Bending Stress and Beam Deflection Calculator: Shear and Moment Diagrams

Free bending stress and beam deflection calculator: simply supported, cantilever, fixed, propped and overhanging beams with point loads, spread loads and moments you drag along the beam. Live shear-force and bending-moment diagrams, the deflected shape, σ = Mc/I for I-beams, channels, tubes and rectangles, safety factor and the L/360 check. SI and US units.

Material

Units

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and

True stress–strain Shade areas Plateau serrations

Speed

Yield point or smooth yielding?

Mild and structural steels (A36, 1020) have a yield point. Carbon and nitrogen atoms pin the dislocations, so the stress climbs a little past the yield strength until a band of metal breaks free: that is the upper yield point. The force then drops to the lower yield point and the bar stretches at nearly constant stress while Lüders bands spread along it (the Lüders strain, about 1 to 3% for A36), with small serrations as each band starts. Only then does strain hardening begin, up to the tensile strength (58 to 80 ksi, 400 to 550 MPa for A36, at about 15 to 20% strain) and necking. The upper yield is usually only a few % to about 10% above the lower one, and its height depends on the test: a stiff machine, a fast pull and a smooth, well-aligned specimen give a taller, sharper peak; a soft machine or a slightly bent bar can hide it. The line also bends a little just before the peak (some grains yield early), so the top is rounded rather than a sharp corner. Codes therefore use the lower yield (or 0.2% offset) as the yield strength: 250 MPa (36 ksi) for A36.

Aluminium alloys (2014-T6, 6061-T6), copper, stainless steel and most other metals yield continuously: no peak, no drop, no plateau. The curve just bends away from the straight line, so the yield strength is defined by the 0.2% offset rule: draw a line with slope E starting at 0.2% strain, and where it meets the curve is the yield strength (414 MPa, 60 ksi, for 2014-T6).

Engineering stress\(\displaystyle \sigma = \frac{F}{A_0},\)\(\displaystyle \varepsilon = \frac{L - L_0}{L_0}\)

True stress\(\displaystyle \sigma_t = \frac{F}{A} = \sigma\,(1+\varepsilon),\)\(\displaystyle \varepsilon_t = \ln(1+\varepsilon)\)

Hardening\(\displaystyle \sigma_t = K\,\varepsilon_p^{\,n}\)

Necking starts\(\displaystyle \frac{d\sigma_t}{d\varepsilon_t} = \sigma_t \;\Rightarrow\; \varepsilon_t \approx n\)

Resilience\(\displaystyle U_r = \frac{\sigma_y^2}{2E}\)

Toughness\(\displaystyle U_t = \int_0^{\varepsilon_f} \sigma\,d\varepsilon\)

The curves are a model fitted to each material's typical Young's modulus, yield strength, tensile strength and elongation (Hollomon / Ramberg–Osgood hardening, a Lüders plateau for mild steel). Specimen: a standard round bar, 12.5 mm across with a 50 mm gauge length (0.500 in and 2.000 in in US units). Real curves also depend on the test speed, temperature and the exact alloy.

Supports

Loads drag them along the beam

Cross-section

Deflection limit

Bending stress (flexure formula)\(\displaystyle \sigma = \frac{M\,c}{I} = \frac{M}{S},\)\(\displaystyle S = \frac{I}{c}\)

Shear stress\(\displaystyle \tau = \frac{V\,Q}{I\,t}\)

Deflection curve\(\displaystyle E I\,\frac{d^2 y}{dx^2} = M(x),\)\(\displaystyle \frac{dM}{dx} = V\)

Rectangle\(\displaystyle I = \frac{b\,h^3}{12},\)\(\displaystyle S = \frac{b\,h^2}{6}\)

Simply supported, load P at the middle\(\displaystyle M_{max} = \frac{P L}{4},\)\(\displaystyle \delta = \frac{P L^3}{48 E I}\)

Simply supported, spread load w\(\displaystyle M_{max} = \frac{w L^2}{8},\)\(\displaystyle \delta = \frac{5 w L^4}{384 E I}\)

Cantilever, load P at the end\(\displaystyle M_{max} = P L,\)\(\displaystyle \delta = \frac{P L^3}{3 E I}\)

Fixed both ends, spread load w\(\displaystyle M_{ends} = \frac{w L^2}{12},\)\(\displaystyle \delta = \frac{w L^4}{384 E I}\)

The beam is solved exactly for any mix of loads (singularity functions: equilibrium plus the deflection conditions at the supports), so the fixed and propped cases, which statics alone cannot solve, come out right too. It assumes a straight, elastic beam with small deflections (Euler–Bernoulli), bending about the strong axis, no buckling sideways (long, slender I-beams without bracing can twist out of the way: lateral-torsional buckling) and no self-weight unless you add it as a spread load.

Words used here

Bending moment M: How much the loads try to bend the beam at a point: force × distance, summed over everything on one side. Sagging (smiling, top squeezed) is drawn upward here; hogging (frowning, top stretched, over supports) downward.
Shear force V: The force trying to slide one part of the beam past the next. It jumps at every point load and support, and the slope of the moment diagram equals it, so M is largest where V crosses zero.
Neutral axis: The line through the section that is neither stretched nor squeezed. For these symmetric sections it sits at mid-depth; the stress grows in a straight line away from it.
Second moment of area I: How the area is spread away from the neutral axis (m⁴, in⁴). Depth counts cubed, which is why I-beams put their metal in the flanges and why a plank is far stiffer on edge.
c and section modulus S = I/c: c is the distance from the neutral axis to the outermost fibre, where the stress is largest. S rolls I and c into one number: σmax = M/S.
Flexure formula σ = Mc/I: The largest bending stress in the section, valid while the material stays elastic.
Q and τ = VQ/(It): Q is the first moment of the area above the neutral axis and t the width there. The shear stress is largest at the neutral axis; in an I-beam the thin web carries nearly all of it.
Deflection δ and L/360: How far the beam sags. Codes limit it as a fraction of the span (L/360 for floors under live load, L/240 or L/180 for roofs) so floors do not feel bouncy and plaster does not crack.
Simply supported, cantilever, fixed, propped, overhang: A pin and a roller at the ends; built into a wall at one end only; built in at both ends; built in at one end and resting on a support at the other; resting on two supports with part of the beam sticking out past one of them.
Point load, spread load (UDL), moment: A force at one spot (kN, kip); a force spread evenly along a length (kN/m, kip/ft), such as a floor's weight; a twisting couple applied at one spot (kN·m, kip·ft).
EI (flexural rigidity): Young's modulus times I: how hard the beam is to bend. Deflection is inversely proportional to it.

For learning and first estimates. Real parts need certified material values, the design code that applies (AISC, Eurocode, ASME…) and a check by a qualified engineer.

Bending stress and beam deflection calculator

Put a beam on its supports, add point loads, spread loads (UDL) and moments, and drag them along the beam: the support reactions, the shear-force and bending-moment diagrams, the deflected shape and the bending stress update as you move them. Pick a rectangle, round bar, pipe, box section, I-beam or channel (or type I and c from a steel table), pick the material, and the calculator checks the stress against the yield strength and the deflection against a limit such as L/360. SI and US units.

How it works

Reactions and diagrams. The beam is solved exactly with singularity (Macaulay) functions: the two equilibrium equations plus the deflection conditions at the supports (no deflection at a support, no slope at a fixed end). That also solves the beams statics alone cannot, the fixed-ended and propped cantilever ones.
Bending stress: \(\sigma = M c / I = M / S\), where M is the bending moment, c the distance from the neutral axis to the outer face and I the second moment of area. The stress is zero at the neutral axis and largest at the top and bottom faces; sagging squeezes the top and stretches the bottom.
Deflection: from \(E I\, y'' = M(x)\), integrated twice. The picture exaggerates the bend and says by how much.
Shear stress at the neutral axis: \(\tau = V Q / (I t)\), 1.5 times the average V/A for a rectangle, carried almost entirely by the web of an I-beam.

Worked examples

Steel beam, 10 kN at the middle of a 4 m span. M = PL/4 = 10 kN × 4 m ÷ 4 = 10 kN·m. For an I-section 200 mm deep with 100 × 8.5 mm flanges and a 5.6 mm web, I = 1,846 cm⁴ and c = 100 mm, so σ = 10,000,000 N·mm × 100 mm ÷ 18,456,000 mm⁴ = 54 MPa: a safety factor of 4.6 against A36's 250 MPa. The deflection is PL³/48EI = 10,000 × 4³ ÷ (48 × 200 GPa × 1.846 × 10⁻⁵ m⁴) = 3.6 mm, well under L/360 = 11.1 mm.

Same beam with a spread load of 5 kN/m. M = wL²/8 = 5 × 16 ÷ 8 = 10 kN·m (the same moment as the point load: a spread load of wL = 20 kN gives the moment of a 10 kN point load at the middle), and δ = 5wL⁴/384EI = 4.5 mm.

Cantilever, 500 N at the end of a 300 mm aluminium bar, 10 × 40 mm on edge. M = PL = 150 N·m at the wall, I = 10 × 40³ ÷ 12 = 53,333 mm⁴, σ = 150,000 × 20 ÷ 53,333 = 56 MPa, δ = PL³/3EI = 500 × 0.3³ ÷ (3 × 68.9 GPa × 5.33 × 10⁻⁸) = 1.2 mm.

Questions

Why is a plank so much stiffer on edge than flat?

I = bh³/12: the depth counts cubed. A 50 × 200 mm board on edge has I = 33.3 × 10⁶ mm⁴; laid flat it has 2.1 × 10⁶ mm⁴, 16 times less, so it sags 16 times more and is 4 times more stressed.

Where is the bending moment largest?

Where the shear force crosses zero (because V = dM/dx), or at a support or a point load. For a simply supported beam that is in the span; for a cantilever, at the wall; for a beam fixed at both ends, at the walls (hogging) under a spread load.

What does L/360 mean?

A deflection limit: the beam may sag at most its span divided by 360 (11 mm on a 4 m span). Building codes use L/360 for floors under live load so they do not feel bouncy or crack the ceiling, and L/240 or L/180 for roofs. For a cantilever use its length; some codes double it.

Is the I of a real steel beam the same as here?

Nearly. Rolled sections have rounded fillets where the web meets the flanges, which add a few % to I. Use the I and c from the steel table (choose "I know I and c") for a real W or IPE shape.

What is not included?

The beam's own weight (add it as a spread load: the Section tile shows it), plastic bending past the yield point, lateral-torsional buckling of long unbraced I-beams, local buckling of thin flanges and webs, shear deflection of deep short beams, and dynamic loads.

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