Fatigue loading of springs - application of Goodman analysis

If a compression spring is designed with the yield limit above the solidity limit and manufactured correctly then the only way it can fail is through fatigue. The leftmost photograph of a typical fatigue failure surface reveals a crack source near the inner more highly stressed surface of the wire. If stress raisers occur due to poor manufacture or to corrosion as illustrated in the other photographs, then fracture is likely to emanate from the stress raisers thereby reducing fatigue life.

In the absence of such stress raisers, correlation of the repeated loading on a spring with its material's properties in order to ascertain safety is usually carried out via a Goodman analysis. Given the spring material and wire diameter, the shear ultimate   S_us and fatigue strength in reversed shear   S_es may be found from the literature, or as explained above. The conservatively straight Goodman failure locus is thus defined in   τ_a - τ_m space, as shown on the diagram ( A)   { nomenclature explanation}.

Spring loading is almost invariably unidirectional, so   τ_a ≤ τ_m and loading states occur only to the right of the 45^o stress equality line. One such load state is illustrated, lying on a line which corresponds to a safety factor of   n, ie. the line joins the points   ( S_us/n, 0), (0, S_es/n) and so lies parallel to the failure locus. From similar triangles the Goodman fatigue equation is :

( 4a)         τ_m /S_us + τ_a /S_es =   1/n       in which the stress components are given by
τ_m =   K_s 8 F_mC / πd² where   F_m =   ( F_hi + F_lo ) /2     and
τ_a =   K_h 8 F_a C / πd² where   F_a =   ( F_hi - F_lo ) /2

The full stress concentration factor   K_h is applied to the alternating component   τ_a in ( 4a). Stress redistribution is presumed to follow localised yielding, with the result that stress concentration is usually ignored in figuring the steady component of fatigue loading. The mean stress   τ_m is therefore based on the factor   K_s, which accounts only for direct stresses in the torsion equation.

The safe operating window of the Goodman diagram is completed by the yield limit. If   n_y is the factor of safety against the yield being exceeded, then :

( 4b)         τ_m + τ_a =   S_ys/n_y since   ( τ_m + τ_a ) represents the maximum stress.

A fatigue analysis of the preceding example using ( 4a) is carried out in this example.

The traditional method of presenting allowable fatigue stresses is to plot the maximum shear stress   τhi against the minimum   τlo, rather than the amplitude versus the mean as above. Although strictly not correct, the full stress concentration factor   Kh is applied conservatively to both components which are inserted into the Goodman expression ( 4a) to give the failure locus:

( 4c)         τ_hi =   { 2 S_us S_es / ( S_us + S_es ) }   + { ( S_us - S_es ) / ( S_us + S_es ) } τ_lo where
τ_hi = K_h 8 C F_hi / πd² and     τ_lo = K_h 8 C F_lo / πd²
Thus for the wire of the foregoing example with stresses in MPa :

1. ( 4a)   Goodman failure locus           τ_m/888 + τ_a /183 = 1
2. ( 4b)   yield failure criterion             τ_hi = 677
3. ( 4c)   modified Goodman locus     τ_hi = 303 + 0.658 τlo

Though ( i) and (iii) appear identical, since   τhi = τm + τa and   τlo = τm - τa, the four stress components are defined with different stress factors in ( 4a) and ( 4c). The safe operating window from the last two relations is shown here. States below the equality line are impossible as this would mean reversal of the minimum / maximum roles. The material strengths of Table 2 were deduced via ( 4c) from plots such as this in Godfrey.

The fatigue strength in repeated (one-way) shear   S'_es forms the basis of a Goodman approach, diagram ( B) which is less conservative than that based on reversed shear, diagram ( A). If the test stress varies between zero and   S'_es, then   τ_a = τ_m = S'_es/2, and the failure line joins the points ( S_us, 0), ( S'_es/2, S'_es/2 ). The resulting design equation is :

( 4d)         τ_m /S_us + τ_a ( 2/S'_es - 1/S_us )   =   1/n

Since it is rare that the fatigue strength in repeated loading is available, we will henceforth employ ( 4a) and ( 4b) with the fatigue strength corresponding to whatever life is of interest.