Module 15 · Course notes
Slabs: two-way load paths and yield lines
A slab is not just a wide beam. Supported on more than two edges, it carries load in two directions at once, and it fails not at a single section but along lines of plastic hinging — yield lines — that fold it into a collapse mechanism. Reading which way a slab spans, and picturing its yield-line pattern, is the qualitative core of slab behaviour. This is about behaviour, not code design.
A slab supported on all four edges carries its load in two directions at once. The load shares between the two spans in proportion to their stiffness — the shorter span, being stiffer, takes more. Only a slab that is long and narrow (or supported on just two opposite edges) behaves essentially one-way, spanning the short direction like a series of beams.
At collapse the slab develops yield lines: lines along which the reinforcement has yielded and the slab hinges plastically. A valid yield-line pattern divides the slab into rigid regions, each rotating about its supported edge, so the whole slab folds as a mechanism.
Yield-line analysis is an upper-bound method. For an assumed pattern it gives a collapse load that is greater than or equal to the true one — so an arbitrary pattern is unsafe (it overestimates capacity). The real collapse load comes from the pattern that gives the lowest load, so you must search for the most critical pattern.
Worked example
Worked example — one way or two?
Decide how a slab spans, then picture its collapse.
Step 1 — how it spans
Supported on all four edges and roughly square, the slab spans in both directions; the load shares between them by stiffness. If it were long and narrow, it would span one way across the short direction.
Step 2 — the collapse pattern
At collapse, yield lines form — diagonals from the corners meeting a central line — dividing the slab into rigid triangular and trapezoidal regions that fold about the supported edges. The critical pattern is the one that gives the lowest collapse load.
In the exercises, decide which way a slab spans, describe a valid yield-line pattern, and reason about the upper-bound nature of the method.
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
Slabs and yield-line thinking exercises
4 exercises on two-way spanning, yield-line patterns, the upper-bound nature of the method and one-way behaviour. Predict, then reveal an explanation.
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.