Module 9 · Lesson 9.1
One-way and two-way action
Why a slab chooses a direction, and why the choice is decided by the fourth power of the span ratio.
Why this matters
Everyone learns that a slab is one-way if it is more than about twice as long as it is wide. Almost nobody is told where the rule comes from — and it comes from one line of algebra that also tells you exactly how wrong the idealisation is at any other ratio. Once you have that line, the rule stops being arbitrary and becomes something you can reason with.
By the end of this lesson you should be able to
- Derive the load-share relation from compatible deflections
- Explain the fourth-power dependence
- Justify the 2:1 rule quantitatively
- Say why secondary reinforcement is required even in a one-way slab
What you should already know
- Deflection of a uniformly loaded beam (Structural Analysis Fundamentals, Module 13)
- Flexural design (Module 4)
From first principles
How a two-way slab divides its load
We want to show: To find what fraction of a uniformly distributed load a slab sends in each direction, using only compatibility and a standard deflection formula.
Imagine cutting two strips out of the slab: one running the short way, one the long way, crossing at the centre. The slab is one piece of concrete, so at the point where the strips cross they must deflect by the same amount. They cannot separate. Each strip carries whatever share of the load it needs to reach that common deflection. A short strip is stiff, so it takes a large share; a long strip is flexible, so it takes little. The whole result follows from making those two deflections equal.
Worked example
A two-way floor slab
Given
- Panel 5.0 m × 7.0 m, simply supported on all four edges
- 200 mm thick, C30/37, 25 mm cover, H12 bars
- Finishes 1.5 kN/m², imposed load 3.0 kN/m²
Find
The moments in each direction and the reinforcement.
Assumptions
- Simply supported, corner restraint neglected — conservative for the span moments
- Partial factors 1.35 and 1.5 — nationally determined, requiring verification
Practice
A slab panel is 4.0 m × 6.0 m. What fraction of a uniform load is carried by the SHORT span? Give the answer as a decimal fraction.
Practice
For a panel with a span ratio of exactly 2.0, what fraction goes the short way? Give the decimal fraction.
Check yourself
Why does load share depend on the FOURTH power of the span ratio rather than the square?
Summary
- Equal deflection at the crossing point gives ws Ls⁴ = wl Ll⁴
- So the share is r⁴/(1 + r⁴), and it moves very fast with r
- 1:1 → 50%, 1.5:1 → 83.5%, 2:1 → 94.1%
- The 2:1 rule is a consequence, not a convention
- The model neglects corner torsion, so it overestimates span moments — safe, not accurate
- Corner reinforcement exists precisely because of the torsion this model ignores
- Secondary steel is required for distribution and crack control, not for strength
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint