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Title: Finite Element Methods Finite Element Analysis


1
Finite Element MethodsFinite Element Analysis
FEM or FEA
  • Original lecture material by Major Pat Heffernan,
    CD, PhD, PEngDepartment of Mechanical
    EngineeringRMC
  • Adapted for Mech422 Nov 2002

2
References
  • Cook, R.D., Malkus, D.S., Plesha, R.E., and Witt,
    R.J., 2002. Concepts and Applications of Finite
    Element Analysis, 4th Ed, Wiley, NY, USA.
  • Huebner K.H., et al., 2001. The Finite Element
    Method For Engineers, 4th Ed, Wiley, NY, USA.

3
Introduction
  • What is the Finite Element Method (FEM) and how
    does it relate to CAD/CAM?
  • What is CAD/CAM?
  • Design
  • Manufacturing
  • Why use CAD/CAM?
  • Design Manufacture New Products
  • Less Time
  • Lower Cost
  • Enhance Overall Productivity

4
MCAE
Mechanical Computer Aided Engineering
  • Allows for concurrent engineering
  • Analysis/Manufacturing Processes/Jigs can be done
    in parallel
  • More design alternatives can be explored
  • Reduces life-cycle costs
  • Founded on Solid Modelling (not 2-D drawings)
  • Solid Model used for
  • Mass Property calculations
  • Interference Studies
  • Stress Analysis
  • Manufacturing Planning
  • If necessary, creation of conventional 2D drawings

5
Product Cycle
Design and Manufacturing
with CAD/CAM overlaid
FEM
6
Finite Element Method
A numerical analysis technique for obtaining
approximate solutions to engineering problems.
  • Discrete elements approximating a continuum
  • Used when the geometry is too complex for an
    analytical solution
  • Continuum is divided into a finite number of
    elements
  • Each element represents a portion of the
    structure
  • Compatibility must be maintained at inter-element
    boundaries
  • Equations are generated to represent the system
    and a numerical solution is determined
  • Accuracy is largely dependent on mesh size

7
Finite Element Method (contd)
  • 2D or 3D Elements
  • 2D or 3D Problems
  • Ideal for complex problems that are difficult to
    solve analytically
  • Computation time is a large concern
  • Suitable for many applications (stress analysis,
    dynamic analysis, fluid flow, heat transferetc.)

8
Approximation to Complex Shapes
  • A fairly simple analytical solution but how much
    more complex of a geometry can be reasonably
    handled?
  • What if we grooved the shaft to handle a rubber
    o-ring?

P
x,u
P
x,u
Complexities may be geometric or material in
nature.
9
The Analysts Dilemma
Approximate the problem
Simplifying Assumptions
or
Approximate the solution
Numerical (FEM) Solution
10
Numerical Solutions
(among many)
  • Finite Difference
  • an array of gridpoints
  • write difference equations between points
  • pointwise approximation
  • Finite Elements
  • an array of small, interconnected subregions
  • system is an assemblage of finite elements
  • piecewise approximation

11
FEM A Brief History
(a very brief history)
  • Mid-1800s basic concepts
  • 1920, Maney (USA) and Ostenfeld (Denmark) basic
    ideas of truss and frame analysis using
    displacements as unknowns.
  • 1932, Cross Moment Distribution Method.
  • 1950s, Digital Computer Suited matrix format
    methods. Allowed solution of a large number of
    simultaneos equations.1960, Clough - first paper
    published using term Finite Element Method
  • 1965, First conference on finite elements.

12
FEM A Method
  • Steps
  • Discretize the Continuum
  • Select Interpolation Functions
  • Find the Element Properties
  • Assemble the Element Properties to Obtain the
    System of Equations
  • Impose the Boundary Conditions
  • Solve the System of Equations
  • Make Additional Computation if Desired

Already donefor most commonelements
Formulate a solution for the element and then
assemble it to find the global stiffness
13
General-Purpose Programs
  • Products ANSYS, ADINA, SAP, EASE, I-DEAS,
    Unigraphics
  • Modular Construction
  • Pre-processor definition of geometry, materials,
    loading, boundary conditions.
  • Processor construction and solution of
    mathematical model.
  • Post-processor display of predicted
    displacements and stresses.
  • Between steps 1 and 2, the program will access a
    library of elements to generate mathematical
    models for the elements and applied loadings.

14
Types of Elements
  • General-purpose programs contain huge libraries
    of elements. Elements can be derived for special
    functions. Some typical elements incl
  • Frame Elements
  • Basic Plane Stress Elements
  • Plate Bending Element
  • Thin Shell Element
  • Axisymmetric Element

15
SAP 2000 Student
  • 3D models
  • 2D elements
  • Static Analysis
  • Great Learning Tool
  • In Real Life mostly Civil applications.

16
SDRC I-DEAS
(or any real life CAD/CAM Integrated Package)
  • Integrated Design Engineering Analysis Software
  • US 5k/yr academic/US 40k/yr commercial
  • To facilitate concurrent engineering
  • focus on Mechanical Engineering Product
  • Multiple Modules
  • Design (Solid Modelling)
  • Drafting (Seen in 2nd Yr)
  • Simulation (FEM)
  • Manufacturing

17
Displacement Stress Analysis
  • Finite Element Modelling (Simulation)
  • Stress/Deflections given loads/constraints
  • Also Heat Transfer and Potential Flow
  • Optimization sub-module
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