An exact static analysis of rigidly-jointed coplanar beam structures subject to distributed loading

A simple matrix-displacement method is presented here to facilitate the static analysis of rigidly-jointed coplanar beam frames. The geometric arrangement of the frame, the boundary conditions, the material properties, and the external loading, are allowed complete generality, thereby enabling a large range of 2D structures to be modelled. Point loads, self-weight, inertial and distributed loads may all be accommodated in the analysis; emphasis is placed on the inclusion of the distributed loads, since these prove a more significant challenge to represent accurately in a matrix-displacement type of analysis. The chief novelty of this work is that for such structures exact solutions may be obtained with great ease for the displacement, reactions, and the individual member bending moment, shear force, and axial force, through the inclusion of suitable serendipity functions in the element formulation. The computational effort is the minimum possible for this class of problem, requiring the solution to just 3S simultaneous equations, where S represents the total number of junctions or nodes used to define the frame. Such solutions can easily be implemented using desktop computing, and five examples of increasing complexity are presented herein to demonstrate the efficacy of the method.