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Civil

Cantilever with a tip load

Tip deflection and root moment of a beam fixed at one end.

Ref: CIV-04
Rev: v1
Field: Civil
Result: δ_tip [mm]

δ_{tip} = \frac{P L^{3}}{3 E I}, M_{root} = P L

P the transverse load at the free end, L the length from root to tip, E and I as before. The calculator uses a rectangular section b × h.

A beam fixed at one end and loaded at the other droops at the free tip. The root carries the full moment, and the tip deflection grows with the cube of the length.

Calculator

Solve for

δ_tipTip deflectionComputed

PTip load

0.150

LLength

0.36

EYoung’s modulus

GPa

10210

bSection width

10200

hSection depth

20400

Root moment PL3 kN·m
Peak stress at root36 MPa

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.

Tip deflection (mm) current value 2.7 mm

hSection depthKey driver

0.0422338

LLength

0.0216173

PTip load

0.13567.5

EYoung’s modulus

2.5754

bSection width

0.67513.5

A cantilever deflects 16 times as much as a simply supported beam of the same span under the same central load.
Both stress and deflection peak at or beyond the root, so check the root joint and fixings.
Shortening the overhang by 20% cuts the deflection almost in half (0.8³ = 0.51).

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (mm)	Local step	Change in result
h Section depth	20 to 400 mm	0.04219	337.5	337.5	+10%	−24.9 %
L Length	0.3 to 6 m	0.0216	172.8	172.8	+10%	+33.1 %
P Tip load	0.1 to 50 kN	0.135	67.5	67.36	+10%	+10 %
E Young’s modulus	10 to 210 GPa	2.571	54	51.43	+10%	−9.09 %
b Section width	10 to 200 mm	0.675	13.5	12.82	+10%	−9.09 %

Sweep of Length

Length (m)	Tip deflection (mm)
0.3	0.0216
0.4363	0.06643
0.6344	0.2043
0.9226	0.6282
1.342	1.932
1.951	5.941
2.837	18.27
4.126	56.19
6	172.8

Sweep one variable

Vary

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

Variables and how each one contributes

δ_tip

Tip deflection

Result · mm

What the equation gives you. Everything else feeds it.

Tip load

kN · typical 0.1 to 50

Transverse force at the free end. Directly proportional.

Length

m · typical 0.3 to 6

Length from the fixed root to the load. The dominant driver: deflection grows with L³.

Young’s modulus

GPa · typical 10 to 210

Material stiffness. Inversely proportional.

Section width

mm · typical 10 to 200

Width of the section. Inversely proportional.

Section depth

mm · typical 20 to 400

Depth in the bending direction. Inversely proportional to h³.

Worked example with the values above

Root moment: M = PL = 2 × 1.5 = 3 kN·m. I = bh³/12 = 4,167,000 mm⁴.
δ = PL³/(3EI) with P in N, L in mm, E in N/mm².
δ_tip = 2.7 mm.

Assumptions and limits

Prismatic, slender, linear-elastic beam, small deflection.
Perfectly rigid root.
Load acts at the tip, transverse to the beam.

Common mistakes

Forgetting the root is more flexible than assumed.
Using L to the load instead of to the tip when the load is not at the tip.
Mixing units of L and E.

Statistics and uncertainty

The cube on length and on depth amplifies measurement errors by three: a 2% error in either becomes about 6% in the deflection. In real structures the “fixed” root is never perfectly rigid, which usually adds more deflection than the formula predicts.

Where it is used

Antenna and sensor booms.
Gun barrels and mast supports.
Brackets and balcony slabs.

Maths you need

Bending moment: moment grows towards the root
Second moment of area: I = bh³/12

Related formulas

Practise this formula