Module 6 · Course notes
Continuity: one beam, many spans
A continuous beam runs over its interior supports without a break. That single fact — the beam does not stop and restart at each support — couples the spans together. Load one span and its neighbours respond; drop a support and the whole beam redistributes. This module teaches you to see those couplings before you calculate them.
Over an interior support the beam is continuous: it shares one deflection and one rotation there. Because it cannot kink, a span that rotates the joint one way forces the next span to follow. Two consequences run through everything below:
Hogging appears over the interior supports. The beam is held up at the support while the spans either side sag away from it, so the top fibre stretches over the support — hogging. Each span still sags in its middle. The moment diagram becomes the familiar wave: sagging humps in the spans, hogging peaks over the supports.
The interior supports carry more than their share. Continuity leans each span on the support between them, so an interior support attracts more load than a simple tributary-area guess. For two equal spans under a uniform load the central support carries 1.25 wL — a quarter more than the wL a naive split would give. Sizing that support or its foundation on the tributary figure under-designs it.
A support that moves generates force with no applied load. Because the beam is over-constrained, forcing a support out of position bends it. Settlement, temperature and lack of fit all do this. A determinate beam would simply follow the movement; a continuous one fights it.
Worked example
Worked example — only one span is loaded
Two equal spans, continuous over the central support, but only the left span carries load. Watch how continuity makes the unloaded span respond.
Step 1 — points of certainty
All three supports hold their points on the beam line. Only the left span is loaded, so it will sag. The question is what the right span does — and continuity is the key: the beam shares one rotation over the central support.
Step 2 — deflected shape and reactions
The loaded left span sags, rotating the central support. Because the beam is continuous, that rotation carries into the right span and lifts it — the unloaded span bows upward in reverse curvature. The central support still takes the most load; the far right support takes very little, and if the loading were heavier it could even be pulled down.
Step 3 — bending moment on the tension side
The left span sags (moment below the beam) and hogging develops over the central support (above), reaching into the unloaded right span. The moment is zero at the two outer simple supports (rule 1) and crosses the baseline in the left span at the point of contraflexure. The all-important feature is that the right span is not blank — continuity has given it a moment even though nothing sits on it.
Every exercise below is one of these couplings on a different beam — three spans, unequal spans, a support added or removed, a support that settles, an internal hinge, or an end that lifts off. Predict the shape, the reactions and the moment, and check that continuity is honoured over every interior support.
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
Continuous-beam exercises
17 exercises — three-span beams, unequal spans, added and removed supports, settlement, an internal hinge and end-bearing lift-off. Predict, then reveal an 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.