## Bnb

The integral part of the expression is the second moment of area of the shaded portion of Fig. 3.24 about the vertical axis. Thus, determination of this quantity for a given >'max value yields the corresponding value of the applied torque T.

As for the case of inelastic bending, the form of the shear stress-strain curve, Fig. 3.24, is identical to the shear stress distribution across the shaft section with the y axis replaced by radius r.

3.16. Residual stresses - strain-hardening materials

The procedure for determination of residual stresses arising after unloading from given stress states is identical to that described in §3.9 and §3.14.

For example, it has been shown previously that the stress distribution across a beam section in inelastic bending will be similar to that shown in Fig. 3.23(a) with the beam depth corresponding to the strain axis. Application of the elastic unloading stress distribution as described in §3.9 will then yield the residual stress distribution shown in Fig. 3.25. The same procedure should be adopted for residual stresses in torsion situations, reference being made to §3.14.

Beam cross-section Residual stresses

Fig. 3.25. Residual stresses produced in a beam constructed from a strain-hardening material.

Beam cross-section Residual stresses

Fig. 3.25. Residual stresses produced in a beam constructed from a strain-hardening material.

3.17. Influence of residual stresses on bending and torsional strengths

The influence of residual stresses on the future loading of members has been summarised by Juvinall^ into the following rule:

An overload causing yielding produces residual stresses which are favourable to future overloads in the same direction and unfavourable to future overloads in the opposite direction.

This suggests that the residual stresses represent a favourable stress distribution which has to be overcome by any further load system before any adverse stress can be introduced into the member of structure. This principle is taken advantage of by spring manufacturers, for example, who intentionally yield springs in the direction of anticipated service loads as part of the manufacturing process. A detailed discussion of residual stress can be found in the Handbook of Experimental Stress Analysis of Hetenyi.^"

t R. C. Juvinall, Engineering Considerations of Stress, Strain and Strength, McGraw-Hill, 1967 t M. Hetenyi, Handbook of Experimental Stress Analysis, John Wiley, 1966.

3.18. Plastic yielding in the eccentric loading of rectangular sections

When a column or beam is subjected to an axial load and a B.M., as in the application of eccentric loads, the elastic stress distribution is as shown in Fig. 3.25(a), the N.A. being displaced from the centroidal axis of the section. As the load increases the yield stress will be reached on one side of the section first as shown in Fig. 3.26(b) and, as in the case of the partially plastic bending of unsymmetrical sections in §3.4, the N.A. will move as plastic penetration proceeds. In the limiting case, when plasticity has spread across the complete section, the N.A. will be situated at a distance h from the centroidal axis (the axis through the centroid of the section) (Fig. 3.26(c)). The precise position of the N.A. is related to the excess of the total tensile force over the total compressive force, i.e. to the area shown shaded in Fig. 3.26(c). In simple bending, for example, there is no resultant force across the section and the shaded area reduces to zero. Thus, the magnitude of the axial load for full plasticity as given by the shaded area

Beam or column (a) Max. elastic (b) Partially (c) Fully plastic section stress distribution plastic

Fig. 3.26. Plastic yielding of eccentrically loaded rectangular section, where B is the width of the section,

Beam or column (a) Max. elastic (b) Partially (c) Fully plastic section stress distribution plastic

Fig. 3.26. Plastic yielding of eccentrically loaded rectangular section, where B is the width of the section, i.e. h = ^ (3.24)

2Bay

The fully plastic load is sometimes written in terms of a load factor N defined as fully plastic load Ppp load factor N

axial load P

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