An improved response spectrum method for non-classically damped systems

Abstract

A complex modal truncation augmentation method is proposed in the present study for the non-classically damped systems. Compared to the traditional mode displacement superposition approach, this method considers the contributions from the high vibration modes and can therefore increase the prediction accuracy of the structural responses. It can be regarded as an improvement of the traditional method. Based on this method, the conventional CCQC (Complex Complete Quadratic Combination) modal combination rule for the non-classically damped systems is extended to take into account the contributions of the truncated high vibration modes and the effects of narrow-band inputs on the modal cross-correlation coefficients. Moreover, a practical method is developed to estimate the velocity response spectrum that is required in the CCQC rule utilizing the commonly used displacement response spectrum based on the random vibration theory. Numerical results show that the extended CCQC rule can result in more accurate structural response estimations especially when the contributions from the high vibration modes to the structural responses cannot be neglected or when the structure is subjected to the seismic inputs with narrow band widths.

Appendix

The first n modal dynamic equations of Eq. (2) can be written in the form of matrix as follows

$${\dot{\mathbf{Z}}}_{L} (t) - {\varvec{\Lambda}}_{L} {\mathbf{Z}}_{L} (t) = - {\mathbf{a}}_{L}^{ - 1} {\varvec{\Psi}}_{L}^{\text{T}} {\mathbf{A}}{\varvec{\upiota}}\ddot{u}_{g} (t)$$

(32)

in which ${\mathbf{Z}}_{L} (t)$ is the modal coordinate vector and the subscript “L” denotes low modes. ${\varvec{\Lambda}}_{L}$ and ${\varvec{\Psi}}_{L}$ are spectrum matrix and the corresponding eigenvector matrix of the first n modes, respectively. ${\mathbf{a}}_{L} = {\varvec{\Psi}}_{L}^{\text{T}} {\mathbf{A}}{\varvec{\Psi}}_{L}$ is the diagonal matrix due to the orthogonality conditions of eigenvectors with respect to ${\mathbf{A}}$. Utilizing the eigenvalue equations, i.e. ${\mathbf{B}}{\varvec{\Psi}}_{L} = - {\mathbf{A}}{\varvec{\Psi}}_{L} {\varvec{\Lambda}}_{L}$, the premultiplication of Eq. (32) by ${\mathbf{A}}{\varvec{\Psi}}_{L}$ yields

$${\mathbf{A}}{\dot{\mathbf{v}}}_{L} + {\mathbf{Bv}}_{L} = - {\mathbf{A}}{\varvec{\Psi}}_{L} {\mathbf{a}}_{L}^{ - 1} {\varvec{\Psi}}_{L}^{\text{T}} {\mathbf{A}}{\varvec{\upiota}}\ddot{u}_{g} (t)$$

(33)

The above equations of motion are associated with the first n modes, and its distribution vector to the seismic input is ${\mathbf{f}}_{L} { = }{\mathbf{A}}{\varvec{\Psi}}_{L} {\mathbf{a}}_{L}^{ - 1} {\varvec{\Psi}}_{L}^{\text{T}} {\mathbf{A}}{\varvec{\upiota}}$. Thus, the spatial distribution vector of earthquake loading corresponding to the high vibration modes can be expressed by

$${\mathbf{f}}_{H} = \left( {{\mathbf{I}} - {\mathbf{A}}{\varvec{\Psi}}_{L} {\mathbf{a}}_{L}^{ - 1} {\varvec{\Psi}}_{L}^{\text{T}} } \right){\mathbf{A}}{\varvec{\upiota}}$$

(34)

or

$${\mathbf{f}}_{H} = \left[ {{\mathbf{I}} - {\mathbf{A}}\sum\limits_{i = 1}^{n} {\left( {\frac{{{\varvec{\uppsi}}_{i} {\varvec{\uppsi}}_{i}^{\text{T}} }}{{a_{i} }} + \frac{{{\bar{\varvec{\uppsi}}}_{i} {\bar{\varvec{\uppsi}}}_{i}^{\text{T}} }}{{\bar{a}_{i} }}} \right)} } \right]{\mathbf{A}}{\varvec{\upiota}}$$

(35)