## R

which can be reduced to a form similar to the standard bending eqn. (4.3)t M = EI/R i.e. M = ^ (3.21)

with Er known as the reduced modulus and given by:

The appropriate value of the reduced modulus Er for any particular curvature is best obtained from a curve of Er against ej. This is constructed rather laboriously by determining the relevant values of £| and ei for a set of assumed gy values using the condition of equal positive and negative areas for each ej value and then evaluating the integral of eqn. (3.22). Having found Er, the value of the bending moment for any given curvature R is found from eqn. (3.21).

It is sometimes useful to remember that, because strains are linear with distance from the neutral axis, the distribution of bending stresses across the beam section will take exactly the same form as that of the stress-strain diagram of Fig. 3.23(a) turned through 90° with sj replaced by the beam depth d. The position of the neutral axis indicated by eqn. (3.19) is then readily observed.

(b) Inelastic torsion

A similar treatment can be applied to the torsion of shafts constructed from materials which exhibit strain hardening characteristics. Figure 3.24 shows the shear stress-shear strain curve for such a material.

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Fig. 3.24. Shear stress-shear strain curve for torsion of materials exhibiting strain-hardening characteristics.

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Fig. 3.24. Shear stress-shear strain curve for torsion of materials exhibiting strain-hardening characteristics.

Once again it is necessary to assume that cross-sections of the shaft remain plane and that the radii remain straight under torsion. The shear strain at any radius r is then given by eqn. (8.9)+ as:

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