For cases which are more complex as a result of shape, support conditions, the presence i openings, or loading conditions it may be worthwhile adopting an ultimate analysis ethod. The two principal approaches are the yield line method, which is particularly litable for slabs with a complex shape or concentrated loading, and the strip method hich is valuable where the slab contains openings.
These methods have been the subject of research, and are well documented although hey are of a relatively specialised nature. A brief introduction is included here to lustrate the general principles and features of the methods, which are particularly • aluable in assisting an understanding of failure mechanisms. In practical design tuations care must be taken to allow for the effects of tie-down forces at corners and •rsion at free edges of slabs.
8.9.1 Yield line method rhe capacity of reinforced concrete to sustain plastic deformation has been described in -•ction 3.6. For an under-reinforced section the capacity to develop curvatures between -.e first yield of reinforcement and failure due to crushing of concrete is considerable, f- t a slab which is subjected to increasing load, cracking and reinforcement yield will rst occur in the most highly stressed zone. This will then act as a plastic hinge as ibsequent loads are distributed to other regions of the slab. Cracks will develop to form . pattern of 'yield lines' until a mechanism is formed and collapse is indicated by -creasing deflections under constant load. To ensure that adequate plastic deformation ..in take place the Code specifies that slabs designed by the yield line method must be reinforced with Class B or C (medium or high) ductility steel and the ratio x/d should >t exceed 0.25 for concrete up to Class C50/60.
For continuous slabs, the intermediate support moment should also lie between half and twice the magnitude of the span moments.
It is assumed that a pattern of yield lines can be superimposed on the slab, which will cause a collapse mechanism, and that the regions between yield lines remain rigid and •lcracked. Figure 8.21 shows the yield line mechanism which will occur for the simple . .ise of a fixed ended slab spanning in one direction with a uniform load. Rotation along :he yield lines will occur at a constant moment equal to the ultimate moment of
Development of yield lines
Plastic hinges resistance of the section, and will absorb energy. This can be equated to the energy expended by the applied load undergoing a compatible displacement and is known as the virtual work method.
Considerable care must be taken over the selection of likely yield line patterns, since the method will give an 'upper bound' solution, that is, either a correct or unsafe solution. Yield lines will form at right angles to bending moments which have reached the ultimate moment of resistance of the slab, and the following rules may be helpful:
1. Yield lines are usually straight and end at a slab boundary.
2. Yield lines will lie along axes of rotation, or pass through their points of intersection.
3. Axes of rotation lie along supported edges, pass over columns or cut unsupported edges.
In simple cases the alternative patterns to be considered will be readily determined on the basis of common sense, while for more complex cases differential calculus may be used. The danger of missing the critical layout of yield lines, and thus obtaining an incorrect solution, means that the method can only be used with confidence by experienced designers.
A number of typical patterns are shown in figure 8.22.
A yield line caused by a sagging moment is generally referred to as a 'positive' yield line and is represented by a full line, while a hogging moment causing cracking on the top surface of the slab causes a 'negative' yield line shown by a broken line.
The basic approach of the method is illustrated for the simple case of a one-way spanning slab in example 8.11
f EXAMPLE 8.11 Simply supported, one-way spanning rectangular slab
The slab shown in figure 8.23 is subjected to a uniformly distributed load u- per unit area. Longitudinal reinforcement is provided as indicated giving a uniform ultimate moment of resistance m per unit width.
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