Page 39 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

26 • Using ansys for finite element analysis
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1.3 general stePs of fem

The objective of structural analysis is usually to determine the displace-
ments and stresses throughout the structure, which is in equilibrium and
is subjected to applied loads. At any point in the continuum body, there
are 15 unknowns (three displacements, six stresses, and six strains). To
determine these 15 unknowns, we have 15 equations in three-dimensional
case shown as follows:

Unknowns Unknowns governing equations
Displacements (u,v,w) 3 Stress equilibrium equations 3
Stresses 6 Compatibility equations 6
xz 
 e  x e y e z g xy g yz g 
Strains 6 Stress–strain equations 6
 e  x e y e z g xy g yz g 
xz 
15 15
There are two general approaches associated with the finite element
method to solve the governing equations: force (or flexibility) method and
displacement (or stiffness) method. The force method uses internal forces
as the unknown of the problem, whereas the displacements are the sys-
tem variable in displacement method. The displacement method is more
desirable because its formulation is simpler for most structural analysis
problems. Furthermore, a vast majority of general-purpose finite element
programs have incorporated the displacement formulation for solving
structural problems. Consequently, only the displacement method will be
used throughout this course.
The basic steps involved in any FEA consist of the following:
Preprocessing phase (build the FE model, loads, and constraints)
1. Discretize and select element type.
2. Select a displacement function.
3. Define strain/displacement and stress/strain relationships.
4. Derive element stiffness matrix and equations.
5. Assemble equations and introduce Boundary Conditions (BCs).

Solution phase: (assemble and solve the system of equations)
6. Solve for the unknown degrees of freedom.