Steel–concrete composite beams in partial interaction: Closed-form “exact” expression of the stiffness matrix and the vector of equivalent nodal forces

The present paper deals with steel–concrete composite beams in partial interaction as a result of the interface relative displacements, possibly occurring between the concrete slab and the steel beam. In particular, after a short outline of the basic assumptions of the well-known Newmark’s Theory, the closed-form analytical expressions of the corresponding stiffness matrix and the vector of equivalent nodal forces for beam in bending are presented. They completely define an “exact” 1D finite element (FE) for composite beams in partial interaction. The possibility of performing linear design-oriented analyses of composite beams by using only one element per member is the most important feature of such an “exact” FE. The closed-form expressions of both the stiffness matrix and the vector of equivalent nodal forces are generally defined as the product of constant coefficients, corresponding to the stiffness terms of simple Bernoulli beams, multiplied by various functions depending on two non-dimensional parameters which completely cover the effect of partial interaction.A simple application is proposed for pointing out the advantages of the proposed “exact” solution. Final comments on numerical instabilities possibly affecting the proposed solution, their practical significance and the way for fixing them are also reported at the end of the paper, leading to a complete formulation of an “exact” finite element which could be easily implemented within the usual numerical codes to analyze steel–concrete composite beams in partial interaction.