Standard solutions to plastic problems 143

7.2 Standard solutions to plastic problems 143

y y Tension

(a) Elastic

Plastic zone

y Tension

(b) Onset of plasticity

Compression

Plastic zone

y Tension

(c) Full plasticity Compression

Figure 7.1

A beam loaded in bending. The stress state is shown on the right for purely elastic loading (a), the onset of plasticity (b), and full plasticity (c).

where y is the distance from the neutral axis, and the influence of the cross- section shape is captured by I, the second moment of area. For elastic deflec- tion, we were interested in the last term in the equation—that containing the curvature κ. For yielding, it is the first term. The maximum longitudinal stress σ max occurs at the surface (Figure 7.1(a)), at the greatest distance y m from the neutral axis

The quantity Z e ⫽ I/y m is called the elastic section modulus (not to be confused with the elastic modulus of the material, E). If σ max exceeds the yield strength σ y of the material of the beam, small zones of plasticity appear at the surface where the stress is highest, as in Figure 7.1(b). The beam is no longer elastic and, in this sense, is damaged even if it has not failed completely. If the moment is increased further, the linear profile is truncated—the stress near the surface

144 Chapter 7 Bend and crush: strength-limited design

FF

Figure 7.2 The plastic bending of beams.

remains equal to σ y and plastic zones grow inwards from the surface. Although the plastic zone has yielded, it still carries load. As the moment increases fur- ther the plastic zones grow until they penetrate through the section of the beam, linking to form a plastic hinge (Figure 7.1(c)). This is the maximum moment that can be carried by the beam; further increase causes it to collapse by rotating about the plastic hinge.

Figure 7.2 shows simply supported beams loaded in bending. In the first, the maximum moment M is FL, in the second it is FL/4 and in the third FL/8. Plastic hinges form at the positions indicated in red when the maximum

moment reaches the moment for collapse. This failure moment, M f , is found by integrating the moment caused by the constant stress distribution over the sec- tion (as in Figure 7.1 (c), compression one side, tension the other)

M f ⫽ ∫ byy () σ section y d y ⫽ Z py σ

where Z p is the plastic section modulus. So two new functions of section shape have been defined for failure of beams: one for first yielding, Z e , and one for full plasticity, Z p . In both cases the moment required is simply Zσ y . Values for both are listed in Figure 7.3. The ratio Z p /Z e is always greater than 1 and is a measure of the safety margin between initial yield and collapse. For a solid rec- tangle, it is 1.5, meaning that the collapse load is 50% higher than the load for initial yield. For efficient shapes, like tubes and I-beams, the ratio is much closer to 1 because yield spreads quickly from the surface to the neutral axis.