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Civil

Simply supported beam: central point load

Midspan deflection and maximum moment of a beam with a central load.

Ref: CIV-02
Rev: v1
Field: Civil
Result: δ_max [mm]

δ_{\max} = \frac{P L^{3}}{48 E I}, M_{\max} = \frac{P L}{4}

δ_max midspan deflection, P the central load, L the span, E Young’s modulus and I the second moment of area. M_max is the midspan moment. The calculator uses a rectangular section b × h.

A beam resting on two supports with a load in the middle sags most at the centre. The sag grows rapidly with span (cubed) and falls with the beam’s stiffness EI.

Calculator

Solve for

δ_maxMidspan deflectionComputed

PCentral load

0.5100

LSpan

112

EYoung’s modulus

GPa

10210

bSection width

30400

hSection depth

50600

Maximum moment PL/48 kN·m
Span ÷ deflection5,000

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 0.8 mm

hSection depthKey driver

0.029651.2

LSpan

0.012521.6

EYoung’s modulus

0.76216

PCentral load

0.0510

bSection width

0.22.67

Both span and depth enter with cubes. A 10% longer span needs about 10% more depth to keep the same deflection.
A common serviceability limit is span/360 to span/250, usually checked even when strength is adequate.
Self-weight adds a uniform load; check it separately with the uniform-load formula.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (mm)	Local step	Change in result
h Section depth	50 to 600 mm	0.02963	51.2	51.17	+10%	−24.9 %
L Span	1 to 12 m	0.0125	21.6	21.59	+10%	+33.1 %
E Young’s modulus	10 to 210 GPa	0.7619	16	15.24	+10%	−9.09 %
P Central load	0.5 to 100 kN	0.05	10	9.95	+10%	+10 %
b Section width	30 to 400 mm	0.2	2.667	2.467	+10%	−9.09 %

Sweep of Span

Span (m)	Midspan deflection (mm)
1	0.0125
1.364	0.03174
1.861	0.08059
2.539	0.2046
3.464	0.5196
4.726	1.319
6.447	3.35
8.796	8.507
12	21.6

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.

Central load

kN · typical 0.5 to 100

Point load at midspan. Directly proportional.

Span

m · typical 1 to 12

Distance between supports. The dominant driver: deflection grows with L³, so +10% span gives about +33%.

Young’s modulus

GPa · typical 10 to 210

Steel 200, aluminium 70, timber 10. 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: very strong.

Worked example with the values above

Support reactions are P/2 = 4 kN each, and M_max = PL/4 = 8 kN·m.
I = bh³/12 = 6.667×10⁷ mm⁴. Then δ = PL³/(48EI) with P in N, L in mm, E in N/mm².
δ_max = 0.8 mm.

Assumptions and limits

Prismatic, slender, linear-elastic beam with small deflection.
Point load exactly at midspan, simple supports.
Shear deformation neglected.

Common mistakes

Using the moment formula for deflection.
Mixing m and mm, or GPa and N/mm².
Applying the central-load formula to an off-centre load.

Statistics and uncertainty

Cubes amplify errors: the relative error in deflection is about √[(δP/P)² + (3δL/L)² + (δE/E)² + (δb/b)² + (3δh/h)²]. A 2% error in depth becomes 6% in deflection.

Where it is used

Checking deck, ramp and bridge planks.
Floor and platform design.
Test-rig frames carrying instruments.

Maths you need

Second moment of area: I = bh³/12
Support reactions: the load is shared between two supports

Related formulas

Practise this formula