Buckling and Prestressed Vibrations of Periodic Nonprismatic Beams Using Integral Equation Approach

Abstract

This paper presents the use of Fredholm–Volterra integro-differential equation approach for determining the critical buckling load and natural frequencies of periodically supported nonprismatic Euler–Bernoulli beams. Neumann series solution is used to solve the integro-differential equation by using the orthogonal Chebyshev polynomials, to avoid ill-conditioning. Gauss quadrature is used to perform various needed integrals. Critical buckling load, natural frequencies, and the corresponding mode shapes are compared with those available in the literature, those obtained using the principle of virtual work, and the ones obtained using NASTRAN®. The effect of variation of taper ratio, both in width and height of a rectangular cross section, on the nondimensional critical buckling load and the nondimensional natural frequency, is also investigated. A singularity is observed near the taper ratio of unity, which impacts the accuracy of both the finite element and the principle of virtual work results. The integral equation approach converges with a lesser number of terms than the principle of virtual work, especially for higher modes for beams approaching the taper ratio of unity. Benchmark examples for free vibration of a multiple-bay beam and of inflated tube are also studied, and the results are compared with those presented in the literature.