Chapter 7, Problem 7.66DP

Interpretation Introduction

Interpretation:

The shaft design using the appropriate material for given condition should be determined.

Concept introduction:

Endurance strength is the value of maxium reveresed bending stress that a material can withstand without failure for finite number of cycle.

The following condition are specified based on number of cycle, If number of cycles are more than ${10}^6$, the corresponding endurance is known as Endurance limit.If number of cycles are less than ${10}^6$, the corresponding endurance is known as Endurance Strength.

Endurance Ratio:

It is defined as the ratio of endurance limit to tensile strength, the relationship is given below:

  $\text{Endurance}\text{Ratio}\text{=}\frac{\text{Endurance}\text{Limit}}{\text{Tensile}\text{Strength}}$

It helps to estimate the properties of fatigue using tensile test. The value of endurance limit for metallic material is 0.3 to 0.4.

Relationship for endurance limit is shown below:

  $\pm \sigma =5.092\frac{\text{FL}}{{\text{d}}^{\text{3}}}$

Where,

$\sigma$ is fatigue strength in psi.

F is the applied load in pounds.

L is the length I inch.

D is the diameter in inch.

Answer to Problem 7.66DP

Magnesium-Manganese alloy is best suitable material for shaft designing based on the value of fatigue strength.

Explanation of Solution

Given Information:

The value of load is $15000\text{lb}$.

The material used along with the values of fatigue strength is shown in table below:

  

The required values are shown in table,

The relationship for Endurance limit is,

  $\pm \sigma =5.092\frac{\text{FL}}{{\text{d}}^{\text{3}}}$

Considering the case for Aluminum manganese alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $\text{110}\text{MPa}$.

Value of fatigue strength in psi is $\text{15954}\text{.15 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}15954.15=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}7.8229\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}7.8229\text{inch}$.

Considering the case for Aluminum magnesium and zinc alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $\text{225}\text{MPa}$.

Value of fatigue strength in psi is $\text{32633}\text{.5 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}32633.5=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}6.162\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}6.162\text{inch}$.

Considering the case for Aluminum- Beryllium alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $\text{295}\text{MPa}$.

Value of fatigue strength in psi is $\text{42786}\text{.1 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}42786.1=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}5.630\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}5.630\text{inch}$.

Considering the case for Magnesium- Manganese alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $80\text{MPa}$.

Value of fatigue strength in psi is $\text{11603 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}11603=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}8.69\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}8.69\text{inch}$.

Considering the case for Beryllium alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $180\text{MPa}$.

Value of fatigue strength in psi is $\text{26106}\text{.8 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}26106.8=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}6.638\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}6.638\text{inch}$.

Considering the case for Tungsten alloy,

The amount of load applied is $15000\text{lb}$.

The value of fatigue strength for aluminum- manganese alloy is $320\text{MPa}$.

Value of fatigue strength in psi is $\text{46412}\text{.1 psi}$.

Assuming the value of length as $100\text{inch}$.

Substituting the values in the above formula,

  $\begin{array}{l}46412.1=5.092\frac{\text{15000}\times 100}{{\text{d}}^{\text{3}}}\\ \text{d}\text{=}5.480\text{inch}\end{array}$

Thus, the required value of diameter for shaft designing is $\text{d}\text{=}5.480\text{inch}$.

The required values are shown table,

Alloys	Fatigue Strength (MPa)	Calculated diameter in inch
Al-Mg	110	7.8229
Al-Mg-Zn	225	6.162
Cu-Be	295	5.630
Mg-Mn	80	8.69
Be	180	6.638
W	320	5.480

Designing of shaft is based on the stress level. On comparing, the values of fatigue strength the alloys of magnesium and manganese gives the lower value of fatigue strength as compared to other.

Thus, based on the fatigue strength value, Mg-Mn is suitable for designing of the shaft.

Conclusion

The best suitable material for shaft designing based on the value of fatigue strength is Magnesium-Manganese alloy.