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During the unloading process a moment of equal value but opposite sense is applied to the beam assuming it to be completely elastic. Thus the equivalent maximum elastic stress a' introduced at the outside surfaces of the beam by virtue of the unloading is given by the simple bending theory with M = Mpp = 10.6 kNm,

, _My _ 10.6 x 103 x 40 x 10~3 x 12 i e' ° ~~T ~ 30 X 803 X 10-'2

The unloading, elastic stress distribution is then linear from zero at the N.A. to ±330 MN/m2 at the outside surfaces, and this may be subtracted from the partially plastic loading stress distribution to yield the residual stresses as shown in Fig. 3.40.

Beam cross- Loading and unloading Residual stresses section stress distributions

Beam cross- Loading and unloading Residual stresses section stress distributions

(c) The residual stress distribution of Fig. 3.40 indicates that the central portion of the beam, which remains elastic throughout the initial loading process, is subjected to a residual stress system when the beam is unloaded from the partially plastic state. The beam will therefore be in a deformed state. In order to remove this deformation an external moment must be applied of sufficient magnitude to return the elastic core to its unstressed state. The required moment must therefore introduce an elastic stress distribution producing stresses of ±75 MN/m2 at distances of 20 mm from the N.A. Thus, applying the bending theory, al 75 x 106 30 x 803 x 10"12

Alternatively, since a moment of 10.6 kNm produces a stress of 165 MN/m2 at 20 mm from the N.A., then, by proportion, the required moment is

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