# Euler-Bernoulli Beam Equation

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## Theorem

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- $q = \map {\dfrac {\d^2} {\d x^2} } {E I \dfrac {\d^2 w} {\d x^2} }$

where:

- $q$ is the bending moment
- $E$ is Young's modulus
- $\dfrac {\d^2 w} {\d x^2}$ is the curvature
- $I$ is the moment of inertia of the cross-section about an axis through the center of mass and perpendicular to the plane of the couple.

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## Proof

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## Also known as

The **Euler-Bernoulli Beam Equation** is also known as the **Euler-Bernoulli Law**.

## Source of Name

This entry was named for Leonhard Paul Euler and Daniel Bernoulli.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text A$: Euler - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Euler-Bernoulli law** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.21$: Euler ($\text {1707}$ – $\text {1783}$)