Module 5 · Course notes

Reading a beam before you calculate it

A structural drawing is a language. Before touching a calculator, a good engineer can already see how a beam will move, which way its supports push, and where it bends the most. That skill is what catches an analysis that has quietly solved the wrong structure. This module builds it one beam at a time.

Numerical analysis gives you magnitudes. What it does not give you is judgement — the ability to look at a result and know whether it is even possible. That judgement comes from reading the deflected shape, the reactions and the bending moment as three descriptions of the same response. If they do not agree, one of them is wrong.

The method below is the order to think in. It is the same order whether the beam is a simple span or a continuous multi-span, and it never needs a number.

Step 1 — start with the shape. Mark the points you are certain about: the deflected curve must pass through every support, and it must move down under a downward load. At a fully fixed support the beam leaves the wall with zero rotation — flat against it. At a pin or roller the beam can rotate freely but cannot move vertically. Sketch a smooth curve that honours all of those.

Both ends are fully fixed, so the beam leaves each wall flat — zero rotation at the face. That forces double curvature: hogging (concave down) near each support and sagging (concave up) in the middle. A pin at either end would let the beam rotate there instead, and the shape would change.

Step 2 — get the reactions by notional removal. To find which way a support pushes, imagine it is not there. Whichever direction the structure would move, the reaction is the force that pushes it back. Usually that is upward — but not always. On an overhang the far support can be pulled down (a hold-down), and a bearing that cannot pull can lift off entirely.

Step 3 — draw the bending moment on the tension side, and check it. The moment is drawn on the face that is in tension: sagging moments below the beam, hogging moments above it. Then check it against the shape you already drew — they must tell the same story.

Zero at both simple supports (rule 1), straight lines because the load is a point load (rule 3), and a single sagging peak under the load — drawn below the beam because the bottom fibre is in tension (rule 6). Because the load is off-centre the triangle is lopsided, and its peak sits under the load, not at mid-span.

Hogging at the fixed end (above the beam), sagging under the load (below), and the diagram crosses the baseline once — that crossing is the point of contraflexure (rule 2), where the curvature reverses. Straight lines here, because the load is a point load rather than distributed.

Worked example

Worked example — a load on an overhang

This beam runs on a pin and a roller with a length hanging past the roller, carrying a distributed load out on that overhang. It is the classic case where a reaction reverses. Follow the three steps and watch the reasoning build.

Step 1 — points of certainty
The pin at the left end and the roller at 6 m hold those two points on the beam line. The load sits out on the overhang, beyond the roller, so the roller acts as a pivot the load levers over. Nothing else is certain yet — but that pivot is the key.
Step 2 — deflected shape and reactions
The overhang droops under its load, and levering over the roller lifts the main span between the two supports — it bows upward. Now remove the pin in your mind: that end would spring up, so the pin reaction must pull it back down. The pin reaction is a hold-down (it acts downward); the roller reaction is large and upward.
Step 3 — bending moment on the tension side
The beam is concave-down along its whole length — the main span and the overhang both hog — so the moment is drawn above the beam, curved (the load is distributed), peaking over the roller. It is zero at the pin and at the free tip (rule 1). There is no contraflexure here, because nothing makes this beam sag: a point of contraflexure appears only when part of the span sags while another part hogs, as in the propped-cantilever figure above. The all-hogging moment agrees with the single hogging curvature you drew in step 2.

That is the whole method. Every exercise below is the same three steps on a different beam — a cantilever, a propped cantilever, a fixed-ended beam, a continuous beam, one with an internal hinge. Predict the shape, the reactions and the moment before you reveal the answer, and check that all three agree.

How to read these problems

The three-step method

1Points of certainty. The deflected curve must pass through every support and deflect downward under the load. Mark what each support prevents before drawing anything.
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).
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.The bending moment is zero at a simple support and at an internal pin or hinge.
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.If a part of the structure stays straight after loading, it carries no bending moment there.
6.The moment is drawn on the tension side: sagging below the member, hogging above it.

Now predict for yourself

Beam behaviour exercises

24 exercises, each a different beam. Predict the deflected shape, reactions or bending moment, then reveal a full explanation checked against a real elastic solve.

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.