$\text { A monel shell } \left( E _ { m } = 170 \mathrm { GPa } , d _ { 3 } = 12 \mathrm {~mm} , d _ { 2 } = 8 \mathrm {~mm} \right) \text { encloses a brass core } \left( E _ { b } = 96 \mathrm { GPa } , d _ { 1 } = 6 \mathrm {~mm} \right)$ Initially, both shell and core have a length of 100 mm. A load P is applied to both shell and core through a cap plate. The load P required to compress both shell and core by 0.10 mm is approximately:
A) 10.2 kN
B) 13.4 kN
C) 18.5 kN
D) 21.0 kN

Question

$\text { A monel shell } \left( E _ { m } = 170 \mathrm { GPa } , d _ { 3 } = 12 \mathrm {~mm} , d _ { 2 } = 8 \mathrm {~mm} \right) \text { encloses a brass core } \left( E _ { b } = 96 \mathrm { GPa } , d _ { 1 } = 6 \mathrm {~mm} \right)$ Initially, both shell and core have a length of 100 mm. A load P is applied to both shell and core through a cap plate. The load P required to compress both shell and core by 0.10 mm is approximately: 
A) 10.2 kN 
B) 13.4 kN 
C) 18.5 kN 
D) 21.0 kN

Quizplus · Accepted Answer

We can use the following equation for the deformation of a composite cylinder under axial load:

$$\frac{\Delta L}{L} = \frac{P}{(D_1^2-D_2^2)E_1+(D_3^2-D_2^2)E_2}$$

where $\Delta L/L$ is the axial strain (deformation), $P$ is the applied load, $D_1$ and $D_2$ are the outer and inner diameters of the brass core, $D_3$ is the outer diameter of the monel shell, and $E_1$ and $E_2$ are the moduli of elasticity of brass and monel, respectively.

Substituting the given values, we get:

$$\frac{0.1 \mathrm{~mm}}{100 \mathrm{~mm}} = \frac{P}{(6^2-0^2)96 \mathrm{~GPa} + (12^2-8^2)170 \mathrm{~GPa}}$$

Solving for $P$, we get:

$$P \approx 13.4 \mathrm{~kN}$$

Therefore, the answer is B: 13.4 kN.

Initially, Both Shell and Core Have a Length of 100