# Bilinear Shape Functions

The shape functions defining the bilinear element used in the Mindlin plate are:(1)
${\Phi }_{I}\left(\xi ,\eta \right)=\frac{1}{4}\left(1+{\xi }_{I}\xi \right)\left(1+{\eta }_{I}\eta \right)$
or, in terms of local coordinates:(2)
${\Phi }_{I}\left(x,y\right)={a}_{I}+{b}_{I}x+{c}_{I}y+{d}_{I}xy$
It is also useful to write the shape functions in the Belytschko-Bachrach 1 mix form:(3)
${\Phi }_{I}\left(x,y,\xi \eta \right)={\text{Δ}}_{I}+{b}_{xI}x+{b}_{yI}y+{\gamma }_{I}\xi \eta$

with

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

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

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

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

$A$ is the area of the element.

The velocity of the element at the mid-plane reference point is found using the relations:(4)
${v}_{x}=\sum _{I=1}^{4}{\Phi }_{I}{v}_{xI}$
(5)
${v}_{y}=\sum _{I=1}^{4}{\Phi }_{I}{v}_{yI}$
(6)
${v}_{z}=\sum _{I=1}^{4}{\Phi }_{I}{v}_{zI}$

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:(7)
${\omega }_{x}=\sum _{I=1}^{4}{\Phi }_{I}{\omega }_{xI}$
(8)
${\omega }_{y}=\sum _{I=1}^{4}{\Phi }_{I}{\omega }_{yI}$

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:(9)
$\frac{\partial {v}_{x}}{\partial x}=\sum _{I=1}^{4}\frac{\partial {\Phi }_{I}}{\partial x}{v}_{xI}$
(10)
$\frac{\partial {v}_{x}}{\partial y}=\sum _{I=1}^{4}\frac{\partial {\Phi }_{I}}{\partial y}{v}_{xI}$
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.