Introduction to Design with Finite Element Approach and Applications to Engineering

Problems  

Dr. K. Padmanabhan FIIPE, FIE, CE(I),FISME

ProfessorManufacturing Division School of MBSVIT-UniversityVellore 632014

February 2013

FEA Introduction

• Numerical method used for solving problems that cannot be solved analytically (e.g., due to complicated geometry, different materials)

• Well suited to computers• Originally applied to problems in solid

mechanics• Other application areas include heat transfer,

fluid flow, electromagnetism

Finite Element Method Phases• 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)

FEA Discretization Process - Meshing

• Continuous elastic structure (geometric continuum) divided into small (but finite), well-defined substructures, called elements

• Elements are connected together at nodes; nodes have degrees of freedom

• Discretization process known as meshing

Element Library

Spring Analogy

, ,

, similar to

F lE

A lEA

F l F kxl

, ,

, similar to

F lE

A lEA

F l F kxl

Elements modelled as linear springs

Matrix Formulation• Local elastic behaviour of each element

defined in matrix form in terms of loading, displacement, and stiffness– Stiffness determined by geometry and material

properties (AE/l)

Global Matrix Formulation

• Elements assembled through common nodes into a global matrix

• Global 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

Solution

• Matrix operations used to determine unknown dof’s (e.g., nodal displacements)

• Run time proportional to #nodes or elements

• Error messages– “Bad” elements– Insufficient disk space, RAM– Insufficiently constrained

Postprocessing

• Displacements used to derive strains and stresses

FEA Prerequisites

• First Principles (Newton’s Laws)– Body under external loading

• Area Moments of Inertia• Stress and Strain

– Principal stresses– Stress states: bending, shear, torsion, pressure,

contact, thermal expansion– Stress concentration factors

• Material Properties• Failure Modes• Dynamic Analysis

Theoretical Basis: Formulating Element Equations

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

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

• If the physical problem can be formulated as the minimization of a functional, then the Variational Formulation is usually used.

Theoretical Basis: MWR

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

• In the MWR, an approximate solution is substituted into the differential equation. Since the approximate solution does not identically satisfy the equation, 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 then gives Dh’’(x) + Q = R, where R is a nonzero residual. The MWR then requires that

Wi(x)R(x) = 0 (2)

where Wi(x) are the weighting functions. The number of weighting functions equals the number of unknown coefficients in the approximate solution.

Theoretical Basis: Galerkin’s Method

• There are several choices for the weighting functions, W i.

• In the Galerkin’s method, the weighting functions are the same functions that were used in the approximating equation.

• The Galerkin’s method yields the same results as the variational method when applied to differential equations that are self-adjoint.

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

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

Theoretical Basis: Variational Method

• The variational method involves the integral of a function that produces a number. Each new function produces a new number.

• 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 specific equation y = f(x). Variational calculus shows that the particular equation y = g(x) which yields the lowest numerical value for is the solution to the differential equation

Dy’’(x) + Q = 0. (2)

Theoretical Basis: Variational Method (cont.)

• In solid mechanics, the so-called Rayeigh-Ritz technique uses the Theorem of Minimum Potential Energy (with the potential energy being the functional, ) to develop the element equations.

• The trial solution that gives the minimum value of is the approximate solution.

• In other specialty areas, a variational principle can usually be found.

Sources of Error in the FEM• The three main sources of error in a typical FEM solution are

discretization errors, formulation errors and numerical errors.

• Discretization error results from transforming the physical system (continuum) into a finite element model, and can be related to modeling the boundary shape, the boundary conditions, etc.

Discretization error due to poor geometry representation.

Discretization error effectively eliminated.

Sources of Error in the FEM (cont.)

• 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 they are not suited are sometimes referred to as ill-conditioned or mathematically unsuitable elements.

• For example a particular finite element might be formulated on the assumption that displacements vary in a linear manner over the domain. Such an element will produce no formulation error when it is used to model a linearly varying physical problem (linear varying displacement field in this example), but would create a significant formulation error if it used to represent a quadratic or cubic varying displacement field.

Sources of Error in the FEM (cont.)

• Numerical error occurs as a result of numerical calculation procedures, and includes truncation errors and round off errors.

• Numerical error is therefore a problem mainly concerning the FEM vendors and developers.

• The user can also contribute to the numerical accuracy, for example, by specifying a physical quantity, say Young’s modulus, E, to an inadequate number of decimal places.

Advantages of the Finite Element Method

• Can readily handle complex geometry:• The heart and power of the FEM.

• Can handle complex analysis types:• Vibration • Transients • Nonlinear • Heat transfer • Fluids

• Can handle complex loading:• Node-based loading (point loads). • Element-based loading (pressure, thermal, inertial

forces). • Time or frequency dependent loading.

• Can handle complex restraints:• Indeterminate structures can be analyzed.

Advantages of the Finite Element Method (cont.)

• Can handle bodies comprised of nonhomogeneous materials:

• Every element in the model could be assigned a different set of material properties.

• Can handle bodies comprised of nonisotropic materials:

• Orthotropic

• Anisotropic

• Special material effects are handled:

• Temperature dependent properties.

• Plasticity

• Creep

• Swelling

• Special geometric effects can be modeled:

• Large displacements.

• Large rotations.

• Contact (gap) condition.

Disadvantages of the Finite Element Method

• A specific numerical result is obtained for a specific problem. A general closed-form solution, which would permit one to examine system response to changes in various parameters, is not produced.

• The FEM is applied to an approximation of the mathematical model of a system (the source of so-called inherited errors.)

• Experience and judgment are needed in order to construct a good finite element model.

• A powerful computer and reliable FEM software are essential.

• Input and output data may be large and tedious to prepare and interpret.

Disadvantages of the Finite Element Method (cont.)

• Numerical problems:• Computers only carry a finite number of significant

digits. • 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 Buckling • Hybrid composites. • Nanomaterials modelling .• Multiple simultaneous causes.

Coupled Field Analysis

Module 6

• In this, we will briefly describe how to do 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.

Coupled Field AnalysisOverview

Thermally Induced Stress• When a structure is heated or cooled, it

deforms by expanding or contracting.• If the deformation is somehow

restricted — by displacement constraints or an opposing pressure, for example — thermal stresses are induced in the structure.

• Another cause of thermal stresses is non-uniform deformation, due to different materials (i.e, different coefficients of thermal expansion).

Thermal stressesdue to constraints

Thermal stressesdue to different

materials

Coupled Field Analysis…Overview

• There are two methods of solving thermal-stress problems using ANSYS. Both methods have their advantages.– Sequential coupled field

- Older method, uses two element types mapping thermal results 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

Coupled Field Analysis…Overview

• The Sequential method involves two analyses:1. First do a steady-state (or transient)

thermal analysis.• Model with thermal elements.• Apply thermal loading.• Solve and review results.

2. Then do a static structural analysis.• Switch element types to structural.• Define structural material properties,

including thermal expansion coefficient.• Apply structural loading, including

temperatures from thermal analysis.• Solve and review results.

ThermalAnalysis

StructuralAnalysis

jobname.rth

jobname.rst

Temperatures

Coupled Field AnalysisA. Sequential Method

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

1. First prepare the model and mesh using one of the following coupled field element types.• PLANE13 (plane solid).• SOLID5 (hexahedron).• SOLID98 (tetrahedron).

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

3. Solve and review both thermal and structural results.

Combined

ThermalAnalysis

StructuralAnalysis

jobname.rst

Coupled Field AnalysisB. Direct Method

Coupled Field Analysis

Sequential vs. Direct Method• Direct

– Direct coupling is advantageous when the coupled-field interaction is highly nonlinear and is best solved in a single solution using a coupled formulation.

– Examples of direct coupling include piezoelectric analysis, conjugate heat transfer with fluid flow, and circuit-electromagnetic analysis.

• Sequential– For coupling situations

which do not exhibit a high degree of nonlinear interaction, the sequential method is more efficient and flexible because you can perform the two analyses independently of each other.

– You can use nodal temperatures from ANY load step or time-point in the thermal analysis as loads for the stress analysis. .

Case Study 1: Composites in Microelectronic Packaging

The BOM includes Copper lead frame,Gold wires for bonding, Silver –epoxy for die attach, Silicon die and Epoxy mould composite with Phenolics, Fusedsilica powder and Carbon black powderas the encapsulant materials. Electrical-Thermal and thermal-structural analyses.

Thermal – Structural Results

Displacement Vector sum Von mises stress

Stress intensity XY Shear stress

Case Study 2: Composites in Prosthodontics

Tooth is a functionally gradedcomposite material with enameland dentin. In the third maxillary molar the occlusal stress can be 2-3 MPa.

The masticatory heavy chewing stress will be around 193 MPa.A composite restorative must with stand this with an FOS and with constant hygrothermal attack.

Case study 3: Various Buckling Analyses

LINEAR

Eccentric Column

Eccentric Column-FEM MODEL

x: 0-0.13

y: 0-0.15

x: 0-0.12

y: 0-0.15FEM METHOD

Load-Deflection Plots

FEM MODEL OF HOLLOW CYLINDER

Outer diameter = 158mmInner diameter = 138mmHeight = 900mmPoisson’s ratio = 0.29Young’s Modulus = 2.15e5 N/mm2

The element used for this model is Solid 186.The

applied pressure is 0.430N/m2. For this analysis large deformation was set ON and also Arc length solution was turned ON.

Hollow Cylinder Dimensions

FEM METHOD

x: 0-2,y: 0-2.5

TOPOLOGICAL METHOD

x: 0-2, y: 0-2.5

Non-linear

00.5

11.5

22.5

3

0 1 2 3x

y

x=0.1 1.25y=0.5x (4-x)

BI-MODAL BUCKLING

Two coaxial tubes, the inner one of steel and cross-sectional area As, and the outer one of

Aluminum alloy and of area Aa,

are compressed between heavy, flat end plates, as shown in figure. Assuming that the end plates are so stiff that both tubes are shortened by exactly the same amount.

Compression of a Pipe

Pipe-FEM MODEL

BI-MODAL BUCKLING

x: 0.2-1

y: 0-0.32

x: 0.2-1

y: 0-0.19FEM METHOD

HINGED SHELL

A hinged cylindrical shell is subjected to a vertical point load (P) at its center.

Snap buckling of a hinged shell

Hinged cylindrical shell-FEM MODEL

x: 0-1.65

y: 0-1 FEM METHOD

Snap-back buckling of a hinged shell

SNAP-THROUGH BUCKLING

x: 0-1.3

y: 0-1.6FEM METHOD

Case Study 4: Vibration of Composite Plates

• Vibration studies in composites are important as the composites are increasingly being used in automotive, aerospace and wind energy applications.

• The combined effect of vibrations and fatigue can degrade a composite further that is already hygrothermal in affinity.

• The different modes of vibrations are discussed here.

Element selection for ANSYS SOLID 46

3D LAYERED STRUCTURAL SOLID ELEMENTElement definition

─ Layered version of the 8-node, 3D solid element, solid 45 with three degrees of 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.

Contd…

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 for our problem

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

SOLID46 3-D 8-Node Layered Structural Solid

Finding Storage Modulus (E’)

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

Natural frequency F = C√ (gEI/wL4)

where

F – Nodal Frequency

C – Constant

g – Acceleration due to gravity

E – Modulus of elasticity

I – Moment of inertia

L – Effective specimen length

w – Weight of the beam

ANSYS MODE SHAPE FOR CARBON FIBRE/EPOXY COMPOSITE

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

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

ANSYS MODE SHAPE FOR GLASS FIBRE/EPOXY COMPOSITE

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

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

ANSYS MODE SHAPE FOR GLASS/POLYPROPYLENE COMPOSITE

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

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

Contd…

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

1.93017.3176 9.736013.733

1.8558.009.84614.22

2.7691.010.230.11

2.511.210.230.12

GF-PP

IIIIIIIV

1.9135.7339.628113.588

1.91046.409.90

12.799

1.140.260.110.06

1.140.320.100.05

CF-E

IIIIIIIV

1.72705.17938.704812.295

1.735.1208.0011.81

3.620.840.300.15

3.660.820.250.14

Determination of Loss Modulus (E”) and Loss Factor (tan δ)

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

damping results obtained for composite materials studied

Specimen Inertia (m)4 E’ (Gpa) Tan δ 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

Case Study 5: Stabilizer Bars for Four Wheelers

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

Deflection Plot for Stabilizer Bar

Deflection Plot for Stabilizer Tube

Equivalent Stresses for Bar

Equivalent Stresses for Tube

Case Study 6: LCA Generator• The study deals with modeling, analysis and performance

evaluation of 5kW DC generator assembly. The complete solid model of the generator with its accessories was modelled using Pro-Engineer. This paper deals with the structural analysis of the DC generator casing to find stress and deflection in the generator casing due to load factor of 9g to which it is designed. The effect of vibration of generator casing and hollow shaft with mounting are investigated through detailed finite element analysis. The bending and torsional natural frequencies of the hollow shaft are estimated to find the critical speeds. Torsional frequency of the hollow shaft is estimated by considering the mass moment of inertias of the rotating masses. For critical speed analysis of the hollow shaft, it is considered as simply supported beam with the required masses and inertias. Then the influence of the critical speeds due to the casing stiffness is found out analyzing the casing with the shaft together.

Model of LCA Generator

Cross-section of the Model

Total Deflection at 9g

Maximum deflection of the generator will be 4.761 microns, with-in limits !

Von Mises Stresses at 9g

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

Mode Shape of Generator Shaft

Mode shape corresponding to the flexural critical speed (54,972 rpm) (using solid element TET10 approximation)

Conclusions

• The lecture introduced the subject `Introduction to Finite Element Analysis (FEA) ’ to the undergraduate audience. The basics, different approaches and the formulations were outlined in the lecture. Emphasis was laid on solving structural, mechanical and multiphysics problems. Understanding the material behaviour that is a prerequisite to the correct modelling of the problem was also discussed. Some engineering applications of the FE approach as investigated by the speaker were illustrated for the benefit of the student society and to enable them to appreciate the depth of the subject field and take it up as their career .

Rig Veda on Infinity

pûrnamadah pûrnamidam pûrnât pûrnamudacyate pûrnâsya

pûrnamadaya pûrnamevâvasishyate

From infinity is born infinity.When infinity is taken out of infinity,

only infinity is left over.