Constructing the S-N Diagram for a Spring-Wire Material Problem Create the torsional-shear S-N diagrams for a range of spring-wire sizes. Given ASTM A228 music wire, unpeened. Assumptions Three diameters will be used: 0.010 in (0.25 mm), 0.042 in (1.1 mm), and 0.250 in (6.5 mm).

Question

Constructing the S-N Diagram for a Spring-Wire Material

Problem Create the torsional-shear S-N diagrams for a range of spring-wire sizes.

Given ASTM A228 music wire, unpeened.

Assumptions Three diameters will be used: 0.010 in (0.25 mm), 0.042 in (1.1 mm), and 0.250 in (6.5 mm).

Accepted Answer

See Figure 14-15.

1 The tensile strength of each wire size is found from equation 14.3 in combination with the coefficient and exponent from Table 14-4 for this material.

[latex] S_{u t} \cong A d^b [/latex] (14.3)

[latex] \begin{aligned} S_{u t} & \cong 184649 d^{-0.1625} \ &=184649(0.010)^{-0.1625}=390239 psi \ &=184649(0.042)^{-0.1625}=309071 psi \ &=184649(0.250)^{-0.1625}=231301 psi \end{aligned} [/latex] (a)

2 These values are converted to shear strengths at 1 000 cycles using equation 14.14:

[latex] S_{m s} \cong 0.9 S_{u s} \cong 0.9\left(0.67 S_{u t} ight) \cong 0.6 S_{u t} [/latex] (14.14)

[latex] \begin{array}{ll} & S_{m s} \cong 0.6 S_{u t} \ d=0.010: & S_{m s} \cong 0.6(390239)=234143 psi \ d=0.042: & S_{m s} \cong 0.6(309071)=185443 psi \ d=0.250: & S_{m s} \cong 0.6(231301)=138781 psi \end{array} [/latex] (b)

3 The torsional fatigue strengths [latex]S_{fw}[/latex] at three values of [latex]N[/latex] are provided as percentages of the tensile strength in Table 14-9 for unpeened A228 music wire.

[latex] \begin{array}{l} d=0.010 @ N=1 E 5: S_{f w} \cong 0.36(390239)=140486 psi \ d=0.010 @ N=1 E 6: S_{f w} \cong 0.33(390239)=128779 psi \ d=0.010 @ N=1 E 7: S_{f w} \cong 0.30(390239)=117072 psi \end{array} [/latex] (c)

These values are plotted in combination with the result from equation 14.14 to generate the [latex]S-N[/latex] curves.

4 Figure 14-15 shows the [latex]S-N[/latex] curves. There are two separate portions to each [latex]S-N[/latex] curve: the [latex]1E3 ≤ N < 1E5[/latex] segment and the segment for [latex]N ≥ 1E5[/latex]. The unpeened wire endurance limit for infinite life [latex]S_{ew}[/latex] is also shown at 45 000 psi (equation 14.13).

[latex] \begin{array}{l} S_{e w^{\prime}} \cong 45.0 kpsi (310 MPa ) \quad ext {for unpeened springs} \ S_{e w} \cong 67.5 kpsi (465 MPa ) \quad ext {for peened springs} \end{array} [/latex] (14.13)

5 If desired, any of these [latex]S-N[/latex] curves can be fitted to an exponential equation (equations 6.10 on p. 338) by the method shown in Section 6-6. Evaluating the coefficients and exponents separately for the two pieces of the [latex]S-N[/latex] curve allows the estimated wire fatigue strength [latex]S_{fw}[/latex] to be easily found for any number of cycles.

[latex] S(N)=a N^b [/latex] (6.10a)

[latex] \log S(N)=\log a+b \log N [/latex] (6.10b)

[latex] \begin{aligned} b &=\frac{1}{z} \log \left\lgroup \frac{S_m}{S_e} ight group \quad ext { where } \quad z=\log N_1-\log N_2 \ \log (a) &=\log \left(S_m ight)-b \log \left(N_1 ight)=\log \left(S_m ight)-3 b \end{aligned} [/latex] (6.10c)

[latex] \begin{array}{c} z_{@ 5 E 8}=\log (1000)-\log (5 E 8)=3-8.699=-5.699 \ b_{@ 5 E 8}=-\frac{1}{5.699} \log \left\lgroup \frac{S_m}{S_f} ight group \quad ext { for } S_f @ N_2=5 E 8 ext { cycles } \end{array} [/latex] (6.10d)

6 It is important to remember that the [latex]S_{fw}[/latex] data in Table 14-9 are for a repeated-stress state, not a fully reversed stress condition, which means that this [latex]S-N[/latex] diagram is taken at some point along the [latex]\sigma _{m}[/latex] axis in Figure 6-43 (p. 362).

Table 14-4 Coefficients and Exponents for Equation 14.3 Source: Reference 1
ASTM#	Material	Range		Exponent [latex]b[/latex]	Coefficient [latex]A[/latex]		Correlation Factor
ASTM#	Material	mm	in	Exponent [latex]b[/latex]	MPa	psi	Correlation Factor
A227	Cold drawn	0.5–16	0.020–0.625	–0.182 2	1 753.3	141 040	0.998
A228	Music wire	0.3–6	0.010–0.250	–0.1625	2 153.5	184 649	0.9997
A229	Oil tempered	0.5–16	0.020–0.625	–0.183 3	1 831.2	146 780	0.999
A232	Chrome-v	0.5–12	0.020–0.500	–0.145 3	1 909.9	173 128	0.998
A401	Chrome-s.	0.8–11	0.031–0.437	–0.093 4	2 059.2	220 779	0.991

Table 14-9 Maximum Torsional Fatigue Strength [latex]S_{fw}[/latex]' for Round-Wire Helical Compression Springs in Cyclic Applications (Stress Ratio, R = 0) No Surging, Room Temperature, and Noncorrosive Environment. Source: Ref. 1
	Percent of Ultimate Tensile Strength
Fatigue Life (cycles)	ASTM 228, Austenitic Stainless Steel and Nonferrous		ASTM A230 and A232
Fatigue Life (cycles)	Unpeened	Peened	Unpeened	Peened
[latex]10^5[/latex]	36%	42%	42%	49%
[latex]10^6[/latex]	33	39	40	47
[latex]10^7[/latex]	30	36	38	46

Question 14.1: Constructing the S-N Diagram for a Spring-Wire Material Prob...

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