Abstract:To derive the closed-form solution for the in-plane free vibration of non-circular arches, an approximate analytical method for the solution derivation of the variable coefficient differential equation is proposed. Based on linear strains of non-circular arches in the Cartesian coordinate system and the Hamilton principle, a variable coefficient equilibrium differential equation for the in-plane free vibration of non-circular arches is derived. The analytical general solution of the corresponding constant coefficient differential equation is substituted into the variable coefficient differential equation to yield the unbalanced deviation following by that the in-plane free vibration modes of non-circular steep arches and shallow arches are almost the same. The practical closed-form solution for the in-plane natural frequency is derived by setting the integration of the unbalanced deviation of the variable coefficient differential equation along the span being zero. The analytical solutions of the pin-ended and fixed arch in the Cartesian coordinate system are derived based on the proposed method, and the theoretical relationship of the in-plane frequency between non-circular steep arches, shallow arches and the straight beams with the same parameter are demonstrated. The numerical results of non-circular pin-ended and fixed arches show that, the basic assumptions has been strictly verified; the proposed method agrees well with the finite element method, the maximum relative error of the former tenth natural frequency are 7.71% and 4.34% for pin-ended arches and fixed arches, respectively; the theoretical relationship of natural frequency between non-circular arches and straight beam with the same parameter can be used in the code revision of arch bridges.
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