Seismic Assessment of RC Shear Wall Structure under Real Seismic Sequences Using Equivalent Shell-Wire Model

Abstract

Shear walls are commonly used in areas with moderate to high seismic hazards. They are cost-effective, simple, and significantly reduce the relative displacement. Accurate non-linear assessments of shear wall structures require highly detailed models and finite element (FE) analyses. The inclusion of analyses on the cumulative damage and damage evolution like seismic sequences adds a higher level of complexity to the assessments. Here, an equivalent shell-wire model is used to propose a simple, accurate technique with less computational cost. The areas under the capacity curves were modeled using 3D solid and equivalent shell-wall models and revealed only a 4.48% difference with a 50% difference in computational cost. The hysteresis curves of the experimental and equivalent shell-wire models showed good agreement. To examine the cumulative damage, damage evolution, strength and stiffness degradation of the model, a six-story structure was designed and analyzed under seven mainshocks-aftershocks in both cases of 3D solid and equivalent shell-wire models. The results showed that increasing the plastic strain or damage increased the difference between the results of the models, but this was negligible. Besides the accurate and appropriate results of the equivalent shell-wire model, it required about 39% less computational cost than the 3D solid model.

1. Introduction

Concrete shear walls are reinforced vertical structural elements that are designed to resist lateral forces, increase structural stiffness, and prevent high inter-story drift ratios [1]. Because of the relatively simple construction procedure, high seismic performance, high energy dissipation, and ductile behavior, shear wall systems have become popular for use in mid- to high-rise structures in cities with moderate-to-high seismic hazards [2, 3]. This makes accurate seismic behavior by shear-wall structures of importance.

Shear walls in structural systems should be able to tolerate either axial or lateral loads to increase the structural system performance [4]. Study of the seismic behavior of such structures in past earthquakes has shown that they are susceptible to flaws in the design procedure. In cases such as in Nicaragua (1972), Mexico (1985), and Armenia (1988), the lack of an appropriate design caused shear-wall collapse [2, 5]. Shear walls should exhibit ductile behavior under large structural forces without failing [6]. Design problems, the location of the shear walls, and openings in the plans as well as construction problems can decrease the efficiency of the shear walls under seismic loads and should be accurately considered during the design [7].

The behavior of shear wall structures under seismic loading has been investigated in a variety of studies. However, these generally have focused on single-degree-of-freedom (SDOF) systems or 2D frames of low height. Generally, 3D structures have been neglected because of the complexity of analysis, design factors, and the computational cost (simply, the computational cost is the time required to perform an analysis) [8, 9, 10].

Experimental studies also have usually been limited to samples having low heights using simple cyclic tests [11]. This could be because of the complex behavior, size of the shear-wall elements, and the interactions between the concrete, steel rebars, shear wall, and main structure in the 3D solid models that increase the modeling complexity and computational cost. In some previous studies, structures have exhibited different seismic behaviors under seismic sequences depending on the use of 2D or 3D modeling; thus, the 3D models have been preferred [12]. The lack of an appropriate technique for modeling shear wall structures at less computational cost with accurate results is evident. This study used an equivalent shell-wire technique to decrease the computational cost in the 3D modeling of shear-wall structures having high accuracy.

A mainshock is usually accompanied by a group of ground motions. Such aftershocks can increase structural damage and cause failure and even building collapse [13, 14]. The causes for these include damage evolution, damage accumulation, strength, and stiffness degradation under seismic sequences or mainshock-aftershocks. Many studies have evaluated the effect of the seismic sequences on the structures; however, because of the high computational cost and analysis complexity, they usually have been limited to short structures or 2D models.

Studying the seismic behavior of the shear wall structures under seismic sequences could be a big challenge that required high computational costs. The complexity of the structural elements and the shear wall, the size of the shear wall elements and their interactions as well as consideration of the cumulative damage, damage evolution, strength, and stiffness degradation under seismic sequences increases the complexity of studying the seismic behavior of shear wall structures. The shell-wire model is a possible alternative to reducing the computational cost of such analyses. To evaluate the accuracy of the shell-wire technique over 3D solid models, it was compared with a full-scale experimental model under cyclic loading. This model also was investigated by design analysis of a six-story medium-height shear wall structure under real seismic sequences.

An alternative analytical model with appropriate accuracy (equivalent shell-wire model) is presented to analyze shear-wall structures. This model can consider damage evolution, cumulative damage and strength and stiffness degradation under arbitrary dynamic loading at a less computational cost compared to usual analytical models such as 3D solid models. The approach makes it possible to analyze taller structural models of greater complexity or with finer mesh to increase accuracy compared to a 3D solid model. Figure 1 lists the aspects of this chapter.

Figure 1.

Research strategy.

2. General equations, laws, explanations, and theoretical basis

2.1 3D shell element

It is possible to model structural elements using 3D solid or shell elements; however, the selection between 2D or 3D shell elements depends on the relative thickness of the structure. In other words, the third dimension could be neglected. Although 3D elements can be suitable for any geometry, for shell elements, the ratio of the thickness to the other dimensions should not be greater than 1 to 10 or less than 1 to 1000 [15]. For shear walls, the ratio of the thickness to height and width is the same for shell elements; thus, it is possible to consider the shear walls as shells or 3D solid elements.

The selected shell element was a 3D shell with a mesh element type of 4-node doubly curved general-purpose shell, reduced integration with hourglass control, and finite membrane strains (S4R) [16, 17]. This is a robust quadrilateral element for general-purpose applications and is suitable for thick and thin walls depending on the thickness of the elements. Each node has three degrees of freedom, as for 3D solid elements (Figure 2). It should be noted that because of the equality of the degree of freedom, it is easy to combine and tie a 3D shell (shear wall) and 3D solid elements (beams and columns). Thus, the 3D shell was selected.

Figure 2.

The S4R element and integration points.

2.2 Wire element

Concrete shear walls contain concrete and rebars. In order to reduce the computational cost, the steel rebars were modeled as 3D wire elements. Ideally, wire elements are used when both their thicknesses and depths are considered to be small compared to their lengths. A 2-node linear 3D truss mesh element (T3D2) was considered for the wires that only support axial loading along the element [18]. It has been assumed that the steel rebars do not support the moments or forces perpendicular to the centerline.

It should be noted that because of the degree of freedom equality (shell shear wall, 3D solid columns, and beams), the 3D wire element was selected.

2.3 Shell-wire model

As the goal was the analysis of shear walls at less computational cost, the alternative model should be able to consider the cumulative damage, damage evolution, strength, and stiffness degradation under arbitrary loads. These results must be in good agreement with the complex 3D solid and experimental models. The equivalent shell-wire model considers shear walls as a combination of a shell (concrete) and wires (rebars) and uses equivalent concrete and rebars to provide accurate results and decrease the computational cost. Figure 3a and b show a simple 3D solid model and an equivalent shell-wire model, respectively.

Figure 3.

Shear wall section: (a) 3D solid model; (b) equivalent shell-wire model scheme.

2.4 Finite element analysis

The type of time integration method and material modeling have an intensive effect on the accuracy of the analysis [19]. For an accurate dynamic analysis, the time integration method should be chosen in accordance with the numerical dissipation, dispersion, wave propagation, excitation frequency content, convergence, desirable behavior and accuracy of the model, and computational cost [16]. Considering the frequency content of the excitation (earthquake acceleration in this case), desirable accuracy, and structural models the implicit dynamic time integration with the application of moderate dissipation was chosen for this study. This analysis covers more general dynamic events with appropriate accuracy [16, 18].

ABAQUS software was used to perform dynamic implicit analysis. Nonlinear implicit time-history analysis has been employed based on the unconditionally stable Hilber Hughes Taylor (HHT) method, with appropriate time increments and a full Newton–Raphson solution [18, 20, 21]. The size of time increments is fully automatic and is calculated based on an algorithm, rate of plasticity, and convergence [18]. The HHT method is a second-order accurate and unconditionally stable method that showed appropriate accuracy and convergence in general dynamic problems [22]. The numerical implementation of the time integration method is explained in Eqs. (1)–(8) [18, 23, 24].

The concrete damage plasticity model was used to define and model the concrete behavior. This model is based on the assumption of isotropic damage and is an analytical model used for the analysis of materials with concrete-like properties subjected to arbitrary loads [18]. Also, this model has a good ability to consider the interaction between rebars and concrete like the current study.

It should be noted that the ability to accurately consider the damage and damage evolution is one of the characteristics of the material model (Section 2.5).

The answer for every chosen time step and at any desired degree of freedom can be obtained as:

$$\mathit{\Delta U}{\left|\tau ={\tau }^3\mathit{\Delta U}\right|}_{\mathrm{t}+\mathrm{\Delta t}}+\tau \left(1-{\tau }^2\right)\mathit{\Delta t}{\dot{U\mid }}_{\mathrm{t}}+{\tau }^2\left(1-\tau \right)\frac{\Delta t^2}{2}\ddot{U\mid }t$$E1

$${\dot{U\mid }}_{\mathrm{t}}=\frac{{\rm Y}}{\mathit{\text{βτΔt}}}\mathit{\Delta U}\mid \mathrm{\tau }+\left(1-\frac{{\rm Y}}{\beta }\right){\dot{U\mid }}_{\mathrm{t}}+\left(1-\frac{{\rm Y}}{2\beta }\right)\tau \mathit{\Delta t}\ddot{U\mid }t$$E2

$${\left.\ddot{U\mid }\tau =\frac{1}{\beta {\tau }^2\Delta t^2}\mathit{\Delta U}\right|}_{\mathrm{\tau }}-\frac{1}{\mathit{\text{βτΔt}}}{\dot{U\mid }}_{\mathrm{t}}+\left(1-\frac{1}{2\beta }\right)\ddot{U\mid }t$$E3

$$\mathit{\Delta U}\mid \mathrm{t}+\mathrm{\Delta t}=U\mid \mathrm{t}+\mathrm{\Delta t}-U\mid$$E4

$$\beta =\frac{1}{4}{\left(1-\alpha \right)}^2$$E5

$${\rm Y}=\frac{1}{2}-\alpha$$E6

$$-\frac{1}{2}\le \alpha \le 0$$E7

$$0⩽\tau ⩽1$$E8

where $\ddot{U}$ is acceleration in an implicit dynamic step, $\dot{U}$ is the velocity, and $U$ is the displacement for a degree of freedom.

2.5 Cumulative damage (damage evolution)

A comprehensive model should consider the damage, cumulative damage, damage evolution, strength, and stiffness degradation for the concrete and steel rebars. Herein, the damage and damage evolution was characterized by the degradation in strength and stiffness of the materials. The initial damage criteria of the material (concrete or steel rebars) depended on the specific strength. For ductile material (steel rebars), when the stress reached its limits, the evolution of the damage variable took an exponential form according to the fraction strain, strain rate, and displacement at failure [18, 25, 26, 27]. Experimental studies have shown that brittle material (concrete) under either compression or tension, the initial damage, damage evolution, or cumulative damage could be linked to the model provided by Lubliner et al. [28]. Thus, it was assumed that the compressive stiffness recovered entirely when the load changed from tension to compression. Under tensile loading, however, the stiffness did not recover because of the existence of small cracks [25, 26, 29, 30].

3. Experimental test

To investigate the accuracy of the shell-wire equivalent model, the capacity curve and cyclic behavior of a shear wall modeled using an equivalent shell-wire model were compared to the results of complex 3D solid and experimental models. The computational costs of the models also were compared. The accuracy of the 3D solid model has been previously investigated and the results have shown good agreement between the complex 3D solid and experimental models [25].

3.1 Capacity curve

The experimental shear wall used was the RW1 model as tested by Thomson and Wallace [11], the details of which are shown in Figure 4a. According to previous studies, there is a good agreement between the experimental test and the 3D solid model [25]. The capacity curves of the 3D solid model and the equivalent shell-wire were obtained using monotonic load and nonlinear FE analysis (∆target = 6.5 cm). Figure 4b shows the equivalent shell-wire model of the RW1 shear wall.

Figure 4.

RW1 model: (a) 3D solid and experimental models [25]; (b) equivalent shell-wire model scheme.

Figure 5 compares the accuracy of the equivalent shell-wire model by comparing the area under the displacement-base shear force curves (capacity curves), which are equal to the absorbed energy. The models (equivalent shell-wire and 3D solid models) differed by only 4.48%, which indicates good agreement between them. The equivalent shell-wire model required about 50% less time to complete this task, which suggests that the computational cost also decreased by 50% when using the equivalent shell-wire model compared to the 3D solid model. It should be noted that the size of the mesh elements was the same for both models.

Figure 5.

Capacity curve comparison. 3D solid model. Equivalent shell-wire model.

3.2 Cyclic behavior

The cyclic behavior of the equivalent shell-wire model was assessed to obtain a hysteresis loop using a displacement protocol. Hysteresis curves provide valuable information about the nonlinear behavior of the models under cyclic loading, including the strength and stiffness degradation [31, 32, 33]. Comparison of the hysteresis curves of the equivalent shell-wire model with the 3D solid and experimental models showed good agreement on both the compressive and tensile sides (Figure 6). This indicates that the equivalent shell-wire is an appropriate model that considers the strength and stiffness degradation under arbitrary loading.

Figure 6.

Hysteresis curve comparison. 3D solid model [25]. Equivalent shell-wire model. Experimental model [11].

4. Shear wall design

The accuracy of the equivalent shell-wire model, including damage evolution, cumulative damage, strength and stiffness degradation, and computational cost were investigated under seismic loading. To this end, a shear wall structure was designed and analyzed under mainshock-aftershock seismic sequences. The building was a six-story reinforced structure with a height of 18 m that was intended for residential use in an area with a high seismic hazard level. The shear walls were designed in accordance with the simple tension and compression boundary element method (Simple T & C) [25, 34, 35, 36].

The pier of the wall was designed for a factored axial force ($P_{f-\mathit{top}}$) and moment ($M_{\mathrm{f}-\mathrm{top}}$) based on the loading combinations. The applied moment and axial force can be converted to the tantamount force applied to the bottom of the boundary elements Eqs. (9)–(10) as:

$$P_{\mathit{\text{left}}-\mathit{top}}=\frac{P_{f-\mathit{top}}}{2}+\frac{M_{f-\mathit{top}}}{(L_p-0.5B_{1-\mathit{\text{left}}}-0.5B_{1-\mathit{\text{right}})}}$$E9

$$P_{\mathit{\text{right}}-\mathit{top}}=\frac{P_{f-\mathit{top}}}{2}-\frac{M_{f-\mathit{top}}}{(L_p-0.5B_{1-\mathit{\text{left}}}-0.5B_{1-\mathit{\text{right}})}}$$E10

where $L_p$ is the wall length and $B_{1-\mathit{\text{left}}}$ and $B_{1-\mathit{\text{right}}}$ are the width of the left and right boundary members, respectively.

Under tensile or compression, the area required for rebars can be calculated using Eqs. (11) and (12), respectively, as:

$$A_{\mathit{st}}=\frac{P}{{\mathrm{\varphi }}_s{f}_y}$$E11

$$A_{\mathit{sc}}=\frac{\frac{\mathit{Abs}\left(P\right)}{P_{max\mathit{\text{factor}}}}-{\mathrm{\varphi }}_c{f}_c^{\prime }{A}_g}{{\mathrm{\varphi }}_s{f}_y-{\mathrm{\varphi }}_c{f}_c^{\prime }}$$E12

where $P$ is either $P_{\mathit{\text{left}}-\mathit{top}}$ or $P_{\mathit{\text{right}}-\mathit{top}}$, $A_{\mathit{st}}$ is the area required for the steel reinforcements in the concrete design, ${\mathrm{\varphi }}_s$ is the steel resistance factor, $f_y$ is the steel yield strength, $P_{max\mathit{\text{factor}}}$ is defined by the shear wall design preferences (default is 0.80), ${\mathrm{\varphi }}_c$ is the concrete resistance factor for compression, $f_c^{\prime }$ is the concrete compressive strength, $A_g$ is the gross area and is equal to the total area (ignoring reinforcements), and $A_{\mathit{sc}}$ is the area required for steel reinforcements in the concrete design.

It should be noted that for negative values of $A_{\mathit{sc}}$, no compressive reinforcement is required; however, for all cases, the reinforcing ratio should be less than the maximum allowable ratio. Table 1 presents the details of the shear walls, boundary members, and equivalent shell-wire model scheme of the six-story structure.

Table 1.

Details of shear walls, boundary members, and equivalent shell-wire model scheme.

5. Seismic sequence selection

The seismic behavior of the shear wall structure in Section 4 was tested under seven real mainshock-aftershock seismic sequences. It is known that near-field and far-field ground motions have different effect on the seismic behavior of a structure [37]; thus, near-field mainshock-aftershock records were selected from the PEER database [38]. The seismic declustering method (independent-dependent) was used to choose the seismic sequences. In this method, the sequences were empirically determined using the data sequences from previous earthquakes and by measurement of the space–time history [39]. The magnitude of the mainshock should be greater than that of the aftershock and there are no limitations for peak ground acceleration (PGA) but the ratio of aftershock PGA to mainshock PGA of the sequences was considered to be less than 1 for this study [25, 26].

The shear wave velocity of the ground motion records agreed with the seismic design assumptions (375 < $V_s$ (m/s) < 750). The specifications of the mainshock-aftershock records are listed in Table 2.

No.	Event	Station	Date	Magnitude (${\mathit{M}}_{\mathit{W}}$)	PGA(g)
1	Friuli-Italy (Mainshock)	Tolmezzo	1976/05/06	6.5	0.35
	Friuli-Italy (Aftershock)		1976/05/07	5.2	0.11
2	Cape Mendocino (Mainshock)	Petrolia	1992/04/25	7.1	0.66
	Cape Mendocino (Aftershock)		1992/04/26	6.6	0.49
3	Mammoth Lakes (Mainshock)	Convict Creek	1980/05/25	6.6	0.42
	Mammoth Lakes (Aftershock)		1980/05/25	5.7	0.37
4	Imperial Valley (Mainshock)	El Centro Array #4	1979/10/15	6.5	0.32
	Imperial Valley (Aftershock)		1979/10/15	5.0	0.23
5	Northridge (Mainshock)	Newhall-Fire Station	1994/01/17	6.6	0.58
	Northridge (Aftershock)		1994/01/18	5.2	0.20
6	Northridge (Mainshock)	Rinaldi Receiving Station	1994/01/17	6.6	0.86
	Northridge (Aftershock)		1994/01/17	5.2	0.52
7	Whittier Narrows (Mainshock)	LA-Obregon Park	1987/10/01	5.9	0.42
	Whittier Narrows (Aftershock)		1987/10/01	5.2	0.34

Table 2.

Specifications of mainshock-aftershock records [25].

Note that an appropriate zero acceleration interval was considered between the mainshock and aftershock to prevent the undesirable vibrations [40].

6. Analytical results

6.1 Inter-story drift ratio

The inter-story drift ratio is a fundamental structural parameter that can be used for the study of seismic behavior of structural [41]. It presents valuable data that can be used for structural judgment and analysis. The use of seismic sequences made it possible to compare the inter-story drift ratio of the mainshock and mainshock-aftershock for both the 3D solid and equivalent shell-wire models of the six-story structure. The behavior of the 3D solid model under seismic sequences was previously investigated [25].

The use of seismic sequences made the cumulative damage easily recognizable during this process. Figure 7 shows the drift ratio for both the 3D solid and equivalent shell-wire models for both the mainshock and mainshock-aftershock. This section examined the accuracy of the equivalent shell-wire model under the uncertainty of cyclic loading (seven real mainshock-aftershock) while considering the cumulative damage, damage evolution, strength, and stiffness degradation.

Figure 7.

Comparison of inter-story drift ratio: (a) Friuli; (b) Cape Mendocino; (c) Mammoth Lakes; (d) Imperial Valley; (e) Northridge (Newhall-Fire Station); (f) Northridge (Rinaldi Receiving Station); (g) Whittier narrows.

As expected, the seismic sequences caused plastic strain as well as an increase in the drift ratio (cumulative damage), chiefly in story one [25, 26, 42]. Story one experienced the highest drift ratio and plastic strain and showed the greatest difference between the 3D solid model and equivalent shell-wire model results. On average, there was a difference of about 2% and 3% in the inter-story drift ratio between the 3D solid model and equivalent shell-wire models under the mainshock and mainshock-aftershock, respectively.

Note that the size of the mesh elements was the same for both models.

6.2 Changes in inter-story drift ratio (range of inter-story drift ratio)

The previous section (Section 6.1) showed that there is a good agreement between the inter-story drift ratios of the 3D solid model and the equivalent shell-wire model (under a 3% difference). It can be seen from Figure 7 that higher plastic strain (high inter-story drift ratio) increases the difference between the 3D solid model and the equivalent shell-wire model. Also, seismic sequences caused plastic strain growth (inter-story drift ratio growth) in some stories. So, it is possible to define the range of inter-story drift ratio that shows changes in drift ratio under mainshock (lower range) and mainshock-aftershock (higher range). This parameter would be valuable to examine the accuracy of the proposed equivalent shell-wire model under high plastic strain. In other words, the equivalent shell-wire model can consider accurately cumulative damage, strength, and stiffness degradation if there is a good agreement between the range of inter-story drift ratio (drift ratio of mainshock and mainshock-aftershock) of the 3D solid model and the equivalent shell-wire model.

Figure 7 shows that the calculated inter-story drift ratios for both the mainshock and mainshock-aftershock obtained from the 3D solid and equivalent shell-wire models were almost similar. The inter-story drift ratios changed (range of the inter-story drift ratio) under the mainshocks and mainshocks-aftershocks for the models, as can be seen in Figure 8. Story one recorded the highest inter-story drift ratio, plastic strain, and damage under the mainshocks and mainshocks-aftershocks; thus, this story was selected for Figure 8.

Figure 8.

Range of story one inter-story drift ratios for mainshock and mainshock-aftershock.

Figure 8 shows that, as the plastic strain and inter-story drift ratio increased, the difference between the results of models increased. For example, the ranges of the drift ratios for the Friuli mainshock and mainshock-aftershock for both models were similar (roughly 1% difference). However, for the Cape Mendocino mainshock-aftershock, there was about a 3.14% difference. Table 3 lists the differences in the mainshock-aftershock inter-story drift ratios of story one. As seen, there was a less than 4% difference between the 3D solid and the equivalent shell-wire model in all cases, which is not remarkable (2.44% on average). Thus, the equivalent shell-wire model provided the appropriate results.

1	2	3	4
Mainshock-aftershock	3D solid model (%)	Equivalent shell-wire model (%)	Difference bet. 3 and 2 (%)
Friuli	1.21	1.23	1.47
Cape Mendocino	1.85	1.91	3.14
Mammoth Lakes	1.42	1.46	2.43
Imperial Valley	1.60	1.64	2.44
Northridge (Newhall-Fire Station)	1.66	1.71	2.53
Northridge (Rinaldi Receiving Station)	1.90	1.97	3.66
Whittier Narrows	1.40	1.42	1.40

Table 3.

Inter-story drift ratio comparison of the story one.

6.3 Computational cost

The computational cost could be defended as “the amount of time required to complete certain operation” [43]. In FE analysis, the computational cost is the execution time per increment or the total time required to analyze a job (sum of execution time for all increments). This study tried to compare the computational cost of the 3D solid model and the equivalent shell-wire model under certain operations (under mainshock-aftershocks). Thus, the total time required to analyze the 3D solid model and equivalent shell-wire model under each seismic sequence was considered as the computational cost in the current study. For each seismic sequence, the computational costs of the 3D solid model and equivalent shell-wire model were compared.

As stated, because of the complexity of 3D solid models and their high computational cost, studies have generally focused on SDOF systems or 2D frames having low heights. Similar 3D structures have been neglected. It is also known that structures can show different seismic behaviors under seismic sequences depending on the 2D or 3D modeling and that 3D models are preferred [12]. Sections 3, 6.1, and 6.2 demonstrated that the equivalent shell-wire model was able to provide analytical results that were similar to those of 3D solid complex models that had high computational costs. Equivalent shell-wire and 3D solid models also showed similar results for cumulative damage, damage evolution, strength, and stiffness degradation. This makes it necessary to determine the relative computational costs of the equivalent shell-wire and 3D solid models.

Table 4 lists and compares the time required for analysis using the equivalent shell-wire and 3D solid models. As it is known that the computational cost is equivalent to the time required for analysis, on average, the equivalent shell-wire model had a 39% lower computational cost than the 3D solid model, which is remarkable. As the inter-story drift ratio or plastic strain increased, the computational cost efficiency of the equivalent shell-wire model over 3D solid model increased. It can be said that, considering the accuracy and computational cost, the equivalent shell-wire model performed better at a less computational cost. This approach also can be used to study complex and taller structures and even use finer mesh elements to increase accuracy.

Mainshock-aftershock	Comparison of time required for analyses (%)
Friuli	−38%
Cape Mendocino	−40%
Mammoth Lakes	−39%
Imperial Valley	−39%
Northridge (Newhall-Fire Station)	−39%
Northridge (Rinaldi Receiving Station)	−40%
Whittier Narrows	−39%

Table 4.

Computational cost comparison.

7. Conclusions

The realistic seismic behavior of structures, chiefly shear wall structures, under seismic sequences is of major concern in seismic engineering. In general, 3D solid structural models show more accurate results than 2D models; however, because of cumulative damage, damage evolution, and strength and stiffness degradation, analysis of 3D solid models under seismic sequences accrue higher computational costs, which means that previous studies have generally only investigated 2D models or low-height 3D solid models. The use of an equivalent shell-wire model is an appropriate technique for generating accurate results while decreasing the computational cost for shear wall structures. The results of this study could be summarized as follows:

• Comparison of the area under the capacity curve of a shear wall modeled using the 3D solid model and equivalent shell-wire model, which is equal to absorbed energy, showed that there was only a slight difference between the curves. The displacement-based shear curves showed less than a 5% difference; however, compared with the 3D solid model, the equivalent shear-wire model required only about 50% of the computational cost.

• The cyclic behavior of the RW1 shear wall modeled using the equivalent shell-wire model was compared with the results of an experimental study. The generated hysteresis curves showed good agreement between the equivalent shell-wire model and the experimental model, including for cyclic strength and stiffness degradation.

• In order to assess the accuracy of the equivalent shell-wire model, including the cumulative damage, damage evolution, strength, and stiffness degradation, a six-story shear wall structure was designed and modeled using the 3D solid model and equivalent shell-wire model in Abaqus. As expected, the seismic sequences increased the drift ratio and damage in the intended structure. Comparison of the inter-story drift ratios under the mainshock and mainshock-aftershock showed good agreement between the 3D solid model and equivalent shell-wire models. For example, in story one, on average, there was about a 1.5% and 2.5% difference in the inter-story drift ratio between the 3D solid and equivalent shell-wire model under mainshock and mainshock-aftershock, respectively.

• Assessment of the inter-story drift ratio showed that an increase in the plastic strain or damage increased the difference between the results from the 3D solid and equivalent shell-wire wire models. For example, under the Friuli seismic sequence, there was about a 1% difference in the inter-story drift ratio between the model for story one. This difference was 3.14% under the Cape Mendocino seismic sequences, which could be because of the higher drift ratio (plastic strain) experienced by story one under the Cape Mendocino mainshock-aftershock (1.91%) compared to the Friuli mainshock-aftershock (1.23%).

• Comparison of the story one inter-story drift ratio for the 3D solid and equivalent shell-wire models showed less than a 4% difference between models for all sequences used (almost 2.5% on average). This indicates that the equivalent shell-wire model provided accurate results and could appropriately consider the cumulative damage, damage evolution, strength, and stiffness degradation.

• Along with the accurate and appropriate results of the equivalent shell-wire model, on average, this model required 39% less computational cost than the 3D solid model. This suggests that it is possible to analyze structural models of greater height, more complexity, or with finer mesh to increase accuracy at the same computational cost and accuracy of a 3D solid model.

Acknowledgments

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Conflict of interest

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