---
title: "Euler's formula: both ends hinged"
subject: "Strength of Materials"
chapter: 8
key: "semester_3/strength_of_materials:euler-both-ends-hinged"
source: https://www.bctengineer.com/q/semester_3/strength_of_materials/euler-both-ends-hinged
site: BCT Engineering
---

# Euler's formula: both ends hinged

Strength of Materials · Chapter 8: Column Theory

- Asked in: 2076 Chaitra, 6 marks; 2073 Shrawan, 8 marks; 2071 Chaitra, 12 marks; 2068 Chaitra, 6 marks (asked 4 times)
- Video: [EULER'S FORMULA OF COLUMN /Euler's Theory Buckling Load (for COLUMN BOTH END HINGED)](https://www.youtube.com/watch?v=p5S_v-_sFo4) — TECHNICAL CLASSES
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/strength_of_materials/euler-both-ends-hinged
- Whole chapter: https://www.bctengineer.com/md/semester_3/strength_of_materials/chapter-8

## Question

Derive the Euler's formula for critical load for a strut with both end hinged.

## Short answer

From EI y'' = -Py, the solution with hinged ends yields the critical load P_cr = π²EI/L². The effective length equals the actual length L, and buckling occurs in a half-sine wave.

## Answer

**Euler's critical (crippling) load** is the smallest axial load at which an initially straight, slender column becomes unstable and buckles sideways.

#### Assumptions
The column is perfectly straight, of uniform section and homogeneous material, the load is perfectly axial, the material obeys Hooke's law, deflections are small, and the effect of shear and axial shortening is ignored.

#### Derivation
Let the column AB of length $L$ be hinged at both ends, with the load $P$ producing a small deflection $y$ at distance $x$ from A.

```text
   P
   |  B
   |  /\
   |  \ y
   |  /
   A /
   P
 (A origin, x along column)
```

The bending moment at the section is $M = -Py$ (the moment of $P$ tends to reduce the deflection, so it is opposite in sign to $y$). Using the differential equation of the elastic curve:

$$
EI\frac{d^2y}{dx^2} = -Py \;\Rightarrow\; \frac{d^2y}{dx^2} + k^2 y = 0,\qquad k^2 = \frac{P}{EI}
$$

The solution is

$$
y = C_1\cos kx + C_2\sin kx
$$

Boundary conditions:
- At $x = 0$, $y = 0 \Rightarrow C_1 = 0$.
- At $x = L$, $y = 0 \Rightarrow C_2\sin kL = 0$.

For a buckled shape $C_2 \ne 0$, so $\sin kL = 0$ and $kL = n\pi$. The smallest non-zero load is for $n = 1$:

$$
k L = \pi \;\Rightarrow\; \frac{PL^2}{EI} = \pi^2
$$

$$
P_{cr} = \frac{\pi^2 EI}{L^2}
$$

Here $I$ is the least moment of inertia, since the column buckles about the axis of least resistance. The shape is a half sine wave, $y = C_2\sin(\pi x/L)$, with maximum deflection at mid-height.

## Check yourself

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

1. Which is an assumption of Euler's column theory?
   A. Column has initial crookedness
   B. Load is applied with eccentricity
   C. Material is stressed beyond the proportional limit
   D. Perfectly straight column, axial load and Hooke's law
   Answer: D. Euler's theory assumes an ideal column and elastic material. (Key point: Assumptions)

2. What equation is used for the buckled hinged column?
   A. EI y'' = -P y, giving y'' + k^2 y = 0
   B. EI y'' = P
   C. EI y'' = -P x
   D. EI y'' = -M0 - Py
   Answer: A. Moment at a section is P times the lateral deflection y. (Key point: Differential equation)

3. Which boundary conditions give kL = n pi?
   A. Slope zero at both ends
   B. y = 0 at x = 0 and x = L, so C1 = 0 and sin kL = 0
   C. y = 0 at x = 0 only
   D. Moment zero at mid span
   Answer: B. Hinged ends have zero deflection, so sin kL = 0. (Key point: Boundary conditions)

4. What is the critical load and which I is used?
   A. 4 pi^2 EI/L^2 with greatest I
   B. pi^2 EI/(2L)^2
   C. pi^2 EI/L^2 with the least moment of inertia
   D. pi^2 EI/L with I about any axis
   Answer: C. Smallest n = 1 gives P_cr = pi^2EI/L^2, and the column buckles about the weaker axis. (Key point: Critical load)

## Related in Strength of Materials

- [Euler's formula both ends hinged, with limitations](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-hinged-limitation) — Strength of Materials, chapter 8
- [Euler's formula: fixed-hinged strut with limitation](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-one-fixed-one-hinged-limitation) — 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
- [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: one end fixed, other hinged](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-one-fixed-one-hinged) — Strength of Materials, chapter 8, asked 2×
- [Buckling load, effective length, Euler hinged](https://www.bctengineer.com/question.md?key=semester_3%2Fstrength_of_materials%3Aeuler-both-ends-hinged-define-buckling) — Strength of Materials, chapter 8

## Related in other subjects

- [Euler's buckling load for columns](https://www.bctengineer.com/question.md?key=semester_4%2Fstrength_of_materials_mechanical%3Aeuler-crippling-load-pinned-column) — Strength of Materials, chapter 10