Plastic behaviour

For structures to collapse ‘gradually’ they have to be behaving plastically on some part of the load path and this plastic behaviour must cause the structure to become a mechanism.

To see how this happens look again at the beam shown in Fig. 2.26.

Safe structures and failure 179

Fig. 6.33

In Chapter 3, the linear elastic stress distribution (Fig. 3.39) was given. It was noted that there was a point of maximum stress at the top and bottom faces at the point of maximum bending moment (Fig. 3.50). As the load on the beam is increased, the stress at these points will eventually reach point B of Fig. 5.6. That is the maximum elastic stress; this is often called the elastic limit.

Fig. 6.34

The stress f_p (force per unit area) is the stress at which the structural material starts to act plastically. As the load is further increased, the point becomes a zone of plastic stress. This zone occurs as parts of the beam adjacent to the original point of maximum stress reach the elastic limit and become plastic.

Fig. 6.35

Because the stress cannot exceed the elastic limit, the stress distribution in the plastic zone changes from that shown in Fig. 3.49.

180 Building Structures: From Concepts to Design

Fig. 6.36

As the load is increased further, the depth of the plastic zone increases until the beam achieves full plasticisation.

Fig. 6.37

When full plasticisation is reached the beam cannot be stressed further and a plastic hinge has formed. The beam now collapses ‘gradually’ as it becomes a mechanism rotating about the plastic hinge.

Fig. 6.38

The bending moment at the formation of the plastic hinge is called the plastic moment.

The ratio between the elastic moment, M_e, the moment at the elastic limit, and the plastic moment, M_p, varies with the cross-sectional shape. For a rectangular cross-section the ratio is 1.5. The behaviour of the beam through the loading range can be illustrated by drawing a graph of the bending moment plotted against the central deflection.

Safe structures and failure 181

Fig. 6.39

What happens to the structure is that a local failure of an element in the load path causes the structure to become a plastic mechanism. The prediction of the plastic mechanism forms the basis of the collapse design method. For the simple beam the elastic behaviour directly predicts the plastic mechanism.

Fig. 6.40

But for slightly more complicated structures, such as a two-span beam, the formation of one plastic hinge will not cause the structure to become a plastic mechanism.

Fig. 6.41

Here the first plastic hinge forms at the central support but the structure is not yet a mechanism.

182 Building Structures: From Concepts to Design

Fig. 6.42

The load on the structure can be increased until one of the span moments reaches the plastic moment. A second plastic hinge now forms and the structure becomes a mechanism and collapses.

Fig. 6.43

For a pitched portal frame loaded both horizontally and vertically there are three different possi-ble collapse mechanisms. Which one will form depends on the rates of loading for each load.

Fig. 6.44

This idea of plastic hinges can be used for laterally loaded two-dimensional structures to predict collapse mechanisms. The plastic moment, instead of being at a point forming a plastic hinge, is along a line. This ‘line of plastic moment’ is usually called a yield line. For a rectangular slab spanning between opposite supports the yield line position is similar to that of the hinge in the beam shown in Fig. 6.38.

Safe structures and failure 183

Fig. 6.45

For this simple case, the position of maximum bending moment is in a straight line across the slab which gives the position of the yield line. A plan of the slab showing the yield line (or lines) is called the yield line pattern. For the slab shown in Fig. 6.45 the yield line pattern has just one line.

Fig. 6.46

A’free edge’ of a laterally loaded slab is one that is unsupported. If the rectangular slab is supported on all sides then it will span two ways (Fig. 2.44). This will cause bending moments in two directions (Fig. 2.47).

Fig. 6.47

But what is the yield line pattern? Whilst the slab is acting elastically the maximum bend-ing moment will be at the centre, but as the load is increased the slab will become plastic at this point, the moment cannot be increased (see Fig. 6.39) and a yield line begins to form.

184 Building Structures: From Concepts to Design

Fig. 6.48

But how will the yield lines ‘grow’ into a yield line pattern that allows the slab to collapse?

For the one-dimensional structure shown in Figs. 6.43 and 6.44, the hinges allowed the structure to ‘fold’ into a collapse mechanism. Similarly the slab must be able to fold to be able to collapse. As the supported sides must remain level, the fold lines (yield lines) must go to the corners.

Fig. 6.49

The fact that the yield lines go diagonally across the slab is because it is necessary for the fold pattern, but these are also lines of principal moments. The idea of principal moments is not described here, but it is similar to the idea of principal stresses (Section 4.1). Moments are applied to the ‘sides of a small element’ and this is rotated in plan to find the maximum and minimum moments on each side.

Fig. 6.50

For a square slab with a uniform load, the yield line pattern may be ‘obvious’ but a rectan-gular slab can be folded in several different ways.

Safe structures and failure 185

Fig. 6.51

These three different foldings of the slab give three different yield line patterns and three different collapse loads.

Fig. 6.52 Yield line patterns

Because the mathematical prediction of the elastic behaviour of slabs (usually called

‘plates’ in the technical literature) is difficult or often impossible, the mathematical elastic analysis of slab structures is not usually carried out as part of structural design. In contrast, yield line analysis, developed for reinforced concrete by the Danish engineer FW Johans-sen, is relatively simple to carry out. Of course the correct yield line pattern must be chosen to make sure that the lowest collapse load is calculated.

The collapse mechanisms for these one- and two-dimensional structures rely on the forma-tion of plastic hinges (yield lines) at posiforma-tions of maximum bending moments. The formaforma-tion of these hinges allows geometrically simple foldings of the structure into a collapse configuration.

This means that these simple types of structural collapse are only possible if internal axial forces are absent or negligible and the geometry of the structure allows a simple folding.

For instance the ideas of plastic hinges and folding do not give any guidance on how a simple column collapses. Again the ideas of yield lines give no guidance on how the curved shell shown in Fig. 4.25 collapses. To see how these structures collapse, the effect of axial forces must be examined.