The stresses and deflections of a uniform cross-section beam in bending
A beam or as it is called flexural member can be often met in mechanical structures and machines and presents one of the most interesting examples of the mechanics of materials. In mechanics it is an object of bending which characterizes the structural element behavior which is subject to an external pressure. Bending impacts the beam by inducing reactive forces. There are three the most notable internal forces resulting from loads: shear parallel, compression alongside the top, and tension alongside the bottom (of the beam).
Different forces induced on the beam in bending include stresses, deflections and strains. Stresses are induced on the beam by tensile and compressive forces. The maximum stress of compression can be found on the uppermost margin of a beam and the maximum stress of tension is situated on the lower margin of a beam. For these stresses between opposing sides vary in a linear manner that is why there exist a point between them where the stress is not registered. The place of these particular points presents neutral axis. This characteristic of the cross sectional uniform beams makes them inefficient for big loads and continuous tensions.
Strain represents the geometrical definition of deformation effect of particular stress on a body. It is assessed by calculating the change between the states of two bodies in the beginning of stress action and in the end (Johnson and Sherwin 1996). The difference in placements of two separate points in a body in these states then expresses the numerical values of the strain. Therefore in bending stress may be determined as change in shape or size of the body. If particular strain equals itself over all locus of the body than it is termed as homogeneous strain; if not it is referred to as inhomogeneous strain. Therefore, in its general form strain can be described as symmetric tensor.
Strain is a quantity without dimension. It doesn’t have any units of measures in its formula; the units regulating length are “cancel out”.
Though strain is frequently expressed in terms of metre/metres or inches to remind that the given number represents a certain change in length. But these units are omitted. Therefore, strain can be expressed as pure number. In general solid materials a change in length is frequently not so significant thus strain is measured in micrometre/metre.
In contrary to stresses and strains, deflection in bending describes certain degree of the displacement of beam by the load. It is directly directed to the slope of the shape deflected and can be easily calculated by means of integrating mathematically described function of the slope. It can be measured by standard formula (it will give common case configuration) or by utilization of such specific methods as “virtual work’, “Castigliano’s method etc (Rahman, Kowser, and Hossain S. M 2006). The deflection is often used in architecture. The engineers use material for various purposes and the deflection is the most important factor among them.
Elastic deflection f and the angle of deflection φ (radian angle) in a uniform cross-section beam are calculated by using next formula:
fB = F·L3 / (3·E·I)
φB = F·L2 / (2·E·I)
F = force acting on the tip of the beam
L = length of the beam (span)
E = modulus of elasticity
I = area moment of inertia
To sum it up, stresses, strains and deflection represent main functions in uniform cross-section beam bending and are important for mechanical engineering theory and practice.
Johnson, A. and Sherwin K. 1996, ‘Foundations of Mechanical Engineering.’ London: Chapman & Hall.
Rahman A., Kowser A., and Hossain S. M. 2006, ‘Large Deflection of the Cantilever Steel Beams of Uniform Strength-Experiment and Nonlinear Analysis’. International Journal of Theoretical and Applied Mechanics vol.1, no. 1, pp. 21–36