Page 35 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

22 • Using ansys for finite element analysis
1.2.3 solution of set of linear algeBraic
equations

1.2.3.1 cramer’s rule

Consider a set of linear algebraic equations written as follows:
In matrix notation: [a]{x} = {c}

Or, in index notation: ∑ n j=1 ax = c i
j
ij

Let matrix d i ()  be matrix [a] with column i replaced by [c]. Then:
 
d i ()
x = a
i

Example:
Consider a set of three linear algebraic equations given as follows:

−x + x − x
3
2
1 2 3
2x − 4x + 2x
1 2 3

4x + x
2 3
In matrix form:

2
2
−  1 3 −  x 1   
   x   
1
 2 −4 2   2  =  

 

  0 4 1  x   
3
  3
Solution using Cramer’s rule:
−1 3 −2
− ) +− ( )(
0
− −16 10
( 4 8
1
3
a = 2 −4 2 = − ( ) −− ) −()(20 28 − ) = 126 =
0 4 1
d 1 () 2 3 − 2 −
x = = 1 1 − 4 2 = 41 = 41 .
1 a − ( 10) − 10
3 4 1