简支薄壁箱梁自由振动的摄动法解析解

为研究简支薄壁箱梁中剪力滞效应对结构动力特性的影响,基于模态摄动法提出了一种求解薄壁箱梁自由振动的新方法.该方法以相同跨度等截面欧拉梁的频率和模态为Ritz基函数,将箱梁自由振动的微分方程组转化为一组非线性代数方程进行求解.对于简支箱梁,可进一步将代数方程组简化为一元二次方程,从而得到精确的特征值和特征向量.在此基础上,利用所得箱梁振动模态的解析表达式,提出了模态剪力滞系数的概念,从而建立了箱梁固有频率和剪力滞效应之间的关系.随后研究了模态剪力滞系数随跨宽比、翼板抗弯刚度和截面抗弯刚度之比的变化规律.计算结果表明:简支梁腹板处剪力滞系数最大且大于1,为正剪力滞效应;随着模态阶数的增加、跨宽比的减小和翼板抗弯刚度和截面抗弯刚度之比的增大,剪力滞效应呈现增大的趋势.

Analytic solution of free vibration of simply-supported thin-walled box girder by perturbation method

To investigate the shear lag effect of thin-walled box girder on the dynamic characteristics, a new approach was developed to analyze the free vibration of box girders based on the modal perturbation method. The natural modes of vibration of the corresponding prismatic Euler beam with the same length and boundary conditions were used as Ritz base functions. Then, the new method can transform the set of partial differential equations governing the transverse vibration of the box girder into a set of nonlinear algebraic equations. For the simply-supported beams, the algebraic equations were further simplified as quadratic equation with one unknown, so that the exact eigenvalues and eigenvectors could be obtained. The analytical vibration modes of the box girder were used to propose the shear lag coefficients of modes, which illustrates the relationship between the natural frequency and shear lag effect. Numerical examples were used to analyze the shear lag coefficients of modes varying with the ratio between span and width, the second moment of area ratio between flange slab and the full section. The numerical results show that the maximum shear lag coefficients of modes located at the web of the box girder are greater than 1, which are positive shear lag effect. As the increase of modes order, the reduction of the ratio between span and width and the increase of the second moment of area ratio between flange slab and the full section, the shear lag coefficients and shear lag effect would be more remarkable.