--- title: "Truss joint E: temperature and lack of fit" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:truss-defl-e-temp-lackfit-2075b" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/truss-defl-e-temp-lackfit-2075b site: BCT Engineering --- # Truss joint E: temperature and lack of fit Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2075 Baisakh, 12 marks - Video: [Unit load method for truss with temperature and lack of fit effect](https://www.youtube.com/watch?v=HNprIHQrq7k) — Vasudeo Chaudhari - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/truss-defl-e-temp-lackfit-2075b - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Determine the vertical deflection of joint E. All the top chord members are subjected to temperature rise 30°C and the members AE and EC are 5 mm too long while fabrication. Take coefficient of thermal expansion as $12\times10^{-6}/^\circ$C, modulus of elasticity as 200 kN/mm², cross sectional area of each member is 1500 mm². [Figure: truss with fixed wall D-C of 8 m on the left; C at the bottom, D at the top; bottom chord C-B-A with CB = BA = 8 m; E is the top middle joint, members DE, EA, EC and EB; loads of 8 kN at B and 4 kN at A.] ## Short answer Vertical deflection of joint E is 1.10 mm upward. Load gives 0.89 mm down, 30°C temperature rise in top chords adds 3.60 mm down, and 5 mm lack-of-fit in AE and EC contributes 5.59 mm up, net upward. ## Answer #### Given data and assumed geometry Wall supports at C (0,0) and D (0,8), both pinned. Bottom chord C-B-A with $CB = BA = 8$ m (B at (8,0), A at (16,0)); E is the top middle joint at (8,4) on the sloping top chord D-E-A. Members: CB, BA, DE, EA, EC, EB. Loads: 8 kN at B and 4 kN at A (downward). $A = 1500$ mm$^2 = 1.5\times10^{-3}$ m$^2$, $E = 200$ kN/mm$^2 = 2\times10^8$ kN/m$^2$, $\alpha = 12\times10^{-6}$/$^\circ$C. A unit vertical load (down) is applied at E. Lengths: $DE = EA = EC = \sqrt{8^2+4^2} = 8.944$ m, $EB = 4$ m. #### Member forces (tension +) | Member | L (m) | F (kN) | k | kFL (kN m) | Temp. term (mm) | Fit term (mm) | |---|---|---|---|---|---|---| | CB | 8.000 | -8.00 | 0 | 0 | 0 | 0 | | BA | 8.000 | -8.00 | 0 | 0 | 0 | 0 | | DE | 8.944 | 17.89 | 1.118 | 178.885 | 3.6000 | 0 | | EA | 8.944 | 8.94 | 0 | 0 | 0 | 0 | | EC | 8.944 | -8.94 | -1.118 | 89.443 | 0 | -5.5902 | | EB | 4.000 | 8.00 | 0 | 0 | 0 | 0 | #### (i) Loading $$ \Delta_1 = \frac{\sum kFL}{AE} = \frac{268.328}{1.5\times10^{-3}\times2\times10^8} = 0.894\ \text{mm}\ (\downarrow) $$ #### (ii) Temperature rise of 30$^\circ$C in the top chord (DE and EA) $$ \Delta_2 = \sum k\alpha\Delta T L = (1.118)(12\times10^{-6})(30)(8944) = 3.600\ \text{mm}\ (\downarrow) $$ (EA has $k = 0$, so it contributes nothing.) #### (iii) AE and EC fabricated 5 mm too long $$ \Delta_3 = \sum k\,e = (0)(5) + (-1.118)(5) = -5.590\ \text{mm}\ (\text{up}) $$ #### Total $$ \Delta_E = 0.894 + 3.600 - 5.590 = -1.096\ \text{mm} $$ **Answer:** vertical deflection of E $= 1.10$ mm upward (net). Individual effects: load 0.89 mm down, temperature 3.60 mm down, lack of fit 5.59 mm up. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. The vertical deflection at E is found with: A. A unit downward load at E and the virtual work sum for loading, temperature and lack of fit B. A unit load at C C. Real work method D. Only loading effects Answer: A. A unit vertical load at E with truss virtual work gives all three effects. (Key point: Method) 2. A temperature rise in the top chord DE and EA contributes only through DE. Why? A. k for EA is zero B. EA is not heated C. k for DE is zero D. Temperature has no effect Answer: A. Only members with non-zero k contribute: DE has k = 1.118, EA has k = 0. (Key point: Temperature effect) 3. EC is 5 mm too long, with k = -1.118. The contribution is: A. -5.59 mm (upward) B. +5.59 mm C. +5 mm D. 0 Answer: A. k e = (-1.118)(5) = -5.59 mm, meaning E moves up. (Key point: Lack of fit sign) 4. The net vertical deflection at E is: A. 1.10 mm upward B. 1.10 mm downward C. 9.5 mm downward D. 5.59 mm upward Answer: A. 0.894 + 3.600 - 5.590 = -1.096 mm, i.e. 1.10 mm upward. (Key point: Total) ## Related in Theory of Structures I - [Truss joint B: temperature and vertical members long](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-b-temp-lackfit-2073) — Theory of Structures I, chapter 3, asked 2× - [Truss joint H: temperature and long vertical members](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-h-temp-lackfit-2069b) — Theory of Structures I, chapter 3, asked 2× - [Truss joint D: loading, lack of fit, temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-d-2074) — Theory of Structures I, chapter 3, asked 3× - [Truss joint E: loading, lack of fit, temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-e-2068) — Theory of Structures I, chapter 3, asked 2× - [Truss deflection at B: load, temperature, lack of fit](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-b-loading-temp-lackfit-2081) — Theory of Structures I, chapter 3 - [Truss joint E: temperature and lack of fit](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-e-2072m) — Theory of Structures I, chapter 3, asked 3× ## Related in other subjects - [Truss: loads, temperature rise, lack of fit](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-loads-temp-lack-of-fit) — Theory of Structures II, chapter 2 - [Truss: lack of fit, temperature, bars BC BF](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-lack-of-fit-temp-bc) — Theory of Structures II, chapter 2 - [Truss forces: member AC 10 mm too long](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-lack-of-fit-ac-10mm) — Theory of Structures II, chapter 2 - [Truss: temperature decrease in vertical members](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Atruss-temp-decrease-vertical-ab-ef) — Theory of Structures II, chapter 2