Chapter 6
Q. 6.2
Determining Estimated SN Diagrams for Nonferrous Materials
Problem Create an estimated SN diagram for an aluminum bar and define its equations. What is the corrected fatigue strength at 2E7 cycles?
Given The S_{ut} for this 6061T6 aluminum has been tested at 45 000 psi. The forged bar is 1.5 in round. The maximum operating temperature is 300°F. The loading is fully reversed torsion.
Assumptions A reliability factor of 99.0% will be used. The uncorrected fatigue strength will be taken at 5E8 cycles.
StepbyStep
Verified Solution
1 Since no fatiguestrength information is given, we will estimate S_{f^{\prime}} based on the ultimate strength using equation 6.5c (p. 330).
\text {aluminums :} \left\{\begin{array}{lll} S_{f_{@ 5 E 8}} \cong 0.4 S_{u t} & \text { for } S_{u t}<48 kpsi (330 MPa ) \\ S_{f_{@ 5 E 8}} \cong 19 kpsi (130 MPa ) & \text { for } S_{u t} \geq 48 kpsi (330 MPa ) \end{array}\right\} (6.5c)
\begin{array}{l} S_{f^{\prime}} \cong 0.4 S_{u t} \quad \text { for } S_{u t}<48 ksi \\ S_{f^{\prime}} \cong 0.4(45000)=18000 psi \end{array} (a)
This value is at N = 5E8 cycles. There is no knee in an aluminum SN curve.
2 The loading is pure torsion, so the load factor from equation 6.7a (p. 330) is
\begin{array}{ll} \text { bending: } & C_{\text {load }}=1 \\ \text { axial loading: } & C_{\text {load }}=0.70 \end{array} (6.7a)
C_{\text {load }} = 1.0 (b)
because the applied torsional stress will be converted to an equivalent von Mises normal stress for comparison to the SN strength.
3 The part size is greater than the test specimen and it is round, so the size factor can be estimated with equation 6.7b (p. 331), noting that this relationship is based on steel data:
\text {for} d \leq 0.3 \text {in} (8 mm) : \quad C_{\text {size }}=1 \\ \text {for} 0.3 \text {in} \lt d \leq 10 \text {in} : \quad C_{\text {size }}=0.869 d^{0.097} \\ \text {for} 8 mm \lt d \leq 250 mm : \quad C_{\text {size }}=1.189 d^{0.097} (6.7b)
C_{\text {size }}=0.869\left(d_{\text {equiv }}\right)^{0.097}=0.869(1.5)^{0.097}=0.835 (c)
4 The surface factor is found from equation 6.7e (p. 333) using the data in Table 63 (p. 333) for the specified forged finish, again with the caveat that these relationships were developed for steels and may be less accurate for aluminum.
C_{\text {surf }} \equiv A\left(S_{\text {ut }}\right)^b \quad \text { if } C_{\text {surf }}>1.0 \text {, set } C_{\text {surf }}=1.0 (6.7e)
C_{\text {surf }}=A S_{u t}^b=39.9(45)^{0.995}=0.904 (d)
5 Equation 6.7f (p. 335) is only for steel so we will assume:
\text {for} T \leq 450^\circ C (840^\circ F) : \quad C_{\text {temp }}=1\\ \text {for} 450^\circ C \lt T \leq 550^\circ C : \quad C_{\text {temp }}=10.0058(T450)\\ \text {for} 840^\circ F \lt T\leq 1020^\circ F : C_{\text {temp }}=10.0032(T840) (6.7f)
C_{\text {temp }}=1 (e)
6 The reliability factor is taken from Table 64 (p. 335) for the desired 99.0% and is
C_{\text {reliab }}=0.814 (f)
7 The corrected fatigue strength S_{f} at N = 5E8 can now be calculated from equation 6.6 (p. 330):
\begin{array}{l} S_e=C_{\text {load }} C_{\text {size }} C_{\text {surf }} C_{\text {temp }} C_{\text {reliab }} S_{e^{\prime}} \\ S_f=C_{\text {load }} C_{\text {size }} C_{\text {surf }} C_{\text {temp }} C_{\text {reliab }} S_f^{\prime} \end{array} (6.6)
\begin{aligned} S_f &=C_{\text {load }} C_{\text {size }} C_{\text {surf }} C_{\text {temp }} C_{\text {reliab }} S_{f^{\prime}} \\ &=1.0(0.835)(0.904)(1.0)(0.814)(18000)=11063 psi \end{aligned} (g)
8 To create the SN diagram, we also need a number for the estimated strength S_{m} at 10³ cycles based on equation 6.9 (p. 337). Note that the bending value is used for torsion.
\text {bending :} \quad \quad S_m=0.9 S_{u t} \\ \text {axial loading :} \quad \quad S_m=0.75 S_{u t} (6.9)
S_m=0.90 S_{u t}=0.90(45000)=40500 psi (h)
9 The coefficient and exponent of the corrected SN line and its equation are found using equations 6.10a through 6.10c (p. 338). The value of z is taken from Table 65 (p. 338) for S_{f} at 5E8 cycles.
S(N)=a N^b (6.10a)
\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} b=\frac{1}{5.699} \log \left\lgroup\frac{S_m}{S_f}\right\rgroup=\frac{1}{5.699} \log \left\lgroup\frac{40500}{11063}\right\rgroup=0.0989 \\ \log (a)=\log \left(S_m\right)3 b=\log [40500]3(0.0989): \quad a=80193 \end{array} (i)
10 The fatigue strength at the desired life of N = 2E7 cycles can now be found from the equation for the corrected SN line:
S(N)=a N^b=80193 N^{0.0989}=80193(2 e 7)^{0.0989}=15209 psi (j)
S(N) is larger than S_{f} because it is at a shorter life than the published fatigue strength.
11 Note the order of operations. We first found an uncorrected fatigue strength S_{f^{\prime}} at some “standard” cycle life (N = 5E8), then corrected it for the appropriate factors from equations 6.7 (pp. 330–335). Only then did we create equation 6.10a (p. 338) for the SN line so that it passes through the corrected S_{f} at N = 5E8. If we had
created equation 6.10a using the uncorrected S_{f^{\prime}} , solved it for the desired cycle life (N = 2E7), and then applied the correction factors, we would get a different and incorrect result. Because these are exponential functions, superposition does not hold.
12 The files EX0602 are on the CDROM.
Table 63 Coefficients for SurfaceFactor Equation 6.7e Source: Shigley and Mischke, Mechanical Engineering Design, 5th ed., McGrawHill, New York, 1989, p. 283 with permission  
For S_{ut} in MPa use  For S_{ut} in kpsi
(not psi) use 

Surface Finish  A  b  A  b 
Ground  1.58  –0.085  1.34  –0.085 
Machined or coldrolled  4.51  –0.265  2.7  –0.265 
Hotrolled  57.7  –0.718  14.4  –0.718 
Asforged  272  –0.995  39.9  –0.995 
Table 64 Reliability Factors for S_{d} = 0.08 \mu 

Reliability %  C_{reliab} 
50  1.000 
90  0.897 
95  0.868 
99  0.814 
99.9  0.753 
99.99  0.702 
99.999  0.659 
99.9999  0.620 
Table 65 zfactors for Eq. 610c 

N_{2}  z 
1.0E6  –3.000 
5.0E6  –3.699 
1.0E7  –4.000 
5.0E7  –4.699 
1.0E8  –5.000 
5.0E8  –5.699 
1.0E9  –6.000 
5.0E9  –6.699 