Closed-form beam formula reference covering simply supported, cantilever, fixed-fixed and propped beams under point, uniform, partial, triangular and moment loading. Enter what you know, click a result to make it a known value, and solve for whatever is left blank. Runs entirely in the browser. No account required.
Closed-form solutions for 24 standard beam load cases, taken from Roark's Formulas for Stress and Strain, 8th edition. Pick a case, enter what you know, and the tool returns support reactions, bending moments, maximum deflection and end slopes, showing the formula and the numbers substituted into it for every value it prints.
Click the result you already know, such as a maximum moment or a deflection limit, then leave the variable you want blank. Each case carries its own algebraically inverted formulas, so the answer is exact rather than iterated, and the rearranged expression is shown alongside it.
A single known value can drive different unknowns. With the maximum moment pinned on a simply supported beam with a point load, leaving P blank returns the load, and leaving L blank instead returns the span.
The Combine switch adds a second case on the same span and the same supports, for example self weight as a uniform load plus a point load at midspan.
The combined answer is not produced by adding the two peak values together. The largest moment of a combined load is generally not the sum of the individual largest moments, because they occur at different positions along the beam. Instead the tool rebuilds the whole problem with both load sets applied, solves it, and searches the resulting bending moment and deflection curves for their true peaks.
Problem: a 6 m simply supported steel beam, E = 200 GPa, I = 8.33e-5 m4, carrying a 12 kN point load at midspan. Find the reactions, maximum moment and maximum deflection.
Reactions. By symmetry, RA = RB = P/2 = 6 kN.
Maximum moment. Mmax = PL/4 = (12)(6)/4 = 18 kN·m at midspan.
Maximum deflection. δmax = PL³/(48EI) = (12)(6)³ / (48 × 2×108 × 8.33×10-5) = 0.00324 m, that is 3.24 mm at midspan. Note the load is in kN and the modulus in kN/m² so the units stay consistent.
Every closed form is checked against a separate beam solver written in Python with sympy. It writes the bending moment with singularity functions and solves the unknown reactions together with the two integration constants of EI·y″ = M, so determinate and indeterminate beams go through the same code. The test values are deliberately non-round and asymmetric. 151 checks cover every reaction, moment, slope and deflection formula in all 24 cases, and a further 154 compare the numbers the tool itself prints.
The check changed four formulas before release: the cantilever with a triangular load peaking at the wall, the propped cantilever under uniform load (the tool uses the exact coefficient, 1/184.6, rather than the rounded 1/185 found in many tables), the propped cantilever with a center load, and the end slopes for a partial uniform load. When seven cases were added in September 2026 it caught one more: a sign error in the fixed-fixed triangular load deflection.
Limits: prismatic beams (constant EI), small deflections, one span. Shear deformation is ignored.
Whatever you choose. SI and Imperial presets are one click, and the dropdown beside each input switches that quantity on its own, so you can work in metres and millimetres at the same time. Conversion is handled internally, and switching units back and forth returns the values you originally typed rather than a re-rounded version of them.
Deflection and slope formulas mix quantities whose everyday units are not consistent with each other, such as GPa with cm4 and mm. Those substitution lines are shown in kN and m, marked with a small unit tag, because that is what actually multiplies out to the answer. Reactions and moments are shown in your chosen display units, which are already consistent.
Rectangle of width b and height h: I = bh³/12. Solid circle of diameter d: I = πd⁴/64. For rolled steel sections, look up the tabulated value in a section table or the AISC Steel Construction Manual.
Deflection scales with L³ or L⁴ depending on the case, so span dominates. After that, check I. For a rectangular section I scales with h³, so depth matters far more than width. Also confirm E is in the right units, since mixing GPa with MPa is a factor of a thousand.
No, that is a separate tool. This one returns peak values and the formulas behind them. For full shear and moment diagrams along the beam, use the shear and moment diagram calculator.