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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) BettiRayleigh 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.
 Fall '05
 Thomson

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