A Shear Wall Element for Nonlinear Seismic Analysis of SuperTall Buildings Using OpenSees Xinzheng Lu ^{a,*}, Linlin Xie ^{a}, Hong Guan^{ b}, Yuli Huang^{ c}, Xiao Lu^{ d} ^{a }Department of Civil Engineering, Tsinghua University, Beijing, P.R. China ^{b }Griffith School of Engineering, Griffith University Gold Coast Campus, Queensland 4222, Australia ^{c }Arup, San Francisco, United States of America ^{d }Department of Civil Engineering, Beijing Jiaotong University, Beijing, P.R. China. Finite Elements in Analysis & Design, 2015, 98: 1425.. Related Download: OpenSees Models for a 606m Megatall Building v0.1 Abstract: Numerical simulation has increasingly become an effective method and powerful tool for performancebased earthquake engineering research. Amongst the existing research efforts, most numerical analyses were conducted using generalpurpose commercial software, which to some extent limits indepth investigations on specific topics with complicated nature. In consequence, this work develops a new shear wall element model and associated material constitutive models based on the open source finite element (FE) code OpenSees, in order to perform nonlinear seismic analyses of highrise RC framecore tube structures. A series of shear walls, a 141.8m framecore tube building and a supertall building (the Shanghai Tower, with a height of 632 m) are simulated. The rationality and reliability of the proposed element model and analysis method are validated through comparison with the available experimental data as well as the analytical results of a well validated commercial FE code. The research outcome will assist in providing a useful reference and an effective tool for further numerical analysis of the seismic behavior of tall and supertall buildings. Key words: Framecore tube; nonlinear analysis; OpenSees; multilayer element; supertall building; Shanghai Tower. DOI: 10.1016/j.finel.2015.01.006 If you need the PDF version of this paper, please email to luxinzheng@sina.com 
2 Multilayer shell element model for shear walls 2.1 Theoretical background of multilayer shell element model The multilayer shell formulation [1] is implemented in OpenSees using the ¡°ShellMITC4¡± element, which is a fournode shell element based on the theory of mixed interpolation of tensorial components (MITC) proposed by Dvorkin et al. [19].
Fig. 1 illustrates the local coordinate system adopted for the shell element. Vector r_{1} extends from the midpoint of Nodes 2 and 3 to the midpoint of Nodes 1 and 4. Vector r_{2} extends from the midpoint of Nodes 3 and 4 to the midpoint of Nodes 1 and 2. The GramSchmidt orthonormalization is then adopted to adjust Vector r_{2}. Finally, Vector r_{3} is determined as the cross product of Vectors r_{1} and r_{2} following the right hand rule. Vectors g_{1}, g_{2} and g_{3} are obtained through vector unitization of Vector r_{1}, orthogonal Vector r_{2} and Vector r_{3}, respectively. The strains and curvatures at the inplane integration points are obtained from the interpolation of the nodal displacements and rotations. When large deformations are taken into consideration, the coordinate system is updated based on the current coordinates of the nodes. For each of the inplane integration points, a layered/composite integrated section is then implemented to account for the nonlinear behavior of reinforced concrete (RC). The layered element simplifies the threedimensional nonlinear behavior of the shear walls into a shell situation by discretizing them into several fullybonded layers in the thickness direction. Different material properties and thicknesses can be assigned to each layer according to the size of the wall and the distribution of reinforcing bars, as illustrated in Fig. 2. The bars are smeared into one or more orthotropic layers according to their physical location and direction, as shown in Fig. 3. The stresses over a layer thickness are assumed to be consistent with those at the midsurface of that layer. Therefore, if the shear wall is subdivided into sufficient number of layers, the multilayer shell element can reasonably simulate the actual stress distribution over the thickness of the wall [2021].
The nonlinear behavior of concrete cracking, crushing and aggregate interlocking is incorporated in the planar concrete constitutive model. For steel reinforcement, the nonlinear behavior of yielding and the Bauschinger effect are taken into consideration in its uniaxial model. In the multilayer shell element, the axial strains and curvature of the middle layer are initially calculated; the strains of each layer are then obtained based on the planesection assumption. The stresses of each integration point on each layer are subsequently calculated in accordance with the constitutive model of the corresponding layer. Lastly, the internal forces are calculated using the standard numerical integration method. The multilayer shell element has been proven to be a suitable model that can well capture the coupled inplane/outofplane bending as well as the inplane direct shear and coupled bending/shear behavior of the RC shear walls [1, 6, 2022]. 2.2 Twodimensional material constitutive model Concrete is assumed to be in a state of planar stress in the multilayer shell element. An appropriate, reliable and robust twodimensional concrete constitutive model is required to cater for the complicated mechanical behavior of shear walls and core tubes induced by an earthquake. The analytical model for concrete proposed in this research is based on the concept of damage mechanics and the smeared crack model. Cracks are assumed to form when the principal tensile stress exceeds the specified concrete tensile strength. Once cracking occurs, concrete is then treated as an orthotropic material. The shear stiffness deterioration (in terms of the reduced shear moduli obtained by multiplying the shear retention factor) is incorporated due to the effects of aggregate interlocking. Being simple in formulation, this analytical model also demonstrates a good stability in computation [21, 22]. The constitutive equation (Eq. 1) of concrete is expressed as follows:
where s_{c}^{¡¯} and e_{c}^{¡¯}_{ }represent the stress and strain tensors, respectively, in the principal stress coordinate system before cracking or in the crack coordinate system after cracking. D_{e} is the elastic constitutive matrix, and d_{1} and d_{2} are the damage parameters, which can be respectively calculated by the damage evolution curves under tension as recommended by Løland [23] and under compression as suggested by Mazars [24]. After cracking, the relationship between the shear stress t and shear strain g in the crack coordinates is as follows:
where G is the elastic shear modulus and b is the shear retention factor to account for the shear stiffness deterioration after cracking occurs [25]. Both Eqs. 1 and 2, formulated with either the principal stress coordinate or the crack coordinate, must be transformed to a global coordinate system [26] following Eq. 3.
where D and D^{¡¯} are the constitutive matrices in the global and local coordinates, respectively. R is the transformation matrix and q is the angle between the global and local coordinates. The reinforcing bars are simulated as smeared steel layers of equivalent thicknesses. The material model for reinforcing steel, designated as PlateRebar, is derived from the uniaxial materials for rebars [15] in conjunction with their corresponding angles. Detailed information can be found in Appendix A. 
3 Implementation and validation of the multilayer shell element 3.1 Implementation of the model in OpenSees The multilayer shell element incorporating the twodimensional material constitutive models is implemented in the latest version 2.4.4 of OpenSees (http://opensees.berkeley.edu/), as shown in Fig. 4(a). The overall procedure for defining the multilayer shell element in OpenSees is illustrated in Fig. 4(b), and the corresponding command line scripts are given in Appendix A.
3.2 Finite element discretization Finite element discretization (i.e., the element size) has a direct impact on the accuracy of the analysis solutions. This study adopts the popularly used crack band theory proposed by Bazant [27, 28], in order to eliminate the size effect. To achieve this, the slope of the softening branch of cracked concrete is proportionally adjusted according to the element size, leading to identical fracture energy for all elements. To verify the effectiveness of the crack band theory, a typical rectangular wall panel SW11 tested by Lu et al. [29] is discretized with three different mesh sizes (40, 160 and 360 elements), as shown in Fig. 5 (ac). The experimental and simulation results are compared in Fig. 6 with close agreement. This comparison indicates that the effect of mesh size on the predicted behavior can be effectively eliminated using the crack band theory, and little improvement can be achieved by introducing additional nodes and elements. In consequence, the crack band theory is recommended and adopted in the following study of both shear walls and supertall buildings.
3.3 Validation of the model with a series of test results To validate the reliability and accuracy of the proposed model, a series of 12 rectangular walls [29] with different axial load ratios, aspect ratios, boundary element widths, reinforcement ratios and stirrup ratios in boundary elements are simulated initially. To further validate the versatility of the model, six shear walls with different sections (i.e., an Hshaped wall, Tshaped wall [30], coupled wall [31] and symmetric double shortlimb walls [32]) are also simulated. To compare the performance of the multilayer shell element and the traditional fiber beam model for shear walls, both are used to simulate the rectangular, Hshaped and Tshaped walls. This comparison reveals that to simulate twin walls connected by coupling beams, the multilayer shell model is much easier to use than the fiber beam modeling. In consequence, only the multilayer shell model is adopted to simulate the coupled walls. For other types of walls including rectangular and flanged shear walls, on the other hand, both the multilayer shell element and the fiber beam model can be adopted. The differences in their respective modelling strategies are discussed below. When using the fiber beam model, the shear walls are modeled with five nonlinear beamcolumn elements using the compliance method and each element has five integration points [17], as shown in Fig. 5(d). The concrete fibers in the fiber section are assumed to follow the KentScottPark stressstrain model and the corresponding hysteretic rules available in OpenSees through Concrete01 [15]. The confinement model proposed by Mander et al. [33] is adopted to calculate the characteristic parameters of the unconfined and confined concrete. The stressstrain model proposed by Filippou et al. [34] known as Steel02 in OpenSees is used to simulate the steel reinforcement. As for the multilayer shell model, the vertical steel reinforcement within the wall web and all the stirrups are smeared into one or more layers, while the longitudinally concentrated steel reinforcement within the boundary elements are simulated with truss elements that are incorporated into the shell element by sharing the same nodes. Taking into account the outofplane mechanical behavior, the flanges (outofplane) and webs (inplane) of the H or Tshaped walls are both simulated using the multilayer shell elements. The characteristic parameters of concrete and steel reinforcements are determined from the tensile tests of rebars and the compressive tests of standard concrete cylinders or cubes. Specifically, based on the compressive strength of concrete, the characteristic parameters of both unconfined concrete in wall webs and confined concrete in boundary elements are calculated using the concrete constitutive model proposed by Mander et al. [33]. The tensile strength, elastic modulus and fracture strain obtained from the tensile tests of rebars are used to determine the characteristic parameters of reinforcements. A comparison between the experimental yield load, yield displacement and peak load and the simulation results, including the results using the multilayer shell elements and fiber elements, is presented in Table 1. It is evident that the majority of the predictions using the multishell element agree well with the experimental data, thus validating the reliability and versatility of the multilayer shell element. Table
1
Amongst the 18 shear walls given in Table 1, only four typical ones are discussed in detail in this article, the dimensions and reinforcement details obtained from corresponding tests are illustrated in Fig. 7. The comparisons between the simulation results and experimental data on the lateral forcetop displacement hysteretic curves of the selected shear walls are presented in Fig. 8.
For rectangular walls with sufficient stirrups and moderate aspect ratios such as SW11 in Fig. 8(a), both the fiber beam model and the multilayer shell model can well capture the characteristic behaviors of shear walls predominately governed by flexural behavior. In contrast, for specimens exhibiting shear behavior such as SW21 in Fig. 8(b), the multilayer shell model better predicts the experimental data than the fiber beam model. The flange wall presented in Fig. 8(c) is also dominated by flexural behavior. Due to relatively small flanges, both the fiber element and the multilayer shell models can accurately capture the characteristic behavior, which also validates the reliability of the modeling approach for flange walls with shell elements. For the coupled shear wall, CW3 shown in Fig. 8(d), the hysteretic loaddisplacement relation curve predicted by the multilayer shell element model only also agrees well with the experimental data.
Based on the above, it can be concluded that the multilayer shell element developed in OpenSees is capable of replicating the complex behavior of various types of shear walls. 
4 Modeling techniques and selection of analysis domain for supertall buildings 4.1 Modeling techniques In modeling a supertall building, the fiber beam element based on the direct stiffness method, i.e., the DispBeamColumn element in OpenSees, is adopted to simulate the beams and columns [35]. The multilayer shell element is used to simulate the shear walls, core tubes, coupling beams and slabs. For these elements, similar element discretization strategies and identical calculation methods for characteristic material parameters as mentioned above are adopted. The absence of a graphical user interface (GUI) makes it difficult to establish and evaluate a complicated supertall building model in OpenSees. In consequence, a conversion program, from MSC.Marc (which has a convenient GUI) to OpenSees, is developed to promote the modeling efficiency. Note that in this conversion program, the fiber beam elements and the multilayer shell elements used in the MSC.Marc model are converted to the OpenSees model either fiberbyfiber or layerbylayer to ensure consistency of these two models and the subsequent analysis. This conversion facilitates convenient and reliable validation of the OpenSees simulation results with those of MSC.Marc. The corresponding conversion relationship is illustrated in Fig. 9.
4.2 Selection of analysis domain in OpenSees The analysis domain in OpenSees is an object composed of several component objects, including the Constraint Handler (determines how the constraint equations are enforced in the analysis), solution Algorithm, System of Equations (constructs a solver to store and solve the system of equations in the analysis) and Eigenvalue Solver (conducts eigenvalues analysis) [15]. Selection of these component objects determines the feasibility, stability and efficiency of the simulation [16]. Note that previous research [3639] primarily focused on specimens and smallscale structures, and limited work is available on the selection of the analysis domain for largescale structures. Note also that, the selected component objects suitable for small structures may not be applicable or efficient for largescale structures. When conducting a nonlinear analysis of supertall buildings in this study, most solvers and numerical algorithms are found to be ineffective in largescale computations. Through a series of analyses of supertall buildings, some efficient component objects are proposed as presented in Table 2. Table
2
As for a Constraint Handler, the transformation method [15, 16] performs well in the following analyses of supertall buildings. Due to their stability and efficiency, the solvers for sparse systems are recommended for the System of Equations command instead of those for band or profile systems. In particular, the solvers for the sparse symmetric system (i.e., SparesSYM and CuSP) are recommended. To conduct an eigenvalue analysis, the solver for a symmetric positive definite profile (i.e., ProfileSPD) is suggested. 
5 Nonlinear dynamic analysis of supertall buildings 5.1 A fortytwo story RC framecore tube building 5.1.1 Overview of the structure To evaluate and improve the performancebased seismic design of tall buildings, the Pacific Earthquake Engineering Research Center (PEER) launched the Tall Buildings Initiative (TBI) research program. One of the case study buildings investigated in this program is Building 2A, a 42story RC framecore tube structure with a total above ground height of 141.8 m [40]. Based on the design information of Building 2A, a new building named Building 2N was redesigned by Lu et al. [41] according to the Chinese building design codes. A threedimensional view and floor plan of Building 2N are shown in Fig. 10. Specific design details can be found in Lu et al. [41].
The entire model of Building 2N, including 8,469 nodes, 9,744 fiber beam elements defined by 8,244 RC fiber sections and 4,704 multilayer shell elements defined by 177 shell sections, was initially constructed in MSC.Marc and subsequently converted to OpenSees. The OpenSees model is freely assessable [42], which can be conveniently shared and reused in the research community. It is very difficult to validate a numerical model for such a tall building by means of experiment. The reliability of the model can however be verified through comparisons between the analytical results of OpenSees and MSC.MARC. This is because MSC.Marc has been widely used in nonlinear analyses of tall buildings and adequately validated in terms of rationality and accuracy [16, 18]. 5.1.2 Basic dynamic properties A gravity analysis and a modal analysis are conducted prior to the nonlinear analysis. A comparison between the fundamental periods predicted by MSC.Marc and OpenSees is presented in Table 3. The results indicate a strong agreement in the basic dynamic properties, which validates the rationality of the model at the elastic stage. Table
3
5.1.3 Pushover analysis The inverted triangular distribution of lateral load pattern is adopted to initiate the pushover analysis using both software packages. It is worth mentioning that the PD effects are considered in the pushover analysis and the subsequent dynamic analysis. The base shear force versus the roof displacement and the distribution of interstory drift ratios are plotted in Fig. 11 and Fig. 12, respectively. Good agreements are evident between the MSC.Marc and OpenSees predictions. The slight discrepancy is attributable to different concrete models adopted by these two packages. While both employ the smeared crack model, the multishell element implemented in OpenSees is based on damage mechanics whereas that in MSC.Marc is based on plastic mechanics. Nonetheless, the differences between the two simulation results are considered acceptable.
Fig. 11. Comparison of the base shear force roof displacement curves
Fig. 12. Comparison of the interstory drift ratio distribution 5.1.4 Nonlinear dynamic analysis The ChiChi ground motion, which is scaled to a value of peak ground acceleration (PGA) of 110 gal and 300 gal according to the Chinese Code, is initially used as the seismic input to the structure along the Xaxis after the gravity analysis is conducted. To study the strong nonlinear behavior of Building 2N under an extreme earthquake, a dynamic analysis with a PGA = 1000 gal is also performed. During the nonlinear dynamic timehistory analysis, a 5% damping ratio suggested in Section 5.3.4 of the Specification for the Design of SteelConcrete Hybrid Structures in Tall Buildings (CECS 230; 280) [43] is adopted. Considering the contribution of higherorder modes to the seismic response of tall buildings, the 1^{st} and 9^{th} modes are chosen to calculate the coefficients of Rayleigh damping which is recommended by Chopra for most practical analyses [44]. The comparisons of the timehistory curves of the roof are shown in Figures 13(a)(c). In addition, the ElCentro and Northridge ground motions, which are scaled to PGA=1000 gal, are also adopted to conduct the dynamic analysis. The timehistory curves of the roof are compared in Figures 13(d)(e). The envelope of interstory drift curves under the Northridge ground motion is presented in Fig. 13(f) because it has the maximum interstory drift ratio amongst the three ground motions. Roof displacements represent the overall response of the entire structure which is nonsensitive to the local damage occurred on each story. In contrast, interstory drift ratios are more sensitive. Due to different concrete constitutive models adopted by OpenSees and MSC.Marc, some discrepancies do exist between the local damage predictions of these two software packages. Such discrepancy is more obvious in the interstory drift ratio curve than in the roof displacement curve. Similar phenomena can also be found in many other researches [4547], in which the differences presented in the interstory drift ratios are more significant than those in the roof displacements. Nonetheless, the overall differences between the simulation results of OpenSees and MSC.Marc are considered acceptable. Based on the above comparisons, it can be concluded that OpenSees is feasible and reliable for conducting nonlinear analyses of tall RC framecore tube structures even under strong nonlinear conditions.
(a) ChiChi 110 gal
(b) ChiChi 300 gal
(c) ChiChi 1000 gal
(d) ElCentro 1000 gal
(e) Northridge 1000 gal
(f) Envelope curves of interstory drift ratio under 1000 gal Northridge ground motion Fig. 13. Comparison of the time history analysis results 5.2 The Shanghai Tower The Shanghai Tower is a 124story multifunctional office building located in Shanghai, China. The total height of the main tower is 632 m. The lateral force resisting system of the building is composed of a square RC tube, 12 shapedsteel RC columns and outriggers, defined as a ¡°megacolumn/coretube/outrigger¡± system. Details of the Shanghai Tower can be obtained from Lu et al. [3]. The design intensity of the Shanghai Tower is 7 degree according to the Chinese code. The corresponding PGA values for earthquakes with 63%, 10% and 2% of exceedance in 50 years are 35, 100 and 220 gal, respectively. A dynamic analysis with a PGA = 220 gal is initially conducted. To study the seismic behavior of the Shanghai Tower under an extreme earthquake and further validate the nonlinear computational capacity of the proposed model, a dynamic analysis with a PGA = 400 gal is also conducted subsequently. The modeling strategy proposed by Lu et al. [3] is adopted to establish the entire model using MSC.Marc. Subsequently, this model is converted to OpenSees, including 53,006 nodes, 48,774 fiber beam elements defined by 81 fiber Hsections and 39,315 multilayer shell elements defined by 148 shell sections, as shown in Fig. 14.
Fig. 14. FE model of Shanghai Tower Similarly, a gravity analysis and modal analysis are initially conducted; the fundamental periods predicted by MSC.Marc and OpenSees are also found to agree well with each other, with only 1.33% of difference. The ElCentro EW ground motion, which is scaled to PGA = 220 gal and 400 gal, is adopted as a typical ground motion input to the structure along the Xaxis after conducting the gravity analysis. A 5% Rayleigh damping ratio is also adopted to perform the nonlinear dynamic timehistory analysis. The comparisons of the timehistory curves of the roof and the envelope of the interstory drift curves are illustrated in Fig. 15 and Fig. 16, respectively. Evidently, the results of OpenSees are strongly compatible with those of MSC.Marc. Further, the nonlinear dynamic analysis of Shanghai Tower performed in OpenSees consumes 27.5 hours using the GPU solver, while it takes 24 hours using the parallel CPU solver when the analysis is conducted with MSC.Marc, conclusively validating the feasibility, reliability and efficiency of OpenSees in simulating complex supertall buildings.
(a) Comparison of the timeroof displacement curves
(b) Comparison of the displacement envelope curves of each story
(c) Comparison of the distribution of interstory drift ratio Fig. 15. Comparison of the time history analysis results under 220 gal
(a) Comparison of the timeroof displacement curves
(b) Comparison of the displacement envelope curves of each story
(c) Comparison of the distribution of interstory drift ratio Fig. 16. Comparison of time history analysis results under 400 gal 
6 Conclusions A new numerical model, viz. the multilayer shell element, is developed and implemented in OpenSees for the simulation of shear walls in supertall buildings. A series of shear walls with various types of sections are simulated, through which the reliability and versatility of the new element model is validated. Subsequently, a numerical model system, integrating the fiberbeam elements for beams/columns and the multilayer shell elements for shear walls/core tubes, is proposed to simulate tall and supertall buildings using OpenSees. Gravity analysis, modal analysis, pushover analysis and dynamic analysis under specified ground motions of a tall RC framecore tube building are conducted. A dynamic analysis of an actual supertall building is also performed. For identical models and analyses, the predicted results of OpenSees and MSC.Marc are compared with close agreement, thus conclusively validating the reliability and rationality of the proposed elements and the analysis method. The research findings of this study will assist in providing an effective tool and a useful reference for further research on the seismic behavior of tall and supertall buildings based on numerical analysis. Acknowledgements The authors are grateful for the financial support received from the National Key Technology R&D Program (No. 2013BAJ08B02), the National Natural Science Foundation of China (No. 51222804, 51378299) and the Beijing Natural Science Foundation (No. 8142024). References [1] X. Lu, X.Z. Lu, H. Guan, L.P. Ye, Collapse simulation of reinforced concrete high¨Crise building induced by extreme earthquakes, Earthquake Engineering & Structural Dynamics, 42(5) (2013) 705723. [2] X.Z. Lu, X. Lu, H. Guan, W.K. Zhang, L.P. Ye, Earthquakeinduced collapse simulation of a supertall megabraced framecore tube building, Journal of Constructional Steel Research, 82 (2013) 5971. [3] X. Lu, X.Z. Lu, W.K. Zhang, L.P. Ye, Collapse simulation of a super highrise building subjected to extremely strong earthquakes, Science China Technological Sciences, 54(10) (2011) 25492560. [4] P.Q. Ren, Y. Li, H. Guan, X.Z. Lu, Progressive collapse resistance of two typical highrise RC frame shear wall structures, Journal of Performance of Constructed FacilitiesASCE, 2014, DOI:10.1061/(ASCE)CF.19435509.0000593. [5] Q. Jiang, X.Z. Lu, H. Guan, X.G. Ye, Shaking table model test and FE analysis of a reinforced concrete megaframe structure with tuned mass dampers, The Structural Design of Tall and Special Buildings, 23 (2014) 14261442. [6] Z.W. Miao, L.P. Ye, H. Guan, X.Z. Lu, Evaluation of modal and traditional pushover analyses in frameshearwall structures, Advances in Structural Engineering, 14(5) (2011) 815836. [7] O. Esmaili, S. Epackachi, R. Mirghaderi, A.A.T. Behbahani, S. Vahdani, Rehabilitation of a high¨Crise coupled shear wall system in a 56¨Cstorey residential reinforced concrete building (Tehran Tower), based on nonlinear dynamic timehistory analyses, The Structural Design of Tall and Special Buildings, 20(8) (2011) 10351047. [8] D. Poon, L. Hsiao, Y. Zhu, L. Joseph, S. Zuo, G. Fu, Ihtiyar, Nonlinear time history analysis for the performance based design of Shanghai Tower, Structures Congress, (2011) 541551. [9] H.J. Jiang, B. Fu, L. Liu, X.W. Yin, Study on seismic performance of a supertall steel¨Cconcrete hybrid structure, The Structural Design of Tall and Special Buildings, 23(5) (2014) 334349. [10] X.L. Lu, N.F. Su, Y. Zhou, Nonlinear time history analysis of a super¨Ctall building with setbacks in elevation, The Structural Design of Tall and Special Buildings, 22(7) (2013) 593614. [11] G. Michaloudis, G. Blankenhorn, S. Mattern, K. Schweizerhof, Modeling structural failure with finite element analysis of controlled demolition of buildings by explosives using LSDYNA, High Performance Computing in Science and Engineering'09. Springer Berlin Heidelberg, (2010) 539551. [12] Marc MSC, Volume A, Theory and user information, MSC Corp, 2007. [13] B. Patz¨¢k, Z. Bittnar, Design of object oriented finite element code, Advances in Engineering Software, 32(10) (2001) 759767. [14] Y.J. Park, A.M. Reinhorn, S.K. Kunnath, IDARC: Inelastic damage analysis of reinforced concrete frameshearwall structures, Department of Civil Engineering State University of New York at Buffalo, 1987. [15] S. Mazzoni, F. McKenna, M.H. Scott, G.L. Fenves, OpenSees command language manual, Pacific Earthquake Engineering Research (PEER) Center, 2006. [16] F. McKenna, M.H. Scott, G.L. Fenves, Nonlinear finiteelement analysis software architecture using object composition, Journal of Computing in Civil Engineering, 24(1) (2009) 95107. [17] J.W. Wallace, Modelling issues for tall reinforced concrete core wall buildings, The Structural Design of Tall and Special Buildings, 16(5) (2007) 615632. [18] Y. Li, X.Z. Lu, H. Guan, L.P. Ye, An improved tie force method for progressive collapse resistance design of reinforced concrete frame structures, Engineering Structures, 33(10) (2011) 29312942. [19] E.N. Dvorkin, D. Pantuso, E.A. Repetto, A formulation of the MITC4 shell element for finite strain elastoplastic analysis, Computer Methods in Applied Mechanics and Engineering, 125(1) (1995) 1740. [20] P. Hallinan, H. Guan, Layered finite element analysis of oneway and twoway concrete walls with opening, Advances in Structural Engineering, 10 (1) (2007) 5572. [21] H. Guan, Y.C. Loo, Flexural and shear failure analysis of reinforced concrete slabs and flat plate, Advances in Structural Engineering, 1 (1997) 7185. [22] Y.C. Loo, H. Guan, Cracking and punching shear failure analysis of RC flat plates, Journal of Structural Engineering, 123(10) (1997) 13211330. [23] K.E. Løland, Continuous damage model for loadresponse estimation of concrete, Cement and Concrete Research, 10(3) (1980) 395402. [24] J. Mazars, A description of microand macroscale damage of concrete structures, Engineering Fracture Mechanics, 25(5) (1986) 729737. [25] J.G. Rots, P. Nauta, M.A. Kusters, J. Blaauwendraad, Smeared crack approach and fracture localization in concrete, HERON, 30(1) (1985) 148. [26] J.G. Rots, J. Blaauwendraad, Crack models for concrete: discrete or smeared? Fixed, multidirectional or rotating?, HERON, 34(1) (1989) 159. [27] Z.P. Bazant, Size effect in blunt fracture: concrete, rock, metal, Journal of Engineering Mechanics, 110(4) (1998) 518535. [28] M. Jir¨¢sek, M. Bauer, Numerical aspects of the crack band approach, Computers & Structures, 110 (2012) 6078. [29] X.L. Lu, Y. Zhou, J.H. Yang, J. Qian, C. Song, Y. Wang, Shear Wall Database, Network for Earthquake Engineering Simulation (database), Dataset, 2010. [30] Q. Chen, Static inelastic analysis of reinforced concrete coupled shear wall, Ph.D. Thesis, Tsinghua University, Beijing, China, 2002. [31] X.L. Lu, Y.T. Chen, Modeling of coupled shear walls and its experimental verification, Journal of Structural Engineering, 131(1) (2005) 7584. [32] D.S. Huang, W.X. Cheng, Study on elastoplastic performances of shear walls with short piers, Journal of Southeast University (Natural Science Edition), 33(2) (2003) 164167. [33] J.B. Mander, M.J. Priestley, R. Park, Theoretical stressstrain model for confined concrete, Journal of Structural Engineering, 114(8) (1988) 18041826. [34] F.C. Filippou, E.P. Popov, V.V. Bertero, Effects of bond deterioration on hysteretic behavior of reinforced concrete joints, Report EERC 8319, Earthquake Research Center, University of California, Berkeley, 1983. [35] E. Spacone, F.C. Filippou, F.F. Taucer, Fiber beamcolumn model for nonlinear analysis of R/C frames: Part I. Formulation, Earthquake Engineering and Structural Dynamics, 25(7) (1996) 711726. [36] P. Martinelli, F.C. Filippou, Simulation of the shaking table test of a sevenstory shear wall building, Earthquake Engineering & Structural Dynamics, 38(5) (2009) 587607. [37] B.F. Maison, K. Kasai, G. Deierlein, ASCE41 and FEMA351 Evaluation of EDefense collapse test, Earthquake Spectra, 25(4) (2009) 927953. [38] A. Kidarsa, M.H. Scott, C.C. Higgins, Analysis of moving loads using forcebased finite elements, Finite Elements in Analysis and Design, 44 (2008) 214224. [39] A. Zona, G. Ranzi, Finite element models for nonlinear analysis of steelconcrete composite beams with partial interaction in combined bending and shear, Finite Elements in Analysis and Design, 47 (2011) 98118. [40] J. Moehle, Y. Bozorgnia, N. Jayaram, Case Studies of the Seismic Performance of Tall Buildings Designed by Alternative Means: Task 12 Report for the Tall Buildings Initiative: Final Report to California Seismic Safety Commission and California Emergency Management Agency, Pacific Earthquake Engineering Research Center, 2011. [41] X.Z. Lu, M.K. Li, H. Guan, X. Lu, L.P. Ye, A comparative case study on seismic design of tall RC framecore tube structures in China and USA, The Structural Design of Tall and Special Buildings, 2015. DOI: 10.1002/tal.1206. [42] URL: http://www.luxinzheng.net/download/OpenSeesTHU.zip; Date Accessed: June 2014. [43] JGJ 32010, Technical specification for concrete structures for tall building, Beijing: Ministry of Housing and UrbanRural Development of the People's Republic of China, 2010 [in Chinese]. [44] A.K. Chopra, Dynamics of structures, Vol. 3. New Jersey: Prentice Hall, 1995. [45] M.E. Rodr¨ªguez, J.I. Restrepo, J.J. Bland¨®n, Shaking table tests of a fourstory miniature steel buildingmodel validation, Earthquake Spectra, 22(3) (2006) 755780. [46] P. Martinelli, F.C. Filippou, Simulation of the shaking table test of a sevenstory shear wall building, Earthquake Engineering and Structural Dynamics, 38 (2009) 587607. [47] D.G. Lignos, Y. Chung, T. Nagae, M. Nakashima, Numerical and experimental evaluation of seismic capacity of highrise steel buildings subjected to long duration earthquakes, Computers & Structures, 80 (2011) 959967. Appendix A. Command lines to implement the multilayer shell element in OpenSees The twodimensional concrete material is introduced to the model through the following command lines as:
The reinforced steel material for the multilayer shell element is introduced to the model through a command line as:
The multilayer shell section is defined by the following command line as:
