Buckling of a column under pre-load restraint has been found to differ from the conventional bifurcation problems and thus requires better understanding. To this end, a simple model consisting of two collinear rigid bars connected by a rotational spring, resting on a smooth surface, loaded by an axial force and restrained by a constant force at the common joint is investigated. The energy method is adopted to establish the equilibrium and stability conditions. The post-buckling response of the simple system indicates that the load-restrained buckling problem is a snap-through problem. The minimum possible buckling force and the buckling force based on the Maxwell construction are found to increase with the restraining force, and can be approximated by power-law functions. It is further shown that an energy barrier exists for the snap-through to occur and the buckling force depends on the perturbation energy. Sensitivity of this system to initial geometric imperfections is also investigated, and an energy barrier is also found to exist for the buckling of systems with imperfection amplitudes smaller than a certain critical value. The characteristic imperfection amplitude depends on the restraining force and, interestingly, transforms the snap-through problem into an eigenvalue problem identical to the conventional one, i.e., the one without imperfection and restraining force.

已经发现，在预载荷约束下，柱的屈曲不同于常规的分叉问题，因此需要更好的理解。为此，研究了一个简单的模型，该模型由两个由旋转弹簧连接的共线刚性杆组成，该刚性弹簧放置在光滑表面上，并受到轴向力的加载，并在公共接头处受到恒定力的约束。采用能量法建立平衡和稳定条件。简单系统的后屈曲响应表明，受载荷约束的屈曲问题是一个直通问题。发现最小可能的屈曲力和基于麦克斯韦构造的屈曲力随约束力而增加，并且可以通过幂律函数近似。进一步表明，存在能量屏障，以使发生突跳，并且屈曲力取决于扰动能量。还研究了该系统对初始几何缺陷的敏感性，并且还发现了对于缺陷幅度小于某个临界值的系统屈曲存在的能量屏障。特征缺陷幅度取决于约束力，并且有趣的是，将捕捉问题转换为与常规问题相同的特征值问题，即没有缺陷和约束力的特征值问题。并且还发现对于缺陷幅度小于某个临界值的系统的屈曲也存在能量障碍。特征缺陷幅度取决于约束力，并且有趣的是，将捕捉问题转换为与常规问题相同的特征值问题，即没有缺陷和约束力的特征值问题。并且还发现对于缺陷幅度小于某个临界值的系统的屈曲也存在能量障碍。特征缺陷幅度取决于约束力，并且有趣的是，将捕捉问题转换为与常规问题相同的特征值问题，即没有缺陷和约束力的特征值问题。