---
title: "Frame deflection due to temperature variation"
subject: "Theory of Structures I"
chapter: 3
key: "semester_4/theory_of_structures_i:frame-temperature-free-end-2076"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/frame-temperature-free-end-2076
site: BCT Engineering
---

# Frame deflection due to temperature variation

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

- Asked in: 2076 Bhadra, 6 marks
- Video: [Deflection Caused by Temperature Change - Problem 3 - Deflection of Statically Determinate Structure](https://www.youtube.com/watch?v=HF4FIQY-lgY) — Ekeeda
- 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/frame-temperature-free-end-2076
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3

## Question

Calculate horizontal and vertical deflection at the free end of the given frame due to temperature variation as stated in figure. The thickness of member is 20 cm and coefficient of linear expansion is $12\times10^{-6}/^\circ$C. [Figure: frame with fixed support A, left column AB, top member BC of 5 m, right column CD hanging 3 m below C (free end D; total frame height 3 m + 3 m); temperature marked +20°C on the outer face of the left column and +10°C inside the frame.]

## Short answer

With outer face +20°C, inner +10°C, α=12×10⁻⁶/°C and depth 0.2 m, the free end deflects 10.8 mm to the left and 24.96 mm downward due to axial expansion and curvature.

## Answer

#### Given data and assumptions
Frame fixed at A: left column AB 6 m, top member BC 5 m, right column CD hanging 3 m (D free). Take A(0,0), B(0,6), C(5,6), D(5,3). Temperature: $+20^\circ$C on the outer face and $+10^\circ$C on the inner face of all members (assumed uniform along each member). Depth $d = 0.20$ m, $\alpha = 12\times10^{-6}$/$^\circ$C.

#### Principle
In each member the temperature varies linearly across the depth:

- **Mean temperature rise** $T_m = \dfrac{20 + 10}{2} = 15^\circ$C. This causes an axial strain $\varepsilon = \alpha T_m = 1.8\times10^{-4}$ (each element lengthens by $\alpha T_m\,ds$).
- **Temperature difference** $\Delta T = 20 - 10 = 10^\circ$C. This causes a curvature $\kappa = \dfrac{\alpha\,\Delta T}{d} = \dfrac{12\times10^{-6}\times10}{0.2} = 6\times10^{-4}$ per m, with the hotter (outer) fibres longer.

By the virtual work (unit load) method, with virtual axial force $n$ and moment $m$ from a unit load at D:

$$
\Delta = \int n\,\alpha T_m\,ds + \int m\,\frac{\alpha\,\Delta T}{d}\,ds
$$

(The equivalent geometric approach: each element's curvature rotates the rest of the frame about that element, and each element's axial elongation moves D along the member.)

#### Evaluation (outer face hotter in every member)
**Axial (mean temperature) effects on D:** AB (6 m) lifts D by $1.8\times10^{-4}\times6000 = 1.08$ mm; BC (5 m) moves D to the right by $0.90$ mm; CD (3 m) lowers D by $0.54$ mm. Net: $\Delta_h = +0.90$ mm (right), $\Delta_v = +0.54$ mm (up).

**Curvature effects:** with the outer face hotter, every element rotates clockwise by $\kappa\,ds$. The movement of D is (horizontal = $\kappa\int(y_D - y)\,ds$, vertical = $-\kappa\int(x_D - x)\,ds$):

| Member | Horizontal | Vertical |
|---|---|---|
| AB | $\kappa\int_0^6(3-y)\,dy = 0$ | $-\kappa\int_0^6 5\,dy = -30\kappa$ |
| BC | $\kappa\int_0^5(3-6)\,dx = -15\kappa$ | $-\kappa\int_0^5(5-x)\,dx = -12.5\kappa$ |
| CD | $\kappa\int_3^6(3-y)\,dy = -4.5\kappa$ | 0 |
| **Sum** | $-19.5\kappa = -11.70$ mm | $-42.5\kappa = -25.50$ mm |

Total:

$$
\Delta_h = 0.90 - 11.70 = -10.80\ \text{mm},\qquad \Delta_v = 0.54 - 25.50 = -24.96\ \text{mm}
$$

**Answer: horizontal deflection of D $= 10.8$ mm towards the left; vertical deflection $= 24.96$ mm downward.** (The results assume the 20$^\circ$C face is on the outside of the frame for every member.)

## Check yourself

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

1. A member has +20 C on the outer face and +10 C on the inner face. The mean temperature rise T_m is:
   A. 10 C
   B. 15 C
   C. 20 C
   D. 30 C
   Answer: B. Linear variation gives mean (20 + 10)/2 = 15 C, which causes axial elongation. (Key point: Mean temperature)

2. The curvature caused by the temperature difference is:
   A. alpha dT / d = 6e-4 per m
   B. alpha T_m
   C. alpha dT x d
   D. dT / alpha
   Answer: A. kappa = 12e-6 x 10 / 0.2 = 6e-4 per m. (Key point: Curvature from gradient)

3. The deflection due to temperature is:
   A. Integral n alpha T_m ds + integral m (alpha dT / d) ds
   B. Integral M m / EI
   C. Sum F k L / AE
   D. alpha T_m only
   Answer: A. The axial term uses mean temperature and virtual n; the bending term uses curvature and virtual m. (Key point: Virtual work formula)

4. Which effect gives most of the deflection?
   A. Curvature effects, giving -11.70 mm and -25.50 mm
   B. Axial elongation
   C. Both equal
   D. Shear
   Answer: A. Curvature rotates the rest of the frame about each element, giving much larger movement than axial elongation. (Key point: Dominant effect)

## Related in Theory of Structures I

- [Frame deflection at C due to temperature](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-temperature-c-2074) — Theory of Structures I, chapter 3
- [Frame rotation and vertical deflection at free end](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-rotation-vertical-free-end-2073m) — Theory of Structures I, chapter 3
- [Frame horizontal deflection at free end](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-virtual-work-free-end-2077) — Theory of Structures I, chapter 3
- [Frame horizontal deflection at roller support](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-virtual-work-roller-2076) — Theory of Structures I, chapter 3
- [Frame vertical deflection at C](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-vertical-defl-c-2065) — Theory of Structures I, chapter 3
- [Cantilever tip deflection, temperature gradient](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-temperature-gradient-cantilever) — Theory of Structures I, chapter 3

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