By calculating the accurate solution of the governing differential equation for a non-prismatic beam–column, the exact elemental stiffness matrix will be obtained. The Euler–Bernoulli beam theory is used in the proposed formulation. This study considers the second-order effects and flexibility of the connections and supports. The effects of the ends’ flexibilities are investigated by utilising the general boundary conditions of the governing differential equation. In addition, shape factors of 2 and 4 in which the cross-section moment of inertia changes by second- and fourth-order functions through the length, respectively, are studied separately. The suggested stiffness matrix is general and can be utilised in the various analyses for a non-prismatic member belonging to a plane frame with different types of connections and restraints. It should be added that the presented stiffness matrix can be used in the finite-element method. To demonstrate the abilities of the formulation, stability analyses of columns and frames with non-prismatic members are performed. Comparing the responses with the available results shows the accuracy, efficiency and capabilities of the presented matrix.

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