Plates - Mechanics of Structures: Variational and Computational Methods - page 816

where A is the area of the element and I xx , I yy , and I yz are the moments of inertia and
product of inertia of the element with respect to the centroidal axes, i.e.,

12 x 1 +

x 3

A

x 2 dA x 2 + I xx = =

A

12 y 1 +

y 3

A

y 2 dA y 2 +

I yy = =

(13.149)

A

A
12 (

I xy = xy dA = x 1 y 1 + x 2 y 2 + x 3 y 3 )

A

in which x i , y i , i 1 , 2 , 3 , are the coordinates of the nodes.

The next important step in the formulation of the stiffness matrix is the evaluation of
the second integral on the right hand side of Eq. (13.140). The integration on the whole
boundary S

=

= 12 + 23 + 31 can be broken into the sum of the integrals on the sides ij ( ij = 12

or 23 or 31), i.e.,

3

U 2 =

S (w

q a − w ,a m a − w ,s m t ) dS = U Si j

i = 1

where

dS

w
w

=

ij (w

− w − w ) = − −

U Si j q a

,a m a ,s m t

dS [ q a m a m t ] (13.150)

,a

ij

w

,s

is the integration on side ij . The components q a , m a , and m t are given in Eq. (13.141) in
which

c = cos γ =− y ij / s = sin γ = x ij /

ij ij ij ij

where x ij , y ij , and

ij are shown in Fig. 13.20. By means of Eqs. (13.141) and (13.144), the
boundary forces q a , m a and m t can be written as

q a
−

ij =

m a
−

R ij σ (13.151)

m t

where

0 c 0 0 0 s 0 s c

R ij = − cc − ccx − ccy − ss − ssx − ssy − 2 cs − 2 csx − 2 csy

cs csx csy − cs − csx − csy − ( cc − ss ) − ( cc − ss ) x − ( cc − ss ) y

with

x = x i − ξ x ij y = y i − ξ y ij and ξ = s /

ij
in which s is shown in Fig. 13.20. It is seen that R ij is a linear expression in

ξ .

and its derivatives on the boundary,
assumptions on the displacements have to be made. Use the shape function

Since Eq. (13.150) involves the displacements w

w = N B 1 (ξ)w + N B 2 (ξ)w + N B 3 (ξ)w + N B 4 (ξ)w (13.152)

i j ,si ,sj

where

2 3 2 3

N B 1 = 1 − 3 ξ + 2 ξ N B 2 = 3 ξ − 2 ξ

2 3 2 3

N B 3 = ij (ξ − 2 ξ + ξ ) N B 4 = ij ( − ξ + ξ )