Stability of Beams with Tapered I‐Sections

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Abstract
The purposes of the paper are: (1) To derive differential equations of equilibrium for a tapered I‐beam; and (2) to formulate a finite element for the beam that takes into account the effect of nonuniform torsion. In the virtual work formulation, the updated Lagrangian approach is adopted, in which the effect of geometric nonlinearity is considered. The present formulation requires obtaining a rigorous expression for the strains based on the membrane theory of shells, through which the effect of tapering is considered. The displacements of each cross section are determined with Vlasov's thin‐walled beam assumptions. The derived finite element model, in terms of the linear and geometric stiffness matrices, is useful in a buckling or an incremental large displacement analysis. Using the present theory, one is able to investigate various torsional‐flexural instability problems. Examples are prepared and comparisons are made with existing solutions.

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