# Beam Properties#

## Inertial properties#

Table 31 Inertial properties#

Keyword

Description

mu

Mass per unit length

mmoi1 | mmoi2 | mmoi3

Mass moment of inertia about x1/x2/x3 axis

msijo (i, j are numbers 1 to 6)

Entry (i, j) of the 6x6 mass matrix at the origin

msijc (i, j are numbers 1 to 6)

Entry (i, j) of the 6x6 mass matrix at the mass center

mcy | mc2

y (or x2) component of the mass center

mcz | mc3

z (or x3) component of the mass center

$\begin{split}\begin{bmatrix} \mu & 0 & 0 & 0 & \mu x_{M3} & -\mu x_{M2} \\ 0 & \mu & 0 & -\mu x_{M3} & 0 & 0 \\ 0 & 0 & \mu & \mu x_{M2} & 0 & 0 \\ 0 & -\mu x_{M3} & \mu x_{M2} & i_{22}+i_{33} & 0 & 0 \\ \mu x_{M3} & 0 & 0 & 0 & i_{22} & i_{23} \\ -\mu x_{M2} & 0 & 0 & 0 & i_{23} & i_{33} \end{bmatrix}\end{split}$

## Stiffness properties#

Table 32 Stiffness properties#

Keyword

Description

ea

Axial stiffness

gj

Torsional stiffness

eiyy | ei22

Principal bending stiffness around the $$y$$ ($$x_2$$) axis (flapwise)

eizz | ei33

Principal bending stiffness around the $$z$$ ($$x_3$$) axis (chordwise or lead-lag)

gayy | ga22

Principal shear stiffness in along the $$y$$ ($$x_2$$) axis

gazz | ga33

Principal shear stiffness in along the $$z$$ ($$x_3$$) axis

stfijc (i, j are numbers 1 to 4)

Entry (i, j) of the 4x4 classical stiffness matrix ($$C^b_{ij}$$)

stfijr (i, j are numbers 1 to 6)

Entry (i, j) of the 6x6 refined stiffness matrix ($$C^b_{ij}$$)

cmpijc (i, j are numbers 1 to 4)

Entry (i, j) of the 4x4 classical compliance matrix ($$S^b_{ij}$$)

cmpijr (i, j are numbers 1 to 6)

Entry (i, j) of the 6x6 refined compliance matrix ($$S^b_{ij}$$)

tcy | tc2

$$y$$ ($$x_2$$) component of the tension center

tcz | tc3

$$z$$ ($$x_3$$) component of the tension center

scy | sc2

$$y$$ ($$x_2$$) component of the shear center

scz | sc3

$$z$$ ($$x_3$$) component of the shear center

Constitutive relation of the Euler-Bernoulli beam model:

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

Constitutive relation of the Timoshenko beam model:

$\begin{split}\begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ M_1 \\ M_2 \\ M_3 \end{Bmatrix} = \begin{bmatrix} C^b_{11} & C^b_{12} & C^b_{13} & C^b_{14} & C^b_{15} & C^b_{16} \\ C^b_{12} & C^b_{22} & C^b_{23} & C^b_{24} & C^b_{25} & C^b_{26} \\ C^b_{13} & C^b_{23} & C^b_{33} & C^b_{34} & C^b_{35} & C^b_{36} \\ C^b_{14} & C^b_{24} & C^b_{34} & C^b_{44} & C^b_{45} & C^b_{46} \\ C^b_{15} & C^b_{25} & C^b_{35} & C^b_{45} & C^b_{55} & C^b_{56} \\ C^b_{16} & C^b_{26} & C^b_{36} & C^b_{46} & C^b_{56} & C^b_{66} \\ \end{bmatrix} \begin{Bmatrix} \gamma_{11} \\ \gamma_{12} \\ \gamma_{13} \\ \kappa_{11} \\ \kappa_{12} \\ \kappa_{13} \end{Bmatrix}\end{split}$
$\begin{split}\begin{Bmatrix} \gamma_{11} \\ \gamma_{12} \\ \gamma_{13} \\ \kappa_{11} \\ \kappa_{12} \\ \kappa_{13} \end{Bmatrix} = \begin{bmatrix} S^b_{11} & S^b_{12} & S^b_{13} & S^b_{14} & S^b_{15} & S^b_{16} \\ S^b_{12} & S^b_{22} & S^b_{23} & S^b_{24} & S^b_{25} & S^b_{26} \\ S^b_{13} & S^b_{23} & S^b_{33} & S^b_{34} & S^b_{35} & S^b_{36} \\ S^b_{14} & S^b_{24} & S^b_{34} & S^b_{44} & S^b_{45} & S^b_{46} \\ S^b_{15} & S^b_{25} & S^b_{35} & S^b_{45} & S^b_{55} & S^b_{56} \\ S^b_{16} & S^b_{26} & S^b_{36} & S^b_{46} & S^b_{56} & S^b_{66} \\ \end{bmatrix} \begin{Bmatrix} F_1 \\ F_2 \\ F_3 \\ M_1 \\ M_2 \\ M_3 \end{Bmatrix}\end{split}$