Question 14.1: Constructing the S-N Diagram for a Spring-Wire Material Prob...
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).
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.
S_{u t} \cong A d^b (14.3)
\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} (a)
2 These values are converted to shear strengths at 1 000 cycles using equation 14.14:
S_{m s} \cong 0.9 S_{u s} \cong 0.9\left(0.67 S_{u t}\right) \cong 0.6 S_{u t} (14.14)
\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} (b)
3 The torsional fatigue strengths S_{fw} at three values of N are provided as percentages of the tensile strength in Table 14-9 for unpeened A228 music wire.
\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} (c)
These values are plotted in combination with the result from equation 14.14 to generate the S-N curves.
4 Figure 14-15 shows the S-N curves. There are two separate portions to each S-N curve: the 1E3 ≤ N < 1E5 segment and the segment for N ≥ 1E5. The unpeened wire endurance limit for infinite life S_{ew} is also shown at 45 000 psi (equation 14.13).
\begin{array}{l} S_{e w^{\prime}} \cong 45.0 kpsi (310 MPa ) \quad \text {for unpeened springs} \\ S_{e w} \cong 67.5 kpsi (465 MPa ) \quad \text {for peened springs} \end{array} (14.13)
5 If desired, any of these S-N 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 S-N curve allows the estimated wire fatigue strength S_{fw} to be easily found for any number of cycles.
S(N)=a N^b (6.10a)
\log S(N)=\log a+b \log N (6.10b)
\begin{aligned} b &=\frac{1}{z} \log \left\lgroup \frac{S_m}{S_e} \right\rgroup \quad \text { where } \quad z=\log N_1-\log N_2 \\ \log (a) &=\log \left(S_m\right)-b \log \left(N_1\right)=\log \left(S_m\right)-3 b \end{aligned} (6.10c)
\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}\right\rgroup \quad \text { for } S_f @ N_2=5 E 8 \text { cycles } \end{array} (6.10d)
6 It is important to remember that the S_{fw} data in Table 14-9 are for a repeated-stress state, not a fully reversed stress condition, which means that this S-N diagram is taken at some point along the \sigma _{m} axis in Figure 6-43 (p. 362).
| Table 14-4 Coefficients and Exponents for Equation 14.3 Source: Reference 1 |
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| ASTM# | Material | Range | Exponent b |
Coefficient A | Correlation Factor | ||
| mm | in | MPa | psi | ||||
| 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 S_{fw}‘ for Round-Wire Helical Compression Springs in Cyclic Applications (Stress Ratio, R = 0) No Surging, Room Temperature, and Noncorrosive Environment. Source: Ref. 1 |
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| Percent of Ultimate Tensile Strength | ||||
| Fatigue Life (cycles) |
ASTM 228, Austenitic Stainless Steel and Nonferrous |
ASTM A230 and A232 | ||
| Unpeened | Peened | Unpeened | Peened | |
| 10^5 | 36% | 42% | 42% | 49% |
| 10^6 | 33 | 39 | 40 | 47 |
| 10^7 | 30 | 36 | 38 | 46 |
