180 Mechanics of Materials $8.3 For a solid shaft, -[ 2πR4 元D4 or 4 32 (8.6) For a hollow shaft of internal radius r, =R-ror五0-的 (8.7) For thin-walled hollow shafts the values of D and d may be nearly equal,and in such cases there can be considerable errors in using the above equation involving the difference of two large quantities of similar value.It is therefore convenient to obtain an alternative form of expression for the polar moment of area. Now 2πr3dr=Σ(2πrdr)r2 =2Ar2 where A(=2xr dr)is the area of each small element of Fig.8.3,i.e.J is the sum of the Ar2 terms for all elements. If a thin hollow cylinder is therefore considered as just one of these small elements with its wall thickness t=dr,then J=Ar2 =(2nrt)r2 =2r3t(approximately) (8.8) 8.3.Shear stress and shear strain in shafts The shear stresses which are developed in a shaft subjected to pure torsion are indicated in Fig.8.1,their values being given by the simple torsion theory as Ge -TR Now from the definition of the shear or rigidity modulus G, t=Gy It therefore follows that the two equations may be combined to relate the shear stress and strain in the shaft to the angle of twist per unit length,thus G0 t =Gy (8.9)180 For a solid shafi, Mechanics of Materials nD* or - 4 32 2n~4 =- For a hollow shaft of internal radius r, J=2n r3dr=2n - i [:I: x x = -(R4-r4) or -(D4-d*) 2 32 $8.3 For thin-walled hollow shafis the values of D and d may be nearly equal, and in such cases there can be considerable errors in using the above equation involving the difference of two large quantities of similar value. It is therefore convenient to obtain an alternative form of expression for the polar moment of area. Now J = 2nr3dr = C(2nrdr)r’ = AY’ 0 i where A ( = 2nr dr) is the area of each small element of Fig. 8.3, i.e. J is the sum of the Ar2 terms for all elements. If a thin hollow cylinder is therefore considered as just one of these small elements with its wall thickness t = dr, then J = Ar’ = (2nrt)r’ = 2xr3t (approximately) (8.8) 8.3. Shear stress and shear strain in shafts The shear stresses which are developed in a shaft subjected to pure torsion are indicated in Fig. 8.1, their values being given by the simple torsion theory as GO L 7=-R Now from the definition of the shear or rigidity modulus G, r = Gy It therefore follows that the two equations may be combined to relate the shear stress and strain in the shaft to the angle of twist per unit length, thus