Beam-Column Equation |

The out-of-plane transverse displacement where If |

Origin of the Beam-Column Equation |

Similar to the Euler-Bernoulli beam equation, the beam-column equation arises from four distinct subsets of beam-column theory: kinematics, consitutive, force resultants, and equilibrium.
The outcome of each of these segments is summarized as follows: |

Kinematics: |
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Constitutive: |
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Resultants: |
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Equilibrium: |

In the equilibrium equations, N is the axial force resulting acting in a tensile manner (opposite in direction to the compressive resultant f).
To relate the beam's out-of-plane displacement |

Kinematics | => | Constitutive | => | Resultants | => | Equilibrium | => | Beam-Column Equation |

This hierarchy can be demonstrated by working backwards. First combine the two equilibrium equations to eliminate
Next replace the moment resultant
Use the constitutive relation to eliminate s in favor of the strain e, and then use kinematics to replace e in favor of the normal displacement
As a final step, recognizing that the integral over
We arrive at the beam-column equation based on the Euler-Bernoulli beam theory,
Since columns are usually used as compression members, engineers may be more familiar with the axial |