Chapter 3, Problem 3.43P

(a)

To determine

The theoretical shear strength for aluminum, plain-carbon steel, and tungsten.

Answer to Problem 3.43P

The theoretical shear strength for aluminum, plain-carbon steel, and tungsten are $4.69\text{GPa}$ , $11.96\text{GPa}$ , and $25.06\text{GPa}$ respectively.

Explanation of Solution

Formula used:

The expression for the modulus of rigidity is given as,

  $G=\frac{E}{2\left(1+v\right)}$

Here, $E$ is the young’s modulus, and $v$ is the Poisson’s ratio.

The expression for the theoretical shear strength is given as,

  $\left({\tau }_{theo}\right)=\frac{\left(G\right)}{2\pi }$

Calculation:

The value of Young’s modulus and Poisson’s ratio for aluminum is,

  $\begin{array}{c}{\left(E\right)}_{Al}=79\text{GPa}\\ v=0.34\end{array}$

The value of modulus of rigidity of aluminum can be calculated as,

  $\begin{array}{c}{\left(G\right)}_{Al}=\frac{{\left(E\right)}_{Al}}{2\left(1+v\right)}\\ =\frac{79\text{GPa}}{2\left(1+0.34\right)}\\ =29.47\text{GPa}\end{array}$

The value of Young’s modulus and Poisson’s ratio for steel is,

  $\begin{array}{c}{\left(E\right)}_{steel}=200\text{GPa}\\ v=0.33\end{array}$

The value of modulus of rigidity of steel can be calculated as,

  $\begin{array}{c}{\left(G\right)}_{steel}=\frac{{\left(E\right)}_{steel}}{2\left(1+v\right)}\\ =\frac{200\text{GPa}}{2\left(1+0.33\right)}\\ =75.18\text{GPa}\end{array}$

The value of Young’s modulus and Poisson’s ratio for tungsten is,

  $\begin{array}{c}{\left(E\right)}_W=400\text{GPa}\\ v=0.27\end{array}$

The value of modulus of rigidity of tungsten can be calculated as,

  $\begin{array}{c}{\left(G\right)}_W=\frac{{\left(E\right)}_W}{2\left(1+v\right)}\\ =\frac{400\text{GPa}}{2\left(1+0.27\right)}\\ =157.48\text{GPa}\end{array}$

The value of theoretical shear strength of aluminum can be calculated as,

  $\begin{array}{c}{\left({\tau }_{theo}\right)}_{Al}=\frac{{\left(G\right)}_{Al}}{2\pi }\\ =\frac{29.47\text{GPa}}{2\pi }\\ =4.69\text{GPa}\end{array}$

The value of theoretical shear strength of steel can be calculated as,

  $\begin{array}{c}{\left({\tau }_{theo}\right)}_{steel}=\frac{{\left(G\right)}_{steel}}{2\pi }\\ =\frac{75.18\text{GPa}}{2\pi }\\ =11.96\text{GPa}\end{array}$

The value of theoretical shear strength of tungsten can be calculated as,

  $\begin{array}{c}{\left({\tau }_{theo}\right)}_W=\frac{{\left(G\right)}_W}{2\pi }\\ =\frac{157.48\text{GPa}}{2\pi }\\ =25.06\text{GPa}\end{array}$

Conclusion:

Therefore, the theoretical shear strength for aluminum, plain-carbon steel, and tungsten are $4.69\text{GPa}$ , $11.96\text{GPa}$ , and $25.06\text{GPa}$ respectively.

(b)

To determine

The theoretical tensile strength for aluminum, plain-carbon steel, and tungsten.

The ratios of their theoretical strength to actual strength.

Answer to Problem 3.43P

The theoretical tensile strength for aluminum, plain-carbon steel, and tungsten are $7.9\text{GPa}$ , $20\text{GPa}$ , and $40\text{GPa}$ respectively.

The ratios of theoretical strength to actual strength of aluminum, plain-carbon steel, and tungsten are $0.87$ , $0.8$ , and $0.95$ respectively.

Explanation of Solution

Formula used:

The expression for the theoretical tensile strength is given as,

  ${\sigma }_{theo}=\frac{E}{10}$

The expression for the ratio of theoretical tensile strength to actual tensile strength is given as,

  $R=\frac{\left({\sigma }_{theo}\right)}{\left({\sigma }_{actual}\right)}$

Here, $\left({\sigma }_{actual}\right)$ is the actual tensile strength.

Calculation:

The value of theoretical tensile strength of aluminum can be calculated as,

  $\begin{array}{c}{\left({\sigma }_{theo}\right)}_{Al}=\frac{{\left(E\right)}_{Al}}{10}\\ =\frac{79\text{GPa}}{10}\\ =7.9\text{GPa}\end{array}$

The value of theoretical tensile strength of steel can be calculated as,

  $\begin{array}{c}{\left({\sigma }_{theo}\right)}_{steel}=\frac{{\left(E\right)}_{steel}}{10}\\ =\frac{200\text{GPa}}{10}\\ =20\text{GPa}\end{array}$

The value of theoretical tensile strength of tungsten can be calculated as,

  $\begin{array}{c}{\left({\sigma }_{theo}\right)}_W=\frac{{\left(E\right)}_W}{10}\\ =\frac{400\text{GPa}}{10}\\ =40\text{GPa}\end{array}$

The actual tensile strength of the aluminum is,

  ${\left({\sigma }_{actual}\right)}_{Al}=9\text{GPa}$

The actual tensile strength of the steel is,

  ${\left({\sigma }_{actual}\right)}_{steel}=25\text{GPa}$

The actual tensile strength of the steel is,

  ${\left({\sigma }_{actual}\right)}_W=42\text{GPa}$

The value of the ratio of theoretical tensile strength to actual tensile strength for aluminum can be calculated as,

  $\begin{array}{c}{R}_{Al}=\frac{{\left({\sigma }_{theo}\right)}_{Al}}{{\left({\sigma }_{actual}\right)}_{Al}}\\ =\frac{7.9\text{GPa}}{9\text{GPa}}\\ =0.87\end{array}$

The value of the ratio of theoretical tensile strength to actual tensile strength for steel can be calculated as,

  $\begin{array}{c}{R}_{steel}=\frac{{\left({\sigma }_{theo}\right)}_{steel}}{{\left({\sigma }_{actual}\right)}_{steel}}\\ =\frac{20\text{GPa}}{25\text{GPa}}\\ =0.8\end{array}$

The value of the ratio of theoretical tensile strength to actual tensile strength for tungsten can be calculated as,

  $\begin{array}{c}{R}_W=\frac{{\left({\sigma }_{theo}\right)}_W}{{\left({\sigma }_{actual}\right)}_W}\\ =\frac{40\text{GPa}}{42\text{GPa}}\\ =0.95\end{array}$

Conclusion:

Therefore, the theoretical tensile strength for aluminum, plain-carbon steel, and tungsten are $7.9\text{GPa}$ , $20\text{GPa}$ , and $40\text{GPa}$ respectively.

Therefore, the ratios of theoretical strength to actual strength of aluminum, plain-carbon steel, and tungsten are $0.87$ , $0.8$ , and $0.95$ respectively.