The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 393

and

t

N0
0N

T N0

0N

T

p i = p V dA + p dS (6.94b)

A S p

Note that in k i , the terms assume forms such as A ( dA, A ( N T x ∂) N T y ∂)

dA ,
etc., where E ij are the elements in E . These expressions are constants and can be obtained
by performing the manipulations indicated. From Eq. (6.75), N can be written as

E ij (∂ x N ) E ij (∂ x N )

α α α

1 2 3

1
2 A [1

N = [ L 1 L 2 L 3 ] = x ] β β β (6.95)

1 2 3

γ γ γ

1 2 3

A typical derivative of N is

α 1 α 2 α 3

1
2 A ∂ x [1

1
2 A [

=

∂ x N = x ] β 1 β 2 β 3 β 1 β 2 β 3 ] (6.96)

γ 1 γ 2 γ 3

Similarly

1
2 A [

∂ y N = γ 1 γ 2 γ 3 ] (6.97)

Then the terms in the stiffness matrix can be evaluated as

L 1
L 2
L 3

x

N T x

A (

∂) (∂ ) = ∂ ∂

E ij

x N

dA E ij

x [ L 1

L 2 L 3 ] dA

A

β 1
β 2
β 3

E ij
4 A 2

= [ β 1 β 2 β 3 ] dA

A

1

β β 1 β 2 β 1 β 3

E ij
4 A

2

= β 2 β 1 β β 2 β 3 (6.98)

3

β 3 β 1 β 3 β 2 β

γ 1 β 1 γ 1 β 2 γ 1 β 3

E ij
4 A

N T y ∂)

A (

E ij (∂ x N ) dA = γ 2 β 1 γ 2 β 2 γ 2 β 3 (6.99)

γ 3 β 1 γ 3 β 2 γ 3 β 3

Similar expressions can be obtained for the remaining terms in k i such as

N T x

A (

∂) E ij (∂

y N

) dA (6.100)

Thus, the stiffness matrix k i 6 matrix.

The lo ad ing vectors are calculated similarly. If the bo dy forces p V =

can be formed as a 6 ×

[ p Vx p Vy ] T , in which
p Vx and p Vy are assumed to have a constant value p V , are to be included, the part of the
loading vector due to these forces is

p Vx
p Vy

t

N0
0N

T

p v = p V dA = (6.101)

A

with

p V t p V t

L 1
L 2
L 3

1
1
1

p V At
3

dA

N T dA

p Vx = p Vy = = = (6.102)

A A