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