Page 22 - Finite Element Modeling and Simulations with ANSYS Workbench

Introduction 7

at node 2,

f 2 1 2 f 1 2

F 2
F 2 = f 2 + f 1 2
1

and, at node 3,

F 3 = f 2 2

f 2 2 3 F 3

Using Equations 1.4 and 1.5, we obtain

F 1 = k u 1 − k u 2
1
1
F 2 =− k u 1 + ( k 1 + ku 2 − ku
)
2
1
23
F 3 =− k u 2 + k u 3
2
2
In matrix form, we have
 k 1 − k 1 0  u 1   F 1 
      

 − k 1 k 1 + k 2 − k 2  u 2 =   (1.6)
F 2

 0 − k 2    

 k 2 u 3   F 3 
or
Ku = F (1.7)
in which, K is the stiffness matrix (structure matrix) for the entire spring system.
1.2.2.1.1 An Alternative Way of Assembling the Whole Stiffness Matrix
“Enlarging” the stiffness matrices for elements 1 and 2, we have

 k 1 − k 1 0 u 1   f 1 1 
     1 

 − k 1 k 1 0  u 2 =  f 2 

 0 0 0  u 3   0  
   
and

0  0 0  1 u   0 
     2 
 0 2 k − 2 k    2 u  =  1 f 
2



 0  − 2 k 2 k   3 u   2 f 