3. Thermal Stresses and Strains
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Whenever there is an increase or decrease in the temperature of a bar, it expands or contracts.
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If the bar is allowed to expand or contract freely, no stresses are induced in the bar.
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If free expansion or contraction of the bar is prevented, thermal stresses are induced in the bar. The corresponding strain is called thermal strain.
| S. No. | Material | Coefficient of linear expansion \(/^oC (\alpha)\) |
|---|---|---|
1. |
Steel |
\(11.5 \times 10^{-6}\ to\ 13 \times 10^{-6}\) |
2. |
Wrought iron, Cast iron |
\(11 \times 10^{-6}\ to\ 12 \times 10^{-6}\) |
3. |
Aluminium |
\(23 \times 10^{-6}\ to\ 24 \times 10^{-6}\) |
4. |
Copper, Brass, Bronze |
\(17 \times 10^{-6}\ to\ 18 \times 10^{-6}\) |
3.1. Stress not to be exceeded
|
Example 1 |
A brass rod \(2.5\ m\) long is fixed at both ends. If the thermal stress is not to exceed \(77\ MPa\), calculate the temperature through which the rod should be heated. Take the values of \(\alpha\) and \(E\) as \(17 \times 10^{-6}/K\) and \(90\ GPa\) respectively. |
|
Solution:
\[\begin{align*}
\sigma &= \epsilon \times E \\
&= \alpha t \times E \\
t &= \frac{\sigma}{\alpha \times E} \\
\end{align*}\]
|
3.2. Stress when temperature falls
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Example 2 |
Two parallel walls \(6.5\ m\) apart are held together by a steel rod \(26\ mm\) diameter passing through metal plates and nuts at each end. The nuts are tightened when the rod is at a temperature of \(98^oC\). Determine the stress in the rod, when the temperature falls to \(60.5^oC\), if (a) the ends do not yield, and |
|
Solution:
\[\begin{align*}
\text{(a) } \sigma &= \alpha t \times E \\
\text{(b) } \Delta L &= L \alpha t - \Delta \\
\sigma &= \epsilon \times E \\
&= \big( \alpha t - \frac{\Delta}{L} \big) \times E
\end{align*}\]
|
3.3. Thermal stress of a tapered bar
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Example 3 |
A rigidly fixed circular bar \(1.5\ m\) long uniformly tapers from \(124\ mm\) diameter at one end to \(95\ mm\) diameter at the other. If the maximum stress in the bar is not to exceed \(108\ MPa\), find the temperature through which it can be heated. Take \(E\) and \(\alpha\) for the bar material as \(100\ GPa\) and \(18 \times 10^{-6}/ K\) respectively. |
|
Solution:
\[\begin{align*}
\Delta L &= L \alpha t \\
\Delta L &= \frac{4FL}{\pi E d D} \\
L \alpha t &= \frac{4FL}{\pi E d D} \\
F &= \alpha t \times \frac{\pi E d D}{4} \\
\sigma &= \frac{F}{A} \\
\sigma_{max} &= \frac{F}{(\frac{\pi}{4}d^2)} \\
\sigma_{max} &= \frac{ \alpha t \times \frac{\pi E d D}{4}}{(\frac{\pi}{4}d^2)} \\
\sigma_{max} &= \frac{ \alpha t E \times D}{d} \\
t &= \frac{\sigma_{max} d}{ \alpha E \times D}
\end{align*}\]
|
3.4. Stress, strain, and modulus of elasticity
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Assignment 1 |
(a) A test piece is cut from a brass bar and subjected to a tensile test. With a load of \(6.4\ kN\) the test piece, of diameter \(11.28\ mm\), extends by \(0.04\ mm\) over a gauge length of \(50\ mm\). Determine: (i) the stress, (b) A spacer is turned from the same bar. The spacer has a diameter of \(28\ mm\) and a length of \(250\ mm\). both measurements being made at \(20\ ^oC\).The temperature of the spaceris then increased to \(100\ ^oC\),the natural expansion being entirely prevented. Taking the coefficientof linear expansion to be \(18 \times 10^{-6}/^oC\) determine: (i) the stress in the spacer, |
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Solution:
Please attempt this assignment. |
3.5. Stress when temperature rises
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Assignment 2 |
A steel rod of cross-sectional area \(600\ mm^2\) and a coaxial copper tube of cross-sectional area \(1000\ mm^2\) are firmly attached at their ends to form a compound bar. Determine the stress in the steel and the copper when the temperature of the bar is raised by \(80\ ^oC\) and an axial tensile force of \(60\ kN\) is applied. For steel, \(E = 200\ GN/m^2\) with \(\alpha = 11 \times 10^{-6}/^oC\) |
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Solution:
Please attempt this assignment. |
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Comments (7)
Ae we doing chapters 6, 7, 8 & 9?
No. That's for Solid II.
Kindly confirm for us the formula for volumetric strain
Hello, are the notes here all we will have for SSM2 or is there a possibility for addition?
Solid Mechanics II starts from chapter 6.
Could you confirm if in example 1 the shear force diagram is correct since from my shear force calculations the forces are positive meaning the diagram ought to be upwards
Yes, in example 1, the shear force diagram is correct. There are two point loads acting downwards, which tend to shear the bar in the negative direction.