Bilinear Shape Functions

The shape functions defining the bilinear element used in the Mindlin plate are:(1)
Φ I ( ξ , η ) = 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 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:(4)
v x = I = 1 4 Φ I v x I
(5)
v y = I = 1 4 Φ I v y I
(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:(7)
ω x = I = 1 4 Φ I ω x I
(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:(9)
v x x = I = 1 4 Φ I x v x I
(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.