Modern Trends in Structural and Solid Mechanics 2. Группа авторов

Modern Trends in Structural and Solid Mechanics 2 - Группа авторов


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of DEE when decaying solutions cannot be constructed for some wave numbers (Bolotin 1984). The resolution of this problem was proposed in Elishakoff (1974) and Elishakoff and Wiener (1976). The solution of the original problem is represented as a sum of solutions of two subproblems. Each of these solutions satisfy the boundary conditions at two opposite boundaries only. The matching conditions described in Bolotin’s original papers (Bolotin 1960a, 1960b, 1961a, 1961b, 1961c) are not used, and it gives us the possibility to avoid difficulties caused by the degeneracy of DEEM.

      Other generalizations of DEEM are described in the following sections.

      To describe the generalization of DEEM to the nonlinear case, we use the nonlinear Kirchhoff beam equation (Kauderer 1958):

      Let the beam be elastically supported:

      where c*= c/EI, c is the coefficient characterizing elastic support.

      We search a generating solution in the form

      where

image

      [1.23] image

      has the solution

      [1.24] image

      where cn(σt, k) is the Jacobi cosine elliptic functions with period T = 4K, in9-1.gif is the complete elliptic integral of the first kind with modulus in9-2.gif (Abramowitz and Stegun 1965).

      The solution to the problem far from the edges is

      where in9-3.gif.

      [1.28] image

      Restricting ourselves to the term of order (π/λ)2 ≫ 1 in equation [1.27], we reduce it to the form:

      where in10-1.gif.


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