A simple element for static and dynamic response of beams with material and geometric nonlinearities

A unified simple 6 degrees‐of‐freedom beam finite element and the associated computational procedures have been developed for the fast and efficient solution of a wide class of static and dynamic response problems of the beam type with material and/or geometrical nonlinearities. The material nonlinearity is treated by including its effect in the governing equations by forming the stiffness matrix of each element using a two‐dimensional grid of Gauss points and using the material properties at each point corresponding to the uniaxial strain at that point. Examples are provided for metal and reinforced concrete beams. A powerful yet straightforward method for the solution of elastica problems of beams and frames, using the beam element developed by the senior author, has been extended for determining the dynamic response of beams undergoing large displacements, including large rotations. The solution procedure involves piecewise linearization of response equations and iterations at each incremental step to achieve equilibrium. The solution procedure is simple and easy to apply. A variety of problems is solved to determine the applicability of the proposed simple formulations. Excellent agreement with existing analytical solutions which employ higher order elements demonstrates the efficiency and versatility of the present simple beam element in nonlinear analysis.