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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.

Checks needed for project use

  • An independent structural reviewer must check the illustrated models, prediction feedback, geometry and full interactive input ranges.
  • Confirm diagram sign conventions and mathematical model assumptions. A pin support under a continuous beam does not make internal moment zero. Numeric convergence is a tolerance test, not proof of correct arithmetic or model.

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.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

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Module 1 · Course notes

A drawing is a structural claim

Before you can predict a structure, you have to learn to read one. Every symbol on a structural drawing — a support, a hinge, an arrow, a shaded diagram — is a claim about what the structure does. This module builds the habit that runs through the whole course: read the movement, the reactions and the internal forces as three descriptions of one response, and check that they agree.

Deflected shape, reactions and bending moment describe the response to the same load. Check that equilibrium, compatibility and the member stiffness relate them consistently. Moment sign follows the local curvature and the declared sign convention; downward displacement alone does not imply sagging moment. A true moment release transmits no bending moment. A continuous member may carry moment over a support that restrains translation.

That is the coherence check, and it is the most powerful hand tool you have. A result that looks polished but fails it is wrong, however it was produced.

deflected shape

the same beam, read as a shape
The beam sags in single curvature and returns to zero at both supports. A coherent bending-moment diagram for it must be sagging (drawn below), zero at the two simple supports, and peaking under the load — the shape and the moment tell the same story.

The second habit is to predict before you calculate. Numbers tell you how much; they do not tell you whether the answer is even possible. If you have already pictured the response, a wrong analysis stands out immediately — a moment appearing at a pin, a reaction pointing the wrong way, a span deflecting upward under downward load.

The third habit follows from it: a converged solver is not a validated model. A program can solve the wrong structure perfectly. Convergence means the arithmetic is consistent, not that the topology, supports and releases are what you intended.

Worked example

Worked example — is this a coherent set?

A beam with an internal hinge is offered with a deflected shape and a moment diagram. Read them together and check the hinge.

  1. Step 1 — read the symbols

    A beam fixed at the left, propped at the right, with an internal hinge at mid-span carrying a uniform load. The fixing prevents rotation; the hinge releases moment. Those two facts constrain everything.

  2. Step 2 — the moment must be zero at the hinge

    A genuine hinge cannot transfer bending moment. So a coherent bending-moment diagram must pass through zero exactly at the hinge — hogging towards the fixing, sagging towards the prop. Any diagram showing moment at the hinge fails the coherence check and must be rejected, no matter how neat.

    bending moment (tension side)

The exercises below give you sets of claims — some coherent, some not. Your job is not to calculate, but to read them and decide whether the movement, the reactions and the internal forces can all be true at once.

How to read these problems

The three-step method

  1. 1Points of certainty. Match the displacement and rotation that each support prescribes, including any settlement. Infer the displacement direction from the whole structure and load case; it need not be downward at every loaded point.
  2. 2Deflected shape and reaction directions. Sketch the compatible deflected shape. To find a reaction's direction, imagine removing that support: the direction that pushes the structure back to its place is the reaction's sense (it may be a hold-down).
  3. 3Bending moment and contraflexure. Draw the bending-moment diagram on the tension side and check it against the shape: hogging where the curve is convex-up, sagging where convex-down, zero at pins and at every contraflexure.

Rules that must always hold

  • 1.Moment is zero at a genuine moment release and at an unloaded simple end. A continuous member can carry moment over a translational support.
  • 2.A bending-moment diagram crosses the baseline exactly at a point of contraflexure.
  • 3.Under a distributed load the bending-moment diagram is curved; under point loads alone it is straight lines.
  • 4.At a fully fixed support the deflected shape leaves the support with zero rotation (tangent along the member).
  • 5.For an initially straight elastic beam without imposed curvature, zero curvature implies zero bending moment.
  • 6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Visual-language exercises

5 exercises on reading structures and checking that a set of diagrams is coherent — including a converged software result you should not trust.

Start the exercises →

This lesson is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions.