--- title: "Temperature stresses in bars with a gap" subject: "Strength of Materials" chapter: 3 key: "semester_3/strength_of_materials:gap-bars-temperature-stress" source: https://www.bctengineer.com/q/semester_3/strength_of_materials/gap-bars-temperature-stress site: BCT Engineering --- # Temperature stresses in bars with a gap Strength of Materials · Chapter 3: Simple Stress and Strain - Asked in: 2081 Bhadra, 8 marks; 2066 Bhadra (old course), 8 marks (asked 2 times) - Video: [At room temperature (20 C), a 0.5 mm gap exists between the ends of the rods shown. At a later time.](https://www.youtube.com/watch?v=kwIJBK69XYQ) — Engineering Made Possible - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/gap-bars-temperature-stress - Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-3 ## Question The gap between bar A of cross-sectional area $1000\ \text{mm}^2$ and bar B of cross-sectional area $800\ \text{mm}^2$ is $0.26\ \text{mm}$ at room temperature. What are the stresses induced in the bars if the temperature is raised to $40^\circ\text{C}$? Given: $E_A = 2\times10^5\ \text{N/mm}^2$, $\alpha_A = 12\times10^{-6}/^\circ\text{C}$ and $E_B = 1\times10^5\ \text{N/mm}^2$, $\alpha_B = 23\times10^{-6}/^\circ\text{C}$. [Figure: bar A (length 400 mm) fixed to the left wall and bar B (length 300 mm) fixed to the right wall, with a gap of 0.26 mm between them.] ## Short answer Free thermal expansions total 0.468 mm; excess 0.208 mm causes compressive force. From axial stiffnesses, P =36.2 kN. Stress σ_A = 36.2 N/mm², σ_B = 45.2 N/mm²; both compressive. ## Answer Take bar A as the left bar and bar B as the right bar, and take the temperature rise as $\Delta T = 40^\circ\text{C}$. The bars first expand freely; if the free expansion exceeds the gap, the walls prevent the excess and a compressive force $P$ (equal in both bars) develops. #### Free thermal expansion $$ \begin{aligned} \delta_A &= \alpha_A\,\Delta T\,L_A = 12\times10^{-6}\times40\times400 = 0.192\ \text{mm}\\ \delta_B &= \alpha_B\,\Delta T\,L_B = 23\times10^{-6}\times40\times300 = 0.276\ \text{mm}\\ \delta_{total} &= 0.468\ \text{mm} > \text{gap} = 0.26\ \text{mm} \end{aligned} $$ The expansion that is prevented is $\delta_{prevented} = 0.468 - 0.26 = 0.208$ mm, so the bars come into contact and stresses are induced. #### Compatibility The compressive force $P$ shortens the bars by exactly the prevented expansion: $$ \frac{P L_A}{A_A E_A} + \frac{P L_B}{A_B E_B} = 0.208 $$ $$ P\left(\frac{400}{1000\times2\times10^5} + \frac{300}{800\times1\times10^5}\right) = P\,(5.7500\times10^{-6}) = 0.208 $$ $$ P = 36173.9\ \text{N} = 36.17\ \text{kN} $$ #### Stresses $$ \sigma_A = \frac{P}{A_A} = \frac{36173.9}{1000} = 36.17\ \text{N/mm}^2,\qquad \sigma_B = \frac{P}{A_B} = \frac{36173.9}{800} = 45.22\ \text{N/mm}^2 $$ **Answer: both bars are in compression: $\sigma_A = 36.17$ N/mm² (bar A) and $\sigma_B = 45.22$ N/mm² (bar B); common force $P = 36.17$ kN.** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Why do stresses develop in the bars? A. The bars are heated unevenly B. The gap closes by itself C. Total free expansion 0.468 mm exceeds the 0.26 mm gap D. Bar B expands more than A Answer: C. Only expansion in excess of the gap (0.208 mm) is prevented and produces stress. (Key point: Contact check) 2. Which compatibility equation is used? A. PLA/AAEA = PLB/ABEB B. PLA/(AAEA) + PLB/(ABEB) = 0.208 mm C. P*(LA+LB)/E = 0.26 D. Free expansion = 0.26 Answer: B. Both bars carry the same P and their combined shortening equals the prevented expansion. (Key point: Compatibility) 3. Which stress is larger and why? A. Bar B (45.22 N/mm^2) since the force is common and its area is smaller B. Bar A since E is larger C. Both equal D. Bar A since its α is smaller Answer: A. Same P = 36.17 kN acts on both, so the stress is higher in the smaller area, 800 mm^2. (Key point: Force and stress) 4. What is the nature of the stresses? A. Tensile in both B. Tensile in A, compressive in B C. Zero D. Compressive in both bars Answer: D. The walls resist expansion, so both bars are compressed. (Key point: Nature) ## Related in Strength of Materials - [Composite bar: total change in length and stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acomposite-bar-aluminium-copper-steel) — Strength of Materials, chapter 3 - [Steel-aluminium bar with temperature rise: stresses](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-aluminium-bar-temperature-supports) — Strength of Materials, chapter 3 - [Round bar fixed-free (effective length 2 m): buckling load](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Around-bar-clamped-free-buckling-2m) — Strength of Materials, chapter 8, asked 2× - [Bar fixed at both ends: load shares, displacements](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acircular-bar-fixed-both-ends-50kn) — Strength of Materials, chapter 3 - [Round bar fixed-free: buckling load from lateral deflection](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Around-bar-fixed-free-buckling-3m) — Strength of Materials, chapter 8, asked 2× - [Total elongation of stepped-load steel bar](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Asteel-bar-total-elongation-abcd) — Strength of Materials, chapter 3 ## Related in other subjects - [Bar: nodal displacements, stresses, reactions](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Aaxial-bar-stepped-p300) — Computational Techniques in Civil Engineering, chapter 4 - [Air gap and rotor bars, 10 kW motor](https://www.bctengineer.com/question.md?key=semester_5%2Felectric_machine_design%3Acage-rotor-10kw-design-data) — Electric Machine Design, chapter 4 - [Cable over 300 m gap: tension and length](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Acable-gap-300m-1-5m-dip) — Theory of Structures I, chapter 7 - [Temperature stresses in rigid pavement; dowel bar design](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Atemperature-stress-rigid-dowel-bar-procedure) — Transportation Engineering II, chapter 2