00029___7d5c422c8a70c8f7eb7d2278bc417063 - =Pa dU d U i i =...

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8 INTRODUCTION Chap. 1 strain energy, we have, from (3) and (4), If we differentiate Eq. (6) with respect to Pi, we obtain But, the right-hand side is precisely ui; hence, we obtain (F) Castigliano’s theorem (7) - =ua, dU dPa i=1,2 ,.*., 72. i=l, ..., n. In other words, if a set of loads PI,. . . , P, is applied on a perfectly elastic body as described above and the strain energy is expressed as a function of the set PI,. . , , P,, then the partial derivative of the strain energy, with respect to a particular load, gives the corresponding displacement at the point of application of that particular load in the direction of that load. (E) The principle of virtual work On the other hand, for a body in equilibrium under a set of external forces, the principle of virtual work can be applied to show that, if the strain energy is expressed as a function of the corresponding displacements, then
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Unformatted text preview: - =Pa, dU d U i i = 1, ..., n The proof consists in allowing a virtual displacement 6u to take place in the body in such a manner that Su is continuous everywhere but vanishes at all points of loading except under P,. Due to 6u, the strain energy changes by an amount 6U, while the virtual work done by the external forces is the product of Pi times the virtual displacement, i.e., Pihi. According to the principle of virtual work, these two expressions are equal, SU = Pidui. On rewriting it in the differential form, the theorem is established. The important result (8) is established on the principle of virtual work as applied to a state of equilibrium under the additional assumption that a strain energy function that is a function of displacement exists. It is a p plicable also to elastic bodies that follow the nonlinear load-displacement relationship....
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