基于多维激励应变能响应Rayleigh阻尼系数计算方法

Rayleigh阻尼模型广泛应用于结构地震响应计算，致使Rayleigh阻尼系数计算方法将显著影响结构响应，而地震多维激励是同时作用于结构，且应变能响应能较好地反映结构力学行为，因此，本文对基于多维激励应变能响应的Rayleigh阻尼系数计算方法进行了研究。根据单维地震激励时应变能响应的计算公式，构建出用于计算应变能响应的合理虚拟激励形式，应用于多维激励体系，提出计算多维激励时应变能响应的虚拟激励法，并构建出振型阻尼比变化量平方的新型权重系数，提出基于多维激励应变能响应的Rayleigh阻尼系数计算方法。分别以不带楼板的双向对称、单向偏心和双向偏心的大跨空间钢结构，以及带楼板的对称双塔、非对称双塔和非对称三塔钢筋混凝土结构为工程案例，以多条实际地震波作为激励，对提出的算法进行验证。结果表明：对于双向对称空间钢结构，基于各类阻尼系数计算方法求得的结构响应误差均较小，对于单向偏心和双向偏心空间钢结构，基于二阶振型阻尼系数计算方法求得的结构响应误差最小值和最大值分别为5.89%和19.64%，基于多维激励应变能响应阻尼系数计算方法求得的结构响应误差最小值和最大值分别为0.00%和6.10%，对于对称双塔、非对称双塔和非对称三塔结构，基于二阶振型阻尼系数计算方法求得的结构响应误差最小值和最大值分别为5.76%和51.17%，基于多维激励应变能响应计算求得的结构响应误差最小值和最大值分别为0.07%和4.48%，证明该计算方法是合理的。

Calculation Method for Rayleigh Damping Coefficients Based on Strain Energy Response Under Multi-dimensional Excitation

It is necessary to use a rational calculation method for Rayleigh damping coefficients, because the Rayleigh damping model is widely used in structural seismic response and the influence of the calculation method for Rayleigh damping coefficients on structural response is significant. With the necessity and the reality that the multi-dimensional earthquakes along different directions act on structures simultaneously and the strain energy response can well reflect the structural mechanical behavior, the calculation method for Rayleigh damping coefficients based on strain energy response under multi-dimensional excitation was conducted in the paper. The rational pseudo excitation formula was constructed used to calculate the strain energy response based on the calculation formula for the response under one direction seismic excitation. Through the application of the pseudo excitation formula to the system under multi-dimensional excitation, the pseudo excitation method for the strain energy of the system was proposed. Based on the pseudo excitation method, the novel weight coefficient for the error square of the mode damping ratio and the corresponding calculation method for Rayleigh damping coefficients were obtained. Taking three kinds of large-span spatial structures without slab and three kinds of tower structures for examples. The large-span spatial structures included the symmetric structure, the eccentric structure in one direction and the eccentric structure in two directions, while the tower structures included the symmetric double towers building, the eccentric double towers building in one direction and the eccentric three towers building in two directions. The proposed calculation method for Rayleigh damping coefficients and the existed calculation method considering the influence of only two modes were put into the application with four strong ground motions. The results showed that the errors of the structural response of the symmetric large-span spatial structure from both the method for Rayleigh damping coefficients were both small, but for the other spatial structures, the minimum value and the maximum value of the errors of the response from the existed calculation method were 5.89% and 19.64%, while that values from the proposed calculation method were 0.00% and 6.10%, and for all the tower structures, that values from the existed calculation method were 5.76 and 51.17%, while that values from the proposed calculation method were 0.07% and 4.48%. The comparison showed the proposed calculation method was rational.