Table Of ContentPreface
This textbook represents the Finite Element Analysis lecture course given
to students in the third year at the Department of Engineering Sciences (now
F.I.L.S.), English Stream, University Politehnica of Bucharest, since 1992.
It grew in time along with a course taught in Romanian to students in the
Faculty of Transports, helped by the emergence of microcomputer networks and
integration of the object into mechanical engineering curricula. The syllabus of the
28-hour course, supplemented by 28-hour tutorial and lab. classes, was structured
along the NAFEMS recommendations published in the October 1988 issue of
BENCHmark. The course represents only an introduction to the finite element
analysis, for which we wrote simple stand-alone single-element programs to assist
students in solving problems as homework. It is followed by an advanced course in
the fourth year at F.I.L.S., called Computational Structural Mechanics, where
students are supposed to use commercial programs.
In designing the course, our aim was to produce students capable of: (a)
understanding the theoretical background, (b) appreciating the structure of finite
element programs for potential amendment and development, (c) running packages
and assessing their limitations, (d) taking a detached view in checking output, and
(e) understanding failure messages and finding ways of rectifying the errors.
The course syllabus was restricted to 2D linear elastic structural problems.
It has been found advantageous to divide the finite element analysis into two parts.
Firstly, the assembly process without any approximations (illustrated by
frameworks) followed by the true finite element process which involves
approximations. This is achieved starting with trusses, then with beams and plane
frames, and progressively dealing with membrane and plate-bending elements.
Solid elements and shells are not treated. Our objective was to ensure that students
have achieved: (a) a familiarity in working with matrix methods and developing
stiffness matrices, (b) an understanding of global versus local coordinate systems,
(c) the abilty to use the minimum potential energy theorem and virtual work
equations, (d) the mapping from isoparametric space to real geometrics and the
need for numerical integration, (e) an insight in numerical techniques for linear
equation solving (Gauss elimination, frontal solvers etc), and (f) the use of
equilibrium, compatibility, stress/strain relations and boundary conditions.
As a course taught for non-native speakers, it has been considered useful to
reproduce as language patterns some sentences from English texts.
November 2006 Mircea Radeş
Prefaţă
Lucrarea reprezintă cursul Analiza cu elemente finite predat studenţilor
anului III al Facultăţii de Inginerie în Limbi Străine, Filiera Engleză, la
Universitatea Politehnica Bucureşti, începând cu anul 1992.
Conţinutul cursului s-a lărgit în timp, fiind predat din 1992 şi studenţilor de
la facultatea de Transporturi, favorizat de apariţia reţelelor de calculatoare şi de
includerea sa în planul de învăţământ al facultăţilor cu profil mecanic. Programa
cursului, care prevede 28 ore de curs şi 28 ore de seminar/laborator, a fost
structurată în conformiatate cu recomandările NAFEMS publicate în numărul din
Octombrie 1988 al revistei BENCHmark. Cursul reprezintă doar o introducere în
analiza cu elemente finite, pentru care am scris programe simple, cu un singur tip
de element finit, care să fie utilizate de studenţi la rezolvarea unor teme de casă. Nu
se tratează învelişuri şi elemente tridimensionale. În anul IV, planul de învăţământ
de la F.I.L.S. conţine cursul Computational Structural Mechanics, la care studenţii
aprofundează modelarea cu elemente finite şi utilizează un program de firmă.
La structurarea cursului am avut în vedere necesitatea formării unor
studenţi capabili: (a) să înţeleagă baza teoretică, (b) să desluşească structura
programelor cu elemente finite pentru eventuale corecţii şi dezvoltări, (c) să ruleze
programe şi să recunoască limitele acestora, (d) să poată verifica rezultatele şi (e)
să înţeleagă mesajele de eroare şi să găsească modalităţi de corectare a erorilor.
Programa cursului a fost limitată la structuri elastice liniare
bidimensionale. S-a considerat potrivit să se prezinte analiza cu elemente finite în
două etape: întâi procesul de asamblare fără nici o aproximare (aplicat la grinzi cu
zăbrele), apoi modelarea cu elemente finite, care presupune aproximarea câmpului
de deplasări, de la triunghiul cu deformaţii specifice constante la elemente
patrulatere izoparametrice, incluzând integrarea numerică. S-a urmărit ca studenţii
să dobândească: (a) familiaritate cu metodele matriciale şi calculul matricelor de
rigiditate; (b) înţelegerea utilităţii coordonatelor locale şi globale; (c) abilitatea
folosirii principiului energiei potenţiale minime şi a principiului lucrului mecanic
virtual; (d) trecerea de la coordonate naturale la coordonate fizice şi necesitatea
integrării numerice; (e) o vedere de ansamblu asupra rezolvării sistemelor algebrice
liniare (eliminarea Gauss, metoda frontală etc.) şi (f) utilizarea celor patru tipuri de
ecuaţii – echilibru, compatibilitate, constitutive şi condiţii la limită.
Fiind un curs predat unor studenţi a căror limbă maternă nu este limba
engleză, au fost reproduse expresii şi fraze din cărţi scrise de vorbitori nativi ai
acestei limbi.
Noiembrie 2006 Mircea Radeş
Contents
Preface i
Contents iii
1. Introduction 1
1.1 Object of FEA 1
1.2 Finite element displacement method 3
1.3 Historical view 4
1.4 Stages of FEA 5
2. Displacement Method 9
2.1 Equilibrium equations 9
2.2 Conditions for geometric compatibility 10
2.3 Force/elongation relations 11
2.4 Boundary conditions 12
2.5 Solving for displacements 12
2.6 Comparison of the force method and displacement method 13
3. Direct Stiffness Method 17
3.1 Stiffness matrix for a bar element 17
3.2 Transformation from local to global coordinates 19
3.2.1 Coordinate transformation 19
3.2.2 Force transformation 20
3.2.3 Element stiffness matrix in global coordinates 21
3.2.4 Properties of the element stiffness matrix 22
3.3 Link’s truss 25
3.4 Direct method 26
3.5 Compatibility of nodal displacements 28
3.6 Expanded element stiffness matrix 29
3.7 Unreduced global stiffness matrix 30
iv FINITE ELEMENT ANALYSIS
3.8 Joint force equilibrium equations 31
3.9 Reduced global stiffness matrix 33
3.10 Reactions and internal forces 35
3.11 Thermal loads and stresses 36
3.12 Node numbering 37
Exercises 41
4. Bars and shafts 47
4.1 Plane bar elements 47
4.1.1 Differential equation of equilibrium 47
4.1.2 Coordinates and shape functions 48
4.1.3 Bar not loaded between ends 49
4.1.4 Element stiffness matrix in local coordinates 51
4.1.5 Bar loaded between ends 52
4.1.6 Vector of element nodal forces 55
4.1.7 Assembly of the global stiffness matrix and load vector 56
4.1.8 Initial strain effects 59
4.2 Plane shaft elements 60
Exercises 63
5. Beams, frames and grids 79
5.1 Finite element discretization 79
5.2 Static analysis of a uniform beam 81
5.3 Uniform beam not loaded between ends 83
5.3.1 Shape functions 84
5.3.2 Stiffness matrix 86
5.3.3 Physical significance of the stiffness matrix 88
5.4 Uniform beam loaded between ends 89
5.4.1 Consistent vector of nodal forces 89
5.4.2 Higher degree interpolation functions 92
5.4.3 Bending moment and shear force 95
5.5 Basic convergence requirements 96
5.6 Frame element 97
5.6.1 Axial effects 97
5.6.2 Stiffness matrix and load vector in local coordinates 98
CONTENTS v
5.6.3 Coordinate transformation 98
5.6.4 Stiffness matrix and load vector in global coordinates 100
5.7 Assembly of the global stiffness matrix 100
5.8 Grids 111
5.9 Deep beam bending element 116
5.9.1 Static analysis of a uniform beam 117
5.9.2 Shape functions 118
5.9.3 Stiffness matrix 121
6. Linear elasticity 123
6.1 Matrix notation for loads, stresses and strain 123
6.2 Equations of equilibrium inside V 125
6.3 Equations of equilibrium on the surface S 126
σ
6.4 Strain-displacement relations 127
6.5 Stress-strain relations 128
6.6 Temperature effects 130
6.7 Strain energy 130
7. Energy methods 131
7.1 Principle of virtual work 131
7.1.1 Virtual displacements 131
7.1.2 Virtual work of external forces 133
7.1.3 Virtual work of internal forces 133
7.1.4 Principle of virtual displacements 134
7.1.5 Proof that PDV is equivalent to equilibrium equations 137
7.2 Principle of minimum total potential energy 139
7.2.1 Strain energy 139
7.2.2 External potential energy 140
7.2.3 Total potential energy 140
7.3 The Rayleigh-Ritz method 143
7.4 FEM – a localized version of the Rayleigh-Ritz method 148
7.4.1 FEM in Structural Mechanics 148
7.4.2 Discretization 149
7.4.3 Principle of virtual displacements 149
vi FINITE ELEMENT ANALYSIS
7.4.4 Approximating functions for the element 149
7.4.5 Compatibility between strains and nodal displacements 150
7.4.6 Element stiffness matrix and load vector 151
7.4.7 Assembly of the global stiffness matrix and load vector 151
7.4.8 Solution and back-substitution 152
8 Two-dimensional elements 153
8.1 The plane constant-strain triangle (CST) 153
8.1.1. Discretization of structure 153
8.1.2 Polynomial approximation of the displacement field 154
8.1.3 Nodal approximation of the displacement field 155
8.1.4 The matrix [B] 158
8.1.5 Element stiffness matrix and load vector 159
8.1.6 Remarks 160
8.2 Rectangular elements 176
8.2.1 The four-node rectangle (linear) 176
8.2.2 The eight-node rectangle (quadratic) 178
8.3 Triangular elements 180
8.3.1 Area coordinates 180
8.3.2 Linear strain triangle (LST) 182
8.3.3 Quadratic strain triangle 185
8.4 Equilibrium, convergence and compatibility 187
8.4.1 Equilibrium vs. compatibility 187
8.4.2 Convergence and compatibility 188
9 Isoparametric elements 191
9.1 Linear quadrilateral element 191
9.1.1 Natural coordinates 192
9.1.2 Shape functions 193
9.1.3 The displacement field 194
9.1.4 Mapping from natural to Cartesian coordinates 195
9.1.5 Element stiffness matrix 198
9.1.6 Element load vectors 199
9.2 Numerical integration 200
9.2.1 One dimensional Gauss quadrature 200
CONTENTS vii
9.2.2 Two dimensional Gauss quadrature 203
9.2.3 Stiffness integration 204
9.2.4 Stress calculations 207
9.3 Eight-node quadrilateral 208
9.3.1 Shape functions 209
9.3.2 Shape function derivatives 210
9.3.3 Determinant of the Jacobian matrix 211
9.3.4 Element stiffness matrix 211
9.3.5 Stress calculation 213
9.3.6 Consistent nodal forces 214
9.4 Nine-node quadrilateral 219
9.5 Six-node triangle 221
9.6 Jacobian positiveness 223
10 Plate bending 225
10.1 Thin plate theory (Kirchhoff) 225
10.2 Thick plate theory (Reissner-Mindlin) 229
10.3 Rectangular plate-bending elements 232
10.3.1 ACM element (non-conforming) 232
10.3.2 BFS element (conforming) 238
10.3.3 HTK thick rectangular element 239
10.4 Triangular plate-bending elements 244
10.4.1 Thin triangular element (non-conforming) 245
10.4.2 Thick triangular element (conforming) 248
10.4.3 Discrete Kirchhoff triangles (DKT) 250
References 257
Index 265
1.
INTRODUCTION
Finite Element Analysis (FEA) as applied to structures is a
multidisciplinary technique, based on knowledge from three fields: (1) Structural
Mechanics, encompassing elasticity, strength of materials, dynamics, plasticity, etc,
(2) Numerical Analysis, involving approximation methods, solving linear sets of
equations, eigenproblems, etc, and (3) Applied Computer Science, dealing with the
development and maintenance of large computer codes.
FEA is used to solve large-scale analytical problems. Its task is to model
and describe the mechanical behaviour of geometrically complex structures. The
procedure is a discretized approach: the geometric shape or the internal stress-
strain-displacement field are described by a series of discrete quantities (like
coordinates) distributed through the structure. This requires a matrix notation. The
tools are the computers, able to store long lists of numbers and manipulate them.
1.1 Object of FEA
The object of FEA is to replace the infinite degree of freedom system in
continuum applications by a finite system exhibiting the same basis as discrete
analysis.
The aim is finding an approximate solution to a boundary- and initial-
value problem by dividing the domain of the system into a set of interconnected
finite-sized subdomains of different size and shape, and defining the unknown state
variable approximately, within each element, by means of a linear combination of
trial functions. The subdomains are called finite elements, the set of finite elements
is known as the mesh and the trial functions are referred to as interpolation
functions. With the individually defined functions matching each other at certain
points called nodes, the unknown function is approximated over the entire domain.
The primary difference between the FEA and other approximate methods
for the solution of boundary-value problems (finite-difference, weighted-residual,
2 FINITE ELEMENT ANALYSIS
Rayleigh-Ritz, Galerkin) is that in the FEA the approximation is confined to
relatively small subdomains.
FEA is a localized version of the Rayleigh-Ritz method. Instead of finding
an admissible function satisfying the boundary conditions for the entire domain,
which is often difficult, in the FEA the admissible functions (called shape
functions) are defined over element domains with simple geometry and pay no
attention to complications at the boundaries.
Since the entire domain is divided into numerous elements and the function
is approximated in terms of its values at the element nodes, the evaluation of such a
function will require the solution of simultaneous equations. This was possible only
at the time the computers became available. The outstanding success of the finite
element method can be attributed to a large extent to timing. While the finite
element method was being developed, so were increasingly powerful digital
computers, which led to automation. The computer is not only able to solve the
discretized equations of equilibrium, but also to carry out such diverse tasks as the
formulation of equations, by making decisions concerning the finite element mesh
and the assembly of stiffness matrices.
Perhaps more important is the fact that the finite element method can
accommodate systems with complicated geometries and parameter distributions.
The wide use of the classical Rayleigh-Ritz method has been limited by the
inability to generate suitable admissible functions for a large number of practical
problems. Indeed, systems with complex boundary conditions or complex
geometry cannot be described easily by global admissible functions, which tend to
have complicated expressions, difficult to handle on a routine basis. In turn, in the
FEA an approximate solution is constructed using local admissible functions,
defined over small subdomains of the structure. In order to match a given irregular
boundary, or to handle parameter non-uniformities, the FEA can change not only
the size of the finite elements but also their shape. This extreme versatility, coupled
with the development of powerful computer codes based on the method, some of
them made available as open source free software, has made the FEA the method
of choice for the analysis of structures.
In FEA, the equations of equilibrium are obtained from variational
principles implying the stationarity of the functional defined by the total potential
energy. While solving differential equations with complicated boundary conditions
may be difficult, integrating known polynomial functions, even approximately,
[ ]{ } { }
should be easier. Mathematically, solving A x = b is equivalent to
minimizing P(x)= 1{x}T[A]{x}−{x}T {b}. This is the heart of the FEA when
2
applied to structures.
