Bending Moment Calculator

Bending moment is the internal couple at a cross section that resists the tendency of applied loads to rotate one part of a beam relative to another. It equals the algebraic sum of the moments of all forces acting on one side of the cut, and its diagram along the span shows the designer where the member works hardest. For a simply supported beam under uniform load the diagram is parabolic and peaks at midspan with wL^2/8. A single concentrated load P placed at distance a from one support produces a triangular diagram peaking at the load with Pab/L, which reduces to PL/4 when the load sits at midspan. End restraint redistributes moment rather than eliminating it. Fixing both ends of a uniformly loaded beam moves the peak to the supports at wL^2/12 and drops midspan to wL^2/24, a 33 percent reduction in the controlling value that in exchange demands a connection capable of developing that negative moment. A cantilever concentrates everything at one point: wL^2/2 for uniform load, four times the simply supported case, and PL for a tip load. Moment scales with the square of span, so a beam 25 percent longer carries about 56 percent more moment under the same load intensity. These closed-form results assume idealized restraint, no axial force interaction, and no differential support settlement. Fixed-end values are especially sensitive to real joint behavior; a partially restrained connection produces moments somewhere between the fixed and simply supported cases.

The bending resistance of a beam section was the first problem of modern structural mechanics. Galileo Galilei devoted the second day of his 1638 Dialogues Concerning Two New Sciences to a cantilever breaking at a wall, correctly deducing that strength grows with the square of depth but wrongly assuming uniform tension across the section. Edme Mariotte revised the stress distribution in 1686, Antoine Parent located the neutral axis correctly in 1713, and Charles-Augustin Coulomb in his 1773 memoir set out the internal force equilibrium that yields the moment of resistance directly. Claude-Louis Navier brought these results into a coherent theory of bending in his 1826 Resume des lecons, making the moment diagram a routine design tool. The nineteenth century then extended the analysis to indeterminate structures: Benoit Clapeyron published the theorem of three moments in 1857, and Otto Mohr and Carlo Alberto Castigliano developed geometric and energy methods through the 1860s and 1870s. Hardy Cross of the University of Illinois published moment distribution in 1930, an iterative hand technique that finally let engineers analyze multi-story continuous frames, and it dominated practice until computer matrix analysis displaced it in the 1960s. The standard beam formula tables printed in the AISC Steel Construction Manual, which supply the coefficients used in this calculator, are the distilled product of that entire lineage.