Shear force and bending moment diagrams

By the end of this topic you will be able to

Explain what shear force and bending moment are and state a sign convention.
Draw shear force and bending moment diagrams for simply supported beams and cantilevers.
Use the relationships between load, shear and moment to locate the maximum bending moment.

Internal forces in a beam

A beam carries load across a span by developing internal forces. Imagine cutting the beam at a section: to keep the left part in equilibrium, the cut face must supply

a vertical shear force $V$ that balances the net vertical load on that part, and
a bending moment $M$ that balances the moment of those loads about the cut.

$V$ and $M$ change from section to section. Plotting them against position gives the shear force diagram (SFD) and bending moment diagram (BMD), the starting point for sizing every beam.

Sign convention

Shear $V$ is positive when it tends to rotate the part of the beam it acts on clockwise. Equivalently, at a section, $V$ equals the sum of the upward forces minus the downward forces to the left of the section.
Bending moment $M$ is positive when the beam sags (curves like a smile), putting the top in compression and the bottom in tension.

Method

Find the support reactions (see Equilibrium and support reactions).
Move from left to right. A point load makes $V$ jump by the size of the load (down for a downward load); a UDL makes $V$ slope downward at $w$ per metre.
$M$ is the area under the shear diagram. Where $V = 0$ , $M$ has a maximum or minimum.
Check: $V$ and $M$ must return to zero at a free end or simple support (unless a moment is applied there).

Load, shear and moment are linked

\frac{\mathrm{d}V}{\mathrm{d}x} = -w, \qquad \frac{\mathrm{d}M}{\mathrm{d}x} = V

So a uniform load gives a sloping shear line and a parabolic moment curve; a point load gives a jump in shear and a kink in the moment curve.

Interactive modelAdd a point load, a UDL, or both. Switch to a cantilever to see how the diagrams change.

This interactive model needs JavaScript. The theory, worked example and problems on this page work without it.

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Worked example 1: UDL on a simply supported beam

A 6 m simply supported beam carries 10 kN/m over its whole length. By symmetry $R_A = R_B = 30$ kN.

At distance $x$ from A:

V = 30 - 10x, \qquad M = 30x - 5x^2

$V = 0$ at $x = 3$ m, where $M_{\text{max}} = 30\times3 - 5\times9 = 45$ kN·m. This agrees with $wL^2/8 = 10\times36/8 = 45$ kN·m. The model is set to this case.

Practice problems

A simply supported beam of span 8 m carries a 40 kN point load at 3 m from A. Sketch V and find the maximum moment.

Show answer

$R_A = 25$ kN, $R_B = 15$ kN. $V = +25$ kN from A to the load, then $-15$ kN. $M_{\text{max}}$ is under the load: $25\times3 = 75$ kN·m.
A cantilever of length 2 m carries a 15 kN load at the free end. Find the moment at the wall.

Show answer

$M = -15\times2 = -30$ kN·m (hogging, tension in the top fibres). Shear is a constant 15 kN.
A cantilever of length 3 m carries 10 kN/m over its length. Find $V$ and $M$ at the wall.

Show answer

$V = 30$ kN, $M = -wL^2/2 = -45$ kN·m.
Where is the bending moment largest on a beam, and how do you find it from the shear diagram?

Show answer

Where the shear force crosses zero (or changes sign), since $\mathrm{d}M/\mathrm{d}x = V$ .

Key formulae

Shear force at a section: $V = \sum F_{\text{up, left}} - \sum F_{\text{down, left}}$
Load–shear relation: $\frac{\mathrm{d}V}{\mathrm{d}x} = -w$
Shear–moment relation: $\frac{\mathrm{d}M}{\mathrm{d}x} = V$
Simply supported beam, central point load: $M_{\text{max}} = \frac{PL}{4}$
Simply supported beam, full-span UDL: $M_{\text{max}} = \frac{wL^2}{8}$
Cantilever, end load or full UDL: $M_{\text{max}} = PL \;\text{or}\; \frac{wL^2}{2}$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (55 min). 10 min: cut a foam beam to show internal forces; 10 min: sign convention (spend time here); 15 min: step-by-step worked example 1; 10 min: model, then example 2; 10 min: students draw problem 1 and check against the model.

Activity. "Predict then confirm": show the loading, have students sketch V and M, then reveal the model.

Discussion. Why does the maximum moment occur where shear is zero?

Assessment idea. A beam with overhang or with a UDL over part of the span (extend with the problems above).