Euler-Bernoulli Beam Theory formula |
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| \( \dfrac{ d^2 }{ dx^2 } \cdot \left( E \cdot I \cdot \dfrac{ d^2 \cdot w(x) }{ dx^2 } \right) \;=\; q(x) \) (Euler-Bernoulli Beam Theory) | ||
| Symbol | English | Metric |
| \( q(x) \) = Distributed Load per Unit Length | \(lbf\;/\;ft\) | \(N\;/\;m\) |
| \( d \) = Derivative Operator | \(dimensionless\) | \(dimensionless\) |
| \( x \) = Position Along the Beam's Axis | \(in\) | \(mm\) |
| \( dx \) = Derivative Change in x | \(in\) | \(mm\) |
| \( \dfrac{d}{dx} \) = Derivative Rate of Change | \(in\) | \(mm\) |
| \( E \) = Young's Modulus of the Material | \(lbf\;/\;in^2\) | \(Pa\) |
| \( I \) = Area Moment of Inertia of the Beam's Cross-section | \(in^4\) | \(mm^4\) |
| \( w(x) \) = Transverse Deflection at Position x | \(in\) | \(mm\) |
Euler–Bernoulli beam theory, a theory in structural and mechanical engineering, is used to describe how slender beams bend under applied loads. It assumes that plane cross-sections of the beam remain plane and perpendicular to the neutral axis after deformation, meaning shear deformation is neglected. This theory relates the bending moment in the beam to its curvature through the material’s elastic modulus and the beam’s second moment of area, allowing engineers to predict deflections, slopes, and stresses in beams subjected to forces. Euler–Bernoulli beam theory is most accurate for long, thin beams where bending effects dominate and shear effects are relatively small.
