The Finite Element Method in Mechanical Engineering

1
The Finite Element Method in Mechanical
Engineering

• MENG 4326

2
MENG 4326

• An introduction to the finite element method in
  mechanical engineering. Emphasizes linear stress
  and strain analysis, but includes other field
  problems. Utilizes commercial computer codes to
  solve stress analysis, heat transfer, and other
  engineering related problems. (Catalog
  description)
• Text Fundamentals of Finite Element Analysis, by
  David V. Hutton, McGraw-Hill, 2004
• Class meeting times
• Lecture MW 100 - 150 pm
• Computer Lab F 900 am - 1145 am

3
Course Objectives

• Demonstrate an understanding of the fundamental
  concepts of the finite element method form basic
  matrix equations
• (stiffness) x (displacement) (load)
• Select appropriate FE element for the physical
  model
• Know how to apply loads and boundary conditions
• Preprocess, solve, and postprocess 2- and
  3-dimensional, linear problems using ALGOR
  software
• Understand importance of checking solutions with
  back-of-envelope calculations/engineering
  judgment

4
MENG 4326

• Topics covered
• Basic concepts of the Finite Element Method (FEM)
• Stiffness matrices, spring and bar elements
• Truss structures the direct stiffness method
• Flexure elements
• Method of weighted residuals
• Interpolation functions for general element
  formulation
• Applications in solid mechanics
• Applications in heat transfer
• Structural dynamics

5
Course Grading
(Note Late homework and lab assignments will
receive a reduced grade no grade after one
week.)
6
What is Finite Element Analysis?

• The finite element method is a general technique
  for constructing approximate solutions to
  boundary-value problems. The method involves
  dividing the domain of the solution into a finite
  number of simple sub domains, the finite
  elements, and using variational concepts to
  construct an approximation of the solution over
  the collection of finite elements.
• Becker, Carey, and Oden, Finite Elements, vol. 1
  An Introduction, Prentice-Hall, 1981

7
What is Finite Element Analysis?

• Finite element analysis is a method for
  numerical solution of field problems. A field
  problem requires that we determine the spatial
  distribution of one or more dependent variables
  (e.g., temperature, displacement, stress).
  Mathematically, a field problem is described by
  differential equations or by an integral
  expression. Either description may be used to
  formulate finite elements.
• Cook, Malkus, Plesha, and Witt, Concepts and
  Applications of Finite Element Analysis, 4th ed.,
  John Wiley, 2001

8
What is Finite Element Analysis?

• The finite element method (FEM), sometimes
  referred to as finite element analysis (FEA), is
  a computational technique used to approximate
  solutions of boundary value problems in
  engineering.
• Hutton, Fundamentals of Finite Element Analysis,
  McGraw-Hill, 2004 (Our text)

9
What is a Boundary Value Problem?

• A boundary value problem is a mathematical
  problem in which one or more dependent variables
  must satisfy a differential equation everywhere
  within a known domain of independent variables
  AND satisfy specific conditions on the boundary
  of the domain.

10
The Finite Element Method

• The FE method works by breaking a real object
  down into a large number of elements, such as
  little cubes
• The behavior of each little element is readily
  predicted by set mathematical equations.
• The "finite" in finite element analysis comes
  from the idea that there are a finite number of
  elements in a finite element model (as opposed to
  the infinite number of infinitesimal elements of
  integral calculus).
• The computer adds up all the individual behaviors
  to predict the behavior of the actual object.

11
2D Elements, Nodes, and Meshes
Nodes
12
3D Elements, Nodes, and Meshes
13
The FEA Modeling Process
Physical Reality
Model of the Physical Reality
Good Predictor?
Mathematical (FEA) Model
Solution of the Mathematical (FEA) Model
14
Best Practices

• FEA requires engineering judgment. In the best
  case, you should know the approximate answer
  before you begin.
• Proper selection of elements, materials, loads,
  constraints and analysis parameters comes from
  experience.

15
Best Practices

• Understand that the computer model never matches
  reality (its only an approximation).
• The surest route to failure in FEA is to
  underestimate the complexity of the technology.

16

Some Example Applications
17
Ski Resort Chairlift Grip Redesign
Source Algor Customer Application Stories,
Algor, Inc.
18
Female Adhesive Anchor Analysis
Deflected Shape
Source Algor Customer Application Stories,
Algor, Inc.
19
FEA in Different Industries

• Orbital Technologies Corporation
• The above illustration shows how engineers
  analyzed a Biomass Production System to conduct
  biotechnology plant research.

20
FEA in Different Industries
Automotive Industry

• Danly Engineering Services, Division of Enprotech
  Mechanical Services, Inc.
• The above illustration shows how engineers
  analyzed a power press with additional cutouts.

21
FEA in Different Industries
Biomedical Industry

• Ophthalmic Consultants of Boston and the Tufts
  University School of Medicine
• The above illustration shows stresses on an eye
  as it underwent a 30 saccadic eye movement.
  This was modeled to help understand why retinal
  detachments occur.

22
FEA in Different Industries
Power/Utility Industry

• Cronulla Sewage Treatment Plant
• The above illustration shows how engineers
  modeled a piping system to verify that the number
  of bellows could safely be reduced by using
  lightweight, spiral-wound stainless steel. This
  allowed them to keep a 90 million sewage
  treatment plant upgrade on budget.

23
Application of the FEM

• FEM is employed to predict the behavior of
  virtually all physical phenomena in engineering
• Mechanical stress (stress analysis)
• Mechanical vibration
• Heat transfer - conduction, convection, radiation
  
• Fluid Flow - both liquid and gaseous fluids
• Many other field phenomena

24
Finite Element Analysis (FEA)

• Typical Steps in FEA
• Create a mesh (a grid of nodes and elements) that
  represents the model
• Apply the loads
• Define the boundary conditions which prevent the
  model from moving
• Define the element properties
• Assemble the element stiffness matrices
• Solve the system of linear algebraic equations
• Calculate the stresses
• Review deflections and stresses results

Solve
Preprocess
Postprocess
25
ALGOR FEA Software
General FEA Process
Solve
Preprocess
Postprocess
Basic ALGOR FEA Software Modules
Solve
Superdraw
Superview
26
Additional ALGOR Modules
Solid Modeler
InCAD
FEM Pro
Solve
Report
Superdraw
Superview
IGES
27

Summary
28
MENG 4326

• An introduction to the finite element method in
  mechanical engineering. Emphasizes linear stress
  and strain analysis, but includes other field
  problems. Utilizes commercial computer codes to
  solve stress analysis, heat transfer, and other
  engineering related problems.
• Class meeting times
• Lecture MW 100 - 150 pm
• Computer Lab F 900 am - 1145 am

29
What is Finite Element Analysis?

• Finite element analysis is a method for numerical
  solution of boundary value problems.
• The method involves dividing the domain of the
  solution into a finite number of simple sub
  domains, the finite elements, and using
  variational concepts to construct an
  approximation of the solution over the collection
  of finite elements.

30
Finite Element Analysis (FEA)

• Typical Steps in FEA
• Create a mesh (a grid of nodes and elements) that
  represents the model
• Apply the loads
• Define the boundary conditions which prevent the
  model from moving
• Define the element properties
• Assemble the element stiffness matrices
• Solve the system of linear algebraic equations
• Calculate the stresses
• Review deflections and stresses results

Solve
Preprocess
Postprocess
31
The Modeling Process
Physical Reality
Model of the Physical Reality
Good Predictor?
Mathematical (FEA) Model
Solution of the Mathematical (FEA) Model
32
ALGOR FEA Software
General FEA Process
ALGOR FEA Software Modules
33
The End