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F/dx)(u, 0) - (ZF/dx)(O, 0) = - ~P,/u. /2u)[P,y2 + Pyx2]. 45) 51 Buckling Behauior qf Elastic Structures Assuming that P, and P, are fixed in magnitude, we look for the critical value of the scalar multiplier A. x2) + P,(d2w,/dy2)] = 0, V4F1 = 0. ~/a)sin(n~~/a), F , 2, = = O,? 47) ~TC’D/LI(P, P,), ~ ~t. The bifurcation is clearly symmetric, so where we have made ( w ~ ) , , ,= A 1 = 0. 48), for r r ~ j orthogonality ) condition ( i v 2 , w l ) w2 = 0, F, = (Et3/32)[cos(2nx/a) + cos(2ny/a)]. 49) To calculate A,, we note that all of the simplifying assumptions (i)-(iv) discussed earlier are satisfied, and we can use Eqs.
53) and the condition &'u: < 0 previously mentioned, A - A. > 0. These results are, again, similar to those given in Eqs. 13) for the simple model (in which there was no prebuckling shortening Ao). It is worth emphasizing that they are quite general, including cases in which the prebuckling variation of A. with 2 is not linear. relations to Some typical initial A-A curves, together with post-buckling jL-< which they correspond, are sketched in Fig. 10. I P E E (C) y - A & A A E A (d) FIG. 10. General load-shortening relatiom.
9b,d) the same conclusion clearly applies for A < A,. 8 1) ;-ro where L' is the displacement increment along the bifurcated path originally defined in Eq. 3). 79) for A > A, is now about u = uo u and LI = 0, and this gives + $"(6U)Z = (b"[uo + v(A); )L](6t/)2 + $""uo + L'(A); 149 - + ... I~)(dU)Z + ~ ~ " * [+u uo , o ; A];l(6U)' + . ' = 0. 83) z. + = (du, 6 u ) = I l f i ~ (and / ~ AuT = 1, with respect to an admiswhere sible (6u). 83) be stationary with respect to variations in w implies that w and p are solutions of the eigenvalue problem 6''~ 6~ - j 3 A 6~ ~ = 0 on the bifurcated path.
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