Question 7.5: Safety Factor against Fatigue Failure of a Tensile Link A te...
Safety Factor against Fatigue Failure of a Tensile Link
A tensile link of thickness t with two fillets is subjected to a load fluctuating between P_{\min} and P_{\max} (Figure 7.15). Calculate the factor of safety n for unlimited life based on the Goodman criterion.
Given: D = 120 mm, d = 80 mm, r = 16 mm,
t = 15 mm, P_{\min} = 90 kN, P_{\max} = 210 kN.
Design Decisions: The link is made of steel with S_{u} = 700 MPa. The fillets and adjacent surfaces are ground. A reliability of 99.9% is desired.
The minimum cross-sectional area equals A = 80 × 15 = 1200 mm². The maximum mean and range stresses are
\begin{array}{c} \sigma_m=\frac{(210+90)\left(10^3\right)}{2\left(1200 \times 10^{-6}\right)}=125 MPa \\ \sigma_a=\frac{(210-90)\left(10^3\right)}{2\left(1200 \times 10^{-6}\right)}=50 MPa \end{array}
Referring to Figure C.1, for D/d = 1.5 and r/d = 0.2, we obtain K_{t} = 1.72. Inasmuch as the given 16 mm fillet radius is large, we use the value of the notch sensitivity for the steel having S_{u} = 700 MPa, corresponding to r = 4 mm in Figure 7.9a; that is, q = 0.85. Hence, K_f=1+0.85(1.72-1)=1.61 .
For axial loading, there is no size factor, C_{s} = 1.0. Corresponding to a round surface finish, from Equation 7.7 and Table 7.2,
C_f=A S_u^b (7.7)
\begin{aligned} C_f &=A\left(S_u\right)^b \\ &=1.58(700)^{-0.085}=0.91 \end{aligned}
By Table 7.3, for 99.9% material reliability, C_{r} = 0.75. The temperature is not elevated, C_{t} = 1.0. The modified endurance limit, using Equations 7.3 and 7.6, is
\begin{aligned} S_e &=C_f C_r C_s C_t\left(1 / K_f\right)\left(0.45 S_u\right) \\ &=(0.91)(0.75)(1.0)(1.0)(1 / 1.61)(0.45 \times 700) \\ &=133.5 MPa \end{aligned}
The factor of safety, applying Equation 7.22, is therefore
n=\frac{S_u}{\sigma_m+\frac{S_u}{S_e} \sigma_a} (7.22)
n=\frac{700}{125+\frac{700}{133.5}(50)}=1.81Comments: If the load is well controlled and there is no impact, this factor guards the link against the fatigue failure.
| Table 7.2 Surface Finish Factors C_{f} |
|||
| A | |||
| Surface Finish | MPa | ksi | b |
| Ground | 1.58 | 1.34 | −0.085 |
| Machined or cold drawn | 4.51 | 2.7 | −0.265 |
| Hot rolled | 57.7 | 14.4 | −0.718 |
| Forged | 272.0 | 39.9 | −0.995 |
| Table 7.3 Reliability Factors |
|
| Survival Rate (%) | C_{r} |
| 50 | 1.00 |
| 90 | 0.89 |
| 95 | 0.87 |
| 98 | 0.84 |
| 99 | 0.81 |
| 99.9 | 0.75 |
| 99.99 | 0.70 |
