A solid cylinder of steel is placed inside a copper tube. The assembly is compressed between rigid plates by forces P. What is the value of increase in temperature so that all the load is carried by the copper tube ? (Let the parameters with suffix c represent copper and suffix s represent steel)
- At = 4P / (As Es (αc − αs))
- Bt = 2P / (Ac Ec (αs − αc))
- Ct = P (αc − αs) / (Ac Ec)
- Dt = P / (Ac Ec (αc − αs))Correct
Solution
Concept
Compatibility in a composite bar with temperature change
Formula
αs t = αc t − P / (Ac Ec)
Step-by-step solution
1. Both materials have the same total length change (rigid plates). 2. If the steel carries no load, its strain is αs t. 3. The copper carries all of P: strain = αc t − P/(Ac Ec). 4. Equate: t (αc − αs) = P / (Ac Ec), so t = P / [Ac Ec (αc − αs)].
Explanation
Copper expands more than steel on heating, which relieves the compressive force in the copper only if the steel is stretched; the load transfers entirely to copper at this temperature.
Why the other options are wrong
The other expressions have wrong areas, wrong factors (2, 4) or an inverted difference of expansion coefficients.
Shortcut
t = P / [A
Correct answer: D) t = P / (A_c E_c (α_c − α_s))