00027_687c22be2d77 - easily seen to be(4 1 2 W =(C11P,” c22P3 Cl2PlP2 When the order of application of the forces is interchanged the work done

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6 INTRODUCTION Chap, 1 If we multiply the first equation by PI, the second by P2, etc., and add, we obtain (3) 9.1 + P2u2 + . . . + Pnu, = CllP? + C12PlP2 + . . * + ClnPlP, + C21PlP2 + c22Pz” + . . . + c,,P,2. The quantity above is independent of the order in which the loads are applied. It is the total work done by the set of forces. (C) Maxwell’s reciprocal relation The influence coeficients for corresponding forces and displacements are symmetric. c.. %J - c,, 32 * In other words, the displacement at a point i due to a unit load at another point j is equal to the displacement at j due to a unit load at a, provided that the displacements and forces ‘%orrespond,” i.e., that they are measured in the same direction at each point. The proof is simple. Consider two forces P1 and P2 (Fig. 1.1:l). When the forces are applied in the order PI, P2, the work done by the forces is
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Unformatted text preview: easily seen to be (4) 1 2 W = -(C11P,” + c22P3 + Cl2PlP2. When the order of application of the forces is interchanged, the work done is 1 2 W’ = -(C22r)22 + CllP,”) + C21PlPZ. But according to (B) above, W = M” for arbitrary PI, P2. Hence, c12 = c21, and the theorem is proved. (D) Betti-Rayleigh reciprocal theorem Let a set of loads P I , P2, . . . , P, produce a set of corresponding displace- ments u1, u2,. . . ,u,. Let a second set of loads Pi,Ph,. . . ,PL, acting in the same directions and having the same points of application as those of the first, produce the corresponding displacements ui, ui, . . . , uk. Then (5) Plu: + P2Uk + . . . + P,.:, = P[u, + * . . + P{u,, In other words, in a linear elastic solid, the work done by a set of forces acting through the corresponding displacements produced by a second set of...
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This note was uploaded on 01/04/2012 for the course ENG 501 taught by Professor Thomson during the Fall '05 term at MIT.

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