Question 14.2: Creating the Modified-Goodman Diagram for a Helical Spring P...

See Figure 14-16.

1    The material’s ultimate tensile strength from Figure 14-3 or equation 14.3 (p. 792), converted to ultimate torsional strength with equation 14.4 (p. 793) using data from Table 14-4 (p. 792) allows one point on the Goodman line to be determined.

S_{u t} \cong A d^b     (14.3)

S_{u s} \cong 0.67 S_{u t}       (14.4)

S_{u t} \cong 184649(0.042)^{-0.1625}=309071  psi       (a)

This value is plotted as point A on the diagram in Figure 14-16.

2    The S-N diagrams each provide one data point (S_{fw} or S_{ew}, depending on whether for finite or infinite life) on the modified-Goodman line for a material/size combination in pure torsional loading. The fatigue strength S_{fw} for that wire material and condition is taken from the S-N line of Figure 14-15 or calculated from the data in Table 14-9 (p. 813) as

@  N = 1E6  :  S_{f w} \cong 0.33(309071)=101993 psi      (c)

The x and y intercepts are 0.5S_{fw} = 50 996 psi. This is plotted as point B on the diagram in Figure 14-16. Note that for infinite life the value of S_{ew} from equation 14.13 would be plotted at B instead of this value of S_{fw} for a finite life.

3    Note in Figure 14-16 that the wire fatigue strength S_{fw} is plotted at point B (\tau _{a} = \tau _{m} = 0.5 S_{fw}) corresponding to the test conditions of equal mean and alternating stress components (stress ratio R = \tau _{min}/\tau _{max} = 0). Point B is then connected with the ultimate shear strength S_{us} on the mean-stress axis at point A to draw the Goodman line, which is extended to point C.

4    We can now find the value of the fully reversed fatigue strength (R = –1), which is point C on the diagram. This value can be found from the equation for the Goodman line, defined in terms of its two known points, A and B:

5    This use of the Goodman line is conservative for stress ratios R ≥ 0, and its use is justified in this case because springs should always be loaded in the same direction. Helical compression springs tend to have stress ratios between 0 and 0.8, which puts their stress coordinates to the right of the 45° line in the figure, where the Goodman line is more conservative than the Gerber line.

6    Any other combination of mean and alternating stress with a stress ratio R ≥ 0 for this material and number of cycles can now be plotted on this diagram to obtain a safety factor.