Bending Stress Calculator

The flexure formula, sigma = Mc/I, distributes a bending moment into normal stresses across a cross section. Its derivation rests on three assumptions: plane sections remain plane after bending, the material behaves linear elastically, and bending occurs about a principal axis. Stress varies linearly from zero at the neutral axis to a maximum at the extreme fiber, compression on one face and tension on the other. Because I and the extreme fiber distance c always appear together, engineers collapse them into the section modulus S = I/c, reducing the check to sigma = M/S. Geometry dominates efficiency. A rectangle has I = bh^3/12 and S = bh^2/6, so doubling the depth quadruples I and doubles S while only doubling the weight, which is why joists stand on edge. Wide-flange shapes push material out to the flanges where the lever arm is greatest, achieving section moduli far above a rectangle of equal area. Allowable stresses span a wide range: A36 steel at 36 ksi yield permits roughly 21.6 ksi under older allowable stress design, A572 Grade 50 permits about 30 to 33 ksi for compact sections, while visually graded dimensional lumber carries reference bending values from roughly 575 psi for No. 3 to 1,500 psi for select structural before NDS adjustment factors are applied. The formula loses validity near concentrated loads and holes, in deep beams where shear deformation matters, and once yielding begins. It also ignores lateral-torsional buckling, which frequently governs an unbraced steel beam long before the extreme fiber reaches yield.

The flexure formula took nearly two centuries to reach its familiar form. Galileo Galilei framed the beam strength problem in 1638 but assumed the entire cross section resisted in tension, overestimating the capacity of a rectangle by a factor of three. Edme Mariotte recognized in 1686 that fibers stretch in proportion to their distance from a pivot, and Antoine Parent showed in 1713 that equilibrium requires both a tension zone and a compression zone separated by a neutral axis carrying no stress. Charles-Augustin Coulomb restated the correct result in his 1773 memoir to the French Academy of Sciences, though it attracted little attention at the time. Claude-Louis Navier consolidated the theory in his 1826 Resume des lecons donnees a l Ecole des Ponts et Chaussees, presenting the elastic bending relation in terms of the modulus named for Thomas Young, whose 1807 lectures had introduced that measure of stiffness. The section modulus emerged as a practical shortcut once rolled iron and then steel shapes were mass produced and their geometric properties needed tabulating. Steel mills published handbooks of section properties from the 1870s onward, and the American Institute of Steel Construction, founded in 1921, issued the first Steel Construction Manual in 1927, standardizing the S and I values engineers still look up today. Timber design values were similarly standardized when the first National Design Specification appeared in 1944.