Euler–Bernoulli beam equation

Assumptions

• Beam cross-sections remain perpendicular to the beam axis after the deformation takes place.

Internal Forces

${\frac {dV}{dx}}=-w(x)$

${\frac {dM}{dx}}=V$

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

Elastic Curve

$d\theta ={\frac {M}{EI}}dx$

${\frac {d^{2}v}{dx^{2}}}={\frac {M}{EI}}$

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

References

• Russell C. Hibbeler: Structural Analysis, Third Edition, 1995 (p. 345, Elastic Beam Theory)

External Links

• Euler–Bernoulli beam equation at Wikipedia