Plates - Mechanics of Structures: Variational and Computational Methods - page 814

assumed displacement field inside the element, with stresses assumed along interelement
boundaries. See Pian and Tong (1969) for more details.
In the formulation of the HSM element, the classical Kirchhoff thin plate theory is em-
ployed. The shear stresses γ

xz and

γ

yz are assumed to vanish everywhere in the plate, i.e.,

θ

,y . The HSM element uses the same nodal DOF as the DKT element.
The extended complementary hybrid functional of Chapter 2, Eq. (2.104) is

=− w

,x and

θ =− w

x y

∗ H =− U 0 ( p T u dS u T σ ) dV + + ( p − p ) dS (13.137)

V S u S p

where V U 0 ( σ )

dV is the complementary energy, which can be expressed as

t / 2

1
2 σ T E − B σ dV

1
2 σ T

U 0 ( σ ) E − B σ dA d z

− dV =− =− (13.138)

V V − t / 2 A

Substitute the expressions for σ

x ,

σ

y , and

τ

xy of Eq. (13.27) into Eq. (13.138) to obtain

1
2

U 0 ( E − 1

B

s T

− σ ) dV =− s dA (13.139)

V A

[ m x m y m xy ] T

where s = . For a homogeneous isotropic plate, E B from Eq. (13.35) leads to

Et 3 (

m x m y )

1
2 s T

12

E − 1
B

2

m xy −

s = m x + m y ) + 2 ( 1 + ν)(

The second and the third terms on the right hand side of Eq. (13.137) can be rearranged as

p T u dS u T p T u dS p T u dS

+ ( − ) = −

p p dS

S u S p S S p

where S = S u + S p . Then Eq. (13.137) becomes

1
2 s T

∗ H =− E − 1

B

p T s dA +

S (w

q a − w

,a m a

− w

,s m t

) dS − u dS

A S p

p T u dS

= U 1 + U 2 − (13.140)

S p

where U 1 =− U 0 ( σ ) =− A s dA and U 2 = S (w

1

2 s T E − 1

The
stress resultants m a and m t are shown in Fig. 13.20 and q a is the shear force on the boundary.
These quantities are expressed as

dV q a − w ,a m a − w ,s m t ) dS .

B

q a = c ( m x,x + m xy, y ) + s ( m y, y + m xy,x )

m a = ccm x + 2 csm xy + ssm y (13.141)

m t =− scm x + ( cc − ss ) m xy + scm y

where c and s abbreviate cos denotes the angle between

the outer normal of the boundary and the x axis. The shape functions for s ,

γ and sin γ , respectively, in which γ

w , w ,a , and w ,s

will be given later.
The last term of Eq. (13.140) represents the loading vector. It is assumed that uniform
loading can be represented by lumped loads in the form of Eq. (13.136).
The independent quantities subject to variation in the hybrid stress functional are the
moment components m x , m y , and m xy inside the element and the displacements w and w ,a

along the element boundary S with the subsidiary conditions

+ + =

m x,xx 2 m xy,xy m y, yy 0in A (13.142)