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This paper develops a mathematical model for fixed-end moments for two different types of loads on beams with a parabolic shaped variable rectangular cross section. The loads applied on beam are: 1) a uniformly distributed load and 2) a concentrated load located anywhere along the beam length. The properties of the rectangular cross section of the beam varies along its axis, i.e., the width “b” is constant and the height “h” varies along the beam, this variation follows a parabolic form. The consistent deformation method based on the superposition of the effects is used to solve these problems. The deformation anywhere along the beam is obtained by using the Bernoulli-Euler theory. Traditional methods used to obtain deflections of variable cross section members are any techniques that perform numerical integration, such as Simpson's rule. Tables presented by other authors are restricted to certain relationships. Beyond the effectiveness and accuracy of the developed model, a significant advantage of it is the moments are calculated at any cross section of the beam using the respective integral representations as mathematical formulas.

A mathematical model for fixed-end moments for two types of loads for a parabolic shaped variable rectangular cross section