CHAPTER 8 TORSION Summary For a solid or hollow shaft of uniform circular cross-section throughout its length,the theory of pure torsion states that Tt G0 万=R=D where T is the applied external torque,constant over length L; J is the polar second moment of area of shaft cross-section .πD4 32 for a solid shaft and (dfor a hollow shat; 32 D is the outside diameter;R is the outside radius; d is the inside diameter; t is the shear stress at radius R and is the maximum value for both solid and hollow shafts; G is the modulus of rigidity(shear modulus);and 6 is the angle of twist in radians on a length L. For very thin-walled hollow shafts J=2nr3t,where r is the mean radius of the shaft wall and t is the thickness. Shear stress and shear strain are related to the angle of twist thus: t=- -R=Gy Strain energy in torsion is given by T2L GJ82 U= 2G= 2L 4Gx volume for solid shafis For a circular shaft subjected to combined bending and torsion the equivalent bending moment is M.=[M+√/(M2+T)] and the equivalent torque is T。=√/(M2+T2) where M and T are the applied bending moment and torque respectively. The power transmitted by a shaft carrying torque T at o rad/s Tw. 176CHAPTER 8 TORSION Sommary For a solid or hollow shft of uniform circular cross-section throughout its length, the theory of pure torsion states that T T GO J R=L -=- where Tis the applied external torque, constant over length L; J is the polar second moment of area of shaft cross-section x(D4 - d 4, for a hollow shaft; xD4 32 32 = - for a solid shaft and D is the outside diameter; R is the outside radius; d is the inside diameter; T is the shear stress at radius R and is the maximum value for both solid and hollow shafts; G is the modulus of rigidity (shear modulus); and 8 is the angle of twist in radians on a length L. For very thin-walled hollow shafts J = 2nr3t, where T is the mean radius of the shaft wall and t is the thickness. Shear stress and shear strain are related to the angle of twist thus: GB L T=-R=G~ Strain energy in torsion is given by U=- x volume for solid shafis 2GJ 2L For a circular shaft subjected to combined bending and torsion the equivalent bending moment is Me = i[M + J(Mz +T ')I and the equivalent torque is where M and T are the applied bending moment and torque respectively. The pa~er transmitted by a shaft carrying torque Tat o rad/s = To. T, = +J( M +T 2, 176