Cantilever — Concentrated Load

Calculate support reaction, shear force, and bending moment for a cantilever beam with a concentrated end load

Inputs

W

Load

l

Span

Formula Interpretation

Support Reaction

R = W \quad \text{(N)}

By vertical equilibrium, the fixed support at A must provide an upward reaction equal to the applied load W. This is the only external reaction for a cantilever.

Shear Force

F = -W \quad \text{(N)}

Cutting a free body from the right side of any section: only the external force W acts on it, producing a constant shear of −W throughout the entire span.

Bending Moment

M_x = Wx \quad (0 \le x \le l) \qquad M_{\max} = Wl

The bending moment varies linearly from zero at the free end (x = 0) to the maximum value Wl at the fixed support (x = l). The fixed end is the critical section.

Knowledge Points

Critical Section

The cross-section where the bending moment is maximum experiences the greatest normal stress. For a cantilever with a tip load, this section is always at the fixed support. Failure typically initiates here.

Constant Shear Force Diagram

Because there are no intermediate loads, the shear force is constant (−W) across the entire beam. The SFD is a horizontal straight line parallel to the beam axis, a hallmark of end-loaded cantilevers.

Linear Bending Moment Diagram

The bending moment increases linearly from the free end (M = 0) to the fixed end (M = Wl). The BMD is a right triangle. This linear distribution guides where reinforcement or cross-section enlargement is most beneficial.

Worked Example

A cantilever beam of span l = 1000 mm carries a concentrated load W = 2000 N at its free end. Draw the SFD and BMD, and find the maximum bending moment.

Knowns

• Span: l = 1000 mm
• Concentrated load at free end: W = 2000 N

Solution

Step 1 — Support Reaction (Formula ①)

R = W = 2000 \text{ N}

Step 2 — Shear Force (Formula ②)

F = -W = -2000 \text{ N} \quad \text{(constant, parallel to beam axis)}

Step 3 — Bending Moment (Formula ③)

M_x = Wx = 2000x \quad (0 \le x \le 1000 \text{ mm})

Step 4 — Maximum Bending Moment

M_{\max} = Wl = 2000 \times 1000\,\text{N·mm} = 2000\,\text{N·m}

Result: R = 2000 N (upward), F = −2000 N (constant), Mmax = 2000 N·m at the fixed support A.

Extended Knowledge

•Both are real-world cantilevers with tip loads. Engineers size the cross-section at the fixed end to resist Mmax, and check shear capacity against the constant force F.
•A crane arm is essentially a cantilever — the lifted load W acts at the tip. The bending moment at the mast connection equals W × reach, which is why crane ratings decrease as reach increases.
•Since the bending moment varies linearly, an optimised cantilever uses a tapered cross-section — deeper near the support, shallower near the free end. This saves material while maintaining equal stress throughout.
•At the critical section, the maximum normal stress is σmax = Mmax / W_section where W_section is the section modulus. Fibres on the tension side elongate; fibres on the compression side shorten.
•When a load is dropped onto the free end, the dynamic bending moment can far exceed the static value (by a factor of 2 or more for suddenly applied loads). Impact factors must be included in the design.