Application of an efficient stochastic calculation method on the seismic analysis of an isolated structure

Abstract

An isolated structure often possesses distinct non-proportional damping characteristics. However, traditional seismic calculation theory and methods are derived based on the assumption that damping is proportional. Based on this drawback, a new, more efficient stochastic calculation method, an improvement on the pseudo-excitation method, is introduced. This method is then applied to the seismic analysis of an isolated structure. By comparing it with the forced decoupling, matrix inversion and iteration methods, it is shown that the presented method can produce accurate results while increasing the efficiency of the stochastic analysis. Moreover, the calculation process of the seismic response of an isolated structure is convergent. Based on the results of the example presented in this paper, the given method is applicable to the seismic analysis of an isolated structure and can be utilized in practice.

Keywords

isolated structure / stochastic response / non-proportional damping / efficient / accurate / pseudo-excitation method

Introduction

Ground motion during an earthquake is a random process; therefore, the record of a single earthquake can neither accurately reflect past situations nor predict future ones. To solve this problem, a large number of earthquake records are generally required to perform a single statistical analysis. Another routine strategy to represent ground motion is to apply the random vibration theory, from which the corresponding results can directly reflect the stochastic response of the structure. However, the application of random vibration theory in practical engineering has declined from the 1970s to the 1990s. There are two primary factors that are inhibiting its development: the first is that traditional random vibration theory requires great computational effort, which makes its application difficult; the second problem is that the earthquake random model has yet to be perfected. To address the first issue, Lin [1] developed the pseudo-excitation method to improve the computational efficiency of stochastic analysis. Compared with traditional theory, this new method has obvious computational advantages that can greatly improve the prospects of applying random vibration theory in both industry and academia. The pseudo-excitation method has been applied in the design and analysis processes of many large engineering projects, such as the seismic analysis of the Nanjing Yangtze River Bridge, the Shanghai Donghai Bridge, and Shenzhen Xinxing Square, along with a wind-induced vibration analysis of the Tsing Ma suspension Bridge in Hong Kong, China.

Nonlinearity is a basic characteristic of our world. Therefore, applications of structural damping should also be nonlinear. Proportional damping methods are simplifications of the real damping characteristics, the applications of which are not always accurate in that the damping is not proportional to the mass and stiffness of the existing structure. This is called non-proportional damping for real characteristics. There are several situations in which proportional damping are not an accurate approximation, such as the structure-damper system and the soil-structure interaction system. When a structure has non-proportional damping, the traditional method based on the proportional damping assumption may cause large errors; therefore, it is essential to develop corresponding stochastic methods for non-proportionally damped structures. There are several strategies to deal with non-proportional damping characteristics. The first, the forced decoupling method, directly ignores non-diagonal elements of the damping matrix, which is generalized by the mode matrix. This method adopts proportional damping instead of real non-proportional damping and can only produce approximate results. Although non-proportional damping is obvious, this method can lead to inaccurate results. The second method is matrix inversion, which is the most original and accurate option; however, matrix inversion require substantial computational effort, which may lead to an error when an ill-conditioned matrix is included. By introducing the pseudo-excitation method, Lin [2] proposed a new method for non-proportionally damped structures in which matrix inversion is required. The complex mode method utilizes the complex mode of a constructed state equation to create the decoupling calculation. The state equation and complex mode can be adopted to decouple the dynamic equilibrium equation of a non-proportionally damped structure, but it must also calculate complex eigenvalues and eigenvectors of an extended-order matrix, again requiring significant computational effort. Scholars [3-5] have proposed several strategies to simplify the solution of these complex eigenvalues. Furthermore, Zhang [6] introduced the complex mode into the pseudo-excitation method and proposed a method to ensure high computational efficiency. A fourth method is the iteration method, which uses an iterative process to avoid matrix inversion and the use of the complex mode. Jandid [7] proposed an iterative method to obtain the transfer function of non-proportionally damped structures, and if the iterative process converges, the method can avoid both matrix inversion and complex operations. This method has been further extended to primary-secondary systems [8]. Guo et al. [9] proposed an iterative expression based on the pseudo-excitation method, and the whole iteration process focuses on pseudo response, which is more efficient than the transfer function iteration. Moreover, the iteration of pseudo response has been improved, and an iteration matrix reconstruction strategy has been proposed [10]. By using matrix reconstruction, the iteration process can be ensured to be convergent.

Isolation is an effective vibration control technique for the seismic response of a structure in an earthquake, and it has a high potential for application in practical engineering. Obviously, an isolated structure possesses a non-proportional damping characteristic; however, the forced decoupling method is usually adopted in seismic analysis and design to be consistent with existing theory and methods. This forced decoupling operation for non-proportional damping might also lead to some error. To reduce the potential for error, an accurate and efficient method previously introduced by the authors is applied in a stochastic seismic response calculation of an isolated structure. Additionally, its applicability in an isolated structure is studied through comparisons with the forced decoupling method, matrix inversion and the iteration method. The comparisons show that the new method is both highly accurate and efficient, whereas the calculation process is convergent. It is therefore suggested that this method be adopted in the implementation of stochastic analyses of isolated structure affected by earthquakes.

Fast stochastic calculation method

The forced decoupling method, matrix inversion and the iteration method are all described in reference [10], so these strategies will not be further explained. Only the fast stochastic calculation method previously proposed by the authors [10] is reintroduced here. First, the ground acceleration during the earthquake is assumed to be {\stackrel{\mathrm{..}}{U}}_g={\left[{\stackrel{\mathrm{..}}{u}}_{g\mathrm{1}},{\stackrel{\mathrm{..}}{u}}_{g\mathrm{2}},\mathrm{...},{\stackrel{\mathrm{..}}{u}}_{gm}\right]}^T, which represents m excitations. The dynamic equilibrium equation of a non-proportionally damped structure subjected to earthquake loads can be written as

(1)\mathit{M}\ddot{\mathit{U}}+\mathit{C}\dot{\mathit{U}}+\mathit{K}\mathit{U}=-\mathit{M}\mathit{E}{\ddot{\mathit{U}}}_g,

(2)\ddot{\mathit{q}}\mathit{+}\overline{\mathit{C}}\ddot{\mathit{q}}\mathit{+}\overline{\mathit{k}}\mathit{q}\mathit{=}\mathit{\Gamma }\ddot{\mathit{U}}\mathit{,}

in which \mathit{M}, \mathit{C} and \mathit{K} are the mass, damping and stiffness matrices, respectively, whereas U={[u_{\mathrm{1}},u_{\mathrm{2}},\cdots ,u_n]}^{\mathrm{T}} is the relative or absolute displacement of the structure. The superscript dots on \mathit{U} represent the associated time derivative. \mathit{E}\in {\mathit{R}}^{n\times m} represents the acting position vector of the earthquake load. \overline{C}={\mathit{\Phi }}_c^{\mathrm{T}}C{\mathit{\Phi }}_c\in R^{n_c\times n_c} and \overline{K}={\mathit{\Phi }}_c^{\mathrm{T}}K{\mathit{\Phi }}_c\in R^{n_c\times n_c} are the modal damping and stiffness matrices, respectively, in which \mathit{\Phi }=[{\mathit{\varphi }}_{\mathrm{1}},{\mathit{\varphi }}_{\mathrm{2}},\cdots ,{\mathit{\varphi }}_n] and {\mathit{\Phi }}_c=[{\mathit{\varphi }}_{\mathrm{1}},{\mathit{\varphi }}_{\mathrm{2}},\cdots ,{\mathit{\varphi }}_{n_c}] denote the original and the truncated modal matrix of the structure, respectively. \mathit{q}\in {\mathit{R}}^{n\times \mathrm{1}} is the vector of the normal coordinate, where \mathit{\Gamma }=-{\mathrm{\Phi }}_c^{\mathrm{T}}ME. Both the modal stiffness matrix \overline{\mathit{\kappa }} and the modal damping matrix \overline{\mathit{C}} are diagonal during a proportional damping situation; however, for a non-proportional damping situation, the modal damping matrix \overline{\mathit{C}} includes non-diagonal elements. Therefore, Eq. (2) is a coupled equation. Matrix \overline{\mathit{C}} is divided into two parts, \mathit{A}=\alpha {\overline{\mathit{C}}}_d, and \mathit{B}=(\mathrm{1}-\alpha ){\overline{\mathit{C}}}_d+{\overline{\mathit{C}}}_f, in which {\overline{\mathit{C}}}_{\mathit{d}} denotes the corresponding diagonal matrix of \overline{\mathit{C}}, {\overline{\mathit{C}}}_{\mathit{f}} denotes the non-diagonal matrix of \overline{\mathit{C}} and parameter a is a scalar quantity. Assuming that the power spectrum density function matrix of the seismic excitations is {\mathit{S}}_{{\ddot{U}}_g}, and based on the pseudo excitation method in reference [2], the pseudo excitation vector {\widetilde{\ddot{\mathit{U}}}}_{g\mathrm{1}}^{},{\widetilde{\ddot{\mathit{U}}}}_{g\mathrm{2}}^{},\cdots ,{\widetilde{\ddot{\mathit{U}}}}_{\mathit{g}\mathit{b}} can be constructed by

(3){\widetilde{\ddot{\mathit{U}}}}_{\mathit{g}\mathit{j}}(t)={\varphi }_j\sqrt{s_j}{e}^{r\omega t}

,

(4)\begin{array}{ll}{S}_{{\ddot{\mathit{U}}}_g}={\displaystyle \sum _{j=\mathrm{1}}^b{s}_j{\varphi }_j^{*}{\varphi }_j^T},b\le m,& j=\mathrm{1},\mathrm{2},\cdots ,b\end{array},

where the superscript \sim refers to the pseudo excitation or response, b is the rank of power spectrum density function matrix {\mathit{S}}_{{\ddot{\mathit{U}}}_g} and {\mathit{s}}_j and {\varphi }_j are the corresponding eigenvalue and eigenvector of power spectrum density function matrix {\mathit{S}}_{{\ddot{\mathit{U}}}_g}, respectively. Substituting Eq. (3) into Eq. (2), it can be seen that

(5){\widetilde{\ddot{\mathit{q}}}}_j+\mathit{A}{\widetilde{\dot{\mathit{q}}}}_j+\overline{\mathit{K}}{\tilde{\mathit{q}}}_j=\mathit{\Gamma }{\varphi }_j\sqrt{s_j}{e}^{r\omega t}-\mathit{B}{\widetilde{\dot{\mathit{q}}}}_j,

in which {\widetilde{\ddot{\mathit{q}}}}_j, {\widetilde{\dot{\mathit{q}}}}_j and {\tilde{\mathit{q}}}_j represent the pseudo acceleration, velocity and displacement responses, respectively. The iterative expression of the normal coordinate vector q can be established, and through rigorous theoretical derivation, the iterative solution of the pseudo response can be given by

(6){\tilde{\mathit{q}}}_j^{(k)}={\tilde{\mathit{q}}}_{j,\omega }^{(k)}\sqrt{s_j}{e}^{r\omega t},

(7){\tilde{q}}_{j,\omega }^{(k)}={\tilde{H}}_d\left(\mathrm{\Gamma }{\varphi }_j-r\omega B{\tilde{q}}_{j,\omega }^{(k-\mathrm{1})}\right),

in which {\tilde{\mathit{H}}}_d={\left(-\mathit{I}{\omega }^{\mathrm{2}}+r\omega \mathit{A}+\overline{\mathit{K}}\right)}^{-\mathrm{1}}, {\tilde{\mathit{q}}}_j^{(k)} represents the pseudo-response time history of the structure, and {\tilde{\mathit{q}}}_{j,\omega }^{(k)} represents the time-independent portion of pseudo response {\tilde{\mathit{q}}}_j^{(k)}. The initial pseudo responses are defined as {\mathit{q}}_j^{(\mathrm{0})}=\mathrm{0} and {\mathit{q}}_{j,\omega }^{(\mathrm{0})}=\mathrm{0}. Then, the power spectrum density function of structural response {\mathit{S}}_{\ddot{\mathit{U}}} can be obtained according to the theory of the pseudo- excitation method,

(8){\mathit{S}}_{\ddot{\mathit{U}}}={\tilde{\mathit{U}}}^{*}{\tilde{\mathit{U}}}^{\mathrm{T}},

(9)\tilde{\mathit{U}}={\displaystyle \sum _{j=\mathrm{1}}^b{\tilde{\mathit{U}}}_j}={\displaystyle \sum _{j=\mathrm{1}}^b{\mathit{\Phi }}_c{\tilde{\mathit{q}}}_j}={\displaystyle \sum _{j=\mathrm{1}}^b{\mathit{\Phi }}_c{\tilde{\mathit{q}}}_j^{(k)}},

in which ^* represents the conjugate operation and T represents the transpose operation of a matrix. \tilde{\mathit{U}} is the pseudo response of the structure, and {\tilde{\mathit{U}}}_j is the jth modal pseudo response of structure. Furthermore, parameter a can be given as an optimal value to ensure the convergence and efficiency of the iteration process,

{\alpha }_{\mathrm{o}\mathrm{p}\mathrm{t}}=\left[{\lambda }_{\max }\left({\overline{\mathit{C}}}_d^{-1}\overline{\mathit{C}}\right)+{\lambda }_{\min }\left({\overline{\mathit{C}}}_d^{-1}\overline{\mathit{C}}\right)\right]/2,

where {\lambda }_{\max }\left({\overline{\mathit{C}}}_{\mathit{d}}^{\mathrm{-1}}\overline{\mathit{C}}\right) and {\lambda }_{\min }\left({\overline{\mathit{C}}}_d^{-\mathrm{1}}\overline{\mathit{C}}\right) correspond to the minimum and maximum eigenvalues of matrix {\overline{\mathit{C}}}_d^{-\mathrm{1}}\overline{\mathit{C}}, respectively. For the iteration process of response quantity z, the convergence criterion o_k and the real error e_k are given as

(11)o_k=\left|{\sigma }_{z^{(k)}}-{\sigma }_{z^{(k-\mathrm{1})}}\right|/\left|{\sigma }_{z^{(k-\mathrm{1})}}\right|\le tol,

(12)e_k=\left|{\sigma }_{z^{(k)}}-{\sigma }_{z^e}\right|/\left|{\sigma }_{z^e}\right|,

where o_k denotes the difference of the kth and (k-1) th iterative results, {\sigma }_{z^{(k)}} and {\sigma }_{z^{(k-\mathrm{1})}} are the root mean square values of the kth and (k-1)th iterative responses, respectively, and tol is the convergent tolerance. e_k denotes the error of the kth iteration, and {\sigma }_{z^e} is the real response value of z. It should be noted that the convergence criterion o_k cannot reflect a real error situation in the iterative process.

Numerical study

Numerical examples are presented here to demonstrate the effectiveness of the proposed method. As shown in Fig. 1, a simple model of a 3-story structure is given in which the isolation devices are set at the top floor. Here, the simple model is adopted to benefit the verification process. Additionally, this model demonstrates non-proportional damping characteristics. The parameters of the 3-story structure are as follows. Floors 1 and 2 have masses equal to m₁ = m₂ = 1×10⁷ kg and lateral stiffness equal to k₁ = k₂ = 2×10⁸ N/m. Floor 3 has a mass equal to m₃ = 0.5×10⁷ kg and a lateral stiffness equal to k₃ = 1×10⁷ N/m. Rayleigh damping is adopted in the two different regions of the structure. The first region consists of floors 1 and 2, with the first two Rayleigh modal damping ratios equal to 0.05. Floor 3, which contains the isolation devices, is the second region, with the damping ratio set equal to 0.3. The entire damping matrix can then be obtained by combining the two regions of the structure. The Kanai-Tajimi model is adopted here as a stochastic model of ground motion, which can be described by the expression

(13)S_{{\ddot{u}}_g}={\displaystyle \frac{{\omega }_g^{\mathrm{4}}+{(\mathrm{2}{\zeta }_g{\omega }_g\omega )}^{\mathrm{2}}}{{({\omega }_g^{\mathrm{2}}-{\omega }^{\mathrm{2}})}^{\mathrm{2}}+{(\mathrm{2}{\zeta }_g{\omega }_g\omega )}^{\mathrm{2}}}}{S}_{\mathrm{0}},

where ω_g = 13.96 rad/s and ξ_g = 0.72, which correspond to site type 1 and an earthquake classification of type 3, as defined in the seismic code GB50011-2001 of China. The seismic intensity is assumed to be S₀ = 0.06. Here, the relative displacements of floors 1-3 are selected as study indexes, and the forced decoupling method, matrix inversion, and iterative expression by the pseudo-excitation method are adopted to calculate the root mean square value of the displacement.

Figure 2(a) and (b) gives the results calculated by the above-mentioned methods, and the iteration numbers are, set to 1 and 6 for the new method. The results show that the forced decoupling method has an obvious error in Example 1, and if the corresponding results are adopted for analysis, it will lead to an incorrect conclusion. An accurate value of the stochastic response can be instead obtained by matrix inversion, which can be used to evaluate the accuracy of the results by other methods. The transfer function iteration method and the pseudo-response iteration method (α = 1, α = α_opt) can give accurate results in 6 iterations. The methods require only one iteration, with α = 1, and are equivalent to the forced decoupling method; however, when α = α_opt, the pseudo-response iteration method is completely different after only 1 iteration. Figure 3(a) and (b) describes the convergence situation and error of each floor of the structure using two types of iteration methods: the transfer function iteration and the pseudo-response iteration. Because one iteration (α = 1) is equal to the forced decoupling method, the convergence criterion parameter o_k is calculated from two contiguous iterations. From the figure, it can be seen that the two iteration methods both tend to reach an accurate value in four iterations and that the transfer function iteration is actually equal to the pseudo-response iteration (α = 1). As shown in the theoretical derivation, the two methods are completely identical in theory, but the expressions are in different forms, leading to different computational results. Meanwhile, the curve change in Fig. 3(a) and (b) tends to be consistent; thus, it can be stated that when the results converge, the iteration convergence criterion parameter o_k can accurately describe the change in the real error e_k.

To study the convergence situation of the iteration methods, we set the damping ratios of the isolation story at 0.5 and 0.8, indicating that the degree of the non-proportional damping of the structure increases. Figure 4(a) and (b) describes the convergence situation of the methods based on the pseudo-excitation method when the damping ratios of the isolation story are 0.5 and 0.8. When the degree of non-proportional damping increases, the convergence rate of the iteration methods is lower and the error introduced by the initial iteration is greater. However, for Example 1, the solutions determined by the iteration methods all converge to accurate values if the iteration parameters are reasonable.

Conclusions

Considering the inherent non-proportional damping characteristics of isolated structures, and the error produced by the traditional calculation methods based on proportional damping, the method previously proposed by the authors is utilized and studied here to verify its applicability in the stochastic seismic analysis of an isolated structure. From the study, a number of conclusions can be reached. First, it can be seen that the forced decoupling method will lead to a certain degree of error, whereas the direct matrix inversion can produce exact results. This, however, requires a large amount of computational effort. Second, the efficient stochastic calculation method previously discussed in this study is based on the pseudo-excitation method and is essentially an iteration process of pseudo response. It can be used in the stochastic seismic response calculation of an isolated structure. Third, the matrix reconstruction strategy determines the iteration convergence and convergence rate, so the use of a reconstructed matrix during iteration is important. Finally, the iteration of a pseudo response can converge more rapidly than the iteration of a transfer function by matrix reconstruction, while the established method based on the pseudo excitation method is more efficient than traditional random theory. The clear advantages of the given method, such as avoiding matrix inversion and complex operations, as well as a high level of accuracy and efficiency, lead to the recommendation that this method be utilized in the stochastic seismic response calculations of an isolated structure.

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