双集中荷载下一次超静定梁的加载过程分析

Analysis of the Loading Process of a Beam with One Degree of Indeterminacy under Double Concentrated Load

对双集中荷载作用下一次超静定梁的弹塑性加载全过程进行了分析.根据变形特点可把加载过程分成四个阶段.第一阶段是常规的弹性阶段,第二阶段是固支端附近单个塑性变形区扩展的阶段,第三阶段是固支端和一个集中荷载附近两个塑性变形区同时扩展的阶段,第四阶段是固支端保持为塑性铰从而引起固支端附近区域卸载而梁中间的塑性变形区继续扩展直至形成第二个塑性铰的阶段.在第一阶段,弯矩内力和挠度与外荷载均是线性比例递增关系;在第二、第三两个阶段,弯矩和挠度与外荷载是不同的非线性关系;在第四阶段,弯矩与外荷载是非比例的线性关系,但挠度与外荷载却是复杂的非线性关系.给出了加载各阶段的弯矩及挠度计算公式,具有理论意义,也可供对应的结构设计应用.

The entire elastic-plastic loading process of a beam with one degree of indeterminacy under double concentrated load is analyzed. Based on the deformation, the entire loading process can be divided into four stages. The first stage is elastic stage. The second stage is associated with the propagation of the plastic region near the fixed end. The third stage is associated with the propagation of two plastic regions near the fixed end and near the application position of one concentrated load. The fourth stage is associated with the formation of a plastic hinge at the fixed end in which the plastic region near the fixed end is unloaded while the plastic region near the concentrated load is propagating until the emergence of the second plastic hinge. In the first stage, the relations between the moment, the deflection and the load are proportional. In the second and third stages, the relations between the moment, the deflection and the load are nonlinear. In the fourth stage, the relation between the moment and the load is non-proportional linear, while the relation between deflection and the load is highly nonlinear. The formulas of moment and deflection at each stage are derived. These formulas are of theoretical importance and can be applied to engineering structure design.