Inelastic parametric analysis of frequencies and seismic responses of soil-multistorey bidirectional eccentric structure interaction system

2.1. Proposal and derivation of the simplified model for superstructure

A specified $n\times m$ span and $N$-storey reinforced concrete building is shown in Fig. 1. Only the stiffness eccentricities due to the unsymmetrical layout of columns are considered in this study. For ease of conducting a series of parametric analyses with different uncoupled torsion to lateral frequency ratios or stiffness eccentricities, a simplified model of the shear-torsional type series rigid plate layer model is adopted as shown in Fig. 2. In the analysis process, the following assumptions are made:

1) The degrees of freedom of each slab are located at the mass center (CM) of the slab. The slab of each story has three degrees of freedom, namely the translational degrees of freedom in the $x$ and $y$ directions and the torsional degree of freedom around the vertical axis.

2) The mass center of each storey is collinear and is located at the geometric center (GC) of the slab. The radius of gyration of each floor is identical to those of the other floors.

3) The mass of the slab and columns of each storey concentrates at the mass center location of that storey.

4) The slab possesses absolute rigidity in the plane of itself. The out-of-plane stiffness is very small and can be ignored. All the members are assumed to be massless and axially inextensible, and the torsional inertia of each member is assumed to be negligible and thus ignored in the analysis.

Fig. 1Plan view of superstructure investigated

Fig. 2Plan view of the simplified model

The slab of each floor is supported by two massless, axially inextensible load-resisting members in the $x$ and $y$ directions, respectively. The two frames parallel to the $x$ direction are labeled frame A and frame B, which are formed by combining the frames at the $x$ direction of the original structure of floor $j$ towards both sides; the two frames parallel to the $y$ direction are labeled frame C and frame D, which are formed by combining the frames at the $y$ direction of the original structure of floor $j$ towards both sides. Frame A and frame B are located at equal distance $b$ from, but on opposite sides of, $x$-axis, providing the total translational stiffness of floor $j$ in the $x$ direction; Frame C and frame D are located at equal distance $a$ from, but on opposite sides of, $y$-axis, providing the total translational stiffness of floor $j$ in the $y$ direction. $K_{xjb}$, $K_{xja}$, $K_{yjd}$, and $K_{yjc}$ are the lateral stiffness of each frame of floor $j$.

The calculation of simplification model of floor $j$ can be expressed by a simultaneous equation group given in Eq. (1). That is, $K_{xjb}$, $K_{xja}$, $K_{yjd}$, $K_{yjc}$, $a$ and $b$ can be calculated in Eq. (2):

in which, $k_{xji}$ and $k_{yji}$ are the lateral stiffness of the $i$th column of floor $j$ in the $x$ and $y$ directions, respectively; $x_{ji}$ and $y_{ji}$ designate the $x$ and $y$ co-ordinates of the $i$th column of floor $j$; $K_{xj}=\sum k_{xji}$ and $K_{yj}=\sum k_{yji}$ represent the total translational stiffness of floor $j$ in the $x$ and $y$ directions, respectively; $K_{\theta j}=\sum k_{xji}\times y_{ji}^2+\sum k_{yji}\times x_{ji}^2$ is the torsional stiffness about the mass centre of floor $j$. The coordinates of the stiffness center (CS) of floor $j$ are ($e_{xj}$, $e_{yj}$). As in multistorey buildings, the centre of rigidity cannot be exactly defined, an approximate CS of floor $j$ is computed herein for reference purposes, $e_{xj}=\sum k_{yji}·x_{ji}/K_{yj}$, $e_{yj}=\sum k_{xji}·y_{ji}/K_{xj}$. The normalized stiffness eccentricities of floor $j$ are defined as $b_{xj}=e_{xj}/r$, $b_{yj}=e_{yj}/r$.

It is clear from Eqs. (1) and (2) that $k_{xji}$ and $k_{yji}$ of floor $j$ of the original model do not vary during the elastic stage, thus, $K_{xj}$ and $K_{yj}$ also do not vary. As a result, $b$ and $a$ of floor $j$ of the simplified model remain constant. When the structure enters the inelastic stage, $k_{xji}$ and $k_{yji}$ vary with the external load, and correspondently, $b$ and $a$ will also vary. It is evident that when conducting inelastic analysis by using a simplified model in replace of the original structure, the values of $b$ and $a$ can be adjusted so that the simplified model can be used to accurately replace the original model for inelastic analysis.

2.2. Simplified model of soil-eccentric structure interaction system

As shown in Fig. 3, the soil-structure interaction of torsionally coupled system consists of an $N$-storey three-dimensional eccentric superstructure founded on a rigid foundation, which in turn, is supported by the horizontal and rotational springs that correspond to the horizontal and rotational degrees of freedom of the foundation. The stiffness coefficients of the springs represent the corresponding confining stiffness of the supporting soil to the foundation. The mass of the footing is concentrated at the center with its value is $m_0$. Two orthogonal principal axes ($x$ and $y$) of the footing can be defined through the mass center of the footing, and the vertical axis ($z$) passes through the mass center which is taken to be coincident with its geometric center. Five displacement degrees of freedom of the foundation slab are considered in this system. $x_0$ and $y_0$ are the translational degrees of freedom in the x and y directions, respectively; ${\theta }_0$ denotes the twist angle around the vertical axis; ${\phi }_x$ and ${\phi }_y$ designate the rocking rotations in the $xoz$ and $yoz$ planes of the foundation, respectively. The stiffness coefficients of the horizontal springs in the two primary axis directions are $K_{hx}$ and $K_{hy}$, respectively; $K_{rx}$ and $K_{ry}$ represent the stiffness coefficients of the rotational springs in the $xoz$ and $yoz$ planes, respectively. Moreover, $C_{hx}$, $C_{hy}$, $C_{rx}$,and $C_{ry}$ are the damping coefficients of the dampers. The mass moment of inertia of the footing slab about the vertical axis is $J_0$; $I_{x0}$ and $I_{y0}$ denote the mass moment of inertia of the footing slab with respect to the $y$ and $x$ axes, respectively. Furthermore, $m_j$ is the mass of floor $j$; $J_j$ is the mass moment of inertia of the floor being equal to $m_j{r}_j^2$ and $r_j$ denotes the radius of gyration of floor $j$; $x_j$, $y_j$ and ${\theta }_j$ are the $x$-translation and $y$-translation with respect to the two principal horizontal axes and the $\theta$-rotation about the vertical axis of the superstructure, respectively. $I_{xj}$ and $I_{yj}$ are the mass moments of inertia of floor $j$ with respect to the y and x axes, respectively. The height of the mass center of the foundation to the $j$th floor slab is $h_j$.

Fig. 3Schematic diagram of soil-structure interaction of torsionally coupled system

2.3. Motion equation of the soil-eccentric structure interaction system

For the soil-structure interaction system shown in Fig. 3, the motion equations of the entire system can be derived by the Lagrange energy method [21] according to the kinetic energy, potential energy, and non-conservative force-work equations of the system. Limited by space, the motion equation of the soil-three storey bidirectional eccentric structure interaction system under an earthquake ground motion in the $x$ direction is given as:

in which ${\mathbf{M}}_S$ and ${\mathbf{K}}_s$ are respectively the mass matrix and stiffness matrix of the superstructure [23]:

and ${\mathbf{u}}_s={\left(\begin{array}{lll}{\mathbf{u}}_x& {\mathbf{u}}_y& \mathbf{u}{}_{\theta }{}^{\mathrm{}}\mathrm{}\end{array}\right)}^T={\left(\begin{array}{lllllllll}{x}_1& x_2& x_3& y_1& y_2& y_3& {\theta }_1& {\theta }_2& {\theta }_3\end{array}\right)}^T$is the displacement vector of the superstructure which is relative to the foundation displacement ${\mathbf{u}}_d$:

${\mathbf{C}}_s$ and ${\mathbf{C}}_d$ have the same forms with ${\mathbf{K}}_s$ and ${\mathbf{K}}_d$, respectively. And the expressions of ${\mathbf{C}}_s$ and ${\mathbf{C}}_d$ can be obtained by changing the letter “$K$” in ${\mathbf{K}}_s$ and ${\mathbf{K}}_d$ to “$C$”, respectively.

2.4. Solving the equation of motion for the interaction system

In a soil-structure interaction system, the modal damping ratio of foundation soil normally ranges between 15 %-20 %, which is much higher than the damping ratio of the superstructure which is between 3 %-5 %. The damping matrix of the entire system can be constructed from the damping of the foundation soil and the structure obtained through Rayleigh damping. However, this damping matrix cannot be solved through conventional modal analysis method because it is non-proportional damping. Both superstructure and foundation soil undergo proportional damping when they are considered separately. For this reason, the coupled term in the equation of motion for the soil-structure interaction system is moved to the right side of the equation as an external load to decouple the superstructure from the foundation soil using modal analysis. This method is called forced decoupling method which can reduce the number of equations and facilitate finding the superstructural seismic response solution.

The process of solving the equation of motion for the interaction system is as follows:

Eq. (3) can be further written in open form as Eqs. (4) and (5):

Thus, the Fourier transform is applied to Eqs. (4) and (5), and the following results can be obtained:

Based on the assumption that the fixed base structure possesses classical normal modes, the natural frequencies and mode shapes satisfy the following orthogonality conditions [24]:

where ${\omega }_k$ and ${\zeta }_k$ ($k=$ 1-9) are the natural frequencies and damping ratios, respectively.

The structural seismic response in the complex frequency domain can be expressed in terms of the mode shapes as:

Introducing the transformation of Eq. (9) and applying the orthogonality conditions, Eq. (6), when premultiplied by ${\mathbf{\Phi }}^T$, becomes:

where $\mathrm{\Gamma }$ and $H_k\left(\varpi \right)$ are defined as:

$H_k\left(\varpi \right)$ is the modal structure transfer function, the modal displacement can then be solved from Eq. (10) as:

Substituting Eqs. (9) and (13) into Eq. (7), the expression of foundation seismic response can be obtained, and Eq. (7) can be written as:

where:

${\mathbf{U}}_d\left(\varpi \right)$ can be solved from Eq. (14);

Substituting ${\mathbf{U}}_d\left(\varpi \right)$ into Eq. (13), and then substituting Eq. (13) into Eq. (9), the structural seismic response in complex frequency domain can be obtained.

Damping in soil involves multiple problems, and the parametric analysis is the main purpose of this paper. So the soil damping adopts stiffness-proportional damping, defined by $\left[{\mathbf{C}}_d\right]=\alpha \left[{\mathbf{M}}_d\right]+\beta \left[{\mathbf{K}}_d\right]$. Soil damping is assumed to be in direct proportion only to the linear portion of stiffness, denoted $\left[{\mathbf{K}}_d\right]$, i.e. $\alpha =$ 0. $\beta$ is given by $\beta =2\zeta /{\omega }_{d1}$, where ${\omega }_{d1}$ is the first natural frequency of the foundation and $\zeta$, at approximately 0.17, is the corresponding damping ratio [25].

3.1. Selection and calculation of model parameters

The upper model in this study is an imaginary 5×3 span three storey homogenous bidirectional eccentric frame structure with a span of m and m, respectively. The plan view is shown in Fig. 1. The story height is m. The columns in Fig. 1 marked as KZ1 have a dimension of 0.5 m×m, and there are 6 in total. The sectional dimensions of the other frame columns are m×m, and there are 18 in total. Due to the asymmetric distribution of the columns, the distribution of stiffness is unsymmetrical both about the $x$-axis and the $y$-axis, thus the system has biaxial stiffness eccentricities. Moreover, there is no variation in each column along the vertical direction. That is, the superstructure investigated herein is a regular asymmetric building. The corresponding simplified model of the superstructure is acquired according to the approach specified in Section 2.1. The bilinear hysteretic model as shown in Fig. 4 is utilized to simulate the force-displacement relationship of each lateral load-resisting member-both in the $x$ and $y$ directions. $K_1$ denotes the stiffness of the member in the elastic stage; $K_2$ designates the stiffness after yielding, taken as 10 % of the elastic stiffness [26]; and $\mathrm{\Delta }y$ is the yield displacement, which can be calculated according to empirical equations [27].

Fig. 4Bilinear restoring force model for each lateral-load resisting member

Fig. 5Schematic diagram of the analytical model of the foundation and its supporting soil

Three types of soil are considered in this paper. And their physical properties are normally described by the elastic modulus $E$, density $\rho$, and Poisson’s ratio $\mu$. The physical properties of each soil analyzed in this study are listed in Table 1. In order to acquire the confining stiffness of the supporting soil to the foundation, the finite element analysis software ANSYS is used to establish the analytical model of the foundation and its supporting soil using the approach specified in study [21]. The analytical model is shown in Fig. 5. The supporting soil is fixed at the bottom, and the supporting effect of surrounding unbounded soil is simulated by imposing viscous-elastic boundary elements in the perimeter of the soil [28]. The dimension of the supporting soil on each side of the foundation is taken as three times of the foundation dimension in the same direction, and the depth is taken as twice the foundation width, that is, $L=7\times l$, $W=7\times w$, $H=2\times w$. The dead weight of the superstructure is applied on the top surface of the foundation, and the natural settlement of the soil is simulated. The constitutive relation of the soil is simulated by multi-linear kinematic hardening in ANSYS [29]. The hysteresis curve and skeleton curve of the confining stiffness of the supporting soil to the foundation is acquired by a cyclic loading. The least square method is adopted to fit the skeleton curve into a bi-fold line.

Limited by space, the acquired elastic and inelastic confining stiffness of the supporting soil 1 to the foundation and the corresponding yield displacement or yield rotational angle are listed in Table 2, and the bi-fold line of the horizontal confining stiffness in the $x$ direction of the soil 1 to the foundation is given in Fig. 6.

Fig. 6The bi-fold line of the horizontal confines stiffness in the x direction of the soil 1 to the foundation

3.2. Analysis by load incremental method

Before nonlinear static analysis, the free vibration motion equation for the entire system is derived by the Lagrange energy method when the structural displacement is absolute displacement:

in which ${\mathbf{M}}_d=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left[\begin{array}{lllll}{m}_0& m_0& J_0& I_{x0}+\sum _{j=1}^3{I}_{xj}& I_{y0}+\sum _{j=1}^3{I}_{yj}\end{array}\right]$, $\left[{\mathbf{K}}_{ds}\right]={\left[{\mathbf{K}}_{sd}\right]}^T$, ${\delta }_s$ is the structural absolute displacement:

The meanings of other parameters refer to Section 2.3.

The inverted triangular static loads along the $x$ direction are applied at the mass centers of the analytical model in a step-by-step manner using Matlab programming. The stiffness of each member is modified according to its displacement and restoring force model of the lateral load-resisting member at each step, and the confining stiffness of the foundation slab in each direction is also examined in order to see if it yields according to its translational displacement or rotational angle. The stiffness matrix of the system is subsequently modified and extracted for the next step analysis. Under the $t$th step of load increment, the equilibrium equation of the system can be represented as:

${\left[\begin{array}{ll}{\mathbf{K}}_s& {\mathbf{K}}_{sd}\\ {\mathbf{K}}_{ds}& {\mathbf{K}}_d\end{array}\right]}_{t-1}$ is the stiffness matrix of the system calculated according to the displacement and restoring force model of the member and the confining stiffness of the supporting soil to the foundation after the $t-1$ step of load. The initial stiffness matrix of the system is taken during the calculation of the first load step, ${\left\{\begin{array}{c}\mathrm{\Delta }{\mathbf{\delta }}_s\\ \mathrm{\Delta }{\mathbf{u}}_d\end{array}\right\}}_t$ is the incremental displacement vector at the CMs under the $t$th load increment. It is noted that ${\left\{\mathrm{\Delta }{\mathbf{\delta }}_s\right\}}_t$ is the absolute displacement increment, $\left\{\mathrm{\Delta }{\mathbf{P}}_t\right\}={\left\{\begin{array}{lll}\mathrm{\Delta }{P}_{1t},& \mathrm{\Delta }{P}_{2t},& \mathrm{\Delta }{P}_{3t},\end{array}\dots \right\}}^{\mathrm{\text{'}}}$ is the $t$th incremental applied load vector.

The structural relative displacement increment ${\left\{\mathrm{\Delta }{\mathbf{u}}_s\right\}}_t$ at the $t$th step can be calculated by subtracting the structural horizontal displacement due to the translational movement and the rotation of the foundation slab from ${\left\{\mathrm{\Delta }{\mathbf{\delta }}_s\right\}}_t$:

${\left\{{\mathbf{u}}_s\right\}}_t$ is the structural relative displacement at the $t$th step, ${\left\{{\mathbf{u}}_s\right\}}_0$ is the initial structural relative displacement. The calculating equation of the relative displacement of the $i$th member of floor $j$ under the $t$th step is as follows:

Thus, the non-synchronization of the displacement of the members, which is caused by the torsion of the floor, can be considered in detail during the analysis. When the inter-story displacement angle ${\theta }_p$ of any floor reaches 1/50 specified by the Code for Seismic Design of Buildings (GB50011-2010)[8], the loading stops.

3.3. Inter-story restoring force model

For ease of inelastic parametric analysis, the inter-story restoring force model of the system needs to be determined. Using the calculated relationship of inter-story shear $V_j$ – inter-story displacement $u_j$ of floor $j$, the trilinear idealization model is demonstrated in Fig. 7. The bilinear model adopted by the members and the effect of slab rotation on yield of each member make the curve of inter-story shear-inter-story displacement exhibit trilinear characteristics. The parameters to be determined for the trilinear idealization model of floor $j$ include the inter-storey stiffness of the three stages $K_{j1}$, $K_{j2}$ and $K_{j3}$, the first turn $D1$ and the second turn $D2$, which all can be determined by the study [30]. In practice, in order to acquire the inter-story restoring force model of the superstructure, a few more load steps may be required. During parametric analysis hereafter, a 1/50 displacement angle for any floor of the superstructure is still used as the limit value.

Fig. 7Schematic diagram of equivalent three-linear restoring force model

Fig. 8Variation of eccentricities with λ

3.4. Definition of analysis stages

Fig. 8 presents the variation of $b_{xj}$ and $b_{yj}$ ($j=$ 1, 2, 3) with the loading coefficient $\lambda$. $\lambda$ is defined as the ratio of the load already applied on the system after each step to the final total load applied on the system with its maximum being one. The initial normalized stiffness eccentricities of the superstructure are $b_{y1}=b_{y2}=b_{y3}=$ 0.1344 and $b_{x1}=b_{x2}=b_{x3}=$ 0.2082. The variation curves of $b_{xj}$ with $\lambda$ have no mutation points, indicating that the members perpendicular to the load direction from the first floor to the third floor do not yield. Conversely, mutation points appear on the variation curves of $b_{y1}$ and $b_{y2}$ with $\lambda$, suggesting that the members in the load direction on the first and second floors yield in turns. Moreover, $b_{y3}$ remains a straight line with $\lambda$, which indicates that the members parallel to the load direction on the third floor do not yield.

Variation of $x_0$ with $\lambda$ is given in Fig. 9. It is clear from Fig. 9 that the variation curve of the soil 1-structure interaction system has a mutation point, suggesting that the translational confining stiffness in the $x$ direction of the soil 1-structure interaction system’s foundation enters into the inelastic stage. However, the confining stiffnesses in other directions are still in elastic stage when the loads stop. The confining stiffnesses in each direction of the systems situated on other soils are all in elastic stage when the loads stop.

Fig. 10 shows the variation of the first order natural frequency with $\lambda$ for the soil-original model interaction system and the soil-correspondent simplified model interaction system. It is clear from Fig. 10 that during the entire loading process, the variation curve of first order natural frequency of the soil-original model system matches favorably with the soil-simplified model system of the same soil type, and there are only minor differences during the yielding process of some structural lateral load-resisting members. Fig. 10 suggests that the parametric analysis for soil-eccentric structure interaction system using a simplified model for superstructure is both reliable and suitable for inelastic stage analysis. It is indicated that the reduction of the natural frequencies from elastic to inelastic is in a step-shaped manner and divides its variation process into three relatively stable stages. The stable variation of the natural frequencies reflects stable variation of the structural stiffness.

Fig. 9Variation of translational displacement x0 of the foundation with λ

Fig. 10Variation of the first order natural frequency with λ

According to inelastic development of the superstructure, as well as the yield development of the translational confining stiffness in the $x$ direction of the soil 1-structure interaction system, some typical stages are chosen to conduct the parametric analysis.

1) The first stage: none of the members yield. The inter-story stiffness is taken as $K_{j1}$, and the confining stiffness of each soil type is taken as the elastic value.

2) The second stage: the inter-story stiffness in the load direction of floor one reaches $K_{12}$, whereas those of the second and third floors remain as $K_{21}$ and $K_{31}$. The confining stiffness of each soil type is still taken as the elastic value.

3) The third stage: the inter-story stiffnesses parallel to the load direction reach $K_{13}$ and $K_{22}$ of floor one and floor two, respectively. It is still $K_{31}$ for floor three. The translational confining stiffness in the $x$ direction of soil 1-structure interaction system enters into the inelastic stage, while the confining stiffnesses in the other directions remain in the elastic stage.

Here, the following parametric analyses only consider the elastic stage of the members perpendicular to the load direction. Apart from soil 1, the confining stiffness in each direction of other soil conditions remains in the elastic stage. The inter-storey stiffness of each floor in each stage together with other parameters of the entire system in that stage are substituted into the stiffness matrix so as to determine the stiffness matrix in each stage, and parametric analysis of frequencies in each stage can be carried out. The stiffness matrix is substituted again into Eq. (3), and the equation of motion in each stage is achieved. The equation of motion in each stage can be solved by Eqs. (4-14), and parametric analysis of seismic responses in each stage can be carried out.

3.5. Calculation of the mode direction factors

The main vibration mode of lateral-torsional coupling vibration mode can be determined by the mode direction factors when the structure’s bottom is fixed [31]. For a specific vibration mode, it is called the translational vibration mode when the torsional factor is less than 0.5, or it is called the torsional vibration mode when it is larger than 0.5. When the mode direction factor is equal to 1, the vibration mode is the pure translational vibration mode or pure torsional vibration mode. The translational factor $D{X}_i$ in the $x$ direction, the translational factor $D{Y}_i$ in the $y$ direction, and the torsional factor $D{\theta }_i$ for the $i$th vibrating mode of a fixed base torsionally coupled structure can be calculated as:

For a soil-structure interaction of torsionally coupled system analyzed in this study, the above formulae cannot be used directly because the two rotational degrees of freedom of the foundation slab ${\phi }_x$ and ${\phi }_y$ are not taken into considered in them. The following modifications are applied to Eq. (19) in order to incorporate the foundation’s rotational degrees of freedom so that they can be utilized to analyze the soil- eccentric structure interaction system:

in which $x_{ji}$, $y_{ji}$, and ${\theta }_{ji}$ are the displacement components in the $i$th vibration mode of the $j$th mass point in a regularized vibrating mode vector space; $x_{0i}$, $y_{0i}$,${\theta }_{0i}$, ${\phi }_{xi}$ and ${\phi }_{yi}$ are the displacement components in the $i$th vibration mode of the foundation slab; $D{\phi }_{xi}$ and $D{\phi }_{yii}$ are the corresponding rotational factors. Table 3 shows the first three mode direction factors of the vibration mode in a soil-structure interaction of torsionally coupled system.

It can be seen from Table 3 that the modified calculating formulae of the mode direction factors are reasonable and can be adopted for analyzing a soil-eccentric structure interaction system. It is also apparent from Table 3 that the first vibration mode of the system is primarily translational movement in the x direction, coupled with translational movement in the $y$ direction and torsion. The second vibration mode is mainly translational movement in the $y$ direction, coupled with the translational movement in the $x$ direction and torsion. Additionally, the third vibration mode is chiefly torsion coupled with translational movement in the $x$ and $y$ directions. In the below parametric analysis, the uncoupled torsion to lateral frequency ratio $\mathrm{\Omega }$ and the stiff eccentricities are introduced as the two main parameters. $\mathrm{\Omega }={\omega }_{\theta }/{\omega }_x$ is defined as the ratio of the first order pure torsional frequency to the first order pure translational frequency in the $x$ direction of the corresponding fixed base torsionally uncoupled system. Primarily, the effects of uncoupled torsion to lateral frequency ratios and the normalized stiffness eccentricities on the normalized frequencies ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$, ${\omega }_3/{\omega }_{\theta }$ and seismic responses are evaluated. ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ are the first three natural frequencies of the soil-structure interaction of torsionally coupled system. ${\omega }_x$, ${\omega }_y$ and ${\omega }_{\theta }$ are the first order pure translational frequency in the x direction, the first order pure translational frequency in the $y$ direction, and the first order torsional frequency of the soil-structure interaction of torsionally uncoupled system, respectively.

4.1. Parametric analysis of the normalized frequencies

4.1.1. Effects of $\mathbf{\Omega }$ and soil

The effects of $\mathrm{\Omega }$ and soil on ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ are presented in this subsection. $b_x$ and $b_y$ are kept constant at 0.2.

Fig. 11The effects of Ω and soil on ω1/ωx in the three stages

a) The first stage

b) The second stage

c) The third stage

The variations of ${\omega }_1/{\omega }_x\text{,}$${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ with $\mathrm{\Omega }$ are displayed in Figs. 11-13, respectively. It is evident from the figures that increasing $\mathrm{\Omega }$ causes ${\omega }_1/{\omega }_x$ to approach G1 (Fig. 11). This finding demonstrates that increasing $\mathrm{\Omega }$ leads to a weak lateral-torsional coupling effect in the first order mode and makes ${\omega }_1$ approach ${\omega }_x$. That is, the first vibration mode of the system is close to the first pure translational vibration mode in the $x$ direction of the soil-corresponding equivalent symmetric structure interaction system. In other words, the translational deformation component in the $x$ direction in the structural response will gradually increase, and the coupled torsional deformation component will decrease. The structural response will be near the first order pure translational in the $x$ direction of a soil-structure interaction of torsionally uncoupled system.

Fig. 12The effects of Ω and soil on ω2/ωy in the three stages

a) The first stage

b) The second stage

c) The third stage

Fig. 13The effects of Ω and soil on ω3/ωθ in the three stages

a) The first stage

b) The second stage

c) The third stage

Furthermore, the increase of $\mathrm{\Omega }$ causes ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ toapproach G2 and G3, respectively (Figs. 12-13), suggesting that ${\omega }_2$ and ${\omega }_3$ respectively approach ${\omega }_y$ and ${\omega }_{\theta }$ as $\mathrm{\Omega }$ increases. This indicates that the system with a larger $\mathrm{\Omega }$ value has a larger torsional stiffness and a weaker lateral-torsional coupling effect. For systems situated on soil 2 and soil 3, as the soil softens, ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ more closely approach G1, G2, and G3, respectively. This demonstrates that the soil-structure interaction of torsionally coupled system reduces the lateral-torsional coupling effect. The softer the soil of the interaction system is, the weaker the lateral-torsional coupling effect is. As a result, ${\omega }_1$, ${\omega }_2$ and ${\omega }_3$ respectively approach ${\omega }_x$, ${\omega }_y$ and ${\omega }_{\theta }$ as $\mathrm{\Omega }$ increases and as the soil softens.

4.1.2. Effects of ${\mathit{b}}_{\mathit{y}}$ and soil

The effects of $b_y$ and soil on the normalized frequencies ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ are presented in this subsection. $b_x$ and $\mathrm{\Omega }$ are kept constant at 0.2 and 1.4, respectively.

Fig. 14The effects of by and soil on ω1/ωx in the three stages

a) The first stage

b) The second stage

c) The third stage

The variations of ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ with $b_y$ situated in different soil conditions are illustrated in Figs. 14-16, respectively. As it can be seen, increasing $b_y$ causes ${\omega }_1/{\omega }_x$ to deviate from G1 (Fig. 14). Moreover, ${\omega }_2/{\omega }_y$ is close to G2 in the first and second stages (Fig. 15(a) and (b)) and is farther from G2 in the third stage (Fig. 15(c)) with the increase in $b_y$. The increase of $b_y$ causes ${\omega }_3/{\omega }_{\theta }$ to shift away from G3 in the first stage (Fig. 16(a)). In the second stage, increasing b_y leads to ${\omega }_3/{\omega }_{\theta }$ firstly approaching and then shifting away again from G3 for the fixed base condition and makes ${\omega }_3/{\omega }_{\theta }$ deviate from G3 for the interaction system (Fig. 16(b)). In the third stage, ${\omega }_3/{\omega }_{\theta }$ moves nearer to G3 with the increase of $b_y$(Fig. 16(c)). The results reveal that the increased $b_y$ improves the lateral-torsional coupling effect of the first order vibration mode of the soil-torsionally coupled system, and ${\omega }_1$ deviates from ${\omega }_x$. That is, the first vibration mode of the system deviates from the first pure translational vibration mode in the $x$ direction of the soil-corresponding equivalent symmetric structure interaction system. In other words, the translational deformation component in the $x$ direction in the structural response will reduce, and the coupled torsional deformation component will increase. The structural response will be close to the first torsional response of the soil-torsionally uncoupled system. It is worth mentioning that the effects of increasing $b_y$ on ${\omega }_2$ and ${\omega }_3$ are different at the second and third stages. The larger $b_y$ is in the elastic stage, the closer ${\omega }_2$ is to ${\omega }_y$, and the farther ${\omega }_3$ is from ${\omega }_{\theta }$. As inelastic develops, the larger $b_y$ is, the farther ${\omega }_2$ is from ${\omega }_y$, and the closer ${\omega }_3$ is to ${\omega }_{\theta }$. Moreover, the softer the soil of the interaction system is, the closer the variation curve is to 1, which demonstrates that the softer the soil of the interaction system is, the weaker lateral-torsional coupling effect is.

Fig. 15The effects of by and soil on ω2/ωy in the three stages

a) The first stage

b) The second stage

c) The third stage

For systems situated on soil 2, soil 3 and fixed base condition, the effects of structural inelastic development on ${\omega }_1/{\omega }_x$, ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ are identical, and the only differences are the values. It is clear from Figs. 11-16 that ${\omega }_1/{\omega }_x$gets more close to G1 (Figs. 11 and 14), ${\omega }_2/{\omega }_y$ is far from G2 (Figs. 12 and 15), and ${\omega }_3/{\omega }_{\theta }$ is first close to then deviate from G3 (Figs. 13 and 16) during the process from the first stage to the second stage and then to the third stage of a system. The above finding is due to the fact that the yielding of the lateral load-resisting members in the $x$ direction leads to the reduction of the structure’s overall lateral stiffness in the $x$ direction, and the lateral-torsional coupling effect of the first order vibration mode is weakened. As a result, ${\omega }_1$ is close to ${\omega }_x$. A greater $b_y$ value and a smaller $\mathrm{\Omega }$ value lead to a greater reduction in the effects of $b_y$ and $\mathrm{\Omega }$ on ${\omega }_1/{\omega }_x$. The reduction of the overall torsional stiffness is not pronounced in the second stage, and ${\omega }_3$ is slightly closer to ${\omega }_{\theta }$. Nevertheless, the overall torsional stiffness is also significantly reduced in the third stage, which causes ${\omega }_3$ to deviate from ${\omega }_{\theta }$. The lateral stiffness in the $y$ direction does not change because the lateral load-resisting members in this direction are in the elastic stage. With the reduced lateral stiffness in the load direction, the coupled translational component in the $y$ direction in the second vibration mode reduces while the corresponding torsional component increases. In other words, ${\omega }_2$ deviates from ${\omega }_y$.

Fig. 16The effects of by and soil on ω3/ωθ in the three stages

a) The first stage

b) The second stage

c) The third stage

Variations of ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ with $\mathrm{\Omega }$ or $b_y$ of soil 1-structure interaction system in the third stage are significantly different from those of the system situated on other soils. In the third stage, the increase of $\mathrm{\Omega }$ makes ${\omega }_2/{\omega }_y$ and ${\omega }_3/{\omega }_{\theta }$ approach G2 and G3, respectively. However, the variation is less significant than in the cases with other soil types. The deviation distances between ${\omega }_2/{\omega }_y$ and G2 as well as ${\omega }_3/{\omega }_{\theta }$ and G3 are larger than those of other soil conditions when $\mathrm{\Omega }$ is slightly larger than 1.4. However, they do not indicate that the lateral-torsional coupling effects are stronger in the second and third vibration modes of the soil 1-structure interaction system. Instead, they signify the fact that the vibration modes of the soil 1-eccentric structure interaction system change. For example ($b_x=b_y=$0.2), in the third stage of a soil 1-eccentric structure interaction system with $\mathrm{\Omega }=$1.1, the main mode direction factors of the second and third vibration modes are $D{Y}_2=$ 0.0500, $D{\theta }_2=$ 0.9455 and $D{Y}_3=$ 0.9267, $D{\theta }_3=$ 0.0517, respectively. Additionally, they are respectively $D{Y}_2=$ 0.0311, $D{\theta }_2=$ 0.9680 and $D{Y}_3=$ 0.9461, $D{\theta }_3=$ 0.0318 of the system with $\mathrm{\Omega }=$ 1.8. Moreover, in the third stage of a soil 2-structure interaction system with $\mathrm{\Omega }=$1.8, the main mode direction factors of the second and third vibration modes are $D{Y}_2=$ 0.9713, $D{\theta }_2=$ 0.0115 and $D{Y}_3=$ 0.0111, $D{\theta }_3=$ 0.9882, respectively. The above data suggest that the second and third order vibration modes of the soil 1-structure interaction system respectively change to the torsional vibration mode and the translational vibration mode in the $y$ direction in the third stage compared to other soil conditions. This is because that in the third stage, the yielding of the members in the $x$ direction and the translational confining stiffness in the $x$ direction of the soil 1-structure interaction system lead to a considerable reduction in the overall lateral stiffness and overall torsional stiffness of the system. However, the lateral stiffness in the $y$ direction does not change because the lateral load-resisting members in this direction are all in the elastic stage. As a result, the second and third order vibration modes of the soil 1-structure interaction system respectively change to the torsional vibration mode and the translational vibration mode in the y direction. This is obviously different from the systems situated on soil 2 and soil 3. It is also clear that the influences of $\mathrm{\Omega }$ and $b_y$ on the lateral-torsional coupling effects of the vibration modes are considerably reduced in the soil 1-structure interaction system compared to the systems situated on soil 2 and soil 3.

4.2. Parametric analysis of the structural seismic responses

The ratio of the superstructural displacements in complex frequency domain at the mass centre of each floor to Fourier transform of the ground motion displacement, denoted $\left|U_s\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|$, is referred to as the displacement transfer function. As the ratio $\left|U_s\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|$ reflects the dynamic response of the system under unit harmonic excitation, the transfer function can represent the structural dynamic response when given the same input.

$\left|U_{sx1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|$ and $\left|U_{sy1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|$ are respectively the translational displacement transfer functions in the $x$ and $y$ directions of the first floor; $r\left|U_{s\theta 1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|$ is the torsional displacement transfer function of the first floor.

${\left(\left|U_{sx1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\left(\left|U_{sy1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\left(r\left|U_{s\theta 1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are their respectively peak values. And the effects of $\mathrm{\Omega }$ and $b_y$ on${\left(\left|U_{sx1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\left(\left|U_{sy1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$, ${\left(r\left|U_{s\theta 1}\left(\varpi \right)\right|/\left|X_g\left(\varpi \right)\right|\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}$ are mainly studied in this section (Unless otherwise noted, “displacement” in this section means the peak value of the bottom displacement transfer function).