00028___eafe16f172bc3f4c11196b41077912c2 - corresponding...

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Sec. 1.1 HOOKE'SLAW 7 forces is equal to the work done by the second set of forces acting through the corresponding displacements produced by the first set of forces. A straightforward proof is furnished by writing out the u( and u: in terms of Pi and Pi, (i = 1,2,. . . ,n), with appropriate influence coeffi- cients, comparing the results on both sides of the equation, and utilizing the symmetry of the influence coefficients. In the form of Eq. (5), the reciprocal theorem can be generalized to include moments and rotations as the corresponding generalized forces and generalized displacements. An illustration is given in Fig. 1.1:2. These theorems are very useful in practical applications. For the same beom, cPf = ct2. (a) Forces and corresponding displacements. (b) Generolized force (moment) and the
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Unformatted text preview: corresponding generolized displocement (moment -rotation of angle). Fig. 1.1:2. Illustration of the reciprocal theorem. (E) Strain energy Further insight can be gained from the first law of thermodynamics. When a body is thermally isolated and thermal expansions are neglected the first law states that the work done on the body by the external forces in a certain time interval is equal to the increase in the kinetic energy and internal energy in the same interval. If the process is so slow that the kinetic energy can be ignored, the work done is seen to be equal to the change in internal energy. If the internal energy is reckoned as zero in the unstressed state, the stored internal energy shall be called strain energy. Writing U for the...
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