Matrix dynamic and instability analysis with non‐uniform elements

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Abstract
Approximate formulations of non‐uniform beam element stiffness matrices for dynamic and elastic instability analysis are derived. Displacement functions for the uniform beam segment are employed in this development. Moment of inertia and area of the element are prescribed by arbitrary powers of the axial co‐ordinate. Numerical results are obtained and compared with both analytical solutions and numerical solutions based upon stepped representations using uniform section elements. The significance of the inclusion of taper considerations within individual elements upon solution accuracy and convergence characteristics is also examined. This subject problem and solution approach demonstrates that the upper bound character of minimum energy solutions may be difficult to exploit under practical circumstances.