Beam equations

Internal Forces

<tex> \frac{dV}{dx} = - w(x) </tex>

<tex> \frac{dM}{dx} = V </tex>

• These relationships are derived from equilibrium of an infinitesimal portion of the beam.

Elastic Curve

<tex> d\theta = \frac{M}{EI} dx </tex>

<tex> \frac{d^{2}v}{dx^2} = \frac{M}{EI} </tex>

• There relationships are derived from geometry of deformed infinitesimal portion of the beam.

Assumptions

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Derivation

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Notation

• <tex>w(x)</tex> ... distributed load along the girder
• <tex>v</tex> ... deflection
• <tex>\theta</tex> ... slope of elastic curve
• <tex>M</tex> ... moment
• <tex>V</tex> ... shear
• <tex>EI</tex> ... flexural rigidity

See Also

• Euler–Bernoulli beam equation
• Timoshenko beam theory