s12.5 Springs 305 As for the close-coiled spring the total strain energy is given by strain energy U=GJ2EI T2L M2L =L(T'sin2+Tcosa)‖ GJ T2L「sin2x,cos2x 2GJ EI (12.10) and this is equal to the work done by T,namely,T0,where 0 is the angle turned through by one end relative to the other,i.e.the wind-up angle of the spring. 70=T2L and,with L =2Rn sec a as before, wind-up angle 0=2mnRT sec a sin2 a cos2a GJ+EI (12.11) (b)Maximum stress The maximum stress in the spring material will be found by the procedure outlined in 12.3(b)with a bending moment of T'cos a and a torque of T sin a applied to the section. (c）Axial defection Assuming an imaginary axial load W applied to the spring the total strain energy is given by eqn.(11.5)as U=(WRcos+Tsin a)L (Tcosa-WRsina)L 2GJ 2EI Now from Castigliano's theorem the deflection in the direction of W is given by 6= aU ow 「1 =TRLcos a sin GJEI when W=0 deflection 8=2anTR2 sin a 11 GJ EI (12.12) 12.5.Springs in series If two springs of different stiffness are joined end-on and carry a common load W,they are said to be connected in series and the combined stiffness and deflection are given by the following equations.412.5 Springs 305 As for the close-coiled spring the total strain energy is given by T~L M~L strain energy U = __ +- 2GJ 2EI (1 2.10) 1 (Tsinu)’ (TCOS~)~ El + and this is equal to the work done by T, namely, 4 TQ, where 6 is the angle turned through by one end relative to the other, i.e. the wind-up angle of the spring. iTQ =fT2L[7+EI] sin’a cos2 a and, with L = 2zRn sec a as before, sinZa cos’a wind-up angle 0 = 2anRTsec a (12.11) (b) Maximum stress The maximum stress in the spring material will be found by the procedure outlined in tj 12.3(b) with a bending moment of Tcos u and a torque of T sin a applied to the section. (c) Axial deflection Assuming an imaginary axial load W applied to the spring the total strain energy is given by eqn. (1 1.5) as ( WR cos a + Tsin u)’L 2GJ (Tcos u - WR sin a)’ L 2EI U= + Now from Castigliano’s theorem the deflection in the direction of W is given by au aw a=-- =TRLcosusina [jJ --- d,] when W=O deflection 6 = 2xnTR’sin a - - - [iJ A] 12.5. Springs in series (1 2.12) If two springs of different stiffness are joined end-on and carry a common load W, they are said to be connected in series and the combined stiffness and deflection are given by the following equations