300 Mechanics of Materials §12.2 R W Fig.12.1.Close-coiled helical spring subjected to axial load W. are found to be negligible compared with the torsional stresses.Thus the maximum stress in the spring material may be determined to a good approximation using the torsion theory. Tr WRr tma=万-元r产2 2WR 8WD i.e. maximum stress 元r3 nd3 (12.1) (b)Deflection Again,for one half-turn,if one cross-section twists through an angle 6 relative to the other, then from the torsion theory 0= TL_WR(πR)22WR2 G -X- G Gr4 2WR3 But 8=R0=1 Gr total deflection =2n= 4WR3n 8WD3n Gr4 Gd (12.2) W Gd Spring rate=57=8nD3 12.2.Close-coiled helical spring subjected to axial torque T (a)Maximum stress In this case the material of the spring is subjected to pure bending which tends to reduce the radius R of the coils(Fig.12.2).The bending moment is constant throughout the spring and equal to the applied axial torque T.The maximum stress may thus be determined from the bending theory My Tr O max= 1πr4/4300 Mechanics of Materials Q 12.2 i w Fig. 12.1. Closecoiled helical spring subjected to axial load W. are found to be negligible compared with the torsional stresses. Thus the maximum stress in the spring material may be determined to a good approximation using the torsion theory. Tr WRr Tmax= - = __ J xr412 i.e. 2WR 8WD maximum stress = - = - zr3 nd3 (12.1) (b) Dejection Again, for one half-turn, if one cross-section twists through an angle 8 relative to the other, then from the torsion theory TL WR(nR) 2 2WR2 e=-= x-=- GJ G xr4 Gr4 But .. 4WR'n - 8WD3n total deflection 6 = 2nd' = ~ - ~ Gr4 Gd4 (1 2.2) W Gd4 Spring rate = - = - 6' 0n~3 12.2. Close-coiled helical spring subjected to axial torque T (a) Maximum stress In this case the material of the spring is subjected to pure bending which tends to reduce the radius R of the coils (Fig. 12.2). The bending moment is constant throughout the spring and equal to the applied axial torque T. The maximum stress may thus be determined from the bending theory