Euler-Bernoulli Beam Model

BM1. Classical beam theory: transverse shear deformation is neglected, so the sectional constitutive relation is 4x4.

Stiffness

\[\begin{Bmatrix} F_1 \ M_1 \ M_2 \ M_3 \end{Bmatrix} = \begin{bmatrix} C^b_{11} & C^b_{12} & C^b_{13} & C^b_{14} \ C^b_{12} & C^b_{22} & C^b_{23} & C^b_{24} \ C^b_{13} & C^b_{23} & C^b_{33} & C^b_{34} \ C^b_{14} & C^b_{24} & C^b_{34} & C^b_{44} \end{bmatrix} \begin{Bmatrix} \gamma_{11} \ \kappa_{11} \ \kappa_{12} \ \kappa_{13} \end{Bmatrix}\]

Compliance

\[\begin{Bmatrix} \gamma_{11} \ \kappa_{11} \ \kappa_{12} \ \kappa_{13} \end{Bmatrix} = \begin{bmatrix} S^b_{11} & S^b_{12} & S^b_{13} & S^b_{14} \ S^b_{12} & S^b_{22} & S^b_{23} & S^b_{24} \ S^b_{13} & S^b_{23} & S^b_{33} & S^b_{34} \ S^b_{14} & S^b_{24} & S^b_{34} & S^b_{44} \end{bmatrix} \begin{Bmatrix} F_1 \ M_1 \ M_2 \ M_3 \end{Bmatrix}\]