.. _section-beam_properties:


Beam Properties
=================

.. figure:: /figures/beam_prop_frame.png
  :name: fig_beam_prop_frame
  :width: 6in
  :align: center

  Reference frames of beam properties.



Inertial properties
--------------------

..  list-table:: Inertial properties
    :header-rows: 1

    * - 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


..  math::

    \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}



Stiffness properties
---------------------

..  list-table:: Stiffness properties
    :header-rows: 1

    * - Keyword
      - Description
    * - ``ea``
      - Axial stiffness
    * - ``gj``
      - Torsional stiffness
    * - ``eiyy`` | ``ei22``
      - Principal bending stiffness around the |y| (|x2|) axis (flapwise)
    * - ``eizz`` | ``ei33``
      - Principal bending stiffness around the |z| (|x3|) axis (chordwise or lead-lag)
    * - ``gayy`` | ``ga22``
      - Principal shear stiffness in along the |y| (|x2|) axis
    * - ``gazz`` | ``ga33``
      - Principal shear stiffness in along the |z| (|x3|) axis
    * - ``stfijc`` (``i``, ``j`` are numbers 1 to 4)
      - Entry (i, j) of the 4x4 classical stiffness matrix (:math:`C^b_{ij}`)
    * - ``stfijr`` (``i``, ``j`` are numbers 1 to 6)
      - Entry (i, j) of the 6x6 refined stiffness matrix (:math:`C^b_{ij}`)
    * - ``cmpijc`` (``i``, ``j`` are numbers 1 to 4)
      - Entry (i, j) of the 4x4 classical compliance matrix (:math:`S^b_{ij}`)
    * - ``cmpijr`` (``i``, ``j`` are numbers 1 to 6)
      - Entry (i, j) of the 6x6 refined compliance matrix (:math:`S^b_{ij}`)
    * - ``tcy`` | ``tc2``
      - |y| (|x2|) component of the tension center
    * - ``tcz`` | ``tc3``
      - |z| (|x3|) component of the tension center
    * - ``scy`` | ``sc2``
      - |y| (|x2|) component of the shear center
    * - ``scz`` | ``sc3``
      - |z| (|x3|) component of the shear center


Constitutive relation of the Euler-Bernoulli beam model:

..  math::

    \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}

..  math::

    \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}


Constitutive relation of the Timoshenko beam model:

..  math::

    \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}

..  math::

    \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}