Closed-form beam formula reference for simply supported and cantilever beams. Enter any combination of known values and solve for reactions, moments, deflections, or section properties. Runs entirely in the browser. No account required.
This calculator provides closed-form solutions for common beam load cases from Roark's Formulas for Stress and Strain, 8th edition. Select a case, enter the values you know, and the tool solves for the remaining variables including support reactions, maximum bending moment, maximum deflection, and slope.
Additional cases are planned for future versions including partial-span UDL, triangular load, and intermediate fixed supports.
Most beam calculators work in one direction: inputs in, answer out. This tool lets you work backwards. Enter a deflection limit and leave I blank to find the required moment of inertia. Enter a known reaction and leave P blank to back out the applied load. The solver uses numerical root-finding for any valid combination of known and unknown variables.
Problem: A 6 m simply supported steel beam (E = 200 GPa, I = 8.33e-5 m^4) carries a 12 kN point load at midspan. Find maximum deflection and reactions.
Reactions. By symmetry: R_A = R_B = P/2 = 6 kN.
Max moment. M_max = PL/4 = 12(6)/4 = 18 kN·m at midspan.
Max deflection. y_max = PL^3/(48EI) = 12000(216)/(48 x 200e9 x 8.33e-5) = 3.25 mm at midspan.
Any consistent set. If you enter load in kN and length in m, results come back in kN and m. There is no built-in unit conversion; stay consistent throughout.
Rectangle (width b, height h): I = bh^3/12. Solid circle (diameter d): I = pi*d^4/64. For standard steel sections, look up the tabulated I value in your textbook appendix or AISC Steel Manual.
Deflection scales with L^3 or L^4 for most cases, so span has a large effect. Also check your I value. Increasing beam depth has a significant effect since I scales with h^3 for a rectangular cross-section.