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Bending stress in a rectangular beam

Peak stress at the outer fibre of a beam under a bending moment.

Ref: CIV-01
Rev: v1
Field: Civil
Result: σ_max [MPa]

σ = \frac{M y}{I}, I = \frac{b h^{3}}{12}

σ in MPa, M bending moment, y distance from the neutral axis, I the second moment of area of the section, b width and h depth. The calculator gives the peak stress at the extreme fibre, y = h/2, so σ_max = 6M/(bh²).

When a beam bends, the top and bottom fibres carry the most stress, tension on one side and compression on the other. A deeper section resists bending far better than a wider one.

Calculator

Solve for

σ_maxPeak bending stressComputed

MPa

MBending moment

kN·m

0.5200

bSection width

30400

hSection depth

50800

Second moment of area I33,750 cm⁴
Section modulus Z = I/y2,250 cm³

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Peak bending stress (MPa) current value 3.556 MPa

hSection depthKey driver

0.5128

MBending moment

0.22288.9

bSection width

1.3317.8

Depth is squared and width is not, so deepening a beam is far more efficient than widening it, which is why joists are tall and thin.
Stress varies linearly from zero at the neutral axis to its peak at the outer fibre.
Compare the result against the material’s yield stress divided by a safety factor.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
h Section depth	50 to 800 mm	0.5	128	127.5	+10%	−17.4 %
M Bending moment	0.5 to 200 kN·m	0.2222	88.89	88.67	+10%	+10 %
b Section width	30 to 400 mm	1.333	17.78	16.44	+10%	−9.09 %

Sweep of Section depth

Section depth (mm)	Peak bending stress (MPa)
50	128
70.71	64
100	32
141.4	16
200	8
282.8	4
400	2
565.7	1
800	0.5

Sweep one variable

Vary

Peak bending stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_max

Peak bending stress

Result · MPa

What the equation gives you. Everything else feeds it.

Bending moment

kN·m · typical 0.5 to 200

Internal moment at the section considered. Directly proportional.

Section width

mm · typical 30 to 400

Width of the rectangle, parallel to the neutral axis. Inversely proportional.

Section depth

mm · typical 50 to 800

Depth in the direction of bending. The dominant driver: stress falls with h², so +10% depth cuts stress by about 17%.

Worked example with the values above

I = bh³/12 = 150 × 300³ / 12 = 3.375×10⁸ mm⁴. Outer fibre y = h/2 = 150 mm.
σ = My/I = 8,000,000 N·mm × 150 / 3.375×10⁸.
σ_max = 3.556 MPa.

Assumptions and limits

Elastic bending about a principal axis of a prismatic beam.
Plane sections remain plane; small deflection.
Moment at the section is known.

Common mistakes

Swapping b and h: the depth goes in the cube.
Mixing kN·m and N·mm.
Using it past yield.

Statistics and uncertainty

Dimensional errors matter: δσ/σ ≈ √[(δM/M)² + (δb/b)² + (2δh/h)²]. A 2 mm error on a 100 mm depth (2%) gives 4% in stress. Material variability then enters when comparing to strength.

Where it is used

Sizing floor beams, bridge girders and launcher rails.
Checking a bracket or support arm.
Comparing section shapes.

Maths you need

Bending moment: moment at a section from the loads
Second moment of area: I measures the stiffness of the shape

Related formulas

Practise this formula