---
title: "Castigliano: fixed-end moment, 15 kN load"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:castigliano-fixed-moment-15kn-2m-3m"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/castigliano-fixed-moment-15kn-2m-3m
site: BCT Engineering
---

# Castigliano: fixed-end moment, 15 kN load

Theory of Structures II · Chapter 2: Force method

- Asked in: 2066 Jestha, 10 marks
- Video: [CASTIGLIANO'S THEOREM in Just Over 10 Minutes!](https://www.youtube.com/watch?v=JH6iOhOe8h0) — Less Boring Lectures
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/castigliano-fixed-moment-15kn-2m-3m
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Use Castigliano's theorem and find the moment at the fixed end of the propped cantilever loaded as shown below. EI is constant. [Figure: beam AB, A fixed, B roller; 15 kN at 2 m from A, B at 5 m (2 m + 3 m).]

## Short answer

Take R_B as redundant and set ∂U/∂R_B=0. This gives R_B=3.12 kN upward, hence the fixed-end moment is M_A=14.40 kNm hogging.

## Answer

**Method:** the propped cantilever is indeterminate to the 1st degree. Take the roller reaction $R_B$ (upward) as the redundant. B does not move, so by Castigliano's theorem $\partial U/\partial R_B=0$.

Data: $L=5$ m, 15 kN at 2 m from the fixed end A (so 3 m from B), EI constant.

#### Bending moment (x measured from B towards A, sagging +)
- $0\le x\le3$: $M=R_Bx$, $\;\partial M/\partial R_B=x$
- $3\le x\le5$: $M=R_Bx-15(x-3)$, $\;\partial M/\partial R_B=x$

#### Condition $\partial U/\partial R_B=0$
$$
\int_0^3R_Bx\cdot x\,dx+\int_3^5\left[R_Bx-15(x-3)\right]x\,dx=0
$$
$$
R_B\left[\frac{x^3}{3}\right]_0^5-15\int_3^5(x^2-3x)dx=0
\;\Rightarrow\; \frac{125}{3}R_B-15(8.667)=0
\;\Rightarrow\; R_B=\frac{130\times3}{125}=3.12\ \text{kN}\uparrow
$$

#### Fixed-end moment
At A ($x=5$):
$$
M_A=R_B(5)-15(5-3)=15.60-30=-14.40\ \text{kNm}
$$
The negative sign means hogging (tension at the top).

Check with the standard formula: $M_A=\dfrac{Pab(L+b)}{2L^2}=\dfrac{15\times2\times3\times(5+3)}{2\times25}=14.40$ kNm ✓. Also $R_A=15-3.12=11.88$ kN.

**Answer: fixed-end moment $M_A=14.40$ kNm (hogging); $R_B=3.12$ kN, $R_A=11.88$ kN.**

## Check yourself

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

1. Which quantity is taken as the redundant and which condition applies?
   A. R_B, with dU/dR_B = 0 since B does not move
   B. M_A, with dU/dM_A = 15
   C. R_A, with U = 0
   D. R_B, with dU/dR_B = 15
   Answer: A. The roller reaction is the single redundant; B has no deflection so dU/dR_B = 0. (Key point: Choice of redundant)

2. What is R_B?
   A. 3.12 kN up
   B. 11.88 kN up
   C. 7.5 kN up
   D. 15 kN
   Answer: A. (125/3) R_B = 15(8.667) gives R_B = 3.12 kN. (Key point: Value of the redundant)

3. What is the fixed-end moment M_A?
   A. 14.40 kNm hogging
   B. 15.60 kNm sagging
   C. 30 kNm hogging
   D. 6 kNm hogging
   Answer: A. M_A = 3.12(5) - 15(2) = -14.40 kNm, matching Pab(L+b)/2L^2. (Key point: Fixed end moment)

## Close variants set in other exams

- [Castigliano: moment at C of propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-moment-c-propped-30knm) — Theory of Structures II, chapter 2, asked 2×

## Related in Theory of Structures II

- [Castigliano: fixed-end moment, 15 kN/m, L = 5 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-propped-udl-15knm-5m) — Theory of Structures II, chapter 2, asked 2×
- [Castigliano: fixed moment, propped cantilever UDL](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-fixed-moment-udl-5knm-6m) — Theory of Structures II, chapter 2, asked 2×
- [Castigliano: fixed moment, propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-fixed-moment-propped-100kn-50kn) — Theory of Structures II, chapter 2
- [Castigliano: propped cantilever, moment and rotation](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-propped-30knm-50kn-moment-rotation) — Theory of Structures II, chapter 2
- [ILD: fixed-end moment of propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-reaction-moment-fixed-propped-10m) — Theory of Structures II, chapter 4, asked 2×
- [ILD: moment at support B, fixed-propped beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-moment-b-propped-2l-l) — Theory of Structures II, chapter 4

## Related in other subjects

- [Propped cantilever with UDL, two elements](https://www.bctengineer.com/question.md?key=semester_8%2Fcomputational_techniques_in_civil_engineering%3Apropped-cantilever-two-elements) — Computational Techniques in Civil Engineering, chapter 4
- [Propped cantilever by strain energy method](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Apropped-cantilever-redundant-reaction) — Mechanics of Solids, chapter 5
- [Moment area: overhang beam free-end deflection](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Amoment-area-overhang-10kn-m-2078) — Theory of Structures I, chapter 4
- [Propped cantilever with UDL by integration](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Apropped-cantilever-udl-prop-reaction) — Strength of Materials, chapter 7