# Bilinear Mindlin Plate Element

Most of the following explanation concerns four node plate elements, 図 1. 3-Node Shell Elements explains the three node plate element, shown in 図 2.

Plate theory assumes that one dimension (the thickness, z) of the structure is small compared to the other dimensions. Hence, the 3D continuum theory is reduced to a 2D theory. Nodal unknowns are the velocities $\left({v}_{{x}^{\prime }}{v}_{{y}^{\prime }}{v}_{z}\right)$ of the midplane and the nodal rotation rates $\left({\omega }_{{x}^{\prime }}{\omega }_{y}\right)$ as a consequence of the suppressed z direction. The thickness of elements can be kept constant, or allowed to be variable. This is user defined. The elements are always in a state of plane stress, that is ${\sigma }_{zz}=0$ , or there is no stress acting perpendicular to the plane of the element. A plane orthogonal to the midplane remains a plane, but not necessarily orthogonal as in Kirchhoff theory, (where ${\epsilon }_{xz}={\epsilon }_{yz}=0$ ) leading to the rotations rates ${\omega }_{x}=-\frac{\partial {v}_{z}}{\partial y}$ and ${\omega }_{y}=\frac{\partial {v}_{z}}{\partial x}$ . In Mindlin plate theory, the rotations are independent variables.