Dongwei Wang

The complex damping model can only be used to calculate the steady-state responses, while the transient responses are divergent. Based on the complex damping model, the Hilbert transform is introduced to establish a hysteretic damping model, eliminating the divergence phenomenon. However, with the increase of the loss factor, the damped natural frequency also increases. To overcome this shortcoming, a frequency-independent damping model is proposed based on the hysteretic damping model. However, traditional time-domain methods are no longer applicable to frequency-independent damping models. Therefore, the transient response and steady-state response are separated, and the assumption of external excitation acceleration is introduced. Time-domain methods-based linear polynomial assumption, quadratic polynomial assumption and trigonometric function assumption are proposed, respectively. Numerical examples show that the time-domain methods based on linear polynomial assumption and quadratic polynomial assumption have high computational efficiency. But these two methods cannot take into account the vibration frequency of external excitation acceleration. Hence, the computational accuracy is low. Compared with them, the time-domain method based on trigonometric function assumption has the lowest computational efficiency and the highest computational accuracy.

Due to modal truncation, the dynamic responses calculated by the mode superposition method have some errors. In this paper, the source of truncation error of the complex mode superposition method for hysteretic damped systems with non-proportional characteristics is analyzed. It includes truncation errors of external excitation load and Hilbert transform term. With the increase of mode order, the ratio of excitation frequency to natural frequency gradually approaches zero. Combined with the expression of structural modal response caused by harmonic wave, the corresponding approximate expression of high-order modal response is obtained. Through quasi-static analysis, a complex mode superposition method of non-proportionally damped systems for mode static correction is proposed. Numerical examples show that when the ratios of the domain frequencies of the external excitation to the natural frequency of some high-order modes are small, the complex mode superposition method for mode static correction has a good effect.

The free vibration responses are divergent based on a complex time-domain damping model. The traditional step by step integration method cannot be used to calculate the time-domain responses. Based on the theoretic solution that eliminates the divergent term for the complex damping model, the equivalent viscous damping model is proposed and the corresponding average acceleration method is realized in this paper. The numerical cases show that the calculated results of the equivalent viscous damped system are convergent, which are equal to the ones of the deleted divergent term based on the complex damping model. The correctness of the proposed method is verified, and the computational efficiency is high.

The quantitative analysis of seismic performance under small and moderate earthquakes is of great significance. However, only using the stiffness distribution index is no longer applicable to seismic performance evaluation, and the influence of mass distribution must be introduced. In this study, under the action of small and medium earthquakes, the equation of motion can be simplified into the equilibrium equation of the dynamic matrix by the quasi-static method. Furthermore, a condition number index of dynamic matrix is proposed to evaluate the seismic performance of truss structures, which can consider both stiffness distribution and mass distribution. Then, the correctness of the proposed index is verified by comparing with the displacement and strain indexes. In addition, the performance of truss structure is studied by the condition number of stiffness matrix, mass matrix and dynamic matrix. It is found that since the condition number of stiffness matrix and the condition number of mass matrix cannot consider both stiffness and mass, it is not comprehensive to use these two indexes to evaluate the seismic performance of truss structures. The evaluation result based on the condition number of dynamic matrix is more reasonable under small and medium earthquakes.

A damping model is one of the key factors in dynamic analysis. Viscous damping and hysteretic damping models are commonly used in structural damping models. In this study, transient and steady responses are analyzed for a single degree of freedom system based on the two damping models. The attenuation coefficient and damped natural frequency are important parameters of the transient response. In addition, the vibration amplitude is an important parameter of the steady response. When the relative errors of the parameters for the two damping models are less than 10%, the threshold of the damping ratio is selected as 0.1736 and the threshold of the loss factor is 0.3472. The numerical examples show that the dynamic responses based on the viscous damping model are approximately equal to those based on the hysteretic damping model in small damping cases. With the increase in the damping ratio, the difference between the dynamic responses calculated by the two damping models gradually increases. In large damping cases, the two damping models must be distinguished, and the choice of the damping model depends on the characteristic of dissipate energy.

Identifying relatively weak areas is of great significance for improving the seismic reliability of structures. In this paper, a modal strain energy decomposition method is proposed, which can realize the decoupling of the comprehensive modal strain energy of a planar structure into three basic modal strain energies. According to the decomposition results, the modal strain energy decomposition diagram and the modal strain energy cloud diagram can be drawn so as to realize the quantitative and visual analysis of the vibration modes. The method is independent of load cases and can identify relatively weak areas of a structure from the perspective of inherent characteristics. The comparison with the shaking table test results of the two-story shear wall shows that the modal strain energy decomposition method can effectively identify the type of the relatively weak area of a structure and locate the position of the relatively weak area. Finally, the 6-story shear wall is analyzed by the modal strain energy decomposition method, and the relatively weak areas under the first two vibration modes are identified.

In order to realize the time-domain analysis based on hysteretic damping model, the frequency-independent time-domain damping model of single degree of freedom (SDOF) system is constructed. Based on the assumed relationship of vibration responses, the equivalent frequency-independent time-domain damping model in complex domain and real domain are proposed. The characteristic that the dissipated energy in each cycle is not related to the vibration frequency of external excitation is retained for the two equivalent damping models. Combined with Newmark- β method, the corresponding numerical methods are obtained. The numerical examples show that the free vibration responses are stably convergent based on equivalent damping models. The numerical results of vibration responses of SDOF system due to earthquake wave have high calculation accuracy. Compared with equivalent frequency-independent time-domain damping model in real domain, the computational accuracy of equivalent frequency-independent time-domain damping model in complex domain is higher, and the computational efficiency is lower.

Trusses are among the basic components of large-span bridges and large-space structures. A method is proposed to conduct a comprehensive deformation analysis of a truss in terms of the basic rigid body displacements and the tension and compression deformation based on complete mathematical orthogonality and mechanical equilibrium. The correctness of the proposed method is verified by comparison with a traditional strain analysis. Furthermore, a relative deformation decomposition of the mode shape is proposed to analyse in detail its relative displacement and deformation. The correctness and superiority of the proposed method are verified by comparison with the modal mass participation coefficient method and the animation from observation method. Additionally, the relative deformation decomposition of a plane truss structure is realized under any load conditions based on the superposition of mode shapes. The quantitative analysis of the basic deformation performance of a plane truss structure can also be conducted by countable mode shapes, which do not involve load conditions. Finally, the number of mode shapes that must be considered differs when using the maximum displacement and the tension and compression deformation analysis indicators.