考虑地基摩阻时Timoshenko梁弯曲的Fourier级数解答

The Fourier Series Solutions of Timoshenko Beam on Elastic Foundation Considering Longitudinal Frictional Resistances

假定纵向摩擦力与梁底面的纵向位移成正比,引入广义剪力,得到广义Winkler型地基上Timoshenko梁的位移型平衡方程。将位移及荷载展开为带附加项的Fourier级数,利用平衡方程和边界条件对弹性地基梁的一般弯曲进行分析。该解答退化后可用来研究传统的Winkler地基上Bernoulli-Euler梁的弯曲问题。分析表明,纵向摩擦力对梁的位移和内力均有影响。梁的最大挠度、转角、弯矩及剪力随着地基纵向摩擦力的增大而减小,而梁的轴向位移和轴力则随着地基纵向摩擦力的增大而增大。

The equilibrium equations of beam on the general Winkler elastic foundation were established with considering longitudinal friction. Supposing the longitudinal friction is proportional to the beam＇s base longitudinal displacement and introducing a general shearing force, the governing equations, in the forms of displacement, of Timoshenko beam on the general Winkler elastic foundation were obtained. Ex- panding the displacements and loads to Fourier series with additional terms and using the method of dis- placement, the bending problems of elastic foundation beam were analyzed in terms of the equilibrium e- quations and boundary conditions. The results indicate that the beam＇s displacements and internal forces are related to the longitudinal friction. When the friction force coefficient is increased, the maximums of deflection, slope, bending moment and shear force are decreased but the maximums of axial displcement and axial force are increased.