---
title: "Frame vertical deflection at C"
subject: "Theory of Structures I"
chapter: 3
key: "semester_4/theory_of_structures_i:frame-vertical-defl-c-2065"
source: https://www.bctengineer.com/q/semester_4/theory_of_structures_i/frame-vertical-defl-c-2065
site: BCT Engineering
---

# Frame vertical deflection at C

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

- Asked in: 2065 Chaitra, 8 marks
- Video: [Deflection of Frames using Principle of Virtual Work - Intro to Structural Analysis](https://www.youtube.com/watch?v=JAD-8ATyu6k) — Structures with Prof. H
- 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-vertical-defl-c-2065
- Whole chapter: https://www.bctengineer.com/md/semester_4/theory_of_structures_i/chapter-3

## Question

Determine the vertical deflection of point C of the frame shown in fig-1. $E = 200\ \text{kN/mm}^2$, $I = 30\times10^6\ \text{mm}^4$. [Figure: frame with fixed support A at the base of the column AB (4 m high, 2I) and horizontal member BC (3 m, I); 10 kN downward load at the free end C.]

## Short answer

Using virtual work with EI = 6000 kN·m²: the vertical deflection at point C is 270/(EI) = 0.045 m or 45 mm downward.

## Answer

#### Given data
Frame: fixed at A (base of column AB), column AB: 4 m, $2I$; horizontal member BC: 3 m, $I$; 10 kN downward at the free end C. $E = 200$ kN/mm$^2 = 2\times10^8$ kN/m$^2$, $I = 30\times10^6$ mm$^4 = 3\times10^{-5}$ m$^4$.

$$
EI = 2\times10^8\times3\times10^{-5} = 6000\ \text{kNm}^2
$$

#### Virtual work (unit vertical load at C)
Distances $s$ from C along CB, then $s$ from B down BA.

| Member | $s$ (m) | $M$ (kNm) | $m$ | $\int Mm\,ds$ | stiffness | contribution |
|---|---|---|---|---|---|---|
| CB | 0-3 | $10s$ | $s$ | $10\times9 = 90$ | $EI$ | 90 |
| BA | 0-4 | $30$ | 3 | $30\times3\times4 = 360$ | $2EI$ | 180 |

Hence

$$
\Delta_{Cv} = \frac{90 + 180}{EI} = \frac{270}{6000} = 0.045\ \text{m}
$$

**Answer: vertical deflection at C $= 45$ mm downward.**

## Check yourself

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

1. With distance s from C, what are M and m for member CB?
   A. M = 10s and m = 1
   B. M = s and m = 10
   C. M = 30 and m = 3
   D. M = 10s and m = s
   Answer: D. The 10 kN load at C gives moment 10s, and the unit load gives s. (Key point: Moments in CB)

2. What are M and m in column BA?
   A. M = 10s, m = s
   B. M = 0
   C. M = 30 kNm and m = 3, both constant
   D. M = 40 and m = 4
   Answer: C. The vertical loads act at a 3 m horizontal distance from column BA, giving constant moments 10 x 3 = 30 and 1 x 3 = 3. (Key point: Moments in BA)

3. Why is the BA integral 360 divided by 2EI = 180?
   A. The column is ignored
   B. The column is twice as long
   C. The load is halved
   D. The column has stiffness 2EI, so its integral of Mm is divided by 2EI
   Answer: D. Each member's integral is divided by its own EI. (Key point: Stiffness)

4. What is the deflection of C?
   A. 4.5 mm downward
   B. 45 mm downward
   C. 270 mm
   D. 0.45 mm
   Answer: B. Delta = 270/EI = 270/6000 = 0.045 m, downward. (Key point: Result)

## Related in Theory of Structures I

- [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 D](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aframe-vw-d-vertical-2071o) — Theory of Structures I, chapter 3
- [Overhang beam free end deflection by virtual work](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Abeam-vw-free-end-d-2065) — Theory of Structures I, chapter 3
- [Conjugate beam: slope and deflection at free end](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-beam-24kn-10knm) — Theory of Structures I, chapter 4

## Related in other subjects

- [Slope deflection: beam, 360 kN load, fixed end](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aslope-deflection-beam-10kn-20knm-360kn) — Theory of Structures II, chapter 3
- [Rotation and deflection at free end, BMD](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Abeam-fem-rotation-deflection-free-end) — Computational Techniques in Civil Engineering, chapter 4
- [Frame AFD, SFD, BMD (5 and 10 kN/m, hinge)](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aframe-afd-sfd-bmd-5knm-10knm) — Strength of Materials, chapter 1, asked 2×
- [Round bar fixed-free: buckling load from lateral deflection](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Around-bar-fixed-free-buckling-3m) — Strength of Materials, chapter 8, asked 2×