ABSTRACT

Elastic beams are important components in many types of structures. Consequently methods to analyze the shear, moment, slope, and deßection in beams with complex loading and general cross section variation are of signiÞcant interest. A typical beam of the type considered is shown in Figure 11.1. The study of Euler beam theory is generally regarded as an elementary topic dealt with in undergraduate engineering courses. However, simple analyses presented in standard textbooks usually do not reveal difÞculties encountered with statically indeterminate problems and general geometries [115]. Finite element approximations intended to handle arbitrary problems typically assume a piecewise constant depth proÞle and a piecewise cubic transverse deßection curve. This contradicts even simple instances such as a constant depth beam subjected to a linearly varying distributed load which actually leads to a deßection curve which is a Þfth order polynomial. Exact solutions of more involved problems where the beam depth changes linearly, for example, are more complicated. Therefore, an exact analysis of the beam problem is desirable to handle depth variation, a combination of concentrated and distributed loads, and static indeterminacy providing for general end conditions and multiple in-span supports. The current formulation considers a beam carrying any number of concentrated loads and linearly varying distributed loads. The equations for the shear and moment in the beam are obtained explicitly. Expressions for slope and deßection are formulated for evaluation by numerical integration allowing as many integration steps as necessary to achieve high accuracy. A set of simultaneous equations imposing desired constraints at the beam ends and at supports is solved for support reactions and any unknown end conditions. Knowledge of these quantities then allows evaluation of internal load and deformation quantities throughout the beam. The analytical formulation is implemented in a program using a concise problem deÞnition specifying all loading, geometry, and constraint conditions without reference to beam elements or nodal points as might be typical in a Þnite element formulation. The program and example problem are discussed next.