## W

Other structures may require even more plastic hinges depending on their particular support conditions and degree of redundancy, but these need not be considered here. It should be evident, however, that there is now even more strength or load-carrying capacity available beyond that suggested by the shape factor, i.e. with a knowledge of the yield stress and hence the maximum elastic moment for any particular cross-section, the shape factor determines the increase in moment required to produce a fully plastic section or plastic hinge; depending on the support and loading conditions it may then be possible to increase the moment beyond this value until a sufficient number of plastic hinges have been formed to produce complete collapse. In order to describe the increased strength available using this "plastic limit" or "collapse load" procedure a load factor is introduced defined as collapse load load factor = —-u -— (3.9)

allowable working load

This is completely different from, and must not be confused with, the safety factor, which is a factor to be applied to the yield stress in simple elastic design procedures.

3.9. Residual stresses after yielding: elastic-perfectly plastic material

Reference to the results of simple tensile or proof tests detailed in §1.7+ shows that when materials are loaded beyond the yield point the resulting deformation does not disappear completely when load is removed and the material is subjected to permanent deformation or so-called permanent set (Fig. 3.15). In bending applications, therefore, when beams may be subjected to moments producing partial plasticity, i.e. part of the beam section remains elastic whilst the outer fibres yield, this permanent set associated with the yielded areas prevents those parts of the material which are elastically stressed from returning to their unstressed state when load is removed. Residual stress are therefore produced. In order to determine the magnitude of these residual stresses it is normally assumed that the unloading process, from either partially plastic or fully plastic states, is completely elastic (see Fig. 3.15). The

unloading stress distribution is therefore linear and it can be subtracted graphically from the stress distribution in the plastic or partially plastic state to obtain the residual stresses.

Consider, therefore, the rectangular beam shown in Fig. 3.16 which has been loaded to its fully plastic condition as represented by the stress distribution rectangles oabc and odef. The bending stresses which are then superimposed during the unloading process are given by the line goh and are opposite to sign. Subtracting the two distributions produces the shaded areas which then indicate the residual stresses which remain after unloading the plastically deformed beam. In order to quantify these areas, and the values of the residual stresses, it should be observed that the loading and unloading moments must be equal, i.e. the moment of the force due to the rectangular distribution oabc about the N.A. must equal the moment

Now, moment due to oabc

ind moment due to oag ind moment due to oag

Thus the residual stresses at the outside surfaces of the beam = |crv. The maximum residual stresses occur at the N.A. and are equal to the yield stress. The complete residual stress

In loading cases where only partial plastic bending has occurred in the beam prior to unloading the stress distributions obtained, using a similar procedure to that outlined above, are shown in Fig. 3.17. Again, the unloading process is assumed elastic and the line goh in

this case is positioned such that the moments of the loading and unloading stress distributions are once more equal, i.e. the stress at the outside fibre ag is determined by considering the plastic moment Mpp applied to the beam assuming it to be elastic; thus ag-a

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