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PAVEMENT DESIGN AND EVALUATION
THE REQUIRED MATHEMATICS
AND ITS APPLICATIONS
F. Van Cauwelaert
Editor: Marc Stet
Federation of the Belgian Cement Industry
B-1170 Brussels, Rue Volta 9.
PAVEMENT DESIGN AND EVALUATION
ii
INTRODUCTION
iii
Pavement Design and Evaluation: The Required Mathematics and Its Applications
Keywords: textbook for pavement engineers, high order mathematical solutions for, rigid and flexible
pavement, mechanistic methods, practical applications in the field of pavement engineering
Preface
This book is intended for Civil Engineers and more specific for Pavement Engineers, who are interested
in the more advanced field of pavement engineering made available through the theory of high
mathematics. In my extensive carrier as a Civil Engineer by profession, I noticed that some of my
colleagues feel uneasy when it comes to high-level theoretical and computational work. Perhaps this is
because of the fact that as soon as they graduated, they are confronted with many and important practical
problems where derivatives and integrals do not appear very useful at fist sight. However, at many
occasions, conferences, seminars and other meetings, I realised that Civil Engineers remain excited about
the mathematical fundamentals of their art. Slow but sure grew the idea of writing a textbook on High
Mathematics conceivable for practising Pavement Engineers. The primary objective of this book is to
serve two purposes: first, to introduce the basic principles which must be known by people dealing with
pavements; and secondly to present the theories and methods in pavement design and evaluation that may
be used by students, designers engineering consultants, highway and airport agencies, and researchers at
universities. In addition, some of the new concepts developed in recent years to improve the methods of
pavement systems are explained. This book is written in a relatively simple way so that it may be
followed by people familiar with basic engineering courses in mathematics and pavement design.
This book consists of 26 chapters and is divided into two parts. The first part, which include chapters 1-9,
covers the mathematics required by most of the problems related to pavement engineering. The assumed
mathematical knowledge is that of high school level plus some basic elements of trigonometry and
analysis. The second part, which includes chapters 10-26, is concerned with practical solutions as faced
by Pavement Engineers in the assessment of rigid and flexible pavements. The rigorous mathematical
solutions are presented in the Appendix, explaining on complex variables.
Grateful acknowledgement is offered to the Fédération de l’Industrie Cimentière, Belgium (the Federation
of the Belgian Cement Industry) for their intellectual and financial support in the process of the
realisation of this book.
Thanks is due to my dear friend, Marc Stet, for proof-reading and editing the manuscript. His helpful
comments to the mathematical and editorial content are highly appreciated.
Frans Van Cauwelaert
Brussels, December 2003.
PAVEMENT DESIGN AND EVALUATION
iv
Part 1: The Required Mathematics
Chapter 1 covers the Laplace differential equation; the solution of for a great series of applications: it
presents the Bessel function of zero order, solution of the Laplace equation in axisymmetric co-ordinates
that as used in great parts of this book. The Laplacian with coefficients different from 1 will be applied in
problems of anisotropic elasticity, the double Laplacian in most of the problems of multi-layer theory and
the extended Laplacian in the problems of rigid pavement on Winkler or Pasternak foundations. Chapter 2
presents the gamma function, the factorial function for non-integers, required for the definition of Bessel
functions of non-integer order. Chapter 3 gives the general solutions of the different Bessel equations, the
Ber and Bei functions, and the modified form of the Bessel equation. Bessel functions are solutions of the
Laplacian in polar or cylinder co-ordinates, used for applications with axial symmetry, e.g. in multi-
layered structures. Trigonometric functions are solutions of the Laplacian in Cartesian co-ordinates,
applied in cases of beams and rectangular slabs. Chapter 4 deals with the most important properties of
Bessel functions:
- derivatives, functions of half order,
- asymptotic values required to express boundary conditions that must remain valid at infinite
distances,
- indefinite integrals, equations between Bessel functions of different kind (required for integrations in
the complex plane).
Chapter 5 presents the beta function required for the resolution of definite integrals of Bessel functions.
Chapter 6 gives the solutions for a series of important definite integrals of Bessel functions; among others
the Poisson integral giving an integral representation of any Bessel function of the first kind. Chapter 7
presents the hypergeometric function of Gauss required for the resolution of the infinite integrals of
Bessel functions. Chapter 8 presents the most important infinite integrals of Bessel functions of direct
application in pavement analysis, especially in multi-layer theory. Chapter 9 presents the most important
infinite integrals of Bessel functions resolved in the complex plane. They are essentially of application in
slab theory.
Part 2: The Applications
The second part focuses on the applications in the field of pavement engineering: rigid en flexible
pavements. Chapter 10 gives the basic solutions (equilibrium equations) for rigid pavements (theory of
strength of materials) and flexible pavements (theory of elasticity) .This chapter gives the basic equations
on continuum mechanics in different co-ordinate systems. Chapter 11 presents the integral transforms,
Fourier’s expansion, Fourier’s integral, Hankel’s integral, required for the expression of discontinuous
functions: the loads in pavement applications.
Chapters 12 to 19 concern mostly rigid pavements. Chapter 12 gives 3 simple applications on an elastic
subgrade: a beam subjected to a single load, a beam subjected to a distributed load, a slab subjected to a
single load. Chapter 13 gives the complete analytical solution for a beam on a Pasternak foundation (both
of infinite and finite extent). Chapter 14 gives the analytical solution for a circular slab on a Pasternak
foundation subjected to an axi-symmetric load (both of infinite and finite extent). Chapter 15 gives the
complete analytical solution for a rectangular slab on a Pasternak foundation (both of infinite and finite
extent). Chapter 16 gives the analytical solution for a superposition of several slabs included the analysis
of the adhesion between the slabs. Chapter 17 gives a back-calculation method for rigid pavement based
on Pasternak’s theory. Chapter 18 presents a solution for the computation of thermal stresses in rigid
slabs on a Pasternak foundation. Chapter 19 presents two practical tests of interest with rigid slabs: the
diametral test and a test for the determination of k and G in situ.
Chapters 20 to 26 concern mostly flexible pavements. Chapters 20 and 21 present the complete theory for
semi-infinite bodies subjected to all sorts of loads. Chapter 20 presents the Boussinesq problem: stresses
INTRODUCTION
v
and displacements in a semi-infinite body under a circular flexible plate uniformly loaded. It generalises
the solution to different vertical loads (isolated, rigid, rectangular) and to orthotropic bodies. Chapter 21
presents the solution a semi-infinite body subjected to shear stresses: radial and one-directional. Chapter
22 gives the analytical solution for a multi-layered structure, included the problem of the adhesion
between the layers, that of an eventual fixed bottom and that of an anisotropic subgrade. Chapter 23
presents the numerical procedure required for the solution of a multi-layered structure. Chapter 24
presents the theory at the base of the back-calculation methods for flexible pavements. Chapter 25
presents the numerical procedure required for the back-calculation method for flexible pavements.
Chapter 26 presents a practical test of interest with both types of road structures (rigid or flexible) multi-
layered structures: the ovalisation test.
The Appendices gives the basic theory of complex numbers, especially the integration in the complex
plane.
PAVEMENT DESIGN AND EVALUATION
vi
.
CONTENTS
vii
Table of Contents
Page
Preface............................................................................................................................................. iii
PART 1: THE REQUIRED MATHEMATICS................................................................................1
Chapter 1 The Laplace Equation....................................................................................................1
1.1 Introductory note................................................................................................................. 1
1.2 Derivation of the Laplace equation in polar co-ordinates from the Laplace equation in Cartesian
co-ordinates........................................................................................................................ 2
1.3 Equations related to the Laplace equation.............................................................................. 3
1.3.1 The Laplacian with coefficients different from 1................................................................ 3
1.3.2 The double Laplacian....................................................................................................... 3
1.3.3 The extended Laplacian................................................................................................... 3
1.4 Resolution of the Laplace equation....................................................................................... 4
1.4.1 Resolution by separation of the variables........................................................................... 4
1.4.2 Resolution by means of the characteristic equation (Spiegel, 1971) ..................................... 5
1.4.3 Resolution by means of indicial equations ......................................................................... 6
Chapter 2 The Gamma Function....................................................................................................9
2.1 Introductory note................................................................................................................. 9
2.2 Helpful relations.................................................................................................................. 9
2.3 Definition of the Gamma Function......................................................................................10
2.4 Values of Γ(1/2) and Γ(-1/2)...............................................................................................11
Chapter 3 The General Solution of the Bessel Equation. ..............................................................15
3.1 Introductory note................................................................................................................15
3.2 Helpful relations.................................................................................................................16
3.3 Resolution of the Bessel equation (Bessel functions of the first kind).....................................17
3.4 Resolution of the Bessel equation for p an integer (Bessel functions of the second and third
kind)..................................................................................................................................19
3.4.1 For n integer, Jn = (-)n
J-n .................................................................................................19
3.4.2 Bessel functions of the second kind..................................................................................19
3.4.3 Bessel functions of the third kind.....................................................................................21
3.5 The modified Bessel equation. ............................................................................................21
3.6 The ber and bei functions....................................................................................................23
3.7 The ker and kei functions....................................................................................................24
3.8 Resolution of the equation ∇2
∇2
w + w =0............................................................................25
3.9 The modified form of the Bessel equation............................................................................25
Chapter 4 Properties of the Bessel Functions...............................................................................27
4.1 Introductory note................................................................................................................27
4.2 Helpful relations.................................................................................................................27
4.3 Derivatives of Bessel functions ...........................................................................................29
4.3.1 Derivative of (rt)p
Jp(rt)....................................................................................................29
4.3.2 Derivative of (rt)-pJp(rt)..................................................................................................29
PAVEMENT DESIGN AND EVALUATION
viii
4.3.3 Derivative of Jp(rt)..........................................................................................................29
4.4 Bessel functions of half order..............................................................................................30
4.4.1 Values of J1/2(rt), J-1/2(rt), J3/2(rt), J-3/2(rt)...........................................................................30
4.5 Asymptotic values..............................................................................................................31
4.5.1 Asymptotic values for Jp and J-p.....................................................................................31
4.5.2 Asymptotic values for Yp and Y-p ....................................................................................33
4.5.3 Asymptotic values for Hp
(1)
and Hp
(2)
................................................................................33
4.5.4 Asymptotic values for Ip and I-p .......................................................................................33
4.5.5 Asymptotic value for Kn..................................................................................................35
4.5.6 Asymptotic values for ber and bei....................................................................................36
4.5.7 Asymptotic values for ker and kei....................................................................................36
4.6 Indefinite integrals of Bessel functions ................................................................................37
4.6.1 Fundamental relations .....................................................................................................37
4.6.2 The integral ∫rn
J0(rt)dr .....................................................................................................37
4.7 Relations between Bessel functions of different kind ............................................................38
4.7.1 Bessel functions with argument –rt..................................................................................38
4.7.2 Relations between the three kinds of Bessel functions .......................................................38
4.7.3 Bessel functions of purely imaginary argument.................................................................39
Chapter 5 The Beta Function.......................................................................................................41
5.1 Introductory note................................................................................................................41
5.2 Helpful relations.................................................................................................................41
5.3 Definition of the beta function.............................................................................................41
5.4 Relation between beta and gamma functions ........................................................................42
5.5 The duplication formula for gamma functions ......................................................................42
Chapter 6 Definite integrals of Bessel functions ...........................................................................45
6.1 Helpful relations.................................................................................................................45
6.2 Gegenbauer’s integral.........................................................................................................45
6.3 Sonine’s first finite integral.................................................................................................46
6.4 Sonine’s second finite integral.............................................................................................47
6.5 Poisson’s integral...............................................................................................................48
Chapter 7 The hypergeometric type of series ...............................................................................49
7.1 Introductory note................................................................................................................49
7.2 Helpful relations.................................................................................................................49
7.3 Definition ..........................................................................................................................50
7.4 Properties of the multiple product (α)m ................................................................................51
7.4.1 A relation for (1 - b - m)m-n ..............................................................................................51
7.4.2 A relation for (a+b+1+m)m/22 m
.........................................................................................51
7.4.3 A relation for Γ(a-n) .......................................................................................................52
7.4.4 The theorem of Vandermonde .........................................................................................52
7.4.5 The product of two Bessel functions with the same argument ............................................53
7.5 The hypergeometric series of Gauss 2F1[a,b;c;z]...................................................................54
7.5.1 Elementary properties.....................................................................................................54
7.5.2 Integral representation of the hypergeometric function ......................................................54
7.5.3 Value of F[a,b;c;z] for z = 1............................................................................................55
7.5.4 Convergence of the series F[a,b;c;z].................................................................................55
7.5.5 The product of two Bessel functions with different arguments...........................................56
CONTENTS
ix
Chapter 8 Infinite Integrals of Bessel Functions...........................................................................57
8.1 Introductory note................................................................................................................57
8.2 Useful relations..................................................................................................................57
8.3 The integral ∫e-at
Jν(bt)tµ-1
dt..................................................................................................60
8.3.1 Resolution of the integral................................................................................................60
8.3.2 Particular value...............................................................................................................61
8.3.3 The integral ∫ e-at
cos(bt)dt................................................................................................63
8.3.4 The integral ∫ e-at
sin(bt)dt.................................................................................................63
8.3.5 The integral ∫e-at
cos(bt)sin(bt) dt ......................................................................................63
8.3.6 The integral ∫e-at
sin(bt)/tdt...............................................................................................63
8.3.7 The integral ∫e-at
sin(bt)tdt................................................................................................64
8.4 The integral ∫e-at
Jν(bt)Jν(ct)tµ-1
dt ..........................................................................................64
8.4.1 Transformation of the integral..........................................................................................64
8.4.2 The integral ∫e-at
sin(bt)sin(ct)t-1
dt.....................................................................................64
8.4.3 The integral ∫e-at
sin(bt)sin(ct)dt........................................................................................65
8.4.4 The integral ∫∫e-am
sin(bt)sin(cs)/(ts)dsdt ............................................................................65
8.4.5 The integral ∫∫me-am
sin(bt)sin(cs)/(ts)dsdt .........................................................................67
8.5 The integral ∫e-at
Jµ(bt)Jν(ct)tλ-1
dt ..........................................................................................68
8.6 The discontinuous integral ∫Jµ(at)Jν(bt)t-λ
dt...........................................................................68
8.6.1 Resolution of the integral................................................................................................68
8.6.2 The integral ∫Jµ(at)Jν(at)t-λ
dt.............................................................................................71
8.6.3 The integral ∫Jµ(at)Jµ-2k-1(bt)dt..........................................................................................72
8.6.4 Particular solutions of the integral ∫Jµ(at)Jν(bt)t-λ
dt ...........................................................72
8.6.5 Particular solution of the integral ∫Jν(at)Jν(bt)/tλ
dt..............................................................73
Chapter 9 Bessel functions in the complex plane ..........................................................................77
9.1 Introductory note................................................................................................................77
9.2 Helpful relations.................................................................................................................77
9.3 Proof of Γ(z)Γ(1-z) = π/sin(πz)...........................................................................................78
9.4 The Hankel’s contour integral for 1/Γ(z)..............................................................................80
9.5 The integral representation of Jν(z) ......................................................................................81
9.6 The integral representation of Iν(z) ......................................................................................83
9.7 The integral representation of Kν(z).....................................................................................83
9.8 The integral representation of Kv(xz)....................................................................................84
9.9 Resolution of ∫xν+1
Jν(ax)/(x2
+k2
)dx.....................................................................................85
9.10 Resolution of ∫xρ-1
Jµ(bx)Jν(ax)/(x2
+k2
)dx.............................................................................89
9.11 Resolution of ∫sin(bx)cos(ax)/x/(x4
+ k4
)dx ..........................................................................90
PART 2: THE APPLICATIONS...................................................................................................93
Chapter 10 Laplace Equation in Pavement Engineering ............................................................93
10.1 Equilibrium equation for beams in pure bending...................................................................93
10.1.1 Sign conventions.........................................................................................................93
10.1.2 Assumptions...............................................................................................................95
10.1.3 Bending moment and bending stress ............................................................................95
10.1.4 The radius of curvature ...............................................................................................96
10.1.5 Equilibrium ................................................................................................................96
PAVEMENT DESIGN AND EVALUATION
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10.2 Equilibrium equation for bent plates....................................................................................97
10.2.1 Bending moment and shear forces................................................................................97
10.2.2 Equilibrium ................................................................................................................99
10.3 Compatibility equation for a homogeneous, elastic, isotropic body submitted to forces applied
on its surface ...................................................................................................................100
10.3.1 Principle of equilibrium.............................................................................................101
10.3.2 The principle of continuity.........................................................................................101
10.3.3 The principle of elasticity ..........................................................................................102
10.3.4 Stress potential..........................................................................................................102
10.4 Compatibility equation for a homogeneous, elastic, anisotropic body submitted to forces applied
on its surface ...................................................................................................................103
10.5 Basic equations of continuum mechanics in different co-ordinate systems ...........................104
10.5.1 Plane polar co-ordinates............................................................................................104
10.5.2 Axi-symmetric Cylindrical Co-ordinates ....................................................................105
10.5.3 Non symmetric cylindrical co-ordinates .....................................................................106
10.5.4 Cartesian volume co-ordinates...................................................................................107
10.5.5 Axi-symmetric Cylindrical Co-ordinates for an orthotropic body..................................109
Chapter 11 The Integral Transforms........................................................................................ 111
11.1 Introductory note..............................................................................................................111
11.2 Helpful relations...............................................................................................................112
11.3 The Fourier expansion (Spiegel, 1971)...............................................................................112
11.3.1 Definitions................................................................................................................112
11.3.2 Proof of the Fourier expansion. ..................................................................................113
11.3.3 Example ...................................................................................................................114
11.4 The Fourier integral..........................................................................................................115
11.4.1 Definition.................................................................................................................115
11.4.2 Proof of Fourier’s integral theorem............................................................................115
11.4.3 Example ...................................................................................................................116
11.5 The Hankel’s transform....................................................................................................117
11.5.1 Definition.................................................................................................................117
11.5.2 Example ...................................................................................................................117
11.5.3 Application of the discontinuous integral of Weber and Schafheitlin ............................118
Chapter 12 Simple Applications of Beams and Slabs on an Elastic Subgrade........................... 121
12.1 The elastic subgrade.........................................................................................................121
12.2 The beam on an elastic subgrade subjected to an isolated load: ∂4
w/∂x4
+Cw=0....................123
12.3 The beam on an elastic subgrade subjected to a distributed load: ∂4
w/∂x4
+Cw=0..................125
12.4 The infinite slab subjected to an isolated load: ∇2
∇2
w +Cw = 0..........................................125
Chapter 13 The Beam Subjected to a Distributed Load and Resting on a Pasternak Foundation
............................................................................................................................... 129
13.1 The basic differential equations.........................................................................................129
13.2 Case of a beam of infinite length .......................................................................................130
13.2.1 Solution of the differential equation............................................................................130
13.2.2 Application...............................................................................................................133
13.3 Case of a beam of finite length with a free edge 133
13.3.1 Solution #1...............................................................................................................133
13.3.2 Application...............................................................................................................135
13.3.3 Solution #2...............................................................................................................137
13.3.4 Application...............................................................................................................137
CONTENTS
xi
13.4 Case of a finite beam with a joint.......................................................................................138
13.4.1 Solution #1...............................................................................................................138
13.4.2 Application...............................................................................................................140
13.4.3 Solution# 2...............................................................................................................141
13.4.4 Application...............................................................................................................142
13.4.5 Proof that, in de case of a Winkler foundation, at a joint Q = γ T..................................143
Chapter 14 The Circular Slab Subjected to a Distributed Load and Resting on a Pasternak
Foundation............................................................................................................. 145
14.1 The basic differential equations.........................................................................................145
14.2 Case of a slab of infinite extent .........................................................................................146
14.2.1 Solution of the differential equation............................................................................146
14.2.2 Application 1............................................................................................................147
14.2.3 Application 2............................................................................................................148
14.3 Case of a slab of finite extent with a free edge ...................................................................148
14.3.1 Solution #1...............................................................................................................148
14.3.2 Solution #2...............................................................................................................151
14.3.3 Application of solution #2.........................................................................................151
14.4 Case of a slab of finite extent with a joint...........................................................................152
14.4.1 Solution #1...............................................................................................................152
14.4.2 Solution #2...............................................................................................................152
14.4.3 Application of solution #2.........................................................................................153
Chapter 15 The Rectangular Slab Subjected to a Distributed Load and Resting on a Pasternak
Foundation............................................................................................................. 155
15.1 The basic differential equations.........................................................................................155
15.2 Resolution of the deflection equation.................................................................................155
15.3 Boundary conditions.........................................................................................................158
15.4 Case of a slab of finite extent with free edge ......................................................................158
15.4.1 Solution #1...............................................................................................................158
15.4.2 Solution #2...............................................................................................................159
15.5 Case of a slab of finite extent with a joint...........................................................................159
15.5.1 Solution #1...............................................................................................................159
15.5.2 Solution # 2..............................................................................................................160
15.5.3 Application...............................................................................................................161
Chapter 16 The Multislab......................................................................................................... 163
16.1 Theoretical justification....................................................................................................163
16.2 General model..................................................................................................................163
16.3 Full slip at each of the interfaces.......................................................................................163
16.4 Full friction at the first interface, full slip at the second interface.........................................164
16.5 Full friction at both interfaces ...........................................................................................165
16.6 Partial friction..................................................................................................................166
16.6.1 Application...............................................................................................................168
Chapter 17 Back-calculation of Concrete Slabs ........................................................................ 171
17.1 Back-calculation of moduli...............................................................................................171
17.2 Case where the load can be considered as a point load........................................................171
17.3 Computations...................................................................................................................172
17.4 Case when the load is considered as distributed..................................................................173
17.5 Comparison of the two methods ........................................................................................175
PAVEMENT DESIGN AND EVALUATION
xii
17.6 Influence of the reference deflection..................................................................................175
17.7 Analysis of field data........................................................................................................176
17.8 Example of back-calculation.............................................................................................177
Chapter 18 Thermal Stresses in Concrete Slabs ....................................................................... 179
18.1 Thermal stresses ..............................................................................................................179
18.2 Slab of great length...........................................................................................................179
18.2.1 Differential equilibrium equation...............................................................................179
18.2.2 Solution of the equilibrium equation for g < 1.............................................................180
18.2.3 Solution of the equilibrium equation for g = 1.............................................................180
18.2.4 Solution of the equilibrium equation for g > 1.............................................................180
18.2.5 Boundary conditions .................................................................................................180
18.2.6 Expression of the moment for g < 1............................................................................180
18.2.7 Expression of the moment for g = 1............................................................................182
18.2.8 Expression of the moment for g > 1............................................................................182
18.2.9 Verification of the expression of the maximum moment for g = 1.................................183
18.2.10 Equation of the thermal stress....................................................................................183
18.2.11 Example ...................................................................................................................184
18.3 Rectangular slab...............................................................................................................184
18.3.1 Differential equation of equilibrium...........................................................................184
18.3.2 Boundary conditions .................................................................................................185
18.3.3 Examples .................................................................................................................185
18.4 Circular slab....................................................................................................................187
18.4.1 Equilibrium equation.................................................................................................187
18.4.2 Solution of the equilibrium equation...........................................................................188
18.4.3 Boundary conditions .................................................................................................188
18.4.4 Resulting moment.....................................................................................................188
18.4.5 Comparison between the models for rectangular and circular slabs...............................189
18.5 Extension to a multi-slab system.......................................................................................189
Chapter 19 Determination of the Parameters of a Rigid Structure ........................................... 193
19.1 Determination of the Young’s modulus of a concrete slab...................................................193
19.1.1 Resolution of the compatibility equation.....................................................................193
19.1.2 Equations for the stresses..........................................................................................193
19.1.3 Boundary conditions .................................................................................................193
19.1.4 Stresses and displacements........................................................................................194
19.1.5 Tangential normal stress along the vertical diameter....................................................195
19.2 Determination of the characteristics k and G of the subgrade ..............................................195
19.2.1 Equilibrium equation for a Pasternak subgrade ...........................................................195
19.2.2 Eccentrically loaded plate-bearing test........................................................................196
19.2.3 Vertical load.............................................................................................................198
19.2.4 Moment....................................................................................................................199
19.2.5 Determination of k and G..........................................................................................201
Chapter 20 The Semi-Infinite Body Subjected to a Vertical Load ............................................ 203
20.1 Introductory note..............................................................................................................203
20.2 The semi-infinite body subjected to a vertical uniform circular pressure ..............................203
20.3 The semi-infinite body subjected to an isolated vertical load...............................................206
20.4 The semi-infinite body subjected to a circular vertical rigid load..........................................207
20.5 The semi-infinite body subjected to a vertical uniform rectangular pressure.........................209
20.6 Comparison between the vertical stresses. Principle of de Saint-Venant..............................211
CONTENTS
xiii
20.7 The orthotropic body subjected to a vertical uniform circular pressure.................................211
20.8 The orthotropic body subjected to a vertical uniform rectangular pressure ...........................215
Chapter 21 The Semi-Infinite Body Subjected to Shear Loads ................................................. 217
21.1 The semi-infinite body subjected to radialshear stresses.....................................................217
21.2 The semi-infinite body subjected to a one-directional asymmetric shear load.......................218
21.3 The semi-infinite body subjected to a shear load symmetric to one of its axis’s ....................221
Chapter 22 The Multilayered Structure ................................................................................... 225
22.1 The multilayered structure................................................................................................225
22.2 Solutions of the continuity equations .................................................................................226
22.3 Boundary conditions.........................................................................................................227
22.4 Determination of the boundary constants...........................................................................228
22.5 The fixed bottom condition...............................................................................................231
22.6 The orthotropic subgrade ..................................................................................................232
Chapter 23 The Resolution of a Multilayered Structure ........................................................... 233
23.1 Choice of the integration formula ......................................................................................233
23.2 Values at the origin...........................................................................................................234
23.3 The geometrical scale of the structure................................................................................234
23.4 Width of the integration steps............................................................................................235
23.4.1 Influence of the moduli on the integration step............................................................235
23.4.2 Influence of the radii of the loads on the integration step.............................................235
23.4.3 Influence of the offset distance on the integration step.................................................236
23.4.4 Modification of the step width ...................................................................................236
23.5 Stresses and displacements at the surface...........................................................................236
23.6 Stresses and displacements in the first layer.......................................................................237
Chapter 24 The Theory of the Back-Calculation of a Multilayered Structure .......................... 241
24.1 The surface modulus.........................................................................................................241
24.2 Equivalent layers..............................................................................................................242
24.3 Equivalent semi-infinite body............................................................................................242
24.4 Analysis of a deflection basin............................................................................................243
24.4.1 Analysis of a three-layer on a linear elastic subgrade...................................................243
24.4.2 Analysis of a three-layer with a very stiff base course .................................................244
24.4.3 Analysis of a three layer with a very weak base course................................................245
24.4.4 Analysis of a two-layer on a subgrade with increasing stiffness with depth ...................246
24.5 Algorithm of Al Bush III (1980)........................................................................................247
Chapter 25 The Numerical Procedure of the Back-calculation of a Multilayered Structure ..... 249
25.1 The analysis of a back-calculation program for a three-layered structure..............................249
25.2 The sensitivity of the back-calculation procedure for a three layer structure.........................251
25.2.1 The sensitivity to rounding off the values of the measured deflections ..........................251
25.2.2 The sensitivity to the presence of a soft intermediate layer...........................................251
25.2.3 The influence of fixing beforehand the value of one modulus.......................................252
25.3 The sensitivity of the back-calculation procedure for a four layer structure ..........................253
25.3.1 Value of the information given by the surface modulus................................................254
25.3.2 The influence of fixing beforehand the value of one modulus.......................................256
25.4 The influence of degree of anisotropy and Poisson’s ratio on the results of a back-calculation
procedure in the case of a semi-infinite subgrade................................................................257
25.4.1 Influence of the degree of anisotropy on the back-calculated moduli.............................257
PAVEMENT DESIGN AND EVALUATION
xiv
25.4.2 Influence of Poisson’s ratio of the subgrade on the deflections (n = 1) ..........................258
25.5 The influence of Poisson’s ratio and degree of anisotropy on the results of a back-calculation
procedure in the case of a subgrade of finite thickness........................................................258
25.5.1 Influence of the degree of anisotropy on the back-calculated moduli.............................258
25.5.2 Influence of Poisson’s ratio of the subgrade on the deflections (n = 1) ..........................259
Chapter 26 The Ovalisation Test.............................................................................................. 261
26.1 Description of the ovalisation test......................................................................................261
26.2 Interpretation of the results of the ovalisation test...............................................................261
26.3 Slab with a cavity on an elastic subgrade subjected to a symmetrical load............................262
26.3.1 Resolution in the case of a plain slab..........................................................................262
26.3.2 Resolution in the case of a slab with a cavity...............................................................263
26.4 Strains in the case of a non-symmetrical load.....................................................................265
26.4.1 Stresses in a hollow cylinder subjected to a uniform external pressure..........................265
26.4.2 Stresses in a hollow plate...........................................................................................266
26.4.3 Application of the ovalisation test..............................................................................271
References ............................................................................................................................... 273
Appendix Complex Functions ..................................................................................................... 275
THE LAPLACE EQUATION
1
PART 1: THE REQUIRED MATHEMATICS
Chapter 1 The Laplace Equation
1.1 Introductory note.
In mechanical engineering and thus also in the mechanics of civil engineering, one always starts with the
fundamental requirement of equilibrium: equilibrium of the normal forces and equilibrium of the
moments acting on the analysed body. Forces are resultants of stresses. In order to locate exactly those
forces and determine, at least analytically, their amplitude, it is convenient to start from an infinitesimal
section of the body. On such an infinitesimal section, the stresses are necessarily infinitely small and,
hence, can be considered as constant and uniformly distributed over the area of the infinitely small
section. Hence the resulting force is simply the product of the constant stress by the area of the section
and its point of application is the centre of gravity of the section, i.e. the midpoint of the section. All the
resulting equations will then necessarily be differential equations and, nearly in almost all applications,
the differential equation expressing equilibrium is a Laplace or an assimilated equation. The development
of these equations are presented in chapter 10, the first chapter of Part 2 “Applications”. As in many
fields of engineering, also in Pavement engineering the differential equation of Laplace is the solution of
a great series of applications. Symbolically, Laplace equation is written as follows:
02
=∇ Φ (1.1)
It is a homogeneous differential equation with second order partial derivatives.
Function of the co-ordinate system, the equation is developed:
in plane Cartesian co-ordinates
0
yx 2
2
2
2
=+
∂
Φ∂
∂
Φ∂
(1.2)
in volume Cartesian co-ordinates
0
zyx 2
2
2
2
2
2
=++
∂
Φ∂
∂
Φ∂
∂
Φ∂
(1.3)
in polar co-ordinates
0
r
1
rr
1
r 2
2
22
2
=++
∂θ
Φ∂
∂
Φ∂
∂
Φ∂
(1.4)
in cylindrical co-ordinates
0
zr
1
rr
1
r 2
2
2
2
22
2
=+++
∂
Φ∂
∂θ
Φ∂
∂
Φ∂
∂
Φ∂
(1.5)
In case of axial symmetry, ∂Φ/∂θ = 0 and equations (1.4) and (1.5) simplify in:
0
rr
1
r2
2
=+
∂
Φ∂
∂
Φ∂
(1.6)
0
zrr
1
r 2
2
2
2
=++
∂
Φ∂
∂
Φ∂
∂
Φ∂
(1.7)
Equation (1.2) is applied in chapter 10.3.4.
Equation (1.4) will be utilised in chapter’s 10.5.1 and 19.1.
Equation (1.7) will be utilised in chapter’s 10.5.2, 10.6, 20.1, 20.2, 20.3 and 20.4.
PAVEMENT DESIGN AND EVALUATION
2
1.2 Derivation of the Laplace equation in polar co-ordinates from the Laplace equation in
Cartesian co-ordinates.
Consider the relations between Cartesian and polar co-ordinates:
θcosrx =
θsinry =
Hence
222
yxr +=
)x/y(tan 1−
=θ
Express the partial derivatives
( )
θ
∂
∂
cos
r
x
yx
x
x
r
2/122
==
+
=
θ
∂
∂
sin
r
y
y
r
==
r
sin
x
y
x
y
1
1
x 2
2
2
θ
∂
∂θ
−=
+
−=
r
cos
x
1
x
y
1
1
y
2
2
θ
∂
∂θ
=
+
=
Then
xx
r
rx ∂
∂θ
∂θ
Φ∂
∂
∂
∂
Φ∂
∂
Φ∂
+=
∂θ
Φ∂θ
∂
Φ∂
θ
∂
Φ∂
r
sin
r
cos
x
−=
xxx
r
xrx2
2
∂
∂θ
∂
Φ∂
∂θ
∂
∂
∂
∂
Φ∂
∂
∂
∂
Φ∂
+=
∂θ
Φ∂
θθ
∂θ∂
Φ∂
θθ
∂θ
Φ∂
θ
∂
Φ∂
θ
∂
Φ∂
θ
∂
Φ∂
cossin
r
2
r
cossin
r
2
sin
r
1
r
sin
r
1
r
cos
x 2
2
2
2
2
2
2
2
2
2
2
2
+−++=
In the same way, obtain:
∂θ
Φ∂
θθ
∂θ∂
Φ∂
θθ
∂θ
Φ∂
θ
∂
Φ∂
θ
∂
Φ∂
θ
∂
Φ∂
cossin
r
2
r
cossin
r
2
cos
r
1
r
cos
r
1
r
sin
y 2
2
2
2
2
2
2
2
2
2
2
2
−+++=
Make the sum:
2
2
22
2
2
2
2
2
r
1
rr
1
ryx ∂θ
Φ∂
∂
Φ∂
∂
Φ∂
∂
Φ∂
∂
Φ∂
++=+ (1.8)
THE LAPLACE EQUATION
3
1.3 Equations related to the Laplace equation.
Besides the properly so called Laplace differential equation, there exists a series of useful differential
equations closely related to the Laplace equation.
1.3.1 The Laplacian with coefficients different from 1.
For example in plane Cartesian co-ordinates
0
yC
1
x 2
2
2
2
=+
∂
Φ∂
∂
Φ∂
(1.9)
Equation (1.9) is utilised in chapter’s 10.3.5 and 20.6.
1.3.2 The double Laplacian
The double Laplacian, or the Laplace operator applied to a Laplace equation, is also a solution of an
important series of applications.
It writes
022
=∇∇ Φ (1.10)
Developed, for example in volume co-ordinates, the double Laplacian is
0
zyxzyx 2
2
2
2
2
2
2
2
2
2
2
2
=








++








++
∂
Φ∂
∂
Φ∂
∂
Φ∂
∂
∂
∂
∂
∂
∂
(1.11)
Equation (1.10) is applied in chapter’s 10.3, 10.5.1,10.5.2, 10.5.3, 10.5.4, 20.1, 20.2, 20.3 and 20.4.
1.3.3 The extended Laplacian
Often the Laplacian equation is completed by derivatives of an order lower than the second. Here in polar
co-ordinates:
0k
r
1
rr
1
r 2
2
22
2
=+++ Φ
∂θ
Φ∂
∂
Φ∂
∂
Φ∂
(1.12)
or more generally
CBA 222
=+∇+∇∇ ΦΦΦ (1.13)
The extended Laplacian is used in chapters 10.1.5, 10.2.2 and in chapter’s 12 to 19.
PAVEMENT DESIGN AND EVALUATION
4
1.4 Resolution of the Laplace equation
Three very general methods are applied in this book.
1.4.1 Resolution by separation of the variables
Solution in volume Cartesian co-ordinates:
Consider equation (1.3) and assume a solution such as Φ = f1(x)f2(y)f3(z).
Applying (1.3) yields
0
z
)z(f
)y(f)x(f)z(f
y
)y(f
)x(f)z(f)y(f
x
)x(f
2
3
2
2132
2
2
1322
1
2
=++
∂
∂
∂
∂
∂
∂
and dividing by f1(x)f2(y)f3(z)
0
)z(f
z
)z(f
)y(f
y
)y(f
)x(f
x
)x(f
3
2
3
2
2
2
2
2
1
2
1
2
=++ ∂
∂
∂
∂
∂
∂
Each of the 3 terms of the sum is a function of one single variable; this results in
)z(fC
z
)z(f
)y(fC
y
)y(f
)x(fC
x
)x(f
332
3
2
222
2
2
112
1
2
===
∂
∂
∂
∂
∂
∂
0CCC 321 =++
a system with a large series of solutions of the differential equation.
For example: f1(x) = cos(x), f2(y) = cos(y), f3(z) = ez√2
.
Solution in axi-symmetric cylinder co-ordinates ( Bowman,1958):
Consider equation (1.7) and apply a solution such as Φ = f(r)g(z).
Applying (1.7) yields
0
z
)z(g
)r(f)z(g
r
)r(f
r
1
)z(g
r
)r(f
2
2
2
2
=+
∂
+
∂
∂∂
∂
∂
and dividing by f(r)g(z)
0
)z(g
z
)z(g
)r(f
r
)r(f
r
1
)r(f
r
)r(f
2
2
2
2
=++ ∂
∂
∂
∂
∂
∂
Thus
)z(Cg
z
)z(g
)r(Cf
r
f
r
1
r
)r(f
2
2
2
2
=−=+
∂
∂
∂
∂
∂
∂
Choose Cg(z) = ez
.
Hence f(r) must be a solution of
0)r(f
r
)r(f
r
1
r
)r(f
2
2
=++
∂
∂
∂
∂
(1.14)
Equation (1.14) is called the Bessel equation of zero order.
THE LAPLACE EQUATION
5
The function known as Bessel’s function of the first kind and of n-th
order and denoted Jn(r) is defined as
follows (Bowman, 1958):
...
)!3n(!3
)2/r(
)!2n(!2
)2/r(
)!1n(!1
)2/r(
!n!0
)2/r(
)r(J
n6n4n2n
n +
+
−
+
+
+
−=
+++
(1.15)
Hence
...
!3!3
)2/r(
!2!2
)2/r(
!1!1
)2/r(
1)r(J
642
0 +−+−= (1.16)
and
...
!4!3
)2/r(
!3!2
)2/r(
!2!1
)2/r(
2
r
)r(J
753
1 +−+−= (1.17)
Differentiating the series for J0(r) and comparing the result with the series for J1(r) results in:
)r(J
dr
)r(dJ
1
0
−= (1.18)
Also, after multiplying the series for J1(r) by r and differentiating:
( ) )r(rJ)r(rJ
dr
d
01 = (1.19)
Using (1.18), (1.19) can be rewritten in the form:
)r(rJ
dr
)r(dJ
r
dr
d
0
0
=





−
0)r(J
dr
)r(dJ
r
1
dr
)r(Jd
0
0
2
0
2
=++ (1.20)
Hence J0(r) is a solution of (1.14) and Φ = J0(r)ez
is a solution of (1.7).
This particular solution was developed to introduce, from the first chapter on, the Bessel functions. Bessel
functions are frequently used and discussed throughout this book.
1.4.2 Resolution by means of the characteristic equation (Spiegel, 1971)
This solution applies to homogeneous linear differential equations with constant coefficients defined as
0a
dx
d
a...
dx
d
a
dx
d
a n1n1n
1n
1n
n
0 =++++ −−
−
Φ
ΦΦΦ
(1.21)
It is convenient to adopt the notations DΦ, D2
Φ, ..., Dn
Φ to denote dΦ/dx, d2
Φ/dx2
, ...dn
Φ/dxn
, where D,
D2
, ..., Dn
are called differential operators.
Using this notation, (1.21) transforms
0aDa...DaDa n1n
1n
1
n
0 =





++++ −
−
Φ (1.22)
Let Φ = emx
, m = constant, in (1.22) to obtain
0a...mama n
1n
1
n
0 =+++ −
(1.23)
that is called the characteristic equation. It can be factored into
0)mm)...(mm)(mm(a n210 =−−− (1.24)
which roots are m1, m2, ...mn.
PAVEMENT DESIGN AND EVALUATION
6
One must consider three cases:
Case 1. Roots all real and distinct.
Then em1x
, em2x
, ... emnx
are n linearly independent solutions so that the required solution is:
xm
n
xm
2
xm
1
n21
eC...eCeC +++=Φ (1.25)
Case 2. Some roots are complex.
If a0, a1, ..., an are real, then when a+bi is a root of (1.23) so also is a-bi (where a and b are real and i =
√(-1) ) . Then a solution corresponding to the roots a+bi and a-bi is:
( )bxsinCbxcosCe 21
ax
+=Φ (1.26)
where use is made of Euler’s formula eiu
= cos u +i sin u (see Appendix).
Case 3. Some roots are repeated.
If m1 is a root of multiplicity k, then a solution is given by:
( ) xm1k
k
2
321
1
exC...xCxCC −
++++=Φ (1.27)
Example
Consider the double Laplacian in plane Cartesian co-ordinates:
0
yyx
2
x 4
4
22
4
4
4
=++
∂
Φ∂
∂∂
Φ∂
∂
Φ∂
(1.28)
Choose as solution Φ = f(x) ey
. Hence (1.28) reduces in
0)x(f
x
)x(f
2
x
)x(f
2
2
4
4
=++
∂
∂
∂
∂
(1.29)
Equation (1.29) is a homogeneous linear differential equation of order 4. The characteristic equation
writes:
01m2m 24
=++ (1.30)
and can be factored in:
0)im()im( 22
=−+ (1.31)
Based on (1.26) and (1.27), the solution of (1.29) becomes:
( )xsinDxcosCxxsinBxcosA)x(f +++=
and the solution of (1.28):
( )[ ]xsinDxcosCxxsinBxcosAe)y,x( y
+++=Φ (1.32)
1.4.3 Resolution by means of indicial equations
Consider the differential equation:
0wk
dx
wd 2
2
2
=+ (1.33)
Assume a solution in the form of a indicial series
L4
4
3
3
2
210 xaxaxaxaa)x(f ++++=
Apply (1.33)
THE LAPLACE EQUATION
7
0xakxakakxa.3.4xa.2.3a.1.2 2
2
2
1
2
0
22
432 =+++++ LL
This equation must be equal zero for all values of x. Hence the sum of the coefficients of each exponent
of x must be individually zero.
0aka.1.2 0
2
2 =+
0aka.2.3 1
2
3 =+
0aka.3.4 2
2
4 =+
…
First let a0 = 1 and a1 = 0
Hence
)!p2(
k
)(a
!4
k
a
!2
k
a
p2
p
p2
4
4
2
2 −==−= (1.34)
0aaa 1p231 LL ==== + (1.35)
The successive terms of (1.34) are the terms of the cosine series. Hence, the first solution of (1.33) is
f(x) = cos(kx). The second solution is obtained by setting a0 = 0 and a1 = k. Obviously, the second
solution of (1.33) is f(x) = sin(kx).
PAVEMENT DESIGN AND EVALUATION
8
THE GAMMA FUNCTION
9
Chapter 2 The Gamma Function
2.1 Introductory note
In chapter 1, we have defined the Bessel function of the first kind and of order zero as:
( )
!k!k
2/r
)()r(J
k2
k
0 ∑ −=
This equation can be generalised, for n an integer, in:
( )
)!nk(!k
2/r
)()r(J
nk2
k
n
+
−=
+
∑
When p is not an integer, the Bessel function of order p writes:
( )
)1pk(!k
2/r
)()r(J
pk2
k
p
++
−=
+
∑ Γ
where Γ(k+p+1) is called the gamma function of k+p+1.
For our purpose, the gamma function is essentially required to express Bessel functions of non-integer
order: it is the factorial function for non-integers. Observe that with these definitions Γ(n+1) = n!
2.2 Helpful relations
∫
∞
−−
=
o
x1p
dxex)p(Γ
)p(p)1p( ΓΓ =+
1)1( =Γ
!n)1n( =+Γ
πΓ =)2/1(
±∞=− )n(Γ






−
+++++−==
1n
1
...
3
1
2
1
1)n()n('
dn
)n(d
γΓΓ
Γ
5772157.0=γ (Euler’s constant)
PAVEMENT DESIGN AND EVALUATION
10
2.3 Definition of the Gamma Function
The gamma function is defined by:
∫
∞
−−
=
o
x1p
dxex)p(Γ (2.1)
which is convergent for p > 0.
Applying the definition
∫
∞
−
=+
o
xp
dxex)1p(Γ ∫
∞
−−∞−
−−−=
o
1px
o
xp
dx)px)(e(ex ∫
∞
−−
=
o
x1p
dxexp )p(pΓ=
one obtains the recurrence formula:
)p(p)1p( ΓΓ =+ (2.2)
By taking (2.1) as the definition of Γ(p) for p > 0, we can generalise the gamma function to p < 0 by use
of (2.2) in the form
p
)1p(
)p(
+
=
Γ
Γ (2.3)
This process is called analytic continuation.
Applying (2.1) we determine the value of Γ(1)
∫
∞ ∞−−
=−==
o
o
xx
1edxe)1(Γ (2.4)
Hence, applying (2.2)
1)1(1)2( == ΓΓ
!212)2(2)3( =•== ΓΓ
!3123)2(23)3(3)4( =••=•== ΓΓΓ
!n)1n( =+Γ
For n being a positive integer.
2.3. Values of gamma functions.
The values of Γ(p) for non-integer values of p must be computed numerically. One obtains for 1 ≤ p ≤ 2
p 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
Γ(p) 1.00 .951 .918 .898 .887 .886 .894 .909 .931 .962 1.00
Table 1 Values of Γ(p) for non-integer values of p
The value of Γ(p) can be computed for any value of p using (2.2)
Γ(4.3) = 3.3 Γ(3.3) = 3.3 * 2.3 * Γ(2.3) =
3.3 * 2.3 * 1.3 * Γ (1.3) = 3.3 * 2.3 * 1.3 * 0.898 = 8.861
THE GAMMA FUNCTION
11
2.4 Values of Γ(1/2) and Γ(-1/2).
Two particular values of the gamma function, Γ(1/2) and Γ(-1/2), appear in many applications. By
definition:
∫
∞
−−
=
o
x2/1
dxex)2/1(Γ
Write x =u2
and dx = 2udu
∫
∞
−
=
o
u
due2)2/1(
2
Γ
Then
[ ]
















= ∫∫
∞
−
∞
−
dve2due22/1(
o
v
o
u2 22
Γ ∫ ∫
∞∞
+−
=
o o
)vu(
dudve4
22
Write u = r cos θ, v = r sin θ, du dv = r dr dθ
[ ] ∫ ∫
∞
−
=
2/
o o
r2
rdrde4)2/1(
2
π
θΓ ∫=
2/
o
d2
π
θ π=
Hence
πΓ =)2/1( (2.5)
and by (2.3)
π
Γ
Γ 2
2/1
)2/1(
)2/1( −=−=− (2.6)
2.5. Values of Γ(n) for n = 0, -1, -2, ...
By definition
∫
∞ −
=
o
x
dx
x
e
)0(Γ
that we write
dx...
x!3
1
x!2
1
edx...
x!3
1
x!2
1
x
1
e)0(
o
32
x
32
o
x
∫∫
∞
−
∞
−






+−+





−+−=Γ
Derive






⋅⋅⋅+−−





⋅⋅⋅−−=





⋅⋅⋅− −−−
32
x
32
x
32
x
x!3
1
x!2
1
x
1
e
x!3
1
x!2
1
e
x!3
1
x!2
1
e
dx
d
Hence
dx...
x!3
1
x!2
1
e...
x!3
1
x!2
1
e)0(
32
o
x
32
x






+−−





+−−= ∫
∞
−−
Γ
PAVEMENT DESIGN AND EVALUATION
12
dx...
x!3
1
x!2
1
e
32
o
x






+−+ ∫
∞
−
∞
−






+−−=
o
32
x
...
x!3
1
x!2
1
e)0(Γ
∞
−






+−+−+−−=
o
32
x
x!1
1
1...
x!3
1
x!2
1
x!1
1
1e)0(Γ
∞=





+−−=
∞
−−
o
x/1x
x!1
1
1ee)0(Γ (2.7)
−∞=
−
=−
1
)0(
)1(
Γ
Γ (2.8)
±∞=− )n(Γ (2.9)
The gamma function is undefined when the value of the argument is zero or a negative integer.
2.6. Derivative of Γ(n).
We call Γ’(n) the derivative of Γ(n) with respect to n. For simplicity we consider only the case of n being
an integer, which besides is the only case required for our purpose.
By definition
∫
∞
−−
=
o
x1n
dxex
dn
d
)n('Γ ( )∫
∞
−−
=
o
x1n
dxex
dn
d
∫
∞
−−
=
o
x1n
dxexlogx
We first compute Γ’(1)
∫
∞
−
=
o
x
dxexlog)1('Γ
Integrating by parts
∫ ∫ −=−= xxlogxdx
x
x
xlogxxdxlog
∫ ∫ −=−=
2
x
xlog
2
x
dx
x
x
2
1
2
x
xdxlogx
2222
……………
∫ ∫ −=−= −−
2
nn
1n
n
1n
n
x
xlog
n
x
dxx
n
1
xlog
n
x
dxxlogx
Further
( )∫ ∫ ∫
−−−−
−+−= dxxedxexlogxexxlogxdxexlog xxxx
∫ ∫ ∫
−−−−
−+








−= dxex
!2.2
1
dxexlogx
!2
1
e
!2.2
x
xlog
!2
x
dxexlogx x2x2x
22
x
THE GAMMA FUNCTION
13
∫ ∫ ∫
−−−−
−+








−= dxex
!3.3
1
dxexlogx
!3
1
e
!3.3
x
xlog
!3
x
dxexlogx
!2
1 x3x3x
33
x2
...
∫ ∫ ∫
−−−−−
−+








−=
−
dxex
!n.n
1
dxexlogx
!n
1
e
!n.n
x
xlog
!n
x
dxexlogx
)!1n(
1 xnxnx
nn
x1n
Assembling:
∫
−−−








++++−








++++= x
n32
x
n32
x
e
!n.n
x
...
!3.3
x
!2.2
x
!1.1
x
exlog
!n
x
...
!3
x
!2
x
!1
x
dxexlog
∫ ∫ ∫ ∫
−−−−
−−−−− dxex
!n.n
1
...dxex
!3.3
1
dxex
!2.2
1
dxxe
!1.1
1 xnx3x2x
( )∫
−−−








++++−−= x
n32
xxx
e
!n.n
x
...
!3.3
x
!2.2
x
!1.1
x
exlog)1edxexlog
∫ ∫ ∫ ∫
−−−−
−−−−− dxex
!n.n
1
...dxex
!3.3
1
dxex
!2.2
1
dxxe
!1.1
1 xnx3x2x
Integrating between 0 and ∞ yields
∫
∞ ∞−∞−
−=
o
o
x
o
x
exlogxlogdxexlog
...
!3.3
)4(
!2.2
)3(
!1.1
)2(
e...
!3.3
x
!2.2
x
!1.1
x
o
x
32
−−−−








+++−
∞
− ΓΓΓ
( ) ( ) ...
3
1
2
1
100exloglimexloglimxloglimxloglimdxexlog
o
x
0x
x
x0xx
x
−−−−+−+−−=∫
∞
−
→
−
∞→→∞→
−
( ) ⋅⋅⋅−−−−−+= −
→
∞
∞→
−
∫ 3
1
2
1
11exloglimxloglimdxxelog x
0x
0
x
x
∫
∞
∞→
−
−=



−−−−−==
o
M
x
M
1
...
3
1
2
1
1Mloglimdxexlog)1(' γΓ (2.10)
where γ = 0.5772157 is called Euler’s constant.
By extent:
( ) ∫ ∫∫
∞ ∞
−−∞−−
∞
−+−=
o o
xx
o
xx
o
dxxedxexlogxexxlogxdxexlog
∫ ∫ ∫
∞ ∞ ∞
−−−
+=
o o o
xxx
dxxedxexlogdxexlogx
PAVEMENT DESIGN AND EVALUATION
14
∫
∞
−
+−==
o
x
)2(dxexlogx)2(' ΓγΓ






++−=
2
1
1)3(!2)3(' ΓγΓ
...






−
+++++−−=
1n
1
...
3
1
2
1
1)n()!1n()n(' ΓγΓ
1n
1
...
3
1
2
1
1
)n(
)n('
−
+++++−= γ
Γ
Γ
(2.11)
THE GENERAL SOLUTION OF THE BESSEL EQUATION
15
Chapter 3 The General Solution of the Bessel Equation.
3.1 Introductory note.
In the way trigonometric functions are solutions of Laplace equation in Cartesian co-ordinates, Bessel
functions are solutions of Laplace equation in polar or cylindrical co-ordinates. One could say that Bessel
functions is a three dimensional function whereas trigonometric functions are two dimensional functions.
The fundamental characteristics of both functions are identical:
- if f(x) = cos(x) )x(f
dx
)x(fd
2
2
−=
- if f(r) = J0(r) )r(f
dr
)r(df
r
1
dr
)r(fd
2
2
−=+
Bessel functions are essentially utilised in problems presenting an axial symmetry: in pavement
engineering, the common case of a circular load on a layered structure, i.e. nearly all applications of
chapters 12 to 26.
Jp and J-p are the two solutions of the first kind of Bessel’s equation
0t
r
p
dr
d
r
1
dr
d 2
22
2
=+−+ φ
φφφ
However, when p = n, Jn = (-)n
J-n.
In that case, Bessel’s equation requires another second solution, a solution of the second kind, Yp , which
will not be utilised in this book.
Ip and I-p are the two modified solutions of the first kind of Bessel’s modified equation:
0t
r
p
dr
d
r
1
dr
d 2
22
2
=−−+ φ
φφφ
These also, aren’t used in this book. However, they allow introducing the Ker and Kei functions,
solutions of:
0it
dr
d
r
1
dr
d 2
2
2
=++ φ
φφ
which is applied in the problem of a slab subjected to an isolated load (chapter 12.4) and in the
ovalisation test (chapter 26.4).
The modified form of Bessel’s equation:
0FrF
r
p
dr
dF
r
1
)21(
dr
Fd )1(222
2
222
2
2
=+
−
+−+ −γ
γβ
γα
α
permits us to illustrate the existence of relations between Bessel functions of order ½ and trigonometric
functions (chapter 4.3) Those relations allow us to establish asymptotic approximations for Bessel
functions (chapter 4.5), required for the numerical computations of functions of high arguments.
PAVEMENT DESIGN AND EVALUATION
16
3.2 Helpful relations
( )
)1kp(!k
2/rt
)()rt(J
k2p
0
k
p
++
−=
+∞
∑ Γ
( )
)1kp(!k
2/rt
)()rt(J
k2p
0
k
p
++−
−=
+−∞
− ∑ Γ
)rt(J)()rt(J n
n
n −=− for n integer
( )[ ] ∑
− −
−−
−+=
1n
0
nk2
nn
!k
)2/rt()!1kn(1
)2/rtlog()rt(J
2
)rt(Y
π
γ
π
)!kn(!k
)2/rt(
kn
1
...
3
1
2
1
1
1
k
1
...
3
1
2
1
1
1
)(
1 nk2
0
k
+





+
+++++++++−−
+∞
∑π
)rt(iY)rt(J)rt(H pp
)1(
p +=
)rt(iY)rt(J)rt(H pp
)2(
p −=
∑
∞ +
−
++
==
0
k2p
p
p
p
)1pk(!k
)2/rt(
)irt(Ji)rt(I
Γ
∑
−
−
−−
−=
1n
0
k2n
k
n
)2/rt(!k
)!1kn(
)(
2
1
)rt(K






++−+−
+
−+ ∑
∞ +
+
)1kn(
2
1
)1k(
2
1
)2/rtlog(
)!kn(!k
)2/rt(
)(
0
k2n
1n
ΨΨ
k
1
...
3
1
2
1
1
1
)1k()1( +++++−=+−= γΨγΨ
)rt(ibei)rt(ber)irt(I)irti(J 00 ±=±=±
∑∑
∞ +∞
++
−=−=
0
2k4
k
0
k4
k
)!1k2()!1k2(
)2/rt(
)()rt(bei
)!k2()!k2(
)2/rt(
)()rt(ber
)rt(kei)rtker()irt(K0 ±=±
[ ] ( ) ...
2
1
1
!2!2
2/rt
)rt(bei
4
2/rtlog()rt(ber)rtker(
4






+−++−=
π
γ
[ ] ( ) ( ) ...
3
1
2
1
1
!3!3
2/rt
!1!1
2/rt
)rt(bei
4
)2/rtlog()rt(bei)rt(kei
62






++−+−+−=
π
γ
THE GENERAL SOLUTION OF THE BESSEL EQUATION
17
3.3 Resolution of the Bessel equation (Bessel functions of the first kind)
Recall the definition of the Laplacian in cylinder co-ordinates given by § 1.5:
0
zr
1
rr
1
r 2
2
2
2
22
2
=+++
∂
Φ∂
∂θ
Φ∂
∂
Φ∂
∂
Φ∂
(3.1)
Consider next solution obtained by the method of separation of the variables
tz
e)pcos()r(F θΦ =
and apply it to (3.1) that transforms into:
0FtF
r
p
r
F
r
1
r
F 2
2
2
2
2
=+−+
∂
∂
∂
∂
(3.2)
The general solution of equation (3.2) can be found by the method of the indicial equations (Wayland,
1970). Therefore assume that the solution can be written as a series such as:
2n
0
nraF
++
∞
∑=
α
(3.3)
Hence
∑
∞
++
+=
0
1n
nra)n(
r
F α
α
∂
∂
(3.4)
∑
∞
+
−++=
0
n
2
2
r)1n)(n(
r
F α
αα
∂
∂
(3.5)
Replace the terms in (3.2) by (3,3), (3.4) and (3.5):
[ ]{ } 0ratrap)n()1n)(n(
0
2n
n
2n
n
2
=+−++−++∑
∞
+++ αα
ααα
Take out the first two terms of the first summation:
ra)1p)(1p(a)p(r 10
22
+++−+



− ααα
α
0ratra)pn)(pn(
0
2n
n
2
2
n
n =




+++−++ ∑∑
∞
+
∞
αα
If the terms in r0
and r1
are equal zero, the relation becomes homogeneous regarding the exponents of r.
Therefore choose a0 = arbitrary, a1 = 0 and α = ± p.
Rearrange the indices
[ ] 0rtaa)p22n)(2n(r
0
2n2
n2n =








+±++∑
∞
+
+
α
(3.6)
If (3.6) has to be identical zero whatever the value of m, each term of the summation has to be equal zero.
Hence the recurrence formula
2
n2n taa)p22n)(2n( −=±++ + (3.7)
and because a1 = 0, all the terms with an odd index are also equal zero.
Finally resulting in:
PAVEMENT DESIGN AND EVALUATION
18
2
0
2
0
2
0
2
2
t
)p1(1
a
)p1(22
ta
)p22(2
ta
a 





±×
−=
±×
−=
±
−=
4
0
2
2
4
2
t
)p2)(p1(21
a
)p24(4
ta
a 





±±×
=
±
−=
6
0
2
4
6
2
t
)p3)(p2)(p1(321
a
)p26(6
ta
a 





±±±××
−=
±
−=
..............
s2
0
s
s2
2
t
)ps)...(p2)(p1(!s
a)(
a 





±±±
−
=
If p is different from zero or not an integer, one obtains two linearly independent solutions
∑∑
∞ +−∞ +
++−
−+
++
−=
0
k2p
k
0
k2p
k
)1kp(!k
)rt(
)(B
)1kp(!k
)2/rt(
)(AF
ΓΓ
(3.8)
The corresponding series are called Bessel functions of the first kind and noted
∑
∞ +
++
−=
0
k2p
k
p
)1kp(!k
)2/rt(
)()rt(J
Γ
(3.9)
∑
∞ +−
−
++−
−=
0
k2p
k
p
)1kp(!k
)2/rt(
)()rt(J
Γ
(3.10)
The Bessel functions of the first kind and of order 0 and 1 are represented in Figure 3.1.
Figure 3.1 Bessel functions of the first kind and of order 0 and 1
THE GENERAL SOLUTION OF THE BESSEL EQUATION
19
3.4 Resolution of the Bessel equation for p an integer (Bessel functions of the second and third
kind)
3.4.1 For n integer, Jn = (-)n
J-n
When p is an integer, let say n, solutions (3.9) and (3.10) are not linearly independent any more. Indeed
when p = -n
∑ ∑∑
∞ ∞ +−− +−+−
−
+−
−+
+−
−=
+−
−=
0 n
k2n
k
1n
0
k2n
k
k2n
k
n
)!kn(!k
)2/rt(
)(
)!kn(!k
)2/rt(
)(
)!kn(!k
)2/rt(
)()rt(J
When 0, 1, 2, 3, ..., k = n-1, (-n+k)! = ± ∞ and
0
)!kn(!k
)2/rt( k2n
=
+−
+−
Write k = n + j
∑ ∑
∞ ∞ +
+
+−
−
+
−=
+−
−=
n 0
j2n
jn
k2n
k
n
!j)!jn(
)2/rt(
)(
)!kn(!k
)2/rt(
)()rt(J
∑
∞ +
− −=
+
−−=
0
n
n
j2n
jn
n )rt(J)(
)!jn(!j
)2/rt(
)()()rt(J
Hence the two solutions are not linearly independent.
3.4.2 Bessel functions of the second kind
In order to find a second linearly independent solution, we define the function
)psin(
)rt(J)pcos()rt(J
)rt(Y
pp
p
π
π −−
= (3.11)
which is valid for p not an integer.
Then we define for p = n
)psin(
)rt(J)pcos()rt(J
lim)rt(Y
pp
np
n
π
π −
→
−
=
We search the solution for n = 0.
)psin(
)rt(J)pcos()rt(J
lim)rt(Y
pp
0p
0
π
π −
→
−
=
Apply de L’Hospital’s rule
)pcos(
p
)rt(J
)psin()rt(J)pcos(
p
)rt(J
)rt(Y
p
p
p
0
ππ
∂
∂
πππ
∂
∂ −
−−
=






−=
−
p
)rt(J
p
)rt(J1
)rt(Y
pp
0
∂
∂
∂
∂
π
Compute the derivatives of Jp(rt) and J-p(rt):
∑
∞ +






++
++
−
++
−=
0
k2p
kp
)1kp(
)1kp('
)2/rtlog(
)1kp(!k
)2/rt(
)(
p
)rt(J
Γ
Γ
Γ∂
∂
PAVEMENT DESIGN AND EVALUATION
20
∑
∞ +−
−






++−
++−
+−
++−
−=
0
k2p
kp
)1kp(
)1kp('
)2/rtlog(
)1kp(!k
)2/rt(
)(
p
)rt(J
Γ
Γ
Γ∂
∂
and for p → 0
∑
∞






+
+
−−=
0
k2
k
0
)1k(
)1k('
)2/rtlog(
!k!k
)2/rt(
)(
2
)rt(Y
Γ
Γ
π
By (2.11)
k
1
...
3
1
2
1
1
1
)1k(
)1k('
+++++−=
+
+
γ
Γ
Γ
Hence
[ ]








−





+++





+−++= ...
3
1
2
1
1
!3!3
)2/rt(
2
1
1
!2!2
)2/rt(
!1!1
)2/rt(
)2/rtlog()rt(J
2
)rt(Y
642
00 γ
π
(3.12)
For n ≠ 0, (3.12) is extended to
[ ]{ } ∑
− −
−−
−+=
1n
0
nk2
nn
!k
)2/rt()!1kn(1
)2/rtlog()rt(J
2
)rt(Y
π
γ
π
(3.13)
∑
∞ +
+





+
+++++++++−−
0
nk2
k
)!kn(!k
)2/rt(
kn
1
...
3
1
2
1
1
1
k
1
...
3
1
2
1
1
1
)(
1
π
The functions Yp(rt) are called functions of the second kind.
The Bessel functions of the second kind and of order 0 and 1 are represented in Figure 3.2.
Figure 3.2 Bessel functions of the second kind and of order 0 and 1
THE GENERAL SOLUTION OF THE BESSEL EQUATION
21
3.4.3 Bessel functions of the third kind
Hankel introduced next pair of conjugate complex functions, with i = √(-1):
)rt(iY)rt(J)rt(H pp
)1(
p += (3.14)
)rt(iY)rt(J)rt(H pp
)2(
p −= (3.15)
These functions are called Hankel functions or Bessel functions of the third kind.
3.5 The modified Bessel equation.
If in § 3.3 we chose:
)tzcos()pcos()r(F θΦ =
as solution of the Laplace equation, equation (3.2) would modify into:
0FtF
r
p
r
F
r
1
r
F 2
2
2
2
2
=−−+
∂
∂
∂
∂
(3.16)
Equation (3.16) is called the modified Bessel equation.
Its solution can immediately be deduced from the original solution (3.8)
)irt(BJ)irt(AJ)rt(F pp −+= (3.17)
where
∑
∞ ++
++
−=
0
k2ppk2
k
p
)1kp(!k
)2/rt(i
)()irt(J
Γ
∑
∞ +
++
=
0
k2p
p
p
)1kp(!k
)2/rt(
i)irt(J
Γ
Hence we define the function Ip, modified Bessel function of the first kind, as a real function, solution of
the modified Bessel equation.
∑
∞ +
−
++
==
0
k2p
p
p
p
)1kp(!k
)2/rt(
)irt(JiI
Γ
(3.18)
pp BIAI)rt(F −+= (3.19)
It is easy to deduce from (3.18) that when p is an integer, let say p = n, In = I-n. In that case, the second
solution is usually defined by






−
−−
=
−
→ np
)rt(I)rt(I
2
)(
lim)rt(K
pp
n
np
n (3.20)
which is called the modified Bessel function of the second kind.
The function Kp(rt) is defined for unrestricted values of p by the equation
π
π
psin
)rt(I)rt(I
2
)rt(K
pp
p
−
=
−
(3.21)
Applying de L’Hospital’s rule on (3.20) and (3.21), one verifies that
)rt(Klim)rt(K p
np
n
→
=
In a similar way as given in § 3.4.2 one obtains:
PAVEMENT DESIGN AND EVALUATION
22
∑
−
−
−−
−=
1n
0
k2n
k
n
)2/rt(!k
)!1kn(
)(
2
1
)rt(K
(3.22)






++−+−
+
−+ ∑
∞ +
+
)1kn(
2
1
)1k(
2
1
)2/trln(
)!kn(!k
)2/rt(
)(
0
k2n
1n
ΨΨ
k
1
...
3
1
2
1
1
1
)1k()1( +++++−=+−= γΨγΨ
The modified Bessel functions of the first kind and of order 0 and 1 are represented in Figure 3.3.
Figure 3.3 Modified Bessel functions of the first kind of order 0 and 1
The modified Bessel functions of the second kind and of order 0 and 1 are given in Figure 3.4.
Figure 3.4 Modified Bessel functions of the second kind and of order 0 and 1
THE GENERAL SOLUTION OF THE BESSEL EQUATION
23
3.6 The ber and bei functions.
Consider next modified Bessel equation:
0Ft)i(
r
F
r
1
r
F 2
2
2
=±−+
∂
∂
∂
∂
(3.23)
which solutions of the first kind are
...
!4!4
)2/rt(
!3!3
)2/rt(i
!2!2
)2/rt(
!1!1
)2/rt(i
1)irt(I
8642
0 +−−+=
...
!4!4
)2/rt(
!3!3
)2/rt(i
!2!2
)2/rt(
!1!1
)2/rt(i
1)irt(I
8642
0 ++−−=−
We define:
)rt(ibei)rt(ber)irt(I0 ±=± (3.24)
where
...
!6!6
)2/rt(
!4!4
)2/rt(
!2!2
)2/rt(
1)rt(ber
1284
+−+−= (3.25)
...
!5!5
)2/rt(
!3!3
)2/rt(
!1!1
)2/rt(
)rt(bei
1062
++−= (3.26)
Notice that:
( ) ( ) ( ) ( ) ( ) ( )
2
irtJirtJ
2
irtiJirtiJ
2
irtIirtI
)rt(ber 000000 +−
=
−+
=
−+
= (3.27)
( ) ( ) ( ) ( ) ( ) ( )
i2
irtJirtJ
i2
irtiJirtiJ
i2
irtIirtI
)rt(bei 000000 −−
=
−−
=
−−
= (3.28)
The ber and bei functions can be depicted in Figure 3.5
Figure 3.5 The ber and bei functions
PAVEMENT DESIGN AND EVALUATION
24
3.7 The ker and kei functions.
Again consider equation (3.23), which solutions of the second kind are:
( ) ( ) ...
3
1
2
1
1
!3!3
2
irt
2
1
1
!2!2
2
irt
!1!1
2
irt
2
irt
logirtIirtK
642
00 





++








+





+








+








+





+−= γ
Recall that:
4
i
2
rt
loge
2
1
2
rt
logilog
2
1
2
rt
logilog
2
rt
log
2
irt
log 2/i ππ
+=+=+=+=
and applying (3.24)
( ) [ ] ...
3
1
2
1
1
!3!3
2
rt
i
2
1
1
!2!2
2
rt
!1!1
2
rt
i
4
i
2
rt
log)rt(ibei)rt(berirtK
642
0 





++






−





+






−






+



+++−=
π
γ
( ) [ ] ...
3
1
2
1
1
!3!3
2
rt
i
2
1
1
!2!2
2
rt
!1!1
2
rt
i
4
i
2
rt
log)rt(ibei)rt(berirtK
642
0 





++






+





+






−






−





−+−−=−
π
γ
We define:
( ) ( )
2
irtKirtK
)rtker( 00 −+
= (3.29)
( ) ( )
i2
irtKirtK
)rt(kei 00 −−
= (3.30)
[ ] ( ) ...
2
1
1
!2!2
2/rt
)rt(bei
4
)2/rtlog()rt(ber)rtker(
4






+−++−=
π
γ (3.31)
[ ] ( ) ( ) ...
3
1
2
1
1
!3!3
2/rt
!1!1
2/rt
)rt(ber
4
)2/rtlog()rt(bei)rt(kei
62






++−+−+−=
π
γ (3.32)
The ker and kei functions are given in figure 3.6.
Figure 3.6 The ker and kei functions
THE GENERAL SOLUTION OF THE BESSEL EQUATION
25
3.8 Resolution of the equation ∇2
∇2
w + w =0
Consider the next equation:
0w
dr
dw
r
1
dr
wd
dr
d
r
1
dr
d
2
2
2
2
=+








+








+ (3.33)
The solution can be obtained by splitting (3.33) into two simultaneous differential equations
z
dr
dw
r
1
dr
wd
2
2
=+ (3.34)
w
dr
dz
r
1
dr
zd
2
2
−=+ (3.35)
Both equations are verified together if iwz m= . Indeed, if, for example, z = iw
(3.34) becomes 0iw
dr
dw
r
1
dr
wd
2
2
=−+ , and
(3.35) becomes 0w
dr
dw
r
1
i
dr
wd
i
2
2
=++ or 0iw
dr
dw
r
1
dr
wd
2
2
=−+
Hence the solution of (3.33) is given by the solution of
0w)i(
dr
dw
r
1
dr
wd
2
2
=±−+ (3.36)
The equations (3.25), (3.26), (3.31) and (3.32) give the solution of (3.36), for its two signs
)r(Dkei)rker(C)r(Bbei)r(Aberw +++= (3.37)
3.9 The modified form of the Bessel equation
Consider the next differential equation:
0FrF
r
p
r
F
r
1
)21(
r
F )1(222
2
222
2
2
=+
−
+−+ −γ
γβ
γα
∂
∂
α
∂
∂
(3.38)
which has for solutions, as can be verified by substitution,
)r(JBr)r(JAr)r(F pp
γαγα
ββ −+= (3.39)
or, if p is an integer,
)r(YBr)r(JAr)r(F pp
γαγα
ββ += (3.40)
An interesting application of (3.38) is the resolution of the Laplace equation in two- dimensional
Cartesian co-ordinates
0
yx 2
2
2
2
2
=+=∇
∂
Φ∂
∂
Φ∂
Φ
Consider a solution such as Φ = F(x)ety
PAVEMENT DESIGN AND EVALUATION
26
Hence
0e)x(Ft
x
)x(F ty2
2
2
2
=








+=∇
∂
∂
Φ (3.41)
Comparing (3.41) with (3.38) yields:
t12/1 === βγα
2/1p0p 222
==− γα
Hence
)tx(JBx)tx(JAx)x(F 2/1
2/1
2/1
2/1
−+= (3.42)
However
)txsin(B)txcos(A)x(F += (3.43)
is also a solution of (3.41). Therefore, one must conclude that there exists a relation between Bessel
functions of order ± 1/2 and trigonometric functions.
PROPERTIES OF THE BESSEL FUNCTIONS
27
Chapter 4 Properties of the Bessel Functions.
4.1 Introductory note
This chapter deals with the most important properties of Bessel functions: derivatives, functions of half
order, asymptotic values, indefinite integrals and relations between functions of different kind.
4.2 Helpful relations
[ ] )rt(J)rt(t)rt(J)rt(
dr
d
1p
p
p
p
−=
[ ] )rt(J)rt(t)rt(J)rt(
dr
d
1p
p
p
p
+
−−
−=
)rt(J
r
p
)rt(tJ)rt(J
dr
d
p1pp −= −
)rt(J
r
p
)rt(tJ)rt(J
dr
d
p1pp +−= +
)rt(J
rt
p2
)rt(J)rt(J p1p1p =+ +−
t
)rt(rJ
dr)rt(rJ 1
0
=∫
∫ −=
t
)rt(J
dr)rt(J 0
1
)rtsin(
rt
2
)rt(J 2/1
π
=
)rtcos(
rt
2
)rt(J 2/1
π
=−
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
−=
π
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
− +=
π
rt
2/12/1 e
rt2
KK −
− ==
π






−−≅
2
p
4
rtcos
rt
2
)rt(J p
ππ
π
for high values of rt






++≅−
2
p
4
rtsin
rt
2
)rt(J p
ππ
π
for high values of rt






−−≅
2
p
4
rtsin
rt
2
)rt(Yp
ππ
π
for high values of rt
PAVEMENT DESIGN AND EVALUATION
28






++≅−
2
p
4
rtcos
rt
2
)rt(Y p
ππ
π
for high values of rt
)4/2/prt(i)1(
p e
rt
2
H ππ
π
−−
≅ for high values of rt
)4/2/prt(i)2(
p e
rt
2
H ππ
π
−−−
≅ for high values of rt
( )i)2/1p(rtrt
p ee
rt2
1
)rt(I π
π
++−
+≅ for high values of rt






−≅
82
rt
cos
rt2
e
)rt(ber
2/rt
π
π
for high values of rt






−≅
82
rt
sin
rt2
e
)rt(bei
2/rt
π
π
for high values of rt






+≅ −
82
rt
cose
rt2
)rtker( 2/rt ππ
for high values of rt






+−≅ −
82
rt
sine
rt2
)rt(kei 2/rt ππ
for high values of rt
( ) )z(JezeJ)z(J p
pii
pp ±±± ==− ππ
( ) )z(JezeJ p
pmiim
p ±± = ππ
2
)rt(H)rt(H
)rt(J
)2(
p
)1(
p
p
+
=
)psin(
)pcos()rt(Y)rt(Y
)rt(J
pp
p
π
π−
=
−
)psin(i
e)rt(J)rt(J
)rt(H
ip
pp)1(
p
π
π−
− −
=
)psin(i
e)rt(J)rt(J
)rt(H
ip
pp)2(
p
π
π
−
−
=
−
)rt(J
)psin(
)mpsin(
e2)rt(He)rte(H p
ip)1(
p
impim)1(
p
π
ππππ −−
−=
)rt(J
)psin(
)mpsin(
e2)rt(He)rte(H p
ip)2(
p
impim)2(
p
π
ππππ
+= −
)()( )2/2/ ππ i
p
pi
p rteJirtI −
=
)rti(He
2
i)rt(K )1(
p
2/pi
p
ππ
=
)rti(He
2
i)rt(K )2(
p
2/pi
p
ππ
−=
PROPERTIES OF THE BESSEL FUNCTIONS
29
4.3 Derivatives of Bessel functions
4.3.1 Derivative of (rt)p
Jp(rt)
By definition of the Bessel function
[ ]








++
−= ∑
∞
+
+
0
k2p
k2p2
k
p
p
)1kp(!k2
)rt(
)(
dr
d
)rt(J)rt(
dr
d
Γ
∑
∞
+
−+
++
+
−=
0
k2p
1k2p2
k
)1kp(!k2
t)rt)(kp(2
)(
Γ
)kp(!k2
t)rt(
)(
1k2p
1k2p2
0
k
+
−=
−+
−+∞
∑
Γ
[ ]
∑
∞
+−
+−
++−
−=
0
k2)1p(
k2)1p(
kp
1k)1p(!k2
)rt(
)()rt(t
Γ
[ ] )rt(J)rt(t)rt(J)rt(
dr
d
1p
p
p
p
−= (4.1)
4.3.2 Derivative of (rt)-pJp(rt)
One finds similarly
[ ] )rt(J)rt(t)rt(J)rt(
dr
d
1p
p
p
p
+
−−
−= (4.2)
4.3.3 Derivative of Jp(rt)
Deriving the left member of (4.1) by parts yields
[ ] )rt(J
r
p
)rt(tJ)rt(J
dr
d
p1pp −= − (4.3)
deriving the left member of (4.2) by parts yields
[ ] )rt(J
r
p
)rt(tJ)rt(J
dr
d
p1pp +−= + (4.4)
Particularly
[ ] )rt(tJ)rt(tJ)rt(J
dr
d
110 −== − (4.5)
Adding (4.3) and (4.4) yields
[ ] [ ])rt(J)rt(J
2
t
)rt(J
dr
d
1p1pp +− −= (4.6)
and subtracting yields the recurrence formula for Bessel functions
)rt(J
rt
p2
)rt(J)rt(J p1p1p =+ +− (4.7)
PAVEMENT DESIGN AND EVALUATION
30
4.4 Bessel functions of half order
4.4.1 Values of J1/2(rt), J-1/2(rt), J3/2(rt), J-3/2(rt)
Expanding the Bessel functions in their series and recalling that Γ(1/2) = √π yields:
)rtsin(
rt
2
)rt(J 2/1
π
= (4.8)
)rtcos(
rt
2
)rt(J 2/1
π
=− (4.9)






−= )rtcos(
rt
)rtsin(
rt
2
)rt(J 2/3
π
(4.10)






+−=− )rtsin(
rt
)rtcos(
rt
2
)rt(J 2/3
π
(4.11)
4.3.2. Values of I1/2(rt), I-1/2(rt)
Expanding the Bessel functions in their series yields
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
−=
π
(4.12)
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
− +=
π
(4.13)
4.3.3. Values of K1/2(rt), K-1/2(rt)
By definition (3.21)
rt2/12/1
2/1 e
rt2)2/sin(
)rt(I)rt(I
2
)rt(K −− =
−
=
π
π
π
(4.14)
rt2/12/1
2/1 e
rt2)2/sin(
)rt(I)rt(I
2
)rt(K −−
− =
−
=
π
π
π
(4.15)
Hence
)rt(K)rt(K 2/12/1 −= (4.16)
4.4. Values of K0(rt) and Kn(rt)
By definition (3.21)
π
π
psin
)rt(I)rt(I
2
lim)rt(K
pp
0p
0
−
=
−
→
Applying de l’Hospital’s rule yields and omitting the lim sign






−=
−
=
−
−
dp
)rt(dI
dp
)rt(dI
2
1
pcos
dp
)rt(dI
dp
)rt(dI
2
)rt(K
pp
pp
0
ππ
π
PROPERTIES OF THE BESSEL FUNCTIONS
31








++
−
+−
= ∑∑
+−
)1pk(!k
)2/rt(
dp
d
)1pk(!k
)2/rt(
dp
d
2
1
)rt(K
pk2pk2
0
ΓΓ




+−+−
+−++−−
=
−−
∑ )1pk()1pk(
)1pk(')2/rt()1pk()2/rtlog()2/rt(
!k2
)2/rt(
)rt(K
ppk2
0
ΓΓ
ΓΓ




++++
++−++
−
)1pk()1pk(
)1pk(')2/rt()1pk()2/rtlog()2/rt( pp
ΓΓ
ΓΓ
Letting p → 0






+
+
+−= ∑ )1k(
)1k('
)2/rtlog(
!k!k
)2/rt(
)rt(K
k2
0
Γ
Γ
Developing Γ’(k+1) as in § 3.3.2 yields
[ ] 





++++−=
2
1
1
!2!2
)2/rt(
!1!1
)2/rt(
)2/rtlog()rt(I)rt(K
42
00 γ
⋅⋅⋅





+++
3
1
2
1
1
!3!3
)2/rt( 6
(4.17)
For n ≠ 0, (4.17) is relatively easily extended to
[ ] ∑ −+ −−−
++−=
n
0
nk2
k
n
1n
n )2/rt(
!k
)!1kn()(
2
1
)2/rtlog()rt(I)()rt(K γ
[ ]∑
∞ +
+
++−+
0
nk2
n
)!nk(!k
)2/rt(
)kn()k()(
2
1
ΦΦ (4.18)
4.5 Asymptotic values
4.5.1 Asymptotic values for Jp and J-p
Transform Bessel equation (3.2):
0FtF
r
p
r
F
r
1
r
F 2
2
2
2
2
=+−+
∂
∂
∂
∂
by writing F(rt) = G(rt)/(rt)1/2
0G
r
4/1p
t
r
G
2
2
2
2
2
=







 −
−+
∂
∂
(4.19)
When p = ½, the equation reduces to
0Gt
r
G 2
2
2
=+
∂
∂
(4.20)
whose general solution is given by
)rtsin(B)rtcos(AG +=
Hence
)rtsin()rt(B)rtcos()rt(AF 2/12/1 −−
+= (4.21)
PAVEMENT DESIGN AND EVALUATION
32
but also:
)rt(DJ)rt(CJF 2/12/1 += − (4.22)
Equations (4.21) and (4.22) confirm, as we knew by (4.8) and (4.9), that there exists a relation between
the Bessel functions of half order and the trigonometric functions, and that this relation is valid for all
values of the argument, thus in particular for high values of the argument. Hence the equations:
)rtsin(
rt
2
)rt(J 2/1
π
=
)rtcos(
rt
2
)rt(J 2/1
π
=−
can be considered as the asymptotic equations for the Bessel functions of half order.
Thus (4.19) must have, for p not being an integer, two approximate values for high values of the argument
such as
)rtcos()rt(A)rt(J p
2/1
pp α+≅ −
(4.23)
)rtsin()rt(B)rt(J p
2/1
pp β+≅ −
− (4.24)
The coefficients αp and βp must be determined in such a way that equations (4.23) and (4.24) are linearly
independent and compatible with the definitions of Jp and J-p.
Derive, with respect to r, Jp in (4.23)
)rtsin(t)rt(A)rtcos()rt(A
2
t
)rt(J p
2/1
pp
2/3
p
'
p αα +−+−= −−
For high values of the argument the first term can be neglected against the second
)rtsin(t)rt(A)rt(J p
2/1
p
'
p α+−≅ −
Further (4.3) and (4.4) simplify for high values of the argument
)rt(tJ)rt(J 1p
'
p −≅
)rt(tJ)rt(J 1p
'
p +−≅
Hence
)rtcos()rt(tA)rtsin(t)rt(A 1p
2/1
1pp
2/1
p −
−
−
−
+≅+− αα (4.25)
)rtcos()rt(tA)rtsin(t)rt(A 1p
2/1
1pp
2/1
p +
−
+
−
+−≅+− αα (4.26)
If we choose Ap = Ap-1 = Ap+1 = A and αp = k - pπ/2 in such a way that A and k are independent from p ,
we notice that equations (4.25) and (4.26) are satisfied. Indeed





 −
−+=





−+−
2
)1p(
krtcosA
2
p
krtsinA
ππ
Then






−+≅
−
2
p
krtcos)rt(A)rt(J
2/1
p
π
This equation must be satisfied for all values of p, thus also for p=1/2 for which






−−==
44
rtcos
rt
2
)rtsin(
rt
2
)rt(J 2/1
ππ
ππ
Hence A = (2π)1/2
and k = - π/4
PROPERTIES OF THE BESSEL FUNCTIONS
33
Finally






−−≅
2
p
4
rtcos
rt
2
)rt(J p
ππ
π
(4.27)
Replacing p by –p in (4.27) yields






++≅−
2
p
4
rtsin
rt
2
)rt(J p
ππ
π
(4.28)
4.5.2 Asymptotic values for Yp and Y-p
Since Yp and Y-p are the paired second solutions with Jp and J-p we shall admit implicitly the following
asymptotic equations






−−≅
2
p
4
rtsin
rt
2
)rt(Yp
ππ
π
(4.29)






++≅−
2
p
4
rtcos
rt
2
)rt(Y p
ππ
π
(4.30)
4.5.3 Asymptotic values for Hp
(1)
and Hp
(2)
By application of the definitions (3.14) and (3.15), one immediately finds:
)2/p4/rt(i)1(
p e
rt
2
H ππ
π
−−
= (4.31)
)2/p4/rt(i)2(
p e
rt
2
H ππ
π
−−−
= (4.32)
4.5.4 Asymptotic values for Ip and I-p
Transform Bessel equation (3.16)
0FtF
r
p
r
F
r
1
r
F 2
2
2
2
2
=−−+
∂
∂
∂
∂
by writing F(rt) = G(rt)/(rt)1/2
0G
r
4/1p
t
r
G
2
2
2
2
2
=







 −
+−
∂
∂
(4.33)
When p = ½, the equation reduces to
0Gt
r
G 2
2
2
=−
∂
∂
(4.34)
whose general solution is given by
rtrt
BeAeG −
+=
Hence
rt2/1rt2/1
e)rt(Be)rt(AF −−−
+= (4.35)
but also
PAVEMENT DESIGN AND EVALUATION
34
)rt(DI)rt(CIF 2/12/1 −+= (4.36)
Equation (4.35) is valid for all values of the argument, thus in particular for high values of the argument.
Hence the equations:
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
−=
π
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
− +=
π
can be considered as the asymptotic equations for the modified Bessel functions of half order.
Hence (4.33) must have, for p not an integer, two approximate values for high values of the argument
such as






+=
++−++− )2/1p(rt)2/1p(rt2/1
pp ee)rt(A)rt(I
βα
(4.37)






+=
+−+−+−+−
−
)2/1p(rt)2/1p(rt2/1
pp ee)rt(B)rt(I
βα
(4.38)
The coefficients α and β must be determined in such a way that equations (4.37) and (4.38) are linearly
independent and compatible with the definitions of Ip and I-p.
Derive, with respect to r, Ip in (4.37)
[ ])2/1p(rt)2/1p(rt2/3
p
'
p ee)rt(A
2
t
)rt(I ++−++−
+−= βα






−+
++−++− )2/1p(rt)2/1p(rt2/1
p eet)rt(A
βα
For high values of the argument the first term can be neglected against the second






−≅
++−++− )2/1p(rt)2/1p(rt2/1
pp eet)rt(A)rt('I
βα
Further simplify the derivatives of Ip(rt) for high values of the argument
)rt(tJ)rt(I 1p
'
p −≅
)rt(tJ)rt(I 1p
'
p +≅
Hence






−
++−++− )2/1p(rt)2/1p(rt2/1
p eet)rt(A
βα






+=
−+−−+−
−
)2/1p(rt)2/1p(rt2/1
1p eet)rt(A
βα
(4.39)






−
++−++− )2/1p(rt)2/1p(rt2/1
p eet)rt(A
βα






+=
++−++−
+
)2/3p(rt)2/3p(rt2/1
1p eet)rt(A
βα
(4.40)
If we choose Ap = Ap-1 = Ap+1 = A and α = 0, β = πi in such a way that A and β are independent from p ,
we notice that equations (4.39) and (4.40) are satisfied.
PROPERTIES OF THE BESSEL FUNCTIONS
35






−
++−− i)2/1p(rtrt2/1
ee)rt(A
π






+=
−+−− i)2/1p(rtrt2/1
ee)rt(A
π
Hence






+≅
++−− i)2/1p(rtrt2/1
p ee)rt(A)rt(I
π
This equation must be satisfied for all values of p, thus also for p=1/2 for which
[ ]rtrt
2/1 ee
rt2
1
)rt(I −
−=
π
Hence A = (2π)1/2
Finally






+≅
++− i)2/1p(rtrt
p ee
rt2
1
)rt(I
π
π
(4.41)
Replacing p by –p in (4.41) yields






+≅
+−+−
−
i)2/1p(rtrt
p ee
rt2
1
)rt(I
π
π
(4.42)
4.5.5 Asymptotic value for Kn
Consider definition (3.20):
np
)rt(I)rt(I
2
)(
lim)rt(K
pp
n
np
n
−
−−
=
−
→
(4.43)
For large values of the argument
np
eeee
rt2
1
2
)(
)rt(K
i)2/1p(rtrti)2/1p(rtrtn
n
−
−−+−
≅
++−−−− ππ
π
np
eeie
rt2
1
2
)(
)rt(K
ipiprt
n
n
−






−
−
≅
−− ππ
π
Applying de l’Hospital’s rule yields
rt
n
n e)ncos(
rt2
2
2
)(
)rt(K −−
≅ ππ
π
rt
n e
rt2
)rt(K −
≅
π
(4.44)
In this form Kn(rt) may be generalised into
rt
p e
rt2
)rt(K −
≅
π
(4.45)
where p not an integer and also
PAVEMENT DESIGN AND EVALUATION
36
rt
0 e
rt2
)rt(K −
≅
π
(4.46)
4.5.6 Asymptotic values for ber and bei
For high values of the argument (4.27) yields
( ) 





−





≅
4
irtcos
irt
2
irtJ
2/1
0
π
π
With (i)-1/4
= e-iπ/8
and √i = (1 + i)/√2
2
eeee
4
irtcos
2/rt)4/2/rt(i2/rt)4/2/rt(i



 +
=





−
−−−− ππ
π
2
ee
4
irtcos
2/rt)4/2/rt(i π
π −−
≅





−
( ) )8/2rt(i
2/rt
0 e
rt2
e
irtJ π
π
−−
≅
( ) 











−−





−≅
82
rt
sini
82
rt
cos
rt2
e
irtJ
2/rt
0
ππ
π
(4.47)
Similarly
( ) 











−+





−≅−
82
rt
sini
82
rt
cos
rt2
e
irtJ
2/rt
0
ππ
π
(4.48)
Hence
( ) ( ) 





−=
+−
≅
82
rt
cos
rt2
e
2
irtJirtJ
)rt(ber
2/rt
00 π
π
(4.49)
( ) ( ) 





−=
−−
≅
82
rt
sin
rt2
e
i2
irtJirtJ
)rt(bei
2/rt
00 π
π
(4.50)
4.5.7 Asymptotic values for ker and kei
Recall equations (3.29) and (3.30)
( ) ( )
2
irtKirtK
)rtker( 00 −+
= (4.51)
( ) ( )
i2
irtKirtK
)rt(kei 00 −−
= (4.52)
Apply (4.46)
irt
0 e
irt2
)irt(K −
=
π
PROPERTIES OF THE BESSEL FUNCTIONS
37
2/)i1(rt8/i
ee
rt2
+−−
= ππ
)8/2/rt(i2/rt
ee
rt2
ππ +−−
=












+−





+= −
82
rt
sini
82
rt
cose
rt2
)irt(K 2/rt
0
πππ
(4.53)
Similarly obtain












++





+=− −
82
rt
sini
82
rt
cose
rt2
)irt(K 2/rt
0
πππ
(4.54)
Adding and subtracting (4.53) and (4.54) yields






+= −
82
rt
cose
rt2
)rtker( 2/rt ππ
(4.55)






+−= −
82
rt
sine
rt2
)rt(kei 2/rt ππ
(4.56)
4.6 Indefinite integrals of Bessel functions
4.6.1 Fundamental relations
In § 4.2.we have derived the following derivatives of Bessel functions:
[ ] )rt(J)rt(t)rt(rtJ
dr
d
01 =
[ ] )rt(tJ)rt(J
dr
d
10 −=
From those equations we easily deduce next fundamental integrals
∫ =
t
)rt(rJ
dr)rt(rJ 1
0 (4.57)
∫ −=
t
)rt(J
dr)rt(J 0
1 (4.58)
4.6.2 The integral ∫rn
J0(rt)dr
Integrating by parts solves the integral:
∫ ∫
−−
−= dr)rt(Jr
t
)1n(
t
)rt(J
rdr)rt(Jr 1
1n1n
0
n
∫ ∫
−
−
− −
+−= dr)rt(Jr
t
)1n(
)rt(J
t
r
dr)rt(Jr 0
2n
0
1n
1
1n
Hence
∫ ∫
−− −
−
−
+= dr)rt(Jr
t
)1n(
)rt(Jr
t
)1n(
t
)rt(J
rdr)rt(Jr 0
2n
2
2
0
1n
2
1n
0
n
(4.59)
If n is odd, formula (4.59) leads to (4.57). If n is even, formula (4.59) leads to ∫J0(rt)dr, which is
tabulated.
PAVEMENT DESIGN AND EVALUATION
38
4.7 Relations between Bessel functions of different kind
4.7.1 Bessel functions with argument –rt
Since Bessel’s equation is unaltered if rt is replaced by –rt, we must expect the functions J±p(-rt) to be
solutions of the equations satisfied by J±p(rt). Considering the relation eπi
= -1, we may write
)rte(J)rt(J i
pp
π
=−
We can even consider the more general function Jp(emπi
rt) where m is an integer.
( )∑ ++
−=
+
)1pk(!k
2/rte
)()rte(J
k2pim
kim
p
Γ
π
π
Restricting the complex exponent to its principal value we get
( ) pim)mk2pm(ik2pim
eee πππ
== ++
Hence
( ) )rt(Je
)1pk(!k
)2/rt(
)(erteJ p
pim
k2p
kpimim
p
πππ
Γ
∑ =
++
−=
+
(4.60)
4.7.2 Relations between the three kinds of Bessel functions
Consider (3.14) and (3.15)
)rt(iY)rt(J)rt(H pp
)1(
p +=
)rt(iY)rt(J)rt(H pp
)2(
p −=
Hence by multiplying Yp(rt) by cos(pπ) and subtracting from Y-p(rt)
2
)rt(H)rt(H
)rt(J
)2(
p
)1(
p
p
+
= (4.61)
Consider (3.11)
)psin(
)rt(J)pcos()rt(J
)rt(Y
pp
p
π
π −−
=
Replace p by –p
)psin(
)rt(J)pcos()rt(J
)rt(Y
pp
p
π
π +−
=
−
−
Hence by subtracting and dividing by 2
)psin(
)pcos()rt(Y)rt(Y
)rt(J
pp
p
π
π−
=
−
(4.62)
Consider (3.14) together with (3.11)
)rt(iY)rt(J)rt(H pp
)1(
p +=
)psin(
)rt(J)pcos()rt(J
i)rt(J)rt(H
pp
p
)1(
p
π
π −−
+=
PROPERTIES OF THE BESSEL FUNCTIONS
39
[ ]
)psin(i
)rt(J)psin(i)pcos()rt(J
)rt(H
pp)1(
p
π
ππ −+−−
=
)psin(i
e)rt(J)rt(J
)rt(H
ip
pp)1(
p
π
π−
− −
= (4.63)
and similarly
)psin(i
e)rt(J)rt(J
)rt(H
ip
pp)2(
p
π
π
−
−
=
−
(4.64)
Add equations (4.63) and (4.64) together
2
HH
)rt(J
)2(
p
)1(
p
p
+
= (4.65)
Multiply equation (4.63) by eipπ
and (4.64) by e-ipπ
and add together
2
e)rt(He)rt(H
)rt(J
ip)2(
p
ip)1(
p
p
ππ −
−
+
= (4.66)
In equation (4.63) replace rt by rt emπI
)psin(i
e)rte(J)rte(J
)rte(H
ipim
p
im
pim)1(
p
π
πππ
π
−
− −
=
together with equation (4.60)
)sin(
)()(
)( ))1(
π
πππ
π
pi
ertJertJe
rteH
ip
p
imp
p
imp
im
p
−
−
−
−
=
( ) ( )
)psin(i
eee)rt(J
)psin(i
e)rt(J)rt(Je
)rte(H
impimpip
p
ip
pp
imp
im)1(
p
ππ
πππππ
π −
+
−
=
−−−
−
−
applying equation (4.63)
)rt(J
)psin(
)mpsin(
e2)rt(He)rte(H p
ip)1(
p
impim)1(
p
π
ππππ −−
−= (4.67)
and similarly
)rt(J
)psin(
)mpsin(
e2)rt(He)rte(H p
ip)2(
p
impim)2(
p
π
ππππ
+= −
(4.68)
4.7.3 Bessel functions of purely imaginary argument
Consider (3.18)
∑
∞ +
−
++
==
0
k2p
p
p
p
)1kp(!k
)2/rt(
)irt(Ji)rt(I
Γ
Recall that eπi/2
= i, e3πi/2
= -i.
Hence
)irt(Je)rt(I p
2/ip
p
π−
= (4.69)
Apply (3.21)
PAVEMENT DESIGN AND EVALUATION
40
π
π
psin
)rt(I)rt(I
2
)rt(K
pp
p
−
=
−
π
π
ππ
psin
)irt(Je)irt(Je
2
)rt(K
p
2/ip
p
2/ip
p
−
− −
=
π
π
π
π
psini
)irt(Je)irt(J
e
2
i)rt(K
p
ip
p2/ip
p
−
− −
=
Hence by (4.63)
)irt(He
2
i)rt(K )1(
p
2/ip
p
ππ
= (4.70)
and similarly
)irt(He
2
i)rt(K )2(
p
2/ip
p
ππ
−= (4.71)
Pavement design and evaluation
Pavement design and evaluation
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Pavement design and evaluation

  • 1. i PAVEMENT DESIGN AND EVALUATION THE REQUIRED MATHEMATICS AND ITS APPLICATIONS F. Van Cauwelaert Editor: Marc Stet Federation of the Belgian Cement Industry B-1170 Brussels, Rue Volta 9.
  • 2. PAVEMENT DESIGN AND EVALUATION ii
  • 3. INTRODUCTION iii Pavement Design and Evaluation: The Required Mathematics and Its Applications Keywords: textbook for pavement engineers, high order mathematical solutions for, rigid and flexible pavement, mechanistic methods, practical applications in the field of pavement engineering Preface This book is intended for Civil Engineers and more specific for Pavement Engineers, who are interested in the more advanced field of pavement engineering made available through the theory of high mathematics. In my extensive carrier as a Civil Engineer by profession, I noticed that some of my colleagues feel uneasy when it comes to high-level theoretical and computational work. Perhaps this is because of the fact that as soon as they graduated, they are confronted with many and important practical problems where derivatives and integrals do not appear very useful at fist sight. However, at many occasions, conferences, seminars and other meetings, I realised that Civil Engineers remain excited about the mathematical fundamentals of their art. Slow but sure grew the idea of writing a textbook on High Mathematics conceivable for practising Pavement Engineers. The primary objective of this book is to serve two purposes: first, to introduce the basic principles which must be known by people dealing with pavements; and secondly to present the theories and methods in pavement design and evaluation that may be used by students, designers engineering consultants, highway and airport agencies, and researchers at universities. In addition, some of the new concepts developed in recent years to improve the methods of pavement systems are explained. This book is written in a relatively simple way so that it may be followed by people familiar with basic engineering courses in mathematics and pavement design. This book consists of 26 chapters and is divided into two parts. The first part, which include chapters 1-9, covers the mathematics required by most of the problems related to pavement engineering. The assumed mathematical knowledge is that of high school level plus some basic elements of trigonometry and analysis. The second part, which includes chapters 10-26, is concerned with practical solutions as faced by Pavement Engineers in the assessment of rigid and flexible pavements. The rigorous mathematical solutions are presented in the Appendix, explaining on complex variables. Grateful acknowledgement is offered to the Fédération de l’Industrie Cimentière, Belgium (the Federation of the Belgian Cement Industry) for their intellectual and financial support in the process of the realisation of this book. Thanks is due to my dear friend, Marc Stet, for proof-reading and editing the manuscript. His helpful comments to the mathematical and editorial content are highly appreciated. Frans Van Cauwelaert Brussels, December 2003.
  • 4. PAVEMENT DESIGN AND EVALUATION iv Part 1: The Required Mathematics Chapter 1 covers the Laplace differential equation; the solution of for a great series of applications: it presents the Bessel function of zero order, solution of the Laplace equation in axisymmetric co-ordinates that as used in great parts of this book. The Laplacian with coefficients different from 1 will be applied in problems of anisotropic elasticity, the double Laplacian in most of the problems of multi-layer theory and the extended Laplacian in the problems of rigid pavement on Winkler or Pasternak foundations. Chapter 2 presents the gamma function, the factorial function for non-integers, required for the definition of Bessel functions of non-integer order. Chapter 3 gives the general solutions of the different Bessel equations, the Ber and Bei functions, and the modified form of the Bessel equation. Bessel functions are solutions of the Laplacian in polar or cylinder co-ordinates, used for applications with axial symmetry, e.g. in multi- layered structures. Trigonometric functions are solutions of the Laplacian in Cartesian co-ordinates, applied in cases of beams and rectangular slabs. Chapter 4 deals with the most important properties of Bessel functions: - derivatives, functions of half order, - asymptotic values required to express boundary conditions that must remain valid at infinite distances, - indefinite integrals, equations between Bessel functions of different kind (required for integrations in the complex plane). Chapter 5 presents the beta function required for the resolution of definite integrals of Bessel functions. Chapter 6 gives the solutions for a series of important definite integrals of Bessel functions; among others the Poisson integral giving an integral representation of any Bessel function of the first kind. Chapter 7 presents the hypergeometric function of Gauss required for the resolution of the infinite integrals of Bessel functions. Chapter 8 presents the most important infinite integrals of Bessel functions of direct application in pavement analysis, especially in multi-layer theory. Chapter 9 presents the most important infinite integrals of Bessel functions resolved in the complex plane. They are essentially of application in slab theory. Part 2: The Applications The second part focuses on the applications in the field of pavement engineering: rigid en flexible pavements. Chapter 10 gives the basic solutions (equilibrium equations) for rigid pavements (theory of strength of materials) and flexible pavements (theory of elasticity) .This chapter gives the basic equations on continuum mechanics in different co-ordinate systems. Chapter 11 presents the integral transforms, Fourier’s expansion, Fourier’s integral, Hankel’s integral, required for the expression of discontinuous functions: the loads in pavement applications. Chapters 12 to 19 concern mostly rigid pavements. Chapter 12 gives 3 simple applications on an elastic subgrade: a beam subjected to a single load, a beam subjected to a distributed load, a slab subjected to a single load. Chapter 13 gives the complete analytical solution for a beam on a Pasternak foundation (both of infinite and finite extent). Chapter 14 gives the analytical solution for a circular slab on a Pasternak foundation subjected to an axi-symmetric load (both of infinite and finite extent). Chapter 15 gives the complete analytical solution for a rectangular slab on a Pasternak foundation (both of infinite and finite extent). Chapter 16 gives the analytical solution for a superposition of several slabs included the analysis of the adhesion between the slabs. Chapter 17 gives a back-calculation method for rigid pavement based on Pasternak’s theory. Chapter 18 presents a solution for the computation of thermal stresses in rigid slabs on a Pasternak foundation. Chapter 19 presents two practical tests of interest with rigid slabs: the diametral test and a test for the determination of k and G in situ. Chapters 20 to 26 concern mostly flexible pavements. Chapters 20 and 21 present the complete theory for semi-infinite bodies subjected to all sorts of loads. Chapter 20 presents the Boussinesq problem: stresses
  • 5. INTRODUCTION v and displacements in a semi-infinite body under a circular flexible plate uniformly loaded. It generalises the solution to different vertical loads (isolated, rigid, rectangular) and to orthotropic bodies. Chapter 21 presents the solution a semi-infinite body subjected to shear stresses: radial and one-directional. Chapter 22 gives the analytical solution for a multi-layered structure, included the problem of the adhesion between the layers, that of an eventual fixed bottom and that of an anisotropic subgrade. Chapter 23 presents the numerical procedure required for the solution of a multi-layered structure. Chapter 24 presents the theory at the base of the back-calculation methods for flexible pavements. Chapter 25 presents the numerical procedure required for the back-calculation method for flexible pavements. Chapter 26 presents a practical test of interest with both types of road structures (rigid or flexible) multi- layered structures: the ovalisation test. The Appendices gives the basic theory of complex numbers, especially the integration in the complex plane.
  • 6. PAVEMENT DESIGN AND EVALUATION vi .
  • 7. CONTENTS vii Table of Contents Page Preface............................................................................................................................................. iii PART 1: THE REQUIRED MATHEMATICS................................................................................1 Chapter 1 The Laplace Equation....................................................................................................1 1.1 Introductory note................................................................................................................. 1 1.2 Derivation of the Laplace equation in polar co-ordinates from the Laplace equation in Cartesian co-ordinates........................................................................................................................ 2 1.3 Equations related to the Laplace equation.............................................................................. 3 1.3.1 The Laplacian with coefficients different from 1................................................................ 3 1.3.2 The double Laplacian....................................................................................................... 3 1.3.3 The extended Laplacian................................................................................................... 3 1.4 Resolution of the Laplace equation....................................................................................... 4 1.4.1 Resolution by separation of the variables........................................................................... 4 1.4.2 Resolution by means of the characteristic equation (Spiegel, 1971) ..................................... 5 1.4.3 Resolution by means of indicial equations ......................................................................... 6 Chapter 2 The Gamma Function....................................................................................................9 2.1 Introductory note................................................................................................................. 9 2.2 Helpful relations.................................................................................................................. 9 2.3 Definition of the Gamma Function......................................................................................10 2.4 Values of Γ(1/2) and Γ(-1/2)...............................................................................................11 Chapter 3 The General Solution of the Bessel Equation. ..............................................................15 3.1 Introductory note................................................................................................................15 3.2 Helpful relations.................................................................................................................16 3.3 Resolution of the Bessel equation (Bessel functions of the first kind).....................................17 3.4 Resolution of the Bessel equation for p an integer (Bessel functions of the second and third kind)..................................................................................................................................19 3.4.1 For n integer, Jn = (-)n J-n .................................................................................................19 3.4.2 Bessel functions of the second kind..................................................................................19 3.4.3 Bessel functions of the third kind.....................................................................................21 3.5 The modified Bessel equation. ............................................................................................21 3.6 The ber and bei functions....................................................................................................23 3.7 The ker and kei functions....................................................................................................24 3.8 Resolution of the equation ∇2 ∇2 w + w =0............................................................................25 3.9 The modified form of the Bessel equation............................................................................25 Chapter 4 Properties of the Bessel Functions...............................................................................27 4.1 Introductory note................................................................................................................27 4.2 Helpful relations.................................................................................................................27 4.3 Derivatives of Bessel functions ...........................................................................................29 4.3.1 Derivative of (rt)p Jp(rt)....................................................................................................29 4.3.2 Derivative of (rt)-pJp(rt)..................................................................................................29
  • 8. PAVEMENT DESIGN AND EVALUATION viii 4.3.3 Derivative of Jp(rt)..........................................................................................................29 4.4 Bessel functions of half order..............................................................................................30 4.4.1 Values of J1/2(rt), J-1/2(rt), J3/2(rt), J-3/2(rt)...........................................................................30 4.5 Asymptotic values..............................................................................................................31 4.5.1 Asymptotic values for Jp and J-p.....................................................................................31 4.5.2 Asymptotic values for Yp and Y-p ....................................................................................33 4.5.3 Asymptotic values for Hp (1) and Hp (2) ................................................................................33 4.5.4 Asymptotic values for Ip and I-p .......................................................................................33 4.5.5 Asymptotic value for Kn..................................................................................................35 4.5.6 Asymptotic values for ber and bei....................................................................................36 4.5.7 Asymptotic values for ker and kei....................................................................................36 4.6 Indefinite integrals of Bessel functions ................................................................................37 4.6.1 Fundamental relations .....................................................................................................37 4.6.2 The integral ∫rn J0(rt)dr .....................................................................................................37 4.7 Relations between Bessel functions of different kind ............................................................38 4.7.1 Bessel functions with argument –rt..................................................................................38 4.7.2 Relations between the three kinds of Bessel functions .......................................................38 4.7.3 Bessel functions of purely imaginary argument.................................................................39 Chapter 5 The Beta Function.......................................................................................................41 5.1 Introductory note................................................................................................................41 5.2 Helpful relations.................................................................................................................41 5.3 Definition of the beta function.............................................................................................41 5.4 Relation between beta and gamma functions ........................................................................42 5.5 The duplication formula for gamma functions ......................................................................42 Chapter 6 Definite integrals of Bessel functions ...........................................................................45 6.1 Helpful relations.................................................................................................................45 6.2 Gegenbauer’s integral.........................................................................................................45 6.3 Sonine’s first finite integral.................................................................................................46 6.4 Sonine’s second finite integral.............................................................................................47 6.5 Poisson’s integral...............................................................................................................48 Chapter 7 The hypergeometric type of series ...............................................................................49 7.1 Introductory note................................................................................................................49 7.2 Helpful relations.................................................................................................................49 7.3 Definition ..........................................................................................................................50 7.4 Properties of the multiple product (α)m ................................................................................51 7.4.1 A relation for (1 - b - m)m-n ..............................................................................................51 7.4.2 A relation for (a+b+1+m)m/22 m .........................................................................................51 7.4.3 A relation for Γ(a-n) .......................................................................................................52 7.4.4 The theorem of Vandermonde .........................................................................................52 7.4.5 The product of two Bessel functions with the same argument ............................................53 7.5 The hypergeometric series of Gauss 2F1[a,b;c;z]...................................................................54 7.5.1 Elementary properties.....................................................................................................54 7.5.2 Integral representation of the hypergeometric function ......................................................54 7.5.3 Value of F[a,b;c;z] for z = 1............................................................................................55 7.5.4 Convergence of the series F[a,b;c;z].................................................................................55 7.5.5 The product of two Bessel functions with different arguments...........................................56
  • 9. CONTENTS ix Chapter 8 Infinite Integrals of Bessel Functions...........................................................................57 8.1 Introductory note................................................................................................................57 8.2 Useful relations..................................................................................................................57 8.3 The integral ∫e-at Jν(bt)tµ-1 dt..................................................................................................60 8.3.1 Resolution of the integral................................................................................................60 8.3.2 Particular value...............................................................................................................61 8.3.3 The integral ∫ e-at cos(bt)dt................................................................................................63 8.3.4 The integral ∫ e-at sin(bt)dt.................................................................................................63 8.3.5 The integral ∫e-at cos(bt)sin(bt) dt ......................................................................................63 8.3.6 The integral ∫e-at sin(bt)/tdt...............................................................................................63 8.3.7 The integral ∫e-at sin(bt)tdt................................................................................................64 8.4 The integral ∫e-at Jν(bt)Jν(ct)tµ-1 dt ..........................................................................................64 8.4.1 Transformation of the integral..........................................................................................64 8.4.2 The integral ∫e-at sin(bt)sin(ct)t-1 dt.....................................................................................64 8.4.3 The integral ∫e-at sin(bt)sin(ct)dt........................................................................................65 8.4.4 The integral ∫∫e-am sin(bt)sin(cs)/(ts)dsdt ............................................................................65 8.4.5 The integral ∫∫me-am sin(bt)sin(cs)/(ts)dsdt .........................................................................67 8.5 The integral ∫e-at Jµ(bt)Jν(ct)tλ-1 dt ..........................................................................................68 8.6 The discontinuous integral ∫Jµ(at)Jν(bt)t-λ dt...........................................................................68 8.6.1 Resolution of the integral................................................................................................68 8.6.2 The integral ∫Jµ(at)Jν(at)t-λ dt.............................................................................................71 8.6.3 The integral ∫Jµ(at)Jµ-2k-1(bt)dt..........................................................................................72 8.6.4 Particular solutions of the integral ∫Jµ(at)Jν(bt)t-λ dt ...........................................................72 8.6.5 Particular solution of the integral ∫Jν(at)Jν(bt)/tλ dt..............................................................73 Chapter 9 Bessel functions in the complex plane ..........................................................................77 9.1 Introductory note................................................................................................................77 9.2 Helpful relations.................................................................................................................77 9.3 Proof of Γ(z)Γ(1-z) = π/sin(πz)...........................................................................................78 9.4 The Hankel’s contour integral for 1/Γ(z)..............................................................................80 9.5 The integral representation of Jν(z) ......................................................................................81 9.6 The integral representation of Iν(z) ......................................................................................83 9.7 The integral representation of Kν(z).....................................................................................83 9.8 The integral representation of Kv(xz)....................................................................................84 9.9 Resolution of ∫xν+1 Jν(ax)/(x2 +k2 )dx.....................................................................................85 9.10 Resolution of ∫xρ-1 Jµ(bx)Jν(ax)/(x2 +k2 )dx.............................................................................89 9.11 Resolution of ∫sin(bx)cos(ax)/x/(x4 + k4 )dx ..........................................................................90 PART 2: THE APPLICATIONS...................................................................................................93 Chapter 10 Laplace Equation in Pavement Engineering ............................................................93 10.1 Equilibrium equation for beams in pure bending...................................................................93 10.1.1 Sign conventions.........................................................................................................93 10.1.2 Assumptions...............................................................................................................95 10.1.3 Bending moment and bending stress ............................................................................95 10.1.4 The radius of curvature ...............................................................................................96 10.1.5 Equilibrium ................................................................................................................96
  • 10. PAVEMENT DESIGN AND EVALUATION x 10.2 Equilibrium equation for bent plates....................................................................................97 10.2.1 Bending moment and shear forces................................................................................97 10.2.2 Equilibrium ................................................................................................................99 10.3 Compatibility equation for a homogeneous, elastic, isotropic body submitted to forces applied on its surface ...................................................................................................................100 10.3.1 Principle of equilibrium.............................................................................................101 10.3.2 The principle of continuity.........................................................................................101 10.3.3 The principle of elasticity ..........................................................................................102 10.3.4 Stress potential..........................................................................................................102 10.4 Compatibility equation for a homogeneous, elastic, anisotropic body submitted to forces applied on its surface ...................................................................................................................103 10.5 Basic equations of continuum mechanics in different co-ordinate systems ...........................104 10.5.1 Plane polar co-ordinates............................................................................................104 10.5.2 Axi-symmetric Cylindrical Co-ordinates ....................................................................105 10.5.3 Non symmetric cylindrical co-ordinates .....................................................................106 10.5.4 Cartesian volume co-ordinates...................................................................................107 10.5.5 Axi-symmetric Cylindrical Co-ordinates for an orthotropic body..................................109 Chapter 11 The Integral Transforms........................................................................................ 111 11.1 Introductory note..............................................................................................................111 11.2 Helpful relations...............................................................................................................112 11.3 The Fourier expansion (Spiegel, 1971)...............................................................................112 11.3.1 Definitions................................................................................................................112 11.3.2 Proof of the Fourier expansion. ..................................................................................113 11.3.3 Example ...................................................................................................................114 11.4 The Fourier integral..........................................................................................................115 11.4.1 Definition.................................................................................................................115 11.4.2 Proof of Fourier’s integral theorem............................................................................115 11.4.3 Example ...................................................................................................................116 11.5 The Hankel’s transform....................................................................................................117 11.5.1 Definition.................................................................................................................117 11.5.2 Example ...................................................................................................................117 11.5.3 Application of the discontinuous integral of Weber and Schafheitlin ............................118 Chapter 12 Simple Applications of Beams and Slabs on an Elastic Subgrade........................... 121 12.1 The elastic subgrade.........................................................................................................121 12.2 The beam on an elastic subgrade subjected to an isolated load: ∂4 w/∂x4 +Cw=0....................123 12.3 The beam on an elastic subgrade subjected to a distributed load: ∂4 w/∂x4 +Cw=0..................125 12.4 The infinite slab subjected to an isolated load: ∇2 ∇2 w +Cw = 0..........................................125 Chapter 13 The Beam Subjected to a Distributed Load and Resting on a Pasternak Foundation ............................................................................................................................... 129 13.1 The basic differential equations.........................................................................................129 13.2 Case of a beam of infinite length .......................................................................................130 13.2.1 Solution of the differential equation............................................................................130 13.2.2 Application...............................................................................................................133 13.3 Case of a beam of finite length with a free edge 133 13.3.1 Solution #1...............................................................................................................133 13.3.2 Application...............................................................................................................135 13.3.3 Solution #2...............................................................................................................137 13.3.4 Application...............................................................................................................137
  • 11. CONTENTS xi 13.4 Case of a finite beam with a joint.......................................................................................138 13.4.1 Solution #1...............................................................................................................138 13.4.2 Application...............................................................................................................140 13.4.3 Solution# 2...............................................................................................................141 13.4.4 Application...............................................................................................................142 13.4.5 Proof that, in de case of a Winkler foundation, at a joint Q = γ T..................................143 Chapter 14 The Circular Slab Subjected to a Distributed Load and Resting on a Pasternak Foundation............................................................................................................. 145 14.1 The basic differential equations.........................................................................................145 14.2 Case of a slab of infinite extent .........................................................................................146 14.2.1 Solution of the differential equation............................................................................146 14.2.2 Application 1............................................................................................................147 14.2.3 Application 2............................................................................................................148 14.3 Case of a slab of finite extent with a free edge ...................................................................148 14.3.1 Solution #1...............................................................................................................148 14.3.2 Solution #2...............................................................................................................151 14.3.3 Application of solution #2.........................................................................................151 14.4 Case of a slab of finite extent with a joint...........................................................................152 14.4.1 Solution #1...............................................................................................................152 14.4.2 Solution #2...............................................................................................................152 14.4.3 Application of solution #2.........................................................................................153 Chapter 15 The Rectangular Slab Subjected to a Distributed Load and Resting on a Pasternak Foundation............................................................................................................. 155 15.1 The basic differential equations.........................................................................................155 15.2 Resolution of the deflection equation.................................................................................155 15.3 Boundary conditions.........................................................................................................158 15.4 Case of a slab of finite extent with free edge ......................................................................158 15.4.1 Solution #1...............................................................................................................158 15.4.2 Solution #2...............................................................................................................159 15.5 Case of a slab of finite extent with a joint...........................................................................159 15.5.1 Solution #1...............................................................................................................159 15.5.2 Solution # 2..............................................................................................................160 15.5.3 Application...............................................................................................................161 Chapter 16 The Multislab......................................................................................................... 163 16.1 Theoretical justification....................................................................................................163 16.2 General model..................................................................................................................163 16.3 Full slip at each of the interfaces.......................................................................................163 16.4 Full friction at the first interface, full slip at the second interface.........................................164 16.5 Full friction at both interfaces ...........................................................................................165 16.6 Partial friction..................................................................................................................166 16.6.1 Application...............................................................................................................168 Chapter 17 Back-calculation of Concrete Slabs ........................................................................ 171 17.1 Back-calculation of moduli...............................................................................................171 17.2 Case where the load can be considered as a point load........................................................171 17.3 Computations...................................................................................................................172 17.4 Case when the load is considered as distributed..................................................................173 17.5 Comparison of the two methods ........................................................................................175
  • 12. PAVEMENT DESIGN AND EVALUATION xii 17.6 Influence of the reference deflection..................................................................................175 17.7 Analysis of field data........................................................................................................176 17.8 Example of back-calculation.............................................................................................177 Chapter 18 Thermal Stresses in Concrete Slabs ....................................................................... 179 18.1 Thermal stresses ..............................................................................................................179 18.2 Slab of great length...........................................................................................................179 18.2.1 Differential equilibrium equation...............................................................................179 18.2.2 Solution of the equilibrium equation for g < 1.............................................................180 18.2.3 Solution of the equilibrium equation for g = 1.............................................................180 18.2.4 Solution of the equilibrium equation for g > 1.............................................................180 18.2.5 Boundary conditions .................................................................................................180 18.2.6 Expression of the moment for g < 1............................................................................180 18.2.7 Expression of the moment for g = 1............................................................................182 18.2.8 Expression of the moment for g > 1............................................................................182 18.2.9 Verification of the expression of the maximum moment for g = 1.................................183 18.2.10 Equation of the thermal stress....................................................................................183 18.2.11 Example ...................................................................................................................184 18.3 Rectangular slab...............................................................................................................184 18.3.1 Differential equation of equilibrium...........................................................................184 18.3.2 Boundary conditions .................................................................................................185 18.3.3 Examples .................................................................................................................185 18.4 Circular slab....................................................................................................................187 18.4.1 Equilibrium equation.................................................................................................187 18.4.2 Solution of the equilibrium equation...........................................................................188 18.4.3 Boundary conditions .................................................................................................188 18.4.4 Resulting moment.....................................................................................................188 18.4.5 Comparison between the models for rectangular and circular slabs...............................189 18.5 Extension to a multi-slab system.......................................................................................189 Chapter 19 Determination of the Parameters of a Rigid Structure ........................................... 193 19.1 Determination of the Young’s modulus of a concrete slab...................................................193 19.1.1 Resolution of the compatibility equation.....................................................................193 19.1.2 Equations for the stresses..........................................................................................193 19.1.3 Boundary conditions .................................................................................................193 19.1.4 Stresses and displacements........................................................................................194 19.1.5 Tangential normal stress along the vertical diameter....................................................195 19.2 Determination of the characteristics k and G of the subgrade ..............................................195 19.2.1 Equilibrium equation for a Pasternak subgrade ...........................................................195 19.2.2 Eccentrically loaded plate-bearing test........................................................................196 19.2.3 Vertical load.............................................................................................................198 19.2.4 Moment....................................................................................................................199 19.2.5 Determination of k and G..........................................................................................201 Chapter 20 The Semi-Infinite Body Subjected to a Vertical Load ............................................ 203 20.1 Introductory note..............................................................................................................203 20.2 The semi-infinite body subjected to a vertical uniform circular pressure ..............................203 20.3 The semi-infinite body subjected to an isolated vertical load...............................................206 20.4 The semi-infinite body subjected to a circular vertical rigid load..........................................207 20.5 The semi-infinite body subjected to a vertical uniform rectangular pressure.........................209 20.6 Comparison between the vertical stresses. Principle of de Saint-Venant..............................211
  • 13. CONTENTS xiii 20.7 The orthotropic body subjected to a vertical uniform circular pressure.................................211 20.8 The orthotropic body subjected to a vertical uniform rectangular pressure ...........................215 Chapter 21 The Semi-Infinite Body Subjected to Shear Loads ................................................. 217 21.1 The semi-infinite body subjected to radialshear stresses.....................................................217 21.2 The semi-infinite body subjected to a one-directional asymmetric shear load.......................218 21.3 The semi-infinite body subjected to a shear load symmetric to one of its axis’s ....................221 Chapter 22 The Multilayered Structure ................................................................................... 225 22.1 The multilayered structure................................................................................................225 22.2 Solutions of the continuity equations .................................................................................226 22.3 Boundary conditions.........................................................................................................227 22.4 Determination of the boundary constants...........................................................................228 22.5 The fixed bottom condition...............................................................................................231 22.6 The orthotropic subgrade ..................................................................................................232 Chapter 23 The Resolution of a Multilayered Structure ........................................................... 233 23.1 Choice of the integration formula ......................................................................................233 23.2 Values at the origin...........................................................................................................234 23.3 The geometrical scale of the structure................................................................................234 23.4 Width of the integration steps............................................................................................235 23.4.1 Influence of the moduli on the integration step............................................................235 23.4.2 Influence of the radii of the loads on the integration step.............................................235 23.4.3 Influence of the offset distance on the integration step.................................................236 23.4.4 Modification of the step width ...................................................................................236 23.5 Stresses and displacements at the surface...........................................................................236 23.6 Stresses and displacements in the first layer.......................................................................237 Chapter 24 The Theory of the Back-Calculation of a Multilayered Structure .......................... 241 24.1 The surface modulus.........................................................................................................241 24.2 Equivalent layers..............................................................................................................242 24.3 Equivalent semi-infinite body............................................................................................242 24.4 Analysis of a deflection basin............................................................................................243 24.4.1 Analysis of a three-layer on a linear elastic subgrade...................................................243 24.4.2 Analysis of a three-layer with a very stiff base course .................................................244 24.4.3 Analysis of a three layer with a very weak base course................................................245 24.4.4 Analysis of a two-layer on a subgrade with increasing stiffness with depth ...................246 24.5 Algorithm of Al Bush III (1980)........................................................................................247 Chapter 25 The Numerical Procedure of the Back-calculation of a Multilayered Structure ..... 249 25.1 The analysis of a back-calculation program for a three-layered structure..............................249 25.2 The sensitivity of the back-calculation procedure for a three layer structure.........................251 25.2.1 The sensitivity to rounding off the values of the measured deflections ..........................251 25.2.2 The sensitivity to the presence of a soft intermediate layer...........................................251 25.2.3 The influence of fixing beforehand the value of one modulus.......................................252 25.3 The sensitivity of the back-calculation procedure for a four layer structure ..........................253 25.3.1 Value of the information given by the surface modulus................................................254 25.3.2 The influence of fixing beforehand the value of one modulus.......................................256 25.4 The influence of degree of anisotropy and Poisson’s ratio on the results of a back-calculation procedure in the case of a semi-infinite subgrade................................................................257 25.4.1 Influence of the degree of anisotropy on the back-calculated moduli.............................257
  • 14. PAVEMENT DESIGN AND EVALUATION xiv 25.4.2 Influence of Poisson’s ratio of the subgrade on the deflections (n = 1) ..........................258 25.5 The influence of Poisson’s ratio and degree of anisotropy on the results of a back-calculation procedure in the case of a subgrade of finite thickness........................................................258 25.5.1 Influence of the degree of anisotropy on the back-calculated moduli.............................258 25.5.2 Influence of Poisson’s ratio of the subgrade on the deflections (n = 1) ..........................259 Chapter 26 The Ovalisation Test.............................................................................................. 261 26.1 Description of the ovalisation test......................................................................................261 26.2 Interpretation of the results of the ovalisation test...............................................................261 26.3 Slab with a cavity on an elastic subgrade subjected to a symmetrical load............................262 26.3.1 Resolution in the case of a plain slab..........................................................................262 26.3.2 Resolution in the case of a slab with a cavity...............................................................263 26.4 Strains in the case of a non-symmetrical load.....................................................................265 26.4.1 Stresses in a hollow cylinder subjected to a uniform external pressure..........................265 26.4.2 Stresses in a hollow plate...........................................................................................266 26.4.3 Application of the ovalisation test..............................................................................271 References ............................................................................................................................... 273 Appendix Complex Functions ..................................................................................................... 275
  • 15. THE LAPLACE EQUATION 1 PART 1: THE REQUIRED MATHEMATICS Chapter 1 The Laplace Equation 1.1 Introductory note. In mechanical engineering and thus also in the mechanics of civil engineering, one always starts with the fundamental requirement of equilibrium: equilibrium of the normal forces and equilibrium of the moments acting on the analysed body. Forces are resultants of stresses. In order to locate exactly those forces and determine, at least analytically, their amplitude, it is convenient to start from an infinitesimal section of the body. On such an infinitesimal section, the stresses are necessarily infinitely small and, hence, can be considered as constant and uniformly distributed over the area of the infinitely small section. Hence the resulting force is simply the product of the constant stress by the area of the section and its point of application is the centre of gravity of the section, i.e. the midpoint of the section. All the resulting equations will then necessarily be differential equations and, nearly in almost all applications, the differential equation expressing equilibrium is a Laplace or an assimilated equation. The development of these equations are presented in chapter 10, the first chapter of Part 2 “Applications”. As in many fields of engineering, also in Pavement engineering the differential equation of Laplace is the solution of a great series of applications. Symbolically, Laplace equation is written as follows: 02 =∇ Φ (1.1) It is a homogeneous differential equation with second order partial derivatives. Function of the co-ordinate system, the equation is developed: in plane Cartesian co-ordinates 0 yx 2 2 2 2 =+ ∂ Φ∂ ∂ Φ∂ (1.2) in volume Cartesian co-ordinates 0 zyx 2 2 2 2 2 2 =++ ∂ Φ∂ ∂ Φ∂ ∂ Φ∂ (1.3) in polar co-ordinates 0 r 1 rr 1 r 2 2 22 2 =++ ∂θ Φ∂ ∂ Φ∂ ∂ Φ∂ (1.4) in cylindrical co-ordinates 0 zr 1 rr 1 r 2 2 2 2 22 2 =+++ ∂ Φ∂ ∂θ Φ∂ ∂ Φ∂ ∂ Φ∂ (1.5) In case of axial symmetry, ∂Φ/∂θ = 0 and equations (1.4) and (1.5) simplify in: 0 rr 1 r2 2 =+ ∂ Φ∂ ∂ Φ∂ (1.6) 0 zrr 1 r 2 2 2 2 =++ ∂ Φ∂ ∂ Φ∂ ∂ Φ∂ (1.7) Equation (1.2) is applied in chapter 10.3.4. Equation (1.4) will be utilised in chapter’s 10.5.1 and 19.1. Equation (1.7) will be utilised in chapter’s 10.5.2, 10.6, 20.1, 20.2, 20.3 and 20.4.
  • 16. PAVEMENT DESIGN AND EVALUATION 2 1.2 Derivation of the Laplace equation in polar co-ordinates from the Laplace equation in Cartesian co-ordinates. Consider the relations between Cartesian and polar co-ordinates: θcosrx = θsinry = Hence 222 yxr += )x/y(tan 1− =θ Express the partial derivatives ( ) θ ∂ ∂ cos r x yx x x r 2/122 == + = θ ∂ ∂ sin r y y r == r sin x y x y 1 1 x 2 2 2 θ ∂ ∂θ −= + −= r cos x 1 x y 1 1 y 2 2 θ ∂ ∂θ = + = Then xx r rx ∂ ∂θ ∂θ Φ∂ ∂ ∂ ∂ Φ∂ ∂ Φ∂ += ∂θ Φ∂θ ∂ Φ∂ θ ∂ Φ∂ r sin r cos x −= xxx r xrx2 2 ∂ ∂θ ∂ Φ∂ ∂θ ∂ ∂ ∂ ∂ Φ∂ ∂ ∂ ∂ Φ∂ += ∂θ Φ∂ θθ ∂θ∂ Φ∂ θθ ∂θ Φ∂ θ ∂ Φ∂ θ ∂ Φ∂ θ ∂ Φ∂ cossin r 2 r cossin r 2 sin r 1 r sin r 1 r cos x 2 2 2 2 2 2 2 2 2 2 2 2 +−++= In the same way, obtain: ∂θ Φ∂ θθ ∂θ∂ Φ∂ θθ ∂θ Φ∂ θ ∂ Φ∂ θ ∂ Φ∂ θ ∂ Φ∂ cossin r 2 r cossin r 2 cos r 1 r cos r 1 r sin y 2 2 2 2 2 2 2 2 2 2 2 2 −+++= Make the sum: 2 2 22 2 2 2 2 2 r 1 rr 1 ryx ∂θ Φ∂ ∂ Φ∂ ∂ Φ∂ ∂ Φ∂ ∂ Φ∂ ++=+ (1.8)
  • 17. THE LAPLACE EQUATION 3 1.3 Equations related to the Laplace equation. Besides the properly so called Laplace differential equation, there exists a series of useful differential equations closely related to the Laplace equation. 1.3.1 The Laplacian with coefficients different from 1. For example in plane Cartesian co-ordinates 0 yC 1 x 2 2 2 2 =+ ∂ Φ∂ ∂ Φ∂ (1.9) Equation (1.9) is utilised in chapter’s 10.3.5 and 20.6. 1.3.2 The double Laplacian The double Laplacian, or the Laplace operator applied to a Laplace equation, is also a solution of an important series of applications. It writes 022 =∇∇ Φ (1.10) Developed, for example in volume co-ordinates, the double Laplacian is 0 zyxzyx 2 2 2 2 2 2 2 2 2 2 2 2 =         ++         ++ ∂ Φ∂ ∂ Φ∂ ∂ Φ∂ ∂ ∂ ∂ ∂ ∂ ∂ (1.11) Equation (1.10) is applied in chapter’s 10.3, 10.5.1,10.5.2, 10.5.3, 10.5.4, 20.1, 20.2, 20.3 and 20.4. 1.3.3 The extended Laplacian Often the Laplacian equation is completed by derivatives of an order lower than the second. Here in polar co-ordinates: 0k r 1 rr 1 r 2 2 22 2 =+++ Φ ∂θ Φ∂ ∂ Φ∂ ∂ Φ∂ (1.12) or more generally CBA 222 =+∇+∇∇ ΦΦΦ (1.13) The extended Laplacian is used in chapters 10.1.5, 10.2.2 and in chapter’s 12 to 19.
  • 18. PAVEMENT DESIGN AND EVALUATION 4 1.4 Resolution of the Laplace equation Three very general methods are applied in this book. 1.4.1 Resolution by separation of the variables Solution in volume Cartesian co-ordinates: Consider equation (1.3) and assume a solution such as Φ = f1(x)f2(y)f3(z). Applying (1.3) yields 0 z )z(f )y(f)x(f)z(f y )y(f )x(f)z(f)y(f x )x(f 2 3 2 2132 2 2 1322 1 2 =++ ∂ ∂ ∂ ∂ ∂ ∂ and dividing by f1(x)f2(y)f3(z) 0 )z(f z )z(f )y(f y )y(f )x(f x )x(f 3 2 3 2 2 2 2 2 1 2 1 2 =++ ∂ ∂ ∂ ∂ ∂ ∂ Each of the 3 terms of the sum is a function of one single variable; this results in )z(fC z )z(f )y(fC y )y(f )x(fC x )x(f 332 3 2 222 2 2 112 1 2 === ∂ ∂ ∂ ∂ ∂ ∂ 0CCC 321 =++ a system with a large series of solutions of the differential equation. For example: f1(x) = cos(x), f2(y) = cos(y), f3(z) = ez√2 . Solution in axi-symmetric cylinder co-ordinates ( Bowman,1958): Consider equation (1.7) and apply a solution such as Φ = f(r)g(z). Applying (1.7) yields 0 z )z(g )r(f)z(g r )r(f r 1 )z(g r )r(f 2 2 2 2 =+ ∂ + ∂ ∂∂ ∂ ∂ and dividing by f(r)g(z) 0 )z(g z )z(g )r(f r )r(f r 1 )r(f r )r(f 2 2 2 2 =++ ∂ ∂ ∂ ∂ ∂ ∂ Thus )z(Cg z )z(g )r(Cf r f r 1 r )r(f 2 2 2 2 =−=+ ∂ ∂ ∂ ∂ ∂ ∂ Choose Cg(z) = ez . Hence f(r) must be a solution of 0)r(f r )r(f r 1 r )r(f 2 2 =++ ∂ ∂ ∂ ∂ (1.14) Equation (1.14) is called the Bessel equation of zero order.
  • 19. THE LAPLACE EQUATION 5 The function known as Bessel’s function of the first kind and of n-th order and denoted Jn(r) is defined as follows (Bowman, 1958): ... )!3n(!3 )2/r( )!2n(!2 )2/r( )!1n(!1 )2/r( !n!0 )2/r( )r(J n6n4n2n n + + − + + + −= +++ (1.15) Hence ... !3!3 )2/r( !2!2 )2/r( !1!1 )2/r( 1)r(J 642 0 +−+−= (1.16) and ... !4!3 )2/r( !3!2 )2/r( !2!1 )2/r( 2 r )r(J 753 1 +−+−= (1.17) Differentiating the series for J0(r) and comparing the result with the series for J1(r) results in: )r(J dr )r(dJ 1 0 −= (1.18) Also, after multiplying the series for J1(r) by r and differentiating: ( ) )r(rJ)r(rJ dr d 01 = (1.19) Using (1.18), (1.19) can be rewritten in the form: )r(rJ dr )r(dJ r dr d 0 0 =      − 0)r(J dr )r(dJ r 1 dr )r(Jd 0 0 2 0 2 =++ (1.20) Hence J0(r) is a solution of (1.14) and Φ = J0(r)ez is a solution of (1.7). This particular solution was developed to introduce, from the first chapter on, the Bessel functions. Bessel functions are frequently used and discussed throughout this book. 1.4.2 Resolution by means of the characteristic equation (Spiegel, 1971) This solution applies to homogeneous linear differential equations with constant coefficients defined as 0a dx d a... dx d a dx d a n1n1n 1n 1n n 0 =++++ −− − Φ ΦΦΦ (1.21) It is convenient to adopt the notations DΦ, D2 Φ, ..., Dn Φ to denote dΦ/dx, d2 Φ/dx2 , ...dn Φ/dxn , where D, D2 , ..., Dn are called differential operators. Using this notation, (1.21) transforms 0aDa...DaDa n1n 1n 1 n 0 =      ++++ − − Φ (1.22) Let Φ = emx , m = constant, in (1.22) to obtain 0a...mama n 1n 1 n 0 =+++ − (1.23) that is called the characteristic equation. It can be factored into 0)mm)...(mm)(mm(a n210 =−−− (1.24) which roots are m1, m2, ...mn.
  • 20. PAVEMENT DESIGN AND EVALUATION 6 One must consider three cases: Case 1. Roots all real and distinct. Then em1x , em2x , ... emnx are n linearly independent solutions so that the required solution is: xm n xm 2 xm 1 n21 eC...eCeC +++=Φ (1.25) Case 2. Some roots are complex. If a0, a1, ..., an are real, then when a+bi is a root of (1.23) so also is a-bi (where a and b are real and i = √(-1) ) . Then a solution corresponding to the roots a+bi and a-bi is: ( )bxsinCbxcosCe 21 ax +=Φ (1.26) where use is made of Euler’s formula eiu = cos u +i sin u (see Appendix). Case 3. Some roots are repeated. If m1 is a root of multiplicity k, then a solution is given by: ( ) xm1k k 2 321 1 exC...xCxCC − ++++=Φ (1.27) Example Consider the double Laplacian in plane Cartesian co-ordinates: 0 yyx 2 x 4 4 22 4 4 4 =++ ∂ Φ∂ ∂∂ Φ∂ ∂ Φ∂ (1.28) Choose as solution Φ = f(x) ey . Hence (1.28) reduces in 0)x(f x )x(f 2 x )x(f 2 2 4 4 =++ ∂ ∂ ∂ ∂ (1.29) Equation (1.29) is a homogeneous linear differential equation of order 4. The characteristic equation writes: 01m2m 24 =++ (1.30) and can be factored in: 0)im()im( 22 =−+ (1.31) Based on (1.26) and (1.27), the solution of (1.29) becomes: ( )xsinDxcosCxxsinBxcosA)x(f +++= and the solution of (1.28): ( )[ ]xsinDxcosCxxsinBxcosAe)y,x( y +++=Φ (1.32) 1.4.3 Resolution by means of indicial equations Consider the differential equation: 0wk dx wd 2 2 2 =+ (1.33) Assume a solution in the form of a indicial series L4 4 3 3 2 210 xaxaxaxaa)x(f ++++= Apply (1.33)
  • 21. THE LAPLACE EQUATION 7 0xakxakakxa.3.4xa.2.3a.1.2 2 2 2 1 2 0 22 432 =+++++ LL This equation must be equal zero for all values of x. Hence the sum of the coefficients of each exponent of x must be individually zero. 0aka.1.2 0 2 2 =+ 0aka.2.3 1 2 3 =+ 0aka.3.4 2 2 4 =+ … First let a0 = 1 and a1 = 0 Hence )!p2( k )(a !4 k a !2 k a p2 p p2 4 4 2 2 −==−= (1.34) 0aaa 1p231 LL ==== + (1.35) The successive terms of (1.34) are the terms of the cosine series. Hence, the first solution of (1.33) is f(x) = cos(kx). The second solution is obtained by setting a0 = 0 and a1 = k. Obviously, the second solution of (1.33) is f(x) = sin(kx).
  • 22. PAVEMENT DESIGN AND EVALUATION 8
  • 23. THE GAMMA FUNCTION 9 Chapter 2 The Gamma Function 2.1 Introductory note In chapter 1, we have defined the Bessel function of the first kind and of order zero as: ( ) !k!k 2/r )()r(J k2 k 0 ∑ −= This equation can be generalised, for n an integer, in: ( ) )!nk(!k 2/r )()r(J nk2 k n + −= + ∑ When p is not an integer, the Bessel function of order p writes: ( ) )1pk(!k 2/r )()r(J pk2 k p ++ −= + ∑ Γ where Γ(k+p+1) is called the gamma function of k+p+1. For our purpose, the gamma function is essentially required to express Bessel functions of non-integer order: it is the factorial function for non-integers. Observe that with these definitions Γ(n+1) = n! 2.2 Helpful relations ∫ ∞ −− = o x1p dxex)p(Γ )p(p)1p( ΓΓ =+ 1)1( =Γ !n)1n( =+Γ πΓ =)2/1( ±∞=− )n(Γ       − +++++−== 1n 1 ... 3 1 2 1 1)n()n(' dn )n(d γΓΓ Γ 5772157.0=γ (Euler’s constant)
  • 24. PAVEMENT DESIGN AND EVALUATION 10 2.3 Definition of the Gamma Function The gamma function is defined by: ∫ ∞ −− = o x1p dxex)p(Γ (2.1) which is convergent for p > 0. Applying the definition ∫ ∞ − =+ o xp dxex)1p(Γ ∫ ∞ −−∞− −−−= o 1px o xp dx)px)(e(ex ∫ ∞ −− = o x1p dxexp )p(pΓ= one obtains the recurrence formula: )p(p)1p( ΓΓ =+ (2.2) By taking (2.1) as the definition of Γ(p) for p > 0, we can generalise the gamma function to p < 0 by use of (2.2) in the form p )1p( )p( + = Γ Γ (2.3) This process is called analytic continuation. Applying (2.1) we determine the value of Γ(1) ∫ ∞ ∞−− =−== o o xx 1edxe)1(Γ (2.4) Hence, applying (2.2) 1)1(1)2( == ΓΓ !212)2(2)3( =•== ΓΓ !3123)2(23)3(3)4( =••=•== ΓΓΓ !n)1n( =+Γ For n being a positive integer. 2.3. Values of gamma functions. The values of Γ(p) for non-integer values of p must be computed numerically. One obtains for 1 ≤ p ≤ 2 p 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 Γ(p) 1.00 .951 .918 .898 .887 .886 .894 .909 .931 .962 1.00 Table 1 Values of Γ(p) for non-integer values of p The value of Γ(p) can be computed for any value of p using (2.2) Γ(4.3) = 3.3 Γ(3.3) = 3.3 * 2.3 * Γ(2.3) = 3.3 * 2.3 * 1.3 * Γ (1.3) = 3.3 * 2.3 * 1.3 * 0.898 = 8.861
  • 25. THE GAMMA FUNCTION 11 2.4 Values of Γ(1/2) and Γ(-1/2). Two particular values of the gamma function, Γ(1/2) and Γ(-1/2), appear in many applications. By definition: ∫ ∞ −− = o x2/1 dxex)2/1(Γ Write x =u2 and dx = 2udu ∫ ∞ − = o u due2)2/1( 2 Γ Then [ ]                 = ∫∫ ∞ − ∞ − dve2due22/1( o v o u2 22 Γ ∫ ∫ ∞∞ +− = o o )vu( dudve4 22 Write u = r cos θ, v = r sin θ, du dv = r dr dθ [ ] ∫ ∫ ∞ − = 2/ o o r2 rdrde4)2/1( 2 π θΓ ∫= 2/ o d2 π θ π= Hence πΓ =)2/1( (2.5) and by (2.3) π Γ Γ 2 2/1 )2/1( )2/1( −=−=− (2.6) 2.5. Values of Γ(n) for n = 0, -1, -2, ... By definition ∫ ∞ − = o x dx x e )0(Γ that we write dx... x!3 1 x!2 1 edx... x!3 1 x!2 1 x 1 e)0( o 32 x 32 o x ∫∫ ∞ − ∞ −       +−+      −+−=Γ Derive       ⋅⋅⋅+−−      ⋅⋅⋅−−=      ⋅⋅⋅− −−− 32 x 32 x 32 x x!3 1 x!2 1 x 1 e x!3 1 x!2 1 e x!3 1 x!2 1 e dx d Hence dx... x!3 1 x!2 1 e... x!3 1 x!2 1 e)0( 32 o x 32 x       +−−      +−−= ∫ ∞ −− Γ
  • 26. PAVEMENT DESIGN AND EVALUATION 12 dx... x!3 1 x!2 1 e 32 o x       +−+ ∫ ∞ − ∞ −       +−−= o 32 x ... x!3 1 x!2 1 e)0(Γ ∞ −       +−+−+−−= o 32 x x!1 1 1... x!3 1 x!2 1 x!1 1 1e)0(Γ ∞=      +−−= ∞ −− o x/1x x!1 1 1ee)0(Γ (2.7) −∞= − =− 1 )0( )1( Γ Γ (2.8) ±∞=− )n(Γ (2.9) The gamma function is undefined when the value of the argument is zero or a negative integer. 2.6. Derivative of Γ(n). We call Γ’(n) the derivative of Γ(n) with respect to n. For simplicity we consider only the case of n being an integer, which besides is the only case required for our purpose. By definition ∫ ∞ −− = o x1n dxex dn d )n('Γ ( )∫ ∞ −− = o x1n dxex dn d ∫ ∞ −− = o x1n dxexlogx We first compute Γ’(1) ∫ ∞ − = o x dxexlog)1('Γ Integrating by parts ∫ ∫ −=−= xxlogxdx x x xlogxxdxlog ∫ ∫ −=−= 2 x xlog 2 x dx x x 2 1 2 x xdxlogx 2222 …………… ∫ ∫ −=−= −− 2 nn 1n n 1n n x xlog n x dxx n 1 xlog n x dxxlogx Further ( )∫ ∫ ∫ −−−− −+−= dxxedxexlogxexxlogxdxexlog xxxx ∫ ∫ ∫ −−−− −+         −= dxex !2.2 1 dxexlogx !2 1 e !2.2 x xlog !2 x dxexlogx x2x2x 22 x
  • 27. THE GAMMA FUNCTION 13 ∫ ∫ ∫ −−−− −+         −= dxex !3.3 1 dxexlogx !3 1 e !3.3 x xlog !3 x dxexlogx !2 1 x3x3x 33 x2 ... ∫ ∫ ∫ −−−−− −+         −= − dxex !n.n 1 dxexlogx !n 1 e !n.n x xlog !n x dxexlogx )!1n( 1 xnxnx nn x1n Assembling: ∫ −−−         ++++−         ++++= x n32 x n32 x e !n.n x ... !3.3 x !2.2 x !1.1 x exlog !n x ... !3 x !2 x !1 x dxexlog ∫ ∫ ∫ ∫ −−−− −−−−− dxex !n.n 1 ...dxex !3.3 1 dxex !2.2 1 dxxe !1.1 1 xnx3x2x ( )∫ −−−         ++++−−= x n32 xxx e !n.n x ... !3.3 x !2.2 x !1.1 x exlog)1edxexlog ∫ ∫ ∫ ∫ −−−− −−−−− dxex !n.n 1 ...dxex !3.3 1 dxex !2.2 1 dxxe !1.1 1 xnx3x2x Integrating between 0 and ∞ yields ∫ ∞ ∞−∞− −= o o x o x exlogxlogdxexlog ... !3.3 )4( !2.2 )3( !1.1 )2( e... !3.3 x !2.2 x !1.1 x o x 32 −−−−         +++− ∞ − ΓΓΓ ( ) ( ) ... 3 1 2 1 100exloglimexloglimxloglimxloglimdxexlog o x 0x x x0xx x −−−−+−+−−=∫ ∞ − → − ∞→→∞→ − ( ) ⋅⋅⋅−−−−−+= − → ∞ ∞→ − ∫ 3 1 2 1 11exloglimxloglimdxxelog x 0x 0 x x ∫ ∞ ∞→ − −=    −−−−−== o M x M 1 ... 3 1 2 1 1Mloglimdxexlog)1(' γΓ (2.10) where γ = 0.5772157 is called Euler’s constant. By extent: ( ) ∫ ∫∫ ∞ ∞ −−∞−− ∞ −+−= o o xx o xx o dxxedxexlogxexxlogxdxexlog ∫ ∫ ∫ ∞ ∞ ∞ −−− += o o o xxx dxxedxexlogdxexlogx
  • 28. PAVEMENT DESIGN AND EVALUATION 14 ∫ ∞ − +−== o x )2(dxexlogx)2(' ΓγΓ       ++−= 2 1 1)3(!2)3(' ΓγΓ ...       − +++++−−= 1n 1 ... 3 1 2 1 1)n()!1n()n(' ΓγΓ 1n 1 ... 3 1 2 1 1 )n( )n(' − +++++−= γ Γ Γ (2.11)
  • 29. THE GENERAL SOLUTION OF THE BESSEL EQUATION 15 Chapter 3 The General Solution of the Bessel Equation. 3.1 Introductory note. In the way trigonometric functions are solutions of Laplace equation in Cartesian co-ordinates, Bessel functions are solutions of Laplace equation in polar or cylindrical co-ordinates. One could say that Bessel functions is a three dimensional function whereas trigonometric functions are two dimensional functions. The fundamental characteristics of both functions are identical: - if f(x) = cos(x) )x(f dx )x(fd 2 2 −= - if f(r) = J0(r) )r(f dr )r(df r 1 dr )r(fd 2 2 −=+ Bessel functions are essentially utilised in problems presenting an axial symmetry: in pavement engineering, the common case of a circular load on a layered structure, i.e. nearly all applications of chapters 12 to 26. Jp and J-p are the two solutions of the first kind of Bessel’s equation 0t r p dr d r 1 dr d 2 22 2 =+−+ φ φφφ However, when p = n, Jn = (-)n J-n. In that case, Bessel’s equation requires another second solution, a solution of the second kind, Yp , which will not be utilised in this book. Ip and I-p are the two modified solutions of the first kind of Bessel’s modified equation: 0t r p dr d r 1 dr d 2 22 2 =−−+ φ φφφ These also, aren’t used in this book. However, they allow introducing the Ker and Kei functions, solutions of: 0it dr d r 1 dr d 2 2 2 =++ φ φφ which is applied in the problem of a slab subjected to an isolated load (chapter 12.4) and in the ovalisation test (chapter 26.4). The modified form of Bessel’s equation: 0FrF r p dr dF r 1 )21( dr Fd )1(222 2 222 2 2 =+ − +−+ −γ γβ γα α permits us to illustrate the existence of relations between Bessel functions of order ½ and trigonometric functions (chapter 4.3) Those relations allow us to establish asymptotic approximations for Bessel functions (chapter 4.5), required for the numerical computations of functions of high arguments.
  • 30. PAVEMENT DESIGN AND EVALUATION 16 3.2 Helpful relations ( ) )1kp(!k 2/rt )()rt(J k2p 0 k p ++ −= +∞ ∑ Γ ( ) )1kp(!k 2/rt )()rt(J k2p 0 k p ++− −= +−∞ − ∑ Γ )rt(J)()rt(J n n n −=− for n integer ( )[ ] ∑ − − −− −+= 1n 0 nk2 nn !k )2/rt()!1kn(1 )2/rtlog()rt(J 2 )rt(Y π γ π )!kn(!k )2/rt( kn 1 ... 3 1 2 1 1 1 k 1 ... 3 1 2 1 1 1 )( 1 nk2 0 k +      + +++++++++−− +∞ ∑π )rt(iY)rt(J)rt(H pp )1( p += )rt(iY)rt(J)rt(H pp )2( p −= ∑ ∞ + − ++ == 0 k2p p p p )1pk(!k )2/rt( )irt(Ji)rt(I Γ ∑ − − −− −= 1n 0 k2n k n )2/rt(!k )!1kn( )( 2 1 )rt(K       ++−+− + −+ ∑ ∞ + + )1kn( 2 1 )1k( 2 1 )2/rtlog( )!kn(!k )2/rt( )( 0 k2n 1n ΨΨ k 1 ... 3 1 2 1 1 1 )1k()1( +++++−=+−= γΨγΨ )rt(ibei)rt(ber)irt(I)irti(J 00 ±=±=± ∑∑ ∞ +∞ ++ −=−= 0 2k4 k 0 k4 k )!1k2()!1k2( )2/rt( )()rt(bei )!k2()!k2( )2/rt( )()rt(ber )rt(kei)rtker()irt(K0 ±=± [ ] ( ) ... 2 1 1 !2!2 2/rt )rt(bei 4 2/rtlog()rt(ber)rtker( 4       +−++−= π γ [ ] ( ) ( ) ... 3 1 2 1 1 !3!3 2/rt !1!1 2/rt )rt(bei 4 )2/rtlog()rt(bei)rt(kei 62       ++−+−+−= π γ
  • 31. THE GENERAL SOLUTION OF THE BESSEL EQUATION 17 3.3 Resolution of the Bessel equation (Bessel functions of the first kind) Recall the definition of the Laplacian in cylinder co-ordinates given by § 1.5: 0 zr 1 rr 1 r 2 2 2 2 22 2 =+++ ∂ Φ∂ ∂θ Φ∂ ∂ Φ∂ ∂ Φ∂ (3.1) Consider next solution obtained by the method of separation of the variables tz e)pcos()r(F θΦ = and apply it to (3.1) that transforms into: 0FtF r p r F r 1 r F 2 2 2 2 2 =+−+ ∂ ∂ ∂ ∂ (3.2) The general solution of equation (3.2) can be found by the method of the indicial equations (Wayland, 1970). Therefore assume that the solution can be written as a series such as: 2n 0 nraF ++ ∞ ∑= α (3.3) Hence ∑ ∞ ++ += 0 1n nra)n( r F α α ∂ ∂ (3.4) ∑ ∞ + −++= 0 n 2 2 r)1n)(n( r F α αα ∂ ∂ (3.5) Replace the terms in (3.2) by (3,3), (3.4) and (3.5): [ ]{ } 0ratrap)n()1n)(n( 0 2n n 2n n 2 =+−++−++∑ ∞ +++ αα ααα Take out the first two terms of the first summation: ra)1p)(1p(a)p(r 10 22 +++−+    − ααα α 0ratra)pn)(pn( 0 2n n 2 2 n n =     +++−++ ∑∑ ∞ + ∞ αα If the terms in r0 and r1 are equal zero, the relation becomes homogeneous regarding the exponents of r. Therefore choose a0 = arbitrary, a1 = 0 and α = ± p. Rearrange the indices [ ] 0rtaa)p22n)(2n(r 0 2n2 n2n =         +±++∑ ∞ + + α (3.6) If (3.6) has to be identical zero whatever the value of m, each term of the summation has to be equal zero. Hence the recurrence formula 2 n2n taa)p22n)(2n( −=±++ + (3.7) and because a1 = 0, all the terms with an odd index are also equal zero. Finally resulting in:
  • 32. PAVEMENT DESIGN AND EVALUATION 18 2 0 2 0 2 0 2 2 t )p1(1 a )p1(22 ta )p22(2 ta a       ±× −= ±× −= ± −= 4 0 2 2 4 2 t )p2)(p1(21 a )p24(4 ta a       ±±× = ± −= 6 0 2 4 6 2 t )p3)(p2)(p1(321 a )p26(6 ta a       ±±±×× −= ± −= .............. s2 0 s s2 2 t )ps)...(p2)(p1(!s a)( a       ±±± − = If p is different from zero or not an integer, one obtains two linearly independent solutions ∑∑ ∞ +−∞ + ++− −+ ++ −= 0 k2p k 0 k2p k )1kp(!k )rt( )(B )1kp(!k )2/rt( )(AF ΓΓ (3.8) The corresponding series are called Bessel functions of the first kind and noted ∑ ∞ + ++ −= 0 k2p k p )1kp(!k )2/rt( )()rt(J Γ (3.9) ∑ ∞ +− − ++− −= 0 k2p k p )1kp(!k )2/rt( )()rt(J Γ (3.10) The Bessel functions of the first kind and of order 0 and 1 are represented in Figure 3.1. Figure 3.1 Bessel functions of the first kind and of order 0 and 1
  • 33. THE GENERAL SOLUTION OF THE BESSEL EQUATION 19 3.4 Resolution of the Bessel equation for p an integer (Bessel functions of the second and third kind) 3.4.1 For n integer, Jn = (-)n J-n When p is an integer, let say n, solutions (3.9) and (3.10) are not linearly independent any more. Indeed when p = -n ∑ ∑∑ ∞ ∞ +−− +−+− − +− −+ +− −= +− −= 0 n k2n k 1n 0 k2n k k2n k n )!kn(!k )2/rt( )( )!kn(!k )2/rt( )( )!kn(!k )2/rt( )()rt(J When 0, 1, 2, 3, ..., k = n-1, (-n+k)! = ± ∞ and 0 )!kn(!k )2/rt( k2n = +− +− Write k = n + j ∑ ∑ ∞ ∞ + + +− − + −= +− −= n 0 j2n jn k2n k n !j)!jn( )2/rt( )( )!kn(!k )2/rt( )()rt(J ∑ ∞ + − −= + −−= 0 n n j2n jn n )rt(J)( )!jn(!j )2/rt( )()()rt(J Hence the two solutions are not linearly independent. 3.4.2 Bessel functions of the second kind In order to find a second linearly independent solution, we define the function )psin( )rt(J)pcos()rt(J )rt(Y pp p π π −− = (3.11) which is valid for p not an integer. Then we define for p = n )psin( )rt(J)pcos()rt(J lim)rt(Y pp np n π π − → − = We search the solution for n = 0. )psin( )rt(J)pcos()rt(J lim)rt(Y pp 0p 0 π π − → − = Apply de L’Hospital’s rule )pcos( p )rt(J )psin()rt(J)pcos( p )rt(J )rt(Y p p p 0 ππ ∂ ∂ πππ ∂ ∂ − −− =       −= − p )rt(J p )rt(J1 )rt(Y pp 0 ∂ ∂ ∂ ∂ π Compute the derivatives of Jp(rt) and J-p(rt): ∑ ∞ +       ++ ++ − ++ −= 0 k2p kp )1kp( )1kp(' )2/rtlog( )1kp(!k )2/rt( )( p )rt(J Γ Γ Γ∂ ∂
  • 34. PAVEMENT DESIGN AND EVALUATION 20 ∑ ∞ +− −       ++− ++− +− ++− −= 0 k2p kp )1kp( )1kp(' )2/rtlog( )1kp(!k )2/rt( )( p )rt(J Γ Γ Γ∂ ∂ and for p → 0 ∑ ∞       + + −−= 0 k2 k 0 )1k( )1k(' )2/rtlog( !k!k )2/rt( )( 2 )rt(Y Γ Γ π By (2.11) k 1 ... 3 1 2 1 1 1 )1k( )1k(' +++++−= + + γ Γ Γ Hence [ ]         −      +++      +−++= ... 3 1 2 1 1 !3!3 )2/rt( 2 1 1 !2!2 )2/rt( !1!1 )2/rt( )2/rtlog()rt(J 2 )rt(Y 642 00 γ π (3.12) For n ≠ 0, (3.12) is extended to [ ]{ } ∑ − − −− −+= 1n 0 nk2 nn !k )2/rt()!1kn(1 )2/rtlog()rt(J 2 )rt(Y π γ π (3.13) ∑ ∞ + +      + +++++++++−− 0 nk2 k )!kn(!k )2/rt( kn 1 ... 3 1 2 1 1 1 k 1 ... 3 1 2 1 1 1 )( 1 π The functions Yp(rt) are called functions of the second kind. The Bessel functions of the second kind and of order 0 and 1 are represented in Figure 3.2. Figure 3.2 Bessel functions of the second kind and of order 0 and 1
  • 35. THE GENERAL SOLUTION OF THE BESSEL EQUATION 21 3.4.3 Bessel functions of the third kind Hankel introduced next pair of conjugate complex functions, with i = √(-1): )rt(iY)rt(J)rt(H pp )1( p += (3.14) )rt(iY)rt(J)rt(H pp )2( p −= (3.15) These functions are called Hankel functions or Bessel functions of the third kind. 3.5 The modified Bessel equation. If in § 3.3 we chose: )tzcos()pcos()r(F θΦ = as solution of the Laplace equation, equation (3.2) would modify into: 0FtF r p r F r 1 r F 2 2 2 2 2 =−−+ ∂ ∂ ∂ ∂ (3.16) Equation (3.16) is called the modified Bessel equation. Its solution can immediately be deduced from the original solution (3.8) )irt(BJ)irt(AJ)rt(F pp −+= (3.17) where ∑ ∞ ++ ++ −= 0 k2ppk2 k p )1kp(!k )2/rt(i )()irt(J Γ ∑ ∞ + ++ = 0 k2p p p )1kp(!k )2/rt( i)irt(J Γ Hence we define the function Ip, modified Bessel function of the first kind, as a real function, solution of the modified Bessel equation. ∑ ∞ + − ++ == 0 k2p p p p )1kp(!k )2/rt( )irt(JiI Γ (3.18) pp BIAI)rt(F −+= (3.19) It is easy to deduce from (3.18) that when p is an integer, let say p = n, In = I-n. In that case, the second solution is usually defined by       − −− = − → np )rt(I)rt(I 2 )( lim)rt(K pp n np n (3.20) which is called the modified Bessel function of the second kind. The function Kp(rt) is defined for unrestricted values of p by the equation π π psin )rt(I)rt(I 2 )rt(K pp p − = − (3.21) Applying de L’Hospital’s rule on (3.20) and (3.21), one verifies that )rt(Klim)rt(K p np n → = In a similar way as given in § 3.4.2 one obtains:
  • 36. PAVEMENT DESIGN AND EVALUATION 22 ∑ − − −− −= 1n 0 k2n k n )2/rt(!k )!1kn( )( 2 1 )rt(K (3.22)       ++−+− + −+ ∑ ∞ + + )1kn( 2 1 )1k( 2 1 )2/trln( )!kn(!k )2/rt( )( 0 k2n 1n ΨΨ k 1 ... 3 1 2 1 1 1 )1k()1( +++++−=+−= γΨγΨ The modified Bessel functions of the first kind and of order 0 and 1 are represented in Figure 3.3. Figure 3.3 Modified Bessel functions of the first kind of order 0 and 1 The modified Bessel functions of the second kind and of order 0 and 1 are given in Figure 3.4. Figure 3.4 Modified Bessel functions of the second kind and of order 0 and 1
  • 37. THE GENERAL SOLUTION OF THE BESSEL EQUATION 23 3.6 The ber and bei functions. Consider next modified Bessel equation: 0Ft)i( r F r 1 r F 2 2 2 =±−+ ∂ ∂ ∂ ∂ (3.23) which solutions of the first kind are ... !4!4 )2/rt( !3!3 )2/rt(i !2!2 )2/rt( !1!1 )2/rt(i 1)irt(I 8642 0 +−−+= ... !4!4 )2/rt( !3!3 )2/rt(i !2!2 )2/rt( !1!1 )2/rt(i 1)irt(I 8642 0 ++−−=− We define: )rt(ibei)rt(ber)irt(I0 ±=± (3.24) where ... !6!6 )2/rt( !4!4 )2/rt( !2!2 )2/rt( 1)rt(ber 1284 +−+−= (3.25) ... !5!5 )2/rt( !3!3 )2/rt( !1!1 )2/rt( )rt(bei 1062 ++−= (3.26) Notice that: ( ) ( ) ( ) ( ) ( ) ( ) 2 irtJirtJ 2 irtiJirtiJ 2 irtIirtI )rt(ber 000000 +− = −+ = −+ = (3.27) ( ) ( ) ( ) ( ) ( ) ( ) i2 irtJirtJ i2 irtiJirtiJ i2 irtIirtI )rt(bei 000000 −− = −− = −− = (3.28) The ber and bei functions can be depicted in Figure 3.5 Figure 3.5 The ber and bei functions
  • 38. PAVEMENT DESIGN AND EVALUATION 24 3.7 The ker and kei functions. Again consider equation (3.23), which solutions of the second kind are: ( ) ( ) ... 3 1 2 1 1 !3!3 2 irt 2 1 1 !2!2 2 irt !1!1 2 irt 2 irt logirtIirtK 642 00       ++         +      +         +         +      +−= γ Recall that: 4 i 2 rt loge 2 1 2 rt logilog 2 1 2 rt logilog 2 rt log 2 irt log 2/i ππ +=+=+=+= and applying (3.24) ( ) [ ] ... 3 1 2 1 1 !3!3 2 rt i 2 1 1 !2!2 2 rt !1!1 2 rt i 4 i 2 rt log)rt(ibei)rt(berirtK 642 0       ++       −      +       −       +    +++−= π γ ( ) [ ] ... 3 1 2 1 1 !3!3 2 rt i 2 1 1 !2!2 2 rt !1!1 2 rt i 4 i 2 rt log)rt(ibei)rt(berirtK 642 0       ++       +      +       −       −      −+−−=− π γ We define: ( ) ( ) 2 irtKirtK )rtker( 00 −+ = (3.29) ( ) ( ) i2 irtKirtK )rt(kei 00 −− = (3.30) [ ] ( ) ... 2 1 1 !2!2 2/rt )rt(bei 4 )2/rtlog()rt(ber)rtker( 4       +−++−= π γ (3.31) [ ] ( ) ( ) ... 3 1 2 1 1 !3!3 2/rt !1!1 2/rt )rt(ber 4 )2/rtlog()rt(bei)rt(kei 62       ++−+−+−= π γ (3.32) The ker and kei functions are given in figure 3.6. Figure 3.6 The ker and kei functions
  • 39. THE GENERAL SOLUTION OF THE BESSEL EQUATION 25 3.8 Resolution of the equation ∇2 ∇2 w + w =0 Consider the next equation: 0w dr dw r 1 dr wd dr d r 1 dr d 2 2 2 2 =+         +         + (3.33) The solution can be obtained by splitting (3.33) into two simultaneous differential equations z dr dw r 1 dr wd 2 2 =+ (3.34) w dr dz r 1 dr zd 2 2 −=+ (3.35) Both equations are verified together if iwz m= . Indeed, if, for example, z = iw (3.34) becomes 0iw dr dw r 1 dr wd 2 2 =−+ , and (3.35) becomes 0w dr dw r 1 i dr wd i 2 2 =++ or 0iw dr dw r 1 dr wd 2 2 =−+ Hence the solution of (3.33) is given by the solution of 0w)i( dr dw r 1 dr wd 2 2 =±−+ (3.36) The equations (3.25), (3.26), (3.31) and (3.32) give the solution of (3.36), for its two signs )r(Dkei)rker(C)r(Bbei)r(Aberw +++= (3.37) 3.9 The modified form of the Bessel equation Consider the next differential equation: 0FrF r p r F r 1 )21( r F )1(222 2 222 2 2 =+ − +−+ −γ γβ γα ∂ ∂ α ∂ ∂ (3.38) which has for solutions, as can be verified by substitution, )r(JBr)r(JAr)r(F pp γαγα ββ −+= (3.39) or, if p is an integer, )r(YBr)r(JAr)r(F pp γαγα ββ += (3.40) An interesting application of (3.38) is the resolution of the Laplace equation in two- dimensional Cartesian co-ordinates 0 yx 2 2 2 2 2 =+=∇ ∂ Φ∂ ∂ Φ∂ Φ Consider a solution such as Φ = F(x)ety
  • 40. PAVEMENT DESIGN AND EVALUATION 26 Hence 0e)x(Ft x )x(F ty2 2 2 2 =         +=∇ ∂ ∂ Φ (3.41) Comparing (3.41) with (3.38) yields: t12/1 === βγα 2/1p0p 222 ==− γα Hence )tx(JBx)tx(JAx)x(F 2/1 2/1 2/1 2/1 −+= (3.42) However )txsin(B)txcos(A)x(F += (3.43) is also a solution of (3.41). Therefore, one must conclude that there exists a relation between Bessel functions of order ± 1/2 and trigonometric functions.
  • 41. PROPERTIES OF THE BESSEL FUNCTIONS 27 Chapter 4 Properties of the Bessel Functions. 4.1 Introductory note This chapter deals with the most important properties of Bessel functions: derivatives, functions of half order, asymptotic values, indefinite integrals and relations between functions of different kind. 4.2 Helpful relations [ ] )rt(J)rt(t)rt(J)rt( dr d 1p p p p −= [ ] )rt(J)rt(t)rt(J)rt( dr d 1p p p p + −− −= )rt(J r p )rt(tJ)rt(J dr d p1pp −= − )rt(J r p )rt(tJ)rt(J dr d p1pp +−= + )rt(J rt p2 )rt(J)rt(J p1p1p =+ +− t )rt(rJ dr)rt(rJ 1 0 =∫ ∫ −= t )rt(J dr)rt(J 0 1 )rtsin( rt 2 )rt(J 2/1 π = )rtcos( rt 2 )rt(J 2/1 π =− [ ]rtrt 2/1 ee rt2 1 )rt(I − −= π [ ]rtrt 2/1 ee rt2 1 )rt(I − − += π rt 2/12/1 e rt2 KK − − == π       −−≅ 2 p 4 rtcos rt 2 )rt(J p ππ π for high values of rt       ++≅− 2 p 4 rtsin rt 2 )rt(J p ππ π for high values of rt       −−≅ 2 p 4 rtsin rt 2 )rt(Yp ππ π for high values of rt
  • 42. PAVEMENT DESIGN AND EVALUATION 28       ++≅− 2 p 4 rtcos rt 2 )rt(Y p ππ π for high values of rt )4/2/prt(i)1( p e rt 2 H ππ π −− ≅ for high values of rt )4/2/prt(i)2( p e rt 2 H ππ π −−− ≅ for high values of rt ( )i)2/1p(rtrt p ee rt2 1 )rt(I π π ++− +≅ for high values of rt       −≅ 82 rt cos rt2 e )rt(ber 2/rt π π for high values of rt       −≅ 82 rt sin rt2 e )rt(bei 2/rt π π for high values of rt       +≅ − 82 rt cose rt2 )rtker( 2/rt ππ for high values of rt       +−≅ − 82 rt sine rt2 )rt(kei 2/rt ππ for high values of rt ( ) )z(JezeJ)z(J p pii pp ±±± ==− ππ ( ) )z(JezeJ p pmiim p ±± = ππ 2 )rt(H)rt(H )rt(J )2( p )1( p p + = )psin( )pcos()rt(Y)rt(Y )rt(J pp p π π− = − )psin(i e)rt(J)rt(J )rt(H ip pp)1( p π π− − − = )psin(i e)rt(J)rt(J )rt(H ip pp)2( p π π − − = − )rt(J )psin( )mpsin( e2)rt(He)rte(H p ip)1( p impim)1( p π ππππ −− −= )rt(J )psin( )mpsin( e2)rt(He)rte(H p ip)2( p impim)2( p π ππππ += − )()( )2/2/ ππ i p pi p rteJirtI − = )rti(He 2 i)rt(K )1( p 2/pi p ππ = )rti(He 2 i)rt(K )2( p 2/pi p ππ −=
  • 43. PROPERTIES OF THE BESSEL FUNCTIONS 29 4.3 Derivatives of Bessel functions 4.3.1 Derivative of (rt)p Jp(rt) By definition of the Bessel function [ ]         ++ −= ∑ ∞ + + 0 k2p k2p2 k p p )1kp(!k2 )rt( )( dr d )rt(J)rt( dr d Γ ∑ ∞ + −+ ++ + −= 0 k2p 1k2p2 k )1kp(!k2 t)rt)(kp(2 )( Γ )kp(!k2 t)rt( )( 1k2p 1k2p2 0 k + −= −+ −+∞ ∑ Γ [ ] ∑ ∞ +− +− ++− −= 0 k2)1p( k2)1p( kp 1k)1p(!k2 )rt( )()rt(t Γ [ ] )rt(J)rt(t)rt(J)rt( dr d 1p p p p −= (4.1) 4.3.2 Derivative of (rt)-pJp(rt) One finds similarly [ ] )rt(J)rt(t)rt(J)rt( dr d 1p p p p + −− −= (4.2) 4.3.3 Derivative of Jp(rt) Deriving the left member of (4.1) by parts yields [ ] )rt(J r p )rt(tJ)rt(J dr d p1pp −= − (4.3) deriving the left member of (4.2) by parts yields [ ] )rt(J r p )rt(tJ)rt(J dr d p1pp +−= + (4.4) Particularly [ ] )rt(tJ)rt(tJ)rt(J dr d 110 −== − (4.5) Adding (4.3) and (4.4) yields [ ] [ ])rt(J)rt(J 2 t )rt(J dr d 1p1pp +− −= (4.6) and subtracting yields the recurrence formula for Bessel functions )rt(J rt p2 )rt(J)rt(J p1p1p =+ +− (4.7)
  • 44. PAVEMENT DESIGN AND EVALUATION 30 4.4 Bessel functions of half order 4.4.1 Values of J1/2(rt), J-1/2(rt), J3/2(rt), J-3/2(rt) Expanding the Bessel functions in their series and recalling that Γ(1/2) = √π yields: )rtsin( rt 2 )rt(J 2/1 π = (4.8) )rtcos( rt 2 )rt(J 2/1 π =− (4.9)       −= )rtcos( rt )rtsin( rt 2 )rt(J 2/3 π (4.10)       +−=− )rtsin( rt )rtcos( rt 2 )rt(J 2/3 π (4.11) 4.3.2. Values of I1/2(rt), I-1/2(rt) Expanding the Bessel functions in their series yields [ ]rtrt 2/1 ee rt2 1 )rt(I − −= π (4.12) [ ]rtrt 2/1 ee rt2 1 )rt(I − − += π (4.13) 4.3.3. Values of K1/2(rt), K-1/2(rt) By definition (3.21) rt2/12/1 2/1 e rt2)2/sin( )rt(I)rt(I 2 )rt(K −− = − = π π π (4.14) rt2/12/1 2/1 e rt2)2/sin( )rt(I)rt(I 2 )rt(K −− − = − = π π π (4.15) Hence )rt(K)rt(K 2/12/1 −= (4.16) 4.4. Values of K0(rt) and Kn(rt) By definition (3.21) π π psin )rt(I)rt(I 2 lim)rt(K pp 0p 0 − = − → Applying de l’Hospital’s rule yields and omitting the lim sign       −= − = − − dp )rt(dI dp )rt(dI 2 1 pcos dp )rt(dI dp )rt(dI 2 )rt(K pp pp 0 ππ π
  • 45. PROPERTIES OF THE BESSEL FUNCTIONS 31         ++ − +− = ∑∑ +− )1pk(!k )2/rt( dp d )1pk(!k )2/rt( dp d 2 1 )rt(K pk2pk2 0 ΓΓ     +−+− +−++−− = −− ∑ )1pk()1pk( )1pk(')2/rt()1pk()2/rtlog()2/rt( !k2 )2/rt( )rt(K ppk2 0 ΓΓ ΓΓ     ++++ ++−++ − )1pk()1pk( )1pk(')2/rt()1pk()2/rtlog()2/rt( pp ΓΓ ΓΓ Letting p → 0       + + +−= ∑ )1k( )1k(' )2/rtlog( !k!k )2/rt( )rt(K k2 0 Γ Γ Developing Γ’(k+1) as in § 3.3.2 yields [ ]       ++++−= 2 1 1 !2!2 )2/rt( !1!1 )2/rt( )2/rtlog()rt(I)rt(K 42 00 γ ⋅⋅⋅      +++ 3 1 2 1 1 !3!3 )2/rt( 6 (4.17) For n ≠ 0, (4.17) is relatively easily extended to [ ] ∑ −+ −−− ++−= n 0 nk2 k n 1n n )2/rt( !k )!1kn()( 2 1 )2/rtlog()rt(I)()rt(K γ [ ]∑ ∞ + + ++−+ 0 nk2 n )!nk(!k )2/rt( )kn()k()( 2 1 ΦΦ (4.18) 4.5 Asymptotic values 4.5.1 Asymptotic values for Jp and J-p Transform Bessel equation (3.2): 0FtF r p r F r 1 r F 2 2 2 2 2 =+−+ ∂ ∂ ∂ ∂ by writing F(rt) = G(rt)/(rt)1/2 0G r 4/1p t r G 2 2 2 2 2 =         − −+ ∂ ∂ (4.19) When p = ½, the equation reduces to 0Gt r G 2 2 2 =+ ∂ ∂ (4.20) whose general solution is given by )rtsin(B)rtcos(AG += Hence )rtsin()rt(B)rtcos()rt(AF 2/12/1 −− += (4.21)
  • 46. PAVEMENT DESIGN AND EVALUATION 32 but also: )rt(DJ)rt(CJF 2/12/1 += − (4.22) Equations (4.21) and (4.22) confirm, as we knew by (4.8) and (4.9), that there exists a relation between the Bessel functions of half order and the trigonometric functions, and that this relation is valid for all values of the argument, thus in particular for high values of the argument. Hence the equations: )rtsin( rt 2 )rt(J 2/1 π = )rtcos( rt 2 )rt(J 2/1 π =− can be considered as the asymptotic equations for the Bessel functions of half order. Thus (4.19) must have, for p not being an integer, two approximate values for high values of the argument such as )rtcos()rt(A)rt(J p 2/1 pp α+≅ − (4.23) )rtsin()rt(B)rt(J p 2/1 pp β+≅ − − (4.24) The coefficients αp and βp must be determined in such a way that equations (4.23) and (4.24) are linearly independent and compatible with the definitions of Jp and J-p. Derive, with respect to r, Jp in (4.23) )rtsin(t)rt(A)rtcos()rt(A 2 t )rt(J p 2/1 pp 2/3 p ' p αα +−+−= −− For high values of the argument the first term can be neglected against the second )rtsin(t)rt(A)rt(J p 2/1 p ' p α+−≅ − Further (4.3) and (4.4) simplify for high values of the argument )rt(tJ)rt(J 1p ' p −≅ )rt(tJ)rt(J 1p ' p +−≅ Hence )rtcos()rt(tA)rtsin(t)rt(A 1p 2/1 1pp 2/1 p − − − − +≅+− αα (4.25) )rtcos()rt(tA)rtsin(t)rt(A 1p 2/1 1pp 2/1 p + − + − +−≅+− αα (4.26) If we choose Ap = Ap-1 = Ap+1 = A and αp = k - pπ/2 in such a way that A and k are independent from p , we notice that equations (4.25) and (4.26) are satisfied. Indeed       − −+=      −+− 2 )1p( krtcosA 2 p krtsinA ππ Then       −+≅ − 2 p krtcos)rt(A)rt(J 2/1 p π This equation must be satisfied for all values of p, thus also for p=1/2 for which       −−== 44 rtcos rt 2 )rtsin( rt 2 )rt(J 2/1 ππ ππ Hence A = (2π)1/2 and k = - π/4
  • 47. PROPERTIES OF THE BESSEL FUNCTIONS 33 Finally       −−≅ 2 p 4 rtcos rt 2 )rt(J p ππ π (4.27) Replacing p by –p in (4.27) yields       ++≅− 2 p 4 rtsin rt 2 )rt(J p ππ π (4.28) 4.5.2 Asymptotic values for Yp and Y-p Since Yp and Y-p are the paired second solutions with Jp and J-p we shall admit implicitly the following asymptotic equations       −−≅ 2 p 4 rtsin rt 2 )rt(Yp ππ π (4.29)       ++≅− 2 p 4 rtcos rt 2 )rt(Y p ππ π (4.30) 4.5.3 Asymptotic values for Hp (1) and Hp (2) By application of the definitions (3.14) and (3.15), one immediately finds: )2/p4/rt(i)1( p e rt 2 H ππ π −− = (4.31) )2/p4/rt(i)2( p e rt 2 H ππ π −−− = (4.32) 4.5.4 Asymptotic values for Ip and I-p Transform Bessel equation (3.16) 0FtF r p r F r 1 r F 2 2 2 2 2 =−−+ ∂ ∂ ∂ ∂ by writing F(rt) = G(rt)/(rt)1/2 0G r 4/1p t r G 2 2 2 2 2 =         − +− ∂ ∂ (4.33) When p = ½, the equation reduces to 0Gt r G 2 2 2 =− ∂ ∂ (4.34) whose general solution is given by rtrt BeAeG − += Hence rt2/1rt2/1 e)rt(Be)rt(AF −−− += (4.35) but also
  • 48. PAVEMENT DESIGN AND EVALUATION 34 )rt(DI)rt(CIF 2/12/1 −+= (4.36) Equation (4.35) is valid for all values of the argument, thus in particular for high values of the argument. Hence the equations: [ ]rtrt 2/1 ee rt2 1 )rt(I − −= π [ ]rtrt 2/1 ee rt2 1 )rt(I − − += π can be considered as the asymptotic equations for the modified Bessel functions of half order. Hence (4.33) must have, for p not an integer, two approximate values for high values of the argument such as       += ++−++− )2/1p(rt)2/1p(rt2/1 pp ee)rt(A)rt(I βα (4.37)       += +−+−+−+− − )2/1p(rt)2/1p(rt2/1 pp ee)rt(B)rt(I βα (4.38) The coefficients α and β must be determined in such a way that equations (4.37) and (4.38) are linearly independent and compatible with the definitions of Ip and I-p. Derive, with respect to r, Ip in (4.37) [ ])2/1p(rt)2/1p(rt2/3 p ' p ee)rt(A 2 t )rt(I ++−++− +−= βα       −+ ++−++− )2/1p(rt)2/1p(rt2/1 p eet)rt(A βα For high values of the argument the first term can be neglected against the second       −≅ ++−++− )2/1p(rt)2/1p(rt2/1 pp eet)rt(A)rt('I βα Further simplify the derivatives of Ip(rt) for high values of the argument )rt(tJ)rt(I 1p ' p −≅ )rt(tJ)rt(I 1p ' p +≅ Hence       − ++−++− )2/1p(rt)2/1p(rt2/1 p eet)rt(A βα       += −+−−+− − )2/1p(rt)2/1p(rt2/1 1p eet)rt(A βα (4.39)       − ++−++− )2/1p(rt)2/1p(rt2/1 p eet)rt(A βα       += ++−++− + )2/3p(rt)2/3p(rt2/1 1p eet)rt(A βα (4.40) If we choose Ap = Ap-1 = Ap+1 = A and α = 0, β = πi in such a way that A and β are independent from p , we notice that equations (4.39) and (4.40) are satisfied.
  • 49. PROPERTIES OF THE BESSEL FUNCTIONS 35       − ++−− i)2/1p(rtrt2/1 ee)rt(A π       += −+−− i)2/1p(rtrt2/1 ee)rt(A π Hence       +≅ ++−− i)2/1p(rtrt2/1 p ee)rt(A)rt(I π This equation must be satisfied for all values of p, thus also for p=1/2 for which [ ]rtrt 2/1 ee rt2 1 )rt(I − −= π Hence A = (2π)1/2 Finally       +≅ ++− i)2/1p(rtrt p ee rt2 1 )rt(I π π (4.41) Replacing p by –p in (4.41) yields       +≅ +−+− − i)2/1p(rtrt p ee rt2 1 )rt(I π π (4.42) 4.5.5 Asymptotic value for Kn Consider definition (3.20): np )rt(I)rt(I 2 )( lim)rt(K pp n np n − −− = − → (4.43) For large values of the argument np eeee rt2 1 2 )( )rt(K i)2/1p(rtrti)2/1p(rtrtn n − −−+− ≅ ++−−−− ππ π np eeie rt2 1 2 )( )rt(K ipiprt n n −       − − ≅ −− ππ π Applying de l’Hospital’s rule yields rt n n e)ncos( rt2 2 2 )( )rt(K −− ≅ ππ π rt n e rt2 )rt(K − ≅ π (4.44) In this form Kn(rt) may be generalised into rt p e rt2 )rt(K − ≅ π (4.45) where p not an integer and also
  • 50. PAVEMENT DESIGN AND EVALUATION 36 rt 0 e rt2 )rt(K − ≅ π (4.46) 4.5.6 Asymptotic values for ber and bei For high values of the argument (4.27) yields ( )       −      ≅ 4 irtcos irt 2 irtJ 2/1 0 π π With (i)-1/4 = e-iπ/8 and √i = (1 + i)/√2 2 eeee 4 irtcos 2/rt)4/2/rt(i2/rt)4/2/rt(i     + =      − −−−− ππ π 2 ee 4 irtcos 2/rt)4/2/rt(i π π −− ≅      − ( ) )8/2rt(i 2/rt 0 e rt2 e irtJ π π −− ≅ ( )             −−      −≅ 82 rt sini 82 rt cos rt2 e irtJ 2/rt 0 ππ π (4.47) Similarly ( )             −+      −≅− 82 rt sini 82 rt cos rt2 e irtJ 2/rt 0 ππ π (4.48) Hence ( ) ( )       −= +− ≅ 82 rt cos rt2 e 2 irtJirtJ )rt(ber 2/rt 00 π π (4.49) ( ) ( )       −= −− ≅ 82 rt sin rt2 e i2 irtJirtJ )rt(bei 2/rt 00 π π (4.50) 4.5.7 Asymptotic values for ker and kei Recall equations (3.29) and (3.30) ( ) ( ) 2 irtKirtK )rtker( 00 −+ = (4.51) ( ) ( ) i2 irtKirtK )rt(kei 00 −− = (4.52) Apply (4.46) irt 0 e irt2 )irt(K − = π
  • 51. PROPERTIES OF THE BESSEL FUNCTIONS 37 2/)i1(rt8/i ee rt2 +−− = ππ )8/2/rt(i2/rt ee rt2 ππ +−− =             +−      += − 82 rt sini 82 rt cose rt2 )irt(K 2/rt 0 πππ (4.53) Similarly obtain             ++      +=− − 82 rt sini 82 rt cose rt2 )irt(K 2/rt 0 πππ (4.54) Adding and subtracting (4.53) and (4.54) yields       += − 82 rt cose rt2 )rtker( 2/rt ππ (4.55)       +−= − 82 rt sine rt2 )rt(kei 2/rt ππ (4.56) 4.6 Indefinite integrals of Bessel functions 4.6.1 Fundamental relations In § 4.2.we have derived the following derivatives of Bessel functions: [ ] )rt(J)rt(t)rt(rtJ dr d 01 = [ ] )rt(tJ)rt(J dr d 10 −= From those equations we easily deduce next fundamental integrals ∫ = t )rt(rJ dr)rt(rJ 1 0 (4.57) ∫ −= t )rt(J dr)rt(J 0 1 (4.58) 4.6.2 The integral ∫rn J0(rt)dr Integrating by parts solves the integral: ∫ ∫ −− −= dr)rt(Jr t )1n( t )rt(J rdr)rt(Jr 1 1n1n 0 n ∫ ∫ − − − − +−= dr)rt(Jr t )1n( )rt(J t r dr)rt(Jr 0 2n 0 1n 1 1n Hence ∫ ∫ −− − − − += dr)rt(Jr t )1n( )rt(Jr t )1n( t )rt(J rdr)rt(Jr 0 2n 2 2 0 1n 2 1n 0 n (4.59) If n is odd, formula (4.59) leads to (4.57). If n is even, formula (4.59) leads to ∫J0(rt)dr, which is tabulated.
  • 52. PAVEMENT DESIGN AND EVALUATION 38 4.7 Relations between Bessel functions of different kind 4.7.1 Bessel functions with argument –rt Since Bessel’s equation is unaltered if rt is replaced by –rt, we must expect the functions J±p(-rt) to be solutions of the equations satisfied by J±p(rt). Considering the relation eπi = -1, we may write )rte(J)rt(J i pp π =− We can even consider the more general function Jp(emπi rt) where m is an integer. ( )∑ ++ −= + )1pk(!k 2/rte )()rte(J k2pim kim p Γ π π Restricting the complex exponent to its principal value we get ( ) pim)mk2pm(ik2pim eee πππ == ++ Hence ( ) )rt(Je )1pk(!k )2/rt( )(erteJ p pim k2p kpimim p πππ Γ ∑ = ++ −= + (4.60) 4.7.2 Relations between the three kinds of Bessel functions Consider (3.14) and (3.15) )rt(iY)rt(J)rt(H pp )1( p += )rt(iY)rt(J)rt(H pp )2( p −= Hence by multiplying Yp(rt) by cos(pπ) and subtracting from Y-p(rt) 2 )rt(H)rt(H )rt(J )2( p )1( p p + = (4.61) Consider (3.11) )psin( )rt(J)pcos()rt(J )rt(Y pp p π π −− = Replace p by –p )psin( )rt(J)pcos()rt(J )rt(Y pp p π π +− = − − Hence by subtracting and dividing by 2 )psin( )pcos()rt(Y)rt(Y )rt(J pp p π π− = − (4.62) Consider (3.14) together with (3.11) )rt(iY)rt(J)rt(H pp )1( p += )psin( )rt(J)pcos()rt(J i)rt(J)rt(H pp p )1( p π π −− +=
  • 53. PROPERTIES OF THE BESSEL FUNCTIONS 39 [ ] )psin(i )rt(J)psin(i)pcos()rt(J )rt(H pp)1( p π ππ −+−− = )psin(i e)rt(J)rt(J )rt(H ip pp)1( p π π− − − = (4.63) and similarly )psin(i e)rt(J)rt(J )rt(H ip pp)2( p π π − − = − (4.64) Add equations (4.63) and (4.64) together 2 HH )rt(J )2( p )1( p p + = (4.65) Multiply equation (4.63) by eipπ and (4.64) by e-ipπ and add together 2 e)rt(He)rt(H )rt(J ip)2( p ip)1( p p ππ − − + = (4.66) In equation (4.63) replace rt by rt emπI )psin(i e)rte(J)rte(J )rte(H ipim p im pim)1( p π πππ π − − − = together with equation (4.60) )sin( )()( )( ))1( π πππ π pi ertJertJe rteH ip p imp p imp im p − − − − = ( ) ( ) )psin(i eee)rt(J )psin(i e)rt(J)rt(Je )rte(H impimpip p ip pp imp im)1( p ππ πππππ π − + − = −−− − − applying equation (4.63) )rt(J )psin( )mpsin( e2)rt(He)rte(H p ip)1( p impim)1( p π ππππ −− −= (4.67) and similarly )rt(J )psin( )mpsin( e2)rt(He)rte(H p ip)2( p impim)2( p π ππππ += − (4.68) 4.7.3 Bessel functions of purely imaginary argument Consider (3.18) ∑ ∞ + − ++ == 0 k2p p p p )1kp(!k )2/rt( )irt(Ji)rt(I Γ Recall that eπi/2 = i, e3πi/2 = -i. Hence )irt(Je)rt(I p 2/ip p π− = (4.69) Apply (3.21)
  • 54. PAVEMENT DESIGN AND EVALUATION 40 π π psin )rt(I)rt(I 2 )rt(K pp p − = − π π ππ psin )irt(Je)irt(Je 2 )rt(K p 2/ip p 2/ip p − − − = π π π π psini )irt(Je)irt(J e 2 i)rt(K p ip p2/ip p − − − = Hence by (4.63) )irt(He 2 i)rt(K )1( p 2/ip p ππ = (4.70) and similarly )irt(He 2 i)rt(K )2( p 2/ip p ππ −= (4.71)