Page 37 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

24 • Using ansys for finite element analysis
Steps in Gaussian elimination:

1. Eliminate the coefficient of x in every equation except the first one.
1
Select a as the pivot element.
11
• Add the multiple −a / a of the first row to the second row.
21
11
• Add the multiple −a / a of the first row to the third row.
11
31
• Continue this procedure through the nth row
After this step:
a  11 a 12 a   
c
n 1
 ´ ´   x   
1
1

 
 0 a a n  x   

´
c
 22 2   2  =  
2
     

 ´ `   x   
c
  0 a n2 a   n   
n
nn 
2. Eliminate the coefficient of x in every equation below the second
2
′
one. Select a as the pivot element.
22
′
′
• Add the multiple a / a of the second row to the third row.
32
22
′
• Add the multiple a 42 a / ′ 22 of the second row to the fourth row.
• Continue this procedure through the nth row.
After this step:
a  11 a 12 a 13 a   c 
n 1
 ´ ´ ´   x 1   
1
 0 a a a   x  ´
c  
 22 23 2 n   2   


2
 0 0 a '' a   x 3  =  
''
c
'''
n
 33 3     
3
     
 ´  x   

  0 a n2 a '' n3 a   n  c n 
'''
nn 
3. Repeat the process for the remaining rows until we have a triangu-
larized system of equations:
a  a a a a n 1   c 
 11 12 13 14   x 1   1 
 0 a ´ a ´ a ´ a ´  x   c 
´
 22 23 24 2 n  2   2 
 0 0 a '' a '' a '' n x   c 
''
 33 34 3  3  =   3 
x

 0 0 0 a ' ''' 44 a 4 ''' n  4   c 
'''
     4 
     
x
 ´ n 1  n    c n −1

−
  0 a n2 a n3 a ''' n4 a nn    n 
''