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2.8.4.2 The Theorem of Virtual Work

The Theorem of Virtual Work can be derived by considering the beam shown in Figure 2.63. The full curved line represents the equilibrium position of the beam under the given loads. Assume the beam to be given an additional small deformation consistent with the boundary conditions. This is called a virtual deformation and corresponds to increments of deflection Dy1, Ay2,..., Ayn at loads P1, P2,..., Pn as shown by the dashed line. The change in potential energy of the loads is given by n

By the Law of Conservation of Energy this must be equal to the internal strain energy stored in the beam. Hence, we may state the Theorem of Virtual Work as: ''if a body in equilibrium under the action of a system of external loads is given any small (virtual) deformation, then the work done by the external loads during this deformation is equal to the increase in internal strain energy stored in the body.''

### 2.8.4.3 The Theorem of Minimum Potential Energy

Let us consider the beam shown in Figure 2.64. The beam is in equilibrium under the action of loads, P1, P2, P3,..., P,,..., Pn. The curve ACB defines the equilibrium positions of the loads and reactions. Now apply by some means an additional small displacement to the curve so that it is defined by AC'B. Let yi be the original equilibrium displacement of the curve beneath a particular load P,. The additional small displacement is called dy,. The potential energy of the system while it is in the equilibrium configuration is found by comparing the potential energy of the beam and loads in equilibrium and in the undeflected position. If the change in potential energy of the loads is W and the strain energy of the beam is V, the total energy of the system is

If we neglect the second-order terms, then dU = d(W + V)=0 (2 .116)

The above is expressed as the Principle or Theorem of Minimum Potential Energy, which can be stated as, ''if all displacements satisfy the given boundary conditions, those that satisfy the equilibrium conditions make the potential energy a minimum.''

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