$8.4 Torsion 181 or,in terms of some internal radius r, G0 -LI-GY (8.10) These equations indicate that the shear stress and shear strain vary linearly with radius and have their maximum value at the outside radius(Fig.8.4).The applied shear stresses in the plane of the cross-section are accompanied by complementary stresses of equal value on longitudinal planes as indicated in Figs.8.1 and 8.4.The significance of these longitudinal shears to material failure is discussed further in $8.10. Complementary longitudinal shears Fig.8.4.Complementary longitudinal shear stress in a shaft subjected to torsion. 8.4.Section modulus It is sometimes convenient to re-write part of the torsion theory formula to obtain the maximum shear stress in shafts as follows: Tt 万=R TR With R the outside radius of the shaft the above equation yields the greatest value possible for t (Fig.8.4), TR i.e. Tmax= tmu-Z (8.11) where Z =J/R is termed the polar section modulus.It will be seen from the preceding section that: πD3 for solid shafts, Z= (8.12) 16 and for hollow shafts, z=0*-d4 (8.13) 16D$8.4 Torsion 181 or, in terms of some internal radius r, (8.10) These equations indicate that the shear stress and shear strain vary linearly with radius and have their maximum value at the outside radius (Fig. 8.4). The applied shear stresses in the plane of the cross-section are accompanied by complementary stresses of equal value on longitudinal planes as indicated in Figs. 8.1 and 8.4. The significance of these longitudinal shears to material failure is discussed further in 88.10. Fig. 8.4. Complementary longitudinal shear stress in a shaft subjected to torsion. 8.4. Section modulus It is sometimes convenient to re-write part of the torsion theory formula to obtain the maximum shear stress in shafts as follows: TT JR -=- With R the outside radius of the shaft the above equation yields the greatest value possible for T (Fig. 8.4), i.e. TR 7-= - J T Z .. T-=- (8.11) where 2 = J/R is termed the polar section modulus. It will be seen from the preceding section that: nD3 16 for solid shafts, Z=- (8.12) and for hollow shafts, ~(D1-d~) 160 Z- (8.13)