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A Designer’s Guide To Finite Element Analysis

Contents

Contents Introduction ......................................................................................................2 About This Series ...........................................................................................2 About This Book .............................................................................................2 Supporting Material ........................................................................................2 FEA – What It Is … And What It Isn’t..................................................................4 Typical Usage.................................................................................................4 Limitations .....................................................................................................5 Learning FEA..................................................................................................8 The Importance Of Computing Power..............................................................9 Mechanics, Mathematics, And Other Theory ...................................................... 10 Solid Mechanics............................................................................................ 11 Thermal Analysis .......................................................................................... 23 Fatigue And Fracture .................................................................................... 26 Mathematics ................................................................................................ 28 Essential FEA Theory ....................................................................................... 38 From The Differential Equation To A Matrix Equation...................................... 38 Nodes, Elements And Shape Functions .......................................................... 43 Some Common Elements Used In Stress Analysis ........................................... 45 Matrix Solvers .............................................................................................. 47 Some Important Properties Of The FE Solution .............................................. 50 Putting It Together – OptiStruct/Analysis .......................................................... 52 Capabilities .................................................................................................. 52 Setting Up An “Analysis” ............................................................................... 57 Nomenclature and Data Organization ............................................................ 61 Verification And Validation................................................................................ 65 Product Liability Laws ................................................................................... 66 The Seductive Appeal Of Graphics ................................................................. 66 Quick and Basic Checks ................................................................................ 67 Special Topics ................................................................................................. 70 Advanced Materials ...................................................................................... 70 Advanced Dynamics ..................................................................................... 72 Glossary And References.................................................................................. 74 References................................................................................................... 77 Common Material Properties ......................................................................... 77 Useful Data For Heat Transfer....................................................................... 79 Consistent Units ........................................................................................... 81 Lumped Mass Models In Vehicle-Crash Simulation .......................................... 81 Measures Of Element Quality ........................................................................ 82

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Introduction

A Designer’s Guide To Finite Element Analysis

Introduction About This Series To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments. Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book This volume introduces you to Finite Element Analysis, a numerical method that is the cornerstone of most Computer Aided Engineering (CAE) programs in the industry. Design is addressed in the other volumes of this series. This book focuses wholly on Analysis. However the presentation is from a design perspective: FEA is presented with a minimum of mathematics, and with a strong focus on applications. You may choose to treat some chapters as references, depending on your level of familiarity with the underlying theory. If you find any chapter hard to follow, you should read it once for an overview, and refer to it again when doing the assignments if necessary. The various references cited in the book will probably be most useful after you have worked through your project and are interpreting the results.

Supporting Material Your instructor will have the Student Projects and Student Projects Summaries that accompany these volumes – they should certainly be made 2

A Designer’s Guide To Finite Element Analysis

Introduction

use of. Further reading and references are indicated both in this book and in the Projects themselves. If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

Do not be afraid to skip equations (I do this frequently myself). Roger Penrose

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FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t “After working nearly four years with engineers, a mathematician confessed to me that he finally began to understand what engineers meant by finite elements … And another mathematician, attending an engineering conference, seemingly in disbelief begged the author to tell him whether engineers really thought of finite elements as little plate elements connected at their edges …1”. For an engineering tool that dates back to at least 1960 when Clough chose the term “Finite Elements”2, the Finite Element Method has had more than its fair share of misinterpretations. Part of this, of course, is simply because of its success. As a robust, reliable and versatile tool, it has enjoyed unparalleled success over the past several decades. Almost every field of engineering – Aeronautical, Naval, Mechanical, Electrical, Chemical, Civil, … – has seen the method deployed with spectacular success. And, unfortunately, a few spectacular failures. A good place to start the study of this fascinating blend of mathematics and engineering-design is by reviewing some typical applications.

Typical Usage Most design involves one or more of form, function, and fit. The various tasks of an Engineering Analyst revolve around the investigation of the function. Industrial designers usually address form, while CAD modelers address fit. Stress Analysis is easily the most widely known of these. Given a component or an assembly, the analyst assesses the likelihood that the stress in the material will fall within permissible limits. The difficulty lies in the shapes of the parts to be analyzed, the properties of the materials used, and the various sources of loads. In most popular usage, stress analysis is taken to apply only to solids – that is, to the study of Solid Mechanics. 1

“The Mathematical Foundation of Structural Mechanics”, Friedel Hartmann And probably even further back to 1943 when Courant introduced the linear triangular element.

2

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FEA – What It Is … And What It Isn’t

Fluid Mechanics can often be treated independently of Solid Mechanics. An engineer investigating the flow of air around a car or the flow of water around a speedboat focuses on the drag forces, leaving the study of stresses for later investigations. Sometimes, however, solid and fluid mechanics cannot be separated as easily – such coupled analyses are called Fluid Structure Interactions (FSI). Thermal engineering involves the study of the flow of heat, often covered in undergraduate courses on Heat Transfer. The analyst’s job is to predict the temperature distribution in the areas of interest. Mass Transfer is a closely related field, encountered for instance in the mixing of chemicals in a reactor, the seepage of underground water around an oil well, or the effects of tidal flow on the release of effluents. A new entrant to the field can be excused for believing that FEA is the engineering equivalent of a miracle-drug, a cure for every disease known and unknown. Experienced engineers, of course, know that there’s no such thing. Or of there is, it hasn’t been discovered yet! It’s important, therefore, to also remember that there are some fields where FEA can be used only sparingly if at all, and that there are some fields where other tools are better deployed than FEA.

Limitations As we will see later, FEA can be used to investigate the behavior of any process that can be described by a differential equation. However, it’s one of many methods that can be, and have been, used to solve differential equations. Some of the mechanical-engineering disciplines where other methods are more widely used than FEA are the analysis of rigid bodies, the flow of fluids, and acoustics.

Mechanisms One of the first topics a student of mechanical engineering encounters is the difference between structures and mechanisms. Plainly put, a mechanism is 5

FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

a body that is capable of rigid movement. Under the action of a force, the mechanism gains kinetic energy. This is in contrast to structures, which gain strain energy under the action of a force. Finite Element methods have been extended to include the simulation of both behaviors, for often times the analyst cannot be expected to know when which form of energy-transfer dominates. A good example of this is the simulation of a car-crash. At some instants, when the car is in contact with other bodies such as the vehicle or the road, its strain energy is increasing. At other instants, when the car loses contact with other bodies as it flips over or bounces, it behaves as a rigid body. The analyst cannot know before hand which event occurs when: the solution of the equations must provide this information! Situations like these call for multi-disciplinary or multi-physics analyses, and FEA is indeed a vital part of the investigations. But in many other cases, an investigation of the kinematics or dynamics of the mechanism is a prelude to stress analysis. The accelerations calculated as a part of mechanisms-analysis are used to estimate the forces that will be used for subsequent stress-analysis. Some FE solvers today provide access to mechanisms-analyses too. OptiStruct, which we will discuss later, provides such facilities, but it is important to remember that kinematics and dynamics calculations use numerical methods other than FEA3.

Fluid Flow The Navier-Stokes equations have long troubled engineers. Sir Horace Lamb is said to have observed “I am an old man now, and when I die and go to

Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. And about the former I am rather more optimistic4.” The Navier-Stokes equations themselves have various applications – for incompressible flow, for transonic flow, for laminar flow, and so on.

3

See CAE For Rigid Body Mechanics Quoted in Computational Fluid Mechanics and Heat Transfer, by Anderson, Tannehill, and Pletcher, 1984. 4

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A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t

Some forms of fluid flow have been simulated using FEA, but this is more the exception than the rule. Computational Fluid Dynamics (CFD) as a science has tended towards Finite Difference Methods (FDM), Finite Volume Methods (FVM) and Smoothed Particle Hydrodynamics (SPH). The reasons for this are many, and beyond the scope of this book. We will only note that if FEA is used to analyze fluid flows, great care should be exercised both in modeling and interpreting the results.

Acoustics Vibration is an essential part of engineering design. Not only does it cause stress and strain, it causes discomfort and generates noise. All noise, of course, is sound. And all sound is a form of vibration. However the reverse is not true. Engineers, particularly in fields related to vehicle design, use the term NVH (for Noise, Vibration and Harshness) to segregate vibrations based on the frequency of the vibration. FEA has been employed very successfully at the lower end of the frequency-spectrum, typically for excitations that are lower than about 10,000 rpm. There is nothing sacrosanct about that number, of course – FEA has also been used for higher frequency vibrations. But the use of alternative numerical methods (BEM or Boundary Element Methods are quite popular) is more common in the investigation of the generation and transmission of sound or noise.

NVH Characteristics 5 – 25 Hz: Shake

25 – 100 Hz: Harshness, Boom

100 – 150 Hz: Moan

150 – 300 Hz: Noise

Some FE solvers, including OptiStruct, provide acoustic analysis capabilities too.

Time Integration Quantum Mechanicians like to theorize about time travel. But for most engineers, the spatial and temporal domains are firmly separated. The first involves the familiar spatial coordinates – x / y / z in Cartesian coordinates, r / θ / z in Polar coordinates and r / θ / ϕ in Spherical coordinates – and is amenable to the mathematics of Finite Elements. Problems that involve a 7

FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

variation of the parameters of interest in the spatial domain are called Boundary Value Problems. Time is not so accommodating. Problems in which the variables of interest change with time are called Initial Value Problems. These are often solved by applying FEA to the spatial-part of the differential equation and FDM to the timevariation.

Learning FEA Traditionally, authors tended to choose one of two approaches to introduce FEA. One is a purely mathematical approach. This uses functional analysis, which is one way to study the behavior of functions whose arguments are functions. This approach has the advantage that it is very general in its application. There is usually little mention of applications. The other is an application specific approach, where the method is developed entirely in the context of one or more specific design disciplines. The advantage is that the physical significance of the various terms is emphasized right from the start, while the disadvantage is that the logic is not readily transferred to other disciplines. Fortunately, over the years, the method has matured. Mathematical rigor has been added to most engineering discoveries. Examples of usage and relevance illustrate almost every mathematical proof. Like most powerful tools, FEA in the hands of a novice can wreak havoc. For effective use, an understanding of the mechanics is essential. A grasp of the mathematics can help enormously, particularly to diagnose errors, if they should occur. The next chapter presents a quick summary of mechanics, recalling topics you would have covered in your courses on mechanics. This is followed by a brief summary of the theory of Finite Elements, before moving on to applications.

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A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t

The Importance Of Computing Power The popularity of FEA is inextricably tied with the growth of computing power. If FEA is today a part of the computer-desktop of most practicing designers, it is no accident. It is largely due to the current proliferation of desktop computers, several of which are more powerful than decade-old supercomputers. The two most important measures of computational power are storage space and computing speed. The Finite Element method, of course, makes strong demands on both. The level of detail addressed by FE models has grown steadily over the years, making the best use of available computing power5. One of the more important trends in analysis today is the increasing usage of stochastic analyses, which can often require hundred or even thousands of analyses of the same model to yield more reliable designs6. The assignment problems that accompany this book are completely realistic, drawn from actual engineering applications. However they are restricted to linear analysis only. The other volumes in this series address more advanced applications of FEA: optimization, reliability, multi-disciplinary analyses, and manufacturing simulation.

It was absolutely marvelous working for Pauli. You could ask him anything. There was no worry that he would think a particular question was stupid, since he thought all questions were stupid. Frederick Weisskopf

5

The table is from the American Iron And Steel Institute’s Vehicle Crashworthiness

And Occupant Safety 6 See CAE And Design Optimization - Advanced 9

Mechanics, Mathematics, And Other Theory

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory Mathematical physics saw an explosion of activity in the early 18th century, following the invention of calculus. The initial enthusiasm led to the development of numerous partial differential equations as a result of investigations into a wide variety of areas. Developing a differential equation to model a physical phenomenon only addresses half the problem. It shows that the phenomenon is understood. The other half of the problem, of course, is to predict behavior: how parameters of interest will change as controllable parameters are varied. This has been long recognized7: We remember very well the time when almost every author dealing with a non-trivial elasticity problem considered it very nearly a matter of his honour to reduce it by all means to a Fredholm equation of the second kind. After this, he was prone at least to think that his investigation was completed theoretically without concerning himself with the implementation of the solution. (People wonder at it now).

For a long time, the search for solutions to differential equations was an art by itself. Every undergraduate student of engineering is familiar with the wide variety of methods to solve differential equations: integrating factors, separation of variable, partial fractions, substitution, and so on. One method, which proved to be particularly successful, was the use of infinite series to estimate the solutions. This led to another burst of activity in the development of infinite series. As we will see subsequently, Finite Element Analysis can be viewed as yet another series method of solving differential equations. In this chapter, we will review the various terms in mechanics and mathematics. We will need to be familiar with these in order to interpret the results of a Finite Element Analysis. Note that rigor has been omitted, and sometimes even sacrificed, for simplicity and brevity.

7

10

From Integral Equations in Elasticity, V.Z.Parton and P.I.Perlin, 1977

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

Solid Mechanics As outlined in the earlier chapter, FEA is not the preferred method for problems involving fluid flow. The sections below, therefore, address the mechanics of solids only. The emphasis is on introducing the continuum approach. In this, we define most parameters of interest using differential equations rather than difference equations.

Engineering Mechanics It is not always easy to determine whether it is better to use difference or differential equations. A first semester course on engineering mechanics usually adopts the former approach. In this approach, we first define a link as a rigid body that possesses at least two joints, and a joint as a point of attachment between links. A binary link has 2 joints, a ternary link has 3 and a quaternary link has 4. We then use Gruebler’s equation to determine whether the assemblage is a

structure or a mechanism: if there are no unconstrained degrees-of-freedom it is a structure, else it is a mechanism. We further categorize structures as statically determinate and statically indeterminate. The former are structures which have 0 degrees of freedom. All forces can be calculated using only the equilibrium equation: that is, by solving for the force and moment balances.

∑F = ∑F = ∑F = ∑M x

y

z

x

= ∑My = ∑Mz = 0

Statically indeterminate structures are those that are over-determined. They have “negative” degrees of freedom. To solve these, the force and moment equilibrium equations are not enough. The elastic properties of the body must be used. Methods of solution involve the construction of free body diagrams. This involves showing all external forces acting on a body. If the externally applied forces are independent of time, the problem is one of Statics. If, however, the externally applied forces are time dependent, we are faced with a problem in Dynamics. 11

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A dynamic problem can be solved by applying D’Alembert’s principle. In this, we assume the body is in static equilibrium at every instant by applying a D’Alembert force. This is calculated as

F = M ⋅a where a is the acceleration and m is the mass. We assume that F is applied at the center of mass, and is in the direction opposite to the acceleration. The stress in the link is given by

σ=

F a

ε=

∆L L

The strain is defined by

Stress and strain are related by Hooke’s Law,

σ = Eε where E is the Modulus of Elasticity of the Young’s Modulus. Hooke’s Law is sometimes referred to as the constitutive equation, since it involves the material properties, which in turn depend on the constitution of the material.

Continuum Mechanics The difference-equation approach is valid as long as it is reasonable to treat the link as a rigid body. That is, if it is reasonable to assume that there is no variation of stress or deformation within the link itself. In engineering design, these assumptions are relevant if the links are more or less unidimensional – long and thin – and if the links are joined with pins. The study of such assemblies usually centers around “trusses”. Electric-transmission towers are an excellent example. If the areas of interest involve changes in cross-section, in area, in material properties, etc., it is better to use a continuum approach. That is, to assume that the variables of interest vary though the continuum, and to write equations that allow us to track these changes. 12

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The first step is to define the deformation vector. Recall that a body in 3dimensional space has 6 degrees of freedom – translations and rotations about 3 coordinate axes. Remember also that we need to differentiate between rigid movements and flexible deformations. A rigid movement is relevant to a mechanism, while a structure must deform under the action of external forces. By definition, the structure cannot move as a rigid body. The displacement vector has six components. These are decomposed into a deformation vector and a rotation vector. The deformation vector is usually denoted by u. Since continuum mechanics involves extensive use of vectors, columnmatrices or column vectors are frequently employed to represent quantities of interest. Some quantities like stress, which have components that themselves are vectors, are called tensors8. Again, matrices are used to represent the components. In vector form, the deformation vector is

u x u = u y u z Strain is defined as the rate of change of deformation. If we restrict our attention to a single dimension9, we have

ε=

du dx

In the more general case, working in 3-dimensions, the above equation is replaced by

8

Strictly speaking, a tensor is a group of functions that satisfy some conditions. Matrices are a convenient way to represent tensors, but not every matrix is a tensor. 9 And restricting attention to small strains. That is, we assume (du/dx)2 and higher powers are much less than du/dx and can be ignored. 13

Mechanics, Mathematics, And Other Theory

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∂u x ∂x ε xx ∂u x ε ∂y xy ∂u ε xz x ∂z ε = = ∂u ε yy y ∂y ε xz ∂u y ε zz ∂z u ∂ z ∂z The strain tensor is given by

ε xx ε = ε yx ε zx

ε xy ε xz ε yy ε yz ε zy ε zz

Note that the strain tensor is symmetric. That is, εij = εji. Accordingly, there are 6 components for the strain: 3 normal strains and 3 shear strains. Similarly, the stress tensor is

σ xx σ xy σ xz σ = σ yx σ yy σ yz σ zx σ zy σ zz and is also symmetric, with σij = σji. It is useful to remember the physical meaning of the vector notation used for tensors. For instance, σxy is the stress caused by a force in the y direction acting on a face whose normal is in the x direction. The force per unit area is called the traction. When resolved on a particular face, we get the stress component.

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The drawback of the stress and strain tensors is that they are coordinate dependent. In other words, a change of coordinates can change the stresses and strains. This reduces the utility of the mathematical definitions of the stress and the strain. Physically, we expect the stresses and strains to give us an indication as to the state of the body. Obviously a change in axes cannot change the physical state of the body! To get around this, we usually construct strain invariants and stress invariants. These are derived from the strain and stress tensors respectively, and are invariant – that is, they do not change under coordinate transformations. The invariants use principal values: the principal stresses and the principal strains. An easy way to calculate principal stresses is to solve the equation

σ xx − λ σ xy σ xz σ yx σ yy − λ σ yx = 0 σ zx σ zy σ zz − λ for λ. From linear algebra we recall that for the above equation gives us a cubic equation that has 3 roots – these are the 3 principal stresses for that state of stress. Mohr’s Circle is another way to calculate the principal stresses, σ1, σ2 and σ3. Similarly, we can calculate the principal strains. The main use of the principal stresses and strains is to calculate a value that can be used in a failure theory. Various failure theories have been proposed based on empirical observations. For ductile materials like steel, for instance, the Von Mises criterion is often used:

σe =

(σ 1 − σ 2 )2 + (σ

− σ 3 ) + (σ 3 − σ 1 ) 2

2

2

2

As per this theory, a material will yield when the equivalent stress crosses the yield stress. 15

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The yield stress or the yield point is most familiar when considered for a unidimensional experimental specimen:

Not all materials have a clearly defined yield point. For these, a proof stress or offset yield stress is sometimes used. This is obtained by drawing the line as shown, at a strain of 0.002. This strain is called the 0.2% proof strain. For tri-axial states of stress, designers often use the principal stresses as axes. In this depiction, the yield surface marks the limit of the elastic region. Stress and strain are related by the 3-dimensional version of Hooke’s Law. Since the stress and strain each have 6 components, the material properties are often represented by a constitutive matrix.

σ 11 c11 σ 12 c 21 σ 13 c 31 = σ 22 c 41 σ 23 c 51 σ 33 c 61

c12 c 22 c32 c 42 c52 c 62

c13 c 23 c33 c 43 c53 c 63

c14 c 24 c34 c 44 c54 c 64

c15 c 25 c 35 c 45 c 55 c 65

c16 ε 11 c 26 ε 12 c36 ε 13 c 46 ε 22 c56 ε 23 c 66 ε 33

Note that in the above equation the subscripts 1, 2 and 3 have been used instead of x, y and z. Finally, the equilibrium equation is now three differential equations: 16

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∂σ xx ∂σ yx ∂σ zx + + + Bx = 0 ∂x ∂y ∂z ∂σ xy ∂σ yy ∂σ zy + + + By = 0 ∂x ∂y ∂z ∂σ xz ∂σ yz ∂σ zz + + + Bz = 0 ∂x ∂y ∂z where Bx, By and Bz are the body forces in the respective directions x, y and z. A body force is distinct from a boundary force. The former is distributed all through the body – for instance the magnetic attraction an iron bar experiences, the force due to gravity, etc. The latter can be a point, line force, or an area force. Point loads are also called concentrated loads. It is important to remember that point loads and line loads are idealizations – they can never be achieved in practice.

Statics – From Theory To Design In the equations above, time variations were ignored. All derivatives were spatial derivatives only. In other words, the equations defined problems of static analysis. Designers often use a factor of safety to allow for uncertainties in various data. In this case, the permissible stress is a fraction of the yield stress:

σp =

σy k

where k is the stress concentration factor. Design codes use historical data to suggest factors of safety for different applications. In closing, two important aspects of stress analysis are worth noting: stress

concentration factors and St.Venant’s Principle.

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Mechanics, Mathematics, And Other Theory

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If a geometry is complicated, the nominal stress is calculated over a simpler geometry, and then factored up using a stress-concentration factor. Handbooks10 present factors for various configurations. If a load is hard to characterize, we rely on St.Venant’s principle, which tells us that the difference between the stresses caused by statically equivalent

load systems is insignificant at distances greater than the largest dimension of the area over which the loads are acting. This means that we can replace a load by its static equivalent, and is the reason why point and line loads are used even if they are not realistic.

Dynamics - Theory If variables of interest change with time, the differential equations need to be modified to include time derivatives. Conventionally, different symbols are used for spatial-derivatives and time derivatives: a prime for the former and a dot for the latter:

du d 2u = u ′, 2 = u ′′ dx du du d 2u = u&, 2 = u&& dt dt Note that that the first and second time derivatives of deformation (u) give the velocity (v) and the acceleration (a) respectively. Problems in dynamic analysis are harder to characterize than problems in statics. To understand why, it is useful to recall the equilibrium equations for the mass-spring-dashpot, which is a single-degree-of-freedom system:

ma + cv + ku = p The mass and stiffness are relatively easy to measure, but the damping is often difficult to obtain. In the equation above, we assume that the dashpot is a viscous damper. That is, the resisting force depends on the velocity of motion. Viscous

10

18

See, for instance, “Stress Concentration Factors” by R.E.Peterson

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

damping is not always a good model, however. Various alternatives are available, including hysteretic damping, sliding friction, Coulomb damping, etc. Unfortunately damping, like friction, is hard to characterize. We often continue to use the above equation. To maintain accuracy, we modify the damping coefficient c to match experimental results. This is sometimes called the equivalent damping coefficient. Estimating the damping coefficient requires experimentally collected data. For most materials, the damping coefficient depends on the frequency of excitation. A material that behaves in this fashion is characterized by modal damping factors. The dependence of the damping coefficient is not always predictable: higher frequency excitations die away (or attenuate) more rapidly than lower frequency excitations, for most mechanical engineering applications11. To estimate the damping coefficient, an experimental specimen is excited by a source that sweeps through the entire range of frequencies of interest. The response is captured and characterized as the Frequency Response Function (FRF).

H (ω ) =

X (ω ) F (ω )

An FRF is a transfer function that relates the excitation at one point with the response at a point of interest. Note that an FRF is a function of the frequency of excitation, ω12. If there are multiple points of excitation / interest, the FRFs are best treated as a matrix, with

H ij (ω ) =

X i (ω ) F j (ω )

The damping coefficients can be estimated from the FRFs as a function of frequency. The critical damping value ζ, which is the value at which the 11

An important exception is the occurrence of flutter. See the references for more details. 12 ω is usually measured in Hertz (Hz). Beginners often confuse rpm, Hz and radians / sec. 19

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structure loses its oscillatory response to any excitation ω, is used to estimate the modal damping coefficients σ:

ζ =

σ σ + ω2 2

The single-degree-of-freedom system can be readily extended to a multipledegree-of-freedom system using similar equations, with the added complexity of manipulating multiple equations. Often matrices are used to make the manipulation easier. Using matrix notation, the equilibrium equation is written as

[M ]{a}+ [C ]{v} + [K ]{u} = { f } where the square braces represent square matrices, and the curly braces represent column vectors. For a system that has n degrees of freedom, M, C and K are each nxn, while a, v, u and f are each nx1. The system of equations needs to be extended to infinite degrees of freedom if it is to represent a continuum, which is what an ideal elastic body is. The principal is the same even if the mathematics can be quite messy. Recall that we earlier wrote the equilibrium equation in terms of the stress components and the body forces. Dynamics involves accelerations and velocities. These are derived from the deformation vector by taking the time derivative. We can rewrite the equation in terms of the deformation using the stress-strain and strain-deformation equations. One familiar form of this equation is the wave equation. In its simplest form, for example the vibration of a string, the equation is: 2 ∂2 y 2 ∂ y = a ∂t 2 ∂x 2

where a depends on the tension in the string and the mass-per-unit-length (or density) of the string.

Dynamics – From Theory To Design Most design problems involve multiple degrees of freedom. If the structure consists of rigid links, as in a transmission tower, then a model with a finite 20

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number of degrees of freedom is a good representation. If, however, the structure is like a casting or is made from sheet-metal, a continuum representation is more appropriate. In this case, the structure has infinite degrees of freedom. Regardless of which approach is taken, the reality is that most models have more than one degree of freedom. If there are no external forces, the equilibrium equation that we wrote above becomes

[M ]{a} + [C ]{v} + [K ]{u} = 0 One obvious solution to this equation is the trivial solution: an identically zero deformation vector. But could there be other solutions? In other words, is it possible that the structure will deform even in the absence of an external force? The answer is “yes”, and leads to the study of mode-shapes and natural-frequencies, which lie at the very heart of studies of vibration13. If we assume that the deformation is of the form

u ( x, y, z , t ) = uo ( x, y, z ) sin(ωt ) we then have

v = u& = ω.u0 cos(ωt ) and

a = u&& = −ω 2 .u0 sin(ωt ) Of course, there is no necessity that we only use trigonometric functions – we could well use other forms, but this is a very convenient choice. If we neglect damping14, the equilibrium equation now becomes

− w 2 [M ]{u0 } + [K ]{u0 } = 0 13

The Sturm-Liouville equation is usually treated as the basis for all such investigations for partial-differential equations. 14 Only for brevity. Damping need not be excluded. 21

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which can be rewritten as

[A]{u0 } = 0 where

[A] = −w2 [M ] + [K ] If we assume that

λ {uo } is a solution, then we can rewrite the equation as

[A − λI ]{u0 } = 0 where [I] is the identity matrix. This equation will have non-trivial solutions (i.e. solutions where u0 is not identically zero) if the matrix is singular. That is, if the determinant is zero. The equation

A − λI = 0 gives us a polynomial equation in λ, called the characteristic equation. The order of the polynomial equation depends on the size of the matrix A. An nxn matrix gives us an nth order polynomial, which will have n roots. For each of these values of λ, we can find a corresponding {u}. These λ and {u} pairs are called eigen-solutions. λ is called the natural-frequency or eigenvalue and {u} is the corresponding mode shape or eigen-vector. If we include the damping factors, [C], the characteristic equation may have imaginary roots. These imaginary roots can be interpreted physically15: the 15

Interested readers should review de Moivre’s theorem, the Argand diagram, Nyquist and Bode plots 22

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imaginary terms are the equivalent of a phase lag. That is, due to the presence of damping, response lags behind excitation. In the absence of damping, the eigen-values are all real: physically, this means the response and excitation are always in phase. In this case, the eigen-solutions are called normal modes. Complex Modes are what we get if damping is included. Normal modes have stationary nodal lines – that is, they are standing waves. Complex modes are traveling waves. In many physical structures, damping is neglected at the initial stage when estimating natural frequencies, and introduced later when estimating forced response, which is the response to an excitation. This simplified procedure is reliable and very widely used. This is particularly convenient in product design. Remember that the damping factors need to be estimated experimentally, which means a prototype needs to be available for testing. Early in the design cycle, this is unrealistic. However estimation of mass and stiffness requires no experimentation. CAD users will know that solid models readily provide the mass, and we will see later how FE models provide the stiffness. Therefore, early in the design cycle, the concentration is on the calculation of the normal modes. In most cases we want to avoid resonance, but in some cases we may want resonance. Either way, we use the results of the analysis to modify the design. To calculate the forced response, we often use damping coefficients from literature or from databases of earlier designs. When the prototype is built and tested, a quick verification of the earlier design can be carried out.

Thermal Analysis The study of heat, Thermodynamics, is notoriously difficult:

Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more16. 16

Supposed to have been said by Arnold Sommerfeld, when asked why he had not written a book on the subject. 23

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Temperature itself, however, is much easier to deal with than stress, mainly because it’s a scalar quantity. The difficulties in design lie in the estimation of physical quantities required for a reliable analysis. This section briefly reviews the essentials of thermal analysis and thermo-mechanical analysis. The three modes of heat transfer are conduction, convection and radiation. The first is important principally in solids. The second is significant in fluids (liquids and gases). The third, radiation, is significant only at high temperatures or if other modes of transfer are negligible, as in outer space. The heat flux, which is proportional to the gradient of temperature, represents the flow of thermal energy. In a solid it depends on the conductivity, and in a fluid on the convection coefficient. At a steady-state (i.e. when thermal equilibrium has been reached) in one-dimension we have

q = − kA

dT dx

and

q = hA(T − T∞ ) The conduction coefficient, k, is easily measured. It does vary with temperature. For many engineering materials the variance is so small at operating ranges that we assume it is constant. The convection coefficient, h, is rather more difficult to estimate. It is not just a property of the materials that make up the fluid, but also depends on the velocity of the fluid. The study of heat-transfer therefore needs to borrow heavily from the study of fluid-flow. It is customary to estimate nondimensional numbers (the Reynolds Number, the Biot Number, the Prandtl Number, the Nusselt Number, the Grashoff Number, etc.) for a design problem. These numbers are then used to look up literature to estimate the convection coefficient. Radiative heat transfer is governed by the equation

q = fAσ (T14 − T24 ) where f is a factor that depends on the configuration, σ is the StefanBoltzman constant, and T1 and T2 are the temperatures of the two bodies in 24

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question, in degrees-Kelvin. Since σ is 5.669 x 10-8 W/m2.K4, radiative heat transfer is significant only when the temperatures are substantially high (as in the cooling of molten steel) or when other modes of heat transfer are negligible (as in spacecraft traveling through “empty” space). The differential equation that governs heat conduction in a body is easily derived:

∂ ∂T ∂ ∂T ∂ ∂T ∂T + k k + k + q& = ρ .c. ∂x ∂x ∂y ∂y ∂z zx ∂t where ρ is the density of the solid, c is the specific heat capacity, and

q& is

the energy generated per unit volume. If the temperature is dependent on time, this called transient heat transfer. For a steady-state problem the timederivative is zero. If the conductivity is a constant, it can be taken out of the derivatives. In this case, we sometimes define the thermal diffusivity as

α=

k ρc

For most materials, the thermal field is independent of the deformation field. This means the temperature in the body can be solved for without worrying about the stress distribution in the body. Once the temperature field (that is the distribution of temperature in the body) has been obtained, it can be used to calculate the stresses in the body. Remember that a temperature field gives rise to stresses because of differential expansion or if the body is prevented from expanding or contracting freely. The coefficient of thermal expansion is an important property of the material in this regard. The stresses caused by the temperature field are referred to as thermomechanical stresses. They are a significant part of many designs. For instance, it is customary to hold two components in a fixture for welding. Since the welding process involves an inflow of thermal energy and the fixture prevents free expansion, designers have to worry about how the structure will distort after the fixture is removed! 25

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Fatigue And Fracture Describing the investigations of 19th century engineers into the failure of parts, Frost et al write17 “because these failures occurred in a part that had

functioned satisfactorily for a certain time, the general opinion developed that the material had tired of carrying the load or that the continual reapplication of a load had in some way exhausted the ability of the material to carry load. Thus the word ‘fatigue’ was coined to describe such failures”. The study of fatigue is over a century old. But there is still not much fundamental understanding of how to design for fatigue. Most design rules are based on gathered data and presented in the form of curves or tables, as in the Atlas of Fatigue Curves18. What is clear is that the assumption that a continuous body is a flawless continuum is not correct. Various imperfections on the surface of a finished component are thought to contribute to fatigue-failure, and factors are prescribed to modify the permissible stress, which in turn depends on the endurance limit. Usually tabled as S-N curves, the endurance limit is the fatigue strength (S) for a given the number of cycles (N). What is also clear is that metals fail from fatigue if the load is “reversed” or “alternating19”. Given a load that changes magnitude and / or direction, engineers use the stress analysis methods described earlier to calculate:

17 18 19

σmin

the minimum stress

σm

the mean stress

σmax

the maximum stress

σr

the stress range

σa

the stress amplitude

Metal Fatigue by N.E.Frost, K.J.Marsh, L.P.Pook The American Society of Metals’ Atlas of Fatigue Curves, edited by H.E.Boyer

Some materials fail if the load is held steady but is held for a long time. This can be due to creep or viscoelasticity. Several plastics creep at room temperature while metals tend to creep only at elevated temperature. 26

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If there is a static (i.e. non-alternating) load, steady or static stress, σs, is also calculated. Note that if the stress-state is tri-axial, the stresses listed above are equivalent stresses (a von Mises stress, for example). As described earlier, these are calculated from the stress components. Once these components have been calculated, an appropriate failure theory is applied. Commonly used theories include Goodman, Modified Goodman, Soderberg, and Gerber. In case the part is subjected to σ1 for n1 cycles, followed by σ2 for n2 cycles, and so on, cumulative-fatigue theories such as the Palmgren-Miner cycleratio summation theory are employed. It’s important to remember that these are all theories, not yet accepted as laws. To emphasize the fact that Fatigue and its adjunct, Fracture, are not well understood, we will close our brief discussion of fatigue just as we started it: with an extract from two books on the subject. First20, describing the investigation into the failure of a tank in 1919, which was part of a law-suit, the court appointed auditor wrote “amid this swirl of polemical scientific waters, it is not strange that the auditor has at times felt that the only rock to which he could safely cling was the obvious fact that at least one half of the scientists must be wrong.” A description of an accident that occurred in 1988 – almost 8 decades later! - says21

“In 1988 the roof of the forward cabin of a 737 tore away during flight, killing a flight attendant and injuring many passengers. The cause was multiple fatigue cracks linking upto form a large, catastrophic crack. The multitude of cycles accumulated on this aircraft, corrosion and maintenance problems all played a role in this accident. Furthermore, the accident challenged the notion that fracture was well understood and under control in modern structures. “ 20

Fracture and Fatigue Control in Structures, Applications of Fracture Mechanics, by

S.T.Rolfe and J.M.Barsom. 21

Lecture Notes On Fracture Mechanics, Alan T.Zehnder, Department of Theoretical and Applied Mechanics, Cornell University 27

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Mathematics Mathematics is usually easier to deal with than mechanics, if only for the reason that most statements can be accepted as complete and true. There is little room for a subjective interpretation. In fact, it is this mathematical approach that both makes it possible for us to attack Linear Differential Equations with confidence and advises us to treat Non-linear Differential Equations with great care, as we shall see in the following pages.

Differential Equations and Tractability The power of differential equations has been noted already: they give us a ready way to model physical problems. Once modeled, we can understand, investigate and ultimately predict the behavior of the physical world. In our study of mechanics in the previous pages, we have already reviewed several mathematical models. Of course, the models themselves may well be wrong. That aspect is addressed in the chapter on Verification And Validation later in this book. For now, we assume that the mathematical model is what we need to solve. We focus on the tractability of these mathematical models. That is, the ease with which we can understand, evaluate and predict real-world behavior using the model. It is this requirement for tractability that will lead us onto our next chapter, where we will study the theory of the Finite Element Method.

Classes Of Differential Equations Functions of a single variable yield Ordinary Differential Equations (ODEs) while functions of many variables give us Partial Differential Equations (PDEs). PDEs themselves are categorized depending on the coefficients (constant or variable coefficients), on the right-hand-side of the equation (homogeneous and inhomogeneous), on the highest order of derivative (first order, second order, etc.) and so on. Mathematical physics, the study of physical phenomena using mathematical models, is dominated by second order PDEs. A PDE may do an excellent job of characterizing more than one phenomenon. An excellent example of this is Prandtl’s membrane analogy. Prandtl pointed out that the torsional stress 28

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in a non-circular bar is governed by the same equation that describes the deflection of a thin membrane under the action of a pressure. The immediate application of this insight was to allow experimenters to use the easier membrane-inflation experiments to measure torsional stresses! Investigations into 2nd order PDEs led to the comparison of these equations with the general second order analytic curve, the general conic section. Recall that if we write the conic section in its canonical form,

ax2 + bxy + cy2 + dx + ey + f = 0 we can then calculate the discriminant (b2-4ac) to determine whether the curve represents an ellipse (b2-4ac < 0), a parabola (b2-4ac =0) or a hyperbola (b2-4ac > 0). In a similar fashion, a general 2nd order PDE can be written in canonical form and classified as elliptic, parabolic or hyperbolic, depending on the discriminant. An elliptic PDE is said to be a Boundary Value Problem (BVP). All values within the domain (that is, the area over which the DE is valid) are uniquely specified by values on the boundary. These values on the boundary are called Boundary Conditions. Parabolic and Hyperbolic PDEs are encountered in dynamic problems, where values change with time. They require not just Boundary Conditions, but Initial Conditions too. Some problems are diffusive in nature. That is, any errors in initial conditions die away as time progresses. Other problems are convective – initial data propagates without dying away, which means that any errors in initial data continue to affect the solution even as time progresses. Remember, in closing, that initial and boundary conditions can be specified either on the derivatives of the independent variable (for instance the heat flux in a thermal problem, and the stress in a stress-analysis problem) or on the independent variables themselves (the temperature or the deformation)22. Boundary and Initial conditions should be prescribed with care, since they should both mimic the physical world and be consistent with the statement 22

Boundary conditions are often classified as Neumann, Dirichlet and Mixed 29

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of the differential equation. Problems that are thus defined are said to be well-posed. The opposite, of course, is an ill-posed problem.

Linear Differential Equations To a mathematician, a linear differential equation has several pleasing properties. In the first place, the mathematician can prove that the equation has a solution. This proof of existence justifies the hunt for the solution. Next, the mathematician can prove that the solution that exists is unique. This is even better, since it tells us that regardless of what method we use to find the solution, once we have found it it’s the only one. A PDE is linear if •

there are no products of any derivatives in the differential equation itself and in boundary conditions

•

none of the coefficients depend on the dependent variables

•

none of the boundary conditions depend on the dependent variables

Few physical problems, of course, are truly linear. As Mandelbrot23 demonstrated with his remarkable fractal geometry, the deeper we dig into the real world the more we uncover facts that complicate our equations. But just as it is reasonable to depict a planet as a round ball from a macro level, linear models are widely used as a first level of investigation. A deeper investigation is not always required, just as all planets are not explored. If a deeper investigation is required, we often end up with non-linear differential equations. In this book our focus is on linear differential equation. We will quickly review some important aspects of non-linear differential equations.

Non-Linear Differential Equations The general non-linear equation is not blessed with the uniqueness or existence theorems that linear equations boast of.

23

30

The Fractal Geometry of Nature, B.Mandelbrot

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Not only does this mean that the equation you are going to investigate may have no solution, it also means that it may have multiple solutions. The engineering usage of non-linear equations has been steadily increasing over the past several years. As computing power becomes cheaper, and software tools become more powerful, designers are increasingly able to deploy these powerful methods to design better products. In this series, the volume on simulation of manufacturing processes addresses one such application. However non-linear equations should still be addressed with some care. Solutions are better viewed as sources of insight into the mechanics, than as definitive answers to design problems. How do you know whether you should choose a linear model or a non-linear model? The main “sources of non-linearity” are:

Geometric: if the body deforms considerably, the deformed configuration may be markedly different from the un-deformed configuration. Such problems are said to be problems of large deformation. Alternately, there may be a large strain, which violates the assumptions implicit in the strain-displacement equation presented earlier. In some cases, the deformation changes the stiffness of the material because of the geometric properties. For instance, a convex curved plate’s stiffness reduces if the load causes it to first become flat and then concave.

Material: plastic deformation is the most common, but other forms of material behavior include creep, visco-elasticity, etc. In thermal problems, if the conductivity depends on the temperature, the analysis is non-linear.

Boundary: radiative boundary conditions are clearly non-linear since they involve the 4th power of the independent variable) i.e. temperature). In structural problems, gap or contact are nonlinear, since the stiffness of the structure changes with the deformation - depending on whether the gap is open or closed. Determining material data for non-linear analysis can be quite demanding, and requires a more complete study than can even be summarized here. 31

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Series Solutions And PDEs The study of infinite series goes hand in hand with the study of differential equations, and the reason is easy to find. Given the difficulty of solving differential equations, there’s a strong motivation to see if the solution can be constructed out of known and simple functions24. Bessel’s functions will be familiar to any one who has studied the analytical solution to the problem of the cooling of a fin. It was the investigation of heat transfer that also gave rise to the only series that we will review – the Fourier series. No study of mechanical vibrations can be undertaken without this. In its simplest form, the Fourier Series tells us that almost any periodic function can be expanded as a series of sin and cosine functions.

f ( x) =

a0 ∞ + ∑ (an cos(nx) + bn sin(nx) ) 2 n=1

The “almost” contains some important conditions. For example, one of the conditions is that function should have a finite number of finite discontinuities within the range of interest. This means, for example, that the familiar trigonometric function tangent cannot be expanded as a Fourier Series over any interval that includes π/2 or 3π/2, since the function is infinite at those points. The significance of the coefficients of the series is not always apparent. But if the function being investigated is time dependent, as in mechanical vibrations, a simple transformation can inject life into the coefficients of the series!

The Fourier Transform Given a non-periodic function f(t), the Fourier Transform is defined by the equation

1 F (ω ) = 2π 24

∞

∫ x(t )e

−iωt

dt

−∞

See, for example, Fourier Series and Boundary Value Problems, R.V.Churchill and J.W.Brown 32

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Here we assume that the non-periodic function is a function with an infinite period. Note that the independent variable in the “original” function, f(t), was “t”, while the independent variable in the “transformed” function is ω. It is customary to interpret ω as the angular frequency (radians / second). This is natural, since it is the frequency of the sine and cosine terms in the Fourier Series. We say that the transform has taken us from the time domain to the frequency domain. The inverse transform takes us back from the frequency to the time domain. Of the several properties that the Fourier Transform has, one in particular is interesting: the Fourier Transform of f’(t) is given by iωF(ω). This means differential equations can be transformed into algebraic equations25. The transform has a host of other interesting properties, covering areas like time shifting, modulation, etc. which are very useful in signal processing. Theory guarantees us that the two representations are entirely equivalent: there is no loss of information going from one domain to the other. Which form we choose to use depends purely on convenience. It's just a question of which is easier to understand and comprehend. Compare the problem to that of listening to an orchestra and trying to pick out instruments. A trained ear can do it, but for the average ear extracting such info from the time-domain-signal is hard. If, instead, you use the Fourier Transform to break the signal into a spectrum, you can clearly see signatures of different instruments, since each has a different frequency range. To illustrate this, consider the 3-degree-of-freedom system shown below26. Models such as this are not just instructive – they are also used quite widely in practice. Called lumped mass models, they are used extensively in a wide variety of applications. One such example, the simulation of vehicle crash, is included for reference at the end of this book.

25

Recall Laplace transforms – they serve a similar purpose. These figures are from The Fundamentals Of Modal Testing, Application Note 2433, Agilent Technologies

26

33

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Since the system has 3 degrees of freedom, it will have 3 modes of vibration. These are the natural frequencies - ω1, ω2 and ω3 - and the corresponding mode shapes. Our design goal is to control the response of the system. That is, given any source of excitation, we want to first predict, then alter, the response. The first step in this is to understand how the modes of vibration contribute to the response. Since the modes depend on k and m, we can tune the response by making appropriate changes to the structure. The time-response, shown below, doesn’t provide much insight. It does tell us how the response changes with time, but tells us nothing of the relative importance of the 3 modes.

But now let’s look at the Fourier Transform – the frequency domain plot.

34

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Note that the peaks in the plot correspond to ω1, ω2 and ω3, the natural frequencies of the system. The contribution of ω1 is more than that of ω2, so we’ll probably be better off tuning the structure to change ω1. In fact, as the figure below shows, the frequency domain plot can be obtained as the superposition of the plots for each individual mode! Remember this, since we will discuss the Modal Superposition Method later on in our study.

Frequency domain plots can intimidate a beginner, but there’s no reason to be intimidated. While a physical interpretation of the frequency domain plot is not always possible (in fields like electrical engineering, for instance), mechanical vibrations are kinder. As shown in the figure, the frequencydomain and time-domain plots of the modes of vibration of a cantilever beam are very easy to correlate with the physical behavior of the vibrating beam.

35

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While the “actual” beam, being a continuous body, has infinite degrees of freedom, our physical measurement is “reducing” it to a 3-degree of freedom body if we measure just the first 3 modes. The plot along the “frequency axis” is the frequency-domain plot we’ve seen above. Obviously the plot will change depending on the measurement point, just as the time response changes at each measurement point – recall the FRF we saw earlier. Figuring out where to measure (i.e. the choice of the measurement points) is one of the many challenges test-engineers face, and is one reason why the FE method goes so well with testing in the design of vibrating equipment. FE is often used to suggest measurement points. It is harder to visualize the equivalence between the time-domain and frequency-domain forms of a function for complicated structures, but the principle is the same. The Fourier Transform is an invaluable tool in the study of vibrations. Several times, it is implemented in real-time. That is, even as a signal is acquired, its Transform is calculated.

The Fast Fourier Transform The Discrete Fourier Transform is obtained by replacing the integral with a finite sum. This does introduce an error, but this is often tolerable27. The 27

Refer the Nyquist Criterion, taught in most undergraduate courses on Control Systems.

36

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DFT, implemented in hardware or software gives us the FFT – the Fast Fourier Transform. There are several different algorithms that implement the FFT. Common to all of them is that they can generate the frequency domain form of a signal even as the signal is being sampled. Remember the earlier discussion of the damping factors of a structure. We saw that the factors are estimated from actual physical tests. In these tests, the structure is excited at various frequencies and the response logged. From the response, the damping factors are estimated. FFT is very widely used to interpret data collected in such experiments, since the Fourier Transform helps us “spread out” data into peaks. In some cases, log-log plot or log-linear plots are used to present the data. The FFT of data on vibrating equipment also serves as a diagnostic tool. It can help us understand which sources are most important in the response. A detailed discussion of the FFT and the FRF is beyond the scope of this book.

The mathematics, for most structural engineers including myself, is difficult to understand. Also, the solutions are difficult to verify. Ed Wilson describing frequency domain analysis

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Essential FEA Theory Of the several ways to introduce the Finite Element Method, the easiest is to view it as nothing more than a series solution for partial differential equations. This is the approach we will follow in this brief theoretical introduction to the method. The next chapter will put this mathematical approach in the context of specific applications of mechanics. From a designer’s perspective, however, FE is more than just a differentialequation solver. As we will see, it offers us a toolbox of ready-made differential equations that we can put together to describe most physical problems. This is very appealing to a designer, who is familiar with the approach of breaking assemblies down into standard parts or machine elements: gears, chains, belts, beams, trusses, plates, fasteners, and so on. It is this capability to model complicated structures that makes FE so much more valuable to a designer than many other numerical methods.

From The Differential Equation To A Matrix Equation A study of mechanics familiarizes us with the many differential equations that are used to model continuous bodies: there is the differential equation of beam bending, the differential equation for heat conduction, the differential equation for general tri-axial stress, and so on. Solving these equations on simple geometries is hard enough, as anyone who has worked through the temperature distribution in a cooling pin knows. Solving them on general geometries like an engine-block is next to impossible using analytical methods. An obvious alternative is to use numerical methods. This is a well-known approach, well documented in the derivation of the many infinite series, including the Fourier Series. One approach, suggested by B.G.Galerkin28 for boundary-value problems, reduces the differential equation into a problem of linear algebra – that is, a

28

In a paper published in 1915. The method is also called the Bubnov-Galerkin method, and is similar to the Rayleigh-Ritz method.

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problem of simultaneous linear equations that can be solved using matrix methods that any high-school student is familiar with. For the sake of simplicity, we will illustrate the method for a simple second order elliptic differential equation with constant coefficients. We start with the statement of the problem: that is, the differential equation,:

k

d 2u du +c + au = f 2 dx dx

the domain, and the boundary conditions29

for o ≤ x ≤ l subject to

u x =0 = u 0 u x=l = ul Note that the boundary conditions could, in general, specify either the dependent variable u, or it’s first derivative u’, or a combination of these. In a heat transfer problem, the dependent variable is the temperature, the first derivative represents the flux (i.e. heat injected or removed at the boundary) while a combination of both represents a convective boundary.

k, c and a are material constants that could depend on the independent variable x. If they depend on the dependent variable u, the equation is nonlinear. For stress analysis, the dependent variable is the deformation while the first derivative represents the strain or the stress on the boundary. Combinations of the dependent-variable and its first derivative are less common, but can represent spring-supported boundaries.

29

The FE method treats boundary conditions with great elegance, though we do not cover the mathematics here. 39

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Higher order differential equations are encountered in mechanics. For instance the beam differential equation can be written as a 2nd order equation

d2y M =− 2 dx EI but it is usually more convenient to work with the 4th order form

d4y w =− 4 dx EI In the latter case, boundary conditions can be prescribed on the independent variable, its first derivative, its second derivative, and its third derivative, or any combination. Our problem is to find a function u(x) that satisfies the differential equation everywhere in the domain (i.e. for o ≤ x ≤ l ) and also satisfies the boundary conditions. For brevity, and with no loss of generality, we will simplify our second-order equation further, restricting our attention to just two terms:

k.

d 2u = f dx 2

Rather than integrate the differential equation, we will assume the form of the solution, then search for conditions that the parameters in this assumed form must satisfy for the assumed equation to be a solution. We will call our assumed solution u , but before searching for the solution, we will recast the equation in its weak form. Remember that if u is to be a solution of the differential equation, it must have a second derivative. This is obvious: the assumed form will not fit into the differential equation unless it has a second derivative. But also remember that our modeling approach does lead to situations where the solution itself cannot have a second derivative: it may at best 40

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have a first derivative. For example, consider a weight hanging from a cable that is supported at both ends. In actual reality the cable is continuous – that is, its shape changes smoothly all through. The slope of the string may change more rapidly in the vicinity of the weight than at other points, but it is still continuous. However if we model the weight as a point-load30, the string now has a sharp kink at this point. It has a first derivative, but is not differentiable: that is, the slope to the left of the kink is not equal to the slope to the right of the kink. Obviously, there can be no second derivative! Why should we demand that our candidate solution u , which we will call the trial function since we will want to try it out as a possible solution, have a second derivative, when the mathematical model has an exact solution that does not have second derivatives? This is the reason we recast the problem in the weak form. To do this, we say that to qualify as the solution, our trial function will be accepted if it satisfies the equation l

l

d 2u v . ∫0 dx 2 dx = ∫0 vf ( x)dx Remember that requiring that that

∫ g ( x)dx = 0 is not the same as requiring

g ( x) = 0 ! The latter requires that g(x) be zero for all x, while the

former only requires that g(x) be zero on average. The use of v(x) in the above equation compensates for this. If we require that the integral equation be satisfied for all possible functions v(x), we can satisfy ourselves that the integral equation is as demanding as the original differential equation. This approach, called the method of weighted residuals, uses v(x) as a test function. From now on, it’s just a matter of basic calculus and algebra. We integrate by parts to get

30

Such point-functions are defined using the Dirac-delta function. They are not physically realizable, but are used commonly. 41

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v'

du dx

x =l

l

x =0

l

dv du . dx = ∫ vf ( x)dx dx dx 0 0

−∫

This equation is now symmetric. That is, it requires that both the trial function and the test function have the same order of continuity. Since the equation only involves the first derivatives of these two functions, they do not need to have second derivatives at all31. We now assume a series form for both the test and the trial functions, with the constant coefficients a and b: ∞

u ( x) = ∑ ai .φi ( x) i =1

and ∞

v( x) = ∑ b j .ψ j ( x) j =1

As with any other infinite series, we will choose to terminate the series at our convenience. That is, we choose to calculate only a finite sum. How many terms should we include? That depends on how much error we are willing to tolerate. Since this is a critical decision in design, we will return to this aspect in the chapter on Verification and Validation! In other words, the series are summed till N instead of till ∞. You may find it useful to carry out the exercise yourself (for a small N) to show that after substituting the series in the integral equation and rearranging terms and ignoring the boundary terms, we get the matrix equation

31 A C0 function is one whose zero’th derivative (i.e. the function itself) is continuous. A C1 function has a continuous first derivative, a C2 function has a continuous second derivative, and so on. By continuous, we mean the left-limit = the right-limit = the value of the function at the point.

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Since we can cancel the row vector {b} from the equation as it occurs on both sides, we get the matrix equation

{b1

b2

l ' ' ∫ φ1ψ 1 0 l ' ' ∫ φ2ψ 1 . . bN } 0 . . .

l

∫φ ψ ' 1

' 2

. .

' 2

. .

0 l

∫φ ψ ' 2

0

. .

. . . .

.

. .

l ∫ fψ 1 a1 0 a l fψ 2 . 2 ∫ . = {b1 b2 . . bN } 0 . . . . . a N l l ' ' f ψ φ ψ N ∫0 N N ∫0 .

[K ]{u} = { f } Given the similarity to the equation of spring-deflection, it is customary to call the matrix [K] the Stiffness Matrix. Obviously, it may represent other physical quantities too, such as the heat capacity in a thermal problem, but the name is used regardless. While we have not shown how the boundary terms are included, we will only state that they can be very elegantly included in the matrix equation itself. In a few steps, then we have reduced an intractable differential equation to a simple matrix equation. All that’s left is to choose names for the variables introduced above, and we will have completed our introduction to the Finite Element Method!

Nodes, Elements And Shape Functions In the series expansions used above for the test and trial functions, we did not assign any physical significance to the constants ai and bj. Nor did we say anything about how to choose the functions φi(x) and ψj(x). The choice of these functions determines the shape of the trial function, which is nothing but our assumed solution. We therefore call these functions shape functions. It is by no means easy to construct shape functions for all differential equations. Despite long years of research, it continues to be an active area of research! The choice of shape functions depends on the differential equation being solved, and on the order of accuracy required. 43

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The bj, of course, are irrelevant to us since we have cancelled them from the equation – we never actually evaluate them. What of the ai? We can make a choice that is very intuitive and that turns out to be very useful. We ask that the constants ai be the values of the function u at selected locations in the domain. We call these locations the nodes or grids, and since we are constructing the series, it is upto us to decide where to locate these nodes. Without explaining the mathematical basis, we will just state that the nodes should be closely spaced wherever we expect the solution to vary rapidly, while they can be spaced farther apart in areas where the solution is not expected to change rapidly. Again, note that we are making our decision based on our judgment, not on knowledge. We can guess where the solution will vary rapidly, but we cannot know this till we have solved the problem! The chapter on Verification And Validation addresses this seeming paradox. Now remember that each term in the stiffness matrix is an integral. Since we are using computers, we evaluate these integrals numerically. This step, often called quadrature, usually uses the Gauss integration rule. What then is an element? It’s a part of the domain defined by nodes, over which everything (the shape functions, the material constants, the forcing function, etc.) is continuous. Putting a lot of elements together gives us the complete domain. One important consequence of this definition is that we can now evaluate the integral over each element, since the integral-of-a-sum is the sum-ofintegrals. When programming the FE method, we can use this property to calculate local or element stiffness matrices, then assemble these to get the global stiffness matrix. Another is that we can view each element as a black-box. It interacts with its neighbors only by the values of the dependent variable at the nodes: and different elements can even represent different governing differential equations! This last property is extremely useful. Most mechanical components are complex, and it is next to impossible to find a single differential equation that defines the behavior of the dependent variable everywhere over the body. Now, however, we need not worry about this. We can “break” the 44

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component into subsets that are similar to readily recognizable machine elements: beams, trusses, plates, etc., and model our component as a combination of these. In other words, the Finite Element method not only gives us a way to solve differential equations, it also helps us model complicated bodies by piecing together various differential equations in a consistent fashion!

Some Common Elements Used In Stress Analysis Many commercial finite element packages offer a dizzying range of elements, but most are variations on 4 basic types: trusses, beams, plates, and general 3-dimensional solids. If you are familiar with these, you will be able to model a large majority of the problems in solid mechanics. Most commercial elements use polynomial functions as the shape functions. If the polynomial is of the 1st order, we call the element a linear element. A 2nd order polynomial makes the element a quadratic element. Higher order elements are rarely used, since it is generally believed that the increased accuracy is not worth the increase in required computing resources. Remember that a linear shape function in 1D requires 2 points to uniquely specify it, while a quadratic function requires 3. Truss Elements: • •

•

3 dof / node (ux, uy, uz) Data required: Cross-section Area Appropriate for pin-jointed members

Beam Elements: • • • •

• •

6 dof / node (ux, uy, uz, rx, ry, rz) Data required: Cross-section Area, Moments of Inertia Appropriate for slender (uni-dimensional) members that support bending moments Linear Beams have 2 nodes per element 2nd order beams can be curved – they have 3 nodes per element Beam-theory allows several additional constants to be defined. These are important if working with thin-walled sections such as pipes or channels.

Plate Elements: • •

6 dof / node (ux, uy, uz, rx, ry, rz) Data required: Thickness 45

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Use for thin (two-dimensional) members Linear Quadrilateral Plates have 4 nodes per element Linear Triangular Plates have 3 nodes per element 2nd order (quadratic) Quadrilateral Plates have 8 nodes per element (some packages allow 9 nodes per element, with the 9th at the centroid of the element) 2nd order (quadratic) Triangular Plates have 6 nodes per element (some packages allow 7 nodes per element, with the 7th at the centroid of the element)

3-Dimensional Solid Elements: • • • • •

• •

3 dof / node (ux, uy, uz) Data required: none Linear Hexahedral Solids have 8 nodes per element Linear Tetrahedral Solids have 4 nodes per element 2nd order (quadratic) Hexahedral Solids have 20 or 21 nodes per element 2nd order (quadratic) Tetrahedral Solids have 10 or 11 nodes per element Other shapes (pentahedron or wedge, pyramid, etc.) are also offered by some packages

Remember that a degree-of-freedom (often abbreviated to dof) is the unknown at a node. If your model has N nodes, and each node has M degrees of freedom, your stiffness matrix will have NxM rows and NxM columns. The r in the dofs above refers to the slope at the node. For several differential equations, we use both the deformation and its derivative (i.e. the slope) as dofs. It is tempting for a beginner to model all components with general 3dimensional solids because the resulting FE model looks similar to the actual component. This is wrong. Remember that your FE model needs to mimic the behavior, not the appearance, of the component. If bending stress is more important (as for long slender members) a beam element will be much more accurate and effective than a solid element! Triangular and tetrahedral elements are frowned upon in stress analysis since they don’t offer as good accuracy as quadrilaterals or hexahedra. For thermal analysis, however, triangular and tetrahedral elements work just fine. Many packages can automatically generate elements from the geometric model. Called auto-meshing or automatic mesh generation, this is rarely supported for hexahedra.

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Plane stress, plane strain and axi-symmetric behavior are good approximations in specific situations. Elements that offer this type of behavior are very efficient if used in the right situations, but with the drop in the price of computing power, they have largely fallen out of favor. One element that is not an element at all (in that it doesn’t represent any differential-equation) is a “rigid link”. It is just an equation that tells the software that two dofs must be matched – they should have the same values. From an engineering perspective, this is as if the two dofs have been joined by an infinitely rigid link, hence the name for the element. A spring element is sometimes used to represent a spring-support if the spring-stiffness is known. Lumped mass elements are used to represent components whose weight is relevant but whose stiffness can be ignored. The mass is concentrated at a node, so the element is a “0” dimensional element. Finally, you will see from the summary above that not all elements have the same number of dofs / node. Beams and plates have 6 dof / node while trusses and solids have 3. Joining a plate element to a beam is consistent: equilibrium can be maintained since both the elements have the same dofs. But what if you want to connect a beam to a solid, or a plate to a solid? Is it sensible? That depends on the physical situation. If, in the physical problem, there is a hinge at the connection then your mathematical model will be accurate. If, however, your physical problem has a slender member welded to a block, then it is tempting to model the block with solid elements and the slender member with beam elements. But it is important to remember that the physical model is now different from the real model since moments are not transmitted! Building models that mix elements that have different dofs / node should be done with care. It is often resorted to in practice, and works well if safeguards are followed.

Matrix Solvers Understanding how matrix equations are solved is really not important for a designer. If the model is created right, the computer program should take care of solving the matrix equation. 47

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We will briefly review methods of solution, though, for two reasons. First, simply because it is interesting! Second, because if things go wrong (as they inevitably will!) familiarity with the inner workings helps diagnose what went wrong.

High-school courses teach us to solve matrix equations by evaluating the determinant (Cramer’s rule is usually adopted), and calculating the inverse. Anyone who has tried to invert a 4x4 matrix by hand will understand why the FE method languished till computers came along! If you pause to consider that it is quite common to solve a problem with 100,000 dofs on a desktop computer, it is clear that computers too can grind to a halt unless the matrix equation is solved efficiently. Computer programs, in fact, rarely use Cramer’s rule. Most solvers use a variation on Gauss’s method of triangularizing a matrix, called Gaussian Elimination. Such solvers, called Direct Solvers, rely on the fact that subtracting one row of a matrix from another does not alter the determinant32. There are several variations on this method (Cholesky’s method is perhaps the most popular), all of which involve the calculation of a pivot for every row operation. Without going into the mathematics, we will only observe that is a pivot is zero, the matrix is singular – that is, it’s determinant is 0. Obviously this means that the matrix equation cannot be solved. For solid mechanics, where we are guaranteed that the matrix will be symmetric, pivots should never be zero. (In linear algebra, we say the stiffness matrix is Symmetric Positive Definite). If a pivot is zero, it’s probably because you have made a mistake in modeling. The most common mistake is in the application of restraints (which are one type of boundary condition). This is worth discussing, since it can be quite counter-intuitive for a beginner! Given a plate with a force applied at each end, what “boundary conditions” can we apply? 32

Iterative solvers, which are useful for very large problems, use variations on the Gauss-Seidel method.

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The data does not tell us anything about the prescribed deformation, so we can be forgiven for building a finite element model shown below. The problem can easily be solved by hand, so it can be quite confusing if we attempt it on a FE program and are told by the computer that the problem cannot be solved because a “zero or negative pivot was encountered”. Remember that we omitted the treatment of boundary conditions when we derived the matrix equations earlier. A study of that would indicate why an FE program must include restraints, but for now we will only note that an FE model will not be solvable unless all rigid body motions are eliminated. That is, you examine your FE model and check if it will be in static equilibrium regardless of whether the forces are symmetric or not. In the case above, the plate is in equilibrium only if the forces are symmetric. If the forces were unsymmetrical, the plate would move as a rigid body – with no strain33.

Symmetry is often used to introduce restraints (and can also help cut the computational time since it reduces the size of the model!). For the plate, applying symmetry, the model would be

Understanding symmetry-restraints is not always easy, since a physical insight is sometimes hard to gain. Symmetry can be of different types – cyclic, rotational, planar, etc. Planar symmetry is the easiest to model. For the translational degrees of freedom, you should restrain all translations

perpendicular to the plane of symmetry for all nodes that lie on the plane.

33

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Rotational degrees of freedom, if they exist, should be set to the inverse of the corresponding translational dof. For instance, for a plate symmetric about the x-axis, we would set ux = 0 for all nodes on the x-axis, and leave uy and uz free. Since plate elements have 6 dofs / node, we would set ry and rz = 0 for these nodes, and leave rx free, applying the “inverse” rule of the previous paragraph. We will conclude our discussion of matrix solvers for FEM by noting that in general, stiffness matrices tend to be sparse. That is, the number of entries that are 0 is usually pretty high. Various schemes are used to take advantage of this behavior – band-solvers, column- or skyline-solvers and frontal-solvers are some popular methods.

Some Important Properties Of The FE Solution Several FE packages refer to error norms. A norm is just a measure of anything, so the error norm is a measure of the error. It is an interesting aspect of the FE method that we can estimate the error in our solution even without knowing the “correct” answer! There are several different norms that can be used to estimate error, but the error in strain energy is favored since it is consistent with the mathematics of FE. As with any other infinite series, truncating the series is a source of error. Adding more terms reduces the error, but this behavior is asymptotic – the law of diminishing returns applies here too, so after a while the increased cost of including additional terms does not provide an adequate increase in accuracy. The key to efficient usage of FE is to stop increasing terms when the rate of improvement levels off. Increasing terms, of course, means increasing the number of nodes, which can be done in two ways: either by using more (and therefore smaller) elements or by using higher order shape functions. The former is referred to as h-refinement while the latter is called p refinement. If both methods are used simultaneously, we call it h-p refinement. Regardless of how fine the mesh is, the use of the weak form means that the derivative of the dependent variable can never be continuous. In other words, in a stress analysis solution, the deformation will be continuous but 50

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the strain will not be. This behavior is sometimes used to judge whether the solution is adequately accurate or not. Finally, we saw that Galerkin’s method applies to BVPs only (Boundary Value Problems). Remember that we integrated-by-parts. This resulted in the term

du v' dx

x =l

x =0

which has to be evaluated at both “ends” of the independent variable – x = 0 and x = l. What if the independent variable were “time”? There is no known way, outside of science fiction, that we can know the conditions at future time, so obviously we cannot use the same approach for the timederivatives. There is no way we can apply the above integration-by-parts rule to a time derivative. Problems that involve time-derivatives use finite-difference methods to “step forward” in time. The Finite Element mathematics is restricted to the spatial derivatives only.

A theory is a good theory if it satisfies two requirements: it must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations. Stephen Hawking

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Putting It Together – OptiStruct/Analysis OptiStruct can address a variety of design tasks. It can address optimization, as described in another volume in this series. An FEM analysis is an integral part of a lot of design optimization, so OptiStruct includes an embedded FE solver: OptiStruct/Analysis. OptiStruct/Analysis is intended purely for analysis. That is, it can be used to setup the problem, carry out the analysis and review results. Strictly speaking the setting-up and review tasks (often referred to as preprocessing and post-processing) are done using HyperMesh and HyperView. As the assignments illustrate, this distinction is immaterial to the designer, with all tasks carried out under the same interface! In this chapter we •

summarize some of the main capabilities of OptiStruct/Analysis

•

review what a problem consists of – what files are involved in an analysis and how to present results

•

list the nomenclature that OptiStruct/Analysis uses

Capabilities Remember that a model is an idealization of a physical phenomenon. Any results you get are only as good as the model you choose. At the same time, the model should not be so complicated that it cannot be solved in the required time or with available resources. Choosing an appropriate model is not always easy. It is always a good idea to refer to published literature for examples of models that have been used successfully.

Linear, Static This model is used when the response of the body is linear, and there’s no variation with time. In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small and when the loads do not vary with time. 52

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This model is used widely since it’s quick to solve and relatively easy to interpret the results. Very often, even if a non-linear model is more realistic, a linear model is used to investigate likely behavior. Once the options have been narrowed, a full non-linear analysis is resorted to. The equilibrium equation is

[K ]{u} = { f } where K, u and f are functions of x, y and z only – they are independent of

t.

Normal Modes Sometimes our design problem is not just to calculate stresses or deformations. We may be interested in identifying the resonance frequencies of the system. In vehicle design, avoidance of resonance enhances ride comfort by cutting out unwanted rattles. When designing a loudspeaker or a megaphone, on the other hand, you may want resonance to occur. In cases like these, we need to solve the “eigenvalue” problem and evaluate the natural frequencies of the body. The equilibrium equation is

[M ] ∂

u + [K ]{u} = {0} 2 ∂t 2

where K and u are functions of x, y and z only – they are independent of t. The solutions to this equation are pairs of natural frequencies and the corresponding “mode shapes”. Since we omit the damping we call this a normal modes analysis.

Linear, Transient In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small but when the external conditions do vary with time. Either the loads or the restraints or both could be time-dependent. 53

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Transient problems can be solved using either Direct Integration or Modal Superposition. The latter is preferred if the solution needs to be calculated for a large period of time, but is only applicable to linear problems. The former is preferred if the solution is required for short periods of time, and can be applied both to linear and non-linear problems. In either case, we usually perform a normal-modes analysis. If we are using modal-superposition, we will need the mode shapes anyway. If we are using direct-integration, the highest important mode helps us determine the time step to use.

Random Response In some situations, we cannot specify the exact value of the loads as a function of time, but can specify the total energy in these loads. An example would be the forces experienced by a plane when its engines are firing. We know the total energy being transferred from the jet engines to the frame, but cannot claim that we know the loads precisely as functions of time. In cases like these, the loads are characterized by Probability Density Functions, and the behavior is called stochastic. The designer’s goal then is to predict a probability of safety. The several ways to evaluate these responses is beyond the scope of this book.

Inertia Relief Setting up a Finite Element model for static analysis requires that the structure be supported adequately. Consider an aircraft in flight or a spacecraft. There are no supports, but the model can be static if there are no time-varying loads. Obviously there are structures that are not supported explicitly but are still best represented by static-analysis models. Inertia Relief is an approach used to model such problems. The inertial of the structure is used to mimic equilibrium, like a D’Alembert force. In OptiStruct, we use special instructions – SUPORT cards - to eliminate all rigid body motions. Then we tell the solver that we want to perform an Inertia Relief analysis. It doesn’t matter which points you specify 54

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the SUPORTs at. Think of the SUPORT points as reference points with respect to which the deformation is calculated.

Frequency Response In many designs where vibration is important and correlation with testresults is essential, designers have to characterize the response of the structure as a function of frequency-of-excitation instead of as a function of time. In these cases a Fourier Transform converts the equilibrium equation from the “time domain” to the “frequency domain”. In the equilibrium equation, the variables are expressed as functions of ω rather than time. This is called the frequency domain. Of course, the Inverse Fourier Transform can convert the solution back to the time domain. In OptiStruct, we do not evaluate the FRFs – the frequency response functions – explicitly. Instead, we specify the range of frequencies for which we want to excite the structure. This is equivalent to “sweeping” through the excitation range. We then plot the results just as we would in an experimental setup. That is, we view the magnitude and the phase at points of interest. This allows us to correlate with the test setup. Test-correlation is an advanced task, beyond the scope of this book. We will end by pointing out for a multiple-degree-of-freedom structure, the FRFs are a matrix, and the FRFs themselves can be extracted from the FE solution if required.

Linear Buckling Designers sometimes have to take into account the fact that even if stresses are less than permissible values, the structure may fail if it buckles – like a tall column in compression. That is, the deflection continues to increase even if the load is removed. This is sometimes called instability. The equilibrium equation

[K ]{u} = λB [K G ]{u} is similar to that of Normal Modes, but the results are interpreted as a “buckling load factor”. KG is called the incremental stiffness.

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Buckling load factors are often important in the design of aerospace structures, where the quest for a minimal weight and the use of advanced materials leads to the frequent use of thin-walled designs.

Non-linear – Gap / Contact The terms “gap” and “contact” are often used to mean the same thing – an opening in the body that may close or widen under the influence of external factors. Clearly, if a gap closes or opens, the stiffness of the body changes. Since the gap opens or closes depending on the deformation, this means the stiffness depends on the deformation. In other words, since the stiffness and boundary forces depend on the deformation, the equation is non-linear.

Component Mode Synthesis When working with large models, resource constraints sometimes force the analyst to break the problem into smaller parts. In static analysis, this approach is called sub-structuring. When used in dynamic analyses, it is called Component Mode Synthesis. It’s sometimes impossible to treat the product purely as a structure or purely as a mechanism. Consider, for example, the feed mechanism for a highspeed packing machine. The rates of acceleration that the mechanism experiences may be quite high. High enough that the deflection of the levers is large enough, perhaps, for the feed mechanism to jam because of misalignment. Designing such a product requires that the equations of rigid-bodymechanics be coupled with the equations of structural deformation. Component Mode Synthesis also provides a way to do this.

Thermal and Thermo-Mechanical Analysis In some cases we only want to calculate the temperature distribution within our area of interest. We call this a thermal analysis. If we also want to calculate the stresses, we can use the temperature distribution as data for the stress analysis. This is called thermo-mechanical analysis.

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Setting Up An “Analysis” When we reviewed the theory of the Finite Element Method, we saw that a problem consists of •

the domain – the space and / or time over which we want to evaluate behavior

•

the differential equations that govern behavior in the various parts of the domain

•

data available on the spatial boundaries – boundary conditions

•

data on behavior at initial time – initial conditions

•

the order of elements chosen by the analyst – linear, quadratic, etc.

•

the locations of nodes – closely spaced where we expect the dependent variable to vary rapidly

•

material related properties

To solve a problem using OptiStruct / Analysis, we provide exactly the same data to define the problem. Since defining the problem is only the first part, we usually also take some additional steps. Some solution methods require additional data – for instance how we want to solve a direct integration problem, how many modes we want to include in a modal-superposition, etc. Also, the results of the analysis can be presented in a variety of ways. Since we don’t want to be overwhelmed with data, we usually also specify the nature of output we want. If we specify this, HyperView allows us to present the same data in a variety of different ways. Most analysis starts with an import of geometry from a CAD modeler. Geometry is not essential, of course. The FE data can de defined even without geometry as a reference, but it’s usually easier to work with a geometric definition as the starting point.

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It is very rare to run an analysis only once. It’s much more likely that you will want to modify some data and run the analysis again to perform what-if studies34. In general, your flow of work is: 1. Import CAD geometry. Simplify the model to make meshgeneration easier 2. Choose a modeling strategy – the materials, elements, restraints, etc. that you will use. 3. Define collectors to organize data to match your strategy 4. Define material data 5. Create the mesh – the collection of elements 6. Apply body loads and boundary loads 7. Apply restraints 8. Solve 9. Review results The sections below describe these steps in greater detail.

Geometry Preparation While it is possible to build a model directly using elements and nodes, this is not often done today. The geometry that defines the area to be analyzed (also called the “domain”) is usually created first using a CAD program, and elements are created to encompass that boundary or represent the volume. CAD designers create models for manufacture. As many details are included as possible. For a numerical analysis, we often choose to ignore aspects that we think will not significantly affect the solution. For instance, a single hole of 1 mm radius in a plate that is 2 meters wide can probably be ignored safely when calculating the deformation of the plate. It reduces computation time dramatically with no significant loss of accuracy. A finer model of only

34

A structured method to conduct what-if analyses is DOE – short for Design Of Experiments. This is covered in another volume of this series.

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the region around the hole can be used subsequently if the hole is an area of high interest. This approach is called sub-modeling. Therefore the first task that most analysts are faced with is that of preparing the geometry for analysis. This involves tasks like removal of features, extraction of mid-surfaces, extrapolation of surfaces, etc. Further, the CAD world has an abundance of data exchange formats, since most CAD applications use proprietary data storage formats. A transfer of data from the CAD package to the FE preprocessor sometimes results in a loss of accuracy – gaps are introduced during the import process, for example. Also, CAD assembly models are sometimes made up of parts that were created in different CAD applications. Therefore a cleaning-up of the geometry is often required. This involves filling gaps, eliminating small edges or surfaces that will mislead the automatic-mesh-generation routines, eliminating dangling faces, and so on.

Guidelines on Element Choice Learning which element to choose is a little like learning how to drive in heavy traffic. Guidelines exist, but can’t be applied blindly. You need to adapt them to specific situations. Remember this warning! If your product has a region that is long and thin, you can probably model it using beam elements. If this region is connected to the rest of the structure by pin-joints, then you should use truss elements. Regions that are like plates are best modeled using shell elements. Any areas that don’t fall in the earlier categories should be modeled using solid elements. If you have different element types in your model, there are rules that govern the assemblage. For several models, we choose to use just one element type to avoid these complications.

Mesh Creation Once the geometry is more or less ready for discretization, you then start to subdivide the geometry into elements or grid points. The collection of elements is usually referred to as a mesh. Meshes that consist of triangular or quadrilateral elements can often be generated automatically, while tetrahedral or hexahedral meshes usually require considerable manual intervention.

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Mesh Editing Once a mesh has been created, the analyst checks if it meets the specifications – several measures of quality are checked, depending on the analysis requirements. Usually, some editing of the mesh is required. Depending on the complexity of the mesh, this can be done either semiautomatically or manually. Acceptable values for the various element quality indicators, which are summarized in the glossary, are very problem dependent, and are sometimes solver-dependent.

Preparing for Analysis Once the mesh is ready, additional data is specified – the properties of the materials used, the thickness or cross-sectional properties of shell or beam elements, the conditions on the boundaries (restraints, loads or excitations), initial conditions, data for the specific solution algorithm to be employed, kind of output required for text and graphics records, and so on. Once this is done, the data is turned over to the solution program for the next phase – solving. Data is often written out in the form of a text file, which is referred to as a deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line. The procedure of building the Finite Element Model is sometimes referred to as FEM – short for Finite Element Modeling. It’s more common, though, to use FEM to refer to the Finite Element Method itself.

Solving The model created in the earlier steps is now taken up for solution – the computer program reads the data, calculates matrix entries, solves the matrix equations and writes data out for interpretation. This task is CPU-intensive, and is often called processing35. Most of the time, very little interaction from the user is required. In some cases, the analyst periodically monitors results to check that they are indeed on the right track. If the solution seems to be evolving in an unexpected direction, the analyst can stop the solver and modify the model, thereby saving valuable time. 35

Hence the term pre-processing for the preceding steps, and post-processing for the subsequent steps.

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Post-Processing After the program has evaluated the results, the analyst examines and interprets the data – looking for errors or improvements in design. As with pre-processing, this calls for substantial interaction from the analyst.

Nomenclature and Data Organization HyperMesh and OptiStruct/Analysis use several different files to organize data. Data is further organized as collectors within the “input” file. Most of these aspects will become clear to you after you work on one or more of the assignment problems. The summary presented here is useful as a quick-reference when you’re working on the software. More detailed descriptions are available in the on-line documentation.

Files modelname.hm

The model you create using HyperMesh. This is a binary file containing the geometry, analysis model and optimization model.

modelname.fem

This is an intermediate file. It contains the analysis and optimization models only, without any geometry. It is created by HyperMesh and read by OptiStruct. It’s a text file and can be interpreted using the format-definitions listed in the OptiStruct On-line Help.

modelname.out

This is a text file created by OptiStruct. The contents depend on the instructions you specify in HyperMesh when creating the model.

modelname.spcf

This file is created only if you explicitly tell OptiStruct to record all reactions. It is a useful check to ensure that you have applied loads correctly.

modelname.stat

This is a text file created by OptiStruct, containing statistics on CPU usage.

modelname.h3d

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use HyperView Player, freely downloadable from www.altair.com. The Player reads this binary file that is created by OptiStruct.

modelname.html

This is a quick summary of analysis and optimization results. Viewable using any webbrowser.

modelname.mvw

This is a text file, intended for use by HyperView. You will use this file to view stresses, displacements, density, convergence history, constraint violation, etc.

modelname.res

This is a binary file containing the results of the analysis and optimization. It’s readable only by HyperMesh and HyperView.

There are several other files36, most of which you can ignore in the normal course of events.

Terminology Collector

A way to group related items together. For instance all elements that have the same thickness would be in the same collector.

Load

External forces acting on the boundary. Includes concentrated forces, moments, pressures, gravity, etc.

SPC

Short for Single Point Constraint. Refers to restraints applied to the analysis model at locations where the body is supported37.

Subcase

Combination of SPCs and Loads. These are treated separately in an FE model, since they represent values on the boundary, often clubbed together as

Boundary Conditions Card

36

Some data in the analysis model, such as the material properties, cannot be displayed graphically. Such data is entered as a card image by typing in

Detailed in the on-line help documentation Do not confuse these with design constraints, which are applicable to the optimization model.

37

62

A Designer’s Guide To Finite Element Analysis

Putting it Together – OptiStruct/Analysis

text or numerical values. Data is often written out in the form of a text file, which is referred to as a

deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line. Some data can be represented graphically – nodes and pressures, for example. Other data, like the Modulus of Elasticity, is easier viewed as text. Such data is defined using the card editor. The table below lists the names of some of the data types OptiStruct/Analysis and HyperMesh use, along with the relations between them. Use this as a guideline to remember when you need to specify which data. Normally, you would proceed down the table: first the mat collector, then the component collector, then elements (grids or nodes are implicit), and then the load collector. While load collectors can contain both loads and restraints, it’s a good practice to keep them in separate load-collectors so that you can organize them into sub-cases. Data Entity Mat& Component Property% Grid Elem [email protected] Restraints! SubCase$

Notes: &

Refers to Mat Coordsys Grids, Props Grids/ Elems Grids Loads, Spcs

Data 1

Data 2

Data 3

Data 4

Matid CompName PropId Gridid Elemid

E ElemType+

Nu Mat Id

etc. Prop Data#

X Propid

Y Grid Id 1

Z Grid Id 2 etc.

Loadid

Gridid

Value

Spcid SubCaseid

Gridid

Dofs

Values

A Mat needs a card image. Use Mat1 for linear isotropic, Mat8 for orthotropic shells, Mat9 for linear anisotropic.

+

The component collector is either a PSOLID or a PSHELL. Composites use a PCOMP or PCOMPG.

#

Data depends on the element type. For a solid, there’s nothing. For a shell, there’s the thickness. For a composite, thickness is derived from the PCOMP or PCOMPG data. 63

Putting It Together – OptiStruct/Analysis

A Designer’s Guide To Finite Element Analysis

%

A Prop collector is needed only for 1D elements like beams or special elements such as springs, connectors, etc.

@

Forces and Moments need no card image. Loads such as gravity, which cannot be depicted graphically, require card images.

!

Restraints normally will not require a card image. Remember that nonzero displacements may be specified, in which case you will need to enter values. Restraints on non-existent dofs are ignored (for instance, specifying restraints on all the rotational dofs of a solid element).

$

Sub-Case definitions are followed by a set of cards, each with a keyword (Load, Spc, etc.) followed by the relevant id. These can be viewed in the “fem” file, not using the card editor.

Anyone who has never made a mistake has never tried anything new. Albert Einstein

64

A Designer’s Guide To Finite Element Analysis

Verification And Validation

Verification And Validation The “bumble-bee paradox” provides an excellent example of the gap between theory and practice. Whatever the reason, designers have successfully violated mathematicians’ recommendations in several areas. FEM is no exception. An AISI publication on the use of FEM for vehicle-safety investigations38 talks of “many

conscious violations of elementary finite element theory” in the very successful deployment of the method for highly nonlinear analyses. Given all this background, a beginner can be misled into thinking that the method more forgiving than it actually is. Spectacular failures that have been attributed to faulty FE modeling are documented in literature, and should serve as warnings to engineers that the cost of a mistake can be extremely high.

“Conventional aerodynamics seemed to suggest that the insect should not generate enough lift to fly. The bees stayed resolutely airborne and the sums caused consternation. The underlying problem turned out to be treating a wing as if it was fixed, like in an aeroplane and, thanks to studies over the past few years, including the construction of robotic bees, this "bumble-bee paradox" has been solved: extra lift comes when flexible insect wings slice through the air at a high angle of attack, creating a large swirling vortex at their leading edge. In this way, insect wings produce the vortices – spinning masses of air – which generate lift and help them move. Today, Prof Ismet Gursul of the University of Bath will describe another step on the way for engineers to make air vehicles smaller than a human hand that can be used for detecting chemicals leaks and reconnaissance.”

Roger Highfield, Science Editor The Telegraph

Sources of error can range from wrong assumptions of mechanics, as in the case of the aerodynamics of the bumble-bee, to a wrong choice of elements or even a disregard for computer round-off.

Validation consists of asking whether the right equations have been solved. Verification involves checking whether the equations have been solved correctly.

38

Vehicle Crashworthiness And Occupant Protection, American Iron and Steel

Institute 65

Verification And Validation

A Designer’s Guide To Finite Element Analysis

Both are an integral part of the engineer’s responsibility, not least because of statutory requirements.

Product Liability Laws Product Liability Insurance, in several countries, is as essential for an engineer as malpractice insurance is for a Doctor. The interpretation of laws is, of course, open to interpretation by lawyers and courts, but the various laws on the subject are fairly consistent in their spirit: it is the engineer’s responsibility to ensure that the product has been designed with care. Designers should ideally use FEM only as a part of the design process. It is often risky to base the entire design only on an FEM analysis. A NAFEMs39 publication warns against the GIGO approach: Garbage In, Gospel Out. This is particularly important since many analysis results are presented visually, and a picture can hide errors!

The Seductive Appeal Of Graphics Stress distributions are often plotted as contours – regions with the same stress are assigned the same color, and the component is shaded accordingly. This is an extremely useful method. Not only is it easier to assimilate than reading a list of stresses at various points in the component, it is also better than contour lines since it also indicates how the stress varies across the component. Consider the stress in a plate that’s fixed at one end, has a tensile load at the other end, and has a hole. The picture looks convincing – high stress areas, colored red, are around the hole as we’d expect.

39

66

National Agency for Finite Element Methods and Standards

A Designer’s Guide To Finite Element Analysis

Verification And Validation

But remember our earlier discussion on the characteristic of the FE solution. We said that the calculated stress is always discontinuous over element boundaries. Why is this not apparent in the image above? Several FE packages smooth the stress. If we plot the same contours without averaging the stresses, a different picture emerges.

Not only is the discontinuity in stresses brought out starkly, we must now doubt our solution itself! While the highest stress still seems to be around the hole, what is the value of this stress? Should we pick the value denoted by red, or by green? There is a “jump” in the stress at the edge shared by two elements, and we cannot decide which is correct. Refining the mesh and running the analysis again quickly brings home one error: the FE mesh was too coarse.

With the refined mesh, the difference between the smoothed contours and the unaveraged contours has reduced considerably. One rule of thumb is to distrust the model until the “jumps” in stress are within an acceptable value. What value is acceptable is a question of judgment, of course, but this can be a very good indicator of whether or not your mesh is fine enough.

Quick and Basic Checks Lots of other things can go wrong, of course, and catching errors in production-analyses is no easy task. 67

Verification And Validation

A Designer’s Guide To Finite Element Analysis

It’s a good idea to compare the results of an FE analysis with approximate solutions tabulated in handbooks40. Since not all problems can be found in handbooks, comparison with results reported by other investigators is a good idea. Several excellent scientific journals carry articles reporting the results of analyses, and most commercial software providers host annual conferences where users present papers discussing their experiences. The verification of a computer model by comparing it with the performance of a prototype is useful, but should not be treated as complete. There’s always the chance that an experiment has missed out on investigating rare combinations – and as Murphy’s Law41 swings into action, it is these rare combinations that can cause product malfunction. The accompanying volume in this series, CAE And Design Optimization – Advanced, discusses ways to design experiments and to detect rare combinations (called outliers), but such a study is beyond the scope of our introduction to the Finite Element Method. With this background, there are levels at which checks should be performed: at the Pre-processing stage and at the Post-Processing stage. HyperView, the post-processor that comes with HyperMesh, is particularly good at the latter. Checks include

68

•

Element free edge display

•

Element Normal display

•

Vector-arrow plots

•

Load contours

•

Unaveraged Stress contours

•

Ill-conditioning of the stiffness matrix

40

Roark’s Formulas for Stress and Strain is an excellent reference.

41

“If anything can go wrong, it will.”

A Designer’s Guide To Finite Element Analysis

Verification And Validation

•

Comparison of applied forces with reactions-at-supports

•

Comparison of mass of FE model with mass of CAD model

•

Orientation of beam elements

•

Shrunken plots to check for missing elements

Several of these are covered in the assignments that accompany this book.

Physics is very muddled again at the moment; it is much too hard for me anyway, and I wish I were a movie comedian or something like that and had never heard anything about physics! Wolfgang Pauli

69

Special Topics

A Designer’s Guide To Finite Element Analysis

Special Topics Just how much FE-theory do you need to know? To answer that question, it’s interesting to look at commonly used CAD-modelers. In the early days of CAD modeling, users had to know the internal details of splines etc. to effectively model surfaces. Today, the software, the technology and user interfaces are robust enough that the internal functions are invisible to the designer. The designer needs to delve into the inner workings only if things go wrong. The same is true of almost any technology. In the early 1900s car drivers had to be efficient mechanics, but today you could own a car for a decade without ever opening the hood! The same, of course, is also true of Finite Element Analysis. Several physical problems present challenges when the level of detail required from the results increases. For instance, the analysis of bolted structures presents formidable challenges if the area of interest is confined to the bolt itself: a realistic model should include contact, friction, pre-stress, plastic deformation, etc. Use the summary below as an indicator only. If your problem involves any of the characteristics described, then you will certainly need more than the material covered in this book. The references listed at the end of this book are a good place to start advanced research.

Advanced Materials As weight, appearance and cost become increasingly important, engineers are forced to choose from a wide array of materials, not all of which are as easy to characterize as metals. Of the various materials used in engineering design, steel is easily the most researched and most widely documented. Like most metals, it is isotropic, and has long been used to bear load. Much of engineering design is restricted to the linear, elastic range of the stress-strain diagram. Plastic deformation has received considerable attention over the recent past, as for instance in the design of formed components.

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A Designer’s Guide To Finite Element Analysis

Special Topics

For designers, plastics are a formidable challenge. As cheap, easy-tomanufacture materials, they are widely adopted. Unfortunately, an understanding of linear-elastic analysis techniques limits the designer to initial investigations only. To make things more difficult, the properties of plastics are not just a function of the composition. They also depend strongly on the processing conditions. Composites, made up of multiple materials, are more widely used than plastics to bear load. Reinforced Cement Concrete (RCC) is an example of a composite. The main body of material is strong in compression but weak in tension. Steel reinforcements add tensile strength. Mechanical engineers rarely analyze RCC, however. For product design, commonly used Engineering Composites include Fiber Reinforced Plastics (FRPs, of which Glass Fiber Reinforced Plastics and Carbon Fiber Reinforced Plastics are very widely used) and Metal Matrix Composites (MMCs). Composites are extensively used in the aircraft industry to provide acceptable strength at very low weights. Space applications are less cost-sensitive and more performance-intensive than others. Exotic metal alloys are sometimes used, but these are isotropic and relatively easy to model. Honeycomb structures are not as easy to model and design. Rubbery materials, also called elastomers, are usually treated as a category by themselves. They are not plastic. In fact, they are extremely elastic hyperelastic. This means they can carry very large strains without any permanent deformation. Large strains, of course, mean the analysis must be non-linear. Further, rubbers are almost incompressible: they have a Poisson’s Ratio of almost 0.5. Since the Bulk Modulus and Poisson’s Ratio are related by the equation

K=

E 3(1 − 2ν )

this makes them very difficult to deal with numerically (if ν= 0.5, K is indeterminate). Even small round-off errors can make enormous differences in the results of the analysis.

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Advanced Dynamics Transient analysis generates a vast amount of information, so postprocessing results is quite challenging. Stress contours and deformationplots are used to present the results of both static analysis and dynamic analysis. For dynamics, however, animations of deformation and timehistory plots are also used. Dynamic analysis requires considerably more computing power than static analysis. Remember that a symmetric structure can have un-symmetric modes of vibration too! So symmetry cannot be used to reduce the size of models. One method sometimes used to reduce the size of models is called Component Mode Synthesis. In this approach, the FE model is divided into sections considered as black-boxes. That is, interest is restricted to the boundary nodes of each section, thereby reducing the number of active degrees of freedom. Matrices are for constructed each section and the overall structure analyzed by putting these reduced matrices together. One form of excitation that is frequently encountered, particularly in vehicle design, is ground excitation. The structure receives energy because of enforced motion of selected regions, not because of explicitly applied forces. Several different methods are used to carry out such analyses. Two different approaches are used to perform transient analysis: mode superposition and direct integration. In the former, we assume that the response of the structure is a weighted sum of selected mode shapes. This is an approximation, where the error stems from the choice of the subset of mode shapes used. Direct Integration methods use Finite Difference Methods to “step-forwards” in time. Here we use equations of the form

v=

du ∆u ut + ∆t − ut = = dt ∆t ∆t

If we know the velocity at initial time, v0, we can write

v0 .∆t + u0 = ut + ∆t 72

A Designer’s Guide To Finite Element Analysis

Special Topics

In other words, given the values of v and u at t=0 (the “initial conditions”) we have just “stepped-forward” in time to calculate ut + ∆t . Together with the equilibrium equation, recurrence formulae of this type are used to calculate the time-dependent response. Several variations of timestepping methods are used: Central Difference Methods, Backward Difference Methods, Runge-Kutta Methods, etc. The source of error in direct integration is the time-step size, ∆t. Time stepping methods are classified based on whether they are explicit or implicit, as well as on the order of the method. In some conditions, we cannot state with any confidence how the forces themselves vary with time. What we can say with confidence is how the energy supplied to the structure varies with time. As an excellent example, consider the forces experienced by a rocket at launch. The combustion pattern is quite random, but the total energy released by the fuels is reasonably clear. Such problems are called Random Response problems. Response Spectrum methods are usually used to calculate the probabilities of various levels of stresses in the body.

Multi-disciplinary analysis, also called multi-physics analysis, involves the use of different branches of mechanics simultaneously. To understand the forces experienced by an aircraft in flight, for example, we cannot calculate the deformation of the aircraft unless we know the air-pressures. And the airpressures depend on the deformation. Such coupled analyses are still the subject of much active research.

Shall I refuse my dinner because I do not fully understand the process of digestion? Oliver Heaviside

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A Designer’s Guide To Finite Element Analysis

Glossary And References Accuracy

A measure of the effectiveness of a numerical method. Accuracy is valuable if coupled with stability.

Adaptive Refinement

A technique where the software automatically modifies the FE mesh, using measures of the errors in the solution.

Anisotropic

Material whose properties vary with direction, but not necessarily along orthogonal directions. Several fused or sintered materials are anisotropic. 21 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT2 for shell elements.

Bandwidth

The stiffness matrix of a typical FE model has zeroes in most entries except for a band about the diagonal. The bandwidth measures this “spread” of non-zeroes in the matrix. A smaller bandwidth means faster computation.

CAD

Computer Aided Design. Usually means creation of 3D models of parts and assemblies.

CAE

Computer Aided Engineering. Said to have been coined by Dr.Jason Lemon in 1980, and meant to consist of CAD + FE Modeling + FE Analysis + Design. Today includes Multi-Body Dynamics (MBD). Often separated into MCAE (for Mechanical CAE, or the analysis of structures) and FCAE (for Fluid CAE, or the analysis of flow of heat and liquids / gases).

CFD

Computational Fluid Dynamics. The use of computers to solve various forms of Navier-Stokes equations to analyze the flow of fluids.

Compliance

Reflects the reciprocal of the stiffness. Maximizing the compliance is the same as minimizing the stiffness.

Consistent Mass Matrix

This is one way to calculate the Mass Matrix (M) in FEA. The alternative is the lumped mass matrix.

Decibel

Frequently used in sound and vibration measurements. Defined by the ISO R 1683 standard as measured − value N = 20 ⋅ log10 reference − value

74

A Designer’s Guide To Finite Element Analysis

Glossary And References

Degree of Freedom

Variable we want the FE analysis to solve for. In an FE model every node / grid has at least 1 dof, often more. For a stress-analysis problem, each node can have upto 6 dofs – 3 rotations and 3 translations. Other variables such as temperature at the nodes, pressure, etc. are also included in some models.

Discretization Error

Difference between the actual domain of a problem and the domain defined by the FE model.

DOF

See Degree of Freedom

Effective Modal Mass

Characterize the relative contribution of the modes of a structure to an excitation

Hysteresis

Represents the history-dependence of a physical system. If a cyclic load is applied, the original properties are not always recovered at the end of a cycle. Closely related to work-hardening, in stress analysis.

Isoparametric

Most commercial FE programs use elements that are isoparametric. That is, the same shape function is used to approximate both the geometry and the solution.

Isotropic

Material whose properties are independent of direction. Applies to most metals. 2 elasticity constants are required to fully specify the material for stress analysis. The Modulus of Elasticity and the Poisson’s Ratio are most frequently used. In OptiStruct, these materials are of type MAT1.

Jacobian

The “Jacobian Matrix” arises when a change-of-variables is applied to a PDE. The determinant of the Jacobian Matrix is important to FE theory: it should always be positive for an element (strictly speaking it should never change sign). Most FE codes refer to it as the “Jacobian” or “Det J”.

Lanczos Method

A way to calculate the normal modes of a structure. Very efficient and stable.

Large Mass Method

A method to apply support-excitations. Uses the principle that a force applied to a large mass attached to a model is equivalent to applying an acceleration to the model. Given an acceleration you want to apply, Newton’s Second Law is used to calculate the equivalent force that should be applied for a mass you choose.

Lumped Mass Matrix

This is one way to calculate the Mass Matrix (M) in FEA. The alternative is the consistent mass matrix. Computationally more efficient, it is often preferred.

75

Glossary And References

76

A Designer’s Guide To Finite Element Analysis

MDO

Multi-disciplinary Optimization. Used, for example, when your product needs to be designed for optimal performance as a mechanism and as a structure.

Mises-Hencky

One of several failure theories for ductile metals.

MPC

Multi-point Constraint. Used to specify that different dofs are linked in a particular fashion.

Nominal Stress

Stress predicted by an equation. Used mainly in analytical calculations, the working stress is calculated by applying a factor of safety to the nominal stress.

NVH

Noise, Vibration and Harshness.

Orthotropic

Material whose properties vary along principal or orthogonal directions. Applies to many fibrous materials, and to composites that have 2 ply directions. Upto 9 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT8 for shell elements. MAT9 should be used for solid elements.

Post-buckling analysis

A highly-non linear analysis that investigates how structures behave after a buckling-related collapse.

Rigid Body Modes

Unrestrained structures can vibrate as a rigid body. A body can have upto 6 rigid-body modes.

Robust Design

A design method to reduce sensitivity of the design to inherent unpredictability of design parameters.

Spectrum

Term frequently used in dynamic analysis. The Fourier transform of a function gives us the spectrum. The spectrum can be used to reconstruct the function using the inverse transform.

Stability

With specific reference to series solutions, stability is related to the convergence characteristics. A monotonically convergent series allows us to estimate the accuracy of a finite sum.

Stiffness

Usually referred to as the “stiffness matrix” in FE models, relates the applied loads to the deformation of the structure. The matrix is square, with “n” rows and columns. “n” is the number of unknowns (dofs) in the FE model.

Superconvergence

A characteristic of the FE solution. The state variable (deformation, temperature, etc.) is best evaluated at the nodes, while the flux (strain, stress, heat flux, etc.) is best evaluated at the Gauss Integration points.

A Designer’s Guide To Finite Element Analysis

Glossary And References

Integration points.

Support Excitation

In many cases, such as a vehicle rolling over a rough road, the source of excitation is not a force: it is the prescribed deflection, which varies with time. While this can be treated mathematically as an “inverse” of a force-excitation problem, different techniques are employed to solve such problems.

Work-hardening

An increase in strength due to plastic deformation.

References Finite Element Procedures, K.J.Bathe Numerical Methods in Finite Element Analysis, K.J.Bathe and E.L.Wilson The Finite Element Method, O.C. Zienkiwicz Engineering Mechanics of Solids, E.P.Popov Mechanical Engineering Design, J.E.Shigley and L.D.Mitchell Roark’s Formulas for Stress and Strains, W.C.Young Formulas for Natural Frequency and Mode Shape, R.D.Blevins Theory of Vibration With Applications, W.T.Thomson Theory of Elasticity, S.Timoshenko Heat Transfer, J.P.Holman

Other Resources www.altair-india.com/edu, which is periodically updated, contains case studies of actual usage. It also carries tips on software usage.

Common Material Properties Be careful when using these properties. Some properties vary widely with alloying elements or processing parameters, so treat these as indicative. It’s probably safe to use them in exploratory design efforts, but not in designs 77

Glossary And References

A Designer’s Guide To Finite Element Analysis

that will be manufactured. For those, you should look for values from the material supplier. Also remember to check the units in your model – they must be consistent! Material

Modulus of Elasticity

Units

N/m2

Steel

200x109

Aluminum

Poisson’s Ratio

Density

Sample Permissible Stress

Kg/m3

N/m2

0.29

7800

250x106

69 x109

0.33

2700

110 x106

Wood42

13 x109

0.029

480

50 x106

Cast Iron

190 x109

0.21

7150

170 x106

ABS Plastics

2.3 x109

Epoxy

40x106 1790

Sample properties for a Carbon Fiber Composite material: E1 181 GPa E2 10.3 GPa G12 7.17 GPa ν12 0.28 Density 1.60 gm/cm3 For an isotropic material, any two properties are enough – the others can be calculated from these. We usually specify the Elasticity Modulus (E) and the Poisson’s Ration (ν). Other material constants such as the Shear Modulus (G) and the Bulk Modulus (K) can be derived from E and ν using equations such as :

G=

42

78

In compression

E (1 + ν ) 2

A Designer’s Guide To Finite Element Analysis

Glossary And References

E K= 3(1 − 2ν )

Useful Data For Heat Transfer Conduction is the main mode of heat transfer within a solid. The conductivity (k) is a material property that’s quite easy to ascertain for most metals, and can be assumed to be independent of temperature for relatively low temperatures. The thermal conductivity of 0.5%-Carbon Steel, for example, ranges from 55 W/m°C at 0°C to 52 W/m°C at 100°C. Plastics are a little less easy to analyze. The heat conduction coefficient must be measured by a test on a sample, or must be supplied by the manufacturer of the plastic. Also, several plastics are more sensitive to temperature than metals. Heat is transferred away from the solid by convection or radiation. The convective boundary conditions require the specification of the convection coefficient, h. This is a function of the fluid material and the fluid velocity. More precisely, it depends on the boundary layer between the fluid and the solid. Boundary layers depend on several factors – surface roughness, whether the boundary is vertical or horizontal, etc. One way to estimate h is to first calculate dimensionless numbers from available data, and then use empirical equations to calculate a dimensionless number (Nusselt, Stanton, etc.) that involves h. For example, given the flow velocity, the density and the viscosity of the fluid, we estimate the Reynold’s number (Re). Next, given the viscosity of the fluid, the specific heat capacity and the conduction coefficient of the solid, we estimate the Prandtl number (Pr). We then look for an empirical equation that relates Pr, Re and the Nusselt number (Nu). For instance for flow in a pipe, Dittus and Boelter43 suggest that

Nu d = 0.023 Re 0d.8 Pr n 43

See Heat Transfer, J.P.Holman, 7th Edition, Page 284 79

Glossary And References

A Designer’s Guide To Finite Element Analysis

We calculate Nu from this equation, then use the definition of Nu to solve for h. Remember that we want to solve for the temperature distribution. For this we need h. To get h we need the Reynold’s number. The Reynold’s number needs the viscosity. The viscosity is temperature dependent. So we need the temperature to get the viscosity. But we don’t know the temperature! This is why problems in heat transfer are often harder to solve than problems in stress analysis. Iterative methods are often used to arrive at the solution – estimate the numbers, solve for the temperature, use this temperature to re-estimate the numbers, solve again, etc. until there is no change. However choosing characteristic dimensions requires some care. For example, it is not always easy to decide whether to use the width or the length to calculate the Nusselt number for a rectangular plate! A strong physical insight is extremely useful for workable solutions to problems in heat transfer! Some useful dimensionless constants are presented here, but a full discussion is beyond the scope of this book.

80

Dimensionless Number

Relevance

Biot Number (Bi)

Ratio of external heat transfer (through convection) to internal heat transfer (through conduction).

Grashof Number (Gr)

Ratio of buoyancy force to viscous force.

Nusselt Number (Nu)

Ratio of internal heat transfer through convection to internal heat transfer through conduction

Prandtl Number (Pr)

Ratio of momentum diffusivity to thermal diffusivity

Peclet Number (Pe)

Ratio of rate of advection (i.e. diffusion and convection of mass) to rate of thermal diffusion.

Reynolds Number (Re)

Ratio of inertial forces to viscous forces.

Stanton Number (St)

Ratio of heat transferred into a fluid to heat capacity of fluid.

A Designer’s Guide To Finite Element Analysis

ρ µ ν β

Glossary And References

density dynamic viscosity kinematic viscosity volume coefficient of thermal expansion specific heat (at constant pressure) velocity of flow coefficient of heat conduction

cp u k

Consistent Units Mixing up units is one of the most common errors. It’s also the least forgivable if committed by an engineer who is allowed to use the SI system of units. While FPS can be challenging, the SI system is very straightforward. The table below lists some common properties of Steel in consistent units. Mass

Length

Time

Force

Stress

Density

Young’s Modulus

Acceleration due to Gravity

kg kg G G Ton lbfs2/in Slug Kg

M Cm Cm mm mm In Ft mm

s s s s s s s s

N 1.0e-02 N dyne 1.0e-06 N N lbf lbf mN

Pa

7.83e+03 7.83e-03 7.83e+00 7.83e-03 7.83e-09 7.33e-04 1.52e+01 7.83e-06

2.07e+11 2.07e+09 2.07e+12 2.07e+11 2.07e+05 3.00e+07 4.32e+09 2.07e+08

9.806 9.806e+02 9.806e+02 9.806e+03 9.806e+03 386 32.17 9.806e+02

dy/cm² Pa MPa psi psf 1.0e+03 Pa

For a dynamic analysis, OptiStruct lists both the natural frequency and the eigen-value. The former is in cycles / unit time, or Hz if SI units are used. The latter, which represents the square of the angular frequency ω, is related to the former by the equation

eigen-value = (2*p*frequency)2

Lumped Mass Models In Vehicle-Crash Simulation This material, from the American Iron and Steel Institute’s Vehicle Crashworthiness and Occupant Protection”, shows the relevance of discrete, multiple-degree-of-freedom (mdof) models. “In 1970, Kamal [1] developed a relatively simple, but powerful model for simulating the crashworthiness response of a vehicle in frontal impact. This model, known as the Lumped Mass-Spring (LMS) model, became widely used by crash engineers because of its simplicity and relative accuracy. This model is shown in Figure 2.2.2.1. The vehicle is approximated by a one-

81

Glossary And References

A Designer’s Guide To Finite Element Analysis

dimensional lumped mass-spring system, an over simplification that is quite acceptable for modeling the basic crash features in frontal impact. Because of its simplistic representation of the crash event, an LMS model requires a user with extensive knowledge and understanding of structural crashworthiness, and considerable experience in deriving the model parameters and translating the output into design data. The crush characteristics (spring parameters) were determined experimentally in a static crusher, as shown in Figure 2.2.2.2. The configuration of the model is arrived at from the study of an actual barrier test, which

identifies pertinent masses and “springs” and their mode of collapse. The model is “tuned” by adjusting the load-deflection characteristics of the “springs” to achieve the best agreement with test results in the timing of the crash events. Figure 2.2.2.4 compares the simulated acceleration histories with those obtained from the test, and it shows that very good agreement can be achieved. LMS models proved to be very useful in developing vehicle structures for crash, enabling the designer to develop generically similar crash energy management systems, that is, development of vehicle derivatives or structural upgrading for crash. Also, LMS models provide an easy method to study vehicle/powertrain kinematics, and give directional guidance to the designer by establishing component objectives.

Measures Of Element Quality The numerical integration procedure used in FE programs imposes several conditions on the shape of the element. Other conditions stem from the types of shape functions used. A detailed description of these measures is beyond the scope of this book. The on-line documentation presents more detail, in case you want to go beyond the summary below. 82

A Designer’s Guide To Finite Element Analysis

Glossary And References

Remember that an ideal element is rarely encountered. The ideal elements are:

Element Type

Ideal Shape

Quadrilateral Triangle Hexahedron Tetrahedron

Square Equilateral Triangle Cube Regular, equilateral tetrahedron

Aspect Ratio Applicable to all elements, this is the ratio of the longest edge to the smallest edge. The ideal value is 1, while an aspect beyond 5 is not recommended.

Interior Angle Applicable to quadrilateral and triangular elements, the ideal value is 90° for a quadrilateral and 60° for a triangle. The angle should never reach 0° or 180° in any case.

Jacobian Applicable to all elements, this is a measure of the distortion of the element as compared to the ideal element. A Jacobian of 0 or less is not permitted. The closer the value is to 1, the better.

Length There is no absolute measure for the acceptable element length. The smaller the elements, the more elements you require to span your geometry, and the more CPU time required. That apart, using very small elements and very large elements in the same mesh can give rise to round-off error.

Skew Applicable to all quadrilateral and triangular elements, an ideal element has a skew of 0°.

Taper Applicable only to quadrilaterals, a taper of 0 is ideal.

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Warpage Applicable only to quadrilateral elements, warpage is 0 if all 4 nodes lie in the same plane. Upto 5° is considered good.

Collapse Applicable to tetrahedral elements, the perfect value is 1. The value should never reach 0.

Volume Aspect Ratio Used for 3D elements (hexahedra and tetrahedral), 1 is the ideal value.

Volume Skew Applicable only to tetrahedral elements, 0 is the ideal value. The value should never reach 1.

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Contents

Contents Introduction ......................................................................................................2 About This Series ...........................................................................................2 About This Book .............................................................................................2 Supporting Material ........................................................................................2 FEA – What It Is … And What It Isn’t..................................................................4 Typical Usage.................................................................................................4 Limitations .....................................................................................................5 Learning FEA..................................................................................................8 The Importance Of Computing Power..............................................................9 Mechanics, Mathematics, And Other Theory ...................................................... 10 Solid Mechanics............................................................................................ 11 Thermal Analysis .......................................................................................... 23 Fatigue And Fracture .................................................................................... 26 Mathematics ................................................................................................ 28 Essential FEA Theory ....................................................................................... 38 From The Differential Equation To A Matrix Equation...................................... 38 Nodes, Elements And Shape Functions .......................................................... 43 Some Common Elements Used In Stress Analysis ........................................... 45 Matrix Solvers .............................................................................................. 47 Some Important Properties Of The FE Solution .............................................. 50 Putting It Together – OptiStruct/Analysis .......................................................... 52 Capabilities .................................................................................................. 52 Setting Up An “Analysis” ............................................................................... 57 Nomenclature and Data Organization ............................................................ 61 Verification And Validation................................................................................ 65 Product Liability Laws ................................................................................... 66 The Seductive Appeal Of Graphics ................................................................. 66 Quick and Basic Checks ................................................................................ 67 Special Topics ................................................................................................. 70 Advanced Materials ...................................................................................... 70 Advanced Dynamics ..................................................................................... 72 Glossary And References.................................................................................. 74 References................................................................................................... 77 Common Material Properties ......................................................................... 77 Useful Data For Heat Transfer....................................................................... 79 Consistent Units ........................................................................................... 81 Lumped Mass Models In Vehicle-Crash Simulation .......................................... 81 Measures Of Element Quality ........................................................................ 82

1

Introduction

A Designer’s Guide To Finite Element Analysis

Introduction About This Series To make the most of this series you should be an engineering student, in your third or final year of Mechanical Engineering. You should have access to licenses of HyperWorks, to the Altair website, and to an instructor who can guide you through your chosen projects or assignments. Each book in this series is completely self-contained. References to other volumes are only for your interest and further reading. You need not be familiar with the Finite Element Method, with 3D Modeling or with Finite Element Modeling. Depending on the volumes you choose to read, however, you do need to be familiar with one or more of the relevant engineering subjects: Design of Machine Elements, Strength of Materials, Kinematics of Machinery, Dynamics of Machinery, Probability and Statistics, Manufacturing Technology and Introduction to Programming. A course on Operations Research or Linear Programming is useful but not essential.

About This Book This volume introduces you to Finite Element Analysis, a numerical method that is the cornerstone of most Computer Aided Engineering (CAE) programs in the industry. Design is addressed in the other volumes of this series. This book focuses wholly on Analysis. However the presentation is from a design perspective: FEA is presented with a minimum of mathematics, and with a strong focus on applications. You may choose to treat some chapters as references, depending on your level of familiarity with the underlying theory. If you find any chapter hard to follow, you should read it once for an overview, and refer to it again when doing the assignments if necessary. The various references cited in the book will probably be most useful after you have worked through your project and are interpreting the results.

Supporting Material Your instructor will have the Student Projects and Student Projects Summaries that accompany these volumes – they should certainly be made 2

A Designer’s Guide To Finite Element Analysis

Introduction

use of. Further reading and references are indicated both in this book and in the Projects themselves. If you find the material interesting, you should also look up the HyperWorks On-line Help System. The Altair website, www.altair.com, is also likely to be of interest to you, both for an insight into the evolving technology and to help you present your project better.

Do not be afraid to skip equations (I do this frequently myself). Roger Penrose

3

FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t “After working nearly four years with engineers, a mathematician confessed to me that he finally began to understand what engineers meant by finite elements … And another mathematician, attending an engineering conference, seemingly in disbelief begged the author to tell him whether engineers really thought of finite elements as little plate elements connected at their edges …1”. For an engineering tool that dates back to at least 1960 when Clough chose the term “Finite Elements”2, the Finite Element Method has had more than its fair share of misinterpretations. Part of this, of course, is simply because of its success. As a robust, reliable and versatile tool, it has enjoyed unparalleled success over the past several decades. Almost every field of engineering – Aeronautical, Naval, Mechanical, Electrical, Chemical, Civil, … – has seen the method deployed with spectacular success. And, unfortunately, a few spectacular failures. A good place to start the study of this fascinating blend of mathematics and engineering-design is by reviewing some typical applications.

Typical Usage Most design involves one or more of form, function, and fit. The various tasks of an Engineering Analyst revolve around the investigation of the function. Industrial designers usually address form, while CAD modelers address fit. Stress Analysis is easily the most widely known of these. Given a component or an assembly, the analyst assesses the likelihood that the stress in the material will fall within permissible limits. The difficulty lies in the shapes of the parts to be analyzed, the properties of the materials used, and the various sources of loads. In most popular usage, stress analysis is taken to apply only to solids – that is, to the study of Solid Mechanics. 1

“The Mathematical Foundation of Structural Mechanics”, Friedel Hartmann And probably even further back to 1943 when Courant introduced the linear triangular element.

2

4

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FEA – What It Is … And What It Isn’t

Fluid Mechanics can often be treated independently of Solid Mechanics. An engineer investigating the flow of air around a car or the flow of water around a speedboat focuses on the drag forces, leaving the study of stresses for later investigations. Sometimes, however, solid and fluid mechanics cannot be separated as easily – such coupled analyses are called Fluid Structure Interactions (FSI). Thermal engineering involves the study of the flow of heat, often covered in undergraduate courses on Heat Transfer. The analyst’s job is to predict the temperature distribution in the areas of interest. Mass Transfer is a closely related field, encountered for instance in the mixing of chemicals in a reactor, the seepage of underground water around an oil well, or the effects of tidal flow on the release of effluents. A new entrant to the field can be excused for believing that FEA is the engineering equivalent of a miracle-drug, a cure for every disease known and unknown. Experienced engineers, of course, know that there’s no such thing. Or of there is, it hasn’t been discovered yet! It’s important, therefore, to also remember that there are some fields where FEA can be used only sparingly if at all, and that there are some fields where other tools are better deployed than FEA.

Limitations As we will see later, FEA can be used to investigate the behavior of any process that can be described by a differential equation. However, it’s one of many methods that can be, and have been, used to solve differential equations. Some of the mechanical-engineering disciplines where other methods are more widely used than FEA are the analysis of rigid bodies, the flow of fluids, and acoustics.

Mechanisms One of the first topics a student of mechanical engineering encounters is the difference between structures and mechanisms. Plainly put, a mechanism is 5

FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

a body that is capable of rigid movement. Under the action of a force, the mechanism gains kinetic energy. This is in contrast to structures, which gain strain energy under the action of a force. Finite Element methods have been extended to include the simulation of both behaviors, for often times the analyst cannot be expected to know when which form of energy-transfer dominates. A good example of this is the simulation of a car-crash. At some instants, when the car is in contact with other bodies such as the vehicle or the road, its strain energy is increasing. At other instants, when the car loses contact with other bodies as it flips over or bounces, it behaves as a rigid body. The analyst cannot know before hand which event occurs when: the solution of the equations must provide this information! Situations like these call for multi-disciplinary or multi-physics analyses, and FEA is indeed a vital part of the investigations. But in many other cases, an investigation of the kinematics or dynamics of the mechanism is a prelude to stress analysis. The accelerations calculated as a part of mechanisms-analysis are used to estimate the forces that will be used for subsequent stress-analysis. Some FE solvers today provide access to mechanisms-analyses too. OptiStruct, which we will discuss later, provides such facilities, but it is important to remember that kinematics and dynamics calculations use numerical methods other than FEA3.

Fluid Flow The Navier-Stokes equations have long troubled engineers. Sir Horace Lamb is said to have observed “I am an old man now, and when I die and go to

Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. And about the former I am rather more optimistic4.” The Navier-Stokes equations themselves have various applications – for incompressible flow, for transonic flow, for laminar flow, and so on.

3

See CAE For Rigid Body Mechanics Quoted in Computational Fluid Mechanics and Heat Transfer, by Anderson, Tannehill, and Pletcher, 1984. 4

6

A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t

Some forms of fluid flow have been simulated using FEA, but this is more the exception than the rule. Computational Fluid Dynamics (CFD) as a science has tended towards Finite Difference Methods (FDM), Finite Volume Methods (FVM) and Smoothed Particle Hydrodynamics (SPH). The reasons for this are many, and beyond the scope of this book. We will only note that if FEA is used to analyze fluid flows, great care should be exercised both in modeling and interpreting the results.

Acoustics Vibration is an essential part of engineering design. Not only does it cause stress and strain, it causes discomfort and generates noise. All noise, of course, is sound. And all sound is a form of vibration. However the reverse is not true. Engineers, particularly in fields related to vehicle design, use the term NVH (for Noise, Vibration and Harshness) to segregate vibrations based on the frequency of the vibration. FEA has been employed very successfully at the lower end of the frequency-spectrum, typically for excitations that are lower than about 10,000 rpm. There is nothing sacrosanct about that number, of course – FEA has also been used for higher frequency vibrations. But the use of alternative numerical methods (BEM or Boundary Element Methods are quite popular) is more common in the investigation of the generation and transmission of sound or noise.

NVH Characteristics 5 – 25 Hz: Shake

25 – 100 Hz: Harshness, Boom

100 – 150 Hz: Moan

150 – 300 Hz: Noise

Some FE solvers, including OptiStruct, provide acoustic analysis capabilities too.

Time Integration Quantum Mechanicians like to theorize about time travel. But for most engineers, the spatial and temporal domains are firmly separated. The first involves the familiar spatial coordinates – x / y / z in Cartesian coordinates, r / θ / z in Polar coordinates and r / θ / ϕ in Spherical coordinates – and is amenable to the mathematics of Finite Elements. Problems that involve a 7

FEA – What It Is … And What It Isn’t

A Designer’s Guide To Finite Element Analysis

variation of the parameters of interest in the spatial domain are called Boundary Value Problems. Time is not so accommodating. Problems in which the variables of interest change with time are called Initial Value Problems. These are often solved by applying FEA to the spatial-part of the differential equation and FDM to the timevariation.

Learning FEA Traditionally, authors tended to choose one of two approaches to introduce FEA. One is a purely mathematical approach. This uses functional analysis, which is one way to study the behavior of functions whose arguments are functions. This approach has the advantage that it is very general in its application. There is usually little mention of applications. The other is an application specific approach, where the method is developed entirely in the context of one or more specific design disciplines. The advantage is that the physical significance of the various terms is emphasized right from the start, while the disadvantage is that the logic is not readily transferred to other disciplines. Fortunately, over the years, the method has matured. Mathematical rigor has been added to most engineering discoveries. Examples of usage and relevance illustrate almost every mathematical proof. Like most powerful tools, FEA in the hands of a novice can wreak havoc. For effective use, an understanding of the mechanics is essential. A grasp of the mathematics can help enormously, particularly to diagnose errors, if they should occur. The next chapter presents a quick summary of mechanics, recalling topics you would have covered in your courses on mechanics. This is followed by a brief summary of the theory of Finite Elements, before moving on to applications.

8

A Designer’s Guide To Finite Element Analysis

FEA – What It Is … And What It Isn’t

The Importance Of Computing Power The popularity of FEA is inextricably tied with the growth of computing power. If FEA is today a part of the computer-desktop of most practicing designers, it is no accident. It is largely due to the current proliferation of desktop computers, several of which are more powerful than decade-old supercomputers. The two most important measures of computational power are storage space and computing speed. The Finite Element method, of course, makes strong demands on both. The level of detail addressed by FE models has grown steadily over the years, making the best use of available computing power5. One of the more important trends in analysis today is the increasing usage of stochastic analyses, which can often require hundred or even thousands of analyses of the same model to yield more reliable designs6. The assignment problems that accompany this book are completely realistic, drawn from actual engineering applications. However they are restricted to linear analysis only. The other volumes in this series address more advanced applications of FEA: optimization, reliability, multi-disciplinary analyses, and manufacturing simulation.

It was absolutely marvelous working for Pauli. You could ask him anything. There was no worry that he would think a particular question was stupid, since he thought all questions were stupid. Frederick Weisskopf

5

The table is from the American Iron And Steel Institute’s Vehicle Crashworthiness

And Occupant Safety 6 See CAE And Design Optimization - Advanced 9

Mechanics, Mathematics, And Other Theory

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory Mathematical physics saw an explosion of activity in the early 18th century, following the invention of calculus. The initial enthusiasm led to the development of numerous partial differential equations as a result of investigations into a wide variety of areas. Developing a differential equation to model a physical phenomenon only addresses half the problem. It shows that the phenomenon is understood. The other half of the problem, of course, is to predict behavior: how parameters of interest will change as controllable parameters are varied. This has been long recognized7: We remember very well the time when almost every author dealing with a non-trivial elasticity problem considered it very nearly a matter of his honour to reduce it by all means to a Fredholm equation of the second kind. After this, he was prone at least to think that his investigation was completed theoretically without concerning himself with the implementation of the solution. (People wonder at it now).

For a long time, the search for solutions to differential equations was an art by itself. Every undergraduate student of engineering is familiar with the wide variety of methods to solve differential equations: integrating factors, separation of variable, partial fractions, substitution, and so on. One method, which proved to be particularly successful, was the use of infinite series to estimate the solutions. This led to another burst of activity in the development of infinite series. As we will see subsequently, Finite Element Analysis can be viewed as yet another series method of solving differential equations. In this chapter, we will review the various terms in mechanics and mathematics. We will need to be familiar with these in order to interpret the results of a Finite Element Analysis. Note that rigor has been omitted, and sometimes even sacrificed, for simplicity and brevity.

7

10

From Integral Equations in Elasticity, V.Z.Parton and P.I.Perlin, 1977

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

Solid Mechanics As outlined in the earlier chapter, FEA is not the preferred method for problems involving fluid flow. The sections below, therefore, address the mechanics of solids only. The emphasis is on introducing the continuum approach. In this, we define most parameters of interest using differential equations rather than difference equations.

Engineering Mechanics It is not always easy to determine whether it is better to use difference or differential equations. A first semester course on engineering mechanics usually adopts the former approach. In this approach, we first define a link as a rigid body that possesses at least two joints, and a joint as a point of attachment between links. A binary link has 2 joints, a ternary link has 3 and a quaternary link has 4. We then use Gruebler’s equation to determine whether the assemblage is a

structure or a mechanism: if there are no unconstrained degrees-of-freedom it is a structure, else it is a mechanism. We further categorize structures as statically determinate and statically indeterminate. The former are structures which have 0 degrees of freedom. All forces can be calculated using only the equilibrium equation: that is, by solving for the force and moment balances.

∑F = ∑F = ∑F = ∑M x

y

z

x

= ∑My = ∑Mz = 0

Statically indeterminate structures are those that are over-determined. They have “negative” degrees of freedom. To solve these, the force and moment equilibrium equations are not enough. The elastic properties of the body must be used. Methods of solution involve the construction of free body diagrams. This involves showing all external forces acting on a body. If the externally applied forces are independent of time, the problem is one of Statics. If, however, the externally applied forces are time dependent, we are faced with a problem in Dynamics. 11

Mechanics, Mathematics, And Other Theory

A Designer’s Guide To Finite Element Analysis

A dynamic problem can be solved by applying D’Alembert’s principle. In this, we assume the body is in static equilibrium at every instant by applying a D’Alembert force. This is calculated as

F = M ⋅a where a is the acceleration and m is the mass. We assume that F is applied at the center of mass, and is in the direction opposite to the acceleration. The stress in the link is given by

σ=

F a

ε=

∆L L

The strain is defined by

Stress and strain are related by Hooke’s Law,

σ = Eε where E is the Modulus of Elasticity of the Young’s Modulus. Hooke’s Law is sometimes referred to as the constitutive equation, since it involves the material properties, which in turn depend on the constitution of the material.

Continuum Mechanics The difference-equation approach is valid as long as it is reasonable to treat the link as a rigid body. That is, if it is reasonable to assume that there is no variation of stress or deformation within the link itself. In engineering design, these assumptions are relevant if the links are more or less unidimensional – long and thin – and if the links are joined with pins. The study of such assemblies usually centers around “trusses”. Electric-transmission towers are an excellent example. If the areas of interest involve changes in cross-section, in area, in material properties, etc., it is better to use a continuum approach. That is, to assume that the variables of interest vary though the continuum, and to write equations that allow us to track these changes. 12

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

The first step is to define the deformation vector. Recall that a body in 3dimensional space has 6 degrees of freedom – translations and rotations about 3 coordinate axes. Remember also that we need to differentiate between rigid movements and flexible deformations. A rigid movement is relevant to a mechanism, while a structure must deform under the action of external forces. By definition, the structure cannot move as a rigid body. The displacement vector has six components. These are decomposed into a deformation vector and a rotation vector. The deformation vector is usually denoted by u. Since continuum mechanics involves extensive use of vectors, columnmatrices or column vectors are frequently employed to represent quantities of interest. Some quantities like stress, which have components that themselves are vectors, are called tensors8. Again, matrices are used to represent the components. In vector form, the deformation vector is

u x u = u y u z Strain is defined as the rate of change of deformation. If we restrict our attention to a single dimension9, we have

ε=

du dx

In the more general case, working in 3-dimensions, the above equation is replaced by

8

Strictly speaking, a tensor is a group of functions that satisfy some conditions. Matrices are a convenient way to represent tensors, but not every matrix is a tensor. 9 And restricting attention to small strains. That is, we assume (du/dx)2 and higher powers are much less than du/dx and can be ignored. 13

Mechanics, Mathematics, And Other Theory

A Designer’s Guide To Finite Element Analysis

∂u x ∂x ε xx ∂u x ε ∂y xy ∂u ε xz x ∂z ε = = ∂u ε yy y ∂y ε xz ∂u y ε zz ∂z u ∂ z ∂z The strain tensor is given by

ε xx ε = ε yx ε zx

ε xy ε xz ε yy ε yz ε zy ε zz

Note that the strain tensor is symmetric. That is, εij = εji. Accordingly, there are 6 components for the strain: 3 normal strains and 3 shear strains. Similarly, the stress tensor is

σ xx σ xy σ xz σ = σ yx σ yy σ yz σ zx σ zy σ zz and is also symmetric, with σij = σji. It is useful to remember the physical meaning of the vector notation used for tensors. For instance, σxy is the stress caused by a force in the y direction acting on a face whose normal is in the x direction. The force per unit area is called the traction. When resolved on a particular face, we get the stress component.

14

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Mechanics, Mathematics, And Other Theory

The drawback of the stress and strain tensors is that they are coordinate dependent. In other words, a change of coordinates can change the stresses and strains. This reduces the utility of the mathematical definitions of the stress and the strain. Physically, we expect the stresses and strains to give us an indication as to the state of the body. Obviously a change in axes cannot change the physical state of the body! To get around this, we usually construct strain invariants and stress invariants. These are derived from the strain and stress tensors respectively, and are invariant – that is, they do not change under coordinate transformations. The invariants use principal values: the principal stresses and the principal strains. An easy way to calculate principal stresses is to solve the equation

σ xx − λ σ xy σ xz σ yx σ yy − λ σ yx = 0 σ zx σ zy σ zz − λ for λ. From linear algebra we recall that for the above equation gives us a cubic equation that has 3 roots – these are the 3 principal stresses for that state of stress. Mohr’s Circle is another way to calculate the principal stresses, σ1, σ2 and σ3. Similarly, we can calculate the principal strains. The main use of the principal stresses and strains is to calculate a value that can be used in a failure theory. Various failure theories have been proposed based on empirical observations. For ductile materials like steel, for instance, the Von Mises criterion is often used:

σe =

(σ 1 − σ 2 )2 + (σ

− σ 3 ) + (σ 3 − σ 1 ) 2

2

2

2

As per this theory, a material will yield when the equivalent stress crosses the yield stress. 15

Mechanics, Mathematics, And Other Theory

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The yield stress or the yield point is most familiar when considered for a unidimensional experimental specimen:

Not all materials have a clearly defined yield point. For these, a proof stress or offset yield stress is sometimes used. This is obtained by drawing the line as shown, at a strain of 0.002. This strain is called the 0.2% proof strain. For tri-axial states of stress, designers often use the principal stresses as axes. In this depiction, the yield surface marks the limit of the elastic region. Stress and strain are related by the 3-dimensional version of Hooke’s Law. Since the stress and strain each have 6 components, the material properties are often represented by a constitutive matrix.

σ 11 c11 σ 12 c 21 σ 13 c 31 = σ 22 c 41 σ 23 c 51 σ 33 c 61

c12 c 22 c32 c 42 c52 c 62

c13 c 23 c33 c 43 c53 c 63

c14 c 24 c34 c 44 c54 c 64

c15 c 25 c 35 c 45 c 55 c 65

c16 ε 11 c 26 ε 12 c36 ε 13 c 46 ε 22 c56 ε 23 c 66 ε 33

Note that in the above equation the subscripts 1, 2 and 3 have been used instead of x, y and z. Finally, the equilibrium equation is now three differential equations: 16

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

∂σ xx ∂σ yx ∂σ zx + + + Bx = 0 ∂x ∂y ∂z ∂σ xy ∂σ yy ∂σ zy + + + By = 0 ∂x ∂y ∂z ∂σ xz ∂σ yz ∂σ zz + + + Bz = 0 ∂x ∂y ∂z where Bx, By and Bz are the body forces in the respective directions x, y and z. A body force is distinct from a boundary force. The former is distributed all through the body – for instance the magnetic attraction an iron bar experiences, the force due to gravity, etc. The latter can be a point, line force, or an area force. Point loads are also called concentrated loads. It is important to remember that point loads and line loads are idealizations – they can never be achieved in practice.

Statics – From Theory To Design In the equations above, time variations were ignored. All derivatives were spatial derivatives only. In other words, the equations defined problems of static analysis. Designers often use a factor of safety to allow for uncertainties in various data. In this case, the permissible stress is a fraction of the yield stress:

σp =

σy k

where k is the stress concentration factor. Design codes use historical data to suggest factors of safety for different applications. In closing, two important aspects of stress analysis are worth noting: stress

concentration factors and St.Venant’s Principle.

17

Mechanics, Mathematics, And Other Theory

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If a geometry is complicated, the nominal stress is calculated over a simpler geometry, and then factored up using a stress-concentration factor. Handbooks10 present factors for various configurations. If a load is hard to characterize, we rely on St.Venant’s principle, which tells us that the difference between the stresses caused by statically equivalent

load systems is insignificant at distances greater than the largest dimension of the area over which the loads are acting. This means that we can replace a load by its static equivalent, and is the reason why point and line loads are used even if they are not realistic.

Dynamics - Theory If variables of interest change with time, the differential equations need to be modified to include time derivatives. Conventionally, different symbols are used for spatial-derivatives and time derivatives: a prime for the former and a dot for the latter:

du d 2u = u ′, 2 = u ′′ dx du du d 2u = u&, 2 = u&& dt dt Note that that the first and second time derivatives of deformation (u) give the velocity (v) and the acceleration (a) respectively. Problems in dynamic analysis are harder to characterize than problems in statics. To understand why, it is useful to recall the equilibrium equations for the mass-spring-dashpot, which is a single-degree-of-freedom system:

ma + cv + ku = p The mass and stiffness are relatively easy to measure, but the damping is often difficult to obtain. In the equation above, we assume that the dashpot is a viscous damper. That is, the resisting force depends on the velocity of motion. Viscous

10

18

See, for instance, “Stress Concentration Factors” by R.E.Peterson

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

damping is not always a good model, however. Various alternatives are available, including hysteretic damping, sliding friction, Coulomb damping, etc. Unfortunately damping, like friction, is hard to characterize. We often continue to use the above equation. To maintain accuracy, we modify the damping coefficient c to match experimental results. This is sometimes called the equivalent damping coefficient. Estimating the damping coefficient requires experimentally collected data. For most materials, the damping coefficient depends on the frequency of excitation. A material that behaves in this fashion is characterized by modal damping factors. The dependence of the damping coefficient is not always predictable: higher frequency excitations die away (or attenuate) more rapidly than lower frequency excitations, for most mechanical engineering applications11. To estimate the damping coefficient, an experimental specimen is excited by a source that sweeps through the entire range of frequencies of interest. The response is captured and characterized as the Frequency Response Function (FRF).

H (ω ) =

X (ω ) F (ω )

An FRF is a transfer function that relates the excitation at one point with the response at a point of interest. Note that an FRF is a function of the frequency of excitation, ω12. If there are multiple points of excitation / interest, the FRFs are best treated as a matrix, with

H ij (ω ) =

X i (ω ) F j (ω )

The damping coefficients can be estimated from the FRFs as a function of frequency. The critical damping value ζ, which is the value at which the 11

An important exception is the occurrence of flutter. See the references for more details. 12 ω is usually measured in Hertz (Hz). Beginners often confuse rpm, Hz and radians / sec. 19

Mechanics, Mathematics, And Other Theory

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structure loses its oscillatory response to any excitation ω, is used to estimate the modal damping coefficients σ:

ζ =

σ σ + ω2 2

The single-degree-of-freedom system can be readily extended to a multipledegree-of-freedom system using similar equations, with the added complexity of manipulating multiple equations. Often matrices are used to make the manipulation easier. Using matrix notation, the equilibrium equation is written as

[M ]{a}+ [C ]{v} + [K ]{u} = { f } where the square braces represent square matrices, and the curly braces represent column vectors. For a system that has n degrees of freedom, M, C and K are each nxn, while a, v, u and f are each nx1. The system of equations needs to be extended to infinite degrees of freedom if it is to represent a continuum, which is what an ideal elastic body is. The principal is the same even if the mathematics can be quite messy. Recall that we earlier wrote the equilibrium equation in terms of the stress components and the body forces. Dynamics involves accelerations and velocities. These are derived from the deformation vector by taking the time derivative. We can rewrite the equation in terms of the deformation using the stress-strain and strain-deformation equations. One familiar form of this equation is the wave equation. In its simplest form, for example the vibration of a string, the equation is: 2 ∂2 y 2 ∂ y = a ∂t 2 ∂x 2

where a depends on the tension in the string and the mass-per-unit-length (or density) of the string.

Dynamics – From Theory To Design Most design problems involve multiple degrees of freedom. If the structure consists of rigid links, as in a transmission tower, then a model with a finite 20

A Designer’s Guide To Finite Element Analysis

Mechanics, Mathematics, And Other Theory

number of degrees of freedom is a good representation. If, however, the structure is like a casting or is made from sheet-metal, a continuum representation is more appropriate. In this case, the structure has infinite degrees of freedom. Regardless of which approach is taken, the reality is that most models have more than one degree of freedom. If there are no external forces, the equilibrium equation that we wrote above becomes

[M ]{a} + [C ]{v} + [K ]{u} = 0 One obvious solution to this equation is the trivial solution: an identically zero deformation vector. But could there be other solutions? In other words, is it possible that the structure will deform even in the absence of an external force? The answer is “yes”, and leads to the study of mode-shapes and natural-frequencies, which lie at the very heart of studies of vibration13. If we assume that the deformation is of the form

u ( x, y, z , t ) = uo ( x, y, z ) sin(ωt ) we then have

v = u& = ω.u0 cos(ωt ) and

a = u&& = −ω 2 .u0 sin(ωt ) Of course, there is no necessity that we only use trigonometric functions – we could well use other forms, but this is a very convenient choice. If we neglect damping14, the equilibrium equation now becomes

− w 2 [M ]{u0 } + [K ]{u0 } = 0 13

The Sturm-Liouville equation is usually treated as the basis for all such investigations for partial-differential equations. 14 Only for brevity. Damping need not be excluded. 21

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which can be rewritten as

[A]{u0 } = 0 where

[A] = −w2 [M ] + [K ] If we assume that

λ {uo } is a solution, then we can rewrite the equation as

[A − λI ]{u0 } = 0 where [I] is the identity matrix. This equation will have non-trivial solutions (i.e. solutions where u0 is not identically zero) if the matrix is singular. That is, if the determinant is zero. The equation

A − λI = 0 gives us a polynomial equation in λ, called the characteristic equation. The order of the polynomial equation depends on the size of the matrix A. An nxn matrix gives us an nth order polynomial, which will have n roots. For each of these values of λ, we can find a corresponding {u}. These λ and {u} pairs are called eigen-solutions. λ is called the natural-frequency or eigenvalue and {u} is the corresponding mode shape or eigen-vector. If we include the damping factors, [C], the characteristic equation may have imaginary roots. These imaginary roots can be interpreted physically15: the 15

Interested readers should review de Moivre’s theorem, the Argand diagram, Nyquist and Bode plots 22

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imaginary terms are the equivalent of a phase lag. That is, due to the presence of damping, response lags behind excitation. In the absence of damping, the eigen-values are all real: physically, this means the response and excitation are always in phase. In this case, the eigen-solutions are called normal modes. Complex Modes are what we get if damping is included. Normal modes have stationary nodal lines – that is, they are standing waves. Complex modes are traveling waves. In many physical structures, damping is neglected at the initial stage when estimating natural frequencies, and introduced later when estimating forced response, which is the response to an excitation. This simplified procedure is reliable and very widely used. This is particularly convenient in product design. Remember that the damping factors need to be estimated experimentally, which means a prototype needs to be available for testing. Early in the design cycle, this is unrealistic. However estimation of mass and stiffness requires no experimentation. CAD users will know that solid models readily provide the mass, and we will see later how FE models provide the stiffness. Therefore, early in the design cycle, the concentration is on the calculation of the normal modes. In most cases we want to avoid resonance, but in some cases we may want resonance. Either way, we use the results of the analysis to modify the design. To calculate the forced response, we often use damping coefficients from literature or from databases of earlier designs. When the prototype is built and tested, a quick verification of the earlier design can be carried out.

Thermal Analysis The study of heat, Thermodynamics, is notoriously difficult:

Thermodynamics is a funny subject. The first time you go through it, you don't understand it at all. The second time you go through it, you think you understand it, except for one or two small points. The third time you go through it, you know you don't understand it, but by that time you are so used to it, it doesn't bother you any more16. 16

Supposed to have been said by Arnold Sommerfeld, when asked why he had not written a book on the subject. 23

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Temperature itself, however, is much easier to deal with than stress, mainly because it’s a scalar quantity. The difficulties in design lie in the estimation of physical quantities required for a reliable analysis. This section briefly reviews the essentials of thermal analysis and thermo-mechanical analysis. The three modes of heat transfer are conduction, convection and radiation. The first is important principally in solids. The second is significant in fluids (liquids and gases). The third, radiation, is significant only at high temperatures or if other modes of transfer are negligible, as in outer space. The heat flux, which is proportional to the gradient of temperature, represents the flow of thermal energy. In a solid it depends on the conductivity, and in a fluid on the convection coefficient. At a steady-state (i.e. when thermal equilibrium has been reached) in one-dimension we have

q = − kA

dT dx

and

q = hA(T − T∞ ) The conduction coefficient, k, is easily measured. It does vary with temperature. For many engineering materials the variance is so small at operating ranges that we assume it is constant. The convection coefficient, h, is rather more difficult to estimate. It is not just a property of the materials that make up the fluid, but also depends on the velocity of the fluid. The study of heat-transfer therefore needs to borrow heavily from the study of fluid-flow. It is customary to estimate nondimensional numbers (the Reynolds Number, the Biot Number, the Prandtl Number, the Nusselt Number, the Grashoff Number, etc.) for a design problem. These numbers are then used to look up literature to estimate the convection coefficient. Radiative heat transfer is governed by the equation

q = fAσ (T14 − T24 ) where f is a factor that depends on the configuration, σ is the StefanBoltzman constant, and T1 and T2 are the temperatures of the two bodies in 24

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question, in degrees-Kelvin. Since σ is 5.669 x 10-8 W/m2.K4, radiative heat transfer is significant only when the temperatures are substantially high (as in the cooling of molten steel) or when other modes of heat transfer are negligible (as in spacecraft traveling through “empty” space). The differential equation that governs heat conduction in a body is easily derived:

∂ ∂T ∂ ∂T ∂ ∂T ∂T + k k + k + q& = ρ .c. ∂x ∂x ∂y ∂y ∂z zx ∂t where ρ is the density of the solid, c is the specific heat capacity, and

q& is

the energy generated per unit volume. If the temperature is dependent on time, this called transient heat transfer. For a steady-state problem the timederivative is zero. If the conductivity is a constant, it can be taken out of the derivatives. In this case, we sometimes define the thermal diffusivity as

α=

k ρc

For most materials, the thermal field is independent of the deformation field. This means the temperature in the body can be solved for without worrying about the stress distribution in the body. Once the temperature field (that is the distribution of temperature in the body) has been obtained, it can be used to calculate the stresses in the body. Remember that a temperature field gives rise to stresses because of differential expansion or if the body is prevented from expanding or contracting freely. The coefficient of thermal expansion is an important property of the material in this regard. The stresses caused by the temperature field are referred to as thermomechanical stresses. They are a significant part of many designs. For instance, it is customary to hold two components in a fixture for welding. Since the welding process involves an inflow of thermal energy and the fixture prevents free expansion, designers have to worry about how the structure will distort after the fixture is removed! 25

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Fatigue And Fracture Describing the investigations of 19th century engineers into the failure of parts, Frost et al write17 “because these failures occurred in a part that had

functioned satisfactorily for a certain time, the general opinion developed that the material had tired of carrying the load or that the continual reapplication of a load had in some way exhausted the ability of the material to carry load. Thus the word ‘fatigue’ was coined to describe such failures”. The study of fatigue is over a century old. But there is still not much fundamental understanding of how to design for fatigue. Most design rules are based on gathered data and presented in the form of curves or tables, as in the Atlas of Fatigue Curves18. What is clear is that the assumption that a continuous body is a flawless continuum is not correct. Various imperfections on the surface of a finished component are thought to contribute to fatigue-failure, and factors are prescribed to modify the permissible stress, which in turn depends on the endurance limit. Usually tabled as S-N curves, the endurance limit is the fatigue strength (S) for a given the number of cycles (N). What is also clear is that metals fail from fatigue if the load is “reversed” or “alternating19”. Given a load that changes magnitude and / or direction, engineers use the stress analysis methods described earlier to calculate:

17 18 19

σmin

the minimum stress

σm

the mean stress

σmax

the maximum stress

σr

the stress range

σa

the stress amplitude

Metal Fatigue by N.E.Frost, K.J.Marsh, L.P.Pook The American Society of Metals’ Atlas of Fatigue Curves, edited by H.E.Boyer

Some materials fail if the load is held steady but is held for a long time. This can be due to creep or viscoelasticity. Several plastics creep at room temperature while metals tend to creep only at elevated temperature. 26

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If there is a static (i.e. non-alternating) load, steady or static stress, σs, is also calculated. Note that if the stress-state is tri-axial, the stresses listed above are equivalent stresses (a von Mises stress, for example). As described earlier, these are calculated from the stress components. Once these components have been calculated, an appropriate failure theory is applied. Commonly used theories include Goodman, Modified Goodman, Soderberg, and Gerber. In case the part is subjected to σ1 for n1 cycles, followed by σ2 for n2 cycles, and so on, cumulative-fatigue theories such as the Palmgren-Miner cycleratio summation theory are employed. It’s important to remember that these are all theories, not yet accepted as laws. To emphasize the fact that Fatigue and its adjunct, Fracture, are not well understood, we will close our brief discussion of fatigue just as we started it: with an extract from two books on the subject. First20, describing the investigation into the failure of a tank in 1919, which was part of a law-suit, the court appointed auditor wrote “amid this swirl of polemical scientific waters, it is not strange that the auditor has at times felt that the only rock to which he could safely cling was the obvious fact that at least one half of the scientists must be wrong.” A description of an accident that occurred in 1988 – almost 8 decades later! - says21

“In 1988 the roof of the forward cabin of a 737 tore away during flight, killing a flight attendant and injuring many passengers. The cause was multiple fatigue cracks linking upto form a large, catastrophic crack. The multitude of cycles accumulated on this aircraft, corrosion and maintenance problems all played a role in this accident. Furthermore, the accident challenged the notion that fracture was well understood and under control in modern structures. “ 20

Fracture and Fatigue Control in Structures, Applications of Fracture Mechanics, by

S.T.Rolfe and J.M.Barsom. 21

Lecture Notes On Fracture Mechanics, Alan T.Zehnder, Department of Theoretical and Applied Mechanics, Cornell University 27

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Mathematics Mathematics is usually easier to deal with than mechanics, if only for the reason that most statements can be accepted as complete and true. There is little room for a subjective interpretation. In fact, it is this mathematical approach that both makes it possible for us to attack Linear Differential Equations with confidence and advises us to treat Non-linear Differential Equations with great care, as we shall see in the following pages.

Differential Equations and Tractability The power of differential equations has been noted already: they give us a ready way to model physical problems. Once modeled, we can understand, investigate and ultimately predict the behavior of the physical world. In our study of mechanics in the previous pages, we have already reviewed several mathematical models. Of course, the models themselves may well be wrong. That aspect is addressed in the chapter on Verification And Validation later in this book. For now, we assume that the mathematical model is what we need to solve. We focus on the tractability of these mathematical models. That is, the ease with which we can understand, evaluate and predict real-world behavior using the model. It is this requirement for tractability that will lead us onto our next chapter, where we will study the theory of the Finite Element Method.

Classes Of Differential Equations Functions of a single variable yield Ordinary Differential Equations (ODEs) while functions of many variables give us Partial Differential Equations (PDEs). PDEs themselves are categorized depending on the coefficients (constant or variable coefficients), on the right-hand-side of the equation (homogeneous and inhomogeneous), on the highest order of derivative (first order, second order, etc.) and so on. Mathematical physics, the study of physical phenomena using mathematical models, is dominated by second order PDEs. A PDE may do an excellent job of characterizing more than one phenomenon. An excellent example of this is Prandtl’s membrane analogy. Prandtl pointed out that the torsional stress 28

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in a non-circular bar is governed by the same equation that describes the deflection of a thin membrane under the action of a pressure. The immediate application of this insight was to allow experimenters to use the easier membrane-inflation experiments to measure torsional stresses! Investigations into 2nd order PDEs led to the comparison of these equations with the general second order analytic curve, the general conic section. Recall that if we write the conic section in its canonical form,

ax2 + bxy + cy2 + dx + ey + f = 0 we can then calculate the discriminant (b2-4ac) to determine whether the curve represents an ellipse (b2-4ac < 0), a parabola (b2-4ac =0) or a hyperbola (b2-4ac > 0). In a similar fashion, a general 2nd order PDE can be written in canonical form and classified as elliptic, parabolic or hyperbolic, depending on the discriminant. An elliptic PDE is said to be a Boundary Value Problem (BVP). All values within the domain (that is, the area over which the DE is valid) are uniquely specified by values on the boundary. These values on the boundary are called Boundary Conditions. Parabolic and Hyperbolic PDEs are encountered in dynamic problems, where values change with time. They require not just Boundary Conditions, but Initial Conditions too. Some problems are diffusive in nature. That is, any errors in initial conditions die away as time progresses. Other problems are convective – initial data propagates without dying away, which means that any errors in initial data continue to affect the solution even as time progresses. Remember, in closing, that initial and boundary conditions can be specified either on the derivatives of the independent variable (for instance the heat flux in a thermal problem, and the stress in a stress-analysis problem) or on the independent variables themselves (the temperature or the deformation)22. Boundary and Initial conditions should be prescribed with care, since they should both mimic the physical world and be consistent with the statement 22

Boundary conditions are often classified as Neumann, Dirichlet and Mixed 29

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of the differential equation. Problems that are thus defined are said to be well-posed. The opposite, of course, is an ill-posed problem.

Linear Differential Equations To a mathematician, a linear differential equation has several pleasing properties. In the first place, the mathematician can prove that the equation has a solution. This proof of existence justifies the hunt for the solution. Next, the mathematician can prove that the solution that exists is unique. This is even better, since it tells us that regardless of what method we use to find the solution, once we have found it it’s the only one. A PDE is linear if •

there are no products of any derivatives in the differential equation itself and in boundary conditions

•

none of the coefficients depend on the dependent variables

•

none of the boundary conditions depend on the dependent variables

Few physical problems, of course, are truly linear. As Mandelbrot23 demonstrated with his remarkable fractal geometry, the deeper we dig into the real world the more we uncover facts that complicate our equations. But just as it is reasonable to depict a planet as a round ball from a macro level, linear models are widely used as a first level of investigation. A deeper investigation is not always required, just as all planets are not explored. If a deeper investigation is required, we often end up with non-linear differential equations. In this book our focus is on linear differential equation. We will quickly review some important aspects of non-linear differential equations.

Non-Linear Differential Equations The general non-linear equation is not blessed with the uniqueness or existence theorems that linear equations boast of.

23

30

The Fractal Geometry of Nature, B.Mandelbrot

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Not only does this mean that the equation you are going to investigate may have no solution, it also means that it may have multiple solutions. The engineering usage of non-linear equations has been steadily increasing over the past several years. As computing power becomes cheaper, and software tools become more powerful, designers are increasingly able to deploy these powerful methods to design better products. In this series, the volume on simulation of manufacturing processes addresses one such application. However non-linear equations should still be addressed with some care. Solutions are better viewed as sources of insight into the mechanics, than as definitive answers to design problems. How do you know whether you should choose a linear model or a non-linear model? The main “sources of non-linearity” are:

Geometric: if the body deforms considerably, the deformed configuration may be markedly different from the un-deformed configuration. Such problems are said to be problems of large deformation. Alternately, there may be a large strain, which violates the assumptions implicit in the strain-displacement equation presented earlier. In some cases, the deformation changes the stiffness of the material because of the geometric properties. For instance, a convex curved plate’s stiffness reduces if the load causes it to first become flat and then concave.

Material: plastic deformation is the most common, but other forms of material behavior include creep, visco-elasticity, etc. In thermal problems, if the conductivity depends on the temperature, the analysis is non-linear.

Boundary: radiative boundary conditions are clearly non-linear since they involve the 4th power of the independent variable) i.e. temperature). In structural problems, gap or contact are nonlinear, since the stiffness of the structure changes with the deformation - depending on whether the gap is open or closed. Determining material data for non-linear analysis can be quite demanding, and requires a more complete study than can even be summarized here. 31

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Series Solutions And PDEs The study of infinite series goes hand in hand with the study of differential equations, and the reason is easy to find. Given the difficulty of solving differential equations, there’s a strong motivation to see if the solution can be constructed out of known and simple functions24. Bessel’s functions will be familiar to any one who has studied the analytical solution to the problem of the cooling of a fin. It was the investigation of heat transfer that also gave rise to the only series that we will review – the Fourier series. No study of mechanical vibrations can be undertaken without this. In its simplest form, the Fourier Series tells us that almost any periodic function can be expanded as a series of sin and cosine functions.

f ( x) =

a0 ∞ + ∑ (an cos(nx) + bn sin(nx) ) 2 n=1

The “almost” contains some important conditions. For example, one of the conditions is that function should have a finite number of finite discontinuities within the range of interest. This means, for example, that the familiar trigonometric function tangent cannot be expanded as a Fourier Series over any interval that includes π/2 or 3π/2, since the function is infinite at those points. The significance of the coefficients of the series is not always apparent. But if the function being investigated is time dependent, as in mechanical vibrations, a simple transformation can inject life into the coefficients of the series!

The Fourier Transform Given a non-periodic function f(t), the Fourier Transform is defined by the equation

1 F (ω ) = 2π 24

∞

∫ x(t )e

−iωt

dt

−∞

See, for example, Fourier Series and Boundary Value Problems, R.V.Churchill and J.W.Brown 32

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Here we assume that the non-periodic function is a function with an infinite period. Note that the independent variable in the “original” function, f(t), was “t”, while the independent variable in the “transformed” function is ω. It is customary to interpret ω as the angular frequency (radians / second). This is natural, since it is the frequency of the sine and cosine terms in the Fourier Series. We say that the transform has taken us from the time domain to the frequency domain. The inverse transform takes us back from the frequency to the time domain. Of the several properties that the Fourier Transform has, one in particular is interesting: the Fourier Transform of f’(t) is given by iωF(ω). This means differential equations can be transformed into algebraic equations25. The transform has a host of other interesting properties, covering areas like time shifting, modulation, etc. which are very useful in signal processing. Theory guarantees us that the two representations are entirely equivalent: there is no loss of information going from one domain to the other. Which form we choose to use depends purely on convenience. It's just a question of which is easier to understand and comprehend. Compare the problem to that of listening to an orchestra and trying to pick out instruments. A trained ear can do it, but for the average ear extracting such info from the time-domain-signal is hard. If, instead, you use the Fourier Transform to break the signal into a spectrum, you can clearly see signatures of different instruments, since each has a different frequency range. To illustrate this, consider the 3-degree-of-freedom system shown below26. Models such as this are not just instructive – they are also used quite widely in practice. Called lumped mass models, they are used extensively in a wide variety of applications. One such example, the simulation of vehicle crash, is included for reference at the end of this book.

25

Recall Laplace transforms – they serve a similar purpose. These figures are from The Fundamentals Of Modal Testing, Application Note 2433, Agilent Technologies

26

33

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Since the system has 3 degrees of freedom, it will have 3 modes of vibration. These are the natural frequencies - ω1, ω2 and ω3 - and the corresponding mode shapes. Our design goal is to control the response of the system. That is, given any source of excitation, we want to first predict, then alter, the response. The first step in this is to understand how the modes of vibration contribute to the response. Since the modes depend on k and m, we can tune the response by making appropriate changes to the structure. The time-response, shown below, doesn’t provide much insight. It does tell us how the response changes with time, but tells us nothing of the relative importance of the 3 modes.

But now let’s look at the Fourier Transform – the frequency domain plot.

34

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Note that the peaks in the plot correspond to ω1, ω2 and ω3, the natural frequencies of the system. The contribution of ω1 is more than that of ω2, so we’ll probably be better off tuning the structure to change ω1. In fact, as the figure below shows, the frequency domain plot can be obtained as the superposition of the plots for each individual mode! Remember this, since we will discuss the Modal Superposition Method later on in our study.

Frequency domain plots can intimidate a beginner, but there’s no reason to be intimidated. While a physical interpretation of the frequency domain plot is not always possible (in fields like electrical engineering, for instance), mechanical vibrations are kinder. As shown in the figure, the frequencydomain and time-domain plots of the modes of vibration of a cantilever beam are very easy to correlate with the physical behavior of the vibrating beam.

35

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While the “actual” beam, being a continuous body, has infinite degrees of freedom, our physical measurement is “reducing” it to a 3-degree of freedom body if we measure just the first 3 modes. The plot along the “frequency axis” is the frequency-domain plot we’ve seen above. Obviously the plot will change depending on the measurement point, just as the time response changes at each measurement point – recall the FRF we saw earlier. Figuring out where to measure (i.e. the choice of the measurement points) is one of the many challenges test-engineers face, and is one reason why the FE method goes so well with testing in the design of vibrating equipment. FE is often used to suggest measurement points. It is harder to visualize the equivalence between the time-domain and frequency-domain forms of a function for complicated structures, but the principle is the same. The Fourier Transform is an invaluable tool in the study of vibrations. Several times, it is implemented in real-time. That is, even as a signal is acquired, its Transform is calculated.

The Fast Fourier Transform The Discrete Fourier Transform is obtained by replacing the integral with a finite sum. This does introduce an error, but this is often tolerable27. The 27

Refer the Nyquist Criterion, taught in most undergraduate courses on Control Systems.

36

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DFT, implemented in hardware or software gives us the FFT – the Fast Fourier Transform. There are several different algorithms that implement the FFT. Common to all of them is that they can generate the frequency domain form of a signal even as the signal is being sampled. Remember the earlier discussion of the damping factors of a structure. We saw that the factors are estimated from actual physical tests. In these tests, the structure is excited at various frequencies and the response logged. From the response, the damping factors are estimated. FFT is very widely used to interpret data collected in such experiments, since the Fourier Transform helps us “spread out” data into peaks. In some cases, log-log plot or log-linear plots are used to present the data. The FFT of data on vibrating equipment also serves as a diagnostic tool. It can help us understand which sources are most important in the response. A detailed discussion of the FFT and the FRF is beyond the scope of this book.

The mathematics, for most structural engineers including myself, is difficult to understand. Also, the solutions are difficult to verify. Ed Wilson describing frequency domain analysis

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Essential FEA Theory Of the several ways to introduce the Finite Element Method, the easiest is to view it as nothing more than a series solution for partial differential equations. This is the approach we will follow in this brief theoretical introduction to the method. The next chapter will put this mathematical approach in the context of specific applications of mechanics. From a designer’s perspective, however, FE is more than just a differentialequation solver. As we will see, it offers us a toolbox of ready-made differential equations that we can put together to describe most physical problems. This is very appealing to a designer, who is familiar with the approach of breaking assemblies down into standard parts or machine elements: gears, chains, belts, beams, trusses, plates, fasteners, and so on. It is this capability to model complicated structures that makes FE so much more valuable to a designer than many other numerical methods.

From The Differential Equation To A Matrix Equation A study of mechanics familiarizes us with the many differential equations that are used to model continuous bodies: there is the differential equation of beam bending, the differential equation for heat conduction, the differential equation for general tri-axial stress, and so on. Solving these equations on simple geometries is hard enough, as anyone who has worked through the temperature distribution in a cooling pin knows. Solving them on general geometries like an engine-block is next to impossible using analytical methods. An obvious alternative is to use numerical methods. This is a well-known approach, well documented in the derivation of the many infinite series, including the Fourier Series. One approach, suggested by B.G.Galerkin28 for boundary-value problems, reduces the differential equation into a problem of linear algebra – that is, a

28

In a paper published in 1915. The method is also called the Bubnov-Galerkin method, and is similar to the Rayleigh-Ritz method.

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problem of simultaneous linear equations that can be solved using matrix methods that any high-school student is familiar with. For the sake of simplicity, we will illustrate the method for a simple second order elliptic differential equation with constant coefficients. We start with the statement of the problem: that is, the differential equation,:

k

d 2u du +c + au = f 2 dx dx

the domain, and the boundary conditions29

for o ≤ x ≤ l subject to

u x =0 = u 0 u x=l = ul Note that the boundary conditions could, in general, specify either the dependent variable u, or it’s first derivative u’, or a combination of these. In a heat transfer problem, the dependent variable is the temperature, the first derivative represents the flux (i.e. heat injected or removed at the boundary) while a combination of both represents a convective boundary.

k, c and a are material constants that could depend on the independent variable x. If they depend on the dependent variable u, the equation is nonlinear. For stress analysis, the dependent variable is the deformation while the first derivative represents the strain or the stress on the boundary. Combinations of the dependent-variable and its first derivative are less common, but can represent spring-supported boundaries.

29

The FE method treats boundary conditions with great elegance, though we do not cover the mathematics here. 39

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Higher order differential equations are encountered in mechanics. For instance the beam differential equation can be written as a 2nd order equation

d2y M =− 2 dx EI but it is usually more convenient to work with the 4th order form

d4y w =− 4 dx EI In the latter case, boundary conditions can be prescribed on the independent variable, its first derivative, its second derivative, and its third derivative, or any combination. Our problem is to find a function u(x) that satisfies the differential equation everywhere in the domain (i.e. for o ≤ x ≤ l ) and also satisfies the boundary conditions. For brevity, and with no loss of generality, we will simplify our second-order equation further, restricting our attention to just two terms:

k.

d 2u = f dx 2

Rather than integrate the differential equation, we will assume the form of the solution, then search for conditions that the parameters in this assumed form must satisfy for the assumed equation to be a solution. We will call our assumed solution u , but before searching for the solution, we will recast the equation in its weak form. Remember that if u is to be a solution of the differential equation, it must have a second derivative. This is obvious: the assumed form will not fit into the differential equation unless it has a second derivative. But also remember that our modeling approach does lead to situations where the solution itself cannot have a second derivative: it may at best 40

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have a first derivative. For example, consider a weight hanging from a cable that is supported at both ends. In actual reality the cable is continuous – that is, its shape changes smoothly all through. The slope of the string may change more rapidly in the vicinity of the weight than at other points, but it is still continuous. However if we model the weight as a point-load30, the string now has a sharp kink at this point. It has a first derivative, but is not differentiable: that is, the slope to the left of the kink is not equal to the slope to the right of the kink. Obviously, there can be no second derivative! Why should we demand that our candidate solution u , which we will call the trial function since we will want to try it out as a possible solution, have a second derivative, when the mathematical model has an exact solution that does not have second derivatives? This is the reason we recast the problem in the weak form. To do this, we say that to qualify as the solution, our trial function will be accepted if it satisfies the equation l

l

d 2u v . ∫0 dx 2 dx = ∫0 vf ( x)dx Remember that requiring that that

∫ g ( x)dx = 0 is not the same as requiring

g ( x) = 0 ! The latter requires that g(x) be zero for all x, while the

former only requires that g(x) be zero on average. The use of v(x) in the above equation compensates for this. If we require that the integral equation be satisfied for all possible functions v(x), we can satisfy ourselves that the integral equation is as demanding as the original differential equation. This approach, called the method of weighted residuals, uses v(x) as a test function. From now on, it’s just a matter of basic calculus and algebra. We integrate by parts to get

30

Such point-functions are defined using the Dirac-delta function. They are not physically realizable, but are used commonly. 41

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v'

du dx

x =l

l

x =0

l

dv du . dx = ∫ vf ( x)dx dx dx 0 0

−∫

This equation is now symmetric. That is, it requires that both the trial function and the test function have the same order of continuity. Since the equation only involves the first derivatives of these two functions, they do not need to have second derivatives at all31. We now assume a series form for both the test and the trial functions, with the constant coefficients a and b: ∞

u ( x) = ∑ ai .φi ( x) i =1

and ∞

v( x) = ∑ b j .ψ j ( x) j =1

As with any other infinite series, we will choose to terminate the series at our convenience. That is, we choose to calculate only a finite sum. How many terms should we include? That depends on how much error we are willing to tolerate. Since this is a critical decision in design, we will return to this aspect in the chapter on Verification and Validation! In other words, the series are summed till N instead of till ∞. You may find it useful to carry out the exercise yourself (for a small N) to show that after substituting the series in the integral equation and rearranging terms and ignoring the boundary terms, we get the matrix equation

31 A C0 function is one whose zero’th derivative (i.e. the function itself) is continuous. A C1 function has a continuous first derivative, a C2 function has a continuous second derivative, and so on. By continuous, we mean the left-limit = the right-limit = the value of the function at the point.

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Since we can cancel the row vector {b} from the equation as it occurs on both sides, we get the matrix equation

{b1

b2

l ' ' ∫ φ1ψ 1 0 l ' ' ∫ φ2ψ 1 . . bN } 0 . . .

l

∫φ ψ ' 1

' 2

. .

' 2

. .

0 l

∫φ ψ ' 2

0

. .

. . . .

.

. .

l ∫ fψ 1 a1 0 a l fψ 2 . 2 ∫ . = {b1 b2 . . bN } 0 . . . . . a N l l ' ' f ψ φ ψ N ∫0 N N ∫0 .

[K ]{u} = { f } Given the similarity to the equation of spring-deflection, it is customary to call the matrix [K] the Stiffness Matrix. Obviously, it may represent other physical quantities too, such as the heat capacity in a thermal problem, but the name is used regardless. While we have not shown how the boundary terms are included, we will only state that they can be very elegantly included in the matrix equation itself. In a few steps, then we have reduced an intractable differential equation to a simple matrix equation. All that’s left is to choose names for the variables introduced above, and we will have completed our introduction to the Finite Element Method!

Nodes, Elements And Shape Functions In the series expansions used above for the test and trial functions, we did not assign any physical significance to the constants ai and bj. Nor did we say anything about how to choose the functions φi(x) and ψj(x). The choice of these functions determines the shape of the trial function, which is nothing but our assumed solution. We therefore call these functions shape functions. It is by no means easy to construct shape functions for all differential equations. Despite long years of research, it continues to be an active area of research! The choice of shape functions depends on the differential equation being solved, and on the order of accuracy required. 43

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The bj, of course, are irrelevant to us since we have cancelled them from the equation – we never actually evaluate them. What of the ai? We can make a choice that is very intuitive and that turns out to be very useful. We ask that the constants ai be the values of the function u at selected locations in the domain. We call these locations the nodes or grids, and since we are constructing the series, it is upto us to decide where to locate these nodes. Without explaining the mathematical basis, we will just state that the nodes should be closely spaced wherever we expect the solution to vary rapidly, while they can be spaced farther apart in areas where the solution is not expected to change rapidly. Again, note that we are making our decision based on our judgment, not on knowledge. We can guess where the solution will vary rapidly, but we cannot know this till we have solved the problem! The chapter on Verification And Validation addresses this seeming paradox. Now remember that each term in the stiffness matrix is an integral. Since we are using computers, we evaluate these integrals numerically. This step, often called quadrature, usually uses the Gauss integration rule. What then is an element? It’s a part of the domain defined by nodes, over which everything (the shape functions, the material constants, the forcing function, etc.) is continuous. Putting a lot of elements together gives us the complete domain. One important consequence of this definition is that we can now evaluate the integral over each element, since the integral-of-a-sum is the sum-ofintegrals. When programming the FE method, we can use this property to calculate local or element stiffness matrices, then assemble these to get the global stiffness matrix. Another is that we can view each element as a black-box. It interacts with its neighbors only by the values of the dependent variable at the nodes: and different elements can even represent different governing differential equations! This last property is extremely useful. Most mechanical components are complex, and it is next to impossible to find a single differential equation that defines the behavior of the dependent variable everywhere over the body. Now, however, we need not worry about this. We can “break” the 44

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component into subsets that are similar to readily recognizable machine elements: beams, trusses, plates, etc., and model our component as a combination of these. In other words, the Finite Element method not only gives us a way to solve differential equations, it also helps us model complicated bodies by piecing together various differential equations in a consistent fashion!

Some Common Elements Used In Stress Analysis Many commercial finite element packages offer a dizzying range of elements, but most are variations on 4 basic types: trusses, beams, plates, and general 3-dimensional solids. If you are familiar with these, you will be able to model a large majority of the problems in solid mechanics. Most commercial elements use polynomial functions as the shape functions. If the polynomial is of the 1st order, we call the element a linear element. A 2nd order polynomial makes the element a quadratic element. Higher order elements are rarely used, since it is generally believed that the increased accuracy is not worth the increase in required computing resources. Remember that a linear shape function in 1D requires 2 points to uniquely specify it, while a quadratic function requires 3. Truss Elements: • •

•

3 dof / node (ux, uy, uz) Data required: Cross-section Area Appropriate for pin-jointed members

Beam Elements: • • • •

• •

6 dof / node (ux, uy, uz, rx, ry, rz) Data required: Cross-section Area, Moments of Inertia Appropriate for slender (uni-dimensional) members that support bending moments Linear Beams have 2 nodes per element 2nd order beams can be curved – they have 3 nodes per element Beam-theory allows several additional constants to be defined. These are important if working with thin-walled sections such as pipes or channels.

Plate Elements: • •

6 dof / node (ux, uy, uz, rx, ry, rz) Data required: Thickness 45

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Use for thin (two-dimensional) members Linear Quadrilateral Plates have 4 nodes per element Linear Triangular Plates have 3 nodes per element 2nd order (quadratic) Quadrilateral Plates have 8 nodes per element (some packages allow 9 nodes per element, with the 9th at the centroid of the element) 2nd order (quadratic) Triangular Plates have 6 nodes per element (some packages allow 7 nodes per element, with the 7th at the centroid of the element)

3-Dimensional Solid Elements: • • • • •

• •

3 dof / node (ux, uy, uz) Data required: none Linear Hexahedral Solids have 8 nodes per element Linear Tetrahedral Solids have 4 nodes per element 2nd order (quadratic) Hexahedral Solids have 20 or 21 nodes per element 2nd order (quadratic) Tetrahedral Solids have 10 or 11 nodes per element Other shapes (pentahedron or wedge, pyramid, etc.) are also offered by some packages

Remember that a degree-of-freedom (often abbreviated to dof) is the unknown at a node. If your model has N nodes, and each node has M degrees of freedom, your stiffness matrix will have NxM rows and NxM columns. The r in the dofs above refers to the slope at the node. For several differential equations, we use both the deformation and its derivative (i.e. the slope) as dofs. It is tempting for a beginner to model all components with general 3dimensional solids because the resulting FE model looks similar to the actual component. This is wrong. Remember that your FE model needs to mimic the behavior, not the appearance, of the component. If bending stress is more important (as for long slender members) a beam element will be much more accurate and effective than a solid element! Triangular and tetrahedral elements are frowned upon in stress analysis since they don’t offer as good accuracy as quadrilaterals or hexahedra. For thermal analysis, however, triangular and tetrahedral elements work just fine. Many packages can automatically generate elements from the geometric model. Called auto-meshing or automatic mesh generation, this is rarely supported for hexahedra.

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Plane stress, plane strain and axi-symmetric behavior are good approximations in specific situations. Elements that offer this type of behavior are very efficient if used in the right situations, but with the drop in the price of computing power, they have largely fallen out of favor. One element that is not an element at all (in that it doesn’t represent any differential-equation) is a “rigid link”. It is just an equation that tells the software that two dofs must be matched – they should have the same values. From an engineering perspective, this is as if the two dofs have been joined by an infinitely rigid link, hence the name for the element. A spring element is sometimes used to represent a spring-support if the spring-stiffness is known. Lumped mass elements are used to represent components whose weight is relevant but whose stiffness can be ignored. The mass is concentrated at a node, so the element is a “0” dimensional element. Finally, you will see from the summary above that not all elements have the same number of dofs / node. Beams and plates have 6 dof / node while trusses and solids have 3. Joining a plate element to a beam is consistent: equilibrium can be maintained since both the elements have the same dofs. But what if you want to connect a beam to a solid, or a plate to a solid? Is it sensible? That depends on the physical situation. If, in the physical problem, there is a hinge at the connection then your mathematical model will be accurate. If, however, your physical problem has a slender member welded to a block, then it is tempting to model the block with solid elements and the slender member with beam elements. But it is important to remember that the physical model is now different from the real model since moments are not transmitted! Building models that mix elements that have different dofs / node should be done with care. It is often resorted to in practice, and works well if safeguards are followed.

Matrix Solvers Understanding how matrix equations are solved is really not important for a designer. If the model is created right, the computer program should take care of solving the matrix equation. 47

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We will briefly review methods of solution, though, for two reasons. First, simply because it is interesting! Second, because if things go wrong (as they inevitably will!) familiarity with the inner workings helps diagnose what went wrong.

High-school courses teach us to solve matrix equations by evaluating the determinant (Cramer’s rule is usually adopted), and calculating the inverse. Anyone who has tried to invert a 4x4 matrix by hand will understand why the FE method languished till computers came along! If you pause to consider that it is quite common to solve a problem with 100,000 dofs on a desktop computer, it is clear that computers too can grind to a halt unless the matrix equation is solved efficiently. Computer programs, in fact, rarely use Cramer’s rule. Most solvers use a variation on Gauss’s method of triangularizing a matrix, called Gaussian Elimination. Such solvers, called Direct Solvers, rely on the fact that subtracting one row of a matrix from another does not alter the determinant32. There are several variations on this method (Cholesky’s method is perhaps the most popular), all of which involve the calculation of a pivot for every row operation. Without going into the mathematics, we will only observe that is a pivot is zero, the matrix is singular – that is, it’s determinant is 0. Obviously this means that the matrix equation cannot be solved. For solid mechanics, where we are guaranteed that the matrix will be symmetric, pivots should never be zero. (In linear algebra, we say the stiffness matrix is Symmetric Positive Definite). If a pivot is zero, it’s probably because you have made a mistake in modeling. The most common mistake is in the application of restraints (which are one type of boundary condition). This is worth discussing, since it can be quite counter-intuitive for a beginner! Given a plate with a force applied at each end, what “boundary conditions” can we apply? 32

Iterative solvers, which are useful for very large problems, use variations on the Gauss-Seidel method.

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The data does not tell us anything about the prescribed deformation, so we can be forgiven for building a finite element model shown below. The problem can easily be solved by hand, so it can be quite confusing if we attempt it on a FE program and are told by the computer that the problem cannot be solved because a “zero or negative pivot was encountered”. Remember that we omitted the treatment of boundary conditions when we derived the matrix equations earlier. A study of that would indicate why an FE program must include restraints, but for now we will only note that an FE model will not be solvable unless all rigid body motions are eliminated. That is, you examine your FE model and check if it will be in static equilibrium regardless of whether the forces are symmetric or not. In the case above, the plate is in equilibrium only if the forces are symmetric. If the forces were unsymmetrical, the plate would move as a rigid body – with no strain33.

Symmetry is often used to introduce restraints (and can also help cut the computational time since it reduces the size of the model!). For the plate, applying symmetry, the model would be

Understanding symmetry-restraints is not always easy, since a physical insight is sometimes hard to gain. Symmetry can be of different types – cyclic, rotational, planar, etc. Planar symmetry is the easiest to model. For the translational degrees of freedom, you should restrain all translations

perpendicular to the plane of symmetry for all nodes that lie on the plane.

33

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Rotational degrees of freedom, if they exist, should be set to the inverse of the corresponding translational dof. For instance, for a plate symmetric about the x-axis, we would set ux = 0 for all nodes on the x-axis, and leave uy and uz free. Since plate elements have 6 dofs / node, we would set ry and rz = 0 for these nodes, and leave rx free, applying the “inverse” rule of the previous paragraph. We will conclude our discussion of matrix solvers for FEM by noting that in general, stiffness matrices tend to be sparse. That is, the number of entries that are 0 is usually pretty high. Various schemes are used to take advantage of this behavior – band-solvers, column- or skyline-solvers and frontal-solvers are some popular methods.

Some Important Properties Of The FE Solution Several FE packages refer to error norms. A norm is just a measure of anything, so the error norm is a measure of the error. It is an interesting aspect of the FE method that we can estimate the error in our solution even without knowing the “correct” answer! There are several different norms that can be used to estimate error, but the error in strain energy is favored since it is consistent with the mathematics of FE. As with any other infinite series, truncating the series is a source of error. Adding more terms reduces the error, but this behavior is asymptotic – the law of diminishing returns applies here too, so after a while the increased cost of including additional terms does not provide an adequate increase in accuracy. The key to efficient usage of FE is to stop increasing terms when the rate of improvement levels off. Increasing terms, of course, means increasing the number of nodes, which can be done in two ways: either by using more (and therefore smaller) elements or by using higher order shape functions. The former is referred to as h-refinement while the latter is called p refinement. If both methods are used simultaneously, we call it h-p refinement. Regardless of how fine the mesh is, the use of the weak form means that the derivative of the dependent variable can never be continuous. In other words, in a stress analysis solution, the deformation will be continuous but 50

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the strain will not be. This behavior is sometimes used to judge whether the solution is adequately accurate or not. Finally, we saw that Galerkin’s method applies to BVPs only (Boundary Value Problems). Remember that we integrated-by-parts. This resulted in the term

du v' dx

x =l

x =0

which has to be evaluated at both “ends” of the independent variable – x = 0 and x = l. What if the independent variable were “time”? There is no known way, outside of science fiction, that we can know the conditions at future time, so obviously we cannot use the same approach for the timederivatives. There is no way we can apply the above integration-by-parts rule to a time derivative. Problems that involve time-derivatives use finite-difference methods to “step forward” in time. The Finite Element mathematics is restricted to the spatial derivatives only.

A theory is a good theory if it satisfies two requirements: it must accurately describe a large class of observations on the basis of a model that contains only a few arbitrary elements, and it must make definite predictions about the results of future observations. Stephen Hawking

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Putting It Together – OptiStruct/Analysis OptiStruct can address a variety of design tasks. It can address optimization, as described in another volume in this series. An FEM analysis is an integral part of a lot of design optimization, so OptiStruct includes an embedded FE solver: OptiStruct/Analysis. OptiStruct/Analysis is intended purely for analysis. That is, it can be used to setup the problem, carry out the analysis and review results. Strictly speaking the setting-up and review tasks (often referred to as preprocessing and post-processing) are done using HyperMesh and HyperView. As the assignments illustrate, this distinction is immaterial to the designer, with all tasks carried out under the same interface! In this chapter we •

summarize some of the main capabilities of OptiStruct/Analysis

•

review what a problem consists of – what files are involved in an analysis and how to present results

•

list the nomenclature that OptiStruct/Analysis uses

Capabilities Remember that a model is an idealization of a physical phenomenon. Any results you get are only as good as the model you choose. At the same time, the model should not be so complicated that it cannot be solved in the required time or with available resources. Choosing an appropriate model is not always easy. It is always a good idea to refer to published literature for examples of models that have been used successfully.

Linear, Static This model is used when the response of the body is linear, and there’s no variation with time. In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small and when the loads do not vary with time. 52

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This model is used widely since it’s quick to solve and relatively easy to interpret the results. Very often, even if a non-linear model is more realistic, a linear model is used to investigate likely behavior. Once the options have been narrowed, a full non-linear analysis is resorted to. The equilibrium equation is

[K ]{u} = { f } where K, u and f are functions of x, y and z only – they are independent of

t.

Normal Modes Sometimes our design problem is not just to calculate stresses or deformations. We may be interested in identifying the resonance frequencies of the system. In vehicle design, avoidance of resonance enhances ride comfort by cutting out unwanted rattles. When designing a loudspeaker or a megaphone, on the other hand, you may want resonance to occur. In cases like these, we need to solve the “eigenvalue” problem and evaluate the natural frequencies of the body. The equilibrium equation is

[M ] ∂

u + [K ]{u} = {0} 2 ∂t 2

where K and u are functions of x, y and z only – they are independent of t. The solutions to this equation are pairs of natural frequencies and the corresponding “mode shapes”. Since we omit the damping we call this a normal modes analysis.

Linear, Transient In stress analysis, this model is appropriate when operating within the elastic region (i.e. the stress-strain curve is linear) and when the deformations are small but when the external conditions do vary with time. Either the loads or the restraints or both could be time-dependent. 53

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Transient problems can be solved using either Direct Integration or Modal Superposition. The latter is preferred if the solution needs to be calculated for a large period of time, but is only applicable to linear problems. The former is preferred if the solution is required for short periods of time, and can be applied both to linear and non-linear problems. In either case, we usually perform a normal-modes analysis. If we are using modal-superposition, we will need the mode shapes anyway. If we are using direct-integration, the highest important mode helps us determine the time step to use.

Random Response In some situations, we cannot specify the exact value of the loads as a function of time, but can specify the total energy in these loads. An example would be the forces experienced by a plane when its engines are firing. We know the total energy being transferred from the jet engines to the frame, but cannot claim that we know the loads precisely as functions of time. In cases like these, the loads are characterized by Probability Density Functions, and the behavior is called stochastic. The designer’s goal then is to predict a probability of safety. The several ways to evaluate these responses is beyond the scope of this book.

Inertia Relief Setting up a Finite Element model for static analysis requires that the structure be supported adequately. Consider an aircraft in flight or a spacecraft. There are no supports, but the model can be static if there are no time-varying loads. Obviously there are structures that are not supported explicitly but are still best represented by static-analysis models. Inertia Relief is an approach used to model such problems. The inertial of the structure is used to mimic equilibrium, like a D’Alembert force. In OptiStruct, we use special instructions – SUPORT cards - to eliminate all rigid body motions. Then we tell the solver that we want to perform an Inertia Relief analysis. It doesn’t matter which points you specify 54

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the SUPORTs at. Think of the SUPORT points as reference points with respect to which the deformation is calculated.

Frequency Response In many designs where vibration is important and correlation with testresults is essential, designers have to characterize the response of the structure as a function of frequency-of-excitation instead of as a function of time. In these cases a Fourier Transform converts the equilibrium equation from the “time domain” to the “frequency domain”. In the equilibrium equation, the variables are expressed as functions of ω rather than time. This is called the frequency domain. Of course, the Inverse Fourier Transform can convert the solution back to the time domain. In OptiStruct, we do not evaluate the FRFs – the frequency response functions – explicitly. Instead, we specify the range of frequencies for which we want to excite the structure. This is equivalent to “sweeping” through the excitation range. We then plot the results just as we would in an experimental setup. That is, we view the magnitude and the phase at points of interest. This allows us to correlate with the test setup. Test-correlation is an advanced task, beyond the scope of this book. We will end by pointing out for a multiple-degree-of-freedom structure, the FRFs are a matrix, and the FRFs themselves can be extracted from the FE solution if required.

Linear Buckling Designers sometimes have to take into account the fact that even if stresses are less than permissible values, the structure may fail if it buckles – like a tall column in compression. That is, the deflection continues to increase even if the load is removed. This is sometimes called instability. The equilibrium equation

[K ]{u} = λB [K G ]{u} is similar to that of Normal Modes, but the results are interpreted as a “buckling load factor”. KG is called the incremental stiffness.

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Buckling load factors are often important in the design of aerospace structures, where the quest for a minimal weight and the use of advanced materials leads to the frequent use of thin-walled designs.

Non-linear – Gap / Contact The terms “gap” and “contact” are often used to mean the same thing – an opening in the body that may close or widen under the influence of external factors. Clearly, if a gap closes or opens, the stiffness of the body changes. Since the gap opens or closes depending on the deformation, this means the stiffness depends on the deformation. In other words, since the stiffness and boundary forces depend on the deformation, the equation is non-linear.

Component Mode Synthesis When working with large models, resource constraints sometimes force the analyst to break the problem into smaller parts. In static analysis, this approach is called sub-structuring. When used in dynamic analyses, it is called Component Mode Synthesis. It’s sometimes impossible to treat the product purely as a structure or purely as a mechanism. Consider, for example, the feed mechanism for a highspeed packing machine. The rates of acceleration that the mechanism experiences may be quite high. High enough that the deflection of the levers is large enough, perhaps, for the feed mechanism to jam because of misalignment. Designing such a product requires that the equations of rigid-bodymechanics be coupled with the equations of structural deformation. Component Mode Synthesis also provides a way to do this.

Thermal and Thermo-Mechanical Analysis In some cases we only want to calculate the temperature distribution within our area of interest. We call this a thermal analysis. If we also want to calculate the stresses, we can use the temperature distribution as data for the stress analysis. This is called thermo-mechanical analysis.

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Setting Up An “Analysis” When we reviewed the theory of the Finite Element Method, we saw that a problem consists of •

the domain – the space and / or time over which we want to evaluate behavior

•

the differential equations that govern behavior in the various parts of the domain

•

data available on the spatial boundaries – boundary conditions

•

data on behavior at initial time – initial conditions

•

the order of elements chosen by the analyst – linear, quadratic, etc.

•

the locations of nodes – closely spaced where we expect the dependent variable to vary rapidly

•

material related properties

To solve a problem using OptiStruct / Analysis, we provide exactly the same data to define the problem. Since defining the problem is only the first part, we usually also take some additional steps. Some solution methods require additional data – for instance how we want to solve a direct integration problem, how many modes we want to include in a modal-superposition, etc. Also, the results of the analysis can be presented in a variety of ways. Since we don’t want to be overwhelmed with data, we usually also specify the nature of output we want. If we specify this, HyperView allows us to present the same data in a variety of different ways. Most analysis starts with an import of geometry from a CAD modeler. Geometry is not essential, of course. The FE data can de defined even without geometry as a reference, but it’s usually easier to work with a geometric definition as the starting point.

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It is very rare to run an analysis only once. It’s much more likely that you will want to modify some data and run the analysis again to perform what-if studies34. In general, your flow of work is: 1. Import CAD geometry. Simplify the model to make meshgeneration easier 2. Choose a modeling strategy – the materials, elements, restraints, etc. that you will use. 3. Define collectors to organize data to match your strategy 4. Define material data 5. Create the mesh – the collection of elements 6. Apply body loads and boundary loads 7. Apply restraints 8. Solve 9. Review results The sections below describe these steps in greater detail.

Geometry Preparation While it is possible to build a model directly using elements and nodes, this is not often done today. The geometry that defines the area to be analyzed (also called the “domain”) is usually created first using a CAD program, and elements are created to encompass that boundary or represent the volume. CAD designers create models for manufacture. As many details are included as possible. For a numerical analysis, we often choose to ignore aspects that we think will not significantly affect the solution. For instance, a single hole of 1 mm radius in a plate that is 2 meters wide can probably be ignored safely when calculating the deformation of the plate. It reduces computation time dramatically with no significant loss of accuracy. A finer model of only

34

A structured method to conduct what-if analyses is DOE – short for Design Of Experiments. This is covered in another volume of this series.

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the region around the hole can be used subsequently if the hole is an area of high interest. This approach is called sub-modeling. Therefore the first task that most analysts are faced with is that of preparing the geometry for analysis. This involves tasks like removal of features, extraction of mid-surfaces, extrapolation of surfaces, etc. Further, the CAD world has an abundance of data exchange formats, since most CAD applications use proprietary data storage formats. A transfer of data from the CAD package to the FE preprocessor sometimes results in a loss of accuracy – gaps are introduced during the import process, for example. Also, CAD assembly models are sometimes made up of parts that were created in different CAD applications. Therefore a cleaning-up of the geometry is often required. This involves filling gaps, eliminating small edges or surfaces that will mislead the automatic-mesh-generation routines, eliminating dangling faces, and so on.

Guidelines on Element Choice Learning which element to choose is a little like learning how to drive in heavy traffic. Guidelines exist, but can’t be applied blindly. You need to adapt them to specific situations. Remember this warning! If your product has a region that is long and thin, you can probably model it using beam elements. If this region is connected to the rest of the structure by pin-joints, then you should use truss elements. Regions that are like plates are best modeled using shell elements. Any areas that don’t fall in the earlier categories should be modeled using solid elements. If you have different element types in your model, there are rules that govern the assemblage. For several models, we choose to use just one element type to avoid these complications.

Mesh Creation Once the geometry is more or less ready for discretization, you then start to subdivide the geometry into elements or grid points. The collection of elements is usually referred to as a mesh. Meshes that consist of triangular or quadrilateral elements can often be generated automatically, while tetrahedral or hexahedral meshes usually require considerable manual intervention.

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Mesh Editing Once a mesh has been created, the analyst checks if it meets the specifications – several measures of quality are checked, depending on the analysis requirements. Usually, some editing of the mesh is required. Depending on the complexity of the mesh, this can be done either semiautomatically or manually. Acceptable values for the various element quality indicators, which are summarized in the glossary, are very problem dependent, and are sometimes solver-dependent.

Preparing for Analysis Once the mesh is ready, additional data is specified – the properties of the materials used, the thickness or cross-sectional properties of shell or beam elements, the conditions on the boundaries (restraints, loads or excitations), initial conditions, data for the specific solution algorithm to be employed, kind of output required for text and graphics records, and so on. Once this is done, the data is turned over to the solution program for the next phase – solving. Data is often written out in the form of a text file, which is referred to as a deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line. The procedure of building the Finite Element Model is sometimes referred to as FEM – short for Finite Element Modeling. It’s more common, though, to use FEM to refer to the Finite Element Method itself.

Solving The model created in the earlier steps is now taken up for solution – the computer program reads the data, calculates matrix entries, solves the matrix equations and writes data out for interpretation. This task is CPU-intensive, and is often called processing35. Most of the time, very little interaction from the user is required. In some cases, the analyst periodically monitors results to check that they are indeed on the right track. If the solution seems to be evolving in an unexpected direction, the analyst can stop the solver and modify the model, thereby saving valuable time. 35

Hence the term pre-processing for the preceding steps, and post-processing for the subsequent steps.

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Post-Processing After the program has evaluated the results, the analyst examines and interprets the data – looking for errors or improvements in design. As with pre-processing, this calls for substantial interaction from the analyst.

Nomenclature and Data Organization HyperMesh and OptiStruct/Analysis use several different files to organize data. Data is further organized as collectors within the “input” file. Most of these aspects will become clear to you after you work on one or more of the assignment problems. The summary presented here is useful as a quick-reference when you’re working on the software. More detailed descriptions are available in the on-line documentation.

Files modelname.hm

The model you create using HyperMesh. This is a binary file containing the geometry, analysis model and optimization model.

modelname.fem

This is an intermediate file. It contains the analysis and optimization models only, without any geometry. It is created by HyperMesh and read by OptiStruct. It’s a text file and can be interpreted using the format-definitions listed in the OptiStruct On-line Help.

modelname.out

This is a text file created by OptiStruct. The contents depend on the instructions you specify in HyperMesh when creating the model.

modelname.spcf

This file is created only if you explicitly tell OptiStruct to record all reactions. It is a useful check to ensure that you have applied loads correctly.

modelname.stat

This is a text file created by OptiStruct, containing statistics on CPU usage.

modelname.h3d

People who don’t have access to HyperWorks licenses but want to view results of analyses 61

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A Designer’s Guide To Finite Element Analysis

use HyperView Player, freely downloadable from www.altair.com. The Player reads this binary file that is created by OptiStruct.

modelname.html

This is a quick summary of analysis and optimization results. Viewable using any webbrowser.

modelname.mvw

This is a text file, intended for use by HyperView. You will use this file to view stresses, displacements, density, convergence history, constraint violation, etc.

modelname.res

This is a binary file containing the results of the analysis and optimization. It’s readable only by HyperMesh and HyperView.

There are several other files36, most of which you can ignore in the normal course of events.

Terminology Collector

A way to group related items together. For instance all elements that have the same thickness would be in the same collector.

Load

External forces acting on the boundary. Includes concentrated forces, moments, pressures, gravity, etc.

SPC

Short for Single Point Constraint. Refers to restraints applied to the analysis model at locations where the body is supported37.

Subcase

Combination of SPCs and Loads. These are treated separately in an FE model, since they represent values on the boundary, often clubbed together as

Boundary Conditions Card

36

Some data in the analysis model, such as the material properties, cannot be displayed graphically. Such data is entered as a card image by typing in

Detailed in the on-line help documentation Do not confuse these with design constraints, which are applicable to the optimization model.

37

62

A Designer’s Guide To Finite Element Analysis

Putting it Together – OptiStruct/Analysis

text or numerical values. Data is often written out in the form of a text file, which is referred to as a

deck. Each line of text in the deck is commonly referred to as a card. A card image is the format followed by the analysis program to interpret the text on the line. Some data can be represented graphically – nodes and pressures, for example. Other data, like the Modulus of Elasticity, is easier viewed as text. Such data is defined using the card editor. The table below lists the names of some of the data types OptiStruct/Analysis and HyperMesh use, along with the relations between them. Use this as a guideline to remember when you need to specify which data. Normally, you would proceed down the table: first the mat collector, then the component collector, then elements (grids or nodes are implicit), and then the load collector. While load collectors can contain both loads and restraints, it’s a good practice to keep them in separate load-collectors so that you can organize them into sub-cases. Data Entity Mat& Component Property% Grid Elem [email protected] Restraints! SubCase$

Notes: &

Refers to Mat Coordsys Grids, Props Grids/ Elems Grids Loads, Spcs

Data 1

Data 2

Data 3

Data 4

Matid CompName PropId Gridid Elemid

E ElemType+

Nu Mat Id

etc. Prop Data#

X Propid

Y Grid Id 1

Z Grid Id 2 etc.

Loadid

Gridid

Value

Spcid SubCaseid

Gridid

Dofs

Values

A Mat needs a card image. Use Mat1 for linear isotropic, Mat8 for orthotropic shells, Mat9 for linear anisotropic.

+

The component collector is either a PSOLID or a PSHELL. Composites use a PCOMP or PCOMPG.

#

Data depends on the element type. For a solid, there’s nothing. For a shell, there’s the thickness. For a composite, thickness is derived from the PCOMP or PCOMPG data. 63

Putting It Together – OptiStruct/Analysis

A Designer’s Guide To Finite Element Analysis

%

A Prop collector is needed only for 1D elements like beams or special elements such as springs, connectors, etc.

@

Forces and Moments need no card image. Loads such as gravity, which cannot be depicted graphically, require card images.

!

Restraints normally will not require a card image. Remember that nonzero displacements may be specified, in which case you will need to enter values. Restraints on non-existent dofs are ignored (for instance, specifying restraints on all the rotational dofs of a solid element).

$

Sub-Case definitions are followed by a set of cards, each with a keyword (Load, Spc, etc.) followed by the relevant id. These can be viewed in the “fem” file, not using the card editor.

Anyone who has never made a mistake has never tried anything new. Albert Einstein

64

A Designer’s Guide To Finite Element Analysis

Verification And Validation

Verification And Validation The “bumble-bee paradox” provides an excellent example of the gap between theory and practice. Whatever the reason, designers have successfully violated mathematicians’ recommendations in several areas. FEM is no exception. An AISI publication on the use of FEM for vehicle-safety investigations38 talks of “many

conscious violations of elementary finite element theory” in the very successful deployment of the method for highly nonlinear analyses. Given all this background, a beginner can be misled into thinking that the method more forgiving than it actually is. Spectacular failures that have been attributed to faulty FE modeling are documented in literature, and should serve as warnings to engineers that the cost of a mistake can be extremely high.

“Conventional aerodynamics seemed to suggest that the insect should not generate enough lift to fly. The bees stayed resolutely airborne and the sums caused consternation. The underlying problem turned out to be treating a wing as if it was fixed, like in an aeroplane and, thanks to studies over the past few years, including the construction of robotic bees, this "bumble-bee paradox" has been solved: extra lift comes when flexible insect wings slice through the air at a high angle of attack, creating a large swirling vortex at their leading edge. In this way, insect wings produce the vortices – spinning masses of air – which generate lift and help them move. Today, Prof Ismet Gursul of the University of Bath will describe another step on the way for engineers to make air vehicles smaller than a human hand that can be used for detecting chemicals leaks and reconnaissance.”

Roger Highfield, Science Editor The Telegraph

Sources of error can range from wrong assumptions of mechanics, as in the case of the aerodynamics of the bumble-bee, to a wrong choice of elements or even a disregard for computer round-off.

Validation consists of asking whether the right equations have been solved. Verification involves checking whether the equations have been solved correctly.

38

Vehicle Crashworthiness And Occupant Protection, American Iron and Steel

Institute 65

Verification And Validation

A Designer’s Guide To Finite Element Analysis

Both are an integral part of the engineer’s responsibility, not least because of statutory requirements.

Product Liability Laws Product Liability Insurance, in several countries, is as essential for an engineer as malpractice insurance is for a Doctor. The interpretation of laws is, of course, open to interpretation by lawyers and courts, but the various laws on the subject are fairly consistent in their spirit: it is the engineer’s responsibility to ensure that the product has been designed with care. Designers should ideally use FEM only as a part of the design process. It is often risky to base the entire design only on an FEM analysis. A NAFEMs39 publication warns against the GIGO approach: Garbage In, Gospel Out. This is particularly important since many analysis results are presented visually, and a picture can hide errors!

The Seductive Appeal Of Graphics Stress distributions are often plotted as contours – regions with the same stress are assigned the same color, and the component is shaded accordingly. This is an extremely useful method. Not only is it easier to assimilate than reading a list of stresses at various points in the component, it is also better than contour lines since it also indicates how the stress varies across the component. Consider the stress in a plate that’s fixed at one end, has a tensile load at the other end, and has a hole. The picture looks convincing – high stress areas, colored red, are around the hole as we’d expect.

39

66

National Agency for Finite Element Methods and Standards

A Designer’s Guide To Finite Element Analysis

Verification And Validation

But remember our earlier discussion on the characteristic of the FE solution. We said that the calculated stress is always discontinuous over element boundaries. Why is this not apparent in the image above? Several FE packages smooth the stress. If we plot the same contours without averaging the stresses, a different picture emerges.

Not only is the discontinuity in stresses brought out starkly, we must now doubt our solution itself! While the highest stress still seems to be around the hole, what is the value of this stress? Should we pick the value denoted by red, or by green? There is a “jump” in the stress at the edge shared by two elements, and we cannot decide which is correct. Refining the mesh and running the analysis again quickly brings home one error: the FE mesh was too coarse.

With the refined mesh, the difference between the smoothed contours and the unaveraged contours has reduced considerably. One rule of thumb is to distrust the model until the “jumps” in stress are within an acceptable value. What value is acceptable is a question of judgment, of course, but this can be a very good indicator of whether or not your mesh is fine enough.

Quick and Basic Checks Lots of other things can go wrong, of course, and catching errors in production-analyses is no easy task. 67

Verification And Validation

A Designer’s Guide To Finite Element Analysis

It’s a good idea to compare the results of an FE analysis with approximate solutions tabulated in handbooks40. Since not all problems can be found in handbooks, comparison with results reported by other investigators is a good idea. Several excellent scientific journals carry articles reporting the results of analyses, and most commercial software providers host annual conferences where users present papers discussing their experiences. The verification of a computer model by comparing it with the performance of a prototype is useful, but should not be treated as complete. There’s always the chance that an experiment has missed out on investigating rare combinations – and as Murphy’s Law41 swings into action, it is these rare combinations that can cause product malfunction. The accompanying volume in this series, CAE And Design Optimization – Advanced, discusses ways to design experiments and to detect rare combinations (called outliers), but such a study is beyond the scope of our introduction to the Finite Element Method. With this background, there are levels at which checks should be performed: at the Pre-processing stage and at the Post-Processing stage. HyperView, the post-processor that comes with HyperMesh, is particularly good at the latter. Checks include

68

•

Element free edge display

•

Element Normal display

•

Vector-arrow plots

•

Load contours

•

Unaveraged Stress contours

•

Ill-conditioning of the stiffness matrix

40

Roark’s Formulas for Stress and Strain is an excellent reference.

41

“If anything can go wrong, it will.”

A Designer’s Guide To Finite Element Analysis

Verification And Validation

•

Comparison of applied forces with reactions-at-supports

•

Comparison of mass of FE model with mass of CAD model

•

Orientation of beam elements

•

Shrunken plots to check for missing elements

Several of these are covered in the assignments that accompany this book.

Physics is very muddled again at the moment; it is much too hard for me anyway, and I wish I were a movie comedian or something like that and had never heard anything about physics! Wolfgang Pauli

69

Special Topics

A Designer’s Guide To Finite Element Analysis

Special Topics Just how much FE-theory do you need to know? To answer that question, it’s interesting to look at commonly used CAD-modelers. In the early days of CAD modeling, users had to know the internal details of splines etc. to effectively model surfaces. Today, the software, the technology and user interfaces are robust enough that the internal functions are invisible to the designer. The designer needs to delve into the inner workings only if things go wrong. The same is true of almost any technology. In the early 1900s car drivers had to be efficient mechanics, but today you could own a car for a decade without ever opening the hood! The same, of course, is also true of Finite Element Analysis. Several physical problems present challenges when the level of detail required from the results increases. For instance, the analysis of bolted structures presents formidable challenges if the area of interest is confined to the bolt itself: a realistic model should include contact, friction, pre-stress, plastic deformation, etc. Use the summary below as an indicator only. If your problem involves any of the characteristics described, then you will certainly need more than the material covered in this book. The references listed at the end of this book are a good place to start advanced research.

Advanced Materials As weight, appearance and cost become increasingly important, engineers are forced to choose from a wide array of materials, not all of which are as easy to characterize as metals. Of the various materials used in engineering design, steel is easily the most researched and most widely documented. Like most metals, it is isotropic, and has long been used to bear load. Much of engineering design is restricted to the linear, elastic range of the stress-strain diagram. Plastic deformation has received considerable attention over the recent past, as for instance in the design of formed components.

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A Designer’s Guide To Finite Element Analysis

Special Topics

For designers, plastics are a formidable challenge. As cheap, easy-tomanufacture materials, they are widely adopted. Unfortunately, an understanding of linear-elastic analysis techniques limits the designer to initial investigations only. To make things more difficult, the properties of plastics are not just a function of the composition. They also depend strongly on the processing conditions. Composites, made up of multiple materials, are more widely used than plastics to bear load. Reinforced Cement Concrete (RCC) is an example of a composite. The main body of material is strong in compression but weak in tension. Steel reinforcements add tensile strength. Mechanical engineers rarely analyze RCC, however. For product design, commonly used Engineering Composites include Fiber Reinforced Plastics (FRPs, of which Glass Fiber Reinforced Plastics and Carbon Fiber Reinforced Plastics are very widely used) and Metal Matrix Composites (MMCs). Composites are extensively used in the aircraft industry to provide acceptable strength at very low weights. Space applications are less cost-sensitive and more performance-intensive than others. Exotic metal alloys are sometimes used, but these are isotropic and relatively easy to model. Honeycomb structures are not as easy to model and design. Rubbery materials, also called elastomers, are usually treated as a category by themselves. They are not plastic. In fact, they are extremely elastic hyperelastic. This means they can carry very large strains without any permanent deformation. Large strains, of course, mean the analysis must be non-linear. Further, rubbers are almost incompressible: they have a Poisson’s Ratio of almost 0.5. Since the Bulk Modulus and Poisson’s Ratio are related by the equation

K=

E 3(1 − 2ν )

this makes them very difficult to deal with numerically (if ν= 0.5, K is indeterminate). Even small round-off errors can make enormous differences in the results of the analysis.

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Advanced Dynamics Transient analysis generates a vast amount of information, so postprocessing results is quite challenging. Stress contours and deformationplots are used to present the results of both static analysis and dynamic analysis. For dynamics, however, animations of deformation and timehistory plots are also used. Dynamic analysis requires considerably more computing power than static analysis. Remember that a symmetric structure can have un-symmetric modes of vibration too! So symmetry cannot be used to reduce the size of models. One method sometimes used to reduce the size of models is called Component Mode Synthesis. In this approach, the FE model is divided into sections considered as black-boxes. That is, interest is restricted to the boundary nodes of each section, thereby reducing the number of active degrees of freedom. Matrices are for constructed each section and the overall structure analyzed by putting these reduced matrices together. One form of excitation that is frequently encountered, particularly in vehicle design, is ground excitation. The structure receives energy because of enforced motion of selected regions, not because of explicitly applied forces. Several different methods are used to carry out such analyses. Two different approaches are used to perform transient analysis: mode superposition and direct integration. In the former, we assume that the response of the structure is a weighted sum of selected mode shapes. This is an approximation, where the error stems from the choice of the subset of mode shapes used. Direct Integration methods use Finite Difference Methods to “step-forwards” in time. Here we use equations of the form

v=

du ∆u ut + ∆t − ut = = dt ∆t ∆t

If we know the velocity at initial time, v0, we can write

v0 .∆t + u0 = ut + ∆t 72

A Designer’s Guide To Finite Element Analysis

Special Topics

In other words, given the values of v and u at t=0 (the “initial conditions”) we have just “stepped-forward” in time to calculate ut + ∆t . Together with the equilibrium equation, recurrence formulae of this type are used to calculate the time-dependent response. Several variations of timestepping methods are used: Central Difference Methods, Backward Difference Methods, Runge-Kutta Methods, etc. The source of error in direct integration is the time-step size, ∆t. Time stepping methods are classified based on whether they are explicit or implicit, as well as on the order of the method. In some conditions, we cannot state with any confidence how the forces themselves vary with time. What we can say with confidence is how the energy supplied to the structure varies with time. As an excellent example, consider the forces experienced by a rocket at launch. The combustion pattern is quite random, but the total energy released by the fuels is reasonably clear. Such problems are called Random Response problems. Response Spectrum methods are usually used to calculate the probabilities of various levels of stresses in the body.

Multi-disciplinary analysis, also called multi-physics analysis, involves the use of different branches of mechanics simultaneously. To understand the forces experienced by an aircraft in flight, for example, we cannot calculate the deformation of the aircraft unless we know the air-pressures. And the airpressures depend on the deformation. Such coupled analyses are still the subject of much active research.

Shall I refuse my dinner because I do not fully understand the process of digestion? Oliver Heaviside

73

Glossary And References

A Designer’s Guide To Finite Element Analysis

Glossary And References Accuracy

A measure of the effectiveness of a numerical method. Accuracy is valuable if coupled with stability.

Adaptive Refinement

A technique where the software automatically modifies the FE mesh, using measures of the errors in the solution.

Anisotropic

Material whose properties vary with direction, but not necessarily along orthogonal directions. Several fused or sintered materials are anisotropic. 21 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT2 for shell elements.

Bandwidth

The stiffness matrix of a typical FE model has zeroes in most entries except for a band about the diagonal. The bandwidth measures this “spread” of non-zeroes in the matrix. A smaller bandwidth means faster computation.

CAD

Computer Aided Design. Usually means creation of 3D models of parts and assemblies.

CAE

Computer Aided Engineering. Said to have been coined by Dr.Jason Lemon in 1980, and meant to consist of CAD + FE Modeling + FE Analysis + Design. Today includes Multi-Body Dynamics (MBD). Often separated into MCAE (for Mechanical CAE, or the analysis of structures) and FCAE (for Fluid CAE, or the analysis of flow of heat and liquids / gases).

CFD

Computational Fluid Dynamics. The use of computers to solve various forms of Navier-Stokes equations to analyze the flow of fluids.

Compliance

Reflects the reciprocal of the stiffness. Maximizing the compliance is the same as minimizing the stiffness.

Consistent Mass Matrix

This is one way to calculate the Mass Matrix (M) in FEA. The alternative is the lumped mass matrix.

Decibel

Frequently used in sound and vibration measurements. Defined by the ISO R 1683 standard as measured − value N = 20 ⋅ log10 reference − value

74

A Designer’s Guide To Finite Element Analysis

Glossary And References

Degree of Freedom

Variable we want the FE analysis to solve for. In an FE model every node / grid has at least 1 dof, often more. For a stress-analysis problem, each node can have upto 6 dofs – 3 rotations and 3 translations. Other variables such as temperature at the nodes, pressure, etc. are also included in some models.

Discretization Error

Difference between the actual domain of a problem and the domain defined by the FE model.

DOF

See Degree of Freedom

Effective Modal Mass

Characterize the relative contribution of the modes of a structure to an excitation

Hysteresis

Represents the history-dependence of a physical system. If a cyclic load is applied, the original properties are not always recovered at the end of a cycle. Closely related to work-hardening, in stress analysis.

Isoparametric

Most commercial FE programs use elements that are isoparametric. That is, the same shape function is used to approximate both the geometry and the solution.

Isotropic

Material whose properties are independent of direction. Applies to most metals. 2 elasticity constants are required to fully specify the material for stress analysis. The Modulus of Elasticity and the Poisson’s Ratio are most frequently used. In OptiStruct, these materials are of type MAT1.

Jacobian

The “Jacobian Matrix” arises when a change-of-variables is applied to a PDE. The determinant of the Jacobian Matrix is important to FE theory: it should always be positive for an element (strictly speaking it should never change sign). Most FE codes refer to it as the “Jacobian” or “Det J”.

Lanczos Method

A way to calculate the normal modes of a structure. Very efficient and stable.

Large Mass Method

A method to apply support-excitations. Uses the principle that a force applied to a large mass attached to a model is equivalent to applying an acceleration to the model. Given an acceleration you want to apply, Newton’s Second Law is used to calculate the equivalent force that should be applied for a mass you choose.

Lumped Mass Matrix

This is one way to calculate the Mass Matrix (M) in FEA. The alternative is the consistent mass matrix. Computationally more efficient, it is often preferred.

75

Glossary And References

76

A Designer’s Guide To Finite Element Analysis

MDO

Multi-disciplinary Optimization. Used, for example, when your product needs to be designed for optimal performance as a mechanism and as a structure.

Mises-Hencky

One of several failure theories for ductile metals.

MPC

Multi-point Constraint. Used to specify that different dofs are linked in a particular fashion.

Nominal Stress

Stress predicted by an equation. Used mainly in analytical calculations, the working stress is calculated by applying a factor of safety to the nominal stress.

NVH

Noise, Vibration and Harshness.

Orthotropic

Material whose properties vary along principal or orthogonal directions. Applies to many fibrous materials, and to composites that have 2 ply directions. Upto 9 elasticity constants are required to fully specify the material for stress analysis. In OptiStruct, these materials are of type MAT8 for shell elements. MAT9 should be used for solid elements.

Post-buckling analysis

A highly-non linear analysis that investigates how structures behave after a buckling-related collapse.

Rigid Body Modes

Unrestrained structures can vibrate as a rigid body. A body can have upto 6 rigid-body modes.

Robust Design

A design method to reduce sensitivity of the design to inherent unpredictability of design parameters.

Spectrum

Term frequently used in dynamic analysis. The Fourier transform of a function gives us the spectrum. The spectrum can be used to reconstruct the function using the inverse transform.

Stability

With specific reference to series solutions, stability is related to the convergence characteristics. A monotonically convergent series allows us to estimate the accuracy of a finite sum.

Stiffness

Usually referred to as the “stiffness matrix” in FE models, relates the applied loads to the deformation of the structure. The matrix is square, with “n” rows and columns. “n” is the number of unknowns (dofs) in the FE model.

Superconvergence

A characteristic of the FE solution. The state variable (deformation, temperature, etc.) is best evaluated at the nodes, while the flux (strain, stress, heat flux, etc.) is best evaluated at the Gauss Integration points.

A Designer’s Guide To Finite Element Analysis

Glossary And References

Integration points.

Support Excitation

In many cases, such as a vehicle rolling over a rough road, the source of excitation is not a force: it is the prescribed deflection, which varies with time. While this can be treated mathematically as an “inverse” of a force-excitation problem, different techniques are employed to solve such problems.

Work-hardening

An increase in strength due to plastic deformation.

References Finite Element Procedures, K.J.Bathe Numerical Methods in Finite Element Analysis, K.J.Bathe and E.L.Wilson The Finite Element Method, O.C. Zienkiwicz Engineering Mechanics of Solids, E.P.Popov Mechanical Engineering Design, J.E.Shigley and L.D.Mitchell Roark’s Formulas for Stress and Strains, W.C.Young Formulas for Natural Frequency and Mode Shape, R.D.Blevins Theory of Vibration With Applications, W.T.Thomson Theory of Elasticity, S.Timoshenko Heat Transfer, J.P.Holman

Other Resources www.altair-india.com/edu, which is periodically updated, contains case studies of actual usage. It also carries tips on software usage.

Common Material Properties Be careful when using these properties. Some properties vary widely with alloying elements or processing parameters, so treat these as indicative. It’s probably safe to use them in exploratory design efforts, but not in designs 77

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that will be manufactured. For those, you should look for values from the material supplier. Also remember to check the units in your model – they must be consistent! Material

Modulus of Elasticity

Units

N/m2

Steel

200x109

Aluminum

Poisson’s Ratio

Density

Sample Permissible Stress

Kg/m3

N/m2

0.29

7800

250x106

69 x109

0.33

2700

110 x106

Wood42

13 x109

0.029

480

50 x106

Cast Iron

190 x109

0.21

7150

170 x106

ABS Plastics

2.3 x109

Epoxy

40x106 1790

Sample properties for a Carbon Fiber Composite material: E1 181 GPa E2 10.3 GPa G12 7.17 GPa ν12 0.28 Density 1.60 gm/cm3 For an isotropic material, any two properties are enough – the others can be calculated from these. We usually specify the Elasticity Modulus (E) and the Poisson’s Ration (ν). Other material constants such as the Shear Modulus (G) and the Bulk Modulus (K) can be derived from E and ν using equations such as :

G=

42

78

In compression

E (1 + ν ) 2

A Designer’s Guide To Finite Element Analysis

Glossary And References

E K= 3(1 − 2ν )

Useful Data For Heat Transfer Conduction is the main mode of heat transfer within a solid. The conductivity (k) is a material property that’s quite easy to ascertain for most metals, and can be assumed to be independent of temperature for relatively low temperatures. The thermal conductivity of 0.5%-Carbon Steel, for example, ranges from 55 W/m°C at 0°C to 52 W/m°C at 100°C. Plastics are a little less easy to analyze. The heat conduction coefficient must be measured by a test on a sample, or must be supplied by the manufacturer of the plastic. Also, several plastics are more sensitive to temperature than metals. Heat is transferred away from the solid by convection or radiation. The convective boundary conditions require the specification of the convection coefficient, h. This is a function of the fluid material and the fluid velocity. More precisely, it depends on the boundary layer between the fluid and the solid. Boundary layers depend on several factors – surface roughness, whether the boundary is vertical or horizontal, etc. One way to estimate h is to first calculate dimensionless numbers from available data, and then use empirical equations to calculate a dimensionless number (Nusselt, Stanton, etc.) that involves h. For example, given the flow velocity, the density and the viscosity of the fluid, we estimate the Reynold’s number (Re). Next, given the viscosity of the fluid, the specific heat capacity and the conduction coefficient of the solid, we estimate the Prandtl number (Pr). We then look for an empirical equation that relates Pr, Re and the Nusselt number (Nu). For instance for flow in a pipe, Dittus and Boelter43 suggest that

Nu d = 0.023 Re 0d.8 Pr n 43

See Heat Transfer, J.P.Holman, 7th Edition, Page 284 79

Glossary And References

A Designer’s Guide To Finite Element Analysis

We calculate Nu from this equation, then use the definition of Nu to solve for h. Remember that we want to solve for the temperature distribution. For this we need h. To get h we need the Reynold’s number. The Reynold’s number needs the viscosity. The viscosity is temperature dependent. So we need the temperature to get the viscosity. But we don’t know the temperature! This is why problems in heat transfer are often harder to solve than problems in stress analysis. Iterative methods are often used to arrive at the solution – estimate the numbers, solve for the temperature, use this temperature to re-estimate the numbers, solve again, etc. until there is no change. However choosing characteristic dimensions requires some care. For example, it is not always easy to decide whether to use the width or the length to calculate the Nusselt number for a rectangular plate! A strong physical insight is extremely useful for workable solutions to problems in heat transfer! Some useful dimensionless constants are presented here, but a full discussion is beyond the scope of this book.

80

Dimensionless Number

Relevance

Biot Number (Bi)

Ratio of external heat transfer (through convection) to internal heat transfer (through conduction).

Grashof Number (Gr)

Ratio of buoyancy force to viscous force.

Nusselt Number (Nu)

Ratio of internal heat transfer through convection to internal heat transfer through conduction

Prandtl Number (Pr)

Ratio of momentum diffusivity to thermal diffusivity

Peclet Number (Pe)

Ratio of rate of advection (i.e. diffusion and convection of mass) to rate of thermal diffusion.

Reynolds Number (Re)

Ratio of inertial forces to viscous forces.

Stanton Number (St)

Ratio of heat transferred into a fluid to heat capacity of fluid.

A Designer’s Guide To Finite Element Analysis

ρ µ ν β

Glossary And References

density dynamic viscosity kinematic viscosity volume coefficient of thermal expansion specific heat (at constant pressure) velocity of flow coefficient of heat conduction

cp u k

Consistent Units Mixing up units is one of the most common errors. It’s also the least forgivable if committed by an engineer who is allowed to use the SI system of units. While FPS can be challenging, the SI system is very straightforward. The table below lists some common properties of Steel in consistent units. Mass

Length

Time

Force

Stress

Density

Young’s Modulus

Acceleration due to Gravity

kg kg G G Ton lbfs2/in Slug Kg

M Cm Cm mm mm In Ft mm

s s s s s s s s

N 1.0e-02 N dyne 1.0e-06 N N lbf lbf mN

Pa

7.83e+03 7.83e-03 7.83e+00 7.83e-03 7.83e-09 7.33e-04 1.52e+01 7.83e-06

2.07e+11 2.07e+09 2.07e+12 2.07e+11 2.07e+05 3.00e+07 4.32e+09 2.07e+08

9.806 9.806e+02 9.806e+02 9.806e+03 9.806e+03 386 32.17 9.806e+02

dy/cm² Pa MPa psi psf 1.0e+03 Pa

For a dynamic analysis, OptiStruct lists both the natural frequency and the eigen-value. The former is in cycles / unit time, or Hz if SI units are used. The latter, which represents the square of the angular frequency ω, is related to the former by the equation

eigen-value = (2*p*frequency)2

Lumped Mass Models In Vehicle-Crash Simulation This material, from the American Iron and Steel Institute’s Vehicle Crashworthiness and Occupant Protection”, shows the relevance of discrete, multiple-degree-of-freedom (mdof) models. “In 1970, Kamal [1] developed a relatively simple, but powerful model for simulating the crashworthiness response of a vehicle in frontal impact. This model, known as the Lumped Mass-Spring (LMS) model, became widely used by crash engineers because of its simplicity and relative accuracy. This model is shown in Figure 2.2.2.1. The vehicle is approximated by a one-

81

Glossary And References

A Designer’s Guide To Finite Element Analysis

dimensional lumped mass-spring system, an over simplification that is quite acceptable for modeling the basic crash features in frontal impact. Because of its simplistic representation of the crash event, an LMS model requires a user with extensive knowledge and understanding of structural crashworthiness, and considerable experience in deriving the model parameters and translating the output into design data. The crush characteristics (spring parameters) were determined experimentally in a static crusher, as shown in Figure 2.2.2.2. The configuration of the model is arrived at from the study of an actual barrier test, which

identifies pertinent masses and “springs” and their mode of collapse. The model is “tuned” by adjusting the load-deflection characteristics of the “springs” to achieve the best agreement with test results in the timing of the crash events. Figure 2.2.2.4 compares the simulated acceleration histories with those obtained from the test, and it shows that very good agreement can be achieved. LMS models proved to be very useful in developing vehicle structures for crash, enabling the designer to develop generically similar crash energy management systems, that is, development of vehicle derivatives or structural upgrading for crash. Also, LMS models provide an easy method to study vehicle/powertrain kinematics, and give directional guidance to the designer by establishing component objectives.

Measures Of Element Quality The numerical integration procedure used in FE programs imposes several conditions on the shape of the element. Other conditions stem from the types of shape functions used. A detailed description of these measures is beyond the scope of this book. The on-line documentation presents more detail, in case you want to go beyond the summary below. 82

A Designer’s Guide To Finite Element Analysis

Glossary And References

Remember that an ideal element is rarely encountered. The ideal elements are:

Element Type

Ideal Shape

Quadrilateral Triangle Hexahedron Tetrahedron

Square Equilateral Triangle Cube Regular, equilateral tetrahedron

Aspect Ratio Applicable to all elements, this is the ratio of the longest edge to the smallest edge. The ideal value is 1, while an aspect beyond 5 is not recommended.

Interior Angle Applicable to quadrilateral and triangular elements, the ideal value is 90° for a quadrilateral and 60° for a triangle. The angle should never reach 0° or 180° in any case.

Jacobian Applicable to all elements, this is a measure of the distortion of the element as compared to the ideal element. A Jacobian of 0 or less is not permitted. The closer the value is to 1, the better.

Length There is no absolute measure for the acceptable element length. The smaller the elements, the more elements you require to span your geometry, and the more CPU time required. That apart, using very small elements and very large elements in the same mesh can give rise to round-off error.

Skew Applicable to all quadrilateral and triangular elements, an ideal element has a skew of 0°.

Taper Applicable only to quadrilaterals, a taper of 0 is ideal.

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Glossary And References

A Designer’s Guide To Finite Element Analysis

Warpage Applicable only to quadrilateral elements, warpage is 0 if all 4 nodes lie in the same plane. Upto 5° is considered good.

Collapse Applicable to tetrahedral elements, the perfect value is 1. The value should never reach 0.

Volume Aspect Ratio Used for 3D elements (hexahedra and tetrahedral), 1 is the ideal value.

Volume Skew Applicable only to tetrahedral elements, 0 is the ideal value. The value should never reach 1.

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