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Ken Youssefi Mechanical Engineering Dept 1

Basics of F inite Element AnalysisWhat is FEA (Finite Element Analysis)?

A complex problem is divided into a smaller andsimpler problems that can be solved by using the

existing knowledge of mechanics of materials and

mathematical tools

Why FEA ?

Modern mechanical design involves complicated shapes,

sometimes made of different materials that as a whole

cannot be solved by existing mathematical tools.

Engineers need the FEA to evaluate their designs

What is FEM (Finite Element Model)?A 3D model prepared specifically for finite element

analysis

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Basics of F inite Element Analysis

The process of dividing the model into small pieces is called meshing.Thebehavior of each element is well-known under all possible support and load

scenarios. The finite element method uses elementswith different shapes.

Elements share common points called nodes.

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Ken Youssefi Mechanical Engineering Dept., SJSU3

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H istory of F ini te Element AnalysisFinite Element Analysis (FEA) was first developed in 1943 by R.

Courant, who utilized the Ritz method of numerical analysis andminimization of variational calculus.

A paper published in 1956 by M. J. Turner, R. W. Clough, H. C.

Martin, and L. J. Topp established a broader definition of

numerical analysis. The paper centered on the "stiffness anddeflection of complex structures".

By the early 70's, FEA was limited to expensive mainframe

computers generally owned by the aeronautics, automotive,

defense, and nuclear industries. Since the rapid decline in the cost

of computers and the phenomenal increase in computing power,

FEA has been developed to an incredible precision.

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Computer-Aided Analysis

Stress analysis

Deflection (Stiffness) analysis

Non-linear analysis

Thermal analysistemp. distribution

Vibration analysisfrequency and mode shapes

Fatigue analysis (cyclic loading) Buckling failure analysis

Perform drop test

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Size and shape optimization

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Consider a cantilever beam shown.

Finite element analysis starts with an approximation of the region ofinterest into a number of meshes (2D or 3D elements). Each mesh is

connected to associated nodes (black dots) and thus becomes a finite

element.

Node

Element

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After approximating the object by finite elements,

each node is associated with the unknowns to besolved.

For the cantilever beam the displacements inxand

ydirections would be the unknowns (2D mesh).

This implies that every node has two degrees of

freedom and the solution process has to solve 2n

degrees of freedom, nis the number of nodes.

Displacement Strain

Partial derivatives

Stress

Stress &Strain

relationship

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Examplea plate under loadDerive and solve the system of equations for a plate loaded as

shown. Plate thickness is 1 cm and the applied load Pyis constant

using two triangular elements,

Py

Reaction

forces

U1thru U8,displacements

inxandy

directions

.

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Examplea plate under load

Displacement within the triangular element (2D) with three

nodes can be assumed to be linear.

u = 1+

2x+

3y

v = 1 +2x+3y

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Displacement for each node,

Node 1

Node 2

Node 3

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Apply the boundary conditions to determine the constants a,

b, and c for nodes I, 2, and 3 of element 1

2a = 40

Calculations:

a1= 40, a2= 0, a3= 0

b1= - 4, b2= 4, b3= 0

c1= -10, c2= 0, c3= 10

Element 1

(1)

(2)

(3)

Evaluate the constants a, b, and c

10 4 0 0

node 1, x1= 0,y1= 0

node 2, x2=10,y2= 0

node 3, x3= 0,y

3= 4

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Example

u1= U1 + [-1/10 (U1)+ (1/10)U3]x+ [-(1/4) U1+ (1/4) U5 ]y

v1= U2 + [-1/10(U2)+ (1/10) U4]x+ [-(1/4) U2+ (1/4) U6 ]y

Calculation:

u = 1+

2x+

3y

v = 1 +2x+3y

Substitute and to obtain

displacements uand vfor element 1.

1

= (1)U1

2 = -(1/10)U1+ (1/10)U3

3

= -(1/4) U1+ (1/4) U5

1

= (1)U2

2 = -(1/10)U2+ (1/10) U4

3 = -(1/4) U2+ (1/4) U6

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Example

Rewriting the equations in the matrix form,

u1= U1 + [-1/10 (U1)+ (1/10)U3]x+ [-(1/4) U1+ (1/4) U5 ]y

v1= U2 + [-1/10(U2)+ (1/10) U4]x+ [-(1/4) U2+ (1/4) U6 ]y

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Example

Similarly the displacements within

element 2 can be expressed as,

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Example

The next step is to determine the strains using 2D strain-

displacement relations,

Displacement Strain

Partial derivatives

Stress

Stress &Strainrelationship

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Example

Differentiate the displacement equation to obtain the strain

u1= U1 + [-1/10(U1) + (1/10) U3]x+ [-(1/4) U1+ (1/4) U5 ]y

v1= U2 + [-1/10(U2) + (1/10) U4]x+ [-(1/4) U2+ (1/4) U6 ]y

1stelement

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Example

Element 22ndelement

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St & St i R l ti hi

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Stress & Strain Relationship

x =(x/ E)- (y)- (z)=(x/ E)- (y/ E)- (z/ E)

y =(y/ E)- (x)-(z)=(y/ E)- (x/ E)- (z/ E)

z=(z/ E)- (x)- (y)=(z/ E)- (x/ E)- (y/ E)

Uniaxial state of stress

x =(x/ E), y =- x, z=- x

x0, y =0,z=0

Stresses interms of strains

Poisson ratio

Triaxial state of stress

St & St i R l ti hi

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Stress & Strain RelationshipThere are many practical problems where the stress in thez-

direction is zero, this is referred to as the state of Plane Stress,

biaxi lal state of stress

G=E

2(1 + )= xy Gxy

Shear stress

Matrix

form

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FEA Results - Principal Stresses


Normal stresses on planes with noshear stresses are maximum and they

are called principal stresses 1, 2,

and 3,where1> 2 >3

The three non-imaginary roots are the principal stresses

2

2

3- (

x

+y

+z

) 2+ (x

y

+x

z

+y

z

- xy

- xz

- yz

)

-

(xyz- 2xyxzyz- xyz-yxz- zxy) = 02

22

2

3- (x+y)

2+ (xy-xy)= 0

Plane stress, two principal stresses, 3 = 0

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Displacement Strain

Partial derivatives

Stress

Stress &Strainrelationship

Material

DuctileYield strength of the material is used indesigning components

BrittleUltimate strength in tension and

compression is used in designing components

Finite element software provides you with,

maximum normal stress (largest principle stress),

maximum shear stress and von Mises stress

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F il Th i D til M t i l

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Failure TheoriesDuctile Mater ials

Maximum Shear Stress

Maximum shear stress theory (Tresca 1886)

Yield strength of a material is used to design components made of

ductile material

(max)component > ()obtained from a tension test at the yield point Failure

(max)component 2

and 3

= 0

Tension test

Sut2 1Stress state

2. Suc >> Sut1. Sut Syt 3. Percent elongation < 5%

F il Th i B ittl M t i l

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Ken Youssefi Mechanical Engineering Dept. 32

Failure TheoriesBrittle Mater ials

1

Sut

Suc

Sut

Suc

Safe

Safe

Safe Safe

-Sut

Cast iron data

Modified Coulomb-Mohr theory

1

2 or3

Sut

Sut

Suc

-Sut

I

II

III

Three design zones

2 or3

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Ken Youssefi Mechanical Engineering Dept., 33

Failure TheoriesBrittle Mater ials

1

2

Sut

Sut

-Suc

-Sut

I

II

III

Zone I

1>0 , 2>0 and 1>2

Zone II1>0 , 2

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Right click on the Results and

select the Factor of Safety Plot

Safety factor plots

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Max. Shear Stress

theory

Distortion energy

theory using von

Mises stress

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Formulation of the F ini te Element Method The classical finite element analysis code (hversion)

The system equations for solid and structural

mechanics problems are derived using the principle of

virtual displacement and work (Bathe, 1982).

The method of weighted residuals (Galerkin Method)

weighted residuals are used as one method of finiteelement formulation starting from the governing

differential equation.

Potential Energy and Equilibrium; The Rayleigh-Ritz

MethodInvolves the construction of assumed displacement field.

Uses the total potential energy for an elastic body

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Formulation of the F ini te Element Method

Lets denote the displacements of any point (x,y,z) of the object

from the unloaded configuration as UT

UT= [U(x,y,z) V(x,y,z) W(x,y,z)]

The displacement U causes the strains

T

= [x y z xy yz zx ]

and the corresponding stresses

The goal is to calculate displacement, strains, and stresses from

the given external forces.

T = [x y z xy yz zx ]

Formulation of the F ini te Element M ethod

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f BBody forces (forces distributed over the volume of the body:

(gravitational forces, inertia, or magnetic)

fB

=

f Bx

fB

y

fB

z

fS

surface forces (pressure of one body on another, or hydrostaticpressure)

fS=

f Sx

fSy

fS

z

f iConcentrated external forces

fi=

f ix

fi

y

fi

z

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Equilibrium condition and principle of virtual displacements

The left side represents the internal virtual work done, and theright side represents the external work done by the actual

forces as they go through the virtual displacement.

Usdenotes the displacement due to surface forces

U

i

denotes the displacement due to point forces

Work done by

body forces

Work done by

surface forces

Work done by

external forces

VTdV V dVU

Tf

B

S d SUS

fST U i

TFi

+ +=

Internal work

The above equation is used to generate finite element equations.

And by approximating the object as an assemblage of discrete

finite elements, these elements are interconnected at nodal points

F l ti f th F i i t El t M th d

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The displacement at any point measured with respect to a local

coordinate system for an element are assumed to be a function of the

displacement at the nodes.

H(m)

is the displacement interpolation matrix

Displacement interpolation matrix

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B (m)is the rows of the strain-displacement matrix

strain-displacement matrix

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Matrix

form



C (m)is the elasticity matrix of element mand I(m) are theelements initial stresses. The elasticity matrix relates strains

to stress.

Elasticity matrix

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Elasticity matrix

Strain-displacement matrix

Displacement interpolation matrix

Displacement at any nodeDisplacement at any element

Initial stress (residual stress)

Strain at any element

Stress at any element

F l ti f th F i i t El t M th d

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The formula for the principle of virtual displacements can be

rewritten as the sum of integration over the volume and areas

for each finite element,

Where mvaries from 1 to the total number of elements


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The equilibrium equation can be expressed using matrix

notations for m elements.

whereB(m) Represents the rows of the strain displacement matrix

C(m) Elasticity matrix of element m

H(m) Displacement interpolation matrix

U Vector of the three global displacement

components at all nodes

F Vector of the external concentrated forces

applied to the nodes

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Continuing the example ( ) h f h i

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Continuing the example B(m)- Represents the rows of the straindisplacement matrix

C(m

)- Elasticity matrix of element m

dxx

y

y =4- x4

10dA=y dx

E l

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Example

Example

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Example

Calculating the stiffness matrix for element 2.

E l

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Example

The stiffness of the structure as a whole is obtained by combing

the two matrices, K = K1

+K2

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Example

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Example

The following matrix equation can be solved for nodal point

displacements

KU = R

Example

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ExampleThe solution can be obtained by applying the boundary conditions

No deflectionat the supports

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Pre-Processing

Meshing the component or the

assembly

Assigning material with mechanical

and physical properties

Applying boundary conditions (Loadsand Constraints)

SW Simulation

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SW Simulation


FEA P P i

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FEA Pre-Processing

Mesh

Mesh is your way of communicating geometry tothe solver, the accuracy of the solution is primarily

dependent on the quality of the mesh.

The better the mesh looks, the more accurate thesolution is.

A good-looking mesh should have well-shaped

elements (proportional), and the transition betweendensities should be smooth and gradual without

skinny, distorted elements.

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FEA Pre-Processing

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FEA Pre-Processing

The mesh elements supported by most finite-element codes:

FEA P P i El t

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FEA Pre-ProcessingElements

Beam Elements

Beam elements typically fall into two categories; able totransmit moments or not able to transmit moments.

Rod (bar or truss) elementscannot carry moments.

Entire length of a modeled component can be captured with a

single element. This member can transmit axial loads only and

can be defined simply by a material and cross sectional area.

FEA P P i El t

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The most general line element is a beam.

(b) and (c) are higher order line elements.

FEA P P i El t

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Plate and Shell Modeling

Plate and shell are used interchangeably and refer to surface-like elements used to represent thin-walled structures.

A quadrilateral mesh is usually more accurate than a mesh of

similar density based on triangles. Triangles are acceptable in

regions of gradual transitions.

FEA Pre-Processing Elements

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FEA Pre Processing Elements

Solid Element Modeling

Tetrahedral (tet) mesh is the only generally

accepted means to fill a volume, used as auto-

mesh element by many FEA codes.10-node Quadratic

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Effect of Mesh Element

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Model 2 produces maximum Von Mises stress of 32,000 psi.

The mesh on model 2 is similar to that on model 1

but uses second-order solid tetrahedral elements

with coarse mesh.

The mesh is too coarse to model stressdistribution correctly or detect stress

concentrations. Some elements are still

highly distorted.


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Model 3 produces maximum von Mises stress of 49,000 psi

This model uses second-order solid tetrahedral elements and hasfiner mesh to model stress distributions properly.

Stress will increase with each

mesh refinement. Thus, the

process of mesh refining andsolving the refined model must

continue until the increase in

stress between two consecutive

iterations becomes sufficientlysmall (about 10%). Only then

can results be accepted as final.


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CAD Modeling for FEA

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CAD and FEA activities should be coordinated at the early stages

of the design process to minimize the duplication of effort.

There are four situations

Analytical geometry developed by or for analyst for

sole purpose of FEA

CAD models prepared by the design group for

eventual FEA

CAD models unsuitable for use in analysis due to the

amount of rework required.

CAD models prepared without consideration of FEA

needs. Little to moderate modifications are needed


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Solid chunky parts (thick-walled, low aspect ratio)

parts mesh cleanly directly off CAD models.

Clean geometry

geometrical features must not prevent the mesh from

being created. The model should not include buried

features.

Parent-child relationships

parametric modeling allows defining features off otherCAD features.


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Short edges and Sliver surfacesShort edges and sliver surfaces usually accompany each other and on

large faces can cause highly distorted elements or a failed mesh.


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Fillet across shallow angle

Sliver surface caused by a slightly

undersized fillet

Sliver surface caused by

misaligned features.

CAD Modeling for FEA Sliver Surfaces

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The rounded rib on the

inside of the piston has athickness of .30 and a

radius of .145, as a result

a flat surface of .01 by 2.5

is created. A mesh size of

.05 is required to avoid

distortedelements. This

results in a 290,000

nodes. If the radius is

increased to .15, a mesh

size of .12 is sufficient

which results in 33,500nodes.

Flat surface

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CAD Modeling for FEA Sliver edge

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G id li f G Pl i

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Guidelines for Geometry Planning

Delay inclusion of fillets and chamfers as long aspossible.

Where possible, try to use permanent datum as a

reference to minimize dependencies.

Avoid using fillet or draft edges as references for

other features (parent-child relationship)

Never bury a feature in your model. Delete or

redefine unwanted or incorrect features.

Guidelines for Part Simpli f ication

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Outside corner breaks or rounds.

Small inside fillets far from areas of interest.

Screw threads or spline features unless they are

specifically being studied.

Small holes and slots outside the load path.

Decorative or identification features.

Large sections of geometry that are essentially

decoupled from the behavior of interested section

Analyze half of the symmetrical parts

In general, features listed below could be considered for

suppression. But, consider the impact before suppression.

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Model with full detail Model with details suppressed

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Guidelines for Part Simplif ication

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p

Fillet added

to the rib

Holes removed

Fillet

removed

Ribs needed

for casting

removed

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Model Conversion

Try to use the same CAD system for all components

in design. When the above is not possible, translate geometry

through kernel based tools such as ACIS orParasolids. Using standards based (IGES, VDA, ..)translations may lead to problem.

Visually inspect the quality of imported geometry.

Avoid modification of the imported geometry in asecond CAD system.

Use the original geometry for analysis. If notpossible, use a translation directly from the

original model.

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FEA Pre-Processing

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g

Material Properties

The only material properties that are generally required byan isotropic, l inear staticFEA are: Youngs modulus

(E), Poissons ratio (v), shear modulus (G), and yield

strength (or ultimate strength). Strength is needed if the

program provides safety factor or performance result.G = E / 2(1+v)

Provide only two of the three properties.

Thermal expansion and simulation analysis requirecoefficient of thermal expansion, conductivity and

specific heat values.

Material Library

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FEA Pre-Processing

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g

Nonlinear Material Properties

A multi-linear model requires the input of stress-straindata pairs to essentially communicate the stress-strain

curve from testing to the FE model

Highly deformable, low stiffness, incompressible materials,

such as rubber and other synthetic elastomers require

distortional and volumetric constants or a more complete set

of tensile, compressive, and shear force versus stretch curve.

A creep analysis requires time and temperature dependentcreep properties. Plastic parts are extremely sensitive to this

phenomenon

FEA Pre-Processing

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Their properties hold constant throughout the assigned entity.

Average values are used (variation could be up to 15%).

Localized changes due to heat or other processing effects are

not accounted for.

Any impurities present in the parent material are neglected.

The assumption is that there are no defects in the material

Comments

If possible, obtain material property values specific to the

application under analysis.

If you are selecting the property set from the codes library,

be aware of the assumptions made with this selection.

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Boundary ConditionsSW

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y

Loads

Loads are used to representinputs to the system. They

can be in the forms of forces,

moments (torque), pressures,

temperature, or accelerations.

SW

Creo

Boundary Conditions

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Linear Static Analysis

Boundary conditions are assumed constantfromapplication to final deformation of system and all loads

are applied gradually to their full magnitude.

Dynamic Analysis

The boundary conditions (Loads) vary with time.

Non-linear Analysis

The orientation and distribution of the boundary

conditions vary as displacement of the structure iscalculated.

Boundary Conditions

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y

Degrees of Freedom

Spatial DOFs refer to the three translational and three rotationalmodes of displacement that are possible for any part in 3D

space. A constraint scheme must remove all six DOFs for the

analysis to run.

Elemental DOFs refer to the ability of each element to transmitor react to a load. The boundary condition cannot load or

constrain a DOF that is not supported by the element to which it

is applied.

ConstraintsSW

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Constraints are used as reactions to

the applied loads. Constraints can

resist translationalor rotationaldeformation induced by applied

loads.

SW

Creo

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Boundary Conditions

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A solid face should always have at least three points in

contact with the rest of the structure. A solid elementshould never be constrained by less than three points and

only translational DOFs must be fixed.

Accuracy

The choice of boundary conditions has a direct impact

on the overall accuracy of the model.

Over-constrained modelan overly stiff model dueto poorly applied constraints.

Boundary Conditions -Example

Excessive Constraints

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Excessive Constraints

Model of the chair seat with patches representing the tops of

the legs.

Patch 3

Patch 1

Patch 2

Patch 4

Boundary Conditions -Example

It t b t bl t t i h i l t h

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Patch 3

Patch 1

Patch 2

Patch 4

It may appear to be acceptable to constrain each circular patch

in vertical translation while leaving the rotational DOFs

unconstraint. This causes the seat to behave as if the leg-to-

seat interfaces were completely fixed.

A more realistic constraint scheme would be to pin the

center point of each circular patch (translational), allowing

the patch to rotate. Each point should be fixed vertically,

and horizontal constraints should be selectively applied sothat in-plane spatial rotation and rigid body translation is

removed without causing excessive constraints.

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Legs are fixed to seat

2000 Napplied force

distributed over the

surface.

Use On Flat Face restraint

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Fixed legs In plane rotation is allowed

St

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Stress Stress

Displacement

Displacement

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One leg is restrained in

x,y,z, one in y, one in

x,y, one in y,z

Stress

Displacement

Displ. = .016 mmAll patches

(l ) fi d

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Displ. = .06 mm

Displ. = .02 mm

(legs) fixed

All patches (legs) On

Flat Face constrains,

in plane rotation

One leg is

restrained in x,y,z,

one in y, one in x,y,

one in y,z

Stress=5 8x107 N/m2Stress=11.6x107N/m2

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Stress 5.8x10 N/m

Stress=10.4x107N/m2All patches

(legs) fixed

All patches (legs)On Flat Face

constrains, in

plane rotation

One leg is

restrained in x,y,z,

one in y, one in x,y,

one in y,z

Summary of Pre-Processing

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Build the geometry (CAD model for FEA)

Prepare the model for meshing (simplify) Create the finite-element mesh

Add boundary conditions; loads and

constraints Select material or provide properties

Specify analysis type (static or dynamic,

linear or non-linear, thermal, etc.)

These activities are called finite element modeling.

Solving the Model - Solver

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g

Once the mesh is complete, and the properties and

boundary conditions have been applied, it is time to solvethe model. In most cases, this will be the point where you

can take a deep breath, push a button and relax while the

computer does the work for a change.

Multiple Load and Constraint Cases

In most cases submitting a run with multiple load cases will

be faster than running sequential, complete solutions for

each load case.

Final Model Check

Unexpectedly high or low displacements (by order of magnitude)

Post-Processing, Displacement Magni tude

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Unexpectedly high or low displacements (by order of magnitude)

could be caused by an improper definition of load and/or elemental

properties.

Post-Processing, Displacement Animation

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Animation of the model displacements serves as the best means of

visualizing the response of the model to its boundary conditions.

Post-Processing, FEA of a connecting rod

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Ken Youssefi Mechanical Engineering Dept111

Post-Processing, Stress Results

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Second Mode (Twisting)

The magnitude of the stresses should not be entirely unexpected.

F ir st Mode (Bending)

Post-Processing, thermal analysis

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Deformation of a duct under thermal load

Deploy MechanismAssembly Analysis

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Displacement

Stress

Can crusher stress

l i

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analysis

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Use finer mesh size

Right click the

h i d

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Mesh icon and

choose Failure

Diagnostics

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Add fillet to the slot

edges (.1 in.)

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Mesh Quali ty

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The ideal shape of a tetrahedral

element is a regular tetrahedron withthe aspect rat io of 1. Analogously, an

equilateral triangle is the ideal shape

for a shell element.

Sometimes, Irregular tetrahedral

are created by the program.

These distorted elements have

high aspect ratio. An aspect ratio

that is too high causes element

degeneration, which in turn affects

the quality of the results.

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Apply mesh control

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Ken Youssefi Mechanical Engineering Dept

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Without mesh control

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With mesh control

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Automatic transition - the programautomatically applies mesh controls to small

features, holes, fillets, and other fine details of

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your model. Uncheck Automatic transition

before meshing large models with many small

features and details to avoid generating a verylarge number of elements.

Post-ProcessingView (animated)Displacements

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Does the shape of deformations make sense?

View Displacement

Fringe Plot

Yes

Review Boundary

Conditions

No

Are magnitudes in line with your expectations?

View Stress

Fringe Plot

Yes

Is the quality and mag. of stresses acceptable?

Review Load Magnitudes

and Units

No

Review Mesh Density

and Quality of ElementsNo

View Results Specific

To the Analysis

Yes

FEA - Flow Chart

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Last Comment

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and a bad engineer dangerous!

Finite Element Analysis makes a goodengineer great

Robert D. Cook, Professor of Mechanical Engineering at

University of Wisconsin, Madison

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