---
title: "Truss joint E: loading, lack of fit, temperature"
subject: "Theory of Structures I"
chapter: 3
key: "semester_4/theory_of_structures_i:truss-defl-e-2068"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/truss-defl-e-2068
site: BCT Engineering
---

# Truss joint E: loading, lack of fit, temperature

Theory of Structures I · Chapter 3: Analysis by the Virtual Work Method

- Asked in: 2068 Bhadra, 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-2068
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3

## Question

Determine the vertical deflection of joint E of the truss due to (i) loading shown, (ii) members CE and DE being 8 mm too long and (iii) temperature of member CD alone is decreased by 15°C. Given: cross-sectional area of all members = 1000 mm², Young's modulus = $2\times10^5\ \text{N/mm}^2$, and coefficient of thermal expansion = $12\times10^{-6}/^\circ$C. [Figure: truss with A (hinge) and B (roller), E at the bottom middle, AE = EB = 2 m; top joints C and D; angles 60°; 100 kN downward at E.]

## Short answer

The vertical deflection at joint E is the sum of three effects: 1.83 mm from the 100 kN load, 9.24 mm from members CE and DE being 8 mm too long, and 0.21 mm from a 15°C temperature drop in CD. Total downward deflection = 11.28 mm.

## Answer

#### Given data
A(0,0) hinge, E(2,0), B(4,0) roller; top joints C(1, 1.732) and D(3, 1.732); all members 2 m (60$^\circ$). 100 kN down at E. $A = 1000$ mm$^2 = 10^{-3}$ m$^2$, $E = 2\times10^5$ N/mm$^2 = 2\times10^8$ kN/m$^2$, $\alpha = 12\times10^{-6}$/$^\circ$C. Unit vertical load (down) at E.

#### Member forces (tension +)
| Member | L (m) | F (kN) | k | kFL (kN m) | Temp. term (mm) | Fit term (mm) |
|---|---|---|---|---|---|---|
| AE | 2.000 | 28.87 | 0.289 | 16.667 | 0.0000 | 0.0000 |
| EB | 2.000 | 28.87 | 0.289 | 16.667 | 0.0000 | 0.0000 |
| AC | 2.000 | -57.74 | -0.577 | 66.667 | -0.0000 | -0.0000 |
| CE | 2.000 | 57.74 | 0.577 | 66.667 | 0.0000 | 4.6188 |
| CD | 2.000 | -57.74 | -0.577 | 66.667 | 0.2078 | -0.0000 |
| ED | 2.000 | 57.74 | 0.577 | 66.667 | 0.0000 | 4.6188 |
| DB | 2.000 | -57.74 | -0.577 | 66.667 | -0.0000 | -0.0000 |

Reactions: $A_y = B_y = 50$ kN.

#### (i) Loading
$$
\Delta_1 = \frac{\sum kFL}{AE} = \frac{366.667}{10^{-3}\times2\times10^8} = 1.833\ \text{mm}\ (\downarrow)
$$

#### (ii) CE and DE 8 mm too long
$$
\Delta_2 = (0.577)(8) + (0.577)(8) = 9.238\ \text{mm}\ (\downarrow)
$$

#### (iii) Temperature of CD decreased by 15$^\circ$C
$$
\Delta L_{CD} = 12\times10^{-6}\times(-15)\times2000 = -0.36\ \text{mm},\qquad \Delta_3 = k_{CD}\Delta L = (-0.577)(-0.36) = 0.208\ \text{mm}\ (\downarrow)
$$

#### Total
$$
\Delta_E = 1.833 + 9.238 + 0.208 = 11.279\ \text{mm}
$$

**Answer: vertical deflection of E = 11.28 mm downward.**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How is the total deflection of E found?
   A. Use the lack of fit only
   B. Add the effects of loading, lack of fit in CE and DE, and cooling of CD
   C. Use the largest effect
   D. Use loading only
   Answer: B. The same unit load forces k are used for each cause, and the effects are added. (Key point: Three effects)

2. Which expression gives the loading effect?
   A. sum kF / E
   B. sum FL / A
   C. sum kFL / AE = 366.667 / (10^-3 x 2 x 10^8) = 1.833 mm
   D. sum kFL x AE
   Answer: C. The standard virtual work formula for trusses; the result is 1.833 mm downward. (Key point: Loading effect)

3. CE and DE are each 8 mm too long and have k = 0.577. What is the effect?
   A. 0.577 mm
   B. 4.619 mm upward
   C. 9.238 mm downward
   D. 16 mm
   Answer: C. Delta_2 = 0.577 x 8 + 0.577 x 8 = 9.238 mm downward. (Key point: Lack of fit)

4. Why does cooling CD by 15°C move E downward by 0.208 mm?
   A. CD shortens by 0.36 mm and k_CD is negative, so the product is positive
   B. Temperature has no effect on deflection
   C. CD lengthens
   D. CD is in tension
   Answer: A. Delta_3 = k x alpha DeltaT L = (-0.577)(-0.36) = +0.208 mm downward. (Key point: Cooling of CD)

## Close variants set in other exams

- [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×

## Related in Theory of Structures I

- [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×
- [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 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 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

## 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: 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: 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