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• SOIL DYNAMICS

Arnold Verruijt

Delft University of Technology

1994, 2009

• PREFACE

This book gives the material for a course on Soil Dynamics, as given for about 10 years at the Delft University of Technology for students ofcivil engineering, and updated continuously since 1994.

The book presents the basic principles of elastodynamics and the major solutions of problems of interest for geotechnical engineering. Formost problems the full analytical derivation of the solution is given, mainly using integral transform methods. These methods are presentedbriefly in an Appendix. The elastostatic solutions of many problems are also given, as an introduction to the elastodynamic solutions, and aspossible limiting states of the corresponding dynamic problems. For a number of problems of elastodynamics of a half space exact solutionsare given, in closed form, using methods developed by Pekeris and De Hoop. Some of these basic solutions are derived in full detail, to assistin understanding the beautiful techniques used in deriving them. For many problems the main functions for a computer program to producenumerical data and graphs are given, in C. Some approximations in which the horizontal displacements are disregarded, an approximationsuggested by Westergaard and Barends, are also given, because they are much easier to derive, may give a first insight in the response of afoundation, and may be a stepping stone to solving the more difficult complete elastodynamic problems.

The book is directed towards students of engineering, and may be giving more details of the derivations of the solutions than strictly neces-sary, or than most other books on elastodynamics give, but this may be excused by my own difficulties in studying the subject, and by helpingstudents with similar difficulties.

The book starts with a chapter on the behaviour of the simplest elementary system, a system consisting of a mass, suppported by a linearspring and a linear damper. The main purpose of this chapter is to define the basic properties of dynamical systems, for future reference. Inthis chapter the major forms of damping of importance for soil dynamics problems, viscous damping and hysteretic damping, are defined andtheir properties are investigated.

Chapters 2 and 3 are devoted to one dimensional problems: wave propagation in piles, and wave propagation in layers due to earthquakesin the underlying layers, as first developed in the 1970s at the University of California, Berkeley. In these chapters the mathematical methodsof Laplace and Fourier transforms, characteristics, and separation of variables, are used and compared. Some simple numerical models are alsopresented.

The next two chapters (4 and 5) deal with the important effect that soils are ususally composed of two constituents: solid particles and afluid, usually water, but perhaps oil, or a mixture of a liquid and gas. Chapter 4 presents the classical theory, due to Terzaghi, of semi-staticconsolidation, and some elementary solutions. In chapter 5 the extension to the dynamical case is presented, mainly for the one dimensionalcase, as first presented by De Josselin de Jong and Biot, in 1956. The solution for the propagation of waves in a one dimensional column ispresented, leading to the important conclusion that for most problems a practically saturated soil can be considered as a medium in which the

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solid particles and the fluid move and deform together, which in soil mechanics is usually denoted as a state of undrained deformations. For anelastic solid skeleton this means that the soil behaves as an elastic material with Poissons ratio close to 0.5.

Chapters 6 and 7 deal with the solution of problems of cylindrical and spherical symmetry. In the chapter on cylindrically symmetricproblems the propagation of waves in an infinite medium introduces Rayleighs important principle of the radiation condition, which expressesthat in an infinite medium no waves can be expected to travel from infinity towards the interior of the body.

Chapters 8 and 9 give the basic theory of the theory of elasticity for static and dynamic problems. Chapter 8 also gives the solution for someof the more difficult problems, involving mixed boundary value conditions. The corresponding dynamic problems still await solution, at leastin analytic form. Chapter 9 presents the basics of dynamic problems in elastic continua, including the general properties of the most importanttypes of waves : compression waves, shear waves, Rayleigh waves and Love waves, which appear in other chapters.

Chapter 10, on confined elastodynamics, presents an approximate theory of elastodynamics, in which the horizontal deformations areartificially assumed to vanish, an approximation due to Westergaard and generalized by Barends. This makes it possible to solve a variety ofproblems by simple means, and resulting in relatively simple solutions. It should be remembered that these are approximate solutions only,and that important features of the complete solutions, such as the generation of Rayleigh waves, are excluded. These approximate solutionsare included in the present book because they are so much simpler to derive and to analyze than the full elastodynamic solutions. The fullelastodynamic solutions of the problems considered in this chapter are given in chapters 11 13.

Chapter 13 presents the solution for a point load on an elastic half space, a problem first solved analytically by Pekeris. The solution isderived using integral transforms and an elegant transformation theorem due to Bateman and Pekeris. In this chapter numerical values areobtained using numerical integration of the final integrals.

In chapter 14 some problems of moving loads are considered. Closed form solutions appear to be possible for a moving wave load, and for amoving strip load, assuming that the material possesses some hysteretic damping.

Chapter 15, finally, presents some practical considerations on foundation vibrations. On the basis of solutions derived in earlier chaptersapproximate solutions are expressed in the form of equivalent springs and dampings.

This is the version of the book in PDF format, which can be downloaded from the authors website , and can beread using the ADOBE ACROBAT reader. This website also contains some computer programs that may be useful for a further illustration of

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the solutions. Updates of the book and the programs will be published on this website.

The text has been prepared using the LATEX version (Lamport, 1994) of the program TEX (Knuth, 1986). The PICTEX macros (Wichura,1987) have been used to prepare the figures, with color being added in this version to enhance the appearance of the figures. Modern softwareprovides a major impetus to the production of books and papers in facilitating the illustration of complex solutions by numerical and graphicalexamples. In this book many solutions are accompanied by parts of computer programs that have been used to produce the figures, so thatreaders can compose their own programs. It is all the more appropriate to acknowledge the effort that must have been made by earlier authorsand their associates in producing their publications. A case in point is the famous paper by Lamb, more than a century ago, with manyillustrative figures, for which the computations were made by Mr. Woodall.

Thanks are due to Professor A.T. de Hoop for his many helpful and constructive comments and suggestions, and to Dr. C. Cornejo Cordovafor several years of joint research. Many comments of other colleagues and students on early versions of this book have been implemented inlater versions, and many errors have been corrected. All remaining errors are the authors responsibility, of course. Further comments will begreatly appreciated.

Delft, September 1994; Papendrecht, March 2009 Arnold Verruijt

Merwehoofd 13351 NA PapendrechtThe Netherlandstel. +31.78.6154399e-mail : [email protected] : http://geo.verruijt.net

1. Vibrating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1 Single mass system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2 Characterization of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Free vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Forced vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Equivalent spring and damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 Solution by Laplace transform method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.7 Hysteretic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2. Waves in Piles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1 One-dimensional wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Solution by Laplace transform method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4 Solution by characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Reflection and transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.6 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.7 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.8 Modeling a pile with friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3. Earthquakes in Soft Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 Earthquake parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 Horizontal vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.3 Shear waves in a Gibson material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4 Hysteretic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

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4. Theory of Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2 Conservation of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Darcys law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Equilibrium equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 Drained deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6 Undrained deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Cryers problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.8 Uncoupled consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.9 Terzaghis problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5. Dynamics of Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.1 Basic differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.2 Propagation of plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4 Analytical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.5 Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6. Cylindrical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.1 Static problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Dynamic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3 Propagation of a shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6.4 Radial propagation of shear waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

7. Spherical Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.1 Static problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.2 Dynamic problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.3 Propagation of a shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

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8. Elastostatics of a Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.1 Basic equations of elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

8.2 Boussinesq problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.3 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.4 Axially symmetric problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

8.5 Mixed boundary value problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8.6 Confined elastostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

9. Elastodynamics of a Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

9.1 Basic equations of elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

9.2 Compression waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9.3 Shear waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

9.4 Rayleigh waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.5 Love waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

10. Confined Elastodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

10.1 Line load on a half space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

10.2 Line pulse on a half space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

10.3 Point load on a half space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

10.4 Periodic load on a half space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

11. Line Load on Elastic Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.1 Line pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.2 Constant line load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

12. Strip Load on Elastic Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

12.1 Strip pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282

12.2 Strip load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

13. Point Load on Elastic Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.1 Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

13.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338

• 8

14. Moving Loads on Elastic Half Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

14.1 Moving wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

14.2 Moving strip load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

15. Foundation Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

15.1 Foundation response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

15.2 Equivalent spring and damper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

15.3 Soil properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

15.4 Propagation of vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

15.5 Design criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

Appendix A. Integral Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

A.1 Laplace transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

A.2 Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

A.3 Hankel transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402

A.4 De Hoops inversion method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Appendix B. Dual Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

Appendix C. Bateman-Pekeris Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

• Chapter 1

VIBRATING SYSTEMS

In this chapter a classical basic problem of dynamics will be considered, for the purpose of introducing various concepts and properties. Thesystem to be considered is a single mass, supported by a linear spring and a viscous damper. The response of this simple system will beinvestigated, for various types of loading, such as a periodic load and a step load. In order to demonstrate some of the mathematical techniquesthe problems are solved by various methods, such as harmonic analysis using complex response functions, and the Laplace transform method.

1.1 Single mass system

Consider the system of a single mass, supported by a spring and a dashpot, in which the damping is of a viscous character, see Figure 1.1. The

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Figure 1.1: Mass supported by spring and damper.

spring and the damper form a connection between the mass and an immovable base(for instance the earth).

According to Newtons second law the equation of motion of the mass is

md2u

dt2= P (t), (1.1)

where P (t) is the total force acting upon the mass m, and u is the displacement ofthe mass.

It is now assumed that the total force P consists of an external force F (t), andthe reaction of a spring and a damper. In its simplest form a spring leads to a forcelinearly proportional to the displacement u, and a damper leads to a response linearlyproportional to the velocity du/dt. If the spring constant is k and the viscosity of the

damper is c, the total force acting upon the mass is

P (t) = F (t) ku cdudt. (1.2)

Thus the equation of motion for the system is

md2u

dt2+ c

du

dt+ ku = F (t), (1.3)

9

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 10

The response of this simple system will be analyzed by various methods, in order to be able to compare the solutions with various problemsfrom soil dynamics. In many cases a problem from soil dynamics can be reduced to an equivalent single mass system, with an equivalent mass,an equivalent spring constant, and an equivalent viscosity (or damping). The main purpose of many studies is to derive expressions for thesequantities. Therefore it is essential that the response of a single mass system under various types of loading is fully understood. For this purposeboth free vibrations and forced vibrations of the system will be considered in some detail.

1.2 Characterization of viscosity

The damper has been characterized in the previous section by its viscosity c. Alternatively this element can be characterized by a response timeof the spring-damper combination. The response of a system of a parallel spring and damper to a unit step load of magnitude F0 is

u =F0k

[1 exp(t/tr)], (1.4)

where tr is the response time of the system, defined bytr = c/k. (1.5)

This quantity expresses the time scale of the response of the system. After a time of say t 4tr the system has reached its final equilibriumstate, in which the spring dominates the response. If t < tr the system is very stiff, with the damper dominating its behaviour.

1.3 Free vibrations

When the system is unloaded, i.e. F (t) = 0, the possible vibrations of the system are called free vibrations. They are described by thehomogeneous equation

md2u

dt2+ c

du

dt+ ku = 0. (1.6)

An obvious solution of this equation is u = 0, which means that the system is at rest. If it is at rest initially, say at time t = 0, then it remains atrest. It is interesting to investigate, however, the response of the system when it has been brought out of equilibrium by some external influence.For convenience of the future discussions we write

0 =k/m, (1.7)

and2 = 0tr =

c

m0=c0k

=ckm

. (1.8)

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 11

The quantity 0 will turn out to be the resonance frequency of the undamped system, and will be found to be a measure for the damping inthe system.

With (1.7) and (1.8) the differential equation can be written as

d2u

dt2+ 20

du

dt+ 20u = 0. (1.9)

This is an ordinary linear differential equation, with constant coefficients. According to the standard approach in the theory of linear differentialequations the solution of the differential equation is sought in the form

u = A exp(t), (1.10)

where A is a constant, probably related to the initial value of the displacement u, and is as yet unknown. Substitution into (1.9) gives

2 + 20+ 20 = 0. (1.11)

This is called the characteristic equation of the problem. The assumption that the solution is an exponential function, see eq. (1.10), appearsto be justified, if the equation (1.11) can be solved for the unknown parameter . The possible values of are determined by the roots of thequadratic equation (1.11). These roots are, in general,

1,2 = 0 02 1. (1.12)

These solutions may be real, or they may be complex, depending upon the sign of the quantity 21. Thus, the character of the response of thesystem depends upon the value of the damping ratio , because this determines whether the roots are real or complex. The various possibilitieswill be considered separately below. Because many systems are only slightly damped, it is most convenient to first consider the case of smallvalues of the damping ratio .

Small damping

When the damping ratio is smaller than 1, < 1, the roots of the characteristic equation (1.11) are both complex,

1,2 = 0 i0

1 2, (1.13)

where i is the imaginary unit, i =1. In this case the solution can be written as

u = A1 exp(i1t) exp(0t) +A2 exp(i1t) exp(0t), (1.14)

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 12

where1 = 0

1 2. (1.15)

The complex exponential function exp(i1t) may be expressed as

exp(i1t) = cos(1t) + i sin(1t). (1.16)

Therefore the solution (1.14) may also be written in terms of trigonometric functions, which is often more convenient,

u = C1 cos(1t) exp(0t) + C2 sin(1t) exp(0t). (1.17)

The constants C1 and C2 depend upon the initial conditions. When these initial conditions are that at time t = 0 the displacement is given tobe u0 and the velocity is zero, it follows that the final solution is

u

u0=

cos(1t )cos()

exp(0t), (1.18)

where is a phase angle, defined by

tan() =0

1=

1 2

. (1.19)

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0.0

1.0

1.0

2 3 4 5 0t

u/u0

= 0.00.10.20.5

Figure 1.2: Free vibrations of a weakly damped system.

The solution (1.18) is a damped sinusoidal vibration. It is a fluc-tuating function, with its zeroes determined by the zeroes of thefunction cos(1t ), and its amplitude gradually diminishing,according to the exponential function exp(0t).

The solution is shown graphically in Figure 1.2 for various val-ues of the damping ratio . If the damping is small, the frequencyof the vibrations is practically equal to that of the undamped sys-tem, 0, see also (1.15). For larger values of the damping ratiothe frequency is slightly smaller. The influence of the frequencyon the amplitude of the response then appears to be very large.For large frequencies the amplitude becomes very small. If thefrequency is so large that the damping ratio approaches 1 thecharacter of the solution may even change from that of a dampedfluctuation to the non-fluctuating response of a strongly dampedsystem. These conditions are investigated below.

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 13

Critical damping

When the damping ratio is equal to 1, = 1, the characteristic equation (1.11) has two equal roots,

1,2 = 0. (1.20)

In this case the damping is said to be critical. The solution of the problem in this case is, taking into account that there is a double root,

u = (A+Bt) exp(0t), (1.21)

where the constants A and B must be determined from the initial conditions. When these are again that at time t = 0 the displacement is u0and the velocity is zero, it follows that the final solution is

u = u0(1 + 0t) exp(0t). (1.22)

This solution is shown in Figure 1.3, together with some results for large damping ratios.

Large damping

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0.0

1.0

1.0

2 3 4 5 0t

u/u0

= 12

5

Figure 1.3: Free vibrations of a strongly damped system.

When the damping ratio is greater than 1 ( > 1) the character-istic equation (1.11) has two real roots,

1,2 = 0 02 1. (1.23)

The solution for the case of a mass point with an initial displace-ment u0 and an initial velocity zero now is

u

u0=

22 1

exp(1t)1

2 1exp(2t), (1.24)

where

1 = 0( 2 1), (1.25)

and

2 = 0( +2 1). (1.26)

This solution is also shown graphically in Figure 1.3, for = 2 and = 5. It appears that in these cases, with large damping, the system willnot oscillate, but will monotonously tend towards the equilibrium state u = 0.

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 14

1.4 Forced vibrations

In the previous section the possible free vibrations of the system have been investigated, assuming that there was no load on the system. Whenthere is a certain load, periodic or not, the response of the system also depends upon the characteristics of this load. This case of forced vibrationsis studied in this section and the next. In the present section the load is assumed to be periodic.

For a periodic load the force F (t) can be written, in its simplest form, as

F = F0 cos(t), (1.27)

where is the given circular frequency of the load. In engineering practice the frequency is sometimes expressed by the frequency of oscillationf , defined as the number of cycles per unit time (cps, cycles per second),

f = /2. (1.28)

In order to study the response of the system to such a periodic load it is most convenient to write the force as

F =

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 15

and2 =

c

m0=c0k

=ckm

. (1.34)

The quantity 0 is the resonance frequency of the undamped system, and is a measure for the damping in the system.With (1.30) and (1.32) the displacement is now found to be

u = u0 cos(t ), (1.35)

where the amplitude u0 is given by

u0 =F0/k

(1 2/20)2 + (2 /0)2, (1.36)

and the phase angle is given by

tan =2 /0

1 2/20. (1.37)

In terms of the original parameters the amplitude can be written as

u0 =F0/k

(1m2/k)2 + (c/k)2, (1.38)

and in terms of these parameters the phase angle is given by

tan =c/k

1m2/k. (1.39)

It is interesting to note that for the case of a system of zero mass these expressions tend towards simple limits,

m = 0 : u0 =F0/k

1 + (c/k)2, (1.40)

andm = 0 : tan =

c

k. (1.41)

The amplitude of the system, as described by eq. (1.36), is shown graphically in Figure 1.4, as a function of the frequency, and for various valuesof the damping ratio . It appears that for small values of the damping ratio there is a definite maximum of the response curve, which evenbecomes infinitely large if 0. This is called resonance of the system. If the system is undamped resonance occurs if = 0 =

k/m. This

is sometimes called the eigen frequency of the free vibrating system.

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 16

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..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................0

1

2

3

4

1 2 3 4 5/0

u0k/F0

= 0.1

0.2

0.51.0

2.0

Figure 1.4: Amplitude of forced vibration.

One of the most interesting aspects of the solution is thebehaviour near resonance. Actually the maximum responseoccurs when the slope of the curve in Figure 1.4 is horizontal.This is the case when du0/d = 0, or, with (1.36),

du0d

= 0 :

0=

1 22. (1.42)

For small values of the damping ratio this means that themaximum amplitude occurs if the frequency is very closeto 0, the resonance frequency of the undamped system. Forlarge values of the damping ratio the resonance frequency maybe somewhat smaller, even approaching 0 when 22 approaches1. When the damping ratio is very large, the system will nevershow any sign of resonance. Of course the price to be paid for

this very stable behaviour is the installation of a damping element with a very high viscosity.

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0

12

1 2 3 4 5/0

= 0.10.2 0.5 1.0 2.0

Figure 1.5: Phase angle of forced vibration.

The phase angle is shown in a similar way in Figure 1.5.For small frequencies, that is for quasi-static loading, the am-plitude of the system approaches the static response F0/k, andthe phase angle is practically 0. In the neighbourhood of theresonance frequency of the undamped system (i.e. if /0 1)the phase angle is about /2, which means that the amplitudeis maximal when the force is zero, and vice versa. For veryrapid fluctuations the inertia of the system may prevent prac-tically all vibrations (as indicated by the very small amplitude,see Figure 1.4), but the system moves out of phase, as indicatedby the phase angle approaching , see Figure 1.5.

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 17

Dissipation of work

An interesting quantity is the dissipation of work during a full cycle. This can be derived by calculating the work done by the force during afull cycle,

W = 2

t=0

Fdu

dtdt. (1.43)

With (1.27) and (1.35) one obtainsW = F0 u0 sin. (1.44)

Because the duration of a full cycle is 2/ the rate of dissipation of energy (the dissipation per second) is

D = W = 12 F0 u0 sin. (1.45)

This formula expresses that the dissipation rate is proportional to the amplitudes of the force and the displacement, and also to the frequency.This is because there are more cycles per second in which energy may be dissipated if the frequency is higher. The proportionality factor sin,which depends upon the phase angle , and thus upon the viscosity c, see (1.8), finally expresses the relative part of the energy that is dissipated.The maximum of this factor is 1, if the displacement and the force are out of phase. Its minimum is 0, when the viscosity of the damper is zero.

Using the expressions for tan and F0/u0 given in eqs. (1.36) and (1.37) the formula for the energy dissipation per cycle can also be writtenin various other forms. One of the simplest expressions appears to be

W = c u20. (1.46)

This shows that the energy dissipation is zero for static loading (when the frequency is zero), or when the viscosity vanishes. It may be notedthat the formula suggests that the energy dissipation may increase indefinitely when the frequency is very large, but this is not true. For veryhigh frequencies the displacement u0 becomes very small. In this respect the original formula, eq. (1.44), is a more useful general expression.

1.5 Equivalent spring and damping

The analysis of the response of a system to a periodic load, as characterized by a time function exp(it), often leads to a relation of the form

F = (K + iC)U, (1.47)

where U is the amplitude of a characteristic displacement, F is the amplitude of the force, and K and C may be complicated functions of theparameters representing the properties of the system, and perhaps also of the frequency . Comparison of this relation with eq. (1.31) showsthat this response function is of the same character as that of a combination of a spring and a damper. This means that the system can be

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 18

considered as equivalent with such a spring-damper system, with equivalent stiffness K and equivalent damping C. The response of the systemcan then be analyzed using the properties of a spring-damper system. This type of equivalence will be used in chapter 15 to study the responseof a vibrating mass on an elastic half plane. The method can also be used to study the response of a foundation pile in an elastic layer. Actually,it is often very convenient and useful to try to represent the response of a complicated system to a harmonic load in the form of an equivalentspring stiffness K and an equivalent damping C.

In the special case of a sinusoidal displacement one may write

u = ={U exp(it)} = U sin(t), (1.48)

if U is real. The corresponding force now is, with (1.47),

F = ={(K + iC)U exp(it)}, (1.49)

or,F = {K sin(t) + C cos(t)}U. (1.50)

This is another useful form of the general relation between force and displacement in case of a spring K and damping C.

1.6 Solution by Laplace transform method

It may be interesting to present also the method of solution of the original differential equation (1.3),

md2u

dt2+ c

du

dt+ ku = F (t), (1.51)

by the Laplace transform method. This is a general technique, that enables to solve the problem for any given load F (t), (Churchill, 1972). Asan example the problem will be solved for a step load, applied at time t = 0,

F (t) ={

0, if t < 0,F0, if t > 0.

(1.52)

It is assumed that at time t = 0 the system is at rest, so that both the displacement u and the velocity du/dt are zero at time t = 0.The Laplace transform of the displacement u is defined as

u =

0

u exp(st) dt, (1.53)

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 19

where s is the Laplace transform variable. The most characteristic property of the Laplace transform is that differentiation with respect to timet is transformed into multiplication by the transform parameter s. Thus the differential equation (1.51) becomes

(ms2 + cs+ k)u =

0

F (t) exp(st) dt = F0s. (1.54)

Again it is convenient to introduce the characteristic frequency 0 and the damping ratio , see (1.7) and (1.8), such that

k = 20m, (1.55)

andc = 2m0. (1.56)

The solution of the algebraic equation (1.54) is

u =F0/m

s (s+ 1)(s+ 2), (1.57)

where1 = 0( i

1 2), (1.58)

and2 = 0( + i

1 2). (1.59)

These definitions are in agreement with equations (1.25) and (1.26) given above.The solution (1.57) can also be written as

u =F0m

{ 112s

11(2 1)(s+ 1)

+1

2(2 1)(s+ 2)

}. (1.60)

In this form the solution is suitable for inverse Laplace transformation. The result is

u =F0m

{ 112

exp(1t)1(2 1)

+exp(2t)2(2 1)

}. (1.61)

Using the definitions (1.58) and (1.59) and some elementary mathematical operations this expression can also be written as

u =F0k

{1

[cos(0t

1 2) +

1 2sin(0t

1 2)

]exp(0t)

}. (1.62)

• Arnold Verruijt, Soil Dynamics : 1. VIBRATING SYSTEMS 20

This formula applies for all values of the damping ratio . For values larger than 1, however, the formula is inconvenient because then the factor1 2 is imaginary. For such cases the formula can better be written in the equivalent form

u =F0k

{1

[cosh(0t

2 1) +

2 1sinh(0t

2 1)

]exp(0t)

}. (1.63)

For the case of critical damping, = 1, both formulas contain a factor 0/0, and the solution seems to degenerate. For that case a simpleexpansion of the functions near = 1 gives, however,

= 1 : u =F0k

{1 (1 + 0t) exp(0t)

}. (1.64)

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