00025___ddfaf8ac0f0ae87783fa4a4196b3b8a7 - direction be...

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4 INTRODUCTION Chap. 1 (H3) There exists a unique unstressed state of the body, to which the body returns whenever all the external forces are removed. A body satisfying these three hypothesis is called a linear elastic solid. A number of deductions can be drawn from these assumptions. We shall list a few important ones. (A) Principle of superposition By a combination of (H2) and (H3), we can show that Eq. (1) is valid not only for systems of loads for which the ratios PI : P2 : . . . : P, remain fixed as originally assumed, but also for an arbitrary set of loads PI, P2, . . . , P,. In other words, Eq. (1) holds regardless of the order in which the loads are applied. Proof. If a proof of the statement above can be established for an arbi- trary pair of loads, then the general theorem can be proved by mathematical induction. Let P1 and P2 (with magnitudes PI and P2) be a pair of arbitrary loads acting at points 1 and 2, respectively. Let the deflection in a specific
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Unformatted text preview: direction be measured at a point 3 (see Fig. 1.1:l). According to (H2), if P1 is applied alone, then at the point 3 a deflection u3 = ~ 3 1 P1 is produced. If Pz is applied alone, a deflection u3 = c32P2 is produced. If P1 and Pa are applied together, with the ratio PI : Pz fixed, then according to (H2) the deflection can be written as (4 u3 = C k 1 Pl + CL2 P2 . The question arises whether cil = ~ 3 1 , cb2 = ~ 3 2 . The answer is af- firmative, as can be shown as follows. After P1 and P2 are applied, we take away P I . This produces a change in deflection, -c&P1, and the total deflection becomes (b) u3 = C k 1 Pl + Ck2P2 - c;1 Pl . Now only P2 acts on the body. Hence, upon unloading Pz we shall have Now all the loads are removed, and u3 must vanish according to (H3). Rearranging terms, we have Since the only possible difference of cbl and c& must be caused by the action of Pz, the difference cil - cgl can only be a function of P2 (and not...
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