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A numerical formulation was carried out in this paper to produce the tangent stiffness matrix for two-nodal tapered  Timoshenko  beam-column elements for geometrically nonlinear analysis. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. The section properties were presented as exponential functions with tapering indices to illustrate the variations in section properties along the tapered element length. The model is applicable for elements with different solid and hollow cross-sections. The proposed formulation is embedded into a Visual Basic code to carry out the analysis accompanied by many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references that showed high accurate results with a small number of elements.

Finite element formulation of Timoshenko tapered beam-column element for large displacement analysis based on the exact shape functions