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3). For the symmetric modification, the deformation U before bifurcation remains symmetric. 13) with Λc0 = Λc (0). 5. 6, respectively. 13) represents the 2/3-power law for C1 = 0 but reduces to the linear law for C1 = 0 and C2 = 0. 5. 17) Thus Λc is differentiable with respect to ξ. 18) c Thus the sensitivity coefficient Λ for a minor modification takes a finite value C2 , and, therefore, is called regular sensitivity. In the following sections, several numerical methods to evaluate the sensitivity coefficient C2 compatibly with a large-scale finite element model are presented focusing on minor modifications, for which Λc exists.

10) where δij is the Kronecker delta. Note that so-called stability coefficients D,ii (i = 1, . . , n) are nonzero at regular points, but some of them become zero at a critical point. t. = 0, (i = 1, . . 3 V -formulation A nonlinear governing equation of a structure subjected to buckling in general involves a large number of independent variables and nonlinear terms and, hence, is highly complex. In the general nonlinear theory of stability [166], to obtain asymptotic general forms of imperfection sensitivity laws, the nonlinear governing equation was simplified on the basis of the following two steps: Step 1: The governing equation is reduced to the bifurcation equation with only a few active independent variables by the Liapunov–Schmidt–Koiter reduction [50, 99, 266].

Instead of this, Λc is to be approximated from the two loads ΛI and ΛII at which λ1 0. 7) for r = 1 with respect to ξ for a specified value λ∗1 of λ1 , and by use of λ1 = 0 that arises from λ1 = 0, we arrive at the following set of relations [227] n S,ij Uj∗ + S,iξ + S,iΛ Λ∗ = 0, (i = 1, . . , n) j=1 n n j=1 k=1 S,ijk φ1j Uk∗ + S,ijξ φ1j + S,ijΛ φ1j Λ∗ + S,ij φ∗1j (i = 1, . . 28) j=1 Here ( · )∗ denotes the sensitivity coefficient at λ1 = λ∗1 . 27), as S,ijk amounts to the third-order differentiation of the total potential energy.

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