Section Modulus & Bending Stress Calculator
Section modulus, moment of inertia and plastic modulus for rectangular, round, tube, box, I-beam, channel and T sections about either axis — then bending stress, shear stress and deflection for simply supported, cantilever and fixed beams under point or uniform loads, with moment, shear and deflection diagrams.
How it works
σ = M ÷ S, S = I ÷ c · rectangle I = b h³/12, Z = b h²/4 · round I = π d⁴/64 · tube I = π(D⁴ − d⁴)/64 · RHS I = (b h³ − bᵢ hᵢ³)/12 · I-beam and channel I = (b h³ − (b − tw)(h − 2tf)³)/12 · T and weak-axis shapes: I = Σ(b h³/12 + A d²) about the centroid · τ = V Q ÷ (I t) · simply supported: point P at midspan M = PL/4, δ = PL³/48EI; uniform w: M = wL²/8, δ = 5wL⁴/384EI · cantilever: M = PL, δ = PL³/3EI; M = wL²/2, δ = wL⁴/8EI · fixed both ends: M = PL/8, δ = PL³/192EI; M = wL²/12, δ = wL⁴/384EI
Bending stress in a beam is the bending moment divided by the section modulus S = I/c, a geometric property that says how well the cross-section places material far from the neutral axis. Depth matters enormously: a rectangle’s modulus grows with the square of its depth, so a 50 × 150 mm timber on edge is three times as strong as the same piece laid flat (choose the weak axis to compare), and an I-beam puts most of its area in the flanges where it does the most work. The calculator finds the area, centroid, second moment of area I, elastic modulus S (for both faces of a T or a channel on its weak axis), plastic modulus Z and radius of gyration for seven section types. You can enter the moment directly, or pick a support and load case — simply supported, cantilever or fixed at both ends, with a point load or a uniform load. It then works out the reactions, maximum moment and shear, bending stress, shear stress at the neutral axis (τ = VQ/It), and deflection from the material’s elastic modulus. It checks the stress against your allowable value and the deflection against a span limit such as L/360, and gives the largest load the beam can take, the section modulus and I you would need, and moment, shear and deflection diagrams. Strength is rarely the only limit: slender, unbraced beams can buckle sideways (lateral-torsional buckling), and thin hollow sections need a local buckling check.
Worked example
A 50 × 150 mm solid rectangle: I = 50 × 150³ ÷ 12 = 1,406 cm⁴ and S = 187.5 cm³. A 3 kN·m moment gives 3,000,000 ÷ 187,500 = 16 MPa — trivial for steel (allowable 165 MPa, maximum 30.9 kN·m) but at the limit for softwood (about 8–10 MPa). Over a 3 m span the steel section could carry 41 kN at midspan on strength. As a simply supported steel beam with a 5 kN/m uniform load: M = 5 × 3² ÷ 8 = 5.63 kN·m, σ = 30 MPa, and δ = 5 × 5 × 3,000⁴ ÷ (384 × 200,000 × 14,062,500) = 1.88 mm, which is L/1,600 — well within L/360.
Sources: Euler–Bernoulli beam theory — Wikipedia · Simply supported beam formulas — CalcResource
Frequently asked questions
What is the difference between elastic and plastic section modulus?
Elastic S = I/c gives the moment at first yield; plastic Z (used in limit-state design) is the moment at a full plastic hinge — 1.5 × S for rectangles and about 1.1–1.2 × S for I-beams. The ratio Z/S is the shape factor shown with the result.
How do I find S for a standard steel beam?
Look it up — section tables list I, S (Wel), Z (Wpl) and radius of gyration for every rolled size. The I-beam and channel options here use sharp-cornered plates without root fillets, so they come out a few percent below the table values.
Why does the weak axis give such a different answer?
Stiffness and strength depend on how far the material sits from the bending axis. A 50 × 150 mm board on edge has I = 1,406 cm⁴; laid flat it has only 156 cm⁴ — nine times less stiff and three times weaker.
Which deflection limit should I use?
Building codes set them by use: L/360 is common for floors under live load, L/240 for total load, L/180 for roofs without ceilings, and Eurocode often uses L/250. Cantilever limits are usually applied to twice the cantilever length, as here. Follow your local code.
Does the calculator include the beam’s own weight?
Only if you switch on “Add the beam’s own weight” under More options. It then adds area × density × g as a uniform load along the span. For long steel or concrete beams this can matter.
What does “fixed both ends” mean?
Both ends are held against rotation, as with a beam welded into stiff columns or built into a wall. That cuts the maximum moment of a uniform load from wL²/8 to wL²/12 and the deflection to one-fifth, but real connections are rarely fully fixed, so simply supported is the safer assumption.
Related calculators
Engineering estimate for education and preliminary design. Use manufacturer data, applicable standards and appropriate safety factors, and have critical designs checked by a qualified engineer.