Mechanics of deformable solids Definition

Euler–Bernoulli beam equation

Builds on Beam theory Bending moment Second moment of area Young's modulus

Definition

Let a straight beam occupy $0 \le x \le L$ along its axis. Let $E(x) > 0$ be the Young's modulus of its material and $I(x) > 0$ the second moment of area of its cross-section about the neutral axis. Let $q(x)$ be the transverse load per unit length and $w(x)$ the deflection of the axis, both measured in the same direction. The Euler–Bernoulli beam equation is $\frac{d^2}{dx^2}\left( E I \frac{d^2 w}{dx^2} \right) = q.$ The bending moment and the shear force are $M = -E I w''$ and $V = M'$ . For a beam with mass $\mu(x)$ per unit length the dynamic equation is $\frac{\partial^2}{\partial x^2}\left( E I \frac{\partial^2 w}{\partial x^2} \right) + \mu \frac{\partial^2 w}{\partial t^2} = q(x, t).$

Remarks

Example: a cantilever with constant $EI$ , clamped at $x = 0$ and loaded by a point force $P$ at $x = L$ , has $w(x) = P x^2 (3L - x) / (6EI)$ . Check: $w'''' = 0$ on $(0, L)$ , $w(0) = w'(0) = 0$ , $w''(L) = 0$ so $M(L) = 0$ , and $V = -EI w''' = P$ . The tip deflection is $w(L) = PL^3/(3EI)$ . A simply supported beam with constant $EI$ under a uniform load $q$ has the largest deflection $5qL^4/(384EI)$ at mid-span.

Non-example: $w(x) = x^2$ with $q = 0$ and constant $EI$ satisfies the equation. It fails the condition $w(L) = 0$ of a simply supported beam, so it is not the deflection of such a beam.

The usual boundary conditions are $w = w' = 0$ at a clamped end, $w = w'' = 0$ at a simply supported end, and $w'' = w''' = 0$ at a free end when $EI$ is constant. The equation assumes small deflections and neglects shear deformation; Timoshenko–Ehrenfest beam theory adds the shear.

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Beam theory concept
Bending moment concept
Second moment of area concept
Young's modulus concept
Euler–Bernoulli beam equation definition

Tag	`CPMD36AH43`
URI	`m2://CPMD36AH43`
Kind	definition
Writing	W0 stub
MSC	74K10, 74B05, 34B60
Defines	Euler–Bernoulli beam equation
Uses	Beam theory, Bending moment, Second moment of area, Young's modulus
Changed	3 November 2026

Euler–Bernoulli beam equation

Definition

Remarks

Reading path

Further reading

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