Page 121 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

STEADY STATE HEAT CONDUCTION IN ONE DIMENSION
4x
N j =
l
l
2x 2 − 4x 2 2 113
x
N k = 2 − (4.43)
l l
From Chapter 3, the stiffness matrix is defined as

T
[K] = [B] [D][B]d

 
14 −16 2
Ak
=  −16 32 −16  (4.44)
6l
2 −16 14
where
4x 3 4 8x 4x 1

[B] = 2 − − 2 2 − (4.45)
l l l l l l
The loading vector is

T
{f}= G[N] d

 

 L i (2L i − 1) 
= G 4L i L j A dx
l  
L j (2L j − 1)
 
GAl 1
= 4 (4.46)
1
6  
In the above equation, the shape functions N i , N j and N k are expressed in terms of the
local coordinate system L i and L j , the use of which will facilitate the integration process
by using

a b a!b!
N N dl = l (4.47)
i
j
l (a + b + 1)!
Example 4.2.4 We shall now solve Example 4.2.2 using one quadratic element only as
shown in Figure 4.9.
As before, we consider only one half of the wall, where L is equal to 30 mm.
Substituting values into Equations 4.44 and 4.46, we obtain
 
1633.33 −1866.66 233.33
[K] e =  −1866.66 3733.33 −1866.66  (4.48)
233.33 −1866.66 1633.33
and
 
1500
{f} e = 6000 (4.49)
1500
 