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1, but there is n o M? # 0 for which (1 - s)kzd x vanishes. Such situations are unlikely to arise in well-formulated problems of mechanics. 1; Bernard Budiansky 38 Consequently, for sufficiently small <, & ' ( S U ) ~must have the same sign as 4 : ( 6 ~ ) ~which , is positive for A < 4 . Thus, stability up to snapping has been established, for small enoygh imperfections. In the case of an imperfection-insensitive structure (Fig. 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.

52) which says that at the critical load the shortening is stationary with respect to the buckling displacement. With the prebuckling shortening defined by A. = - (d/dA)+[uo(A):A] = -$o we have A - A0 A] = -( d / d A ) { 4 [ ~ ; = -(d/dA){&v - - where u = dv/&,. $bv 401 = + + +&v' - l$,9 + 2 0 .. } 4gv + L2 V0L ? + . ]b, But the bracketed expression is the Taylor expansion of 4"uo + c; A] 4; L', since 4' 6 u = about i i o , and, by equilibrium @b must vanish; so does 4; 6u = 0 all along the fundamental path.

W>/t'yj', and the fundamental state is cO = r x , w,, = 0, this condition is satisfied. 48 Bernard Budiansky where the last equality follows from Eq. 13). 12 + 0 2 1::11:/0, ,:,1: + r T 2 1 : : U ~ / 0 ,E l . 26") The results are essentially those given by Budiansky (1965) and Budiansky and Amazigo (1968). All of the formulas of the preceding section concerning the generalized shortening and the post-buckling stiffness [Eqs. 56)] are immediately transformable to the notation of the present section simply by = -A,= -AAii = -(I/%,)q C,, by replacing &'u: by D.

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