The beam finite element with five nodal degrees of freedom

The article presents comparative calculations of reinforced concrete beams using two types of beam finite elements: with three and five nodal degrees of freedom. Calculations were performed both taking into account the concrete and reinforcement physical nonlinearity, and without taking it into account. The expressions for stiffness matrix elements and the load vector were obtained for the finite element with five nodal degrees of freedom. Calculations taking into account physical nonlinearity were performed using the variable elasticity parameters method. As a structure for comparing solutions obtained by the two types of finite elements, a single-span clamped horizontal and inclined reinforced concrete beam were used. The accuracy of calculating beam axis deformations and curvature depending on the number and type of finite elements was assessed. It was shown that when performing linear calculations, bending moments, longitudinal forces and displacements do not depend on the number of finite elements with five degrees of freedom into which the beam had been divided. When solving physically nonlinear problems, if we refine the finite element mesh, the solutions obtained for elements with three degrees of freedom tend to the solutions obtained for a smaller number of elements with five degrees of freedom. The proposed beam finite element with five nodal degrees of freedom makes it possible to determine more accurately the axis curvature and deformation, which is especially important when solving physically nonlinear problems.