---
title: "Propped cantilever boundary conditions"
subject: "Theory of Structures II"
chapter: 3
key: "semester_5/theory_of_structures_ii:propped-cantilever-boundary-conditions"
source: https://www.bctengineer.com/q/semester_5/theory_of_structures_ii/propped-cantilever-boundary-conditions
site: BCT Engineering
---

# Propped cantilever boundary conditions

Theory of Structures II · Chapter 3: Displacement method

- Asked in: 2071 Chaitra, 2 marks
- Video: [Reaction Calculation in Propped Cantilever](https://www.youtube.com/watch?v=nbCkXQEtBtA) — GATE GURUS Exam Prep: GATE, ESE, SSC-JE & AE
- 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/propped-cantilever-boundary-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-3

## Question

Draw a propped cantilever and write down its boundary conditions.

## Short answer

A propped cantilever has one fixed end and one simply supported end. At the fixed end, deflection y=0 and slope dy/dx=0; at the propped end, deflection y=0 and bending moment M=0 (no moment resistance).

## Answer

A **propped cantilever** is a beam that is fixed at one end and simply supported (propped by a roller or hinge) at the other end. It is statically indeterminate to the first degree.

```text
  fixed |============================== o  (roller/prop)
   A    x = 0                      x = L   B
```

#### Boundary conditions (A fixed at $x=0$, B propped at $x=L$)
| End | Condition | Meaning |
|---|---|---|
| A (fixed) | $y(0)=0$ | no deflection |
| A (fixed) | $\dfrac{dy}{dx}\Big|_{x=0}=0$ | no rotation |
| B (prop) | $y(L)=0$ | the prop prevents deflection |
| B (prop) | $M(L)=EI\dfrac{d^2y}{dx^2}\Big|_{x=L}=0$ | no bending moment at a simple support |

Rotation at B is free and the moment at A is a reaction that is not zero. These four conditions fix the four constants of integration of $EI\,d^4y/dx^4=w(x)$. In the double-integration method ($EI\,d^2y/dx^2=M(x)$) the condition $M(L)=0$ is already built into $M(x)$, and the other three conditions give the two constants of integration and the redundant prop reaction $R_B$. For a UDL $w$ they lead to $R_B=\dfrac{3wL}{8}$ and $M_A=\dfrac{wL^2}{8}$ (hogging).

## Check yourself

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

1. A propped cantilever is?
   A. Free at both ends
   B. Hinged at both ends
   C. Fixed at one end, simply supported at the other
   D. Fixed at both ends
   Answer: C. One fixed end plus one prop gives one degree of indeterminacy. (Key point: Definition)

2. Conditions at the propped end x = L?
   A. y = 0 and slope = 0
   B. M = 0 and V = 0
   C. slope = 0 only
   D. y = 0 and M = 0
   Answer: D. The prop prevents deflection; a simple support carries no moment. (Key point: Prop end)

3. Reaction at the prop under UDL w?
   A. 3wL/8
   B. wL/2
   C. 5wL/8
   D. wL/8
   Answer: A. R_B = 3wL/8 with M_A = wL^2/8 hogging. (Key point: UDL result)

## Related in Theory of Structures II

- [ILD: shear at section 2-2, propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-shear-section-2-2-propped) — Theory of Structures II, chapter 4, asked 3×
- [Castigliano: prop reaction of propped cantilever](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acastigliano-prop-reaction-udl-general) — 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: 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×
- [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: reaction at B, propped cantilever 10 m](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aild-reaction-b-propped-10m-2m) — Theory of Structures II, chapter 4, asked 2×

## Related in other subjects

- [Essential and natural boundary conditions](https://www.bctengineer.com/question.md?key=semester_8%2Ffinite_element_method%3Aessential-natural-boundary-conditions) — Finite Element Method, chapter 2
- [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 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
- [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