Page 178 - Finite Element Analysis with ANSYS Workbench

9.2 Finite Element Method 169

 cAL 2  1  1kA   
1

C
   1 6 2    ; K   1 L   1  
c
QAL 1  qpL 1   hT pL 1 
    ;    s   ;     
Q
Q
Q
Q
2 1  q 2 1  h 2 1 
These closed-form matrices and vectors can be used to develop a
finite element computer program directly.
The three-node triangular element is a simple element
type for learning the finite element method in two dimensions. The
element consists of a node at each corner as shown in the figure.
The finite element matrices and load vectors can be derived in
closed form. Examples of these matrices and load vectors are,
(   ) hT T
q s 3  ( 4 T  4 )T 

Q
2
y
t
x 1

2  1 1  1  
 cAt   QAt  
C
Q
  1 2 1 ;    1  
12   Q 3  
1
1   1 2   
2  1 1  1  
 K  hA  1 2 1  ;    hT A  

Q
1  
h
12   h 3  
1  1 2    
1
 b b  cc b b  1 2 cc b b  1 3 cc
1 3 
1 2
11
11
 K  kt  b b  c c bb  c c bb  c c 
c
4A  1 2 1 2 22 22 2 3 2 3 
3 3 
  bb  13 c c b b  2 3 c c b b  3 3 c c 
13
2 3
where b , c ; i =1, 2, 3 are the coefficients that depend on the
i
i
nodal coordinates x , y and A is the element area. Details for
i
i
determining these coefficients and area are given in chapter 4.