基于压弯耦合效应下预应力梁的竖向自由振动研究

基于大位移广义变分原理,考虑梁的压弯耦合、剪切应变能和转动惯量的影响,建立了预应力梁的不完全广义势能泛函,通过对位移变分,推导出预应力梁自由振动微分方程。并以预应力混凝土简支梁和悬臂梁为例,通过引入边界条件,求出了自由振动频率的解答。对比Bernoulli-Eular梁和Timoshenko梁,详细分析了轴向荷载、剪切效应和转动惯量对自振频率的影响,研究发现,轴向压力荷载可使梁的自振频率降低,反之增大。剪切变形的影响约为转动惯量的3倍,随着主模态阶数的增加和长细比L/r的减小,轴向荷载、剪切变形和转动惯量的影响非常显著。因此,对于预应力混凝土梁,当跨高比L/h≤8,或长细比L/r≤28时,必须考虑轴向荷载、剪切变形和转动惯量的影响,通过与Bernoulli-Eular梁和Timoshenko梁的精确解相比较,证明该文的解答是正确的。

STUDY ON VERTICAL FREE VIBRATION OF PRESTRESSED BEAMS SUBJECTED TO AXIAL AND FLEXURAL LOADINGS

Based on the generalized potential energy variational principle with large deflection, the incomplete generalized potential energy functional for free vibration of prestressed beams is established by considering coupling of axial and flexural actions, shearing strain energy and moment of inertia. By variation of vertical displacement, the vertical free vibrational differential equation is formulated .With the consideration of boundary conditions the accurate solutions for free vibration frequencies of pin-ended beam and cantilever beam are obtained. Compared with the Bernoulli-Eular and Timoshenko beams, the effects on natural frequency due to axial loading, shearing behavior and rotation inertia are discussed in detail. It is found that the free vibration frequencies decreased due to compressive axial load, the effects caused by shearing deformation is about three times as large as that caused by rotation inertia, and with the increase in the orders of vibration modals, the effects due to axial load, shearing deformation and rotation inertia become remarkable. Therefore, if L / h≤8, or L / r≤28, such effects must be considered in natural frequencies solution for prestressed concrete beams. By comparison with Bernoulli-Eular and Timoshenko beams, the solution in this paper is proved to be correct.