# Bilinear Shape Functions

The shape functions defining the bilinear element used in the Mindlin plate are:

or, in terms of local coordinates:

It is also useful to write the shape functions in the
Belytschko-Bachrach ^{1} mix form:

with

${\text{\Delta}}_{I}=\left[{t}_{I}-\left({t}_{I}{x}^{I}\right){b}_{xI}-\left({t}_{I}{y}^{I}\right){b}_{yI}\right]\text{\hspace{0.17em}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}t=\left(1,1,1,1\right)$

${b}_{xI}=\left({y}_{24}{y}_{31}{y}_{42}{y}_{13}\right)/A\text{\hspace{0.17em}};\text{\hspace{1em}}\left({f}_{ij}=\left({f}_{i}-{f}_{j}\right)/2\right)$

${b}_{yI}=\left({x}_{42}{x}_{13}{x}_{24}{x}_{31}\right)/A$

${\gamma}_{I}=\left[{\Gamma}_{I}-\left({\Gamma}_{J}{x}^{J}\right){b}_{xI}-\left({\Gamma}_{J}{y}^{J}\right){b}_{xI}\right]/4\text{\hspace{0.17em}};\text{\hspace{1em}}\Gamma =\left(1,-1,1,-1\right)$

$A$ is the area of the element.

The velocity of the element at the mid-plane reference point is found using the relations:

Where, ${v}_{xI},{v}_{yI},{v}_{zI}$ are the nodal velocities in the x, y, z directions.

In a similar fashion, the element rotations are found by:

Where, ${\omega}_{xI}$ and ${\omega}_{yI}$ are the nodal rotational velocities about the x and y reference axes.

The velocity change with respect to the coordinate change is given by:

^{1}Belytschko T. and Bachrach W.E.,

Efficient implementation of quadrilaterals with high coarse-mesh accuracy, Computer Methods in Applied Mechanics and Engineering, 54:279-301, 1986.