Shear Force Calculator

Shear force at a section is the algebraic sum of all transverse forces on one side of that cut, and it is the internal action that resists one part of the beam sliding past the other. Its usefulness comes from two differential relationships that hold along any beam: the slope of the shear diagram equals the negative of the distributed load intensity, and the slope of the moment diagram equals the shear. The second relationship is why the maximum bending moment always occurs where shear crosses zero, and it is the reason engineers draw both diagrams together rather than computing isolated values. For a simply supported span under a uniform load w, each reaction is wL/2, shear falls linearly to zero at midspan, and the moment peaks there at wL-squared over 8. A cantilever concentrates everything at the fixed end, with shear wL and moment wL-squared over 2. Support conditions change the picture substantially. A propped or fully fixed beam is statically indeterminate, so reactions depend on member stiffness rather than statics alone; for a fixed-fixed span under uniform load the reactions remain wL/2 by symmetry, but the moment redistributes to wL-squared over 12 at the supports and wL-squared over 24 at midspan. That is a meaningful caveat, because bolted and welded connections rarely deliver the full rotational restraint the idealization assumes, and partial fixity leaves the real structure somewhere between the two cases. These results also presume linear elastic behavior, small deflections, and no axial force. Member self weight is not included unless added to the applied load.

Beam strength was the first problem of modern structural mechanics and the first one to be gotten wrong in an instructive way. Galileo Galilei posed the cantilever fracture problem in his 1638 Discorsi, correctly identifying that resistance depends on the depth of the section but assuming a uniform tensile stress across the break, which overstates capacity. Edme Mariotte revisited the question experimentally in the 1680s, and Charles Augustin Coulomb finally supplied a correct equilibrium treatment in a memoir presented to the French Academy of Sciences in 1773, one that accounted for both the internal moment and the transverse shear on a cut section. The framework became routine engineering in the nineteenth century. Claude Louis Navier consolidated elastic beam theory in his 1826 lectures on the strength of materials, giving the profession a systematic way to relate loads, internal forces, and stresses. The distribution of shear stress through the depth of a section, the VQ over Ib relationship, was worked out by the Russian engineer Dmitrii Zhuravskii, often transliterated as Jourawski, during his railway bridge work in the 1840s and 1850s. The diagrams themselves came from the graphical statics movement: Karl Culmann published Die graphische Statik in 1866, and Otto Mohr developed complementary graphical constructions in the following decade. Their shear and moment diagrams proved durable enough that engineers still sketch them by hand today, even with matrix analysis software available.