Three Dimensional Seismic Simulation of a Two-Bay, Two-Story Reinforced Concrete Building

Transcription

1 Three Dimensional Seismic Simulation of a Two-Bay, Two-Story Reinforced Concrete Building Catherine Joseph, REU Student Rachel Howser, Graduate Advisor Dr. Y.L Mo, Faculty Advisor Final Report Department of Civil and Environmental Engineering University of Houston Houston, TX Civil Infrastructure Engineering REU Sponsored by National Science Foundation July 2009 Page 1 of 25

2 Abstract: As part of a Research Experience for Undergraduates program, the University of Houston(UH) was awarded funding by the National Science Foundation to further its research on the seismic analysis of a two-story, two-bay reinforced concrete structure. The project consisted of developing a mathematical model to simulate the experimental results collected in previous work. Using UH researchers simulation programs SCS and SRCS within the Open System for Earthquake Engineering Simulation (OpenSees) framework, a three dimensional simulation of the structure was attempted. Using a unique combination of two-dimensional and threedimensional models within the same code, the researcher attempted to model the reaction of reinforced concrete plane stress wall elements. A three-dimensional building frame and non plane stress elements were created. The reinforced concrete plane stress elements were created in two dimensions due to element specifications. The modified OpenSees framework presented some difficulties however, as apparent numerical errors occurred within the SCS and SRCS programs. The simulation codes also present problems with the interaction of the twodimensional and three-dimensional models created to simulate the entire structure. A successful three dimensional simulation would provide a reliable method of predicting the seismic reaction of reinforced concrete structures, guaranteeing higher levels of safety and a longer life of the structure. Page 2 of 25

3 Introduction: Over the past twenty years, the Department of Civil and Environmental Engineering at the University of Houston has made significant progress in the field of earthquake engineering. Using various methods, researchers have developed accurate methods of analyzing the effects of seismic activity on reinforced concrete structures. The research performed to simulate shear behavior of reinforced concrete structures produced constitutive models. These models include: Rotating-angle softened truss model(ra-stm); Fixed-angle softened truss model (FA- STM); Softened membrane model (SMM); Cyclic softened membrane model (CSMM). The most recent of these models is the CSMM, which has been proven to be the most accurate in the prediction of significant shear force-displacement behavior of reinforced concrete walls[1]. Prediction of structural damage caused by earthquakes has been completed using both artificial neural networks (ANN) and nonlinear finite element analysis (FEM). ANNS are developed to mimic the organizational principles observed in the brain, such as recognizing patterns and approximating functions. ANNs are used to analyze complex responses to seismic loading with numerous parameters. Their use in the field of civil engineering is widespread, but their application to the prediction of structural damage caused by earthquakes is limited. In many cases, nonlinear FEM is utilized in cases in which structural damage need only be analyzed for a specific structure, or a small number of similar structures. Although nonlinear FEM is applicable to many different cases of structural damage, a single simulation is not easily applied to numerous situations[2]. Because reliable ANNs have not yet been developed for threedimensional structural simulation, nonlinear FEM will be used in the development of the seismic simulation of this three-dimensional structure. Researchers have implemented the constitutive models developed at the University of Houston in the OpenSees framework. OpenSees, an acronym representing Open System for Earthquake Engineering Simulation, was developed at the Pacific Earthquake Engineering Center (PEER) and the University of California Berkeley[3]. It uses finite element methods for seismic simulations within an object-oriented framework that dictates the interaction of various abstract classes within the code. This generates programs for a specific problem subset, such as finite element analysis. OpenSees is an open source, giving it unique recognition of being a community code for earthquake engineering. It facilitates sharing and communication of research accomplishments, allowing for greater development in the field of earthquake engineering[1]. By modifying the OpenSees framework using the constitutive models developed, researchers at the University of Houston have been able to create extremely accurate simulations of structural damage caused by seismic loading. Researchers have created a modified version of OpenSees that specializes in modeling the effect seismic activity has on shear walls. The model, Page 3 of 25

4 Simulation of Concrete Structures (SCS), was developed using material modules created by PhD students Laskar[4] and Zhong[5] using OpenSees as the finite element framework. SCS was created due to the inability of the original OpenSees framework to analyze reinforced concrete plane stress structures. The main additions to the framework that were needed to be able to complete this type of analysis were appropriate uniaxial materials of steel and concrete and a reinforced concrete plane stress material. The new steel and concrete materials (SteelZ01, ConcreteZ01, ConcreteZ02), developed by Zhong, are described in following paragraphs. A reinforced concrete plane stress material was also developed in OpenSees so that analysis on reinforced concrete plane stress structures, such as panels and walls, could be completed. The development of the uniaxial steel and concrete materials by Zhong, and the creation of a Quadrilateral Reinforced Concrete Plane Stress Element by University of Houston researchers helped to create SCS, the modified version of OpenSees capable of analyzing reinforced concrete plane stress structures. SCS performs nonlinear finite element analysis on such structures under static, dynamic, and reversed cyclic loading. Another program, Simulation of Reinforced Concrete Structures (SRCS) was also developed by Zhong. The abilities of SRCS are more restricted. The longitudinal steel ratio must be approximately equal to the transverse steel ratio of the concrete structure[4]. The creation of nodes and elements within the code is organized according to their type and location in the structure. Material constitutive matrices were required for the creation of the beam, column, wall and floor elements. A material constitutive matrix, also called a material stiffness matrix relates the status of stresses and strains for an element. Although this can be expressed as either secant or tangent formulations, because the OpenSees framework uses the tangent stiffness formulation, SCS and SRCS also use the tangent material constitutive matrix. This relates the relationship between the increments of the elemental stresses and strains, while the secant formulation works with absolute values of the stresses and strains. All elements were defined using the same uniaxial concrete and steel models developed by Zhong ConcreteZ01, ConcreteZ02, and SteelZ01. ConcreteZ01 was designed with similarities to the uniaxial concrete model, Concrete01, in CSMM. The envelopes in compression and tension are the same in both models. The ConcreteZ01 model accounts for the softening effect of concrete struts caused by perpendicular tensile strain. This model also simplifies the unloading and reloading paths used in the concrete constitutive model of CSMM. Equations 1-5 are the basis for the ConcreteZ01 model. These equations are also the foundation for the Concrete01 constitutive model in CSMM and for the ConcreteZ02 discussed in the next paragraph. The main advantage of ConcreteZ01 is the simplification of the unloading and reloading paths. Figure 1 shows the material module for Concrete Z01[5]. Page 4 of 25

5 Compression: (Stage C1) = 2. 0 (1) =. ( ) / 0.9. (2) (Stage C2) = 1. > (3) where σ c = concrete stress D = tangent material stiffness matrix ζ = softened coefficient of concrete in compression when the peak stress-softened coefficient is equal to strain-softened coefficient f c = cylinder compressive strength of concrete = smeared uniaxial concrete strain = concrete cylinder strain corresponding to peak cylinder strength f c = uniaxial tensile strain normal to the compression direction being considered η = parameter defined as ( )/( ) Tension: (StageT1) = if 0 <. (4) (Stage T2) =. if >. (5) where E c = modulus of elasticity of concrete in compression ε cr = cracking strain of concrete f cr = cracking tensile stress of concrete Page 5 of 25

6 Figure 1: ConcreteZ01 Material Module[5]. Another concrete constitutive model that was used in this simulation is ConcreteZ02. This model differs from ConcreteZ01 in that the compressive envelope of ConcreteZ01 has been modified. This envelope uses a slope of when incorporating the initial linear path. Figure 2 shows the material module of ConcreteZ02[5]. Page 6 of 25

7 Figure 2: ConcreteZ02 Material Module[5]. The simulations created for this study use the model SteelZ01 to define the reinforcing rebar. The envelope and reloading patterns included in SteelZ01 are the same as those defined in CSMM for embedded mild steel. A simplification was incorporated to estimate nonlinear curves using straight line segmentation. This modification bypasses the iteration intended to calculate stress based on a given strain. Equations 6-10 demonstrate the foundation for the SteelZ01 constitutive model. Figure 3 depicts the material module for Steel Z01[5]. Envelope: = ( ). (6) where f s = stress in mild steel E s = modulus of elasticity of steel = smeared strain of steel bars embedded in concrete = smeared tensile strain of mild steel bars embedded in concrete at first yielding Page 7 of 25

8 = ( ) ( ). (7) where f y = yielding strength in bare steel bars B = width = smeared biaxial strain in the y-direction Nonlinear unloading and reloading paths: = ( ). (8) =. (9) = 1+. (10) where = smeared uniaxial strain of rebars in i th direction f i = concrete strength A = coefficient of prestressed tendons 1 R = coefficient of prestressed tendons 2 Page 8 of 25

9 Figure 3: SteelZ01 Material Module[5]. In addition to the development of these new uniaxial materials, a quadrilateral reinforced concrete plane stress element, quad, was created by UH researchers and implemented in SCS and SRCS. It was developed to better simulate plane stress panels and wall elements. The quad element is an eight degree of freedom rectangular plane stress and plane strain finite element. This is the simplest plane stress finite element and is adequate for the structural simulations being performed. As demonstrated in Figure 4, the element has length a, width b, and also has a constant thickness t, which is not shown in the figure. Each corner node possesses 2 degrees of freedom, so the entire element has eight degrees of freedom. Thus, the element has eight nodal forces (F x, F y ) and eight nodal displacements (u, v)[6]. The quad element modeled is in two dimensions and has not yet been tested in three-dimensional simulations. The focus of this study is attempting to create a three-dimensional simulation using the two dimensional plane stress element. Page 9 of 25

10 Figure 4: Eight degree of freedom plane stress element Equations 11 and 12 represent the displacement functions of the plane stress element. The constants c can be determined by substituting the nodal displacement conditions (Equations 13-16) back into the displacement functions. This yields Equations 17 and 18, from which the four shape functions, Equations 19-22, are obtained[6]. Some assumptions can be made from the displacement functions describing the plane stress element. First, the boundary lines of the element remains straight after shear deformation. The shearing strain γ xy varies linearly with x and y while the strain, ε x (or ε y ), varies linearly with just y (or x) and is independent of x (or y). Also, statically equivalent or work equivalent nodal forces can replace the element stresses. The u and v displacement distribution along any of the edges of the plane stress element is linear. It is only affected by the displacement of the corner nodes by which it is connected. The afore mentioned displacement functions guarantee the compatibility of these boundary displacement[6]. Displacement Functions: (, )= (11) (, )= (12) where v = displacement in y-direction u = displacement in x-direction Page 10 of 25

11 Nodal Displacement Conditions: = = (0,0). (13) = = (,0). (14) = = (, ). (15) = = (0, ). (16) Updated Displacement Functions: (, )= (, ) + (, ) + (, ) + (, ). (17) (, )= (, ) + (, ) + (, ) + (, ). (18) where f 1 = Force 1 f 2 = Force 2 f 3 = Force 3 f 4 = Force 4 Shape Functions: (, )= 1 1. (19) (, )= 1. (20) (, )=. (21) (, )= 1. (22) where a = length of quadrilateral plane stress element b = width of quadrilateral plane stress element The other specifications used in the simulation code written for this study are defined within the original OpenSees source code. These specifications, which include analysis methods, iteration schemes, and the overall organization of the codes, are covered in the next section. Page 11 of 25

12 Experimental Method: During summer 2008, graduate researchers from the University of Houston and from the National Center for Research on Earthquake Engineering (NCREE) in Taipei, Taiwan performed experiments on a two-story, two-bay asymmetrical reinforced concrete structure. A full-scale structure was built at NCREE that included various structural elements, including: flexurecritical columns, shear-critical columns, low-rise and medium-rise walls. A full-scale rendering is depicted in Figure 5. The model was subjected to five cyclic tests simulating seismic activity. Tests 1 and 2 were uniaxial displacement controlled tests and Tests 3 through 5 were biaxial displacement controlled tests. During Test 1, forces were only applied in the east-west direction and during Test 2, only in the north-south direction. Tests 3 and 4 aided in determining the torsional stiffness of the building. During Test 5, as much torsion as possible was created and resulted in structural failure[7]. The experiments performed on this specific structure are the main focus of the OpenSees simulation outlined in this paper. Figure 5: Rendering of Two-Story, Two-Bay Reinforced Concrete Building (created using Google SketchUp) Page 12 of 25

13 Method 1 To simulate the building in OpenSees, three different structural models were defined within the simulation code. Based on the requirements of the model materials used in creating the structure, one two-dimensional model and two three-dimensional models were defined. Figure 6 shows the organization of the models within the simulation code, listing the dimensions, degrees of freedom per node, type of element created, and type of analysis used. Model 1 3-Dimensions 6 Degrees of Freedom Building Frame Gravity Analysis Model 2 3-Dimensions 3 Degrees of Freedom Non-Plane Stress Wall and Floor Elements Gravity Analysis Model 3 2-Dimensions 2 Degrees of Freedom Plane Stress Elements Cyclic Analysis Figure 6: Description of Method 1 Simulation Models Model 1 is a three-dimensional model that assigns six degrees of freedom per node. It creates the building frame using beams and columns to connect the nodes. The nonlinearbeamcolumn element is used to create the beams and columns that compose the structure s frame. Although the element itself is two-dimensional, it is used in this model to create the threedimensional building frame. Each model within a simulation code must be analyzed with applied loads. The weight of the beams, columns, floors, and plane stress wall are applied to Model 1 nodes. The LoadControl integrator is used in specifying the application of the gravity loads. Model 1 uses the Newton-Raphson algorithm to analyze the structure s gravity loads. The Newton-Raphson algorithm uses straight-line approximations to create the Force vs. Displacement curve that should result from the simulation. Model 2 is also a three-dimensional model with three degrees of freedom per node. The nonplane stress wall elements and floor elements are created within this model, using the OpenSees stdbrick element. The command creates an eight-node, three-dimensional brick Page 13 of 25

14 element object. Model 2 also uses the Newton-Raphson algorithm in the gravity analysis. The loads applied to Model 2 nodes are the weights of the non-plane stress wall elements. The LoadControl integrator is also used here to apply the loads. Model 3 is a two dimensional simulation of the plane stress wall elements with two degrees of freedom per node. The quad element, developed at UH, is a quadrilateral reinforced concrete plane stress element, used within this simulation to create the wall. It creates an eight-node plane stress element. The element can only be created in two-dimensions, but, because the equaldof command is used to link Model 3 nodes to Model 1 nodes, it is connected to the three-dimensional building structure. Model 3 performs a cyclic analysis with cyclic loads applied. The Newton-Raphson algorithm is used in this Model. Cyclic forces are applied in either the x-direction of the y-direction in addition to the displacement implemented using the DisplacementPath integrator. This integrator is in general more stable than LoadControl and demonstrates faster and more successful convergence. Because each model was created with slightly different specifications, the equaldof command was used to connect multiple nodes assigned to the same coordinate. This command was also used to place the nodes and connectivity of the two-dimensional plane stress elements in the correct location on the three-dimensional structure. This was achieved by attaching the twodimensional nodes of Model 3 to three-dimensional nodes created in Model 2 that are at the same location on the structure. Figure 7: Rendering of structure created in full2dofcrj.tcl Page 14 of 25

15 Figure 8: Finite Element Mesh for full2dofcrj.tcl Figure 9: Rendering of structure created in full1dofcrj.tcl Page 15 of 25

16 Figure 10: Finite Element Mesh for full1dofcrj.tcl Figure 7 and Figure 8 show the location of the quadrilateral reinforced concrete plane stress elements (shaded blue in rendering) and the applied forces on the structure created in full2dofcrj.tcl. Figure 9 and Figure 10 depict the location of the plane stress wall (shaded blue in rendering) and applied forces on the structure created in full1dofcrj.tcl. The displacement applied in the cyclic analysis can only occur in one direction. Because of this, the plane stress elements had to be simulated separately. Since the location of the plane stress elements changes between the two simulations, two codes were written. One code, referred to as full2dofcrj.tcl, contains plane stress elements in the y-z coordinate plane, forming a 150mm thick mid-rise wall. The cyclic loads are applied in the y-direction. The plane stress elements in the other code, called full1dofcrj.tcl, are located in the x-z coordinate plane, forming a 100mm thick low-rise wall. The cyclic loads are applied in the x-direction. The numbering of nodes and elements occurs in the same order in each code, but the location of each is slightly different. Method 2 In a second attempt to create a working three-dimensional simulation code, the researcher eliminated the non-plane stress wall elements and floor elements. The building frame, constructed using nonlinearbeamcolumn elements, and the reinforced concrete plane stress wall remained. Thus, only two models were necessary within the simulation code one threedimensional model and one two-dimensional model. Figure 11 shows the organization of the models within the simulation code. Page 16 of 25

17 Model 1 3 Dimensions 6 Degrees of Freedom Building Structure Gravity Analysis Model 2 2 Dimensions 2 Degrees of Freedom Plane Stress Elements Cyclic Analysis Figure 11: Description of Method 2 Simulation Models Model 1 is a near copy of the first model in full2dofcrj.tcl and full1dofcrj.tcl. It is created in three-dimensions with six degrees of freedom per node. The nonlinearbeamcolumn element is used to create the building frame by creating beams and columns. A gravity analysis is performed at the end of the model, using the Newton-Raphson algorithm and LoadControl integrator. The loads applied are the weights of the beams, columns, and plane stress elements. Model 2 is extremely similar to Model 3 in full2dofcrj.tcl and full1dofcrj.tcl. This model is created in two-dimensions with two degrees of freedom per node. The quad element is used to create the quadrilateral reinforced concrete plane stress elements that compose the wall. The equaldof command is used to link Model 2 nodes to Model 1 nodes in the same location to attach the plane stress wall to the building structure. A cyclic analysis is applied to this model. The Newton-Raphson method and DisplacementPath integrator are used in the analysis. Page 17 of 25

18 Figure 12: Rendering of structure created in plstr2dofcrj.tcl Figure 13: Finite Element Mesh for plstr2dofcrj.tcl Page 18 of 25

19 Figure 14: Rendering of structure created in plstr1dofcrj.tcl Figure 15: Finite Element Mesh for plstr1dofcrj.tcl Figure 12 and Figure 13 depict the location of the quadrilateral reinforced concrete plane stress elements (shaded blue in rendering) and the forces applied to the structure created in plstr2dofcrj.tcl. Figure 14 and Figure 15 show the location of the plane stress elements (shaded blue in rendering) and applied forces for the structure created in plstr1dofcrj.tcl. As with the Page 19 of 25

20 first method attempted, two simulation codes were necessary because the location of the plane stress elements changed. The first code, named plstr2dofcrj.tcl, contains plane stress elements in the y-z coordinate plane, creating a 150mm thick mid-rise wall. The cyclic loads are applied in the y-direction. The second code, called plstr1dofcrj.tcl, has plane stress elements in the x-z coordinate plane, forming a 100mm thick low-rise wall. The cyclic loads are applied in the x-direction. The order of the numbering of nodes and elements occurs in the same order in each code, but the coordinates of the Model 2 nodes change. Results: During the summer research session, no results that modeled the experimental data were obtained. A three-dimensional structure was constructed within each of the simulation codes, and reinforced concrete plane stress elements were implemented. The attempt by the researcher to produce a viable three-dimensional structural simulation with seismic loading has proven unsuccessful thus far. Each of the four simulation codes mentioned have been completed to the same extent, with all the structural code written and the cyclic loading applied. An output file is created for each code; however, no data is produced. At least several iterations are being performed by the analysis in each code, but no data is collected in the output file. Because OpenSees is an open source code, there are certain nuances that must be satisfied by a guess-and-check approach to creating the code. In addition, these simulations used special commands developed at the University of Houston, which also had certain unexplainable aspects of incompatibility. During the creation of SCS and SRCS, it was discovered that numerical problems may occur in nonlinear finite element analysis using the programs, especially during reversed cyclic analysis. It was recommended, in most cases, to change the solution algorithm, change the increment size, or increase the iteration number. These numerical problems seemed to be causing the errors in the simulation codes written by the researcher. The errors that occurred stated that OpenSees was unable to calculate the deformation associated with a specific nonlinearbeamcolumn element. Hoever, this error varies when the increment size and number of iterations are changed. The solutions suggested to correct the numerical errors have been tried. The solution algorithm that proved to be most successful was the Newton-Raphson algorithm. The increment size has been changed, as has the iteration number, with some success. However, the number of iterations that the code is designed to perform rises well into the thousands, while the number of iterations actually performed remains below one hundred. Page 20 of 25

21 Conclusion: As there are no final results to this study, no definite conclusion can be drawn. However, it is possible to say that three-dimensional simulation of plane stress elements is viable in the future through the OpenSees framework. The three-dimensional building frame and the threedimensional non-plane stress elements seem to be created correctly. The plane stress wall, composed of quad elements, also seems to be functioning properly. However, the combination of two-dimensional and three-dimensional models doesn t seem to be working correctly. The development of a three-dimensional plane stress element would provide a more direct method of successfully performing this study. In addition, the nuances within the simulation code created by the interaction of the framework models and the additional models developed by the University of Houston s Department of Civil and Environmental Engineering must be determined and accounted for within the simulation code. Although the study has proven inconclusive thus far, it is apparent that more research is needed in determining the interaction of the different models within a three-dimensional simulation. Page 21 of 25

22 Appendix 1: References: [1] Y. L. Mo, et al., "Seismic simulation of RC wall-type structures," Engineering Structures, vol. 30, pp , [2] O. R. de Lautour and P. Omenzetter, "Prediction of seismic-induced structural damage using artificial neural networks," Engineering Structures, vol. 31, pp , Feb [3] G. Fenves, in Annual workshop on open system for earthquake engineering simulation., Berkeley: Pacific Earthquake Engineering Research Center, UC, [4] A. Laskar, "Shear Behavior and Design of Pre-Stressed Concrete Members," PhD, Department of Civil and Environmental Engineering, University of Houston, Houston, TX, [5] J. X. Zhong, "Model-Based Simulation of Reinforced Concrete Plane Stress Structures," PhD, Department of Civil and Environmental Engineering, University of Houston, Houston, TX, [6] T. Y. Yang, "Plane Stress and Plane Strain Finite Elements," in Finite Element Structural Analysis, W. J. H. N. M. Newmark, Ed., ed Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1986, pp [7] R. Howser, et al., "Seismic Simulation of a Two-Bay Two-Story RC Building," presented at the Texas Section ASCE Conference, Page 22 of 25

23 Appendix 2: Notation: a = length of quadrilateral plane stress element A = coefficient of prestressed tendons 1 b = width of quadrilateral plane stress element B = width D = tangent material stiffness matrix E c = modulus of elasticity of concrete in compression E s = modulus of elasticity of steel f c = cylinder compressive strength of concrete f cr = cracking tensile strength of concrete f i = concrete strength f s = stress in mild steel f y = yielding strength in bare steel bars f 1 = force 1 f 2 = force 2 f 3 = force 3 f 4 = force 4 R = coefficient of prestressed tendons 2 u = displacement in x-direction v = displacement in y-direction = smeared uniaxial concrete strain ε cr = cracking strain of concrete = plastic strain of bars embedded in concrete Page 23 of 25

24 = smeared tensile strain of mild steel bars embedded in concrete at first yielding = smeared strain of steel bars embedded in concrete = smeared uniaxial strain of rebars in i th direction = uniaxial tensile strain normal to the compression direction being considered = smeared biaxial strain in the y-direction = concrete cylinder strain corresponding to peak cylinder strength f c η = parameter defined as ( )/( ) σ c = concrete stress ζ = softened coefficient of concrete in compression when the peak stress-softened coefficient is equal to strain-softened coefficient Page 24 of 25

25 Appendix 3: Acknowledgment: 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. Page 25 of 25