrete. Scott, Park, and Priestly (1982) investigated different arrangements of transverse steel in short columns under axial loading. They 2

3 concluded that an increase in the volume ratio of transverse steel increased the maximum stress attained by the reinforced concrete member. Saatcioglu and Razvi (1992) created an analytical model of stress-strain behavior of confined concrete with ability to adapt to different shapes of confinement. More recently, Danygier (2001) determined the contribution of the transverse reinforcement in shear loading, but only in circular elements. Finite element analysis of prestressed and reinforced concrete structures has been made possible by the development of models that simulate their behavior after peak stress in monotonic loading. The Softened Membrane Model (SMM) was created to model the behavior both before and after peak stress in monotonic loading (Hsu and Zhu, 2002) by considering the Poisson effect of cracked reinforced concrete. This model was then extended by Wang (2006) to consider prestressed concrete under shear loads (SMM-PC). It accomplishes this by considering the effect of initial stresses and strains due to the prestressed tendons and incorporating a new factor for the softening coefficient in the constitutive relationship of concrete in compression The Cyclic Softened Membrane Model (CSMM) predicts the reversed cyclic shear response of reinforced concrete membrane elements. Developed by Mansour and Hsu (2005a, 2005b), this extension of the SMM incorporates the constitutive laws of concrete and steel bars in membrane elements by considering the behavior of embedded mild steel when subjected to uniaxial strains. Simulation of Concrete Structures The analysis of the models involved the use of the program Simulation of Concrete Structure (Laskar 2009), or SCS, which is incorporated into OpenSees (Fenves 3

4 2005). OpenSees, Open System for Earthquake Engineering Simulation, is a framework developed at the Pacific Earthquake Engineering Center (PEER) at the University of California in Berkeley. It is an object-oriented, open source software that uses finite element methods to simulate dynamic and monotonic loading, and is able to perform both linear and nonlinear analyses. As it is open source, it allows users to modify the code and add modules to improve and expand the analytical capabilities. This makes it very desirable for research applications. OpenSees consists of four objects that work together to perform an analysis: the ModelBuiler, the Domain, the Analysis, and the Recorder. A diagram of these is shown in Figure 1. The ModelBuilder constructs the model through the user inputs. This Figure 1. Main Objects of OpenSees. includes the nodes and elements of the finite element mesh as well as their constraints and connectivity. Loads, material properties, and the placement of each are also incorporated within this object. Once the model is built, this information is added to the Domain. The Domain object holds the current state of the model. It also serves as a base for the Recorder and Analysis objects as it passes information to each during the process. 4

5 The Recorder monitors specified components in the Domain through the analysis and records the state of these to an external file for further analysis by the user. Examples of parameters commonly specified are time, force, and displacement. The fourth object, the Analysis, is responsible for performing the user-specified analysis on the model. Specifically, this object receives model information from the Domain and moves the model from one time step to the next. The Analysis object contains many sub-objects used to complete this task. One such sub-object is the Integrator. It determines the predictive and corrective step in each time step. The type of Integrator used is determined by the user and the type of analysis, such as static, dynamic, or cyclic. For the static analyses performed in this project, load control and displacement control were used. Another important sub-object determines the solution algorithm. This is specified by the user and can be linear or nonlinear. For nonlinear analysis, as used in this investigation, a combination of an incremental and iterative procedure is used. This minimizes the build up of error. The present project used the modified Newton algorithm that incorporated Krylov subspace acceleration. As useful of a tool as OpenSees is, it is unable to perform a nonlinear analysis on a prestressed concrete beam without further modification. It lacks a proper plane stress material for reinforced concrete. Additionally, the included uniaxial material models are too simplified to properly represent the material behavior that affects the performance of prestressed concrete. Therefore, new finite element models of reinforced concrete plane stress elements and more sophisticated materials need to be implanted into OpenSees. SCS achieves this through the development of a prestressed concrete plane stress material class, PCPlaneStress, and more appropriate materials derived from the Cyclic Softened 5

6 Membrane Model (CSMM) and the Softened Membrane Model for Prestressed Concrete (SMM-PC). Below is a discussion of the PCPlaneStress material and the other material classes found in SCS that were used in this investigation. A PCPlaneStress (Laskar 2009) material is a two-dimensional class of material that models reinforced and prestressed concrete plane stress elements using material relationships from the CSMM. It incorporates concrete, steel, and tendon objects and allows for steel and tendon objects to be oriented in any direction. The first material, Steel01, is included as a module in OpenSees. It creates a uniaxial bilinear steel object with the typical envelope curve. This material has an elastic region and a plastic region with kinematic hardening. Use of this material requires the user to specify the elastic modulus, yield strength, and a hardening ratio. This material was used to model the reinforcing bars and strands contained in the flanges. The second steel material, SteelZ01 (Zhong 2005), includes the unloading/reloading pattern of embedded mild steel from the CSMM in addition to the bilinear envelope curve. Use of this material requires the user to specify the elastic modulus, the yield strength, compressive strength of the concrete, and steel ratio. This material was used to model the steel embedded in the web of the beam and used the transverse steel ratio. The third steel material, TendonL01 (Laskar 2009), is derived from the uniaxial constitutive relationships of embedded prestressing tendons in the SMM-PC. These relationships differ from those used in SteelZ01 only in the tensile envelope. Use of TendonL01 requires the user to input values for elastic modulus, the yield strength, compressive strength of concrete, steel ratio, and initial strain in the prestressing tendons. 6

7 This material was used to model the prestressing tendons in the web of the beam and used the prestressing tendon ratio. The first concrete model used, Concrete01, is an included module in OpenSees. It creates a uniaxial Kent-Scott-Park concrete object with a degraded linear unloading and reloading stiffness as according to Karsan-Jirsa. It also assumes tensile strength to be zero. Use of this material requires specifying compressive strength, strain at compressive strength, crushing strength, and strain at crushing strength. This material was used to model the concrete in the flanges. The second concrete module, ConcreteL01 (Laskar 2009), was developed using the uniaxial concrete model from both the SMM-PC and the CSMM. It incorporates the initial stress due to prestressing as well as the softening effect due to perpendicular tensile strain on the concrete struts. Use of this material requires the user to input values for the ultimate compressive stress and the corresponding compressive strain. This material was used to model the concrete in the web of the beam. Modeling and Analysis Both the web shear and the flexure shear beams were modeled after TxDOT Type-A beams commonly used as highway bridge girders. These beams feature a fortyfive degree slope that connects each flange to the web in the cross section. In order to account for this region in the model, half of this region was incorporated into the flange and the other half into the web. Figure 2 illustrates this adjustment. Both models have a 7

8 Figure 2 (a). Cross Section Model Adjustment and (b). Superimposed Actual and Model Cross Section beam span of 7.3 meters and a height of approximately 0.55 meters. One end of each beam was constrained with a pin and the other with a roller. All concrete had a compressive strength of 72.4 MPa. Other material properties specified for each material model are contained in Appendix A. Additionally, detailed drawings of each crosssection, including the location of the reinforcement, are included in Appendix B. Shear beam The shear beam model discretized the structure to include 17 nodes along the length and two along the height. Between nodes along the length, nonlinear beam column objects were created. The top flange was approximated by a rectangle 304 mm by 140 mm and consisted of 40 fibers of concrete, 10 along the width and 4 along the height. Two steel fibers, each with an area of 200 mm 2, were layered in to simulate the #5 reinforcing bars. The bottom flange, 406 mm by 190 mm, was similarly discretized into 40 fibers of concrete. Additionally, two fibers of steel with an area of 99 mm 2 were 8

9 layered in. These two fibers represent the prestressing strands located in this flange. Illustrations of the discretized flanges are shown in Figure 3(b) and (c). The web itself consisted of quad elements, with four nodes serving as the boundaries. These elements were given a thickness of mm. Each element was modeled with a PCPlaneStress material. This involved creating and layering ConcreteL01, TendonL01, and SteelZ01 materials to represent the reinforcement and tendons located in the web. The orientation of the reinforcement was set along the local x- and y- axes. A tendon steel ratio of 0.55% was used. The transverse steel ratio was set as 1.00%, 0.17%, or 0.00%, depending on the simulation. Once the model was built, the loads were applied in two parts. First, the prestress load of 1.65 MN was applied along at the nodes indicated in Figure 3(a). This load was analyzed using a Krylov-Newton solution algorithm and a load control integrator. After the first analysis was complete, the shear load was applied. Two 1000 N concentrated loads were distributed over three nodes each at a distance of about 0.9 meters from the supports. This analysis also used a Krylov-Newton solution algorithm. In order to obtain a full force/displacement graph, a displacement path integration scheme was used instead of a load control. Flexure beam The flexure beam model discretized the structure to include 16 nodes along the length and two along the height. Between nodes along the length, nonlinear beam column objects were created. The top flange was approximated by a rectangle 304 mm by 140 mm and consisted of 40 fibers of concrete, 10 along the width and 4 along the height. Two steel fibers, each with an area of 200 mm 2, were layered in to simulate the #5 9

10 Figure 3(a) Finite Element Mesh and Loading Scheme of the Shear Beam. Figure 3(b). Top Flange Discretization Figure 3(c). Bottom Flange Discretization. 10

11 reinforcing bars. The bottom flange, 406 mm by 190 mm, was similarly discretized into 40 fibers of concrete. Additionally, ten fibers of steel with an area of 99 mm 2 were layered in. These ten fibers represent the prestressing strands located in this flange. When compared to the shear model, the bottom flange of the flexure model considers more of the prestressing strands here due to the experimental observations of Laskar (2009). Illustrations of the discretized flanges are shown in Figure 4(b) and (c). The web itself consisted of quad elements, with four nodes serving as the boundaries. These elements were given a thickness of mm. Each element was modeled with a PCPlaneStress material. This involved creating and layering ConcreteL01, TendonL01, and SteelZ01 materials to represent the reinforcement and tendons located in the web. The orientation of the reinforcement was set along the local x- and y- axes. A tendon steel ratio of 0.11% was used. The transverse steel ratio was set as 1.00%, 0.17%, or 0.00%, depending on the simulation. Once the model was built, the loads were applied in two parts. First, the prestress load of 1.32 MN was applied at the nodes indicated in Figure 4 (a). This load was analyzed using a Krylov-Newton solution algorithm and a load control integrator. After the first analysis was complete, the shear load was applied. Two 1000 N concentrated loads were distributed over three nodes each at a distance of about 2.4 meters from the supports. This analysis also used a Krylov-Newton solution algorithm. In order to obtain a full force/displacement graph, a displacement path integration scheme was used instead of a load control. 11

12 Figure 4(a). Finite Element Mesh and Loading Scheme of the Flexure Beam. Figure 4(b). Top Flange Discretization. Figure 4(c). Bottom Flange Discretization. 12

13 Results and Discussion Simulations of each beam with varying amounts of transverse steel were completed. Below are discussions of the results in both shear and flexure. Shear The results of the shear loading scheme are plotted below in Figure 5. Each of the Force (N) Displacement (mm) 1.00% Steel 0.17% Steel 0.00% Steel Figure 5. Force vs. Displacement in Shear Loading. graphs displays near-identical behavior for the first five millimeters. This can be contributed to the concrete acting under the increasing load. Past this region, however, is where the effect of the transverse steel can be seen. The 1.00% steel displays the highest displacement and force withstood before failure. When reducing the transverse steel ratio to 0.17% steel, a significant decrease in the withstood displacement and maximum force is seen. Also of note, in order for the 0.17% steel to reach its maximum force, a larger displacement was required as compared to the 1.00% steel at that same force. The 0.00% steel displays the lowest force and displacement of the three simulations, including a very 13

14 short response after the initial region shared by all three. After the initial, very little gain in force is seen until breaking, and even the gain in displacement is small compared to the other percentages. These observations follow what is to be expected. The confining pressure of transverse steel forces the concrete into a higher peak strength (while these graphs are not in terms of stress, as there is no change in area it is safe to assume that an increase in force leads to an increase in strength). A decrease in the confinement from transverse steel leads to a decrease of both withstood force and displacement until failure. Flexure The results of the flexure loading scheme are plotted below in Figure 6. Again, an Force (N) % Steel 0.17% Steel 0.00% Steel Displacement (mm) Figure 6. Force vs. Displacement in Flexure. initial region of near-identical behavior is shared by the three graphs for the first seven millimeters. This can be contributed to the concrete acting under the increasing load. Past this point, the three graphs display a similar shape. Each shows increasing force and displacement, although at a lower rate than seen in the shear results. A decrease in the 14

15 maximum force as well as the maximum displacement with a decrease in transverse steel can be seen. A complete removal of the transverse steel leads to a significant reduction in the force and displacement attained. These observations agree with what is to be expected. Confining concrete has been shown to significantly increase its ductility. The results here indicate that even a small amount of transverse steel greatly improves the ductility of the beam. For reference, Figure 7, which shows all six plots together, is included in Appendix B. Conclusions This study concludes that transverse steel plays an important role in confining concrete and thus significantly aids in the ductility and overall strength of the prestressed reinforced concrete beam. This can be seen in the strength obtained in shear and ductility obtained in flexure. Even a small difference in the reinforcement ratio can have huge results. By agreeing with previous observations, this shows the viability of SCS and OpenSees for modeling such behavior. However, it is important to realize that in an analytical model such as this, experimental results are needed for further validation. Acknowledgements The research study described herein was sponsored by the National Science Foundation under the Award No. EEC The opinions expressed in this study are those of the authors and do not necessarily reflect the views of the sponsor. 15

16 Bibliography Dancygier, A. N. "Shear Carried by Transverse Reinforcement in Circular RC Elements." Journal of Structural Engineering (2001): Fenves, G. L. (2005). Annual Workshop on Open System for Earthquake Engineering Simulation, Pacific Earthquake Engineering Research Center, UC Berkeley, Hsu, T. T. C. and Zhu, R. R. H. (2002). Softened Membrane Model for Reinforced Concrete Elements in Shear, ACI Structural Journal 99.4: Laskar, A. (2009). Shear Behavior and Design of Prestressed Concrete Members. Dissertation. Department of Civil and Environmental Engineering, University of Houston, Houston, TX, Mansour, M. and Hsu, T. T. C. (2005a). Behavior of Reinforced Concrete Elements under Cyclic Shear: Part 1 Experiments, Journal of Structural Engineering, ASCE, (2005): Mansour, M. and Hsu, T. T. C. (2005b). Behavior of Reinforced Concrete Elements under Cyclic Shear: Part 2 - Theoretical Model, Journal of Structural Engineering, ASCE, (2005): Saatcioglu, Murat, and Salim R. Razvi (1992). "Strength and ductility of confined concrete." Journal of Structural Engineering New York (1992): Scott, B. D., R. Park, and M. J. N. Priestley. "Stress-Strain Behavior of Concrete Confined by Overlapping Hoops at Low and High Strain Rates." ACI Journal (1982): Wang, J. (2006). Constitutive Relationships of Prestressed Concrete Membrane Elements. Dissertation. Department of Civil and Environmental Engineering, University of Houston, Houston, TX, Zhong, J. X. (2005). Model-Based Simulation of Reinforced Concrete Plane Stress Structures. Dissertation. Department of Civil and Environmental Engineering, University of Houston, Houston, TX,

17 Appendix A: Physical and Material Properties of the Models Object Property Shear Flexure Unit Model Concrete01 ConcreteL01 Steel01for bars Steel01for strands SteelZ01 TendonL01 Span mm Height mm Nodal Spacing mm compressive strength MPa strain at compressive strength mm/mm crushing strength MPa strain at crushing strength mm/mm compressive strength of concrete MPa strain at compressive strength mm/mm yield stress MPa elastic modulus MPa hardening ratio yield stress MPa elastic modulus MPa hardening ratio yield strength of steel MPa elastic modulus MPa compressive strength of concrete MPa ratio of transverse steel 0/.0017/0.01 0/.0017/0.01 yield strength of strands MPa elastic modulus MPa stress in prestressing tendons MPa steel ratio initial strain mm/mm 17

18 Appendix B: Additional Figures Figure A. Shear Model Cross Section. 18

19 Figure B. Flexure Cross Section. 19

20 Force (N) Displacement (mm) 1.00% Steel, Flexure 0.17% Steel, Flexure 0.00% Steel, Flexure 1.00% Steel, Shear 0.17% Steel, Shear 0.00% Steel, Shear Figure C. Force vs. Displacement, Shear and Flexure. 20