Bilinear Shape Functions

The shape functions defining the bilinear element used in the Mindlin plate are:
図 1.
Φ I ( ξ , η ) = 1 4 ( 1 + ξ I ξ ) ( 1 + η I η )
or, in terms of local coordinates:
図 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 1 mix form:
図 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 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWG0baaaa@39D0@ is the area of the element.

The velocity of the element at the mid-plane reference point is found using the relations:
図 4.
v x = ∑ I = 1 4 Φ I v x I
図 5.
v y = ∑ I = 1 4 Φ I v y I
図 6.
v z = ∑ 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:
図 7.
ω x = ∑ I = 1 4 Φ I ω x I
図 8.
ω y = ∑ 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:
図 9.
∂ v x ∂ x = ∑ I = 1 4 ∂ Φ I ∂ x v x I
図 10.
∂ v x ∂ y = ∑ I = 1 4 ∂ Φ I ∂ 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.