Compression spring shear stress and stiffness

Stresses and stiffness

The free body ( a) of the lower end of a spring whose mean diameter is   D :

• embraces the known upward load   F applied externally and axially to the end coil of the spring, and
• cuts the wire transversely at a location which is remote from the irregularities associated with the end coil and where the stress resultant consists of an equilibrating force   F and an equilibrating rotational moment   FD/2.

The wire axis is inclined at the helix angle   α at the free body boundary in the side view ( b)   (Note that this is first angle projection). An enlarged view of the wire cut conceptually at this boundary ( c) shows the force and moment triangles from which it is evident that the stress resultant on this cross-section comprises four components - a shear force   (F cosα), a compressive force   (F sinα), a torque   (1/2FD cosα) and a bending moment   (1/2FD sinα).

Assuming the helix inclination   α to be small for close- coiled springs approaching solidity ( when working loads are critical ) then   sinα ≈ 0, cosα ≈ 1, and the significant loading reduces to   torsion plus direct shear. The maximum shear stress at the inside of the coil will be the sum of these two component shears : 
             τ   =   τ_torsion + τ_direct   =   Tr/J + F/A 
                  =   (FD/2) (d/2)/(πd⁴/32) + F/(πd²/4) = (1 + 0.5d/D) 8 FD/πd³

( 1)     τ   =   K 8FC /πd² 
                         in which the   stress factor, K assumes one of three values, either . . .

• K = 1     when torsional stresses only are significant - ie. the spring behaves essentially as a torsion bar, or
• K = K_s   ≡   1 + 0.5/C     which accounts approximately for the relatively small direct shear component noted above, and is used in static applications where the effects of stress concentration can be neglected, or
• K = K_h   ≈   ( C + 0.6)/( C - 0.67)     which accounts for direct shear and also the effect of curvature- induced stress concentration on the inside of the coil (similar to that in curved beams). K_h should be used in fatigue applications; it is an approximation for the   Henrici factor which follows from a more complex elastic analysis as reported in   Wahl op cit. It is often approximated by the Wahl factor   K_w = ( 4C - 1)/( 4C - 4) + 0.615/C.

The factors increase with decreasing index as shown here :-

The deflection   δ of the load   F follows from Castigliano's theorem. Neglecting small direct shear effects in the presence of torsion : 
             δ   =   ∂U/∂F   =   ∂/∂F [ ∫_length (T²/2GJ) ds ]       where   T = FD/2 
                   =   ∫_length (T/GJ) (∂T/∂F) ds   =   (T/GJ) (D/2)*(wire length) 
                   =   (FD/2GJ) (D/2) n_aπD       which leads to

( 2)     k   =   F/δ   =   Gd / 8n_aC³       in which   n_a is the number of active coils (Table 1).

Despite many simplifying assumptions, equation ( 2) tallies well with experiment provided that the correct value of rigidity modulus is incorporated, eg.   G = 79 GPa for cold drawn carbon steel.

Standard tolerance on wire diameters less than   0.8mm is 0.01mm, so the error of theoretical predictions for springs with small wires can be large due to the high exponents which appear in the equations. It must be appreciated also that flexible components such as springs cannot be manufactured to the tight tolerances normally associated with rigid components. The spring designer must allow for these peculiarities. Variations in length and number of active turns can be expected, so critical springs are often specified with a tolerance on stiffness rather than on coil diameter. The reader is referred to BS 1726 or AE-11 for practical advice on tolerances.