基础竖向多频参数激励下圆弧拱平面内动力失稳研究

地震波、冲击波、环境振动激励会通过地基基础传递到拱上，致使拱发生动力失稳失去承载能力。为深入研究拱在基础竖向激励下的动力稳定性，该文基于能量法，建立了基础竖向激励下圆弧拱平面内动力稳定能量方程，利用哈密顿原理得到了拱面内径向和切向振动的耦合控制方程，求解了圆弧拱平面内失稳前的动轴力与动弯矩解析解。引入拱轴线不可压缩假设，解决了圆弧拱平面内动力控制方程的解耦问题。利用伽辽金法建立了基础竖向多频激励下圆弧拱平面内二阶常微分动力稳定方程，运用多尺度法推导了基础竖向多频激励下圆弧拱平面内动力失稳的临界激励频率解析公式，得到了圆弧拱同时发生一阶反对称参数共振和二阶正对称共振失稳的动力不稳定域，并利用有限元数值分析验证了理论解析解的正确性。进一步分析了拱矢跨比、长细比和圆心角对动力不稳定域的影响。

STUDIES ON IN-PLANE DYNAMIC INSTABILITY OF VERTICALLY AND PARAMETRICALLY BASE MULTI-FREQUENCY EXCITED CIRCULAR ARCHES

Seismic wave, shock wave and the ambient vibration excitation may be transmitted to an arch through the base, resulting in the dynamic instability and the loss of bearing capacity of the arch. In order to deeply study the dynamic stability of the arch under a vertical base excitation, the energy equations of the in-plane dynamic stability of the circular arch under a vertical base excitation are established based on the energy method. The coupled governing equations of the in-plane radial and tangential vibration of the investigated arch are obtained by using Hamilton principle. The analytical solutions of the dynamic axial forces and the dynamic bending moments prior to the in-plane dynamic instability are solved. For decoupling the in-plane dynamic governing equation, it is assumed that the arch is incompressible. Using the Galerkin method, the in-plane second-order ordinary differential dynamic stability equations of the circular arch under a vertical base multi-frequency excitation are established. The analytical equations of critical excitation frequencies of the circular arch under a vertical base multi-frequency excitation are derived via the multi-scale method, and the dynamic instability regions for the simultaneous first-order antisymmetric parametric resonance and second-order symmetric resonance instability of the circular arch verified by FEA are determined accordingly. Furthermore, the influences of span ratio, slenderness ratio and central angle on the dynamic instability region are analyzed.