The Principle of Virtual Work for a Continuum: (c) Internal virtual work during the application of the virtual displacement. (b) Application of an arbitrary differentiable (small) virtual displacement. (a) Equilibrium position with the external force. The principle of virtual work in a mass-spring system. From an equilibrium position, the external work done by the external forces during the application of a small virtual displacement is equal to the internal work done by the spring force during the application of that small virtual displacement ( Figure 1).įigure 1. The above equation represents the statement of the principle of virtual work in a single degree of freedom system. If the position of the spring is perturbed by an arbitrary small displacement ( Figure 1) and if the equilibrium equation above is multiplied by, then: Let the position of equilibrium be at ( Figure 1). At equilibrium, the sum of the vertical forces is equal to zero, and the force in the spring is equal to the applied external load. Assume that the force in the spring is a function of the displacement of the spring, i.e. The Principle of Virtual Work for a Single Degree of Freedom:Ĭonsider the static equilibrium of a mass spring system. A system that is in equilibrium should satisfy the statement of the principle of virtual work and vice versa. The principle of virtual work itself is a different manifestation of the equilibrium equations, as it is an equivalent form of equilibrium. In the second and third, we will apply the principle to the continuum and to beams. In the first, we will investigate the principle in its simplest form as it applies to a single degree of freedom. In this section, the principle of virtual work will be presented for three applications. However, as will be shown in the following sections, the statement itself can be derived solely from mathematical, rather than physical explanations. The statement of the principle of virtual work usually involves the phrase “virtual displacement field,” which is designed to engage the intuition by attempting to give a physical explanation to a mathematical statement. The principle of virtual work is widely used to solve a variety of continuum and solid mechanics problems.
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