Ken YoussefiMechanical Engineering Dept 1 History of Finite Element Analysis Finite Element Analysis...

Ken Youssefi Mechanical Engineering Dept 1

History of Finite Element Analysis Finite Element Analysis (FEA) was first developed in 1943 by R. Courant, who utilized the Ritz method of numerical analysis and minimization 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 and deflection 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.


Basics of Finite Element Analysis

Why FEM ?

• Modern mechanical design involves complicated shapes, sometimes made of different materials.

• Engineers need to use FEM to evaluate their designs.



FEA Applications

• Evaluate the stress or temperature

distribution in a mechanical component.

• Perform deflection analysis.

• Analyze the kinematics or dynamic response.

• Perform vibration analysis.


Finite element analysis starts with an approximation of the region of interest into a number of meshes (triangular elements). Each mesh is connected to associated nodes (black dots) and thus becomes a finite element.


Consider a cantilever beam shown.


Basics of Finite Element Analysis• After approximating the object by finite elements, each

node is associated with the unknowns to be solved.• For the cantilever beam the displacements in x and y

would be the unknowns.• This implies that every node has two degrees of

freedom and the solution process has to solve 2n degrees of freedom.

• Once the displacements have been computed, the strains are derived by partial derivatives of the displacement function and then the stresses are computed from the strains.


Example – a plate under load Derive and solve the system of equations for a plate loaded as

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

using two triangular elements,

Py


Example – a plate under loadDisplacement within the triangular element with three nodes

can be assumed to be linear.

u = α1 + α2 x + α3 y

v = β1 + β2 x + β3 y


Example – a plate under loadDisplacement for each node,


Example – a plate under load

Solve the equations simultaneously for α and β,


Example – a plate under loadSubstitute x1= 0, y1= 0, x2=10, y2= 0, x3= 0, y3=4 to obtain displacements

u and v for element 1.

2a = 40Calculations:

a1 = 40, a2 = 0, a3 = 0

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

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

Element 1

(1)

(2)

(3)


Example

α1 = (1)U1

2a = 40

a1 = 40, a2 = 0, a3 = 0

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

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

Calculations:

α2 = -(1/10)U1 + (1/10)U3

α3 = -(1/4) U1+ (1/4) U5

β1 = (1)U2

β2 = -(1/10)U2 + (1/10) U4

β3 = -(1/4) U2+ (1/4) U6

u1 = U1, u2 = U3, u3 = U5, v1 = U2, v2 = U4, v3 = U6

Change of notations


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 = α1 + α2 x + α3 y

v = β1 + β2 x + β3 y

Substitute α and β to obtain displacements u

and v for element 1.

α1 = (1)U1

α2 = -(1/10)U1 + (1/10)U3

α3 = -(1/4) U1+ (1/4) U5

β1 = (1)U2

β2 = -(1/10)U2 + (1/10) U4

β3 = -(1/4) U2+ (1/4) U6


ExampleRewriting 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


Example

Similarly the displacements within element 2 can be

expresses as


Example

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

displacement relations,


ExampleDifferentiate 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


Example

Element 2


Example

Using the stress-strain relations for homogeneous,isotropic material and plane-stress,

εx = (σx / E ) - ν (εy) - ν (εz) = (σx / E ) - ν (σy / E ) - ν (σz / E )

εy = (σy / E ) - ν (εx) - ν (εz) = (σy / E ) - ν (σx / E ) - ν (σz / E )

εz = (σz / E ) - ν (εx) - ν (εy) = (σz / E ) - ν (σx / E ) - ν (σy / E )

We have:


Formulation of the Finite Element Method • The classical finite element analysis code (h version)

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 finite

element formulation starting from the governing differential equation.

• Potential Energy and Equilibrium; The Rayleigh-Ritz Method.

Involves the construction of assumed displacement field. Uses the total potential energy for an elastic body


Formulation of the Finite Element Method

f B – Body forces (forces distributed over the volume of the body: gravitational forces, inertia, or magnetic)

f S – surface forces (pressure of one body on another, or hydrostatic pressure)

f i – Concentrated external forces



Let’s denote the displacements of any point (X, Y, Z) of the object from the unloaded configuration as UT

The displacement U causes the strains

and the corresponding stresses

The goal is to calculate displacement, strains, and stresses from the given external forces.


Formulation of the Finite Element MethodEquilibrium condition and principle of virtual displacements

The left side represents the internal virtual work done, and the right side represents the external work done by the actual forces as they go through the virtual displacement. 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.


Formulation of the Finite Element MethodThe equilibrium equation can be expressed using matrix notations for m elements.

where

B(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



The above equation can be rewritten as follows,

The above equation describes the static equilibrium problem. K is the stiffness matrix.


Continuing the example


Example

Calculating the stiffness matrix for element 2.


Example

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


ExampleThe load vector R, equals Rc because only concentrated loads

act on the nodes.

where Py is the known external force and F1x, F1y, F3x, and F3y are

the unknown reaction forces at the supports.

R =

K = UR


ExampleThe following matrix equation can be solved for nodal point displacements

K = UR


ExampleThe solution can be obtained by applying the boundary conditions


ExampleThe equation can be divided into two parts,

The first equation can be solved for the unknown nodal displacements, U3, U4, U7, and U8. And substituting these values into the second equation to obtain unknown reaction forces, F1x, F1y, F3x, and F3y .

Once the nodal displacements have been obtained, the strains and stresses can be calculated.


Finite Element Analysis

• Pre-Processing

• Solving Matrix (solver)

• Post-Processing

FEA requires three steps

FEA is a mathematical representation of a physical system and the solution of that mathematical representation


FEA Pre-ProcessingMesh

Mesh is your way of communicating geometry to the solver, the accuracy of the solution is primarily dependent on the quality of the mesh.

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

A good-looking mesh should have well-shaped elements, and the transition between densities should be smooth and gradual without skinny, distorted elements.


FEA Pre-Processing - meshing

The mesh transition from .05 to .5 element size without control of transition (a) creates irregular mesh around the hole which will yield disappointing results.


FEA Pre-Processing

Finite elements supported by most finite-element codes:


FEA Pre-Processing – Elements

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

Rod (bar or truss) elements cannot 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.



The most general line element is a beam.

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



Plate and Shell ModelingPlate 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 – ElementsSolid Element Modeling

Tetrahedral (tet) mesh is the only generally accepted means to fill a volume, used as auto-mesh by many FEA codes. 10-node Quadratic


CAD Modeling for FEA

• CAD models prepared by the design group for eventual FEA.

• CAD models prepared without consideration of FEA needs.

• CAD models unsuitable for use in analysis due to the amount of rework required.

• Analytical geometry developed by or for analyst for sole purpose of FEA.

CAD and FEA activities should be coordinated at the early stages of the design process to minimize the duplication of effort.



• Solid chunky parts (thick-walled, low aspect ratio)parts mesh cleanly directly off CAD models.

• Clean geometrygeometrical features must not prevent the mesh

from being created. The model should not include buriedfeatures.

• Parent-child relationshipsparametric modeling allows defining features off

other CAD features.


CAD Modeling for FEAShort 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.


CAD Modeling for FEA – Sliver Surfaces

The rounded rib on the inside of the piston has a thickness 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 distorted elements. This results in a 290,000 nodes. If the radius is increased to .15, a mesh size of .12 is sufficient which results in 33,500 nodes.

Flat surface



Fillet across shallow angle

Sliver surface caused by a slightly undersized fillet

Sliver surface caused by misaligned features.


Guidelines for Geometry Planning

• Delay inclusion of fillets and chamfers as long as possible.

• Try to use permanent datums as references where possible 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 Simplification

• 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 outside the load path.• Decorative or identification features.• Large sections of geometry that are essentially

decoupled from the behavior of interested section.

In general, features listed below could be considered for suppression. But, consider the impact before suppression.


Guidelines for Part SimplificationFillet added to the rib

Holes removed

Fillet removed

Ribs needed for casting removed


CAD Modeling for FEAModel 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 or Parasolids. Using standards based (IGES, DXF, or VDA) translations may lead to problem.

• Visually inspect the quality of imported geometry.

• Avoid modification of the imported geometry in a second CAD system.

• Use the original geometry for analysis. If not possible, use a translation directly from the original model.


Example of a solid model corrupted by IGES transfer


FEA Pre-Processing

Material Properties

The only material properties that are generally required by an isotropic, linear static FEA are: Young’s modulus (E), Poisson’s ratio (v), and shear modulus (G).

G = E / 2(1+v)

Provide only two of the three properties.

Thermal expansion and simulation analysis require coefficient of thermal expansion, conductivity and specific heat values.


FEA Pre-Processing

Nonlinear Material Properties

A multi-linear model requires the input of stress-strain data 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 dependent creep properties. Plastic parts are extremely sensitive to this phenomenon


FEA Pre-Processing

• 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.

Comments

If possible, obtain material property values specific to the application under analysis.

If you are selecting the property set from the code’s library, be aware of the assumptions made with this selection.


FEA Pre-ProcessingBoundary Conditions

In FEA, the name of the game is “boundary condition”, that is calculating the load and figuring out constraints that each component experiences in its working environment.

“Garbage in, garbage out”

The results of FEA should include a complete discussion of the boundary conditions.


Boundary Conditions

Loads

Loads are used to represent inputs to the system. They can be in the forms of forces, moments, pressures, temperature, or accelerations.

Constraints

Constraints are used as reactions to the applied loads. Constraints can resist translational or rotational deformation induced by applied loads.


Boundary Conditions

Linear Static AnalysisBoundary conditions are assumed constant from application to final deformation of system and all loads are applied gradually to their full magnitude.

Dynamic Analysis

The boundary conditions vary with time.

Non-linear AnalysisThe orientation and distribution of the boundary conditions vary as displacement of the structure is calculated.


Boundary ConditionsDegrees of Freedom

Spatial DOFs refer to the three translational and three rotational modes 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 transmit or 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.


Boundary ConditionsConstraints and their geometric equivalent in classic beam calculation.

Fixed support

Pin support

Roller support


Boundary Conditions

A solid face should always have at least three points in contact with the rest of the structure. A solid element should 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 model – an overly stiff model due to poorly applied constraints.


Boundary Conditions -ExampleExcessive Constraints

Model of the chair seat with patches representing the tops of the legs.

Patch 3

Patch 1

Patch 2

Patch 4


Patch 3

Patch 1

Patch 2

Patch 4

Boundary Conditions -ExampleIt 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 so that in-plane spatial rotation and rigid body translation is removed without causing excessive constraints.


Boundary Conditions -Example• Constraining the center point of patch 1 in all 3

translational DOFs.

• Constraining x and y translations of the center point of patch 2.

• Constraining z and y translation of the center point of patch 3.

• Constraining just the y translation of the center point of patch 4.

This scheme allows in-plane translation induced by bending of the seat without rigid body translation or rotation.

Patch 3

Patch 1

Patch 2

Patch 4


Summary of Pre-Processing

• Build the geometry

• Make the finite-element mesh

• Add boundary conditions; loads and constraints

• Provide properties of material

• Specify analysis type (static or dynamic, linear or non-linear, plane stress, etc.)

These activities are called finite element modeling.


Solving the Model - SolverOnce the mesh is complete, and the properties and boundary conditions have been applied, it is time to solve the 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) could be caused by an improper definition of load and/or elemental properties.

Post-Processing, Displacement Magnitude


Post-Processing, Displacement Animation

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


Second Mode (Twisting)

The magnitude of the stresses should not be entirely unexpected.

First Mode (Bending)

Post-Processing, Stress Results


Deformation of a duct under thermal load

Post-Processing, thermal analysis


Post-ProcessingView AnimatedDisplacements

Does the shape of deformations make sense?

View DisplacementFringe Plot

Yes

Review BoundaryConditions

No

Are magnitudes in line with your expectations?

View Stress Fringe Plot

Yes

Is the quality and mag. Of stresses acceptable?

Review Load Magnitudesand Units

No

Review Mesh Density and Quality of Elements

No

View Results SpecificTo the Analysis

Yes


FEA - Flow Chart

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Ken YoussefiMechanical Engineering Dept 1 History of Finite Element Analysis Finite Element Analysis...

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