Question 7.1: Endurance Limit of a Torsion Bar A round torsion bar machine...
Endurance Limit of a Torsion Bar
A round torsion bar machined from steel is under reversed torsional loading. Because of the design of the ends, a fatigue stress-concentration factor K_{f} exists. Estimate the modified endurance limit.
Given: The diameter of the bar is d = 1⅝ in. and K_{f} = 1.2. The operating temperature is 500°C maximum.
Assumption: Reliability is 98%.
Design Decision: The bar is made of AISI 1050 cold-drawn steel.
From Table B.3, we find the ultimate strength in tension as S_{u} = 100 ksi. Then, applying Equation 7.4, the endurance limit of the test specimen is
S_{e s}^{\prime}=0.29 S_u=0.29(100)=29 ksi
By Equation 7.7 and Table 7.2, the surface finish factor is
C_f=A S_u^b=2.7(100)^{-0.265}=0.80
The reliability factor corresponding to 98% is C_{r} = 0.84 (Table 7.3). Using Equation 7.9, the size factor C_{s} = 0.85. Applying Equation 7.11,
C_s=\left\{\begin{array}{ll} 0.85 & (13 mm <D \leq 50 mm ) \quad\left(\frac{1}{2}<D \leq 2 in .\right) \\ 0.70 & (D>50 mm ) \quad(D>2 in .) \end{array}\right. (7.9)
C_t=\left\{\begin{array}{cc} 1 \quad T \leq 450 ^{\circ} C & \left(840 ^{\circ} F \right) \\ 1-0.0058(T-450) & 450 ^{\circ} C <T \leq 550 ^{\circ} C \\ 1-0.0032(T-840) & 840 ^{\circ} F <T \leq 1020 ^{\circ} F \end{array}\right. (7.11)
C_t=1-0.0058(500-450)=0.71
Hence, the endurance limit for design is found to be
\begin{aligned} S_{e s} &=C_f C_r C_s C_t\left(1 / K_f\right) S_{e s}^{\prime} \\ &=(0.80)(0.84)(0.85)(0.71)(1 / 1.2)(29) \\ &=9.8 ksi \end{aligned} (b)
| 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 |