--- title: "Truss joint B: temperature and vertical members long" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:truss-defl-b-temp-lackfit-2073" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/truss-defl-b-temp-lackfit-2073 site: BCT Engineering --- # Truss joint B: temperature and vertical members long Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2073 Bhadra, 10 marks - Video: [Virtual Work for Temperature Change Causing Deflection in a Truss](https://www.youtube.com/watch?v=Ufn7V5TysB4) — STATICS THE EASY WAY - 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-b-temp-lackfit-2073 - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Determine the vertical deflection of joint B. All the top chord members are subjected to temperature rise of 20°C and all the vertical members are 10 mm too long. 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 top chord D-E-F-G-H, bottom joints A (hinge), B and C (roller), height 6 m, 4 panels of 5 m; loads 50 kN downward at F, 30 kN upward at H and 20 kN horizontal at H.] ## Short answer Vertical deflection of joint B is 9.07 mm upward. Loading gives 1.93 mm down, 20°C temperature rise in top chords adds 1.00 mm up, and vertical members 10 mm too long contribute 10.00 mm up. ## Answer #### Given data and assumed geometry From the figure: top chord joints D(0,6), E(5,6), F(10,6), G(15,6), H(20,6); bottom joints A(5,0), B(10,0), C(15,0). Members: top chord DE, EF, FG, GH; bottom chord AB, BC; verticals EA, FB, GC; diagonals DA, EB, GB, HC (13 members, 8 joints, 3 reactions: determinate). A hinge, C roller. Loads: 50 kN down at F, 30 kN up and 20 kN to the right at H. $A = 1500$ mm$^2 = 1.5\times10^{-3}$ m$^2$, $E = 2\times10^8$ kN/m$^2$, $\alpha = 12\times10^{-6}$/$^\circ$C. Unit vertical load (down) at B. #### Member forces (tension +); $F$ real, $k$ unit load at B | Member | L (m) | F (kN) | k | kFL (kN m) | Temp. term (mm) | Fit term (mm) | |---|---|---|---|---|---|---| | DE | 5.000 | 0.00 | 0.000 | 0.000 | 0.0000 | 0.0000 | | EF | 5.000 | -23.33 | -0.417 | 48.611 | -0.5000 | -0.0000 | | FG | 5.000 | -23.33 | -0.417 | 48.611 | -0.5000 | -0.0000 | | GH | 5.000 | -5.00 | 0.000 | -0.000 | 0.0000 | 0.0000 | | AB | 5.000 | 20.00 | 0.000 | 0.000 | 0.0000 | 0.0000 | | BC | 5.000 | 25.00 | -0.000 | -0.000 | -0.0000 | -0.0000 | | EA | 6.000 | -28.00 | -0.500 | 84.000 | -0.0000 | -5.0000 | | FB | 6.000 | -50.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | | GC | 6.000 | -22.00 | -0.500 | 66.000 | -0.0000 | -5.0000 | | DA | 7.810 | -0.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | | EB | 7.810 | 36.45 | 0.651 | 185.276 | 0.0000 | 0.0000 | | GB | 7.810 | 28.64 | 0.651 | 145.574 | 0.0000 | 0.0000 | | HC | 7.810 | 39.05 | -0.000 | -0.000 | -0.0000 | -0.0000 | #### (i) Loading $$ \Delta_1 = \frac{\sum kFL}{AE} = \frac{578.073}{1.5\times10^{-3}\times2\times10^8} = 1.927\ \text{mm}\ (\downarrow) $$ #### (ii) Temperature rise of 20$^\circ$C in the top chord $$ \Delta_2 = \sum k\,\alpha\,\Delta T\,L = (-0.417)(12\times10^{-6})(20)(5000)\times2 = -1.000\ \text{mm} $$ (only EF and FG have non-zero $k$). #### (iii) Vertical members 10 mm too long $$ \Delta_3 = \sum k\,e = (-0.5)(10) + (0)(10) + (-0.5)(10) = -10.000\ \text{mm} $$ #### Total $$ \Delta_B = 1.927 - 1.000 - 10.000 = -9.073\ \text{mm} $$ **Answer: vertical deflection of joint B = 9.07 mm upward (net).** Loading gives 1.93 mm down, temperature 1.00 mm up and lack of fit 10.00 mm up. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the total vertical deflection of B obtained? A. By adding the effects of loading, temperature and lack of fit, each found with the unit load forces k B. From the loading effect only C. By taking the largest of the three effects D. By subtracting the lack of fit from the loading effect only Answer: A. Virtual work lets each cause be handled separately with the same k forces, and the three results are added with their signs. (Key point: Method) 2. Which vertical members contribute to the lack of fit term, and what is it? A. None of them, since the lack of fit has no effect on a determinate truss B. EA and GC (k = -0.5 each), giving -10 mm; FB has k = 0 C. EA, FB and GC all equally, giving -15 mm D. Only FB, giving -10 mm Answer: B. Delta3 = sum k e = (-0.5)(10) + 0 + (-0.5)(10) = -10 mm; FB has k = 0 for a unit load at B. (Key point: Lack of fit) 3. The net deflection of B is -9.07 mm. What does this mean? A. B moves 9.07 mm downward B. B moves 9.07 mm upward because the 10 mm lack of fit outweighs the 1.93 mm loading effect C. B does not move D. B moves 1.93 mm upward Answer: B. The loading gives 1.93 mm down, but temperature (1.00 mm up) and lack of fit (10.00 mm up) give a negative net value, meaning upward. (Key point: Final result) ## Close variants set in other exams - [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× ## Related in Theory of Structures I - [Truss joint E: temperature and lack of fit](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-e-temp-lackfit-2075b) — Theory of Structures I, chapter 3 - [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 joint L2 by virtual work and temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-vw-l2-temperature-2069p) — 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× - [Truss joint E: temperature decrease and lack of fit](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Atruss-defl-e-decrease-temp-2075bh) — Theory of Structures I, chapter 3, asked 2× ## Related in other subjects - [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: 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: 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 - [Forces in all members of truss with 10.8 kN loads](https://www.bctengineer.com/question.md?key=semester_1%2Fapplied_mechanics%3Atruss-forces-10-8kn-d-support) — Applied Mechanics, chapter 7