§11.4 Strain Energy 261 If the bending moment is constant this reduces to M2L U= 2EI 11.4.Strain energy-torsion The element is now considered subjected to a torque T as shown in Fig.11.5,producing an angle of twist de radians. ds de., Fig.11.5.Torsion. Strain energy work done =iTde But,from the simple torsion theory, T Gde Tds Jds and do= GJ .'total strain energy resulting from torsion, CT2ds T2L U= (11.8) 2GJ 2GJ since in most practical applications T is constant. For a hollow circular shaft eqn.(11.8)still applies T2L i.e. Strain energy U=2GJ Now,from the simple bending theory Tt Tmax 了，=R where R is the outer radius of the shaft and J= 5(R4-r4) π T-2Rtma (R)91 1.4 Strain Energy 26 1 If the bending moment is constant this reduces to 11.4. Strain energy - torsion The element is now considered subjected to a torque T as shown in Fig. 11.5, producing an angle of twist dO radians. Fig. 11.5. Torsion. Strain energy = work done = 3TdO But, from the simple torsion theory, Tds and dO =- T GdO J ds GJ - .'. total strain energy resulting from torsion, T2ds T2L 2GJ 2GJ 0 since in most practical applications T is constant. For a hollow circular shaji eqn. (1 1.8) still applies T~L Strain energy U = - 2GJ i.e. Now, from the simple bending theory T 7 Tmax - - -- -- JrR where R is the outer radius of the shaft and 7t J=-(R4-r4) 2 (11.8)