Advances in Applied Mechanics, Vol. 14 by Chia-Shun Yih

By Chia-Shun Yih

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Results similar to those given by the dotted curves in Fig. 10 would then be found. , on the fundamental path. [For asymmetric bifurcations, where A - A. ] We find 1/K = (VKO) + “-$34)/2A21 or, since K/Ko = [l + ((h:~:/2A,$~)]-’. 56) This implies the variety of post-buckling behaviors indicated in Fig. 11. The , = 0, and for increasingly posiinitial post-buckling stiffness vanishes for i tive value of I,, approaches the prebuckling stiffness. -A curve becomes vertical. For 2, < I,,&/$A again approaches K O .

2, and I' = 0, and writing A w (h= (w,(h), gives t{&w2 R , &u, q5e,u: - A u , j 6w + + + <2j45:w3 + A, 4:w2 + A 2 & h l + fA:$:u, + (je,u2u, + + A~$;U: + $&"u: - Aw2 /j2Aul) 6w + . . = 0. 86) t In fact, d ' ' ( 6 ~is) ~then positive-definite with respect to 11 11. 39 Buckling Behavior qf Elastic Structures Setting 6w = u 1 in the coefficient of [I1 = < gives A1&hl + qyu:, where use has been made of the fact that Au: = (ul, u l ) [Eq. 21)] it follows that since 4Tu: = - 2 i 1 = 4:~: /(u11)2= 1.

Buckling Behai3ior Qf Elastic Structures 31 P FIG. 12. L 1. (dO2/ds)(dO,/ds)(1s - P j0 0,0, cos 6, (is, ' 0 4""0]0, 0,8, = P ). H , O , H , sin 0 ds, ' 0 @ " [ 0 ] U l 0, u, o4 = P I' 0, H 3 0, cos 0 (1s. 8) becomes @'[0]0, 60 = El [ I. 59) wherein 3, has been identified as P. s = 0 at s = 0, L. L p ) = (2/L) ' 0 we have 0 , = cos(ns/L), where l l H l l ~= 1. 21) vanishes, as it obviously should for the symmetrically bifurcating Euler column. 58) [ K0': = P, 1. : ds $Pc L = '0 and 1. 0: L (jy2 1 - - ds = -1 2L.

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