Bilinear Shape Functions

The shape functions defining the bilinear element used in the Mindlin plate are:(1)

Φ_{I} (ξ, η) = \frac{1}{4} (1 + ξ_{I} ξ) (1 + η_{I} η)

or, in terms of local coordinates:(2)

Φ_{I} (x, y) = a_{I} + b_{I} x + c_{I} y + d_{I} x y

It is also useful to write the shape functions in the Belytschko-Bachrach mix form ¹:(3)

Φ_{I} (x, y, ξ η) = Δ_{I} + b_{x I} x + b_{y I} y + γ_{I} ξ η

with

$Δ_{I} = [t_{I} - (t_{I} x^{I}) b_{x I} - (t_{I} y^{I}) b_{y I}]; t = (1, 1, 1, 1)$

$b_{x I} = (y_{24} y_{31} y_{42} y_{13}) / A; (f_{i j} = (f_{i} - f_{j}) / 2)$

$b_{y I} = (x_{42} x_{13} x_{24} x_{31}) / A$

$γ_{I} = [Γ_{I} - (Γ_{J} x^{J}) b_{x I} - (Γ_{J} y^{J}) b_{x I}] / 4; Γ = (1, - 1, 1, - 1)$

$A$ 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} Φ_{I} v_{x I}

(5)

v_{y} = \sum_{I = 1}^{4} Φ_{I} v_{y I}

(6)

v_{z} = \sum_{I = 1}^{4} Φ_{I} v_{z I}

Where, $v_{x I}, v_{y I}, v_{z I}$ are the nodal velocities in the x, y, z directions.

In a similar fashion, the element rotations are found by:(7)

ω_{x} = \sum_{I = 1}^{4} Φ_{I} ω_{x I}

(8)

ω_{y} = \sum_{I = 1}^{4} Φ_{I} ω_{y I}

Where, $ω_{x I}$ and $ω_{y I}$ 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 Φ_{I}}{\partial x} v_{x I}

(10)

\frac{\partial v_{x}}{\partial y} = \sum_{I = 1}^{4} \frac{\partial Φ_{I}}{\partial y} v_{x I}

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.