Determination of Buckling Criteria By Minimization of Total Energy

Samuel Lubkin

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bn. R 3'b b . C'A, -=; ' f ■ sr'j ^^ kw. ). '.. '^' ■>; +■ ^; * 4- "^ *^' ; -). I -:;-:(;- ^ ^f •^ 27 Hence 2, -1, ,2, -2n . , 2n, -2, 2 ^ (k"-" - k)^ - (k"^''+ k^" - k'" - k") = /;2(k-l. K)2, (k-^-k^)2-. (k-l-k)2 ;^ (..
-•>:{}i^ f D^. I - cf) - '">i( 'iS •! . R > u: '(3^ •"-- -f J f ; I ;■ , -y 1 i •• . -f y I'^^jul nicorir ct^i. '/nv A -. ^-M ^. ', ^1^ I '. I I J. •- J -"MU'S ■;• LWI -^) s • -•xoU f%.^'-'\^xo'i^''- '?"' : /\-- - ■-/ V - «!
•/ -■. - J; r ia t. U- ■':h ■•• . F ■'i r- . -. A 28 ...corresponding to closure of the center to half its original size, since b(l -f A + ^.. ) = b[l + ^^ * ^^"^ A] b^ ^ Thus instability always occurs before compression is this large for any value of k. Furthermore, since s increases monotoni- cally with A, we have, below this limit, s . K^ = tLJ:_lAl2i^lA /_ ^2 \i + [^ + (\+ij, )k jA >. + 2ll, 2 > c ^ _ 1^ 2\ + 3[i - {X + |x)k (X + 2^) + (X + ^)(1 - k^) so that instability due to negative ^ always occurs at a lower 2 deformation than that corresponding to s

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