00026___9a4618c862ee2f5aef2824f7cc2a5ed2 - elastic property...

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Sec. 1.1 HOOKE’SLAW 6 of PI). Similarly, ~32 - cL2 can only be a function of 4. If we write Eq. (d) as then the left-hand side is a function of P2 alone, and the right-hand side is a function of PI alone. Since PI and Pz are arbitrary numbers, the only possibility for Eq. (e) to be valid is for both sides to be a constant k which is independent of both PI and P2. Hence, But a substitution of (f) into (a) yields The last term is nonlinear in PI, P2, and Eq. (g) will contradict (H2) unless k vanishes. Hence, k = 0 and ci2 = ~32. An analogous procedure shows Thus the principle of superposition is established for one and two forces. An entirely similar procedure will show that if it is valid for m forces, it is also valid for m + 1 forces. Thus, the general theorem follows by mathematical induction. Q.E.D. The constants ~31, ~32, etc., are seen to be of significance in defining the
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Unformatted text preview: elastic property of the solid body. They are called influence coeficients or, more specifically, flexibility influence coefficients. C i l = cp1 = c31. (B) Corresponding forces and displacements and the unique meaning of the total work done by the forces. Let us now consider a set of external forces P I , . . . , P, acting on the body and define the set of displacements at the points of application and in the direction of the loads as the displacements “corresponding” to the forces at these points. The reactions at the points of support are considered as external forces exerted on the body and included in the set of forces. Under the loads P I , . . . , P,, the corresponding displacements may be written as u1 = C11Pl + c12P2 +. . * + ClnPn, 212 = C21Pl + c22P2 + ‘ * . + %Pn, 21n=cn1P1 +cn2P2+***+cnnPn. ...... ............................ (2)...
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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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