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Civil

Simply supported beam: uniform load

Midspan deflection and maximum moment of a beam carrying a distributed load.

Ref: CIV-03
Rev: v1
Field: Civil
Result: δ_max [mm]

δ_{\max} = \frac{5 w L^{4}}{384 E I}, M_{\max} = \frac{w L^{2}}{8}

w the uniform load per unit length (kN/m), L the span, E and I as before. The calculator uses a rectangular section b × h.

Spread the load evenly over the span instead of putting it in one spot and the beam sags less at the centre. Span matters hugely: the sag grows with the fourth power.

Calculator

Solve for

δ_maxMidspan deflectionComputed

wLoad per metre

kN/m

0.550

LSpan

112

EYoung’s modulus

GPa

10210

bSection width

30400

hSection depth

50600

Maximum moment wL²/810 kN·m
Total load wL20 kN

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.

Midspan deflection (mm) current value 1.25 mm

LSpanKey driver

0.00488101

hSection depth

0.046380

EYoung’s modulus

1.1925

wLoad per metre

0.12512.5

bSection width

0.3134.17

For the same total load, a uniform load gives a deflection of 5/8 of a central point load (5/384 against 1/48 in the formulas).
The fourth power on span means a slight extension of span has a dramatic effect. Support intermediate points where possible.
Strength is checked with M_max and the bending stress formula; stiffness with this deflection.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (mm)	Local step	Change in result
L Span	1 to 12 m	0.004883	101.3	101.2	+10%	+46.4 %
h Section depth	50 to 600 mm	0.0463	80	79.95	+10%	−24.9 %
E Young’s modulus	10 to 210 GPa	1.19	25	23.81	+10%	−9.09 %
w Load per metre	0.5 to 50 kN/m	0.125	12.5	12.38	+10%	+10 %
b Section width	30 to 400 mm	0.3125	4.167	3.854	+10%	−9.09 %

Sweep of Span

Span (m)	Midspan deflection (mm)
1	0.004883
1.364	0.01691
1.861	0.05859
2.539	0.203
3.464	0.7031
4.726	2.436
6.447	8.438
8.796	29.23
12	101.3

Sweep one variable

Vary

Midspan deflection (mm)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

δ_max

Midspan deflection

Result · mm

What the equation gives you. Everything else feeds it.

Load per metre

kN/m · typical 0.5 to 50

Uniform distributed load over the whole span. Directly proportional.

Span

m · typical 1 to 12

Distance between supports. The dominant driver: deflection grows with L⁴, so +10% span gives about +46%.

Young’s modulus

GPa · typical 10 to 210

Material stiffness. Inversely proportional.

Section width

mm · typical 30 to 400

Width of the rectangular section. Inversely proportional.

Section depth

mm · typical 50 to 600

Depth of the section. Inversely proportional to h³.

Worked example with the values above

Total load wL = 5 × 4 = 20 kN; M_max = wL²/8 = 10 kN·m.
I = bh³/12 = 6.667×10⁷ mm⁴. Use w in N/mm (equal to kN/m numerically), L in mm, E in N/mm².
δ_max = 5wL⁴/(384EI) = 1.25 mm.

Assumptions and limits

Prismatic, slender, linear-elastic beam with small deflection.
Load uniform over the full span.
Self-weight included in w if it matters.

Common mistakes

Using total load in place of load per metre.
Mixing kN/m with N/m.
Using the point-load formula for a distributed load.

Statistics and uncertainty

The fourth power on span means errors in measured length dominate: the relative error in deflection is about √[(δw/w)² + (4δL/L)² + (δE/E)² + (δb/b)² + (3δh/h)²]. A 1% span error becomes 4% in deflection.

Where it is used

Floors, decks and cable trays.
Truck and trailer bodies carrying evenly spread cargo.
Self-weight deflection of bridge beams.

Maths you need

Distributed loads: force per unit length
Beam stiffness EI: product of modulus and second moment of area

Related formulas

Practise this formula