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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 of civil engineering, and updated continuously since 1994.
SOIL DYNAMICS Arnold Verruijt Delft University of Technology 1994, 2008 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 of civil 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 For most problems the full analytical derivation of the solution is given, mainly using integral transform methods These methods are presented briefly in an Appendix The elastostatic solutions of many problems are also given, as an introduction to the elastodynamic solutions, and as possible limiting states of the corresponding dynamic problems For a number of problems of elastodynamics of a half space exact solutions are given, in closed form, using methods developed by Pekeris and De Hoop Some of these basic solutions are derived in full detail, to assist in understanding the beautiful techniques used in deriving them For many problems the main functions for a computer program to produce numerical data and graphs are given, in C Some approximations in which the horizontal displacements are disregarded, an approximation suggested by Westergaard and Barends, are also given, because they are much easier to derive, may give a first insight in the response of a foundation, 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 necessary, or than most other books on elastodynamics give, but this may be excused by my own difficulties in studying the subject, and by helping students 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 linear spring and a linear damper The main purpose of this chapter is to define the basic properties of dynamical systems, for future reference In this chapter the major forms of damping of importance for soil dynamics problems, viscous damping and hysteretic damping, are defined and their properties are investigated Chapters and are devoted to one dimensional problems: wave propagation in piles, and wave propagation in layers due to earthquakes in the underlying layers, as first developed in the 1970’s at the University of California, Berkeley In these chapters the mathematical methods of Laplace and Fourier transforms, characteristics, and separation of variables, are used and compared Some simple numerical models are also presented The next two chapters (4 and 5) deal with the important effect that soils are ususally composed of two constituents: solid particles and a fluid, usually water, but perhaps oil, or a mixture of a liquid and gas Chapter presents the classical theory, due to Terzaghi, of semi-static consolidation, and some elementary solutions In chapter the extension to the dynamical case is presented, mainly for the one dimensional case, as first presented by De Josselin de Jong and Biot, in 1956 The solution for the propagation of waves in a one dimensional column is presented, leading to the important conclusion that for most problems a practically saturated soil can be considered as a medium in which the solid particles and the fluid move and deform together, which in soil mechanics is usually denoted as a state of undrained deformations For an elastic solid skeleton this means that the soil behaves as an elastic material with Poisson’s ratio close to 0.5 Chapters and deal with the solution of problems of cylindrical and spherical symmetry In the chapter on cylindrically symmetric problems the propagation of waves in an infinite medium introduces Rayleigh’s important principle of the radiation condition, which expresses that in an infinite medium no waves can be expected to travel from infinity towards the interior of the body Chapters and give the basic theory of the theory of elasticity for static and dynamic problems Chapter also gives the solution for some of the more difficult problems, involving mixed boundary value conditions The corresponding dynamic problems still await solution, at least in analytic form Chapter presents the basics of dynamic problems in elastic continua, including the general properties of the most important types 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 are artificially assumed to vanish, an approximation due to Westergaard and generalized by Barends This makes it possible to solve a variety of problems 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 solutions are included in the present book because they are so much simpler to derive and to analyze than the full elastodynamic solutions The full elastodynamic solutions of the problems considered in this chapter are given in chapters 11 – 13 In soil mechanics the elastostatic solutions for a line load or a distributed load on a half plane are of great importance because they provide basic solutions for the stress distribution in soils due to loads on the surface In chapters 11 and 12 the solution for two corresponding elastodynamic problems, a line load on a half plane and a strip load on a half plane, are derived These chapters rely heavily on the theory developed by Cagniard and De Hoop The solutions for impulse loads, which can be found in many publications, are first given, and then these are used as the basics for the solutions for the stresses in case of a line load constant in time These solutions should tend towards the well known elastostatic limits, as they indeed An important aspect of these solutions is that for large values of time the Rayleigh wave is clearly observed, in agreement with the general wave theory for a half plane Approximate solutions valid for large values of time, including the Rayleigh waves, are derived for the line load and the strip load These approximate solutions may be useful as the basis for the analysis of problems with a more general type of loading Chapter 13 presents the solution for a point load on an elastic half space, a problem first solved analytically by Pekeris The solution is derived using integral transforms and an elegant transformation theorem due to Bateman and Pekeris In this chapter numerical values are obtained 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 a moving 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 chapters approximate 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 author’s website , and can be read using the ADOBE ACROBAT reader This website also contains some computer programs that may be useful for a further illustration of the solutions Updates of the book and the programs will be published on this website only A The text has been prepared using the L TEX 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 software provides a major impetus to the production of books and papers in facilitating the illustration of complex solutions by numerical and graphical examples In this book many solutions are accompanied by parts of computer programs that have been used to produce the figures, so that readers can compose their own programs It is all the more appropriate to acknowledge the effort that must have been made by earlier authors and their associates in producing their publications A case in point is the paper by Lamb, more than a century ago, with many illustrative 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 Cornjeo C´rdova o for several years of joint research Many comments of other colleagues and students on early versions of this book have been implemented in later versions, and many errors have been corrected All remaining errors are the author’s responsibility, of course Further comments will be greatly appreciated Delft, September 1994; Papendrecht, February 2008 Merwehoofd 3351 NA Papendrecht The Netherlands tel +31.78.6154399 e-mail : a.verruijt@verruijt.net website : http://geo.verruijt.net Arnold Verruijt TABLE OF CONTENTS Vibrating Systems 1.1 Single mass system 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 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 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 Theory of Consolidation 67 4.1 Consolidation 67 4.2 Conservation of mass 69 4.3 Darcy’s law 72 4.4 Equilibrium equations 73 4.5 Drained deformations 76 4.6 Undrained deformations 76 4.7 Cryer’s problem 78 4.8 Uncoupled consolidation 82 4.9 Terzaghi’s problem 85 Plane Waves in Porous Media 90 5.1 Dynamics of porous media 90 5.2 Basic differential equations 93 5.3 Special cases 94 5.4 Analytical solution 97 5.5 Numerical solution 106 5.6 Conclusion 109 Cylindrical Waves 111 6.1 Static problems 111 6.2 Dynamic problems 117 6.3 Propagation of a shock wave 124 6.4 Radial propagation of shear waves 128 Spherical Waves 132 7.1 Static problems 132 7.2 Dynamic problems 137 7.3 Propagation of a shock wave 143 Elastostatics of a Half Space 150 8.1 Basic equations of elastostatics 150 8.2 Boussinesq problems 152 8.3 Fourier transforms 156 8.4 Axially symmetric problems 161 8.5 Mixed boundary value problems 164 8.6 Confined elastostatics 173 Elastodynamics of a Half Space 179 9.1 Basic equations of elastodynamics 180 9.2 Compression waves 181 9.3 Shear waves 181 9.4 Rayleigh waves 182 9.5 Love waves 187 10 Confined Elastodynamics 191 10.1 Line load on a half space 192 10.2 Line pulse on a half space 196 10.3 Point load on a half space 207 10.4 Periodic load on a half space 209 11 Line Load on Elastic Half Space 215 11.1 Line pulse 215 11.2 Constant line load 252 12 Strip Load on Elastic Half Space 280 12.1 Strip pulse 280 12.2 Strip load 302 13 Point Load on Elastic Half Space 333 13.1 Problem 333 13.2 Solution 336 14 Moving Loads on Elastic Half Plane 351 14.1 Moving wave 351 14.2 Moving strip load 365 15 Foundation Vibrations 378 15.1 Foundation response 378 15.2 Equivalent spring and damper 382 15.3 Soil properties 383 15.4 Propagation of vibrations 384 15.5 Design criteria 385 Appendix A Integral Transforms 387 A.1 Laplace transforms 387 A.2 Fourier transforms 391 A.3 Hankel transforms 400 A.4 De Hoop’s inversion method 404 Appendix B Dual Integral Equations 409 Appendix C Bateman-Pekeris Theorem 411 References 415 Author Index 419 Subject Index 422 Chapter VIBRATING SYSTEMS In this chapter a classical basic problem of dynamics will be considered, for the purpose of introducing various concepts and properties The system to be considered is a single mass, supported by a linear spring and a viscous damper The response of this simple system will be investigated, for various types of loading, such as a periodic load and a step load In order to demonstrate some of the mathematical techniques the 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 spring and the damper form a connection between the mass and an immovable base F (for instance the earth) According to Newton’s second law the equation of motion of the mass is • d2 u m = P (t), (1.1) dt Figure 1.1: Mass supported by spring and damper where P (t) is the total force acting upon the mass m, and u is the displacement of the mass It is now assumed that the total force P consists of an external force F (t), and the reaction of a spring and a damper In its simplest form a spring leads to a force linearly proportional to the displacement u, and a damper leads to a response linearly proportional 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 − c du dt (1.2) Thus the equation of motion for the system is m d2 u du +c + ku = F (t), dt2 dt (1.3) A Verruijt, Soil Dynamics : 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 problems from 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 these quantities Therefore it is essential that the response of a single mass system under various types of loading is fully understood For this purpose both 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 time of 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= F0 [1 − exp(−t/tr )], k (1.4) where tr is the response time of the system, defined by tr = 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 equilibrium state, 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 the homogeneous equation d2 u du m +c + ku = (1.6) dt dt 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 at rest 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 ω0 = k/m, (1.7) and 2ζ = ω0 tr = c cω0 c = =√ mω0 k km (1.8) Appendix C BATEMAN-PEKERIS THEOREM This appendix presents the Bateman-Pekeris theorem (Bateman & Pekeris, 1945), which is used in Chapter 13 The theorem is ∞ x f (x) J0 (px) dx = − where p > 0, and f (z) is an analytic function of z in the half plane π ∞ y f (iy) K0 (py) dy, (C.1) (z) > 0, such that f (z) is real if z is real, and satisfies the condition lim z 3/2 f (z) = 0, z→∞ ( (z) > 0) (C.2) In order to prove this theorem the basic integral is written as ∞ N (p) = x f (x) J0 (px) dx (C.3) The Bessel function J0 (z) can be written as the sum of two Hankel functions (Abramowitz & Stegun [1964], eq 9.1.3 and 9.1.4), (1) (2) (C.4) N (p) = N1 (p) + N2 (p), (C.5) J0 (z) = H0 (z) + H0 (z), so that the integral N (p) can be decomposed into two parts where ∞ N1 (p) = x f (x) H0 (px) dx, (1) (C.6) (2) (C.7) and ∞ N2 (p) = x f (x) H0 (px) dx 411 A Verruijt, Soil Dynamics : C BATEMAN-PEKERIS THEOREM 412 (z) C R B α O A Figure C.1: Quarter plane (z) (z) > 0, (z) > (1) The integral N1 (p) will be considered first Because the function f (z), by assumption, is analytic in the half plane (z) > 0, and H0 (z) is an (1) analytic function of z in the entire plane except at infinity, the function z f (z) H0 (pz) is analytic in the quarter plane (z) > 0, (z) > This means that the integral along the contour OABCO in Figure C.1 is zero, (1) z f (z) H0 (pz) dz = 0, (C.8) for every value of the radius R It can be shown that the integral over the arc AB tends towards zero for large values of the radius R For this purpose it may be noted that (1) the asymptotic behaviour of the function H0 (z) is (Abramowitz & Stegun [1964], eq 9.2.3) (1) H0 (z) ≈ ( 1/2 ) exp(ix − y − iπ), πz (−π < arg(z) < 2π) (1) (C.9) (1) Along the arc AB the Bessel function is H0 (pz) ≈ (2/πRp)1/2 The integral of the function z f (z) H0 (pz) along the arc AB is, approximately, (1) z f (z) H0 (pz) dz ≈ R f (A) ( IAB = AB 1/2 ) R α πRp (C.10) A Verruijt, Soil Dynamics : C BATEMAN-PEKERIS THEOREM 413 Because of the condition (C.2) this will tend towards zero if R → ∞ Along the arc BC the integral will also tend towards zero, because there the integral can be overestimated by 2 πR f (R exp(iϕ))( 1/2 ) exp(−αRp) πRp (C.11) For R → ∞ this will certainly tend towards zero, because of the exponential factor, lim IBC = (C.12) R→∞ Because the contour integral is zero, see (C.8), and the integral along the arc AC is zero, it follows, with (C.6), that ∞ N1 (p) = − (1) y f (iy) H0 (ipy) dy (C.13) The Hankel function of imaginary argument can be expressed into the modified Bessel function K0 (z) (Abramowitz & Stegun [1964], eq 9.6.4), (1) K0 (y) = iπ H0 (iy), (C.14) so that the final expression for the integral N1 (p) is N1 (p) = i π ∞ y f (iy) K0 (py) dy (C.15) In a similar way the integral N2 (p) can be transformed into an integral along the imaginary axis Because at infinity the behaviour of the (2) (1) function H0 (z) is different from that of H0 (z), however, (2) H0 (z) ≈ ( 1/2 ) exp(−ix + y + iπ), πz (−2π < arg(z) < π), (C.16) the contour must now be closed by a quarter circle in the lower right half plane This will give ∞ N2 (p) = − (2) y f (−iy) H0 (−ipy) dy (C.17) Again this can be expressed into the modified Bessel function K0 (z), using the formula (Abramowitz & Stegun [1964], eq 9.6.4), (2) K0 (y) = − iπ H0 (−iy) (C.18) A Verruijt, Soil Dynamics : C BATEMAN-PEKERIS THEOREM The final expression for the integral N2 (p) is N2 (p) = − i π 414 ∞ y f (−iy) K0 (py) dy (C.19) Because the function f (z) is real along the real axis, it follows from the reflection principle (Titchmarsh [1948], p 155) that f (−iy) = f (iy) (C.20) N2 (p) = N1 (p) (C.21) Thus it follows that With (C.5) and (C.15) it now follows that N (p) = − This proves the theorem π ∞ y f (iy) K0 (py) dy (C.22) REFERENCES M Abramowitz and I.A Stegun, Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1964 J.D Achenbach, Wave Propagation in Elastic Solids, Elsevier, Amsterdam, 1975 K Aki and P.G Richards, Quantitative Seismology, 2nd ed., University Science Books, Sausalito, 2002 F.B.J Barends, Dynamics of elastic plates on a flexible subsoil, LGM-Mededelingen, 21, 127-134, 1980 D.D Barkan, Dynamics of Bases and Foundations, McGraw-Hill, New York, 1962 H Bateman, Tables of Integral Transforms, vols, McGraw-Hill, New York, 1954 H Bateman and C.L Pekeris, Transmission of light from a point source in a medium bounded by diffusely 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C.H Dix, Reflection and Refraction of Progressive Seismic Waves, McGraw-Hill, New York, 1962 H.S Carslaw and J.C Jaeger, Operational Methods in Applied Mathematics, 2nd ed., Oxford University Press, London, 1948 C.H Chapman, Fundamentals of Seismic Wave Propagation, Cambridge University Press, Cambridge, 2004 R.V Churchill, Operational Mathematics, 3d ed., McGraw-Hill, New York, 1972 J Cole and J Huth, Stresses produced in a half plane by moving loads, ASME J Appl Mech., 25, 433-436, 1958 O Coussy, Poromechanics, Wiley, Chichester, 2004 C.W Cryer, A comparison of the three-dimensional consolidation theories of Biot and Terzaghi, Quart J Mech and Appl Math., 16, 401-412, 1963 B.M Das, Principles of Soil Dynamics, PWS-Kent, Boston, 1993 R De Boer, Theory of Porous Media, Springer, Berlin, 2000 415 REFERENCES 416 A.T De Hoop, A modification of Cagniard’s method for solving seismic pulse problems, Appl Sci Res., B 8, 349-356, 1960 A.T De Hoop, The surface line source problem in 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on consolidation of a massive sphere, Proc 6th Int Conf Soil Mech Montreal, 3, 401-402, 1965 A Verruijt, Elastic storage of aquifers, Flow through Porous Media, (R.J.M De Wiest, editor), Academic Press, New York, 1969 A Verruijt, The theory of consolidation, Fundamentals of Transport Phenomena in Porous Media, (J Bear and Y.M Corapcioglu, editors), Martinus Nijhoff, Dordrecht, 330-350, 1984 A Verruijt, Dynamics of soils with hysteretic damping, Proc 12th Eur Conf Soil Mech and Geotechnical Engineering, vol 1, (F.B.J Barends et al., editors), Balkema, Rotterdam, 3-14, 1999 A Verruijt and C Cornejo C´rdova, Moving loads on an elastic half-plane with hysteretic damping, ASME J Appl Mech., 68, 915-922, 2001 o A Verruijt, R.B.J Brinkgreve and S Li, Analytical and numerical solution of the elastodynamics strip load problem, Int J Numer Anal Meth Geomech., 32, 65-80, 2008 A Verruijt, An approximation of the Rayleigh stress waves generated in an elastic half plane, Soil Dynamics and Earthquake Engineering, 28, 159-168, 2008 H.F Wang, Theory of Linear Poroelasticity, Princeton University Press, Princeton, 2000 G.N Watson, Theory of Bessel Functions, 2nd ed., Cambridge University Press, 1944 E.W Weisstein, The CRC Concise encyclopedia of mathematics, CRC Press, Boca Raton, 1999 H.M Westergaard, A problem of elasticity suggested by a problem in soil mechanics: soft material reinforced by numerous strong horizontal sheets, Contributions to the Mechanics of Solids, Stephen Timoshenko 60th Anniversary Volume, Macmillan, New York, 1938 M.J Wichura, The PiCTeX Manual, TeX Users Group, Providence, 1987 C.R Wylie, Advanced Engineering Mathematics, 2nd ed., McGraw-Hill, New York, 1960 Author Index Abramowitz, M, 175 Abramowitz, M., 26, 57, 118, 120, 121, 123, 162, 163, 184, 400, 401, 415 Achenbach, J.D., 182, 187, 215, 232, 238, 256, 415 Aki, K., 415 Das, B.M., 58, 182, 415 De Boer, R., 104, 415 De Hoop, A.T., 2–4, 179, 198, 215, 221, 279, 280, 387, 404, 406, 416 De Josselin de Jong, G., 2, 90, 104, 416 De Leeuw, E.H., 416 Dix, C.H., 415 Duffy, D.G., 404, 416 Barends, F.B.J., 2, 179, 191, 197, 415 Barkan, D.D., 384, 415 Bateman, H., 411, 415 Bickley, W.G., 415 Biot, M.A., 2, 67, 69, 90, 104, 415 Bishop, A.W., 68, 415 Bornitz, G., 384, 415 Bowles, J.E., 46, 48, 415 Brinkgreve, R.B.J., 48, 415, 418 Erd´lyi, A., 194, 195, 197, 209, 211, 212, 214, 337, 341, 387, 394, 416 e Eringen, A.C., 215, 237, 279, 333, 416 Ewing, M., 187, 416 Flannery, B.P., 417 Flinn, E.A., 415 Foinquinos, R., 333, 416 Fryba, L., 351, 370, 416 Fung, Y.C., 215, 416 Cagniard, L.P.E., 3, 187, 215, 415 Carslaw, H.S., 27, 415 Chapman, C.H., 415 Churchill, R.V., 18, 25–27, 86, 124–126, 144, 163, 192, 209, 216, 334, 340, 387, 389, 390, 394, 398, 402, 415 Cole, J., 351, 355, 370, 373, 415 Cornejo C´rdova, C., 4, 351, 418 o Coussy, O., 69, 415 Cryer, C.W., 415 Gassmann, F., 68, 416 Gazetas, G., 378, 382, 383, 416 Geertsma, J., 68, 416 Gibson, R.E., 56, 82, 416 Goodier, J.N., 77, 152, 155, 163, 164, 167, 203, 252, 256, 266, 370, 374, 375, 377, 382, 417 Grăbner, W., 416 o 419 AUTHOR INDEX Graff, K.F., 215, 416 Green, A.E., 152, 416 Hall, J.R., 182, 383, 417 Hardin, B.O., 22, 351, 416 Harr, M.E., 416 Hofreiter, N., 416 Hopkins, H.G., 138, 416 Huth, J., 351, 355, 370, 373, 415 Idriss, I.M., 50, 58, 416, 417 Ingard, K.U., 404, 417 Jacob, C.E., 85, 416 Jaeger, J.C., 27, 415 Jardetzky, W., 416 Kassir, M.K., 169, 416 Kausel, E., 251, 416 Knight, K., 416 Knuth, D.E., 4, 417 Kolsky, H., 182, 416 Kramer, S.L., 50, 58, 182, 416 Lamb, H., 118, 179, 215, 237, 242, 417 Lambe, T.W., 417 Lamport, L., 4, 417 Li, S., 418 Lysmer, J., 380–382, 417 Magnus, W., 416 Mandel, J., 82, 417 Miklowitz, J., 125, 127, 215, 417 Mooney, M.M., 333, 343, 417 Morse, P.M., 404, 417 420 Oberhettinger, F., 416 Pekeris, C.L., 2, 179, 209, 333, 338, 343, 348, 411, 415, 417 Pilant, W.L., 417 Press, F., 416 Press, W.H., 66, 417 Rayleigh, Lord, 118, 182, 417 Rendulic, L., 84, 417 Richards, P.G., 415 Richart, F.E., 182, 378, 380383, 417 Roăsset, J.M., 333, 416 e Schiffman, R.L., 75, 417 Seed, H.B., 50, 58, 416, 417 Sih, G.C., 169, 416 Skempton, A.W., 68, 417 Smeulders, D.M.J., 100, 109, 417 Smith, E.A.L., 417 Sneddon, I.N., 156, 157, 160, 161, 165–167, 174, 193, 203, 207, 334, 335, 351, 365, 387, 400, 401, 409, 417 Sokolnikoff, I.S., 152, 155, 417 Stam, H.J., 280, 282, 331, 417 Stegun, I.A., 26, 57, 118, 120, 121, 123, 162, 163, 175, 184, 400, 401, 415 Suhubi, E.S., 215, 237, 279, 333, 416 Talbot, A., 415 Taylor, P.W., 416 Terzaghi, K., 2, 67, 69, 85, 417 Teukolsky, S.A., 417 Timoshenko, S.P., 77, 152, 155, 163, 164, 167, 203, 252, 256, 266, 370, 374, 375, 377, 382, 417 Titchmarsh, E.C., 334, 387, 398, 399, 410, 417 Tricomi, F.G., 416 AUTHOR INDEX Van der Grinten, J.G.M., 100, 108, 109, 418 Vermeer, P.A., 48, 415 Verruijt, A., 22, 82, 252, 280, 331, 351, 418 Vetterling, W.T., 417 Wang, H.F., 75, 418 Watson, G.N., 399, 418 Weisstein, E.W., 392, 418 Westergaard, H.M., 2, 150, 173, 179, 191, 418 Whitman, R.V., 417 Wichura, M.J., 4, 418 Willis, D.G., 69, 415 Woods, R.D., 182, 383, 417 Wylie, C.R., 86, 418 Zerna, W., 152, 416 421 Subject Index Abel integral equation, 166, 171 added mass, 92 axial symmetry, 161, 174 constant isotropic total stress, 83 Courant condition, 66 critical damping, 13 critical time step, 46 Cryer’s problem, 78 cylindrical symmetry, 111 Bateman-Pekeris theorem, 338, 411 Bessel function, 118, 124 Biot’s theory, 67, 90 block wave, 106 Boussinesq, 152, 154 bulk modulus, 74 damping, 9, 17, 58, 104, 143, 149, 382 Darcy’s law, 72, 93 dashpot, De Hoop, 215, 404 degree of consolidation, 88 design criteria, 385 Dirac delta function, 216, 249 displacement function, 152 dissipation of work, 17 drained deformations, 76 dual integral equations, 165, 170, 409 dynamic damping, 379 dynamic stiffness, 379 cavity expansion, 116, 136 characteristic equation, 11 characteristic frequency, 30 characteristics, 33 circular area, 155, 162 circular footing, 379 compatibility, 74 compressibility, 92 compression modulus, 74, 94 compression waves, 24, 181, 191, 238 concentrated force, 154 confined elasticity, 173, 191, 209 consolidation, 67 earthquake, 50 effective stress, 69, 74 eigen frequency, 15, 30, 32 422 SUBJECT INDEX eigen functions, 30 elasticity, 150 elastodynamics, 179, 180 elastostatics, 150 elliptic integral, 163, 175 equilibrium, 151 equilibrium equations, 73 equivalent damping, 17, 122, 130, 382 equivalent spring, 17, 122, 130, 382 expansion theorem, 389 finite differences, 64 finite pile, 27, 28 Flamant, 157, 159 forced vibrations, 14 foundation response, 378 foundation vibrations, 378 Fourier integral, 365 Fourier series, 106, 126, 391 Fourier transform, 156, 193, 391 Fredholm integral equation, 410 free vibrations, 10 frequency, 14 FRICTION, 47 friction, 39, 47 Gibbs phenomenon, 392 Gibson material, 56 ground response, 50 half plane, 150, 215, 252, 280, 302 half space, 150, 333 Hankel transform, 161, 207, 400 head waves, 237, 238 Heaviside, 389 423 Heaviside unit step function, 223 hollow cylinder, 114 hollow sphere, 135 Hooke’s law, 74, 111, 132, 151, 180 horizontal normal stress, 269 horizontal vibrations, 52, 53 horizontally confined deformations, 84 hydraulic conductivity, 72 hysteretic damping, 21, 58, 351 IMPACT, 46 impedance, 34, 45 inertia, 122 infinite pile, 25 integral transforms, 387 intergranular stress, 69 inversion theorem, 157 isotropic stress, 253 Lam´ constants, 112, 133, 151, 180, 352 e Lam´ constants, 74 e Lamb, 215 Laplace equation, 154 Laplace transform, 18, 25, 124, 143, 144, 192, 207, 387 leap frog, 43 line load, 157, 178, 192, 215, 252 line pulse, 196, 215 LINELOAD, 215 liquefaction, 50 Love waves, 50, 187 Mandel-Cryer effect, 82, 84 mass density, 94 mass ratio, 380, 383 massive cylinder, 113 SUBJECT INDEX massive sphere, 134 material damping, 384 mixed boundary value problems, 164 moving load, 351 moving strip load, 365 moving wave, 351 Navier equations, 152 numerical solution, 63 Pekeris, 333 penny shaped crack, 169 periodic load, 31, 209 permeability, 72, 92 pile, 24 pile driving, 24 point load, 207, 333, 369 POINTLOAD, 346 Poisson’s ratio, 151 porous media dynamics, 90 pressuremeter, 117 QUAKE, 66 radiation condition, 43, 118 radiation damping, 143, 213, 384 Rayleigh waves, 182, 185, 221, 239, 251, 258, 267, 272, 278, 306, 345, 363 references, 415 reflection, 35 reflection coefficient, 37 relaxation time, 352 resonance, 15, 16, 32, 53 response time, 10 rigid circular inclusion, 115 424 rigid circular plate, 164 rigid soils, 95 rigid spherical inclusion, 135 self-similar solution, 215 separation of variables, 28, 117, 129 settlement, 87 shear modulus, 74 shear stress, 273 shear waves, 53, 56, 128, 181, 191, 213, 238 shock wave, 28, 107, 124, 143 SHOCKWAVE, 106 SHOCKWAVENUM, 108 single mass system, specific discharge, 71, 93 spherical symmetry, 132 spring, 9, 17, 382 spring constant, 141 stability criterion, 66 storage equation, 72 storativity, 72 strains, 151 strip load, 160, 177, 197, 203, 280, 302 strip pulse, 280 STRIPLOAD, 280 subgrade modulus, 40 surface displacements, 239, 251, 338 Terzaghi’s principle, 74 Terzaghi’s problem, 85 tortuosity, 92 transmission, 35 transmission coefficient, 37 uncoupled consolidation, 82 SUBJECT INDEX undrained compression modulus, 77 undrained deformations, 76 undrained waves, 94 unit step function, 26 vertical normal stress, 262 vibrating systems, visco-elastic material, 351 viscosity, 10, 72, 92 viscous damping, 9, 58 volumetric weight, 72 wave equation, 25, 53, 181 wave length, 51 wave period, 51 wave propagation, 90, 124, 128, 138 wave velocity, 95 waves in piles, 24 Young’s modulus, 151 425 ... S[0]=(F/AREA)*sin(PI*T/TT); for (j=1;j