用材料力学公式计算任意截面短梁正应力时的修正项研究

对材料力学中梁的弯曲应力公式增加一修正项,以反映短梁弯剪翘曲变形对应力分布的影响。提出一种根据短梁横截面边界形状及艾瑞应力函数求解应力修正项的方法,应用弹性力学空间问题的一般理论,通过应力平衡方程、应变相容方程及应力边界条件,建立了关于任意截面短梁的应力修正项及剪应力的基本方程。在所建立的基本方程基础上,导出了矩形截面和圆形截面短梁修正应力的具体计算公式,该修正应力与均布荷载大小及弹性模量与剪切模量之比均成正比,但与截面惯性矩成反比。数值算例表明,本文方法计算的应力与通用有限元软件ANSYS计算的结果吻合良好,从而验证了本文方法及其基本公式的正确性。

Study on modification term for normal stress of short beam with arbitrary cross section by applying formula in material mechanics

The formula of bending stress of beam in mechanics of materials is modified by introducing a stress term to reflect the effects of bending-shear warping deformation on the stress distribution.Based on the boundary shape of cross section of a short beam and the Airy stress function,a method for solving the modification stress is proposed.The fundamental equations for determining the stress term and the shear stress of a short beam with arbitrary cross section are established by applying the general theory of spatial problem in Elasticity,where the stress equilibrium equations,the strain compatibility equations and the stress boundary conditions in Elasticity are simultaneously applied.The specific formulas of the stress term of the short beams with rectangular and circular cross section are derived on the basis of the fundamental equations established.The stress term is proportional to the magnitude of uniform load and the ratio of elastic modulus to shear modulus,but inversely proportional to the inertia moment of cross section.Numerical example shows that the stresses calculated by the present method are in a good agreement with those by the general finite element software ANSYS,which validates the correctness of the analytical method and the fundamental equations established.