## Related questions with answers

The 10-mm-diameter steel bolt is surrounded by a bronze sleeve. The outer diameter of this sleeve is 20 mm, and its inner diameter is 10 mm. If the yield stress for the steel is

$(σ_γ)_{st} = 640 MPa,$

and for the bronze

$(σ_γ)_{br} = 520 MPa,$

determine the magnitude of the largest elastic load P that can be applied to the assembly.

$E_{st} = 200 GPa, E_{br} = 100 GPa.$

Solutions

Verified**Given:**

- Diameter of steel bolt: $d_\text{st} = 10 \text{ mm}$;
- Outer diameter of bronze sleeve: $D_\text{br} = 20 \text{ mm}$;
- Inner diameter of bronze sleeve: $d_\text{br} = 10 \text{ mm}$;
- Yield stress for the steel: $(\sigma_Y)_\text{st} = 640 \text{ MPa}$;
- Yield stress for the bronze: $(\sigma_Y)_\text{br} = 520 \text{ MPa}$;
- Modulus of elasticity of the steel: $E_\text{st} = 200 \text{ GPa}$;
- Modulus of elasticity of the bronze: $E_\text{br} = 100 \text{ GPa}$

**Required:**

- Maximum value of force $P$ ;

*Part 1*.

So first of all we take the displacement relationship between the two materials. Since they are considered one body the displacement will be the same meaning:

$\begin{align*} \delta _{br} - \delta _{st} = 0\\ \delta_{br} = \delta _{st} \end{align*}$

From here we write :

$\begin{align*} \dfrac{F_{br}}L{E_{br}A_{br}} &= \dfrac{F_{st}L_{st}}{E_{st}A_{st}}\\ F_{br} &= \dfrac{A_{br}E_{br}}{E_{st}A_{st}}F_{st}\\ &= \dfrac{(20^2-10^2)\frac{\pi}{4}\cdot 100}{(10^2) \cdot \dfrac{\pi}{4} \cdot200}\\ &=\dfrac{3}{2}P_{st} \end{align*}$

So our first equation is :

$F_{br}=\dfrac{3}{2}F_{st} \tag1$

Now taking the equilibrium equation around the y axis :

$F_{br} + F_{st} = P_{max} \tag2$

Substituting (2) into (1) :

$\dfrac{3}{2}F_{st} + F_{st} = P_{max}\tag3$

After rearranging Eq(3) becomes:

$F_{st} = \dfrac{2}{5}P_{max} \tag4$

## Create an account to view solutions

## Create an account to view solutions

## Recommended textbook solutions

#### Fundamentals of Electric Circuits

6th Edition•ISBN: 9780078028229 (9 more)Charles Alexander, Matthew Sadiku#### Physics for Scientists and Engineers: A Strategic Approach with Modern Physics

4th Edition•ISBN: 9780133942651 (8 more)Randall D. Knight#### Advanced Engineering Mathematics

10th Edition•ISBN: 9780470458365 (8 more)Erwin Kreyszig## More related questions

1/2

1/3