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Structural Analysis Fundamentals / Internal forces and structural diagrams

Building the diagrams

Predict the shape, then calculate the values.

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This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.

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These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

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Why this matters

Shear-force and bending-moment diagrams show how internal actions vary along a beam. Sketch their expected shapes before calculating; the sketch helps you check the numerical results.

By the end of this lesson you should be able to

  • Work out reactions, then shear, then moment, in that order
  • Sketch each diagram before finding numbers
  • Read the key values off the diagrams

For the following statically determinate examples, use this sequence. Find the reactions. Then work along the beam from one end building the shear force diagram. Then build the bending moment diagram from the shear. Doing it in any other order makes life much harder.

Two rules cover most of what shear does. At a point load, the shear diagram takes a vertical step equal to that load. Under a UDL, the shear slopes steadily downwards at a rate equal to the load intensity. Where there is no load at all, the shear stays flat.

Predict first

A simply supported beam carries only a uniformly distributed load. What shape is the bending moment diagram?

Try it

Beam laboratory

Change the beam, then reveal one diagram at a time. Predict each shape before you reveal it.

Beam type

m

kN (0 to remove)

m from left

kN/m (0 to remove)

Reveal

Simply supported beam of span 6 metres20 kN6 m
Shear force and bending moment diagrams for an eight metre beam carrying a five kilonewton per metre uniform load and a forty kilonewton point load at three metres.5 kN/m40 kN45.00 kN35.00 kNShear force V (kN)45.00Bending moment M (kN·m)112.50
The completed diagrams for the beam used in the preceding reactions and cut-section lessons. Note the step in shear under the point load, and that the moment peaks exactly where shear crosses zero.

Worked example

Full diagrams for a beam with a UDL and a point load

Full diagrams for a beam with a UDL and a point load5 kN/m40 kN45.00 kN35.00 kN

Given

  • Span 8.0 m, simply supported
  • UDL 5 kN/m over the whole span; point load 40 kN at 3.0 m
  • Reactions already found: RA = 45 kN, RB = 35 kN

Find

The shear force and bending moment diagrams, and the maximum moment.

    Practice

    A simply supported beam spans 8.0 m with a single point load of 32 kN at 3.0 m from the left support. What is the left-hand reaction?

    Practice

    For that same beam, what is the maximum bending moment?

    Practice

    For the same beam, what is the shear force just to the right of the 32 kN load?

    Summary

    • Reactions first, then shear, then moment
    • Point load: step in shear. UDL: sloping shear, curved moment
    • Check moment extrema where shear changes sign, and at member ends and applied couples
    • Moment is zero at a simple end support if no external couple is applied there