new finite element analysis lec1

8/3/2019 New Finite Element Analysis Lec1

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Examples

Compressor valve cover Coal processing plant pipes Mowing blade

Finite Element Analysis (FEA)

Finite Element Method (FEM)


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How do you solve an Engineering Problem?

Engineers are faced with complex problems and use many

methods in order to solve those problems including classic

and numerical methods.

As shown below, the finite element methods is one of several

methods for solving engineering problems.

ENGINEERING ANALYSIS

CLASSICAL METHODS

1. CLOSED-FORM

2. APPROXIMATE

NUMERICAL METHOD

1. FINITE ELEMENT

2. FINITE DIFFERENCE

3. BOUNDARY ELEMENT


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Method of solving Engineering Problems

Historically, two classical approaches have been used:

Classical Method

Closed form solutions are available for simple problems such

as bending of beams and torsion of prismatic bars. (eg:Roarks Formulas for Stress and Stress)

Approximate methods using series solutions to governingdifferential equations are used to analyze more complexstructures such as plate and shells.

If you can solve your problem using a closed form solution

and a classical method, it is probably the best way to do it


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How do you solve a complex Engineering Problem?

When a problem becomes more complex thanwhat can be solved with a closed formsolution, the another approach is to take acomplex problem and break it into simpleproblems.

Those simple problem are solved and thenassembled into a final solution.

A good example is found in the long history ofthe efforts to calculate Pi ()


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People spent centuries breaking the circle up

into smaller triangles to find an accurate value

of Pi ().

Example of FE approaches


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Example of FE

approaches


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Method of solving Engineering Problems

Numerical Methods

FEM

Capable of solving large, complex problems with general

geometry, loading and boundary conditions Increasingly become the primary analysis tools for

designers and analyst

Also known as matrix method of structural analysis in the

literature because it uses matrix algebra to solve thesystem of simultaneous equations.

Real power is in its ability to solve problems that do not fitany standard formula.


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FEA Definition

computational techniques used to obtainapproximate solution of boundary valueproblems in Engineering (Hutton, 2004)

numerical analysis technique for obtainingapproximate solutions to a wide variety ofengineering problems. (Zeid, 2005)

one of several numerical methods that can be

used to solve a complex problem by breaking itdown into a finite number of simple problems(NeiSoftware.com, 2011)


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Before FEA?


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Fundamental Concept

Based on the idea of building a complicated

object into small and manageable pieces.

It is a method of investigating the behavior of

complex and structures by breaking themdown into smaller, simpler pieces.

The smaller pieces are called elements whichare connected by nodes.

The assembly of nodes and elements arecalled finite element model.


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Types of Element

Finite elements have shapes which arerelatively easy to formulate and analyze. The

three basic types of finite elements are beams

(1D), shells (2D) and solids (3D).


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Other examples of 1D elements


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Other examples of 2D

elements


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Other examples of 3D

elements


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How does FEM Work?

The solution of a problem domain by FEMusually follows an orderly, step-by-stepprocess. The following steps show in generalhow the FEM works.

1. Create the Finite Element

2. Develop elemental matrices and equation

3. Generate Global Stiffness Matrix equation

4. Apply Boundary Conditions

5. Solve for the unknown at the nodes


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How does FEM Work?(an example of 2D element)

1. Create the Finite Element

A given problem is discretized by dividing the

original domain into simply shaped element.

Element are connected to each other by

nodes

1 element

4 nodes


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v1D elements

2D elements

3D elements


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2. Develop elemental matrices and equation

The relationship between an element and its

surrounding nodes can be described by the

following equation

*Each of the element will have their own set

of equations.

1

2


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What is [K], {u}, {F} ?

Problem Type


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3. Generate Global Stiffness Matrix equation

The individual Elemental Stiffness Matricesare assembled together into the Global

Stiffness Matrix by summing the equilibrium

equations of the elements. This result in the following Global System

Matrix Equation for the overall structure:


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4. Apply Boundary Conditions

Next, the BCs (Loads and Constraint) is apply

to the model. Mathematically this is achieved

by removing rows and columns corresponding

to the constrained degrees of freedom (DOF)

from the Global System Matrix Equation

ux = 0

uy = 0

uz = 0

F1 = ?

F2 = ?

F3 = ?


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5. Solve for the unknown at the nodes

Finally, the Global System Matrix Equation is

solve to determine the unknown nodal values.

*This requires knowledge in Matrix Algebra.

[K]{u} ={F} {u} = [K]-1{F}


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Advantages of FEA

General enough to handle large class ofengineering problems (stress analysis, heat

transfer, electromagnetism, fluid flow)


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Mistakes by user

Elements are the wrong type (eg: Shell 2D

elements are used where solid elements are

needed.

Distorted elements

Inconsistent units (eg: E = 200GPa, Force =

100lbs)


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Best Practices


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Common misconception

The FE solutions is the most accurate.. The FEsolutions are often approximate. The more refinethe mesh (grid), the more accurate.

Any results obtained is the correct one.. The FE

solution may contain fatal errors as a result ofincorrect modeling of structures, loads andboundary conditions. Even a nice picture can givethe wrong result

FEA replaces testing Depends on theconfidence in the analytical methods used. Thereis no guarantee that the results are accurate.


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Typical FE Procedure by commercial

software

Preprocessing

Process

Postprocessing

user

user

computer

Build FE Model

Conduct

numerical

analysis

See and interpret

the results


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FEA General Procedure

Pre -

Processing

Import / Create Geometry

Define Element Types (1D, 2D, 3D) Define Material Properties (E Youngs Modulus, v Poissons

Ratio etc)

Define geometric properties

Apply Boundary Conditions (Constraint)

Apply Loads

Solution

Solve for displacements

Compute Strains

Compute Stresses

Post-

Processing

Sort element stresses in order of magnitude

Check equilibrium

Plot deformed structural shape

Animate dynamic model behavior

Reports


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Pre Processor

Also know asmodel

definition

This step iscritical a

perfectlycomputed FEsolution is ofabsolutely novalue ifcorresponds towrongproblems.

garbage in,

garbage out

Solution

FE softwareassembles the

algebraic

equations in

matrix form andcomputes the

unknown values. The computed

values then usedby backsubstitution tocompute

additional derived

variables (i.ereaction forces,element stressesetc)

Post-

processor

Analysis andevaluation of thesolution results.

While solution datacan be manipulatedmany ways, the

most importantobjectives is toapply sound

engineering

judgment in

determining

whether the

solution arereasonable or not


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REVIEW OF MATRIX ALGEBRA


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new finite element analysis lec1

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