304 Mechanics of Materials §12.4 The combined twisting moment on the spring cross-section is then T=WR cosa+Tsin a and the combined bending moment M=Tcos a-WR sin a The total strain energy of the system is then T2L M2L U=2GJ2EI (WR cosa+Tsin )L (Tcosa-WR sin)L 2GJ 2EI Now from Castigliano's theorem the angle of twist in the direction of the axial torque T is aU given bynd sinceTall terms includingTmay be ord. 02WR cos asinaL(-2WR sin acosa)L 2GJ 2EI 「11 =WRLcosa sina 0=2πnWR2sina 117 i.e. GJEI」 (12.9) 12.4.Open-coiled helical spring subjected to axial torque T (a)Wind-up angle When an axial torque T is applied to an open-coiled helical spring it has components as shown in Fig.12.5,i.e.a torsional component T sin g about AX and a flexural (bending) component T cos a about AY,the latter tending to increase the curvature of the coils. ①Tcos a Tsin a Fig.12.5.Open-coiled helical spring subjected to axial torque T.304 Mechanics of Materials $12.4 The combined twisting moment on the spring cross-section is then - T= WRcosu+Tsinu and the combined bending moment M =Tcosu- WRsinu The total strain energy of the system is then - - T~L M~L u=-- +- 2GJ 2EI ( WR cos u + Tsin u)’L 2GJ (Tcos u - WR sin u)’ L 2EI + - Now from Castigliano’s theorem the angle of twist in the direction of the axial torque T is given by 0 = - and since T = 0 all terms including T may be ignored. au aT .. i.e. 2WRcosusinuL (-2WRsinucosa)L 2EI + 2GJ e= = WRLcosusinu --- [iJ ;I] 0 = 2xnWR’sina [A A] ( 12.9) 12.4. Open-coiled helical spring subjected to axial torque T (a) Wind-up angle When an axial torque Tis applied to an open-coiled helical spring it has components as shown in Fig. 12.5, i.e. a torsional component T sin u about AX and a flexural (bending) component T cos u about AY the latter tending to increase the curvature of the coils. Fig, 12.5. Opencoiled helical spring subjected to axial torque T