Cantilever Beam — End Load

A beam fixed at one end with a point load at the free end. The classic exhibit of why engineers fear length: deflection grows with the cube of the span.

Visualization

Input

L

Beam length

\mathrm{m}

P

Point load (downward)

\mathrm{N}

a/L

Load position (0 = at the wall, 1 = at the tip)

E

Young's modulus

\mathrm{GPa}

b

Section width

\mathrm{mm}

h

Section height

\mathrm{mm}

Show the engineering view (deflected-shape plot + cross-section)

The setup

A cantilever beam is fixed (built into a wall) at one end and free at the other. Push down on the free end with a load $P$ and the beam droops. The droop at the tip is the tip deflection $\delta_{\max}$ .

For a beam of length $L$ , made of a material with Young's modulus $E$ , and with a cross-section whose second moment of area about the bending axis is $I$ :

\delta_{\max} = \frac{P\,L^{3}}{3\,E\,I}

Source: Wikipedia — Deflection (engineering). The full deflected-shape function — the curve the beam traces — is $y(x) = \frac{P x^{2}}{6 E I}(3L - x)$ , and that's what's drawn in the visualization above as the slider moves.

Why length is so punishing

The headline feature of this formula is the cube on $L$ . Double the length and the deflection multiplies by eight, even with the same load and the same beam. Triple the length and it's twenty-seven times as much.

Try the slider: at $L = 1$ m the steel default barely moves; at $L = 2$ m the same load deflects 8× more; at $L = 3$ m, 27× more. This is why structural engineers obsess about span — it dominates everything else in the equation.

The section is everything else

The other lever you have is $I$ , the second moment of area. For a rectangular section of width $b$ and height $h$ (with the height in the load direction):

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

Source: Wikipedia — Second moment of area, and the same formula derived by integration in LibreTexts: Mechanics of Materials §4.2.

Notice this is also a cube — but on the height $h$ , not the length. Doubling the height of the section makes the beam eight times stiffer. Doubling the width only doubles the stiffness. That's why structural members are tall and thin (I-beams, joists on edge): you get the height-cubed bonus while spending less material.

Stress at the wall

The biggest internal moment in a cantilever with a tip load is at the fixed end, where the entire load $P$ is acting at distance $L$ — so the moment is:

M_{\max} = P \cdot L

This is what the LibreTexts §6.2 Example 1 verifies for the model below — a 5 lb load at 3 ft gives $|M_{\max}| = 15$ ft·lb at the wall.

That moment shows up as bending stress in the beam, biggest at the top and bottom surfaces (the extreme fibres). The Euler–Bernoulli prediction is:

\sigma_{\max} = \frac{M_{\max} \cdot c}{I}

with $c = h/2$ the distance from the neutral axis to the surface. Source: LibreTexts: Mechanics of Materials §4.2 Eq. 4.2.7. Compare $\sigma_{\max}$ to the material's yield stress to know whether your beam will deform plastically.

Show your work

Load position from wall (in metres): $a \;=\; (a/L) \cdot L \;=\; (\textcolor{#dc4124}{\left(2\right)}/\textcolor{#dc4124}{\left(2\right)}) \cdot \textcolor{#dc4124}{\left(2\right)} \;=\; 2\;\mathrm{m}$
Second moment of area (rectangular section): $I \;=\; \dfrac{b\,h^{3}}{12} \;=\; \dfrac{\textcolor{#dc4124}{\left(50\right)}\,\textcolor{#dc4124}{\left(100\right)}^{3}}{12} \;=\; 4.167 \times 10^{-6}\;\mathrm{m^{4}}$
Maximum bending moment (at the fixed end): $M_{\max} \;=\; P \cdot a \;=\; \textcolor{#dc4124}{\left(1000\right)} \cdot \textcolor{#dc4124}{\left(2\right)} \;=\; 2000\;\mathrm{N\cdot m}$
Maximum bending stress (extreme fibre at the wall): $\sigma_{\max} \;=\; \dfrac{M_{\max}\,c}{I} \;=\; \dfrac{M_{\max}\,\textcolor{#dc4124}{\left(0.05\right)}}{\textcolor{#dc4124}{\left(4.167 \times 10^{-6}\right)}} \;=\; 2.4 \times 10^{7}\;\mathrm{Pa}$
Maximum bending stress in MPa: $\sigma_{\max,\text{MPa}} \;=\; \sigma_{\max} \cdot 10^{-6} \;=\; 24\;\mathrm{MPa}$
Tip deflection (load at distance a from the wall): $\delta_{\max} \;=\; \dfrac{P\,a^{2}\,(3L - a)}{6\,E\,I} \;=\; \dfrac{\textcolor{#dc4124}{\left(1000\right)}\,\textcolor{#dc4124}{\left(2\right)}^{2}\,(3\textcolor{#dc4124}{\left(2\right)} - \textcolor{#dc4124}{\left(2\right)})}{6\,\textcolor{#dc4124}{\left(200\right)}\,\textcolor{#dc4124}{\left(4.167 \times 10^{-6}\right)}} \;=\; 0.0032\;\mathrm{m}$
Tip deflection in mm: $\delta_{\max,\text{mm}} \;=\; \delta_{\max} \cdot 10^{3} \;=\; 3.2\;\mathrm{mm}$

Worked Example

LibreTexts Engineering Mechanics: Statics §6.2 Example 1 — a cantilever with a 5 lb point load at 3 ft from the wall has a maximum bending moment of |M_max| = 15 ft·lb (= 20.337 N·m) at the fixed end. Here the load sits at the very tip, so a = L = 0.9144 m.

Quantity	Expected	Computed	Δ
M_max	20.34	20.34	1.704 \times 10^{-4}

Source: LibreTexts, Engineering Mechanics: Statics §6.2, Example 1 (Osgood, Cameron, Christensen)

Worked Example

A 1 m steel cantilever (E = 200 GPa) with a 50×100 mm rectangular section and a 1 kN load placed at the tip (a = L = 1 m). I = bh³/12 = (0.05)(0.1)³/12 = 4.167×10⁻⁶ m⁴; δ = PL³/(3EI) = 1000·1³/(3·2×10¹¹·4.167×10⁻⁶) = 4.0×10⁻⁴ m = 0.40 mm. With a = L the general formula Pa²(3L − a)/(6EI) collapses to PL³/(3EI), which is the Wikipedia tip-deflection formula.

Quantity	Expected	Computed	Δ
I	4.167 \times 10^{-6}	4.167 \times 10^{-6}	3.333 \times 10^{-12}
M_max	1000	1000	0
delta_max	4 \times 10^{-4}	4 \times 10^{-4}	1.084 \times 10^{-19}
delta_max_mm	0.4	0.4	1.11 \times 10^{-16}

Source: Wikipedia — Deflection (engineering); formula δ = PL³/(3EI) applied to chosen inputs

Worked Example

CalcResource cantilever reference: with the same 1 m steel beam (50×100 mm, E = 200 GPa) and a 1 kN load placed at a = L/2 = 0.5 m, the tip deflection is δ = P·a²·(3L − a)/(6EI) = 1000·(0.5)²·(3·1 − 0.5)/(6·2×10¹¹·4.167×10⁻⁶) = 5/16 of the tip-load case = 0.125 mm. The maximum moment is M_max = P·a = 500 N·m.

Quantity	Expected	Computed	Δ
M_max	500	500	0
delta_max	1.25 \times 10^{-4}	1.25 \times 10^{-4}	2.711 \times 10^{-20}
delta_max_mm	0.125	0.125	2.776 \times 10^{-17}

Source: CalcResource — Cantilever beam reference (load at distance a from fixed end)

Sources

[1] Wikipedia — Deflection (engineering), eq. δ_B = FL³/(3EI); δ(x) = Fx²/(6EI)·(3L−x); slender + linear-elastic + small-deflection assumptions (en.wikipedia.org)
[2] Wikipedia — Euler–Bernoulli beam theory, eq. EI d⁴w/dx⁴ = q(x) (en.wikipedia.org)
[3] D. Roylance (MIT). LibreTexts — Mechanics of Materials, §4.2 Stresses in Beams, eq. σ = -My/I (Eq. 4.2.7); I_rect = bh³/12 (eng.libretexts.org)
[4] Osgood, Cameron, Christensen. LibreTexts — Engineering Mechanics: Statics, §6.2 Shear/Moment Diagrams, eq. Cantilever with tip load P at distance L: M_max = -P·L at the wall (Example 1) (eng.libretexts.org)
[5] Wikipedia — Second moment of area, eq. I_x = b h³/12 for a rectangle of base b and height h about its centroid (en.wikipedia.org)
[6] Wikipedia — Bending, eq. σ = M·y/I; assumptions: plane sections remain plane, isotropic linear elastic, slender beam, small deflections (en.wikipedia.org)
[7] CalcResource — Cantilever beam reference, eq. Point load at distance a from the wall: M_max = -P·a at fixed end; tip deflection δ_u = P·a²·(3L − a)/(6EI); piecewise y(x) = P·x²·(3a − x)/(6EI) for x ≤ a, P·a²·(3x − a)/(6EI) for x > a (calcresource.com)

Assumptions

Euler–Bernoulli beam theory: slender beam (L/h ≥ 10), originally straight, isotropic linear-elastic material (Wikipedia: Bending; Wikipedia: Deflection (engineering)).
Cross-sections remain plane during bending (Wikipedia: Bending).
Small deflections only: maximum deflection ≪ L/10 (Wikipedia: Deflection (engineering)).
Rectangular cross-section, bending about the strong axis (b is the width, h is the height in the load direction).
A single point load P applied vertically at distance a from the fixed end (where 0 ≤ a ≤ L); self-weight of the beam is neglected.
Stress formula σ = Mc/I is the Euler–Bernoulli prediction at the extreme fibre (LibreTexts §4.2 Eq. 4.2.7).