Question 9.3: Shaft Design for Repeated Torsion and Bending Power is trans......
Shaft Design for Repeated Torsion and Bending
Power is transmitted from a motor through a gear at E to pulleys at D and C of a revolving solid shaft AB with ground surface. Figure 9.5(a) shows the corresponding load diagram of the shaft. The shaft is mounted on bearings at the ends A and B. Determine the required diameter of the shaft by employing the maximum energy of distortion theory of failure incorporating the Soderberg fatigue relation.
Given: The shaft is made of steel with an ultimate strength of 810 MPa and a yield strength of 605 MPa. Torque fluctuates 10% each way from the mean value. The fatigue stress-concentration factor for bending and torsion is equal to 1.4. The operating temperature is 500°C maximum.
Design Assumptions: Bearings act as simple supports. A factor of safety of n =2 is used. The survival rate is taken to be 50%.
The reactions at A and B , as obtained from the equations of statics, are noted in Figure 9.5(a). The determination of the resultant bending moment of \left(M_y^2+M_z^2\right)^{1 / 2} is facilitated by using the moment diagrams (Figure 9.5(b) and (c)). At point C , we have
M_C=\left[(0.1)^2+(1.5)^2\right]^{1 / 2}=1.503 kN \cdot m
Similarly, at D and E,
M_D=2.121 kN \cdot m \quad M_E=1.304 kN \cdot m
The maximum bending moment is a D. Note from Figure 9.5(d) that the torque is also maximum at D, T_D =1 kN ⋅ m. The exact location along the shaft where the maximum stress occurs, the critical section, is therefore at D . Hence, at point D ,
\begin{array}{ll} M_m=0 & M_a=2.121 kN \cdot m \\ T_m=1 kN \cdot m & T_a=0.1(1)=0.1 kN \cdot m \end{array}Using Equation (7.1), the endurance limit of the material is
\text {Steels} \left(S_e^{\prime}\right)=0.5 S_u \quad\left[S_u<1400 MPa (200 ksi )\right] (7.1)
S_e^{\prime}=0.5\left(S_u\right)=0.5(810)=405 MPa
By Equation (7.7) and Table 7.2, we determine that, for a ground surface,
C_f=A S_u^b (7.7)
C_f=A S_u^b=1.58\left(810^{-0.085}\right)=0.894
For reliability of 50%, we have C_r =1 from Table 7.3. Assuming that the shaft diameter will be larger than 51 mm, _C s=0.70 by Equation (7.9). The temperature factor is found applying Equation (7.11):
C_s= \begin{cases}0.85 & (13 mm <D \leq 50 mm ) \quad\left(\frac{1}{2}<D \leq 2 \text { in. }\right) \\ 0.70 & (D>50 mm ) \quad(D>2 in .)\end{cases} (7.9)
C_t= \begin{cases}1 \quad T \leq 450^{\circ} C \quad\left(840^{\circ} F \right) \\ 1-0.0058(T-450) & 450^{\circ} C <T \leq 550^{\circ} C \\ 1-0.0058(T-450) & 840^{\circ}<T \leq 1020^{\circ} F \end{cases} (7.11)
C_t=1-0.0058(T-450)=1-0.0058(500-450)=0.71
We can now determine the modified endurance limit by Equation (7.6):
S_e=C_f C_r C_s C_t\left(1 / K_f\right) S_e^{\prime} (7.6)
\begin{aligned} S_e=C_f C_r C_s C_t\left(1 / K_f\right) S_e^{\prime} & =(0.894)(1)(0.70)(0.71)(1 / 1.4)\left(405 \times 10^6\right) \\ & =128.5 MPa \end{aligned}
Because the loading is smooth, K_{s b}=K_{s t}=1 from Table 9.1.
Substituting the S_y = 605 MPa for S_u and the numerical values obtained into Equation (9.14), we have
\frac{S_u}{n}=\frac{32}{\pi D^3}\left[K_{s b}\left(M_m+\frac{S_u}{S_e} M_a\right)^2+\frac{3}{4} K_{s t}\left(T_m+\frac{S_u}{S_e} T_a\right)^2\right]^{1 / 2} (9.14)
\frac{605\left(10^6\right)}{2}=\frac{32}{\pi D^3}\left[(1)\left(0+\frac{605 \times 2121}{128.5}\right)^2+(1)\left(\frac{3}{4}\right)\left(1000+\frac{605 \times 100}{128.5}\right)^2\right]^{1 / 2}
Solving,
D=0.0697 m =69.7 mm
Comment: Since this is larger than 51 mm, our assumptions are correct. A diameter of 70 mm is therefore quite satisfactory.
| 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 | 27 | -0.265 |
| Hot rolled | 57.7 | 144 | -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 |
| TABLE 9.1 Shock Factors in Bending and Torsion |
|
| Nature of Loading | K_{s b}, K_{s t} |
| Gradually applied or steady | 1.0 |
| Minor shocks | 1.5 |
| Heavy shocks | 2.0 |