The present paper discusses simple compatibility, equilibrium, and constitutive equations for a nonprismatic planar beam. Specifically, the proposed model is based on standard Timoshenko kinematics (i.e., planar crosssection remain planar in consequence of a deformation, but can rotate with respect to the beam centerline). An initial discussion of a 2D elastic problem highlights that the boundary equilibrium deeply influences the crosssection stress distribution and all unknown fields are represented with respect to global Cartesian coordinates. A simple beam model (i.e. a set of Ordinary Differential Equations (ODEs)) is derived, describing accurately the effects of nonprismatic geometry on the beam behavior and motivating equations terms with both physical and mathematical arguments. Finally, several analytical and numerical solutions are compared with results existing in literature. The main conclusions can be summarized as follows. (i) The stress distribution within the crosssection is not trivial as in prismatic beams, in particular the shear stress distribution depends on all generalized stresses and on the beam geometry. (ii) The derivation of simplified constitutive relations highlights a strong dependence of each generalized deformation on all the generalized stresses. (iii) Axial and shearbending problems are strictly coupled. (iv) The beam model is naturally expressed as an explicit system of six first order ODEs. (v) The ODEs solution can be obtained through the iterative integration of the right hand side term of each equation. (vi) The proposed simple model predicts the real behavior of nonprismatic beams with a good accuracy, reasonable for the most of practical applications.
