Page 139 - Finite Element Modeling and Simulations with ANSYS Workbench

124 Finite Element Modeling and Simulation with ANSYS Workbench

4.4.1 A General Formula for the Stiffness Matrix
Displacements (u, v) in a plane element can be interpolated from nodal displacements
(u, v) using shape functions N as follows:
i
i
i
u 
 1
 

 u  N 1 0 N 2 0   v 1 

  =    2 u u = Nd (4.11)
u  or
v
   0 N 1 0 N 2   
 v 2 


 
where N is the shape function matrix, u the displacement vector, and d the nodal displace-
ment vector. Here, we have assumed that u depends on the nodal values of u only, and v
on nodal values of v only.
From strain–displacement relation (Equation 4.8), the strain vector is
ε= Du = DNd,or ε = Bd (4.12)

where B = DN is the strain–displacement matrix.
Consider the strain energy stored in an element,

1 1
T
U = ∫ σ εdV = ∫ ( σε +σ ε+ τγ ) dV
y
y
x
x
xy
xy
2 2
V V
1 T 1 1
T
T
= ∫ (E ε) εdV = ∫ ε E εdV = d T ∫ B EB dVd
2 2 2
V V V V
1
= dkd
T
2
From this, we obtain the general formula for the element stiffness matrix
T
k = ∫ B EBdV (4.13)
V
Note that unlike the 1-D cases, E here is a matrix which is given by the stress–strain rela-
tion (e.g., Equation 4.5 for plane stress).
The stiffness matrix k defined by Equation 4.13 is symmetric since E is symmetric. Also
note that given the material property, the behavior of k depends on the B matrix only, which
in turn depends on the shape functions. Thus, the quality of finite elements in representing
the behavior of a structure is mainly determined by the choice of shape functions. Most
commonly employed 2-D elements are linear or quadratic triangles and quadrilaterals.
4.4.2 Constant Strain Triangle (CST or T3)
This is the simplest 2-D element (Figure 4.8), which is also called linear triangular element.
For this element, we have three nodes at the vertices of the triangle, which are numbered
around the element in the counterclockwise direction. Each node has two DOFs (can move