Page 42 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

IntroductIon to FInIte element AnAlysIs • 29
The stiffness equation for a single spring element in a local coordinate
system can be written as:

 ∧   ∧ 
∧
 f x 1  k  11 k   d x 1    ∧    
∧
12
  =     ⇒ f   = k d
   
 ∧   k 21 k 22   ∧      
f
 2 x   d x 
2
Step 1: Select the element type
Consider linear spring element subjected to tensile forces
k
1 2 x ˆ
L
1 k 2
T T x ˆ

ˆ ˆ
d 1x d 2x

Step 2: Select a displacement function

• Degree of freedom (DOF) per node = 1
• Number of nodes per element = 2
• Total (DOF) per element = 2 × 1 = 2
• Number of coefficients = 2

1
x y
x 2 x y y 2
2
x 3 x y x y 2 y 3
3
2 2
x 4 x y x y x y 3 y 4
Assume a linear displacement function:

∧
u = a + ax ∧
2
1
Write in matrix form:

a 
∧  ∧  1


u = 1 x  
a
   2 
∧ 
^  ∧ d x2  ).
Express u as a function of nodal displacements ( d x1 , 

