1. Finite Element Primer for Engineers: Part 2
Mike Barton & S. D. Rajan

2. Contents
Introduction to the Finite Element Method (FEM)
Steps in Using the FEM (an Example from Solid Mechanics)
Examples
Commercial FEM Software
Competing Technologies
Future Trends
Internet Resources
References

3. FEM Applied to Solid Mechanics Problems
A FEM model in solid mechanics can be thought of as a system of assembled springs. When a load is applied, all elements deform until all forces balance.
F = Kd
K is dependant upon Young’s modulus and Poisson’s ratio, as well as the geometry.
Equations from discrete elements are assembled together to form the global stiffness matrix.
Deflections are obtained by solving the assembled set of linear equations.
Stresses and strains are calculated from the deflections.
Create elements
of the beam
Nodal displacement and forces
dxi 1
dxi 2
dyi 1
dyi 2 1 2 4 3

4. Classification of Solid-Mechanics Problems
Analysis of solids
Static
Dynamics
Elementary
Advanced
Stress Stiffening
Behavior of Solids
Large Displacement
Geometric
Instability
Linear
Nonlinear
Fracture
Plasticity
Material
Viscoplasticity
Geometric
Classification of solids
Skeletal Systems
1D Elements
Plates and Shells
2D Elements
Solid Blocks
3D Elements
Trusses
Cables
Pipes
Plane Stress
Plane Strain
Axisymmetric
Plate Bending
Shells with flat elements
Shells with curved elements
Brick Elements
Tetrahedral Elements
General Elements

5. Governing Equation for Solid Mechanics Problems
Basic equation for a static analysis is as follows:
[K] {u} = {Fapp} + {Fth} + {Fpr} + {Fma} + {Fpl} + {Fcr} + {Fsw} + {Fld}
[K] = total stiffness matrix
{u} = nodal displacement
{Fapp} = applied nodal force load vector
{Fth} = applied element thermal load vector
{Fpr} = applied element pressure load vector
{Fma} = applied element body force vector
{Fpl} = element plastic strain load vector
{Fcr} = element creep strain load vector
{Fsw} = element swelling strain load vector
{Fld} = element large deflection load vector

6. Six Steps in the Finite Element Method
Step 1 - Discretization: The problem domain is discretized into a collection of simple shapes, or elements.
Step 2 - Develop Element Equations: Developed using the physics of the problem, and typically Galerkin’s Method or variational principles.
Step 3 - Assembly: The element equations for each element in the FEM mesh are assembled into a set of global equations that model the properties of the entire system.
Step 4 - Application of Boundary Conditions: Solution cannot be obtained unless boundary conditions are applied. They reflect the known values for certain primary unknowns. Imposing the boundary conditions modifies the global equations.
Step 5 - Solve for Primary Unknowns: The modified global equations are solved for the primary unknowns at the nodes.
Step 6 - Calculate Derived Variables: Calculated using the nodal values of the primary variables.

7. Process Flow in a Typical FEM Analysis
Problem
Definition
Analysis and
design decisions
Start
Stop
Pre-processor
Reads or generates nodes and elements (ex: ANSYS)
Reads or generates material property data.
Reads or generates boundary conditions (loads and constraints.)
Processor
Generates element shape functions
Calculates master element equations
Calculates transformation matrices
Maps element equations into global system
Assembles element equations
Introduces boundary conditions
Performs solution procedures
Post-processor
Prints or plots contours of stress components.
Prints or plots contours of displacements.
Evaluates and prints error bounds.
Step 6
Step 1, Step 4
Steps 2, 3, 5

8. Step 1: Discretization - Mesh Generation
surface model
airfoil geometry
(from CAD program)
mesh
generator
ET,1,SOLID45
N, 1, , ,
N, 2, , , .
TYPE, 1
E, 1, 2, 80, 79, 4, 5, 83, 82
E, 2, 3, 81, 80, 5, 6, 84, 83
meshed model

9. Step 4: Boundary Conditions for a Solid Mechanics Problem
Displacements DOF constraints usually specified at model boundaries to define rigid supports.
Forces and Moments Concentrated loads on nodes usually specified on the model exterior.
Pressures Surface loads usually specified on the model exterior.
Temperatures Input at nodes to study the effect of thermal expansion or contraction.
Inertia Loads Loads that affect the entire structure (ex: acceleration, rotation).

10. Step 4: Applying Boundary Conditions (Thermal Loads)
Nodes from
FE Modeler
bf, 1,temp,
bf, 2,temp, .
bf, 1637,temp,
bf, 1638,temp,
Temp
mapper
Thermal Soln Files

11. Step 4: Applying Boundary Conditions (Other Loads)
Speed, temperature and hub fixity applied to sample problem.
FE Modeler used to apply speed and hub constraint.
antype,static
omega,10400*3.1416/30
d,1,all,0,0,57,1