基于弹性理论的楔形梁弯曲性能研究

Research on the Bending Performance of TaperedBeams Based on Elastic Theory

基于材料力学平衡原理和弹性力学平面问题几何方程,建立矩形截面悬臂楔形梁挠度微分方程和弯曲应力表达式。对挠度微分方程进行积分,结合弯曲变形边界条件,推导了集中荷载、均匀分布荷载、弯曲力偶等3种简单荷载工况作用下悬臂楔形梁挠度解析解,对弯曲应力求导,可确定任意荷载作用下楔形梁弯曲最大应力值。通过通用结构设计与分析软件SAP2000分别验算了3种简单荷载工况作用下悬臂梁的弯曲问题:在计算弯曲挠度时,发现本文计算方法均与软件计算结果相当,且略大于软件计算结果,这是由于软件计算时未考虑截面转动对挠度的影响所致;在计算最大弯曲应力时,发现当楔形梁截面变换系数小于0.4时,采用本文计算方法具有较高的计算精度,可满足一般土木工程设计要求,具有一定的工程应用价值。

Based on the equilibrium principle of material mechanics and the geometric equation of plane problems in elasticity,the deflection differential equation and bending stress expression of cantilever tapered beams with rectangular cross-section are established.The deflection differential equation is integrated.Combined with the boundary conditions of bending deformation,the analytical solutions of the deflection of cantilever tapered beams under three simple load cases of concentrated load,uniformly distributed load and bending couple are derived.By taking the derivative of bending stress,the maximum bending stress value of tapered beams under arbitrary loads can be determined.The bending problems of cantilever beams under three simple load cases are verified respectively by using the general structural design and analysis software SAP2000.When calculating the bending deflection,it is found that the calculation methods here are all comparable to the software calculation results and are slightly larger than them.This is due to the fact that the influence of section rotation on deflection is not considered in software calculation.When calculating the maximum bending stress,it is found that when the section transformation coefficient of tapered beams is less than 0.4,the calculation method here has high calculation accuracy and can meet the design requirements of general civil engineering and has certain engineering application value.