Question

A hollow right-circular cylinder is made of cast iron and has an outside diameter of 75 mm and an inside diameter of 60 mm. If the cylinder is loaded by an axial compressive force of 50 kN, determine the total shortening (in mm) in a 600 mm length. Also, determine the normal stress (in MPa) under this load. The modulus of elasticity of cast iron is E = 100 GPa.

Answer 1

Answer on Question#50401 - Physics - Mechanics - Kinematics - Dynamics

A hollow right-circular cylinder is made of cast iron and has an outside diameter of $D_{o} = 75 \mathrm{~mm}$ and an inside diameter of $D_{i} = 60 \mathrm{~mm}$. If the cylinder is loaded by an axial compressive force of $F = 50 \mathrm{kN}$, determine the total shortening $\Delta L$ (in mm) in a $L = 600 \mathrm{~mm}$ length. Also, determine the normal stress (in MPa) under this load. The modulus of elasticity of cast iron is $E = 100 \mathrm{GPa}$.

Solution:

The stress of cylinder is given by the Hooke's law

$$\sigma = E \varepsilon,$$

where $\sigma = \frac{F}{A}$ is a normal stress ($F$ is the axial force acting on the cylinder, and $A$ is the cross-sectional area of the cylinder), $E$ is the modulus of elasticity, and $\varepsilon = \frac{\Delta L}{L}$ is the strain of the cylinder.

The cross-sectional area of the cylinder is given by (the difference in the areas of the outer circle of diameter $D_{o}$ and the inner one of diameter $D_{i}$)

$$A = \frac{\pi}{4} \left(D_{o}^{2} - D_{i}^{2}\right)$$

The normal stress of the cylinder

$$\sigma = \frac{F}{A} = \frac{4F}{\pi \left(D_{o}^{2} - D_{i}^{2}\right)} = \frac{4 \cdot 50 \mathrm{kN}}{\pi \left((0.075 \mathrm{m})^{2} - (0.06 \mathrm{m})^{2}\right)} = 31 \mathrm{MPa}$$

The strain of the cylinder

$$\varepsilon = \frac{\sigma}{E}$$

The shortening

$$\Delta L = \frac{\sigma \cdot L}{E} = \frac{31 \mathrm{MPa} \cdot 600 \mathrm{mm}}{100 \mathrm{GPa}} = 0.19 \mathrm{mm}$$

Answer: $\sigma = \frac{4F}{\pi(D_{o}^{2} - D_{i}^{2})} = 31 \mathrm{MPa}$; $\Delta L = \frac{\sigma \cdot L}{E} = 0.19 \mathrm{mm}$.

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