**Introduction**

A slab is a structural member whose depth is considerably smaller than the lateral dimensions. They provide useful surfaces for supporting loads, and may be supported on beams, columns, walls, or masonry. When a slab is supported on two opposite sides, they are referred to as

*one-way*slabs because the loads are being carried by the slab in the direction perpendicular to the beams.

But if a slab is supported on four sides, and the ratio of length to width is greater than 2, most of the load is carried in the short direction to the supporting beams and one-way action is obtained in effect, even though supports are provided on all sides. The structural action of one way slabs may be visualized in terms of the deflected surface. This can be approximated as a cylindrical surface, and curvatures and bending moments are parallel to the short side

*L*x. The slab is normally analysed a beam of unit strip, with the bending moment, shear forces, and reinforcement determined per unit strip.

In many design circumstances, rectangular slabs have dimensions where the ratio of the longer side to the shorter side is greater than 2 (and are also supported in such a way that two-way action results). When loaded, such slabs bend into a dished surface rather than a cylindrical one. This means that at any point the slab is curved in both principal directions, and since bending moments are proportional to curvatures, moments also exist in both directions.

Typical Two Way Action in A Slab |

To resist these moments, the slab must be reinforced in both directions, by at least two layers of bars perpendicular, respectively, to two pairs of edges. The slab must be designed to take a proportionate share of the load in each direction.

**Why Short Span?**

When a civil engineering student enters a structural design classroom for the first time, he is told that the shorter span is more critical in the design of slabs. This is usually source of wonder for the first timer because by mere instincts, the longer span should be more critical. These are very simple proofs to show why the short span is critical.

**(1) Deflection of centre-strip approach**

This offers an extremely simplified concept that shows that the load transmitted to the shorter span is greater than the load transmitted to the longer span.

Let us consider the two way action of the slab shown below;

Let us consider the centre-strips highlighted in red. A little consideration will show that at the intersection point of the strips, the deflection is equal. Logical right?

Let the uniform load on the longer strip be

*q*_{y}
Let the uniform load on the shorter strip be

*q*_{x}
We can therefore say that;

Deflection at centre

*w*= 5*q*_{y}*L*_{y}^{4}/384EI = 5*q*_{x}*L*_{x}^{4}/384EI
5

*q*_{y}*L*_{y}^{4}/384EI = 5*q*_{x}*L*_{x}^{4}/384EI*q*

_{y}

*L*

_{y}

^{4 }=

*q*

_{x}

*L*

_{x}

^{4}

We can verify from the above relation that;

*q*

_{x}

*/*

*q*

_{y}

^{ }=

*L*

_{y}

^{4 }/

*L*

_{x}

^{4}

Since L

_{y}> L_{x}, we can accept that the load on the short span (q_{x}) is greater than the load on the long span (q_{y}) since L_{y}/L_{x}> 1.0.
This is an oversimplification of the behaviour of slabs though.

**(2) Yield Line Approach**

The yield line method was developed to determine the limit state of slabs by considering the yield lines that occur at the slab collapse mechanism. The yield lines are usually approximated to originate at the corners, forming at an angle of 45° until the intersect. These yield lines usually mean trapezoidal loads going to the longer supporting beam of the slab, and triangular loads going to the shorter supporting beam.

L

_{y}= 6m
L

_{x}= 5m
Area of trapezium = 8.6825 m

^{2}
Area of triangle = 6.316 m

^{2}Therefore;

The force parallel to the short span (Px) = 2 × 8.6825m

^{2}× 1 kN/m

^{2}= 17.365 kN

The force parallel to the long span (Py) = 2 × 6.316m

^{2}× 1 kN/m

^{2}= 12.632 kN

Total load on slab = (5 × 6)m

^{2}× 1 kN/m

^{2}= 30 kN

Therefore, you can see why the shorter span is more heavily loaded based on the yield line pattern.

**(3) Finite Element Analysis**

When we carry out finite element analysis on slabs, the result offers an insight;

Let us check out the slab investigated above using finite element analysis from Staad Pro.

Let us assume that the slab is subjected to a unit pressure load (1 kN/m

^{2})
L

_{y}= 6m
L

_{x}= 5m
Thickness of slab = 150mm

*v*= 0.2

E = 21.7 kN/mm

^{2}
From the result, the bending moment parallel to the short span is given below;

In the longer direction;

If you look at the results above, the moments parallel to the the short span is more critical than the moment parallel to the longer side.

**Conclusion**

The load transferred to through the short span of the slab is heavier than the load transferred through the long span of the slab. For one way slab, the slab is analysed as a beam of unit width , and the main reinforcement is provided parallel to the short span, while distribution bars are provided parallel to the long span. Minimum steel can be used as distribution bars. But for two way slabs, the slab is designed for strength in the two principal directions, but the main bars are placed at the bottom, parallel to the short span, while the other is placed near to the bottom (on top of the bottom bar) parallel to the long span.

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