Nonlinear Euler–Bernoulli beam kinematics in progressive collapse analysis based on the Smith's approach

• Nonlinear two-dimensional kinematic model of the Euler-Bernoulli beam is proposed.
• Complete and exact formulations of displacements, curvature and strain are derived.
• All formulations are independent of the displacement/rotation/strain magnitude.
• Green-Lagrange strain represents incomplete strain approximation.
• Smith's approach is valid regardless of the displacement/rotation magnitude.

Within the scope of this article a nonlinear kinematics of the two-dimensional, non-shear-deformable and extensible Euler–Bernoulli beam imposed with the planar flexure and/or lengthening/shortening is considered. The complete and exact formulations of the pertinent kinematic response quantities (displacements, curvature and strain) are derived and discussed. Special emphasis is given to the case of the symmetric bending devoid of the external longitudinal force action, since it represents an appropriate idealization of the realistic load cases for the most of the ship and aircraft structures. The relationship between proposed and conventional formulations, i.e. those commonly accepted in the current structural engineering practice and employed by the current progressive collapse analysis methods based on Smith's approach, is discussed throughout the article and illustratively exemplified through the case of the pure symmetric bending of the Euler–Bernoulli cantilever. Finally, implications of the derived formulations pertinent to the progressive collapse analysis methods based on the Smith's approach are discussed.