intro to fea and applications

8/10/2019 Intro to Fea and Applications

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Numerical method used for solvingproblems that cannot be solved analytically(e.g., due to complicated geometry,

different materials)Well suited to computersOriginally applied to problems in solidmechanics

Other application areas include heattransfer, fluid flow, electromagnetism


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Preprocessing Geometry Modelling analysis type Material properties Mesh Boundary conditions

Solution Solve linear or nonlinear algebraic equations

simultaneously to obtain nodal results(displacements, temperatures etc.)

Postprocessing Obtain other results (stresses, heat fluxes)


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Continuous elastic structure(geometric continuum) dividedinto small (but finite), well-

defined substructures, calledelementsElements are connectedtogether at nodes; nodes havedegrees of freedomDiscretization process knownas meshing


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, ,

, similar to

F l E

A l

EA F l F kx

l

, ,

, similar to

F l E

A l

EA F l F kx

l

Elements modelled as linear springs


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Local elastic behaviour of each elementdefined in matrix form in terms ofloading, displacement, and stiffness Stiffness determined by geometry and material

properties (AE/l)


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Elements assembled through commonnodes into a global matrixGlobal boundary conditions (loads and

supports) applied to nodes (in practice,applied to underlying geometry)

1 1 2 2 1

2 2 2 2

F K K K U

F K K U


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Matrix operations used to determineunknown dofs (e.g., nodaldisplacements)Run time proportional to #nodes orelementsError messages Bad elements Insufficient disk space, RAM Insufficiently constrained


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Displacements used to derive strains andstresses


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First Principles (Newtons Laws) Body under external loading

Area Moments of InertiaStress and Strain Principal stresses Stress states: bending, shear, torsion,

pressure, contact, thermal expansion Stress concentration factors

Material PropertiesFailure ModesDynamic Analysis


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Several approaches can be used to transform the physicalformulation of a problem to its finite element discrete analogue.

If the physical formulation of the problem is described as adifferential equation, then the most popular solution method isthe Method of Weighted Residuals .

If the physical problem can be formulated as the minimizationof a functional, then the Variational Formulation is usuallyused.


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One family of methods used to numerically solve differential equationsare called the methods of weighted residuals (MWR).

In the MWR, an approximate solution is substituted into the differentialequation. Since the approximate solution does not identically satisfy theequation, a residual, or error term, results.

Consider a differential equationDy(x) + Q = 0 (1)

Suppose that y = h(x) is an approximate solution to (1). Substitution thengives Dh(x) + Q = R, where R is a nonzero residual. The MWR thenrequires that

W i(x)R(x) = 0 (2)

where W i(x) are the weighting functions. The number of weightingfunctions equals the number of unknown coefficients in the approximatesolution.


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There are several choices for the weighting functions, W i. In the Galerkins method, the weighting functions are the samefunctions that were used in the approximating equation .

The Galerkins method yields the same results as the variationalmethod when applied to differential equations that are self-adjoint.

The MWR is therefore an integral solution method. The weightedintegral is called the weak form.

Many readers may find it unusual to see a numerical solution thatis based on an integral formulation.


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The variational method involves the integral of a function thatproduces a number. Each new function produces a newnumber.

The function that produces the lowest number has the

additional property of satisfying a specific differential equation. Consider the integral

D/2 * y(x) - Qy]dx = 0. (1)

The numerical value of can be calculated given a specificequation y = f(x). Variational calculus shows that theparticular equation y = g(x) which yields the lowest numericalvalue for is the solution to the differential equation

Dy(x) + Q = 0. (2)


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In solid mechanics, the so-called Rayeigh-Ritz techniqueuses the Theorem of Minimum Potential Energy (with thepotential energy being the functional, ) to develop theelement equations.

The trial solution that gives the minimum value of is theapproximate solution.

In other specialty areas, a variational principle can usuallybe found.


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The three main sources of error in a typical FEM solution arediscretization errors, formulation errors and numericalerrors.

Discretization error results from transforming the physicalsystem (continuum) into a finite element model, and can be

related to modeling the boundary shape, the boundaryconditions, etc.

Discretization error due to poor

geometry representation.

Discretization error effectivelyeliminated.


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Formulation error results from the use of elements that don't

precisely describe the behavior of the physical problem.Elements which are used to model physical problems for which theyare not suited are sometimes referred to as ill-conditioned ormathematically unsuitable elements.For example a particular finite element might be formulated on theassumption that displacements vary in a linear manner over the

domain. Such an element will produce no formulation error when it isused to model a linearly varying physical problem (linear varyingdisplacement field in this example), but would create a significantformulation error if it used to represent a quadratic or cubic varyingdisplacement field.


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Can handle bodies comprised of nonhomogeneous materials:Every element in the model could be assigned a differentset of material properties.

Can handle bodies comprised of nonisotropic materials:Orthotropic

AnisotropicSpecial material effects are handled:

Temperature dependent properties.PlasticityCreep

SwellingSpecial geometric effects can be modeled:

Large displacements.Large rotations.Contact (gap) condition .


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A specific numerical result is obtained for a specific

problem. A general closed-form solution, which wouldpermit one to examine system response to changes in

various parameters, is not produced.

The FEM is applied to an approximation of themathematical model of a system (the source of so-calledinherited errors.)

Experience and judgment are needed in order to constructa good finite element model.

A powerful computer and reliable FEM software areessential.

Input and output data may be large and tedious to prepareand interpret.


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Numerical problems:Computers only carry a finite number of significantdigits.Round off and error accumulation.Can help the situation by not attaching stiff (small)elements to flexible (large) elements.

Susceptible to user-introduced modelling errors:Poor choice of element types.Distorted elements.Geometry not adequately modelled.

Certain effects not automatically included:Complex BucklingHybrid composites.Nanomaterials modelling .Multiple simultaneous causes.


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Module 6


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In this, we will briefly describe how todo a thermal-stress analysis.The purpose is two-fold: To show you how to apply thermal loads in a

stress analysis. To introduce you to a coupled-field analysis.


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Thermally Induced StressWhen a structure is heated or cooled,it deforms by expanding orcontracting.If the deformation is somehowrestricted by displacementconstraints or an opposing pressure,for example thermal stresses areinduced in the structure.

Another cause of thermal stresses isnon-uniform deformation, due todifferent materials (i.e, differentcoefficients of thermal expansion).

Thermal stressesdue to constraints

Thermal stressesdue to different

materials


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There are two methods of solving thermal-stress problemsusing ANSYS. Both methods have their advantages. Sequential coupled field

- Older method, uses two element types mapping thermalresults as structural temperature loads

+ Efficient when running many thermal transient time

points but few structural time points+ Can easily be automated with input files

Direct coupled field+ Newer method uses one element type to solve both

physics problems+ Allows true coupling between thermal and structural

phenomena- May carry unnecessary overhead for some analyses


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The Direct Method usually involves just one analysis that uses a coupled-field elementtype containing all necessary degrees of freedom.

1. First prepare the model and mesh using one of thefollowing coupled field element types.

PLANE13 (plane solid).SOLID5 (hexahedron).SOLID98 (tetrahedron).

2. Apply both the structural and thermal loads andconstraints to the model.

3. Solve and review both thermal and structural results .

Combined

Thermal Analysis

Structural Analysis

jobname .rst


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The BOM includes Copper lead frame,

Gold wires for bonding, Silver epoxyfor die attach, Silicon die and Epoxymould composite with Phenolics, Fusedsilica powder and Carbon black powderas the encapsulant materials. Electrical-Thermal and thermal-structural analyses.


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Tooth is a functionally gradedcomposite material with enameland dentin. In the third maxillarymolar the occlusal stress canbe 2-3 MPa.

The masticatory heavy chewingstress will be around 193 MPa.A composite restorative must withstand this with an FOS and withconstant hygrothermal attack.


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Eccentric Column


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Eccentric Column-FEM MODEL


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x: 0-0.13

y: 0-0.15

x: 0-0.12

y: 0-0.15FEM METHOD


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Outer diameter = 158mmInner diameter = 138mmHeight = 900mmPoissons ratio = 0.29Youngs Modulus = 2.15e5 N/mm2

The element used for this model is Solid 186.Theapplied pressure is 0.430N/m 2. For this analysislarge deformation was set ON and also Arc lengthsolution was turned ON.


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FEM METHOD

x: 0-2,y: 0-2.5

TOPOLOGICAL METHOD

x: 0-2, y: 0-2.5

0

0.5

11.5

2

2.5

3

0 1 2 3x

y

x =0.11.25y=0.5 x (4- x )


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Pipe-FEM MODEL


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x: 0.2-1

y: 0-0.32

x: 0.2-1

y: 0-0.19FEM METHOD


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A hinged cylindrical

shell is subjected to avertical point load (P) atits center.


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Hinged cylindrical shell-FEM MODEL


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x: 0-1.65

y: 0-1 FEM METHOD


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x: 0-1.3

y: 0-1.6FEM METHOD


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Vibration studies in composites areimportant as the composites areincreasingly being used in automotive,

aerospace and wind energy applications.The combined effect of vibrations andfatigue can degrade a composite furtherthat is already hygrothermal in affinity.The different modes of vibrations arediscussed here.


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3D LAYERED STRUCTURAL SOLID ELEMENTElement definition

Layered version of the 8-node, 3D solid element, solid 45 with three degreesof freedom per node(UX,UY,UZ).

Designed to model thick layered shells or layered solids. can stack several elements to model more than 250 layers to allow through-

the-thickness deformation discontinuities.Layer definition

allows up to 250 uniform thickness layers per element. allows 125 layers with thicknesses that may vary bilinearly. user-input constitutive matrix option.

Options Nonlinear capabilities including large strain. Failure criteria through TB,FAIL option.


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Analysis using ANSYS

After making detailed study of the element library of ANSYS it

is decided that SOLID 46 will be the best suited element forour problem

The results obtained from analytical calculation is verifiedusing a standard analysis package ANSYS


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Using the formula taken from PSG Data Book Page 6.14 StorageModulus for the various specimens were determined

Natural frequency F = C (gEI/wL4) where

F Nodal FrequencyC Constantg Acceleration due to gravityE Modulus of elasticityI Moment of inertia

L Effective specimen length w Weight of the beam


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(a) First mode shape (b) second mode shape

(c) Third mode shape (d) Fourth mode shape


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TABLE: Frequency of the material analyzed up to 100Hz

Specimen Mode ShapeNatural Frequency (Hz) Storage Modulus E (GPa)

ANSYS Experiment ANSYS Experiment

GF-E

IIIIIIIV

1.93017.31769.736013.733

1.8558.009.84614.22

2.7691.010.230.11

2.511.210.230.12

GF-PP

IIIIIIIV

1.9135.733

9.628113.588

1.91046.40

9.9012.799

1.140.26

0.110.06

1.140.32

0.100.05

CF-E

IIIIII

IV

1.72705.17938.7048

12.295

1.735.1208.00

11.81

3.620.840.30

0.15

3.660.820.25

0.14


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Following Table shows the values for the loss factor (tan ) of all specimens considered.

damping results obtained for composite materials studied

Specimen Inertia (m)4 E (Gpa) Tan E (Gpa) E (Gpa)a

GE 3.25 10-11 12.05 0.0681 0.822 16.19

GPP 1.33 10-10 11.55 0.051 0.586 8.75

CE 1.66 10-11 50.54 0.095 4.806 14.48

a calculated by composite micromechanics approach


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Anti-roll stabilizer bars for four wheelers. Fatigue lifeof the stabilizer bars was estimated for qualification.


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A stress of about 6.756 MPa is much lesser than the Yield Stress of the material


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The lecture introduced the subject `Introduction toFinite Element Analysis (FEA) to theundergraduate audience. The basics, differentapproaches and the formulations were outlined inthe lecture. Emphasis was laid on solving

structural, mechanical and multiphysics problems.Understanding the material behaviour that is aprerequisite to the correct modelling of theproblem was also discussed. Some engineeringapplications of the FE approach as investigated bythe speaker were illustrated for the benefit of thestudent society and to enable them to appreciatethe depth of the subject field and take it up as theircareer .

intro to fea and applications

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