An approach for dynamic analysis of steel plate shear wall systems

In this paper, the Timoshenko beam model (continuous system model) is originally adapted for dynamic analysis of steel plate shear wall (SPSW) systems. Dynamic characteristics for the first three modes are found by solving differential equation of the equivalent Timoshenko beam model using the differential transformation method. Dynamic characteristics are tabulated for quick and practical calculation. With the help of the dynamic characteristics, the response spectrum analysis of such buildings is performed. Using the approach developed in this study, it is possible to calculate not only natural periods, but also the base shear force, maximum storey displacement, and maximum storey drift ratio. The differential transformation method is used in this study for solving the differential equation written according to the continuous system calculation model. To investigate the suitability of the method presented in the study, an example taken from the literature is solved and the results are evaluated. The results show that the method presented can be used in the preliminary design stage.


Introduction
To prevent building damage under earthquake loads, storey displacements and the storey drift ratios must remain below certain limit values. One of the systems used for limiting the displacement and drift ratio in steel buildings is the steel plate shear wall (SPSW) system. In recent years, many experimental and analytical studies have been conducted to determine behaviour of steel buildings using such systems. Some of these studies are briefly summarized below. Love et al. [1], used SPSW systems to strengthen a steel-framed hospital building damaged in the 1994 Northridge Earthquake. In the study, the authors show that strengthening with SPSW provides the desired level of compliance with relevant standards. Berman et al. [2], investigated damage conditions of SPSW systems and suggested repair methods. As a result of the study, a database was created for systems using steel plate shear walls. Bhowmick et al. [3], proposed a capacity design method for SPSW systems that enables better seismic performance at moderate cost. The method proposed in the study was proven to be compatible with the results of nonlinear analysis. Clayton et al. [4] proposed a design procedure for selfcentring SC-SPSW systems developed using a performancebased design approach. The study shows that SC-SPSW systems perform well at three distinct earthquake levels. Clayton et al. [5], studied behaviour of self-centring SPSW systems under cyclic loading. The experiments carried out in the study show that the SC-SPSW system exhibits a high level of ductility and rigidity. Alavi and Nateghi [6] showed that the response modification factor (R) of diagonally stiffened SPSW systems is approximately 13% greater than that of un-stiffened systems. They observed that especially the use of edge stiffeners in plates improves hysteretic behaviour of SPSW. Nie et al. [7], experimentally investigated seismic behaviour of the perforated and non-perforated stiffened SPSW systems. As a result of the experiments, they recommended the use of stiffened plates to increase the decreasing strength and stiffness of perforated SPSW. Dowden and Bruneau [8] modelled self-centring SPSW systems in full scale and compared them with analytical results. At the end of the study, they demonstrated that the proposed analytical method is compatible with the experimental results. Dowden et al. [9] investigated a self-centring SPSW system on a full scale to limit the damage in SPSW systems. As a result of the experiments carried out in the study, it was established that the self-centring steel plate shear walls systems provided the desired performance. Yu et al. [10], investigated cyclic performance of cross restrained steel plate shear walls with transverse braces. As a result of the experiments, it was established that cyclic performance of the proposed system is higher than that of the classical system. Wang and Xie [11], proposed a concrete-filled steel tubular column (CFST) and cross-reinforced SC-SPSW against the buckling of the H-section steel column base and the pinching effect in hysteretic curves, which are two main problems in thin SC-SPSWs. As a result of the experiments and analytical solutions, it was concluded that the proposed system increased yield load and maximum load. Barua and Bhowmick [12], examined nonlinear seismic performances of perforated SPSW systems designed according to the seismic code. In the study, it was observed that the perforated steel plate shear walls systems performed well in terms of ductility and strength. Curkovic et al. [13], studied behaviour of SPSW with variable flexural stiffness columns experimentally and numerically. At the end of the study, it was emphasized that the minimum column bending stiffnesses in AISC 341-10 may be too conservative. Bai et al. [14], proposed a simplified seismic design method to improve performance for low-rise dual-frame SPSW structures. In the study, low-rise dual-frame SPSW structures are reduced to an equivalent single degree of freedom system. As a result of the analysis, it was established that the proposed single degree of freedom model properly represented behaviour of the system. Azarafrooz and Shekastehband [15] investigated behaviour of the 4-and 8-storey SPSW infill plate connected to the frame in three different ways. The study shows that secondary columns provide a remarkable increase in the shear capacity of the system. Fathy [16] investigated behaviour of frame SPSW buildings without and with outriggers. Nonlinear analyses were carried out in the study using the finite element method. At the end of the study, it was seen that the system with two outriggers performed better than the system with one outrigger. It was also emphasized in the study that lateral displacements decreased with an increase in the number of outriggers. Liu He, and Li [17] experimentally investigated hysteretic behaviour of the SPSW system with self-centring energy dissipation braces. At the end of the study, it was concluded that the compressive bearing capacity of the wall plate should be smaller than the horizontal resting force of the braces to increase the self-centring effect of the system. Jiang et al. [18], proposed a formula for calculating fundamental periods of SPSW systems. In the development of the proposed formula, it was accepted that the steel frame showed shear behaviour, whereas the plate shear walls showed flexural and shear behaviour. The suitability of the analytical method presented at the end of the study was demonstrated through appropriate examples. GRAĐEVINAR 73 (2021) 12, 1195-1207 An approach for dynamic analysis of steel plate shear wall systems Zhao et al. [19], investigated suitability of the empirical period relationship used in the corrugated steel plate shear wall systems and proposed a formula for calculating period of such systems by using the shear-flexural beam model. As a result of the study, it was established that the proposed analytical method gives results close to those obtained by the finite element method. In this study, unlike the literature, the equivalent continuous Timoshenko beam model was used for the dynamic analysis of steel plate shear wall systems, and the differential transformation method was used to solve the differential equation representing dynamic behaviour of the equivalent continuous Timoshenko beam. The development of the presented method assumed that: -The material shows linear elastic behaviour -The displacements are small enough so that geometric nonlinear effects can be neglected -Structural properties can be considered uniform throughout the building height -Torsional effects around the building's vertical axis can be neglected.
An approach based on a continuous system calculation model is developed in this study to determine dynamic characteristics of steel buildings using the SPSW system. In the literature, the continuous system calculation model has been used for free vibration analysis of such systems. In addition to calculation of natural periods, the approach developed in this study also enables calculation of the base shear force, maximum storey displacement, and maximum storey drift ratio.

Mathematical model of the SPSW system
The continuous system calculation model, based on idealization of structures as a cantilever beam, is one of the methods used in the static and dynamic analysis of structures. In the continuous system calculation model, depending on the structural bearing system and the importance of axial displacements, the structures can be idealized as an equivalent flexural-shear beam, shear beam, Timoshenko beam, flexural beam, and sandwich beam. Studies on the modelling of buildings as equivalent Timoshenko beams have been conducted in the literature [20][21][22][23][24][25][26][27][28][29][30]. In these studies, it was concluded that the Timoshenko beam model is suitable for the analysis and identification of regular buildings. In this paper, the typical SPSW system is modelled as an equivalent Timoshenko beam ( Figure 1). According to the literature [21][22][23], the differential equation system of the Timoshenko model shown in Figure 1 is written as follows. (1) where V b is the lateral displacement due to bending, V s is the lateral displacement due to shear, V is the total lateral displacement, z is the vertical axis up to height, (EI) is the equivalent bending stiffness of the Timoshenko beam, ρ is the unit volume weight, A is the cross-sectional area, and (kGA) is the equivalent shear stiffness. The total lateral displacement is the sum of the lateral displacement that occurs with the bending and shear and is calculated by the following relation: Shear force in the Timoshenko beam is calculated by the following equation: Equation (5) can be written with the assumption of harmonic vibration.
where ω is the angular frequency.
If the differential Equation (7) is derived twice according to z, Equation (11) is obtained.
If both sides of the differential Equation (13) are divided into (EI), Equation (14) is written.
Boundary conditions of the linear differential Equation (14) are: a) The displacement and rotation due to bending at the base are zero b) the bending moment and shear force at the top of structures are zero. These expressions are mathematically listed below.
za (19) za (20) za (21) The following transformation can be done to make the given equations non-dimensional.
If the transformation is applied in Equation (14), Equation, (23) is obtained.
If both sides of the differential Equation (23) are multiplied by H 4 , then the differential Equation (24) is obtained.
If the necessary adjustment is made in Equation (24), the differential Equation (25) is written as follows: If dimensionless parameters are assigned to coefficients of the differential Equation (25), the differential Equation (26) is written.
(26) r 2 and a are non-dimensional parameters in the differential Equation (26) and are defined by the following Equations (27) and (28). An approach for dynamic analysis of steel plate shear wall systems (28) Similarly, the boundary condition (15) can be written nondimensionally with Equation (28).
The boundary condition in (19) can be written in non-dimensional form as in Equation (30).

Za (35)
The y function according to the differential transform method is defined as follows [31]: Equation (36) can be written as follows [31].
Here, Y[k] value can be expressed by the following Equation [31]: Based on the above equations, the transformation function is expressed by Equation (39) [31].
Here, y is defined as follows.
Similarly, if DTM is applied to Equations (29),(31), (33) and (35) Using Equation (48), the natural vibration period is found as follows: Equation (50) is obtained from Equation (49). (50) S i values from Equation (50) are calculated for different r 2 values and are presented in Figure 2 and Table 1.
In the lumped mass calculation model, it is accepted that the masses are collected at storey levels, whereas in the continuous system calculation model the masses are considered to be distributed along the height of the building. For this reason, there are significant differences between the two calculation models, especially in low-rise buildings. The correction coefficient (Ж), which varies depending on the number of floors, is defined in order to adapt the continuous system calculation model to the lumped mass system calculation model. Accordingly, the natural period value can be calculated as follows: SAP 2000 was used for the calculation of correction factors. For this, the correction factor is defined through results obtained by solving the structures with varying numbers of floors, using the lumped mass calculation model with SAP 2000. The correction factors for the first three modes are given in Table 2.
Modal participation factor (G) based on dynamic analysis is calculated using the following Equation [32]: The effective mass ratio (em) can be calculated using Equation (53) [32]. where M t represents the total building mass. For the first three modes, em effective mass ratios are presented in Figure 3 and Table 1. The base shear force for i. mode is calculated by the following Equation [32]: The peak point displacement can be calculated with the following Equation [32]: where S ai and S di stand for spectral acceleration and spectral displacement, respectively, and are calculated for the ith mode. For different r 2 values, dep values in the first three modes are calculated and presented in Figure 4 and Table 1.
For the maximum inter-storey drift ratio location found for the corresponding mode, the maximum inter-storey drift ratio values, corresponding to the same place in the other mode, can be calculated using Equation 56.

Numerical example
The plan and section of the 4-storey steel structure taken from the literature [33] are shown in Figure 6. The characteristics of the building are given in Table 3.  Considering the gross moment of inertia of the system, the natural periods of the first three modes are calculated with the method presented in this study (Timoshenko beam model).
The same example is modelled using SAP2000 and ETABS. The An approach for dynamic analysis of steel plate shear wall systems SPSW system is modelled with shell elements in the analysis with SAP2000 and ETABS. The SAP2000 model is shown in Figure 7. Using Equation (27), r 2 is determined as follows.
For r 2 = 0.281 value from Table 1, the values of S 1 S 2 and S 3 are: S 1 = 2.746, S 2 = 0.843 i S 3 = 0.,446 The following Ж values are found in Table 2 for a 4-storey building. The comparison of natural periods calculated using the gross section moment of inertia is given in Figure 8 and Table 4. The response spectrum analysis of the same building is made with the Timoshenko beam model presented in this study in accordance with the Turkish Building Earthquake Code (TBEC 2018). Earthquake parameters used in the calculations are given in Table 5.   Using Equation (27), r 2 can be obtained as follows:

0,287
For r 2 = 0.287 value from Natural periods calculated using the effective cross-section stiffness are given in Table 6 and Figure 10.  The design peak displacement can be calculated according to the square root of the sum of the squares (SRSS) method as follows: [m] For r 2 = 0.287 value from The design maximum drift ratio can be calculated according to the square root of the sum of the squares (SRSS) method as follows: The base shear force, peak displacement, and maximum drift ratio values were calculated using the method presented in this study (Timoshenko beam model) and compared with SAP2000 and ETABS in Table 8 and in figures 11, 12, and 13.

Conclusions
A continuous system calculation model is proposed for spectral analysis of buildings with an SPSW system is presented in this study. The SPSW system is idealized in this study as an equivalent Timoshenko beam. The mathematical model of the equivalent Timoshenko beam was solved by the differential transform method and the result was obtained. The method presented in this study is suitable for regular buildings and cannot be used as such for non-regular buildings. With the presented approach, the natural periods, base shear force, peak displacement, and maximum storey drift ratio are obtained quickly and practically. At the end of the study, it was demonstrated that the approach presented from the solved sample gave results that are sufficiently close to the finite element method. With the presented method, it is possible to obtain information about dynamic behaviour of buildings using a few parameters only. The method presented can be used in the preliminary design stage.