--- title: "Truss joint H: temperature and long vertical members" subject: "Theory of Structures I" chapter: 3 key: "semester_4/theory_of_structures_i:truss-defl-h-temp-lackfit-2069b" source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/truss-defl-h-temp-lackfit-2069b site: BCT Engineering --- # Truss joint H: temperature and long vertical members Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method - Asked in: 2069 Bhadra, 10 marks - 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-h-temp-lackfit-2069b - Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3 ## Question Determine the vertical deflection of joint H. All the top chord members are subjected to temperature rise of 20°C and all vertical members are 10 mm too long. Take $\alpha = 12\times10^{-6}/^\circ$C, $E = 200\ \text{kN/mm}^2$. Cross sectional area of each member is 1500 mm². [Figure: truss with top chord C-D-E-F-G and bottom joints A (hinge), H and B (roller), height 4 m, panels of 3 m; 20 kN downward at E.] ## Short answer Deflection of joint H: 0.45 mm downward due to the 20 kN load, 0.54 mm upward from a 20°C top chord temperature rise, and 10.00 mm upward from all verticals being 10 mm too long. Net deflection = 10.09 mm upward. ## Answer #### Given data and assumed geometry Top chord joints C(0,4), D(3,4), E(6,4), F(9,4), G(12,4); bottom joints A(3,0), H(6,0), B(9,0). Members: top chord CD, DE, EF, FG; bottom chord AH, HB; verticals DA, EH, FB; diagonals CA, DH, FH, GB. A hinge, B roller. 20 kN down at E. $A = 1500$ 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. Unit vertical load (down) at H. #### Member forces (tension +) | Member | L (m) | F (kN) | k | kFL (kN m) | Temp. term (mm) | Fit term (mm) | |---|---|---|---|---|---|---| | CD | 3.000 | 0.00 | 0.000 | 0.000 | 0.0000 | 0.0000 | | DE | 3.000 | -7.50 | -0.375 | 8.438 | -0.2700 | -0.0000 | | EF | 3.000 | -7.50 | -0.375 | 8.438 | -0.2700 | -0.0000 | | FG | 3.000 | 0.00 | 0.000 | 0.000 | 0.0000 | 0.0000 | | AH | 3.000 | 0.00 | 0.000 | 0.000 | 0.0000 | 0.0000 | | HB | 3.000 | -0.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | | DA | 4.000 | -10.00 | -0.500 | 20.000 | -0.0000 | -5.0000 | | EH | 4.000 | -20.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | | FB | 4.000 | -10.00 | -0.500 | 20.000 | -0.0000 | -5.0000 | | CA | 5.000 | -0.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | | DH | 5.000 | 12.50 | 0.625 | 39.062 | 0.0000 | 0.0000 | | FH | 5.000 | 12.50 | 0.625 | 39.062 | 0.0000 | 0.0000 | | GB | 5.000 | -0.00 | -0.000 | 0.000 | -0.0000 | -0.0000 | #### (i) Loading $$ \Delta_1 = \frac{\sum kFL}{AE} = \frac{135.000}{1.5\times10^{-3}\times2\times10^8} = 0.450\ \text{mm}\ (\downarrow) $$ #### (ii) Top chord temperature rise 20$^\circ$C $$ \Delta_2 = \sum k\alpha\Delta T L = 2\times(-0.375)(12\times10^{-6})(20)(3000) = -0.540\ \text{mm} $$ (DE and EF have $k = -0.375$; CD and FG have $k = 0$.) #### (iii) All vertical members 10 mm too long $$ \Delta_3 = (-0.5)(10) + (0)(10) + (-0.5)(10) = -10.000\ \text{mm} $$ #### Total $$ \Delta_H = 0.450 - 0.540 - 10.000 = -10.090\ \text{mm} $$ **Answer: vertical deflection of joint H = 10.09 mm upward (net).** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Where is the unit load applied? A. Horizontal at A B. Downward at H C. Downward at E D. Upward at E Answer: B. The deflection of H is wanted. (Key point: Virtual load) 2. The real force in EH is -20 kN, but it does not contribute to any term. Why? A. Its length is zero B. It is a zero-force member in the real loading C. EH is in tension D. Its virtual force k is zero for a unit load at H Answer: D. The unit load at H passes straight to the joint equilibrium where EH carries no force, so k = 0 and the products vanish. (Key point: Member EH) 3. All vertical members are 10 mm too long. How much does this contribute? A. -5 mm B. +10 mm C. -15 mm D. -10 mm, from DA and FB (k = -0.5 each) Answer: D. Delta_3 = (-0.5)(10) + 0 + (-0.5)(10) = -10 mm. (Key point: Lack of fit) 4. What is the net vertical deflection of H? A. 10.09 mm upward B. 0.45 mm downward C. 10.09 mm downward D. 9.55 mm upward Answer: A. 0.450 - 0.540 - 10.000 = -10.09 mm, meaning upward. (Key point: Net result) ## Close variants set in other exams - [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× ## 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 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 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: 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× - [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 ## 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: 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 - [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: 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