引用本文: 纳新刚. 由结构力学的平面问题例说最小势能原理. 力学与实践, 2023, 45(6): 1409-1413.
Na Xingang. Examplification of the principle of minimum potential energy for plane problems of structural mechanics. Mechanics in Engineering, 2023, 45(6): 1409-1413.
 Citation: Na Xingang. Examplification of the principle of minimum potential energy for plane problems of structural mechanics. Mechanics in Engineering, 2023, 45(6): 1409-1413.

## EXAMPLIFICATION OF THE PRINCIPLE OF MINIMUM POTENTIAL ENERGY FOR PLANE PROBLEMS OF STRUCTURAL MECHANICS

• 摘要: 最小势能原理作为结构力学的提高部分，很少在该课程中看到关于它的完整介绍。本文以等刚度连续梁为研究对象，在只考虑弯曲变形的情况下，讨论了位移法典型方程的适用条件。在此基础上，分析了势能法和位移法之间的对偶关系。得出：等刚度连续梁在平面载荷与支座反力构成的平衡力系作用下，可能的小位移状态由虚力方程控制。若真实位移还能利用叠加原理进行求解，则可能位移状态下的总势能在真实位移处取极小值。等刚度连续梁弯曲变形的分析验证了最小势能原理，有助于对一般情况下最小势能原理的深刻认识。

Abstract: The principle of minimum potential energy is an advanced part of structural mechanics, whose complete introduction is rarely seen in this subject. This paper discussed the application of canonical equations in displacement method when considering the bending deformation for beam with constant stiffness. On this basis, the dual relationship between the potential energy method and the displacement method was analyzed. It is concluded that under the balanced force composed of in-plane loads and support reactions, the possible small displacement of the beam is controlled by the equation of virtual force. Meanwhile, if the real displacement can also be solved by using the superposition principle, then the total potential energy at the real displacement will be the minimum. The principle of minimum potential energy is thereby examined by the analysis of bending deformation for beam of constant stiffness, which is helpful to understand the principle of minimum potential energy in general.

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