---
title: "Castigliano: prop reaction of propped cantilever"
subject: "Theory of Structures II"
chapter: 2
key: "semester_5/theory_of_structures_ii:castigliano-prop-reaction-udl-general"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/castigliano-prop-reaction-udl-general
site: BCT Engineering
---

# Castigliano: prop reaction of propped cantilever

Theory of Structures II · Chapter 2: Force method

- Asked in: 2074 Chaitra, 4 marks; 2070 Chaitra (old course), 6 marks (asked 2 times)
- Video: [Strain Energy Method - Analysis of Propped Cantilever Beam ( With Uniformly Distributed Load )](https://www.youtube.com/watch?v=ZdP5LJ-H_Gk) — Stan Academy
- 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-prop-reaction-udl-general
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-2

## Question

Calculate the reaction at the prop of a propped cantilever with uniform distributed load throughout the span using Castigliano's theorem.

## Short answer

Treat the prop reaction R_B as redundant. Taking origin at B, moment M = R_B x – w x²/2, ∂M/∂R_B = x. Setting ∂U/∂R_B = 0 (zero deflection at B) gives ∫_0^L (R_B x² – w x³/2) dx = 0, yielding R_B = 3wL/8 upward.

## Answer

Take a propped cantilever of span $L$, fixed at A and with a prop (roller) at B, carrying a UDL $w$ per unit length. Treat the prop reaction $R_B$ as the redundant.

```text
 w kN/m
 vvvvvvvvvvvvvvvvvvv
 ||==================o
 A (fixed)           B (prop, R_B)
 |<------- L ------->|
```

Take $x$ from B towards A.

$$
M_x = R_B\,x - \frac{w x^2}{2}, \qquad \frac{\partial M_x}{\partial R_B} = x
$$

The prop is at the same level as the fixed end, so the deflection at B is zero:

$$
\frac{\partial U}{\partial R_B} = \int_0^L \frac{M_x}{EI}\frac{\partial M_x}{\partial R_B}\,dx = 0
$$

$$
\int_0^L \left(R_B\,x^2 - \frac{w x^3}{2}\right)dx = 0
$$

$$
\frac{R_B L^3}{3} - \frac{w L^4}{8} = 0
$$

$$
R_B = \frac{3wL}{8}
$$

#### Other reactions
$$
R_A = wL - \frac{3wL}{8} = \frac{5wL}{8}
$$

$$
M_A = \frac{3wL}{8}\,L - \frac{wL^2}{2} = -\frac{wL^2}{8}
$$

The fixed-end moment is $wL^2/8$ (hogging).

**Answer: prop reaction $R_B = \dfrac{3wL}{8}$ upward.**

## Check yourself

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

1. Which quantity is taken as redundant?
   A. Prop reaction R_B
   B. Fixed-end shear
   C. Load w
   D. Span L
   Answer: A. Releasing the prop leaves a determinate cantilever. (Key point: Redundant)

2. The condition used is:
   A. dU/dR_B = 0 as deflection at the level prop is zero
   B. dU/dw = 0
   C. M = 0 at A
   D. R_B = wL
   Answer: A. The prop is at the same level, so deflection at B is zero. (Key point: Condition)

3. With x from B, M_x and dM/dR_B are:
   A. R_B x - w x^2/2 and x
   B. R_B x and 1
   C. w x and x
   D. -wL and 0
   Answer: A. Moment from the prop reaction minus UDL; derivative with respect to R_B is x. (Key point: Moment expression)

4. Result R_B and fixed-end moment are:
   A. 3wL/8 and wL^2/8
   B. wL/2 and wL^2/12
   C. 5wL/8 and wL^2/8
   D. 3wL/8 and wL^2/2
   Answer: A. R_B = 3wL/8, R_A = 5wL/8, M_A = wL^2/8. (Key point: Result)

## Related in Theory of Structures II

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- [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
- [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×
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- [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×
- [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

## Related in other subjects

- [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
- [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
- [Cantilever deflection and slope by Castigliano](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Acastigliano-cantilever-slope-deflection) — Mechanics of Solids, chapter 5
- [Castigliano's theorem and unit load method](https://www.bctengineer.com/question.md?key=semester_5%2Fmechanics_of_solids%3Acastigliano-unit-load-theorems) — Mechanics of Solids, chapter 5