Understanding Shear Force and Bending Moment Diagrams

Shear force diagrams (SFD) and bending moment diagrams (BMD) are essential tools for beam design. This guide explains what they represent, how to draw them, and how to interpret them for structural design. Generate SFD and BMD instantly with the free beam calculator.

Why SFD and BMD Matter

When a beam carries loads, it develops internal forces that prevent it from breaking apart. These internal forces are not applied externally — they are the beam’s response to the external loads. The two primary internal forces in a beam are shear force and bending moment.

A shear force diagram (SFD) shows how the internal shear force varies along the length of the beam. A bending moment diagram (BMD) shows how the internal bending moment varies. Together, they give a complete picture of the beam’s internal state, which is essential for structural design.

Structural engineers use SFD and BMD to determine the critical design values: the maximum shear force (for web shear capacity checks), the maximum positive moment (for sagging moment capacity), the maximum negative moment (for hogging moment capacity), and the locations where these maxima occur. Without these diagrams, it is impossible to design a safe and efficient beam.

The StructureCalcs Beam Calculator generates both diagrams automatically and interactively. You can hover over any point to read exact values, making it far more practical than hand-drawn diagrams.

What Is Shear Force?

Imagine slicing the beam at any point along its length with an imaginary cut. The shear force at that point is the net vertical force that acts on either side of the cut to maintain equilibrium. It represents the tendency of one part of the beam to slide vertically past the adjacent part.

Mathematically, the shear force V at any section equals the algebraic sum of all vertical forces to the left (or right) of that section: V = ΣF(left). You can calculate it from either side of the cut and get the same answer (by equilibrium).

Inside the beam, shear force creates shear stress (τ) that is distributed across the cross-section. For a rectangular cross-section, shear stress is parabolic, with maximum value at the neutral axis and zero at the top and bottom fibers. For I-beam sections, the web carries most of the shear force (which is why it exists), and the flanges carry very little shear.

Sign convention in StructureCalcs: Positive shear force acts upward on the left face of the cut section. A beam with only downward loads will have positive shear near the left support and negative shear near the right support.

What Is Bending Moment?

The bending moment at any section of the beam is the net rotational effect (torque) of all forces acting to one side of that section. It causes the beam to curve: the fibers on one side of the neutral axis are compressed while those on the other side are stretched in tension.

Mathematically, the bending moment M at any section equals the sum of moments of all forces to the left (or right) of that section about the cut point: M = Σ(F × d).

Sagging moment (positive) curves the beam concave upward, like a smile. The bottom fiber is in tension and the top fiber is in compression. This occurs in the mid-span of simply supported beams under downward loads.

Hogging moment (negative) curves the beam concave downward, like a frown. The top fiber is in tension and the bottom fiber is in compression. This occurs over intermediate supports of continuous beams and along the full length of cantilevers under downward loads.

The bending moment determines the required section modulus of the beam: the maximum bending stress is σ = M divided by the elastic section modulus (the second moment of area divided by the distance to the farthest fiber) — written Z in AS 4100 and NZS 3404, S in AISC, and W_el in Eurocode. For the beam to be safe, this stress must not exceed the material’s allowable bending stress (yield stress divided by a safety factor).

The Relationship Between Load, Shear, and Moment

There is a fundamental mathematical relationship between the distributed load w(x), shear force V(x), and bending moment M(x) at any point along the beam:

dV/dx = −w(x) — The rate of change of shear force equals the negative of the distributed load intensity.
dM/dx = V(x) — The rate of change of bending moment equals the shear force.

These two equations have powerful implications for understanding the shape of SFD and BMD:

Where there is no distributed load, the shear is constant and the moment varies linearly.
Where there is a uniform distributed load, the shear varies linearly and the moment varies parabolically (quadratically).
A concentrated point load causes a sudden jump in the SFD equal to the load magnitude, and a kink (change of slope) in the BMD.
A concentrated moment causes a sudden jump in the BMD but no change in the SFD.
Where the shear force is zero, the bending moment reaches a local maximum or minimum. This is because dM/dx = V, and when V = 0, M has zero slope (a turning point).

This last point is crucial for design: the maximum bending moment always occurs where the shear force is zero or changes sign. Look for zero crossings in the SFD to locate the critical bending moment.

How to Draw an SFD Step by Step

Follow these steps to construct a shear force diagram by hand:

Find support reactions first. Use equilibrium equations (ΣFy = 0 and ΣM = 0) to determine the vertical reaction at each support.
Start at the left end. If there is an upward reaction RA, the shear jumps from 0 to +RA.
Move right along the beam. Between point loads, the shear changes according to the distributed load: constant if no load, linear if uniform load, parabolic if triangular load.
At each point load or support reaction, add or subtract the force from the current shear value. An upward force increases shear; a downward force decreases it.
At the right end, the shear should return to zero (if the right reaction brings it back to zero, your diagram is correct — this is a self-check).

Or simply enter your beam into the Beam Calculator and get the SFD instantly with exact values at every point.

How to Draw a BMD Step by Step

The bending moment diagram can be constructed from the shear force diagram:

Start at the left end. If the left support is a pin or roller, M = 0. If it is fixed, M equals the fixed-end moment reaction.
The change in moment between two points equals the area under the SFD between those points. This follows from dM/dx = V, so ΔM = ∫V dx.
Where shear is positive, the moment is increasing (slope up). Where shear is negative, the moment is decreasing (slope down).
Where shear is zero, the moment has a local peak (maximum sagging or hogging).
At an applied moment, the BMD jumps by the magnitude of the applied moment.
At pin/roller supports, M = 0 (these supports cannot resist moment).

BMD shape rules: Under no-load regions the BMD is linear. Under uniform loads it is parabolic. Under triangular loads it is cubic. These shapes help you sketch the diagram quickly and verify computer output.

Example: Simply Supported Beam with Point Load

Problem: A 6 m simply supported beam carries a 12 kN downward point load at 2 m from the left support.

Reactions:

Taking moments about A: RB × 6 = 12 × 2, so RB = 4 kN. From ΣFy = 0: RA = 12 − 4 = 8 kN.

Shear Force Diagram:

At x = 0: V jumps to +8 kN (RA).
From x = 0 to x = 2: V = +8 kN (constant, no distributed load).
At x = 2: V drops by 12 kN, from +8 to −4 kN.
From x = 2 to x = 6: V = −4 kN (constant).
At x = 6: V jumps by +4 kN (RB), returning to zero.

Bending Moment Diagram:

At x = 0: M = 0 (pin support).
From x = 0 to x = 2: M increases linearly. At x = 2: M = 8 × 2 = 16 kN·m.
From x = 2 to x = 6: M decreases linearly. At x = 6: M = 16 − 4 × 4 = 0 (check).
Maximum moment: 16 kN·m at x = 2 m (where the point load is applied).

Verify this by entering the problem in the Beam Calculator: pin at 0, roller at 6, point load of −12 kN at position 2.

Figure 1.The shear diagram for the problem above, solved: +8 kN for the first 2 m, a vertical drop of the full 12 kN under the load, then −4 kN to the right support. Exactly the hand result — and note the drop equals the load, never anything else.

Figure 2.Its moment diagram: two straight lines meeting in a sharp peak of 16.0 kN·m directly under the load, matching the hand calculation above. Straight, because the shear either side of the load is constant.

Example: Uniform Load on a Simply Supported Beam

Problem: A 10 m simply supported beam carries a uniform load of 5 kN/m over its full length.

Reactions:

By symmetry: RA = RB = 5 × 10 / 2 = 25 kN.

Shear Force Diagram:

Starting at +25 kN (RA), the shear decreases linearly at 5 kN/m. At mid-span (x = 5 m): V = 25 − 5 × 5 = 0. At the right end (x = 10 m): V = 25 − 5 × 10 = −25 kN, then jumps back to 0 by RB. The SFD is a straight line from +25 to −25.

Bending Moment Diagram:

The moment is parabolic (because the shear is linear). Maximum moment occurs where V = 0, at mid-span: Mmax = wL²/8 = 5 × 10² / 8 = 62.5 kN·m. The BMD is a symmetric parabola opening downward (in the engineering convention), peaking at mid-span.

Figure 3.Shear for the uniform load above: a straight slope from +25 to −25 kN, crossing zero at mid-span — the hand result, drawn.

Figure 4.And the moment: a smooth parabola peaking at 62.5 kN·m — wL²/8 — at exactly the point where the shear crossed zero. Integrating a straight line gives a parabola, which is the whole load–shear–moment relationship in one picture.

SFD and BMD for Cantilever Beams

Cantilever diagrams look different from simply supported beam diagrams because the fixed end carries both a reaction force and a reaction moment.

Cantilever with point load P at the free end: The SFD is constant at +P from the fixed end to the free end (or equivalently, constant at −P from free end to fixed end, depending on your sign convention and direction of travel). The BMD is linear, from −PL at the fixed end to zero at the free end. The entire beam is in hogging.

Cantilever with uniform load w: The SFD is linear (from wL at the fixed end to 0 at the free end). The BMD is parabolic, with maximum value −wL²/2 at the fixed end and zero at the free end. Again, the entire beam is in hogging.

The key difference from simply supported beams: the maximum moment is always at the fixed support, not at mid-span. This means the most critical section for design is at the wall or column connection, which must be strong enough to resist the full fixed-end moment.

Figure 5.A cantilever with a 20 kN tip load: the shear is constant along the whole length, because there is nothing between the tip and the support to change it.

Figure 6.Its moment grows linearly from zero at the free end to 80.0 kN·m at the support — hogging the entire way, which is why a cantilever’s tension face is the top one.

SFD and BMD for Continuous Beams

Continuous beams have more complex diagrams because bending moment is transferred across intermediate supports. The BMD typically alternates between sagging (positive, at mid-spans) and hogging (negative, over supports), creating a characteristic wave-like shape.

At each intermediate support, there is a peak hogging moment. Between supports, the moment reverses to sagging, reaching a peak near mid-span. The magnitudes depend on the span lengths, load distribution, and stiffness ratios.

Points of contraflexure (also called inflection points) are locations where the bending moment passes through zero, transitioning from sagging to hogging or vice versa. These are significant for design because they indicate where reinforcement transitions (in concrete beams) or where lateral restraint requirements change (in steel beams).

Continuous beam diagrams are too complex to draw reliably by hand for more than two spans. The StructureCalcs Beam Calculator handles any number of spans and produces exact diagrams automatically, including support reactions, points of contraflexure, and peak values at every critical location.

Figure 7.Two 6 m spans under 12 kN/m. The diagram crosses the axis twice: sagging in each span, hogging over the middle support, with points of contraflexure where it changes sign. Peak magnitude 54.0 kN·m, over the support rather than in a span.

Interpreting Diagrams for Design

Once you have the SFD and BMD, use them to make design decisions:

Maximum shear force determines the required shear capacity of the cross-section. For steel I-beams, check the web thickness and depth. For concrete beams, check stirrup spacing.
Maximum sagging moment determines the required bottom reinforcement (concrete) or bottom flange size (steel) at mid-span.
Maximum hogging moment determines the required top reinforcement or top flange capacity at supports.
Points of zero shear identify the locations of maximum moment — these are the critical sections for bending design.
Points of contraflexure help determine curtailment of reinforcement (concrete) and the unbraced length for lateral-torsional buckling checks (steel).
Diagram shape confirms your model is behaving as expected. If the BMD looks wrong (e.g., hogging where you expected sagging), re-check your support types and load directions.