The Discrete Kirchoff-Mindlin Quadrilateral plate-bending formulation is an isoparametric thick-shell formulation for plate bending behavior.

References

1. Katili, I. "A New Discrete Kirchoff-Mindlin Element Based on Mindlin-Reissner Plate Theory and Assumed Shear Strain Fields - Part I: An Extended DKQ Element for Thick-Plate Bending Analysis." International Journal for Numerical Methods in Engineering. Vol. 36, pp. 1885-1908. (1993)
2. Maknun, I. J., Katili, I., Purnomo, H. "Development of the DKMT Element for Error Estimation in Composite Plate Structures." International Journal of Technology. Vol. 5, pp. 780-789. (2015)

Degrees of Freedom

The DKMQ formulation contains three degrees-of-freedom (DoFs) at each node:

1. z-translation
2. x-rotation
3. y-rotation

There are four vertex nodes for this element. These nodes are numbered from 1 to 4 in a counter-clockwise direction. An additional four mid-side nodes are introduced during the formulation, but they do not appear in the final stiffness matrix.

The local stiffness matrix is formulated with the following degree-of-freedom order:

$\begin{bmatrix}w_1 & \theta_{1,x} & \theta_{1,y} & w_2 & \theta_{2,x} & \theta_{2,y} & w_3 & \theta_{3_x} & \theta_{3,y} & w_4 & \theta_{4,x} & \theta_{4,y} \end{bmatrix}$

Where:

$w_k=$Local z-translation at node $k$

$\theta_{k,x}=$Rotation in the direction of the x-axis (about the local y-axis) at node $k$

$\theta_{k,y}=$Rotation in the direction of the y-axis (about the local x-axis) at node $k$

Rotation and Sign Convention

This element formulation uses a shell rotation and sign convention, where the direction associated with the rotation indicates which direction the bending is towards, NOT the axis about which bending is occurring. Thus, $\theta_x$ denotes rotation in the direction of the x-axis, which is actually rotation about the y-axis. Similarly, $\theta_y$ denotes rotation in the direction of the y-axis, which is actually rotation about the x-axis. Because of this, the sign convention between standard right-hand rule rotations (as is used in the global formulation) and the local convention differ.

The degree-of-freedom mapping at any node between global and local systems is:

$Map=\begin{bmatrix} w \\ \theta_x \\ \theta_y \end{bmatrix}=\begin{bmatrix} U_z \\ Q_y \\ -Q_x \end{bmatrix}$