Beam Bending Stress Calculator

A bent beam stretches on one side and squeezes on the other. Between them runs the neutral axis, where the bending stress is zero, and the stress grows in a straight line to its largest at the two faces farthest from it. This calculator finds that largest stress, where the bending moment peaks.

σ = M c ÷ I = M ÷ S

M is the bending moment, I the section's second moment of area about its neutral axis, c the distance from that axis to the outer face, and S = I ÷ c the section modulus (Machinery's Handbook writes it Z). All five sections here are symmetric about the bending axis, so c is half the depth and the two faces carry the same stress, one in tension and one in compression.

The moment. Type it, or pick a load case and the calculator takes the largest moment from the standard beam formulas, with W the total load and L the span: WL/4 at the center of a simply supported beam with a center load, WL/8 with the load spread evenly, WL at the wall for a cantilever loaded at its end, and WL/2 for a cantilever loaded evenly.

The margin. Give a yield strength and the calculator reports the factor of safety against yield, yield strength over stress. The handbook's general guide puts 1.5 to 2 for reliable materials under loads that are not severe, and lower only where the material is well known and weight matters, so the page calls 1.5 and over a pass, 1 to 1.5 tight, and under 1 a fail. Choose the factor your application and code call for.

Worked example. A 1 in wide, 2 in deep bar of 6061-T6 aluminum, simply supported on a 24 in span, with 1,000 lbf at the center. I = 1 × 2³ ÷ 12 = 0.6667 in⁴ and c = 1 in, so S = 0.6667 in³. M = 1,000 × 24 ÷ 4 = 6,000 lbf·in, and σ = 6,000 ÷ 0.6667 = 9,000 psi, 9.00 ksi. Against a typical yield of 40 ksi the factor of safety is 4.44. In millimeters the example is a 200 mm I-beam, a 3 m cantilever with 10 kN at its end: 143 MPa. Load either with the button under the inputs.

Where it stops. The formula assumes a straight beam of one material that stays elastic, with the same stiffness in tension and compression, sections that stay plane as it bends, and the load in a plane of symmetry. It gives the nominal stress: a hole, notch, shoulder or weld multiplies it by a stress concentration factor this page does not apply. It needs a slender beam: Roark puts the span at 8 depths or more for a compact metal section and 15 or more for one with thin webs, and the page flags a shorter one. It counts the I-beam as thin-webbed, and a rectangular tube whose wall is under a tenth of its depth, since its two side walls are webs; that cut-off is a judgement for your reviewer to confirm. It does not check lateral-torsional buckling of a long I-beam, local buckling of a thin-walled tube, fatigue, or deflection, which often governs first. For a rolled I-beam, use the handbook or catalog properties: its root fillets add a few percent that the parallel-flange formula leaves out. The material values are typical; for a design, use the minimum from the material's own specification.

Reference: Machinery's Handbook, 27th ed.: the flexure formula, p. 213, Table 2, Case 3, equation 11, and p. 214; section properties, pp. 239, 241 and 243; beam cases, Table 1, Cases 1 and 2 (p. 261) and 10 and 11 (p. 265); factors of safety, p. 208; yield strengths, Tables 7b (pp. 580 to 583), 11a (pp. 466 to 467) and 12 (p. 472). A36: ASTM A36 specified minimum. Slenderness: Roark's Formulas for Stress and Strain, §8.1. Reviewer: pending.