Maximum Bending Moment Beam
Computes maximum bending moment for a simply supported beam (UDL or central point load).
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Maximum bending moment in a beam
On a simply supported beam carrying a uniformly distributed load, the largest bending moment lands at midspan and works out to M_max = wL² / 8. Swap that for a concentrated load P at midspan and you get M_max = PL / 4. Plug in w = 12 kN/m and L = 6 m and the moment comes out to 54 kN·m. On the resistance side, a section can carry M_R = W·f_y, where W is the elastic section modulus and f_y the yield strength (around 250 MPa for ASTM A36 or steel grade C-25).
When you design, M_max is what fixes the section you need. Divide it by f_y/γ to find W_req, then grab the smallest I, W, H or channel profile that clears that value. Run the ultimate-limit-state checks under NBR 6118 for concrete and NBR 8800 for steel, keeping the load factors γ_g and γ_q in the mix.
Applications
Sizing floor beams, lintels, roof purlins and pergolas. Quick sanity checks before you open FEM software like SAP2000, Robot or RFEM. Picking steel profiles I, W and H straight from Gerdau or ArcelorMittal tables. Laying out reinforcement in reinforced-concrete beams. And early-stage cost estimates per metre of beam.
FAQ
Why does M_max occur at midspan for UDL? The shear V = wL/2 − wx hits zero exactly at x = L/2, and the moment peaks right where the shear flips sign.
What about cantilevers? A cantilever of length L under UDL reaches M_max = wL²/2 at the fixed end, which is four times what the simply supported beam sees.
How do I get the section modulus W? Use W = I / y_max. A rectangle b×h gives W = b·h²/6, and for rolled profiles the catalogue prints W for you.
Is the formula valid for continuous beams? Not really. Once you have two or more spans you need the three-moment theorem or matrix methods. The midspan moment falls to about wL²/14, and a hogging moment of wL²/8 shows up over the interior supports.
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The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.