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

# Boundary conditions of fixed-fixed beam

Theory of Structures II · Chapter 3: Displacement method

- Asked in: 2072 Kartik, 2 marks
- Video: [3  Boundary Conditions of a beam](https://www.youtube.com/watch?v=b-HkdI-sySU) — NITIANS 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/fixed-beam-boundary-conditions
- Whole chapter: https://www.bctengineer.com/md/semester_5/theory_of_structures_ii/chapter-3

## Question

Write down the boundary conditions for a single span beam fixed at both ends.

## Short answer

For a beam fixed at both ends (x=0 and x=L), the boundary conditions are zero deflection and zero slope at each support: y(0)=0, y'(0)=0, y(L)=0, y'(L)=0. These four conditions determine constants, and end moments correspond to fixed-end moments for the loading.

## Answer

A beam fixed at both ends (A at $x=0$ and B at $x=L$) cannot move or rotate at either support. For the elastic curve $y(x)$:

| End | Deflection | Slope |
|---|---|---|
| A ($x=0$) | $y=0$ | $\dfrac{dy}{dx}=0$ |
| B ($x=L$) | $y=0$ | $\dfrac{dy}{dx}=0$ |

So there are **four boundary conditions**: $y(0)=0$, $y'(0)=0$, $y(L)=0$, $y'(L)=0$. They are enough to evaluate the four constants of integration of $EI\,\dfrac{d^4y}{dx^4}=w(x)$, and they are the compatibility conditions used in the force method. The end moments $M_A$ and $M_B$ and the shears are unknown reactions (the support moments are non-zero).

In the stiffness method the same statement reads $\theta_A=\theta_B=0$ and $v_A=v_B=0$ (no end displacements and no end rotations), so the end moments are the fixed-end moments of the loading, for example $\mp wL^2/12$ for a UDL.

## Check yourself

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

1. Boundary conditions of a beam fixed at both ends?
   A. y = 0 and M = 0 at both ends
   B. y' = 0 only
   C. M = 0 and V = 0
   D. y = 0 and y' = 0 at x = 0 and x = L
   Answer: D. Fixed supports prevent both deflection and rotation. (Key point: Conditions)

2. How many boundary conditions, and why needed?
   A. Four, to find the four integration constants
   B. Two
   C. Three
   D. Six
   Answer: A. EI d^4y/dx^4 = w needs four constants, found from the four conditions. (Key point: Number)

3. The end moments of a fixed beam are?
   A. wL/2
   B. Non-zero reactions, e.g. wL^2/12 for UDL
   C. Zero
   D. wL^2/8
   Answer: B. Fixed supports develop moments; for a UDL the fixed-end moments are wL^2/12. (Key point: End moments)

## Related in Theory of Structures II

- [Slope deflection: three-span beam with settlement](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Aslope-deflection-three-span-fixed-ends) — Theory of Structures II, chapter 3
- [Collapse load of fixed beam with UDL](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-fixed-beam-udl) — Theory of Structures II, chapter 5
- [Fixed beam: end moments from support rotation or settlement](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Afixed-beam-end-moments-rotation-settlement) — Theory of Structures II, chapter 3, asked 4×
- [Propped cantilever boundary conditions](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Apropped-cantilever-boundary-conditions) — Theory of Structures II, chapter 3
- [Three moment equation: derive and apply to fixed beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Athree-moment-derive-fixed-beam-udl) — Theory of Structures II, chapter 2
- [Collapse load of two-span beam](https://www.bctengineer.com/question.md?key=semester_5%2Ftheory_of_structures_ii%3Acollapse-load-two-span-beam-20kn) — Theory of Structures II, chapter 5

## Related in other subjects

- [Euler's formula: both ends fixed](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-fixed) — Strength of Materials, chapter 8, asked 2×
- [Euler's formula both ends fixed, with limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-fixed-limitation) — Strength of Materials, chapter 8
- [Boundary conditions real to conjugate beam](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aconjugate-boundary-conditions) — Theory of Structures I, chapter 4
- [Bar fixed at both ends: load shares, displacements](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Acircular-bar-fixed-both-ends-50kn) — Strength of Materials, chapter 3