Keywords: semi-tangential moments, bi-axial moment buckling, buckling torque, exact dynamic stiffness.

The rigid body concept proposed by Yang and Kuo is a significant contribution to the convergence problem in geometric nonlinear structural analysis so that one can reduce a nonlinear problem to a series of linear problems of initial stress [1,2]. The exact dynamic stiffness matrix is useful in spectral analysis including response analysis using Fourier integral and aerodynamic random analysis using auto-correlation. The paper extends the existing dynamic stiffness to cover all possible types of initial stresses due to axial and shear forces and bi-axial moments and torque. If the initial force and moments are conservative, the matrix is symmetrical. Non-consistent natural boundary conditions lead to non-symmetric matrices events for conservative loadings. Ziegler proposed the concept of semi-tangential and quasi-tangential moments to handle the natural boundary conditions for buckling torque of beams of equal moments of area [3]. Argyris extended to beams of unequal moments of area [4]. Initial torque is categorized as either semi-tangential or quasi-tangential, but nothing in between. The first contribution of the paper is to show that the initial torque can change continuously from semi-tangential to quasi-tangential due to the continuous change of the ratio of moments of area about the two principal axes. The second contribution is to give the explicit form of the governing equations and natural boundary conditions of straight non-uniform beam under all kinds of initial forces and moments and solve them exactly using power series. If the beam is uniform and under uniform initial stresses, the equations are solved more efficiently by exponential functions.

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