Section 5.5 Solid Mechanics Part I Kelly2005.5 Virtual Work Consider a mass attached to a spring and pulled by an applied force aplF, Fig. 5.5.1a. When the mass is in equilibrium, 0=+aplsprFF, where kxFspr−=is the spring force with xthe distance from the spring reference position. Figure 5.5.1: a force extending an elastic spring; (a) block in equilibrium, (b) block not at its equilibrium position In order to develop a number of powerful techniques based on a concept known as virtual work, imagine that the mass is not in fact at its equilibrium position but at an (incorrect) non-equilibrium position xxδ+, Fig. 5.5.1b. The imaginary displacement xδis called a virtual displacement. Define the virtual workWδdone by a force to be the equilibrium force times this small imaginary displacement xδ. It should be emphasized that virtual work is not real work – no work has been performed since xδis not a real displacement which has taken place; this is more like a “thought experiment”. The virtual work of the spring force is then xkxxFWsprsprδδδ−==. The virtual work of the applied force is xFWaplaplδδ=. The total virtual work is ()xFkxWWWaplaplsprδδδδ+−=+=(5.5.1) There are two ways of viewing this expression. First, if the system is in equilibrium (0=+−aplFkx) then the virtual work is zero, 0=Wδ. Alternatively, if the virtual work is zero then, since xδis arbitrary, the system must be in equilibrium. Thus the virtual work idea gives one an alternative means of determining whether a system is in equilibrium. The symbol δis called a variationso that, for example, xδis a variation in the displacement(from equilibrium). Virtual work is explored further in the following section. sprFaplFsprFaplFexxδ)a()b(
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