Yield Line Theory of Slabs

1. Idea

At collapse an under-reinforced slab cracks along lines where the steel has yielded — yield lines. They divide the slab into rigid segments that rotate about their supports, forming a mechanism. Equating the work done by the loads to the energy dissipated in the yield lines gives the collapse load.

2. Assumptions

• The slab is under-reinforced and ductile, so yield lines can rotate freely.
• Segments between yield lines are rigid; all rotation happens at the yield lines.
• The moment along a yield line is constant and equal to the ultimate moment $m$ per unit length.
• Elastic deformations are neglected.

3. Rules for drawing yield-line patterns

1. Yield lines are straight lines (in a plane slab).
2. Yield lines end at a slab boundary or at another yield line.
3. A yield line between two segments passes through the intersection of their axes of rotation (or is parallel to them if the axes are parallel).
4. Axes of rotation lie along supported edges and pass over columns.
5. Negative (hogging) yield lines form along continuous or fixed edges; positive (sagging) yield lines form in the span.

4. The work (virtual work) method

Give the mechanism a small virtual deflection $\delta$ (usually 1 at the point of greatest deflection).

Because any assumed mechanism gives a load equal to or greater than the true collapse load, yield-line analysis is an upper-bound method — the critical pattern is the one giving the least load. (A lower-bound method, by contrast, is the strip method of Hillerborg.)

5. Standard results (isotropic slab, moment $m$ per unit length, UDL $w$)

Rectangular simply supported slab

For an isotropic slab $L_x\times L_y$ ($L_y\ge L_x$) the pattern has diagonal yield lines from the corners meeting a central line parallel to the long side; the result depends on the aspect ratio and reduces to $w{L}^2/24$ for a square.