Keywords

1 Introduction

Reinforced concrete (RC) shear walls play an essential role in lateral load resistance of high-rise buildings, e.g., wind or earthquake loads. Therefore, it is of significant importance to accurately and efficiently simulate the nonlinear behavior of the RC shear walls during the seismic or wind analysis for tall buildings. In the past few decades, great efforts have been taken and various models have been developed for this purpose. These models can be classified into two categories, namely, micro-models and macro-models. The micro-model studies the constitutive behaviors of the RC share wall at material level (e.g., at Gauss point of a finite element), and the cracked concrete is simulated using continuous smeared-crack models; while the macro-models simulate the RC shear wall using a set of discrete components, e.g., trusses, frames and springs. Among the micro-models, the compression field theory (CFT) [1] and modified compression field theory (MCFT) [2] are widely used, in which the shear resistance is provided by concrete struts along the principal stress directions (i.e., the crack grow direction and its normal direction), and the compressive strength of the concrete struts deteriorates with the increasing tensile strain normal to the them for simulating the strength softening effect. Another set of widely used micro-models are presented by Hsu and coworkers, including the rotating-angle softened truss models (RA-STM) [3,4,5,6], the fixed-angle softened truss model (FA-STM) [7,8,9], the softened membrane model (SMM) [10, 11] and the cyclic softened membrane model (CSMM) [12,13,14,15]. These models assume that the direction of cracks is perpendicular to the direction of the principal tensile stress of concrete (RA-STM) or of the reinforced concrete (FA-STM, SMM, CSMM), while the state-dependent Poisson’s effect (i.e., transformation between the biaxial strains and uniaxial strains) is ignored (RA-STM, FA-STM) or studied in great detail using a time-varying Hsu/Zhu ratio (SMM, CSMM). The current micro-models are able to accurately capture the complicated nonlinear behaviors (e.g., stiffness reduction, strength deterioration and pinch effects) of the shear walls, however cannot guarantee the numerical stability of the Newton-Raphson (NR) process when the models are used in large scale nonlinear analysis of realistic tall buildings. Particularly, the iterations used to find the crack grow directions (i.e., the principal stress direction) usually are computational expensive, therefore reducing the computational efficiency.

The second class of models is macro-models. Among these models, the three-vertical-line-element model (TVLEM) [16] simulates the bending and shearing behavior of shear panels using three vertical line (truss) elements, one horizontal truss element and one rotational spring element between two infinitely-rigid beams at top and bottom floor levels. The multi-vertical-line-element model (MVLEM) [17] employs a set of vertical truss elements and a horizontal spring element connecting with two rigid beams at the top and bottom floor levels. The panel-wall macro element (PWME) [18] modifies the TVLEM by replacing the horizontal and rotation truss elements with a plane element. The truss model (TM) [19] discretizes the RC shear wall components into a set of horizontal, vertical and diagonal truss elements, and the strength softening of the diagonal trusses is considered in the similar way as MCFT. Panagiotou and coworkers [20, 21] presented a cyclic beam-truss model for nonplanar reinforced concrete walls, in which a set of nonlinear Euler-Bernoulli fiber-section beam elements are used to model the mechanical behaviors along the vertical direction and nonlinear trusses for horizontal and diagonal directions (the softening effect is simulated similarly with the TM). The macro-models can simulate the nonlinear behavior of shear walls with reasonable accuracy and meet the requirement of numerical stability for nonlinear analysis, therefore is able to be used for large scale nonlinear analysis of high-rise RC buildings. However since the truss and beam components are not physically based, it is not easy to build the FE model and determine the model parameters (e.g., determine the locations and areas of the trusses). Furthermore the micro behavior (e.g., micro-cracking or strain localization) cannot be simulated accurately unless the mesh size is small enough, which may lead to prohibitive computational cost.

Therefore, in spite of the various RC shear wall models in literatures, it remains a challenging task to accurately, efficiently and robustly simulate the complicated nonlinear behaviors of RC shear walls in super high-rise buildings. In this background, the paper presents a novel practical shear wall model, i.e., multi-cross-line model (MCLM), that is accurate, efficient and stable, providing a practically ideal solution for such challenging problems. The MCLM combines the concepts of micro and macro models by simulating the stress-strain relationship at “micro” material level (e.g., at a Gauss point) using a set of “macro” concrete bars or steel bars, and the two-dimensional (2D) membrane stress resultants at a material point are computed using an “equivalent stress” based on the forces of the bars. Furthermore, in order to apply MCLM to realistic three dimensional (3D) problems, a modified shell element is presented that simulates the in-plane nonlinear shearing behavior using the MCLM and out-plane force-deformation responses using linear elastic models. The newly developed MCLM has been implemented into a general finite element software platform, OpenSees (Open System for Earthquake Engineering Simulation), an open source finite element analysis software used to model structural systems and simulates their earthquake responses [22,23,24]. The efficiency and numerical stability of the newly developed MCLM is validated by using two RC shear wall specimens tested in laboratory and a realistic 32-story high-rise RC frame-shear wall building under cyclic loading conditions.

The novel MCLM has distinguishing features particularly for nonlinear seismic analysis of super high-rise buildings: (1) It is robust. The Newton-Raphson (NR) process using MCLM converges systematically, demonstrating the good numerical satiability of the model. (2) It is accurate and efficient. The main features of shear-flexural behavior of RC shear walls (e.g., stiffness reduction, strength deterioration and pinch effects) can be simulated with reasonable computational cost. (3) It is practical. The model setup and parameter selection are relatively easy, making the model practically useful in large scale nonlinear analysis of realistic super high-rise buildings.

2 The Multi-Cross-Line Model (MCLM)

2.1 Model Assumptions

The MCLM presented herein is a 2D plane stress model based on following assumptions:

(i) The 2D membrane stress resultant is computed by combining stresses from two uncoupled layers, i.e., a concrete layer and a reinforcing steel layer. The slippage between reinforcing bars (i.e., rebars) and concrete is not considered (see Fig. 1a). (ii) For the concrete layer, the shear stress is modeled by two physical diagonal concrete bars perpendicular to each other (Fig. 1b). The compressive strength of one concrete bar is reduced, or “softened”, as the tensile strain of the other bar (normal to this bar) increases. The horizontal and vertical normal stresses are computed independently based on the forces of a horizontal and a vertical concrete bars, respectively. (iii) The 2D stress of steel layer is computed from the forces of the rebars using a 1D smeared steel material model.

Fig. 1.
figure 1

The multi-cross-line model

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The MCLM can be considered as a modification or extension of existing micro models (e.g., CSMM model), however employs a set of “physically existing” discrete concrete bars or steel bars to get equivalent shear and normal stresses. The diagonal bars contribute only to shear resistance (i.e., have no contribution to normal stress), while the horizontal and vertical bars contribute to both normal stresses and shear stress. It is important to mention that the MCLM based on the above assumptions has distinguished features by combining the advantages of existing micro and macro models. Firstly, it has an advantage in macro models, i.e., the numerical stability. The MCLM employs four cross concrete bars to simulate concrete behaviors. The concrete bars do not rotate, therefore it avoids the rotation-induced stability problems, e.g., the possible non-convergence of NR process because that the principal stress angle changes sigh when close to horizontal or vertical directions, or because that the difference of the principal directions between the previous time step and current trial step is finite which leads to an abrupt jump of stress. Secondly, the MCLM is able to simulate the main features of shear-flexural behavior of RC shear walls (e.g., stiffness reduction, strength deterioration and pinch effects). The ‘strength softening’ is simulated using two coupled diagonal concrete bars, and the relationship between normal and shear stresses are considered explicitly using a correction factor. In addition, the MCLM can simulate the nonlinear behaviors or local damages of shear walls using moderate-fine mesh and at an acceptable computational cost, therefore is practically efficient. Thirdly, MCLM is a micro model built at material level describing the stress-strain relationships (as opposed to the macro model using a set of discretized beam or truss components), the FE model based on MCLM is relatively easy to build and the model parameters can be directly determined by experiments. There is no need to specify the parameters for each discrete component as in macro models.

2.2 Model Descriptions

According to the above assumptions, the stress resultant at a material (Gauss) point of a RC shear wall (i.e., the horizontal, vertical and shear stresses [\( \sigma_{l} \) \( \sigma_{t} \), \( \tau_{lt} \)] in Fig. 1(a) is calculated from two uncoupled layers, i.e., the stresses of reinforcing steel layer (i.e., [\( \sigma_{l}^{s} \), \( \sigma_{t}^{s} \), \( \tau_{lt}^{s} \)] in Fig. 1(b) and the concrete layer (i.e., [\( \sigma_{l}^{c} \), \( \sigma_{t}^{c} \), \( \tau_{lt}^{c} \)] in Fig. 1(c). The subscripts l and t denote the horizontal and vertical directions respectively, and the superscripts s and c denote the steel and concrete respectively. For the reinforcing steel layer, the stress is computed by transforming the stress of uniformly distributed reinforcement calculated using 1D smeared steel material model (i.e., \( \rho_{1} f_{{\theta_{s1} }}^{s} \) and \( \rho_{2} f_{{\theta_{s2} }}^{s} \) in Fig. 1b, where \( \rho_{i} \) and \( f_{{\theta_{si} }}^{s} \) denote, respectively, the reinforcement ratio and stress along the ith direction, i = 1, 2). For the concrete layer (Fig. 1d), the shear stress \( \tau_{lt}^{c} \) (Fig. 1e) is calculated using two cross diagonal bars perpendicular to each other and having 45 degree with horizontal, which are assumed to have no contribution to the normal stresses (i.e., setting \( \sigma_{t}^{c'} = \sigma_{l}^{c'} = 0 \) in Fig. 1e). The compressive strength of one diagonal bar is reduced by tensile strain of the other bar. The normal stresses are calculated using two uncoupled concrete bars along normal directions (i.e., \( \sigma_{t}^{c} \) and \( \sigma_{l}^{c} \) in Fig. 1f).

2.3 Constitutive Equation of the MCLM

The 2D membrane stress resultant is computed by combing the stresses of concrete layer and reinforcing steel layer (see Fig. 1a–c), i.e.,

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{l} } \\ {\sigma_{t} } \\ {\tau_{lt} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\sigma_{l}^{s} } \\ {\sigma_{t}^{s} } \\ {\tau_{lt}^{s} } \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {\sigma_{l}^{c} } \\ {\sigma_{t}^{c} } \\ {\tau_{lt}^{c} } \\ \end{array} } \right\} \, $$
(1)

The stress of steel layer: the stress of steel can be computed from the stress of each steel bar using the uniform distribution assumption, as,

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{l}^{s} } \\ {\sigma_{t}^{s} } \\ {\tau_{lt}^{s} } \\ \end{array} } \right\} = {\mathbf{T}}\left( { - \theta_{s1} } \right)\left\{ {\begin{array}{*{20}c} {\rho_{1} f_{{\theta_{s1} }}^{s} } \\ 0 \\ 0 \\ \end{array} } \right\} + {\mathbf{T}}\left( { - \theta_{s2} } \right)\left\{ {\begin{array}{*{20}c} {\rho_{2} f_{{\theta_{s2} }}^{s} } \\ 0 \\ 0 \\ \end{array} } \right\} \, $$
(2)

where \( {\mathbf{T}}\left( {\theta_{si} } \right) \) (i = 1, 2) is the coordinate transformation matrix, and \( \theta_{si} \) (i = 1, 2) is the angle of the direction of the ith reinforcing steel with respect to horizontal direction (\( l - \) direction). The parameter \( \rho_{i}^{{}} \) (i = 1, 2) in Eq. (2) is the reinforcement ratio along the i th steel direction, and \( f_{\theta ,i}^{s} \) (i = 1, 2) is the stress of steel. A smeared uniaxial steel material model in literature (Mansour et al. [28]) is employed to compute the stress \( f_{\theta ,i}^{s} \) of the reinforcing bar based on the strain along the i th direction (i.e., \( \varepsilon_{{\theta_{si} }}^{s} \)), which can be obtained by transforming the strain at the Gauss point (i.e., [\( \varepsilon_{l} \), \( \varepsilon_{t} \), \( \gamma_{lt} \)]), as,

$$ \varepsilon_{{\theta_{si} }}^{s} \, = \, \left[ {\begin{array}{*{20}c} 1 & 0 & 0 \\ \end{array} } \right]{\mathbf{T}}\left( {\theta_{si} } \right)\left\{ {\begin{array}{*{20}c} {\varepsilon_{l} } & {\varepsilon_{t} } & {{{\gamma_{lt} } \mathord{\left/ {\vphantom {{\gamma_{lt} } 2}} \right. \kern-0pt} 2}} \\ \end{array} } \right\}^{T} $$
(3)

The stress of concrete layer: the two diagonal bars contribute to shear stress only, therefore,

$$ \left\{ {\begin{array}{*{20}c} {\sigma_{l}^{c} } \\ {\sigma_{t}^{c} } \\ {\tau_{lt}^{c} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \zeta \\ \end{array} } \right]{\mathbf{T}}\left( { - \frac{\pi }{4}} \right)\left\{ {\begin{array}{*{20}c} {f_{1}^{c} } \\ {f_{2}^{c} } \\ 0 \\ \end{array} } \right\} + \left\{ {\begin{array}{*{20}c} {\sigma_{l}^{c} } \\ {\sigma_{t}^{c} } \\ 0 \\ \end{array} } \right\} $$
(4)

Where \( f_{1}^{c} \) and \( f_{2}^{c} \) are the stresses of the concrete bars along two diagonal line directions, respectively. It is worth mentioning that the stresses of the two diagonal bars (i.e., \( f_{1}^{c} \) and \( f_{2}^{c} \)) are assumed to have no contribution to the normal stress (i.e., \( \sigma_{l}^{c} \) and \( \sigma_{t}^{c} \)) since the normal stresses are computed independently by using the horizontal and vertical concrete bars. The diagonal stress \( f_{1}^{c} \) and \( f_{2}^{c} \) are computed from the strains of the two diagonal concrete bars (i.e., \( \varepsilon_{1}^{c} \) and \( \varepsilon_{2}^{c} \)).

In this paper, a correction factor \( \zeta \) is introduced (in Eq. 5) to consider the effective area of the bar and influence of the vertical compression on the shear strength, which is defined as a function of the maximum vertical compressive strain (i.e., \( \varepsilon_{{\rm max }\, c} \)), i.e.,

$$ \zeta = \left\{ {\begin{array}{*{20}l} {k_{1} } \hfill & {\varepsilon_{{{\rm max }\, c}} \ge 0} \hfill \\ {k_{1} \exp \left( { - k_{2} \varepsilon_{{{\rm max }\, c}} } \right) \le 1} \hfill & {\varepsilon_{{{\rm max }\, c}} < 0} \hfill \\ \end{array} } \right. $$
(5)

where k 1 and k 2 are suggested to take 0.25 and 1500 in this paper. The value k1 is the correction due to the area reduction for the diagonal concrete bars. Considering that the MCLM combines the micro and macro models, i.e., simulating the stress-strain relationship by using macro concrete bars, the area reduction approach used in macro models need to be applied, i.e., the equivalent section area of the diagonal concrete bar resisting the shear stress is less than the diagonal section areas. This area reduction is considered in parameter k1. On the other hand, the term \( \exp \left( { - k_{2} \varepsilon_{{{\rm max }\, c}} } \right) \) in Eq. (5) considers the shear strength change due to the vertical compressive forces of the concrete. When the vertical forces (mainly caused by the self-weight) increase, the confinement grows and shear strength enlarges. The relationship between correction factor \( \zeta \) and maximum vertical compressive strain (for k1 = 0.25 and k2 = 1500) is plotted in Fig. 2.

Fig. 2.
figure 2

The correction factor for area reduction and compression effect

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Herein, a smeared concrete model [25] is used to model the nonlinear behavior of the concrete bars.

3 Application Examples

The newly developed MCLM has been implemented in a general finite element software framework, OpenSees. two application examples are used to validate the accuracy and numerical stability of the MCLM, including a RC shear wall specimen test (a flexural-shear specimen test) subjected to quasi-static cyclic loading conditions and a 32-story RC frame-shear wall building structure subject to seismic cyclic loading conditions.

3.1 Application Example 1

The second example is used to validate the MCLM under flexure-shear deformation. A RC frame shear wall specimen subjected to both vertical loading and horizontal pushover forces [26, 27] is re-examined as shown in Fig. 3. The frame elements (Fig. 3b) are modeled using nonlinear displacement-based Euler-Bernoulli elements and fiber sections with uniaxial concrete and steel material models (as shown in Fig. 3c). The constitutive behaviors of the unconfined and confined concretes are modeled using modified Kent-Park-Scott models (Park et al. 1982, see Fig. 3e), where the parameters of unconfined concrete are: compressive peak strength f c  = −42.8 MPa, ε 0  = −0.002, f u  = −8.2 MPa, ε u  = −0.004, f t  = 2.1 MPa, b = 0.1 and E t  = 3.0e3 MPa, while for confined concrete: f c  = −55.6 MPa, ε 0  = −0.006, f u  = −40.8 MPa, ε u  = −0.028, f t  = 2.6 MPa, b = 0.1 and E t  = 3.0e3 MPa. The constitutive behavior of the steel is modeled using a Giuffro-Menegotto-Pinto model (Menegotto and Pinto 1973, see Fig. 3f), where the elastic modulus E = 2.1e6 MPa, strength f y  = 395.2 MPa, and post to initial stiffness ratio b = 0.02. The top frame is modeled using rigid beam element, i.e., the elastic beam element with large elastic modulus. The shear wall is simulated using four-node quad elements with four Gauss points per element. The nonlinear behavior at each Gauss point is modeled using the newly developed MCLM model presented herein (see Fig. 3d). In the steel layer, the constitutive behaviors of reinforcement are simulated using a uniaxial reinforcing steel model [28] with elastic modulus E = 2.1e6 MPa and strength f y  = 448.2 MPa. The steel ratios in longitudinal and vertical directions are 0.32 and 0.32%, respectively. In the concrete layer, the modified smeared-crack concrete model used in MCLM has compressive strength f c  = −47.6 MPa and ε 0  = −0.0025. The strength softening is considered for the diagonal concrete bars.

Fig. 3.
figure 3

The RC frame shear wall specimen in Example 2. (a) Geometry and reinforcement (b) FE model (c) Frame element and section properties (d) Quad element and section properties (e) Concrete constitutive model (f) steel constitutive model

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After applying a uniformly distributed axial load of 378 kN (i.e., the axial compression ratio is equal to 0.07), a static pushover analysis is performed using displacement control method with the displacement increment of 0.1 mm. A tolerance of 1.0e−3 mm on the norm of the incremental displacement is used as the convergence criterion. The detail loading paths refer to [27].

The calculated hysteretic responses between the top displacement and the base shear force are plotted and compared with the experimental results to evaluate the accuracy of MCLM (see Fig. 4). The MCLM can simulate the nonlinear behavior of the shear wall under cyclic loading conditions, e.g., the stiffness degradation, strength reduction and the pinch effects, with reasonable accuracy. The Newton-Raphson iterative algorithm converging quadratically demonstrates the good stability of the MCLM presented herein. It is worth to mention that based on the study of many other experimental results using MCLM, it can be concluded that the numerical errors in term of compressive strength is usually less than 20%, and the numerical stability of the MCLM is practically acceptable.

Fig. 4.
figure 4

Hysteretic curve between shear force and top displacement of the slender wall

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3.2 Application Example 2: A Realistic 32 Story High-Rise Building

In order to verify the applicability of the MCLM in realistic tall building, a 129-m 32-story high-rise RC frame-shear wall building following GB 50011-2010 Code for Seismic Design of Buildings [29] (with seismic fortification intensity larger than 8 degree) subject to earthquake base excitation is studied. The height is 5.0 m in the first story and 4.0 m in the other stories. Both the length and width of the building are 48 m, and remain unchanged with altitude. The building consists of a core tube RC shear wall, outer steel-reinforced concrete frames, outer steel beam and steel braces. The details of component dimensions properties of the steel braces and beams, outer mega columns, shear walls, coupling beams and material properties refer to [30].

A FE model with 1920 nodes and 5696 elements is used to simulate the nonlinear seismic behavior of this tall building. Each shear wall in the core tube is modeled by one modified shell element using MCLM for modeling the in-plane behavior and linear elastic model for the out-plane behaviors. After these loads are applied to the building using a quasi-static analysis, the 1940 EI-Centro earthquake record scaled by 0.19 (the peak ground acceleration, or PGA, is equal to 0.4 g) is employed as the base excitation input of the structural system. The Newmark-β integration scheme is employed with the parameters γ = 0.5 and β = 0.25, and a constant time step of 0.02 s is used. Rayleigh damping is applied to the system such that the damping ratio corresponding to the first frequency and third frequency is 0.025. A tolerance of 1.0e−5 mm on the norm of the incremental displacement is used as the convergence criterion. Figure 5 shows the displacement time histories of three representative floors and the magnitude of top displacement is 884.4 mm.

Fig. 5.
figure 5

Displacement time histories of three representative stories

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The maximum drift ratio envelope of the building during the earthquake is illustrated in Fig. 6. The maximum drift ratio is 0.88%, occurring at the 12th floor. Form the Fig. 6, it is clear that the maximum drift ratio is below the limitation of drift ratio (i.e., 1%) according to the GB 50011-2010 Code for Seismic Design of Buildings.

Fig. 6.
figure 6

Maximum drift ratio envelope of the building during the seismic load

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Figure 7 shows the hysteretic relationship of the base shear force of frame shear wall, outer frame and shear wall with top displacement of the building. From this figure, it is observed that the magnitude of lateral resisting force of the frame shear wall, the frame and the shear wall are 1.883e4, 2.5504e5 and 2.5548e5 kN respectively. It is observed that the lateral resisting force is mainly contributed by the core shear wall. The maximum and minimum top displacements are 583.1 and −884.4 mm, respectively.

Fig. 7.
figure 7

Base shear forces of the frame-shear wall, the frame and the shear wall vs. the top displacement

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The hysteretic responses of the shear forces versus drift ratios for representative shear walls at 1st, 12th, 20th and 30th floors are studied and plotted in Fig. 8 (a) to (d) respectively. It is observed that the maximum shear forces occur at bottom of the structure and the shear forces decrease with altitude. The drift ratios initially increase and then decrease with altitude, and the maximum drift ratio occurs at around 12th floor. Furthermore, the NR process using MCLM and modified shell walls exhibits good stability and converges systematically, indicating that the newly developed shear wall models (i.e., MCLM and modified shell element) are able to be applied to realistic complicated high-rise buildings.

Fig. 8.
figure 8

Shear force versus drift ratio of shear walls in various floors (a) 1st floor (b) 12th floor (c) 20th floor (d) 30th floor

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4 Conclusions

This paper presents a novel RC shear wall model, referred to as multi-cross line model (MCLM) for nonlinear seismic analysis of high-rise buildings. The MCLM combines the advantages of accuracy in the existing micro models and the numerical stability of the macro models, therefore providing a practically ideal solution for nonlinear seismic analysis of tall buildings. The newly developed MCLM has been implemented into a general finite element software platform, OpenSees, and validated by using two RC shear wall specimens tested in laboratory and a realistic 32-story high-rise RC frame-shear wall building under cyclic loading conditions. The MCLM can accurately simulate the experimental results, simulating the main features of the stiffness reduction, strength deterioration and pinch effects. In addition, the Newton-Raphson (NR) process using MCLM converges systematically, demonstrating the numerical satiability of this model. The MCLM provides a practically important simulation tool for nonlinear analysis of realistic high-rise buildings.