Prismatic structural members: Cross section flat and normal to the axis?

Abstract

In the study of prismatic bars subjected to bending forces, the authors of Strength of Materials generally assume the Navier-Bernoulli simplifying hypothesis which states that flat cross sections (CS) normal to their axes before deformation remain flat and normal to their axes. A more detailed study in terms of Elasticity, however, shows how approximate this hypothesis can be for some basic prismatic bar problems in which displacements can readily be obtained. When or whether the surface remains flat, absolutely flat, or not is a point of debate among engineers and architects alike and even for structural specialists, who look deeper into this kind of issues. This paper proposes a detailed study of said problems and clarifies them. Contrary to what should be expected according to well-established literature, the CS of any bar subjected to pure bending forces does not remain flat after deformation. Our analysis revisits accepted displacement solutions for tension, bending and torque and will hopefully remove the misunderstanding leading to a flat geometry for the deformed CS. It also includes the correct interpretation of the warped geometry from the exact equations we obtain in this paper, which we illustrate with results from finite elastic models.