Dyn Soil Prop Imp

RIVAS

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RIVAS

Railway Induced Vibration Abatement Solutions

Collaborative project

TEST PROCEDURES FOR THE DETERMINATION

OF THE DYNAMIC SOIL CHARACTERISTICS

Deliverable D1.1

Submission Date: 22/12/2011

Project coordinator:

Bernd Asmussen

International Union of Railways (UIC)

[email protected]

RIVAS_WP_13_D_11_V06 Page 1 of 107 22 December 2011

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Title RIVAS WP 1.3 Deliverable 1.1

Domain WP 1.3

Date 22 December 2011

Authors Jeroen Houbrechts, Mattias Schevenels, Geert Lombaert, Geert

Degrande, Werner Rücker, Vicente Cuellar, Alexander Smekal

Partners K.U.Leuven, BAM, CEDEX, Trafikverket , ISVR, ADIF, D2S

Document Code rivas_wp_1_3_d1_1_v06

Version 6

Status Final

Dissemination level:

Project co-funded by the European Commission within the seventh framework programme

Dissemination level

PU Public X

PE Restricted to other programme participants (including the Commission Services)

RE Restricted to a group specified by the consortium (including the Commission Services)

CO Confidential, only for members of the consortium (including the Commission Services)

Document history

Revision Date Description

1 01/06/2011 rivas_wp_1_3_d1_1_v01.docx

2 14/07/2011 rivas_wp_1_3_d1_1_v02.docx

3 28/09/2011 rivas_wp_1_3_d1_1_v03.pdf

4 12/10/2011 rivas_wp_1_3_d1_1_v04.pdf

5 21/10/2011 rivas_wp_1_3_d1_1_v05.pdf

6 22/12/2011 rivas_wp_1_3_d1_1_v06.pdf


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EXECUTIVE SUMMARY

Accurate prediction of railway induced vibration and assessment of the efficiency of vibration

mitigation measures within the RIVAS project requires detailed knowledge of the dynamic soil

characteristics.

Within the frame of the project, it is assumed that the soil can be modelled as a layered elastic

halfspace, where the material properties vary only in the vertical direction, and that small strain

behaviour prevails in the case of railway induced vibrations with relatively low amplitude. Within

each layer, linear elastic isotropic constitutive behaviour is assumed; anisotropic constitutive be-

haviour would better represent the formation process, but is not generally used in state-of-the-art

numerical models and geophysical prospection methods.

Apart from the layer thickness, five parameters need to be determined for each layer: the shear

and dilatational wave velocity, the material damping ratios in shear and dilatational deformation,

and the mass density. The depth upto which these parameters should be investigated depends

on the lowest frequency of interest and on the soil profile (stiffness).

After a brief description of wave propagation in elastic media and the dependence of the consti-

tutive soil behaviour on the strain level, the report discusses classical laboratory and in situ tests,

which results can be used for soil characterization and a first estimate of dynamic soil charac-

teristics based on empirical relations. Main emphasis is going to a detailed description of small

strain dynamic laboratory tests and seismic in situ tests that can be used to determine dynamic

soil characteristics.

The report concludes with a recommended course of action to determine dynamic soil charac-

teristics within the frame of the RIVAS project, which is minimally based on a study of geological

maps and historical geotechnical investigations, a first estimate of dynamic soil characteristics

using empirical relations, soil characterization (e.g. mass density) using classical soil mechanics

tests and seismic in situ testing (a combination of surface wave and seismic refraction meth-

ods). If budget permits, it is further recommended to perform an intrusive in situ test (cross-hole,

up-hole, down-hole or SCPT) in order to enhance profiling depth and resolution, as well as to

perform dynamic laboratory tests on undisturbed samples to determine complementary dynamic

soil characteristics and to evaluate their strain dependency.

It is emphasized that, within RIVAS, estimations of dynamic soil characteristics based on empiri-

cal relations cannot replace their determination by means of in situ or laboratory tests. It is further

recommended that impact loads are also measured when performing seismic in situ tests, so that

the transfer functions of the soil are available and can be used for validation of the dynamic soil

characteristics derived from the test.


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TABLE OF CONTENTS

EXECUTIVE SUMMARY 5

TABLE OF CONTENTS 7

1 INTRODUCTION 11

2 WAVE PROPAGATION IN ELASTIC MEDIA 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Elastodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 Dilatational and shear waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 The direct stiffness method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5 Free vibration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.1 Homogeneous halfspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5.2 Layered halfspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6 Forced vibration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.1 Homogeneous halfspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.6.2 Layered halfspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Presence of ground water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 CONSTITUTIVE BEHAVIOUR OF SOILS 27

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2 Small and large strain behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Modelling of the soil behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4 CLASSICAL SOIL MECHANICS TESTS ON (UN)DISTURBED SAMPLES 33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2 Index properties of soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.1 Water content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.2 Bulk and dry density (or unit weight) of an intact soil . . . . . . . . . . . . 34

4.2.3 Soil particles density, particles unit weight and specific gravity of soil solids 35

4.2.4 Void ratio, porosity and relative density . . . . . . . . . . . . . . . . . . . . 35


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4.2.5 Grain size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2.6 Plasticity of soils, Atterberg limits, consistency and plasticity index . . . . . 37

4.2.7 Overconsolidation ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3 Strength of soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.1 Unconfined compression test . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.2 Unconsolidated undrained triaxial compression test . . . . . . . . . . . . . 39

4.3.3 Consolidated drained triaxial compression test . . . . . . . . . . . . . . . 40

4.3.4 Consolidated undrained triaxial compression test . . . . . . . . . . . . . . 40

4.3.5 Direct shear box test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Compressibility and deformation of soils: oedometer testing . . . . . . . . . . . . 41

4.4.1 Consolidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.4.2 Swelling and swelling pressure . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Application in the scope of dynamic soil characteristics . . . . . . . . . . . . . . . 42

5 CLASSICAL IN SITU TESTS 43

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.2 Cone penetration and piezocone penetration tests (CPT, CPTU) . . . . . . . . . . 43

5.3 Standard penetration test (SPT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.4 Comparison of CPT and SPT tests results . . . . . . . . . . . . . . . . . . . . . . 44

5.5 Application in the scope of dynamic soil characteristics . . . . . . . . . . . . . . . 45

6 DYNAMIC LABORATORY TESTS 47

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2 Resonant column test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.1 Physical principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6.2.2 Resonant column device . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2.3 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.2.4 Results of resonant column tests . . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Bender element test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3.1 Physical principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6.3.2 Interpretation of the tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.3.3 Recommended procedure for BE test . . . . . . . . . . . . . . . . . . . . 53

6.4 Cyclic simple shear test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53


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6.4.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6.4.3 Test applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.4.4 Typical test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.5 Cyclic triaxial test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.5.1 Introduction and applications . . . . . . . . . . . . . . . . . . . . . . . . . 59

6.5.2 Laboratory equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

6.5.3 Test procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

7 SEISMIC IN SITU TESTS 63

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

7.2 General guidelines for seismic in situ methods . . . . . . . . . . . . . . . . . . . 63

7.3 Seismic refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

7.4 Down-hole testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

7.5 Up-hole testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.6 Cross-hole testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.7 Seismic cone penetration test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

7.8 Suspension PS logging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

7.9 Spectral Analysis of Surface Waves . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.10 Seismic tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8 DYNAMIC SOIL CHARACTERISTICS FROM EMPIRICAL RELATIONS 87

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

8.2 Dynamic properties from classical soil mechanical properties . . . . . . . . . . . 87

8.2.1 Small strain shear modulus . . . . . . . . . . . . . . . . . . . . . . . . . . 87

8.2.2 Estimation of shear modulus degradation curve . . . . . . . . . . . . . . . 91

8.2.3 Small strain phase velocities . . . . . . . . . . . . . . . . . . . . . . . . . 92

8.2.4 Small strain material damping . . . . . . . . . . . . . . . . . . . . . . . . 94

8.3 Small strain shear wave velocity from CPT and SPT results . . . . . . . . . . . . 96

9 RECOMMENDATION FOR RIVAS TEST PROCEDURE 99

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

9.2 Recommended procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

REFERENCES 101


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1. INTRODUCTION

The aim of the RIVAS project is to reduce the environmental impact of railway induced vibrations

by providing a set of vibration mitigation measures. These measures may affect either the rolling

stock, the track, or the soil below or in the immediate vicinity of the track. Accurate prediction of

railway induced vibration and assessment of the efficiency of vibration mitigation measures re-

quires detailed knowledge of the dynamic soil characteristics. The assessment of the influence of

the dynamic soil properties on the performance of mitigation measures is covered in WP1.3. This

deliverable D1.1 of the RIVAS project identifies the most influential dynamic soil characteristics,

as well as in situ and laboratory test procedures for their determination. Parametric studies on

the influence of dynamic soil characteristics on the performance of vibration mitigating measures

will be performed in WP3 and WP4, considering sites that are representative for different soil

conditions in Europe, as well as test sites that are selected for in situ testing within WP3 and

WP4.

Accurate prediction of railway induced vibration requires detailed knowledge of the dynamic soil

characteristics. The constitutive behaviour of soil under dynamic loading is complex. Soil is a

discontinuous material, where the pores of the solid skeleton can be partly saturated with water.

Laboratory tests show that the soil behaviour is anisotropic and nonlinear. For cohesionless dry

soils, the nonlinear soil behaviour can be neglected when the shear strain γ is smaller than 10−5.

This is the case for free field vibrations in induced by railway traffic.

Soil is frequently modelled as a layered elastic halfspace, where the material properties vary only

in the vertical direction. The assumption of horizontal soil layers is motivated by the fact that the

formation of a soil layer is governed by phenomena affecting large areas of land, such as erosion,

sediment transport, and weathering processes [23]. Within each layer, linear elastic isotropic con-

stitutive behaviour is usually assumed, whereas anisotropic constitutive behaviour would better

represent the formation process, but is not generally used in state-of-the-art numerical models.

This report focuses on laboratory and in situ test methods to determine the dynamic soil charac-

teristics under small strain dynamic loading, as prevailing in the case of railway induced vibrations

with relatively low amplitude. Vibration investigations begin with studies of archive records like

geological maps and results of geotechnical investigations including all available drillings, sam-

plings, laboratory and in situ testing. Although crucial, this task is straightforward and therefore

not covered in this report.

The report is subdivided in 9 sections. Section 2 provides a brief description of wave propagation

in elastic media, resulting in a set of parameters that are needed for the numerical modelling of

railway induced vibrations. Section 3 explains how the constitutive soil behaviour depends on the

strain level. The report subsequently focuses on classical laboratory test in section 4, classical in

situ tests in section 5, dynamic laboratory tests in section 6 and seismic in situ tests in section 7.

Section 8 provides empirical relations that can be used to obtain rough estimates of the dynamic

soil characteristics. Section 9 provides recommendations concerning specific methods to be used

within the frame of the RIVAS project.


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2. WAVE PROPAGATION IN ELASTIC MEDIA

2.1 Introduction

For the prediction of railway induced vibrations in the free field, relatively low strain levels in the

soil prevail, so that the constitutive behaviour can be represented by a linear elastic relation.

The soil is commonly modelled as a horizontally layered halfspace, where the material properties

only vary in the vertical direction. Each layer is assumed to be homogeneous (i.e. with material

characteristics that do not depend on position) and isotropic (i.e. with material properties that do

not depend on direction).

In this section, the elastodynamic equations in a homogeneous linear elastic isotropic medium,

including displacement-strain, constitutive and equilibrium equations are briefly recapitulated, as

to arrive at the Navier equations and the definition of longitudinal and shear waves. The direct

stiffness method is introduced as a very efficient methodology to study wave propagation prob-

lems in a horizontally layered halfspace, with a wide range of applications ranging from the study

of surface waves, forced vibrations and the amplification of seismic waves. The direct stiffness

method is subsequently employed to study surface waves and forced vibrations in a homoge-

neous and a layered halfspace, employing two sites that have been defined as reference sites

within WP4 of RIVAS. Practical guidelines are given on how to model with very good accuracy

wave propagation in saturated porous media media at low excitation frequencies by using an

equivalent dry medium representing the frozen mixture. The section ends with a summary of

dynamic soil characteristics that have to be determined in each layer, as well as an indication of

the required depth and resolution of the soil profile to be determined.

2.2 Elastodynamic equations

In a Cartesian frame of reference, the components of the displacement vector at a position x and

at time t are denoted as ui(x, t). The components ǫij(x, t) of the small strain tensor are related

to the displacements by the following linearized strain-displacement relations:

ǫij =1

2(ui,j + uj,i) (1)

Herein, ui,j denotes the derivative of ui with respect to the j-th spatial coordinate.

The dynamic equilibrium of the elastic medium is expressed as:

σji,j + ρbi = ρui (2)

where ρbi are the body forces and ρ is the density. A dot above a variable denotes differentiation

with respect to time.

For an isotropic linear elastic material, the constitutive relation relating the Cauchy stress tensor

σij to the small strain tensor ǫij , reads as:

σij = λǫkkδij + 2µǫij (3)


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where ǫkk is the volumetric strain, δij is the Kronecker delta and λ and µ are the Lamé constants.

These constants are related to the Young’s modulus E and the Poisson’s ratio ν as follows:

λ =Eν

(1 + ν)(1 − 2ν)(4)

µ =E

2(1 + ν)(5)

When performing calculations in the frequency domain, frequency independent hysteretic ma-

terial damping in the soil can be modelled by means of the correspondence principle [69, 74],

introducing energy dissipation by means of complex Lamé coefficients:

(λ+ 2µ)⋆ = (λ+ 2µ)(1 + 2βpi) (6)

µ⋆ = µ(1 + 2βsi) (7)

where βp and βs represent the frequency independent hysteretic material damping ratio for the

dilatational waves and the shear waves, respectively. This will be further elaborated in section 3.

A linear elastodynamic problem on a domain Ω with a boundary Γ is defined by the linearized

strain-displacement relations (1), the equilibrium equations (2) and the constitutive equations (3).

These equations are complemented with initial and boundary conditions to define the elastody-

namic problem.

Navier’s equations result from the introduction of the constitutive equations (3) and the strain-

displacement relations (1) in the equilibrium equations (2):

(λ+ µ)uj,ij + µui,jj + ρbi = ρui (8)

which represent equilibrium equations in terms of displacements only, and also need to be com-

plemented by initial and boundary conditions. It can be proven that equation (8) can alternatively

been written in vector notation as:

(λ+ 2µ)∇∇ · u− µ∇×∇× u+ ρb = ρu (9)

where the operator ∇ is defined as ∂/∂x, ∂/∂y, ∂/∂zT , and ∇u, ∇ · u, and ∇× u denote the

gradient, the divergence, and the curl of u.

In the following, body forces are not accounted for and the homogeneous Navier equation is

used:

(λ+ 2µ)∇∇ · u− µ∇×∇× u = ρu (10)

In classical elastodynamics [3], it is customary to explain the physical meaning of the Navier

equation by a Helmholtz decomposition of the displacement vector into two parts: the first com-

ponent is the gradient of a scalar function Φ, while the second component is the curl of a vector

function Ψ:

u = ∇Φ +∇×Ψ (11)

As the three displacement components are written in terms of four scalar potential functions Φ,

Ψx, Ψy, and Ψz, an additional relation must hold. According to Achenbach [4], the vector Ψ


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satisfies ∇ ·Ψ = 0. Using the Helmholtz decomposition (11), the homogeneous Navier equation

(10) is transformed in the following set of uncoupled partial differential equations:

(λ+ 2µ)∇2Φ = ρΦ (12)

µ∇2Ψ = ρΨ (13)

defining the propagation of dilatational and shear waves, respectively.

2.3 Dilatational and shear waves

Equation (12) describes the propagation of the dilatational (or: longitudinal, irrotational, primary,

P-) wave in terms of the scalar potential Φ. In the dilational wave, the particles move parallel to

the wave propagation direction (figure 1a). Equation (12) can alternatively be written as:

∇2Φ =1

C2p

Φ (14)

with

Cp =

√

λ+ 2µ

ρ=

√

M

ρ(15)

the dilatational wave velocity and M = λ+ 2µ the constrained modulus.

(a) (b)

Figure 1: (a) Dilatational and (b) shear wave.

Equation (13) describes the propagation of the shear (or: transverse, equivoluminal, rotational,

secondary, S) wave in terms of the vector potential Ψ. In the shear wave, the particles move

perpendicular to the wave propagation direction (figure 1b). Equation (13) can alternatively be

written as:

∇2Ψ =

1

C2s

Ψ (16)

with

Cs =

√

µ

ρ(17)


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the shear wave velocity.

The ratio s of the body wave velocities Cs and Cp only depends on Poisson’s ratio ν:

s =Cs

Cp

=

√

1− 2ν

2− 2ν(18)

The ratio 1/s of the dilatational wave velocity to the shear wave velocity is plotted in figure 2 as a

function of Poisson’s ratio.

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

Poisson ratio [−]

Vel

ocity

rat

io [−

]

Figure 2: Variation of the longitudinal wave velocity (dashed line) and Rayleigh wave velocity in

a halfspace (dashed-dotted line) as a function of Poisson’s ratio, normalized to the shear wave

velocity (solid line).

The use of complex Lamé coefficients according to the correspondence principle, leads to com-

plex phase velocities Cp and Cs and complex wavenumbers kp = ω/Cp and ks = ω/Cs, where ωis the circular frequency. The imaginary part of the wavenumbers corresponds to wave attenua-

tion due to hysteretic material damping. This will be further elaborated in section 3.3.

According to equations (12) and (13), the dilatational motion uncouples from the rotational part of

the disturbance. The uncoupling of dilatational and shear waves only occurs in a homogeneous

medium when the influence of body forces is neglected. In a layered medium, coupling occurs at

the interfaces between layers.

2.4 The direct stiffness method

While analytical solutions have been derived [40] for some problems involving wave propagation

in a homogeneous halfspace, such solutions do not exist for layered media. Numerical tools such

as the direct stiffness method [40, 41] are therefore used. The direct stiffness method is based on

the transfer matrix approach, initially proposed by Thomson [91] and Haskell [30], and recast into

a stiffness matrix formulation by Kausel and Roësset [41]. The method has also been referred to

as a spectral element formulation by Doyle [18, 19, 20] and Rizzi and Doyle [72, 73].

The direct stiffness method is based on a transformation from the time-space domain to the

frequency-wavenumber domain. In the frequency-wavenumber domain, exact solutions can be

obtained for the Navier equations governing wave propagation in a homogeneous layer or a

homogeneous halfspace. These solutions are used to formulate element stiffness matrices for


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homogeneous layer and halfspace elements. Element stiffness matrices express the relation

between the displacements and tractions on the boundaries of an element. The stiffness matrix

of a layered soil is obtained from the assembly of element stiffness matrices. The direct stiffness

method can be regarded as a special form of the finite element method, using exact solutions as

shape functions. Due to the use of these specific shape functions, wave propagation is treated

exactly and there is no need to subdivide homogeneous layers into multiple layer elements.

As an alternative to the direct stiffness method, the thin layer method can be used [41]. The

thin layer method is based on the use of polynomial shape functions to represent the vertical

variation of displacements and tractions. Compared to the direct stiffness method, the thin layer

method leads to mathematically more tractable stiffness matrices involving only polynomial func-

tions instead of transcendental functions. Due to its approximative nature, the thin layer method

requires a small thickness of the layer elements compared to the smallest relevant wavelength.

Furthermore, the method is only applicable to a layered soil supported by a rigid stratum. A hybrid

formulation, where thin layer elements are coupled to a halfspace element, offers a solution, but

again leads to transcendental functions in the stiffness matrix.

The direct stiffness method and the thin layer method can be used to solve a wide variety of

problems, including amplification of waves in layered media, the computation of dispersive wave

modes in layered media, and the computation of the forced response of layered media due to

harmonic or transient loading. Both the direct stiffness method and the thin layer method have

been implemented in the ElastoDynamics Toolbox (EDT) in Matlab [78].

2.5 Free vibration problems

Surfaces waves or Rayleigh waves are the natural modes of vibration of a (homogeneous or

layered) halfspace. While the eigenmodes of a finite structure occur only at certain frequencies,

surface waves in a semi-infinite medium occur at all frequencies at specific wavenumbers or

phase velocities. These phase velocities and the corresponding mode shapes are found as the

solutions of an eigenvalue problem involving the stiffness matrix of the homogeneous or layered

halfspace. The eigenvalue problem is transcendental, has an infinite number of solutions, and

must be solved by search techniques.

2.5.1 Homogeneous halfspace

For a homogeneous halfspace with zero material damping, the characteristic equation reduces

to the classical cubic equation that was first formulated by Rayleigh [68]. In this case, a single

non-dispersive Rayleigh wave exists with a phase velocity CR approximately equal to [3]:

CR ≈ 0.862 + 1.14ν

1 + νCs (19)

Figure 2 also shows the ratio CR/Cs of the Rayleigh wave velocity in a homogeneous halfspace

and the shear wave velocity as a function of Poisson’s ratio. It can be seen that the Rayleigh wave

velocity is very close to the shear wave velocity for a realistic range of Poisson’s ratios. As inferred

from the cubic characteristic equation and the approximation in equation (19), the Rayleigh wave

velocity in a homogeneous halfspace does not depend on the frequency; Rayleigh waves in a

homogeneous halfspace therefore are non-dispersive.


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Layer h Cs Cp βs βp ρ[m] [m/s] [m/s] [-] [-] [kg/m3]

1 ∞ 250 1470 0.025 0.025 1945

Table 1: Dynamic soil characteristics at the site in Horstwalde.

The soil profile of the reference site in Horstwalde (Germany) as proposed in WP4 of RIVAS, is

used to illustrate the free vibration properties of a homogeneous halfspace. Table 1 summarizes

the dynamic soil characteristics at this site. The high value for Cp reflects the saturation of the

soil. The dispersion curve (figure 5a) shows that for a homogeneous halfspace, the Rayleigh

wave velocity does not depend on the frequency. Figures 3 and 4 show the real and imaginary

part of the horizontal and vertical component of the Rayleigh wave mode in the homogeneous

halfspace at frequencies of 20, 40, 60 and 80 Hz. These figures show that the vertical and

horizontal components are 90 out of phase, indicating that particles move on elliptical paths.

The depth upto which the waves have significant motion depends on the frequency and is about

one Rayleigh wavelength λR = CR/f , with f the frequency in Hz.

(a)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(b)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(c)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(d)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

Figure 3: Real (solid line) and imaginary (dashed-dotted line) part of the horizontal component of

the Rayleigh wave mode for the site in Horstwalde at a frequency of (a) 20 Hz, (b) 40 Hz, (c) 60

Hz and (d) 80 Hz.

2.5.2 Layered halfspace

In a layered halfspace, multiple dispersive Rayleigh modes occur. As an example, the phase

velocity and mode shape of the Rayleigh waves at the reference site in Lincent (Belgium), as

proposed in WP4 of RIVAS, are calculated. The dynamic soil characteristics at this site are

summarized in table 2. As the influence of material damping on the phase velocity is negligible,

material damping is not accounted for.

Figure 5b shows the dispersion curves of the layered site in Lincent. Two major differences with

the dispersion curve of a homogeneous halfspace can be observed. Firstly, in a layered soil,


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(a)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(b)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(c)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(d)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

Figure 4: Real (solid line) and imaginary (dashed-dotted line) part of the vertical component of

the Rayleigh wave mode for the site in Horstwalde at a frequency of (a) 20 Hz, (b) 40 Hz, (c) 60

Hz and (d) 80 Hz.

Layer h Cs Cp βs βp ρ[m] [m/s] [m/s] [-] [-] [kg/m3]

1 1.4 128 286 0.044 0.044 1800

2 2.7 176 286 0.038 0.038 1800

3 ∞ 355 1667 0.037 0.037 1800

Table 2: Dynamic soil characteristics at the site in Lincent.

multiple solutions of the eigenvalue problem exist, corresponding to multiple Rayleigh waves.

Secondly, the surface waves are dispersive due to the variation of the dynamic soil characteristics

with depth. A first dispersive Rayleigh mode appears at zero frequency with a phase velocity that

varies between the Rayleigh wave velocity of the underlying halfspace and the Rayleigh wave

velocity of the top layer. At low frequencies, the Rayleigh waves reach very deep and their phase

velocity equals the Rayleigh wave velocity of the halfspace. At high frequencies, the motion is

concentrated near the surface and dominated by the properties of the top layer. Higher order

surface waves appear at higher (cut-on) frequencies with a phase velocity that varies between

the shear wave velocity of the underlying halfspace and the top layer. At this site, for example, a

single Rayleigh wave exists at 10 Hz, while 6 Rayleigh modes appear at 80 Hz. [!htb]

Figures 6 and 7 show the real and imaginary part of the horizontal and vertical components of

the mode shape of the fundamental Rayleigh wave at frequencies of 20, 40, 60 and 80 Hz. It

can be noticed that, at high frequencies, the wave travels only through the top layers. It can also

be observed that soft top layers reduce the total depth of a Rayleigh wave for a given frequency.

At a frequency of 20 Hz, for example, the motion in the layered soil is concentrated between the

surface and a depth of 4 m while for the halfspace this limit would be closer to 20 m.


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(a)0 20 40 60 80 100

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(b)0 20 40 60 80 100

100

200

300

400

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

Figure 5: Phase velocity of the Rayleigh waves at (a) the homogeneous site in Horstwalde and

(b) the layered site in Lincent.

(a)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(b)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(c)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(d)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

Figure 6: Real (solid line) and imaginary (dashed-dotted line) part of the horizontal component of

the Rayleigh wave mode for the site in Lincent at a frequency of (a) 20 Hz, (b) 40 Hz, (c) 60 Hz

and (d) 80 Hz.

2.6 Forced vibration problems

While the propagation of shear and dilatational waves in a homogeneous full space is uncoupled,

interaction between both types of waves occurs at the surface of a halfspace and at the interfaces

between layers. This interaction leads to the emergence of Rayleigh waves that travel along the

surface of a halfspace. This subsection deals with the transient response of a soil medium.

2.6.1 Homogeneous halfspace

The response of a homogeneous halfspace due to a harmonic load is best illustrated by means

of transfer functions. Figure 8 shows the transfer functions between a surface point load and

the vertical displacement at the surface, and this for soils without and with material damping.

These transfer functions are computed for the soil profile from the site in Horstwalde, of which the


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(a)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(b)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(c)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

(d)−1 0 1

0

3

6

9

12

15

Displacement [−]

Dep

th [m

]

Figure 7: Real (solid line) and imaginary (dashed-dotted line) part of the vertical component of

the Rayleigh wave mode for the site in Lincent at a frequency of (a) 20 Hz, (b) 40 Hz, (c) 60 Hz

and (d) 80 Hz.

properties are given in table 1. The material damping ratio (for both shear and dilatational waves)

is equal to 0.025 in the case with damping. The resulting vertical displacement uGzz at the soil’s

surface is computed up to a frequency of 100Hz and a distance of 50m (figure 8). The results are

made dimensionless in such a way that they only depend on the Poisson’s ratio and the material

damping ratio: the dimensionless displacement is defined as ¯uGzz = rρC2

s uGzz and expressed as

a function of a dimensionless frequency defined as ω = ωr/Cs, where r is the source-receiver

distance.

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

Dimensionless frequency [−]

Dim

ensi

onle

ss d

ispl

acem

ent [

− ]

0 20 40 60 80 100 120−1

−0.5

0

0.5

1

Dimensionless frequency [ − ]

Dim

ensi

onle

ss d

ispl

acem

ent [

− ]

(a) (b)

Figure 8: Real (solid line) and imaginary (dashed line) part of the vertical displacement at the

surface of the site in Horstwalde (a) without and (b) with material damping due to a vertical

harmonic point load at the surface.

Since the amplitude of Rayleigh waves decreases exponentially with depth, they only dominate

the response close to the surface. The dominance of Rayleigh waves in the response at the

surface is observed in figure 8. In the case without material damping, the amplitude of the dimen-

sionless displacement ¯uGzz increases proportionally to r0.5 (or that the actual displacement uG

zz

decays proportionally to r−0.5). In the case with material damping, the response is considerably

lower, especially at high dimensionless frequencies (i.e. at large distances and high frequencies).


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The transient response of a homogeneous halfspace due to an impulsive load is illustrated in

figure 9, which shows the norm of the soil displacement vectors due to an impulsive point load

on the surface at x = 0. The dilatational wave has very small displacements due to the low

compressibility of the material and is therefore not distinguishable on this figure. The shear wave

is clearly visible and travels in directions around 45. Near the surface, the Rayleigh wave clearly

dominates the response. It can be observed that the Rayleigh wave velocity is slightly less than

the shear wave velocity.

(a) (b)

(c) (d)

Figure 9: Transient response to an impulsive vertical point load at the surface for the site in

Horstwalde at (a) 0.02 s, (b) 0.04 s, (c) 0.06 s and (d) 0.08 s.

The dominance of Rayleigh waves at large distances can be explained by considering damping

mechanisms. As waves propagate through the medium, their amplitude decreases. This atten-

uation is due to material and geometrical damping. Geometrical or radiation damping is caused

by the expansion of the wavefronts, resulting in the spreading of energy over an increasing area.

Both types of damping are observed in the equation for plane harmonic waves due to a point

source:

u (r, ω) = Ar−n exp

(

− 2πβr

λ

)

exp

(

iω(

t− r

C

)

)

(20)

in which r−n represents the geometrical attenuation of waves in a homogeneous halfspace, with

r the distance traveled and n = 0.5 for Rayleigh waves, n = 1 for body waves at depth, and n = 2for body waves along the surface [70]. Due to the relatively weak geometrical damping of surface

waves, they dominate the wave field at the soil’s surface in the far field. The factor exp (−2πβr/λ)represents the attenuation due to material damping. The relative importance of material and

geometric damping is illustrated in table 3, which compares the amplitude of a Rayleigh wave at

the site in Horstwalde if only material damping is accounted for, with the amplitude of the same


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wave if only geometric damping is accounted for. These amplitudes are valid for a Rayleigh wave

with unit amplitude at r = 1 m. It is inferred that the effect of material damping increases with

frequency and distance, while the effect of geometrical damping only increases with distance.

4 m 16 m 64 m

Geometrical Material Geometrical Material Geometrical Material

10 Hz 0.500 0.981 0.250 0.910 0.125 0.673

50 Hz 0.500 0.910 0.250 0.624 0.125 0.138

100 Hz 0.500 0.828 0.250 0.390 0.125 0.019

Table 3: Amplitude of a Rayleigh wave at the site in Horstwalde at different frequencies and

distances from the point source, if the effect of geometric or material damping is considered

separately. These values are valid for a wave with unit amplitude at 1 m from the source.

2.6.2 Layered halfspace

The transient response of a layered soil due to an impulsive load is illustrated in figure 10, which

shows the norm of the soil displacement vectors due to an impulsive point load on the surface at

x = 0, for the Lincent site. Figure 10a clearly shows the dilatational and shear wave. A reflected

dilatational wave at the second interface and a head wave between the dilatational and shear

wave are also visible. The figure also shows a transmitted dilatational wave in the halfspace,

which travels with a higher velocity than in the top layers.

Figure 10 clearly shows that the motion is concentrated in the soft top layers. This figure also

shows the dominance of dispersive Rayleigh waves in the surface response.


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(a) (b)

(c) (d)

Figure 10: Transient response to an impulsive vertical point load at the surface for the site in

Lincent at (a) 0.02 s, (b) 0.04 s, (c) 0.06 s and (d) 0.08 s.

2.7 Presence of ground water

In the presence of ground water, the pores between the solid skeleton may be completely satu-

rated with water. Wave propagation in saturated (isotropic) poroelastic media can be described

by Biot’s poroelastic equations [11, 12]. Biot’s theory demonstrates the existence of two dilata-

tional waves and a single shear wave in saturated porous media, which are dispersive and involve

coupled motion of the pore fluid and the solid skeleton. The behaviour of a saturated poroelas-

tic medium strongly depends on the excitation frequency, where the transition between low and

high frequency behaviour is defined by means of a characteristic frequency χ that is inversely

proportional to the permeability of the soil.

In the low frequency range, a saturated poroelastic medium behaves as a frozen mixture with-

out relative motion between the solid skeleton and the pore fluid; there is a single dilatational

wave propagating at a velocity Cp0 and a shear wave propagating at a velocity Cs0. In the high

frequency range, there are two propagating dilatational waves (P1 and P2) with velocities Cp1

and Cp2, with in-phase motion between the solid skeleton and the pore fluid in the P1-wave and

out-of-phase motion in the P2-wave; there is also a shear wave propagating at a velocity Cs [77].

At intermediate frequencies, the wave velocities depend on the frequency, varying from Cp0 to

Cp1 for the P1-wave, from 0 to Cp2 for the P2-wave and from Cs0 to Cs for the S-wave.

For typical soils, the characteristic frequency χ is in the order of several kHz, which is much higher

than the frequency range of interest (upto 250 Hz) for railway induced vibrations. Therefore,

the behaviour of a saturated poroelastic medium can be represented by a frozen mixture and

modelled as a mono-phasic or equivalent dry elastic medium, provided that the density and the

incompressibility of the saturated soil layers are accounted for [77]. This is accomplished by using


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equivalent Lamé coefficients µeq and λeq that are defined as follows:

µeq = µ (21)

λeq = λ+Kf

n(22)

with Kf the bulk modulus of the pore fluid and n the porosity, as well as the mixture density

mixture density ρeq:

ρeq = nρf + (1− n) ρs (23)

with ρf the density of the pore fluid and ρs the density of the solid grains. This finally results in the

following shear wave velocity and dilatational wave velocity of the frozen mixture:

Cs0 =

√

µeq

ρeq=

√

µ

nρf + (1− n) ρs(24)

Cp0 =

√

λeq + 2µeq

ρeq=

√

λ+ 2µ+ Kf

n

nρf + (1− n) ρs(25)

The value of Cp0 is close to the longitudinal wave velocity Cf =√

Kf/ρf in water.

2.8 Conclusion

Assuming that the soil can be modelled as a horizontally layered halfspace and that the consti-

tutive behaviour in each layer can be represented by a linear elastic isotropic law, apart from its

thickness, five parameters should be defined for each layer: the Lamé coefficients µ and λ, the

hysteretic material damping ratios βs and βp in shear and dilatational deformation, and the soil

density ρ. Equivalent information is also contained if, in stead of the Lamé coefficients, the shear

wave velocity Cs and the dilatational wave velocity Cp, or any other independent set of two elastic

constants (e.g. Young’s modulus E and Poisson’s ratio ν, or the constrained modulus M and

Poisson’s ratio ν) are defined. Since both in situ tests as laboratory tests usually measure the

velocities directly, it is preferred to certainly report this set of parameters.

The dynamic soil characteristics need to be determined upto a depth that depends on the fre-

quency range of interest and the stiffness of the soil. It is generally recommended that the

dynamic soil characteristics are certainly determined upto a depth of 20 m, and if possible upto

30 m (which also corresponds to the minimum depth of investigation as defined in EC8 for the

seismic analysis of buildings). When low frequencies are of interest, or in the case of stiff soils,

investigation upto larger depths might be necessary. The minimum spatial resolution should cor-

respond with physical interfaces between soil layers. As the stiffness in the soil usually increases

with depth (also in soil layers that look homogeneous), a finer resolution down to 1 m is preferred

(but can only be obtained with intrusive geophysical tests, as will be explained in section 7).


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3. CONSTITUTIVE BEHAVIOUR OF SOILS

3.1 Introduction

Soil behaviour is very complex to model accurately. Most soils consist of nonlinear, elasto-plastic,

anisotropic materials, which slide or crack at large deformations. The behaviour depends on the

rate of deformations, historic loadings, initial stresses, etc.

Section 3.2 discusses the parameters that describe small strain soil behaviour and the effect of

larger strains on these parameters. Subsection 3.3 deals with the modelling of the constitutive

soil behaviour.

3.2 Small and large strain behaviour

Figure 11 shows a typical stress-strain path for a soil under cyclic loading. Hysteresis loops are

observed: the stress-strain path followed in the unloading phase differs from the original loading

path. This hysteresis effect represents the dissipation of energy in the soil. Energy is dissipated

through several mechanisms, such as friction between solid particles in the skeleton and relative

motion between the skeleton and the pore fluid.

Figure 11: Typical stress-strain path for a soil under cyclic loading.

In small strain regime, the stress-strain relation for soils is approximately linear. The small strain

shear modulus µ0 is represented in figure 11 by the slope of the tangent to the stress-strain

curve at a zero strain. It can be seen on this figure that, at small strain levels, the material

indeed behaves linearly as the stress-strain curve closely follows the tangent at zero strain. As a

consequence, almost no hysteresis effect is present at these strain levels.

At strain levels around 10−5, soils typically exhibit a softening nonlinearity, or a decrease in mod-

ulus as strain increases. This degradation can be seen in figure 11 as the decrease of the secant


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modulus with increasing strain. This shear modulus degradation also causes progressively larger

hysteresis in the stress-strain relation, leading to strain dependent material damping. The shear

modulus and damping ratio at larger strains are denoted by µ and β, respectively.

Various authors have investigated the variation of the shear modulus and the material damping

ratio of soil with the strain level under cyclic loading. Seed et al. [81] have presented the modulus

reduction and material damping curves for sandy soils shown in figure 12. The modulus reduction

curve, shown in figure 12a, represents the ratio µ/µ0 of the (equivalent) shear modulus µ and the

small-strain shear modulus µ0 as a function of the shear strain γ. The material damping curve,

shown in figure 12b, represents the material damping ratio β as a function of the shear strain γ.

It is observed that the material damping ratio does not converge to zero for small strain levels,

but that a small strain material damping ratio β0 remains. This small strain material damping is

caused by differential motion between adjacent soil particles.

(a)

10−6

10−5

10−4

10−3

10−2

0

0.2

0.4

0.6

0.8

1

Shear strain [ − ]

Rat

io µ

/µ0 [

− ]

(b)

10−6

10−5

10−4

10−3

10−2

0

0.05

0.1

0.15

0.2

0.25

Shear strain [ − ]

Dam

ping

rat

io [

− ]

Figure 12: (a) Modulus reduction and (b) material damping curves for sandy soils.

It can also be inferred from figure 11 that the stress-strain curve becomes horizontal at very large

strains. This behaviour ultimately leads to failure.

Figure 13 gives the typical strain levels at which the soil behaviour changes. It shows that be-

low strains of 10−5, soils behave elastically. Above this strain level, soils typically start showing

nonlinear elastic behaviour. At these strain levels no residual strains are recorded upon release

of stresses. At strains around 10−4, soils typically start behaving plastic. Strain repetition starts

having an effect at strains over 10−3 and failures are only recorded at strains over 10−2.

Train induced vibrations typically cause strains below 10−4, except in the ballast, subballast and

embankment of the track. It can therefore be concluded that plastic effects, strain repetition

effects and failure are not relevant in the calculation of the surface response at larger distances.

Train induced vibrations can therefore be modelled with the small strain parameters.

3.3 Modelling of the soil behaviour

The effect of energy dissipation is represented by a material damping model. In structural me-

chanics, a viscous damping model is frequently used. The effect of viscous damping can be

explained by means of a Kelvin-Voigt model, which considers a damped sprung mass system. If

a harmonic excitation and response are considered, the Kelvin-Voigt model leads to the following


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Figure 13: Soil behaviour at different strain levels [37].

frequency domain representation:

τ = (µ+ iωµ′) γ, (26)

in which τ and γ represent the complex shear stress and strain, respectively. It can be noted

that both the modulus and phase of the shear modulus depend on the excitation frequency. The

corresponding hysteresis loop is shown in figure 14. The phase difference between strain and

stress determines the width of the loop and hence the energy dissipation. A maximum amount

of energy is dissipated when the phase difference is π/2. According to equation (26), the phase

difference approaches π/2 for high frequencies.

The energy dissipated per unit volume of material during one cycle is equal to:

∆W (ω) =

∫ T

0

Re (τ) Re (γ) dt = πωµ′γ2 (27)

The maximum strain energy stored per unit volume of material is:

W =1

2µγ2 (28)

The energy dissipated by the system per cycle is proportional to the frequency. Viscous damping

is consequently rate dependent: the energy dissipation, the damping ratio and specific damping


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Figure 14: Hysteresis loop as calculated with the Kelvin-Voigt model.

capacity increase with the frequency:

β (ω) =1

4π

∆W

W=

ωµ′

2µ(29)

ηs (ω) = 2β (ω) =ωµ′

µ(30)

In earthquake engineering, material damping is usually assumed to be rate independent in the

low frequency range. Rate independent material damping is sometimes referred to as hysteretic

material damping [39, 47, 48], although viscous damping also involves a hysteresis effect. A

simple modification to the Kelvin-Voigt model leads to rate independent damping. The desired

model has a complex shear modulus, which is independent of the excitation frequency. Con-

sidering equation (26), this can be achieved by introducing a damping constant that is inversely

proportional to the driving frequency:

µ′ =ηsµ

ω(31)

Inserting Eq.(31) in Eq.(26) leads to the correspondence principle:

µ⋆ = µ (1 + iηs) = µ (1 + 2iβs) (32)

The same strategy can be adopted for the behaviour of the soil under longitudinal deformations:

(λ+ 2µ)⋆ = (λ+ 2µ) (1 + 2iβp) (33)


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3.4 Conclusion

In subsection 3.2 it was explained that the small strain soil behaviour can be modelled by two con-

stants, the shear modulus and the damping ratio. For the strain levels that are reached for train

induced vibrations, these small strain characteristics are sufficient to model the soil behaviour. In

subsection 3.3 it was explained that the soil is typically modelled with a Kelvin-Voigt model, which

is adapted with the correspondence principle to result in rate independent behaviour.


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4. CLASSICAL SOIL MECHANICS TESTS ON (UN)DISTURBED

SAMPLES

4.1 Introduction

Historical data about soil properties at a certain site typically concern classical soil mechani-

cal properties. Since such properties can be used to estimate dynamic soil characteristics, this

section provides an overview of the classical soil mechanical properties and corresponding mea-

surement techniques. Moreover, classical soil mechanical tests are necessary to estimate the

mass density of the soil.

The section starts in subsection 4.2 with an overview of index properties of soils and corre-

sponding laboratory tests. Subsection 4.3 then discusses the estimation of the soil strength and

subsection 4.4 discusses the estimation of the soil compressibility.

In section 8, it is discussed how to use these soil mechanical properties to estimate the small

strain dynamic properties.

4.2 Index properties of soils

4.2.1 Water content

The water content is an index parameter that relates the amount of water present in a quantity of

soil to its dry weight. Water content or moisture content w is therefore defined as the ratio of the

weight Ww of pore or free water in a given mass of soil material to the weight Ws of the dry solid

soil particles, and can be written as follows:

w =Ww

Ws

(34)

According to CEN ISO /TS 17892-1:2004, a soil is considered dry when no further water can

be removed at a temperature within the interval of 105 ± 5C. Water content is determined by

drying the soil specimen in an oven at that temperature (or similar, depending on the standard:

110 ± 5C, in ASTM D2216) to a constant weight, that means, as long as water is present to

evaporate.

Table 4 gives recommended minimum soil sample weights that are required to provide reasonably

reliable water content determinations.

For highly organic soils, soils containing appreciable amounts of gypsum or other minerals, cer-

tain clays, and some tropical soils, an oven temperature of 105C is too high, as it may lead to

changes in the soil characteristics. In these cases, a temperature of 50C to 60C is recom-

mended.

According to EN 1997-2, the extent to which the water content measured in the laboratory on the

soil “as received” is representative of the in situ value should be checked to take into account


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Maximum size of soil particles Recommended minimum Balance sensitivity

(95-100% passes the given sample weight [g]

sieve) [g]

N .40 (0.420 mm) 10 to 200 0.01

N 4 (4.75 mm) 300 to 500 0.1

12.5 mm 300 to 1000 0.1

50 mm 1500 to 3000 1.0

Table 4: Recommended minimum wet soil sample weights [14].

the effects of the sampling method, transport and handling, specimen preparation method and

laboratory environment.

4.2.2 Bulk and dry density (or unit weight) of an intact soil

Bulk density is defined as the mass of soil, including the water within the voids of the soil skeleton,

per unit volume. Thus, bulk weight or unit weight γ is the mass density ρ multiplied by gravity and

most of the times refers to the in situ unit weight. Bulk unit weight γ can be expressed as follows:

γ =Ws +Ww

V= gρ (35)

in which V is the total volume. For completely saturated soils, the bulk mass density ρ equals:

ρ = (1− n) ρs + nρw (36)

in which ρw is the mass density of water, ρs is the mass density of the soil particles and n is the

porosity. The latter two are discussed in subsection 4.2.3.

Dry unit density is defined as the mass of soil particles solely, leaving out the water within the

voids, per unit volume. The dry unit weight γd is the mass density multiplied by gravity, and can

be defined as:

γd =Ws

V(37)

Dry unit weight of soils relates to the degree of packing of the particles; thus, it plays an important

role in compaction control and is the proper density parameter to correlate to many other param-

eters, such as friction angle and compressibility. Bulk unit weight is useful in the assessment of

the in situ overburden stresses at a certain depth of an unsaturated soil profile, due the absence

of buoyancy effect of the hydrostatic water pressure.

CEN ISO/TS 17892-2 standard describes three procedures to determine the bulk unit weight of

a soil: on one hand, the linear measurement method, consisting of direct measurements of the

external shape of a specimen (prisms or cylinders); and on the other, the immersion in water and

the fluid displacement methods. The former consists of the determination of the bulk density and

dry density of a specimen by measuring its mass in air and its apparent mass when submerged in

water. The latter consists of the determination of the bulk density and dry density of a specimen

of soil by measuring mass and displacement of water or other appropriate fluid after immersion.

These two methods are suitable for rock specimens, when lumps of material are available.


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4.2.3 Soil particles density, particles unit weight and specific gravity of soil solids

The soil particles density and the particles unit weight γs refer to the average density and unit

weight of the mineral constituents of the soil particles themselves. These parameters are prop-

erties of the constituent minerals and therefore do not depend on packing of the soil skeleton. It

can be expressed as:

γs =Ws

Vs

(38)

The specific gravity of the soil particles Gs, also known as relative density, is a dimensionless

parameter defined as the ratio of the density of the soil particles to the density of water at a

temperature of 4C under atmospheric pressure conditions (101.3 kPa). The definition can be

likewise expressed in terms of the ratio of unit weights instead, as follows:

Gs =γsγw

(39)

Specific gravity is not useful as a criterion for soil classification because the variation is rather

small for most common minerals, such as clay minerals, quartzitic, feldspar or calcareous miner-

als, ranging between 2.60 and 2.80. However, care must be taken when working on gypsiferous

rock masses (gypsum mineral has a specific gravity roughly of 2.3) or on mine engineering, where

rare minerals are present.

The most common method to experimentally determine the specific gravity, known as the pyc-

nometer test, is described in CEN ISO/TS 17892-3:2004 standard. It basically consists of the

determination of the volume of a known mass of soil by the fluid displacement method, using a

pycnometer which is provided with a glass stopper and a capillary rising tube.

4.2.4 Void ratio, porosity and relative density

The void ratio e is defined as the volume of voids Vv within the soil skeleton to the volume of soil

particles Vs, not the volume of the soil skeleton as a whole, and can be written as:

e =Vv

Vs(40)

Quite more commonly in applications involving stiffer porous media than soils, such as rock cores,

the measurement of the ratio of voids is determined by the porosity n, and is defined as the ratio

of volume of voids to the bulk volume of the soil, i.e., the external volume of the soil skeleton as

a whole V . Commonly, porosity can be found as a percent rather than a ratio:

n =Vv

V(41)

Using equation (40), a relation between the void ratio and the porosity can be found:

n =e

1 + e(42)

Typical values of void ratio for granular soils range between 0.35 for well-graded dense and 0.9

for poorly-graded in a loose state. In soft clays from deltaic or marine deposits void ratios can

reach values higher than 2.


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It is of interest to determine the state of density of a soil as it occurs in nature or in a laboratory

experiment with respect to the densest or loosest states in which the soil can exist, i.e., the degree

of packing of the soil skeleton. For granular cohesionless soils, the parameter to describe this

soil state is the relative density DR, which is expressed in terms of the void ratio of the soil in

the state of interest, related to the maximum emax and minimum void ratios emin that the soil can

undergo, and is expressed as:

DR = 100emax − e

emax − emin(43)

The maximum void ratio is determined with the minimum density test and the minimum void ratio

with the maximum density test. ASTM D4254 and ASTM D4253 describe these tests.

4.2.5 Grain size distribution

Due to both physical and chemical weathering processes, rock masses decay on one hand into

granular particles, ranging from boulders, gravels, sands and silts; and on the other hand into

clays, as a result of chemical action that yields new minerals other than the original rock-forming

minerals. Clay minerals are the smallest soil particles, with a planar structural arrangement and

mean sizes ranging from hundredths of a micron to tens of a micron. From a practical point of

view, soil fraction with mean size up to 0.002 mm is considered as clay.

As a result, such a huge gradation of sizes, from large boulders down to tiny particles composed

of clay minerals, plays an important role on the behaviour of the soil. Hydraulic conductivity,

compressibility, and shear strength are related to the grain size distribution of the soil. However,

engineering behaviour is dependent upon many other factors (such as effective stress, stress

history, mineral type, structure, plasticity, and geological origin) and cannot be based solely upon

gradation.

When studying the grain size distribution, a soil can be first split up into two main fractions: coarse

or granular fraction, including boulders, gravels and sands; and fine fraction, with grain size up to

roughly 0.06-0.08 mm, including silts and clays, which exhibits cohesion due to the water affinity

of the particles.

The particle size distribution test is based upon dividing into discrete classes of particle size. It

can be determined by sieving and sedimentation (by hydrometer). For soils with less than 10%of fines, the sieving method is applicable, whereas for soils with more than 10% of fines can

be analyzed by a combination of both sieving and sedimentation. A standard method for the

determination of the grain size distribution can be found in ISO/TS 17892-4:2004. Moreover,

ASTM D6913-04 (2009) standard describes the method using sieve analysis and ASTM D422-

63(2007) describes the method when analyzed by a combination of both.

The determination of soil by grain size distribution by sieving is accomplished by setting up a

stack of sieves in which each is fitted on a second one whose mesh opening is commonly half

the size of the opening of the first. A known weight of soil is added to the top and the set of

sieves is shaken thoroughly for several minutes and the weight of the soil retained on each sieve

is measured. Commonly, the passing of the smallest mesh opening sieve corresponds to the

fines fraction.

Even though sieves with much smaller meshes are available, at those scales the force of gravity


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on particles turns quite small in comparison with electrostatic or adsorption forces. This combi-

nation of factors makes it impractical to use small sieve sizes for fine soils.

Sedimentation (or hydrometer method) consists of mixing a small amount of soil sample with wa-

ter in a long graduated glass cylinder, together with a dispersant to deflocculate the clay particles.

After shaking vigorously, the various sizes of soil particles will settle at rates according to their

sizes (as stated in Stokes law). As a result, the initially uniform density of the suspension will

begin to vary from top to bottom, becoming more dense at the bottom of the cylinder. At any

given level in the suspension, the density, which is measured at intervals by means of a hydrom-

eter, will change with time in a pattern dependent upon the size distribution of the soil. Stokes

law determines a relationship among the size, the particles specific gravity and the steady fall

velocity, so that the percentage of the particles finer than a certain given value can be worked out

by the readings of the hydrometer.

With the joint information provided by the sieve and hydrometer analysis, the distribution by weight

of grain sizes in the soil under study can be plotted in terms of cumulative percentage finer than

each sieve size by weight and on a logarithmic scale to cover the various orders of magnitude.

4.2.6 Plasticity of soils, Atterberg limits, consistency and plasticity index

Unlike granular soils with negligible content of fines, the mechanical behaviour of fine soils is

strongly dependent upon its water content and, as a consequence, it is useful to measure the

qualitative mechanical response of finer-grained soils by means of simple empirical tests, that in

turn, yield an insight of the mineral constituents for the soil under study.

A soil is considered to behave in a plastic way when it can be molded or worked and will maintain

the new shape without either returning to its original shape or fracturing and cracking. At low

water contents states the soil becomes brittle and crumbles when worked, while at higher water

contents the soil turns into a viscous material.

This is a macroscopic characteristic of clays in a certain range of water contents, which is strongly

dependent upon the clay minerals involved. In fact, clay minerals carry a certain net negative

charge in their surface as a result of both isomorphous substitution and of a break in the continuity

of the structure at its edges. The unbalanced charge at the surface causes great affinity for

captions and other polar molecules, such as water, where potential expansivity can be involved.

The Swedish chemist Atterberg devised two simple laboratory method to establish the water

content at a certain state of consistency. These water contents are called the plastic and the

liquid limits. Both CEN ISO/TS 17892-12:2004 and ASTM D4318 describe the procedure for

determining both limits.

The plastic limit wp is the water content at the transition between semi-solid and plastic mechan-

ical behaviour. With a water content equal to the plastic limit, the soil crumbles when rolled into

a cylinder of 3.2 mm in diameter. The plastic limit test consists, therefore, of taking small rolls of

the clay and remould it between the hand and a non-porous plate until the soil becomes difficult

to roll, due to the steady desiccation.

The liquid limit wl is the water content at the transition between plastic and fluid-like behaviour.

Once again, this is a descriptive state rather than an absolute boundary. At the water content

equal to the liquid limit, the soil becomes fluid under a standard dynamic shear stress. In brief,


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a remoulded soil sample is placed in a special cup and divided into two halves with a standard

grooving tool. An arbitrary value of 25 blows to close the sides of the groove is taken to represent

this consistency state.

The liquidity index Il is used to describe the relative consistency state of a natural clay, being to

some extent, analogous to the relative density for granular soils. The liquidity index relates the

existing water content of a soil sample w to the water content of the clay at its liquid and plastic

limits. It is defined as follows:

Il = 100w − wp

wl − wp(44)

Heavily desiccated clays near the surface might well have a negative liquidity index, whereas

marine and deltaic soft young clay deposits might have a liquidity index higher than 100%, with

water contents over the liquid limit.

Accordingly, the consistency index Ic is defined as:

Ic = 100wl − w

wl − wp

(45)

And the plasticity index Ip is defined as:

Ip = wl − wp (46)

4.2.7 Overconsolidation ratio

The overconsolidation ratio (OCR) for a given soil is defined as the ratio of the highest vertical

effective stress at which the soil has been subjected in the past, to the present in situ vertical

effective stress. A clay is said to be normally consolidated if the current effective overburden

pressure is the maximum pressure which the layer has ever been subjected to at any time in its

history, whereas a clay layer is said to be overconsolidated if the layer was subjected at one time

in its history to a greater effective overburden pressure than the present pressure.

Overconsolidation of a clay stratum may have been caused, among others, due to weight of an

overburden of soil which has eroded, weight of a continental ice sheet that melted or desiccation

of layers close to the surface. Experience indicates that the natural moisture content is roughly

close to the liquid limit for normally consolidated clay soil, whereas for the overconsolidated clay

is close to the plastic limit.

4.3 Strength of soils

4.3.1 Unconfined compression test

The unconfined compression strength can be determined on a cylinder of intact soil sample, as

described in ASTM D2166 and CEN ISO/TS 17892-7:2004 standards. After removal from the

sample tube and trimming, an intact soil specimen is placed in a compression loading frame,

and the loading plate is advanced at a rate of 0.5 to 2% of axial strain per minute. The results


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ConsistencyUnconfined compression Undrained

strength (kPa) shear strength (kPa)

Very soft Less than 25 Less than 12.5

soft 25 to 50 12.5 to 25

Medium 50 to 100 25 to 50

Stiff 100 to 200 50 to 100

Very stiff 200 to 400 100 to 200

hard Over 400 Over 200

Table 5: Classification of clays in term of its unconfined compression strength [89].

are presented as a stress-strain curve. The maximum stress is calculated and recorded as the

maximum unconfined compression strength qu.

This test determines the undrained shear strength of saturated clays or cohesive soils of sufficient

low permeability to keep itself undrained during loading. The undrained shear strength cu is

defined as one half of the unconfined compression strength and represents the natural cohesion

of the soils and is strongly dependent on the in situ mean effective stress of the sample. Table 5

shows a classical classification of clays in terms of its unconfined compression strength [89].

4.3.2 Unconsolidated undrained triaxial compression test

This test is mainly recommended for clayey saturated soils. The soil sample is encased with a

rubber membrane and isolated at the base of a cell. The cell is filled up with water and placed

in a compression loading frame. There is a vertical loading piston inserted in the cell cap that

moves frictionlessly. The loading frame plate is adjusted so that the piston is in contact both with

the load cell at the top and the soil sample at the bottom.

The test covers two phases: in the first one, the soil sample is subjected to an isotropic confining

pressure by pressurizing the cell water. However, note that, whenever the soil sample is saturated,

no effective pressure is transmitted to the soil skeleton; in the second phase, a displacement

transducer is attached to the cell and the compression loading frame is turned on at a rate similar

to that of the unconfined compression test. Simultaneous load and displacement readings are

taken and plotted. The test is performed with no drainage during both phases.

The whole test usually comprises a set of three specimens representative of a sample. Theoret-

ically, this test yields the same results as the unconfined compression tests on saturated soils,

as the cell pressure has no effect on the soil skeleton. However, the cell pressure minimizes

the disturbance during sampling, though, leading to more reliable values of the undrained shear

strength.

CEN ISO/TS 17892-8:2004 standard describes broad good practice of this test. Further details

can be found in ASTM D2850 standard as well.


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4.3.3 Consolidated drained triaxial compression test

This test is carried out with the same triaxial equipment as described above and is suitable for all

types of soils. It is aimed at obtaining the strength parameters of the soil, namely, the effective

friction angle and the effective cohesion.

The test procedure covers three phases:

1. Saturation, by means of pressurizing the water in the soil skeleton through the drainage

valve, so the air bubbles get dissolved in the pore water;

2. Consolidation, the cell pressure is raised at a certain value and the soil specimen is allowed

to drain all the excess pore pressure until the specimen is isotropically fully consolidated at

an effective pressure;

3. Failure, the loading frame is turned on to apply the vertical deviator load with the piston.

The higher the consolidation pressure is, the higher the failure load of the sample. The

deviator load is applied at a slow rate so as the excess pore pressure be dissipated through

the drainage valve.

The stress conditions at failure for a number of specimens can be plotted by Mohr circles. The

best linear fit of the circles yields the strength envelope. The intercept of the strength envelope

and the angle of its slope yield, respectively, the effective cohesion and the internal friction angle.

Alternatively, with the help of Lambe or Cambridge parameters, stress paths of the test results

could be used for further assessment of the soil behaviour.

4.3.4 Consolidated undrained triaxial compression test

This test differs from the previous one basically in the way the soil is subjected to failure, as in

the latter is performed under undrained conditions. A pore pressure gauge is attached to the

drainage valve to monitor the development of the pore pressure in the soil skeleton, that in turn,

allows pore pressure corrections when plotting the effective Mohr Circles at failure. Apart from

the strength parameters, this test provides the Skempton parameters and therefore judgment on

the dilatancy behaviour of the soil. Likewise, it is suitable for all types of soils.

CEN ISO/TS 17892-9:2004 standard covers the procedure consolidated triaxial tests, both drained

or undrained conditions, within the scope of geotechnical investigation.

4.3.5 Direct shear box test

The test method consists of placing the test specimen, either cylindrical or square, in a direct

shear device, applying a stress normal to the shearing plane. After unlocking the frames that

hold the specimen, one frame is displaced with respect to the other at a constant rate of shear

deformation. The shearing force and horizontal displacement are recorded at intervals as the

specimen is sheared. If the effective strength parameters are required, the test is performed

under consolidated drained conditions with a shearing load applied slowly enough not to create


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excess pore pressures. in case the undrained shear strength is required, it is performed under

unconsolidated undrained conditions, at a quick shearing rate to prevent drainage.

Details of the test procedure can be found in CEN ISO/TS 17892-10:2004 standard.

4.4 Compressibility and deformation of soils: oedometer testing

4.4.1 Consolidation

When any soil is subjected to an increase in pressure, a readjustment in the soil structure occurs,

leading to much larger deformations than in other common construction materials. Furthermore,

when a load is applied on a saturated fine-grained soil, the strain response is not instantaneous,

but it takes time for the water that takes up space in the voids to flow outwards until the equilib-

rium is gradually regained. It is therefore a transient flow process coupled with the strain-stress

behaviour of the soil, caused by an external load.

This process is called consolidation and is of major engineering concern, as a lack of its assess-

ment might compromise construction deadlines and maximum expected settlements.

The oedometer test is the main laboratory test to determine both the compressibility (how much

a soil reduces its volume) and the consolidation coefficient (how fast the process develops) of

saturated fine-grained soils.

It consists of a circular disk of soil sample fitted to a metal ring and is in direct contact with

two incompressible porous disks at its top and bottom, being the permeability of the disks much

greater than that of the soil. It is a one-dimensional process, as the lateral strain is prevented and

permeable boundaries are at the bases.

The sample is placed inside a cell filled up with water with a cap that keeps from evaporation.

Once the cell is placed in a loading yoke and a displacement transducer is attached to the yoke,

a vertical loading sequence is applied. The consolidation test in an oedometer proceeds by

applying a sequence loads in a roughly geometric progression with a typical load sequence as

follows: 10, 20, 40, 80, 150, 300, 600 and 1000 kPa. Additionally, unloading sequence is tested

as well, but skipping some of the intermediate steps.

Every loading or unloading step consists of measuring at intervals the cumulative settlement just

after the load increment is applied with the loading yoke. The coefficient of consolidation and the

oedometric modulus can be derived from the settlement-log time curve (consolidation curve).

On the other hand, the oedometric curve can be obtained by plotting the final void ratios at each

loading and unloading step. This curve provides rough assessment on the overconsolidation

ratio, compressibility of the soil tested and relationships between the level of stress and the value

of the oedometric modulus.

Fair guidance of the test procedure can be found in CEN ISO/TS 17892-5:2004 and ASTM D2435

standards for one-dimensional consolidation tests.


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4.4.2 Swelling and swelling pressure

Additionally, the oedometer apparatus allows the determination of swelling properties of expan-

sive clayey soils, such as the free swelling index and the swelling pressure. The former is defined

as the percentage of vertical free expansion when an expansive soil is saturated (in fact, a low

vertical load is applied, generally 5 kPa); the latter is defined as the vertical pressure required to

make up for the potential swelling when the sample gets saturated. The standards above cover

much of this procedure.

4.5 Application in the scope of dynamic soil characteristics

As explained in subsection 2.8, the bulk density ρ is one of the five soil parameters that are

necessary to model vibration propagation. This density can only be measured by classical soil

mechanical tests.

ρ bulk density CEN ISO/TS 17892-2

Other parameters can be used to estimate the other dynamic characteristics, through the use of

empirical relations. These empirical relations are discussed in section 8.2 and use the following

mechanical properties:

w water content CEN ISO /TS 17892-1:2004,

wl liquid limit CEN ISO/TS 17892-12:2004

Ip plasticity index CEN ISO/TS 17892-12:2004

e void ratio ASTM D7263 - 09

n porosity ASTM D7263 - 09

cu undrained shear strength CEN ISO/TS 17892-8:2004

OCR overconsolidation ratio -


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5. CLASSICAL IN SITU TESTS

5.1 Introduction

This section deals with in situ tests that are used to obtain estimates for classical soil mechanical

parameters. The most important of such methods are the (piezo-) cone penetration test and the

standard penetration test. The first is discussed in subsection 5.2, the latter in subsection 5.3.

Subsection 5.4 gives a comparison of the results of both tests.

5.2 Cone penetration and piezocone penetration tests (CPT, CPTU)

The cone penetration test (CPT) as a method of ground exploration is only applicable to soft

soils extending, at most, to medium stiff soils, and without any sizeable proportion of medium

to large gravel size particles. In fact, the use of static penetrometers (cone penetrometers) in

parallel to dynamic penetrometers (DPH and DPSH types) showed that static tests may very well

show resistances on the unsafe side, whenever gravels or carbonated nodules are present in the

ground. On the other hand, since the test is carried out “pushing the cone penetrometer into the

soil at a constant rate of penetration” (usually 2± 0.5 cm/s) according to [2], the soil resistance

adequate to perform the test is quite limited.

The cone penetrometer (formerly designed as static penetrometer) comprises the cone and, if

appropriate, a cylindrical shaft or friction sleeve. The penetration resistance of the cone qc, and

whenever appropriate, the local friction on the friction sleeve are measured. Usually the cone

has a 60 angle, with a face area of 10 cm2 (35.7 mm diameter), and it is provided with a mantle

located above the cone with a length of 99 mm (improved Delft cone), forming a so-called “com-

pound cone”. The friction sleeve, located 69 mm above the mantle, with a diameter of 35 mm

and a length of 133 mm, was introduced by Begemann in 1965 [15].

Mechanical and electrical CPTs are in use, depending upon the means of measuring cone resis-

tance and side friction, with the shape of the cone differing according to the method in use, and

this is to be taken into account when it comes to interpret the results of the test in terms of soil

parameters.

The piezocone penetration test (CPTU) is an electrical CPT, which includes additional instrumen-

tation to measure the pore water pressure during penetration at the level of the base of the cone

[2]. Current knowledge concerning cone penetration testing in clays necessitates that measured

qc values be corrected for pore water pressure effects acting in unequal areas of the cone geom-

etry [66], thus obtaining qt. Consequently, piezocones are required if a truly proper assessment

of cone resistance is to be attained. Unfortunately, when it comes to correlate cone penetration

test results to other soil parameters all sources of data do not usually provide such data, and qcvalues have to be used in the statistical analyses.


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5.3 Standard penetration test (SPT)

According to [2] the objectives of the standard penetration test (SPT) are the determination of the

resistance of soil at the base of a borehole to the dynamic penetration of a split barrel sampler

(or solid cone) of standard dimensions and the obtaining of disturbed samples for identification

purposes. The sampler shall be driven into the soil by dropping a hammer of 63.5 kg mass onto

an anvil or drive head from a height of 760 mm. The number of blows (N ) necessary to achieve

a penetration of the sampler of 300 mm (after its penetration under gravity and below a seating

drive) is the penetration resistance.

In [1] specifications are given on how the SPT test shall be carried out and its results reported.

If the results are to be used for quantitative evaluations or for comparison purposes, the energy

ratio Er for the equipment has to be known. Er is defined as the ratio of the actual energy Emeas

(measured energy during calibration) delivered by the drive weight assembly into the drive rod

below the anvil, to the theoretical energy Etheor as calculated for the drive-weight assembly. The

measured number of blows N shall be corrected accordingly [1]. Other corrections, such as the

energy losses due to the rod length and the effect of effective overburden pressure in sands or

those related to the use of liners or a solid cone should also be taken into account [1]. After these

corrections, the normalized value N60 is obtained.

With respect to SPT tests some additional comments may be made:

• Apart from the different corrections that the results of the tests have to be subjected to, as

established in [1], the standard penetration test is highly dependent on the care and quality

of the work effected to clean and stabilize the hole before the test is performed.

• On the other hand, the presence of granular elements, whose size is comparable or larger

than the diameter of the barrel sampler may produce a blow count which is not directly

related to the stiffness or compactness of the material being tested.

It is then possible, when it comes to make statistical analyses of SPT tests performed around the

world that the results are closely dependent upon features like the ability of people in charge of

the borings or the proportion and size of the gravels or nodules within the strata.

5.4 Comparison of CPT and SPT tests results

Typical values of qc/N ratios for different types of soils, using alternatively, Fugro, Delft and

Begemann cones are given in [35]. Nevertheless, a simple relationship has been widely used in

practice, as given in [56]:

qc [kPa] = χN (47)

in which χ is a soil dependant constant which equals 400 for sands and 300 for clays and has

dimensions of kPa over number of hits. Using this relationships, the results of CPT and SPT

tests in clayey and sandy soils can be compared, although it is highly recommendable to carry

out directly the tests in the soil under study and make the appropriate comparison of results.


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5.5 Application in the scope of dynamic soil characteristics

The parameters obtained by classical in situ tests can be used to estimate dynamic character-

istics, through the use of empirical relations. These empirical relations are discussed in section

8.2.


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6. DYNAMIC LABORATORY TESTS

6.1 Introduction

This section provides an overview of laboratory tests that are commonly used to estimate the

dynamic soil properties.

Laboratory tests have some advantages over in in situ tests. Laboratory tests generally enable

to measure properties with a high accuracy, they allow for the controlled drainage of pore water,

they allow to measure properties at other confining stress than those in situ and they can reach

high strain levels, which allows to measure the nonlinearity of the material. On the other hand,

laboratory tests require soil sampling, which inevitably leads to disturbance of the soil. Further-

more, laboratory tests measure soil properties in discrete points only. Due to the heterogeneous

nature of the soil, laboratory tests may yield results that are not representative for the entire test

site.

The most frequently used soil dynamic laboratory tests are the resonant column test (section 6.2),

the bender element test (section 6.3), the cyclic shear test (section 6.4), and the cyclic triaxial test

(section 6.5).

The most important differences between the different dynamic laboratory tests are the stress and

deformation levels that are reached in the soil, as can be seen in figure 15. Since soils typically

behave nonlinear, it is advisable to reproduce the actual stress levels that occur in situ. The in

situ stresses are therefore an indication for what test should be used.

10−4

10−3

10−2

10−1

100

101

WAVE PROPAGATION

RESONANT COLUMN

CYCLIC SIMPLE SHEAR

CYCLIC TRIAXIAL

CYCLIC TORSIONAL SHEAR

Figure 15: Deformation range in dynamic laboratory tests.

6.2 Resonant column test

6.2.1 Physical principles

In the resonant column test, a cylindrical column of soil is tested dynamically. The column of

soil is torsionally excited. The torsional eigenfrequency of the column is observed as well as the


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decay of amplitudes after switching off the excitation. By these experiments the shear modulus

and the shear material damping ratio of the soil can be determined as a function of the confining

pressure and the shear strain amplitude.

6.2.2 Resonant column device

(a) (b)

Figure 16: (a) Resonant column device and (b) open resonant column device, showing the

electro-magnetic driving unit on top of the soil column.

Figure 16 shows a resonant column device. The device consists of:

• a column of soil in a rubber membrane

• a fixed bottom

• a free top plate and permanent magnets

• a triaxial/pressure cell

• coils, as part of the electromagnetic drive system

• a pneumatic air-pressure system

• control devices, measuring equipment, sensors and a computer

Figure 17 indicates these components on a schematic picture of a resonant column device.

6.2.3 Test procedure

A soil sample is prepared and placed on the base plate. The resonant column device is com-

pleted, the pressure cell is closed and air pressure is applied to the soil specimen and the whole

pressure cell. After some time, the dynamic excitation, with a certain voltage and frequency, is ap-

plied. The frequency is varied, either automatically or by hand. Automatically driven, a frequency

response spectrum is obtained. The frequency of the maximum output amplitude is taken as

the resonance frequency or the approximate eigenfrequency. Driven by hand, first the frequency


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Figure 17: Schematic view of the resonant column device.

range of resonance is found by the highest amplitudes. Then the eigenfrequency is found by tun-

ing the excitation to get a 90 phase delay between driving force and head rotation. Both methods

yield the torsional eigenfrequency of the soil column. After the excitation is switched off, the soil

column shows a free decaying vibration which is recorded.

The same procedure is applied for other excitation amplitudes. At the end, a new pressure is

applied to the pressure cell and the soil column. The same experiments are repeated for different

cell pressures.

6.2.4 Results of resonant column tests

The damping can be determined either from the frequency response function by evaluating the

bandwidth of the resonance peak, or from the decay of the free vibration by the logarithmic

decrement method.

The evaluation of the shear modulus requires some more information. The vibrating system is a

column with an end mass. The moment of inertia of the soil column and the top assembly must

be known. The moment of inertia of the top is determined by calibration tests. With all information

of the resonant column device, the measured eigenfrequency is evaluated by using the frequency

equation of the fixed-free torsional column with a top mass (mass momentum). By that the shear

modulus of the soil is obtained for each pressure and each strain level. The shear strain has to be

calculated from the acceleration, the accelerometer position and the dimensions of the column.

Some results of the resonant column tests are shown in figures 18 and 19. The results in figure

18 clearly show the shear modulus degradation with increasing strain. The results in figure 19

show the material damping ratio increase with increasing shear strain.


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Figure 18: The shear modulus degradation, as measured with the resonant column test.

Figure 19: Material damping ratio as a function of the shear strain, as measured with the resonant

column test.


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6.3 Bender element test

The bender element (BE) technique is a widely used method to generate and receive P- and

S-waves in soil specimens which propagate from one end to the other of the specimen parallel to

its length. Figure 20a shows the dimensions of the soil specimen: the diameter D and the length

H (length); and those of the bender elements: their length Lt, penetration length Lc and width

W [98]. Figure 20b shows the two pedestal of a triaxial apparatus for testing 2 inch diameter

specimen, in which two bender elements (transmitter and receiver) are inserted.

(a) (b)

Figure 20: (a) Bender element dimensions [98] and (b) bender elements inserted in the top caps

of 2 inch triaxial equipment.

The use of piezoceramic bender elements for testing soil samples was firstly described by Shirley

and coworkers [84] and [85] and a detailed model for determining shear wave velocity Cs in triaxial

soil samples was presented later on by Dyvik and Madshus [21]. Since then, BEs have been

used to obtain Cs values with other geotechnical apparatus, such as the oedometer, the direct

shear apparatus or the resonant column. A complete list of references covering the use of BE in

different geotechnical laboratory equipments to obtain Cs values, has been elaborated by Viana

da Fonseca et al. [94].

6.3.1 Physical principle

Bender elements are bimorph electric actuators that are polarized in the direction of their thick-

ness (0.5 mm in figure 21) [98]. Two ceramic elements are bounded together with a flexible shim

of metal, such as nickel, acting as an electrode. When a driving voltage is applied on a bimorph

piezoelectric element, one layer elongates and the other shortens, producing a bend in the whole

element. On the other hand, when a deformation is applied, the piezoelectric element generates

a voltage. There are two different types of bender elements (figure 21): parallel connected and

serial connected. In a parallel type connection, polarization direction in both layers of a bimorph


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actuator becomes identical whereas, in a series type, polarization is opposite. The result is such

that a parallel connected element is twice as effective as a serial connected element in generating

mechanical energy and is therefore used for transmission. On the other hand, the generated volt-

age upon deformation is larger in a serial type connection, which is therefore used for reception.

By using serial type connection in parallel benders and parallel type connection in serial benders,

the bender body can be compressed or extended as a whole, thus enabling it to measure P-wave

velocity as shown by Lings and Greening [57].

Figure 21: Bender element actuators [98].

6.3.2 Interpretation of the tests

A bender element test consists of the application of a user-defined input voltage function to the

transmitter to generate a S- or P-wave and the recording of the wave by the receiver, resulting in

an output signal [94]. Most earlier studies using BEs used a single square-wave pulse, as the

one shown in figure 22a. However, sine-wave pulses, as shown in figure 22b, have become more

popular [98].

Currently there are four different approaches to identify the arrival time of the wave in the receiver;

three of them (SS, PP, CC) operate in the time domain and one (FD) in the frequency domain.

The start to start method SS is based on the identification of the instant of the first inflection of

the output signal (figures 22a and 22b). It is the simplest and most commonly used method,

but the interpretation is subjective [94]. In the peak to peak method PP, the time difference

between the first peak of the transmission wave and the corresponding peak of the received

wave is measured. As shown in figure 22b, it tends to give a propagation velocity slightly lower

than the SS method. In the cross-correlation method CC, the cross-correlation between the

transmitted and the received signal is calculated and the position at the maximum amplitude is

taken as the propagation time. In this method, as in the PP method, it is important to ensure that

the transmitted wave retains its shape when passing into the soil. That condition constitutes a

problematic aspect for those techniques. The frequency domain method FD calculates the cross

spectrum of the transmitting and receiving waves and plots the phase angle against frequency.


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(a)

(b)

Figure 22: Examples of time signals and results: (a) a square pulse input and its response to

determine the P-wave velocity in a triaxial specimen with the SS method and (b) a sine pulse and

its response to determine the S-wave velocity in a triaxial specimen by the SS method (270 m/s)

and the PP method (250 m/s).

The arrival time is then calculated from the slope of the phase spectrum. However, because the

scatter in the FD method is quite large [98], it is advised not to interpret the results with only this

method.

6.3.3 Recommended procedure for BE test

In [98], a standard BE test methodology, incorporating the advantages and disadvantages of

the different techniques previously discussed, is proposed. Recommendations are given in that

reference to obtain the same value when time domain methods are used and the conditions that

should be fulfilled to get appropriate values when the FD method is adopted are identified. As

pointed out in [94] the combined use of time domain and frequency domain methods can aid

effectively in the analysis and interpretation of BE tests.

6.4 Cyclic simple shear test

6.4.1 Introduction

The direct simple shear (DSS) apparatus was developed in the 1960’s at the Norwegian Geotech-

nical Institute (NGI) by Landva and Bjerrum. It was designed to simulate the condition in a thin

shear zone separating two essentially rigid masses sliding with respect to each other. This con-

dition approximates the behaviour of some landslides that occur along a planar surface or along

horizontal or gently inclined portions of a slip surface [90]. The simple shear test is also the

laboratory test which best fits an earthquake loading, as shown in figure 23.


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Figure 23: Earthquake loading of a soil element [87].

Silver and Seed [86] used a cyclic simple shear apparatus for low strain amplitude tests to explore

deformational behaviour and the shear stress-shear strain relationship respectively of sand under

cyclic loads.

6.4.2 Apparatus

Figure 24 shows two commercially available cyclic shear testing devices. The simple shear or

direct simple shear apparatus produces an anisotropic normal stress state by external forces and

cell pressure on a rectangular or round specimen. Horizontal movement at the bottom of the

sample relative to the top induces the shear strain. The structure supporting the specimen in

lateral direction follows the movement, allowing horizontal deflection during shear but avoiding

deformation in lateral direction during consolidation. The shear strain over the specimen volume

is considered to be nearly constant. Because the diameter of the sample remains constant, any

change in volume is a result of vertical displacement of the upper pedestal.

(a) (b)

Figure 24: Cyclic shear testing devices at BAM: (a) GIESA mbh and (b) GEONOR, Inc.

Two systems for the lateral support of the specimen are feasible (figure 25). In the first construc-

tion a number of steel, aluminium or brass rings slide across each other during the shearing stage


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of the test. The second option is to support the specimen laterally by a spring construction. Low

internal friction and high stiffness of the spring against changes in diameter allow to deform the

shear frame with the specimen in axial direction and so to avoid wall friction.

(a) (b)

Figure 25: Lateral support of specimen by (a) a spring construction [34] and (b) slip rings [33].

(a) (b)

Figure 26: Example of set-up for simple shear test equipment and concept of direct simple shear

on test specimen.

A schematic picture of the components of a simple shear apparatus is shown in figure 26. Its

main parts are (numbers correspond to the figure):

1. The container for the specimen: a rubber membrane reinforced with a spiral winding or slip

rings. The standard sample is 70 mm in diameter, but tests can also be performed on 50

mm diameter samples. The container is positioned on a pedestal with the same top cap as

used in the triaxial test apparatus.

2. Two actuators, one for the vertical and one for the horizontal load. The horizontal and

vertical actuators are fixed to the frame, which supplies the reaction forces.

3. Load cells to measure the vertical and lateral load.


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4. The fixed upper frame.

5. The lower frame, which can translate quasi-statically or dynamically in the lateral direction.

To avoid sliding between the soil sample and the filters on the top cap or the base pedestal

the arrangement of some pins is required (figure 26b).

6. Two displacement transducers to monitor the vertical and horizontal displacements.

7. The control and data acquisition system.

6.4.3 Test applications

Cyclic simple shear testing is useful in the investigation of stress-strain relationships and the

shear strength for a range of soil types. The simple shear test apparatus allows shear tests

under different conditions. Firstly, tests at a constant height or constant volume are possible.

This means that during shear the specimen volume, and therefore the height, is kept constant

by adjusting the vertical load. The resulting change in vertical stress is equal to the pore water

pressure that would occur in undrained conditions with constant normal stress. The second option

is testing at a constant normal vertical load and thus a constant normal stress (CNL). This kind of

test simulates completely drained conditions. The third option is to drive tests at a constant strain

rate.

The cyclic simple shear apparatus is often used in soil dynamics as it can simulate loading condi-

tions to research the stability under seismic events, the degradation of shear stress under cyclic

loading or the liquefaction potential of non-cohesive soils under cyclic loading. With high accuracy

measurements, the test can also be used to determine the shear modulus and material damping

ratio.

It is not possible to avoid variations from the aspired constant stress state at the border of the

specimen [31]. The irregular distribution of normal and shear stresses in border areas is notably

higher at high deformation levels.

It replicates the in situ conditions in the soil specimen. Samples are inevitably disturbed when

placed in the testing device. Furthermore, it is difficult to obtain constant compaction of granular

samples. For this reason, the simple shear apparatus is often only used in research. Commercial

laboratories generally use the cyclic triaxial apparatus.

When comparing results from the cyclic simple shear tests with results from the cyclic triaxial

tests, the different stress states in both tests have to be considered. At the same normal stress

acting on failure plane the mean principal stress σm differs between both test. In cyclic triaxial

test it equals the consolidation stress σc, but in a cyclic shear test σm = 1/ (3 (σ1 + 2σ2)) with

σ2 = K0σ1 [87].

6.4.4 Typical test results

Typical test results for stress or strain controlled cyclic simple shear test with on ore two way

loading with Berliner sand are shown in the following figures. Figure 27 shows results from a

strain controlled test with strain amplitude γ = ±1.5% on a dense sand sample. One can notice


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a further compaction of the dense sample and a decreasing shear stress over time. The normal

stress is kept constant at 200 kN/m2 during the shear stress application.

Figure 27: Strain controlled test with Berliner sand, drained, two way loading at γ = 1.5%,

compactness D = 60%, normal stress 200 kN/m2.

Figure 28 shows results from a stress controlled, one way test with a shear stress τ between

0 and 40 kN/m2 on a loose sand sample. Shear strains accumulate but the accumulation rate

decreases.

Figure 29 shows a stress controlled, two way test with a shear stress τ between -40 and 40 kN/m2.

Here the sample volume and sample height respectively is kept constant, so the test simulates

undrained conditions. Shear strains accumulate also but the accumulation rate increases with

growing number of cycles. The normal stress tends to zero.


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Figure 28: Stress controlled test with Berliner sand, drained, one-way load τ = 40 kPa, compact-

ness D = 14%, normal stress 200 kN/m2.

Figure 29: Stress controlled test with Berliner sand, drained, two-way load τ = 40 kPa, compact-

ness D = 60%, normal stress 400 kN/m2.


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6.5 Cyclic triaxial test

6.5.1 Introduction and applications

The dynamic triaxial test is one of the most used dynamic laboratory tests due to its versatility and

to its easiness in the reproduction of complex stress paths. The main applications of the dynamic

triaxial tests are the determination of the liquefaction potential, the determination of the dynamic

deformation modulus and damping and the determination of the resilient modulus. Especially the

determination of the strain dependent deformation modulus and damping are important in the

scope of railway induced vibrations.

Determination of the liquefaction potential

This kind of tests must be performed in saturated samples in load-control tests. The liquefac-

tion potential is determined by the analysis of the increase of axial deformations or of the pore

pressure during the test.

Determination of dynamic deformation modulus and damping as a function of deformation

This kind of tests must be performed in saturated samples (both remoulded and unaltered) in

load controlled or axial deformation-control tests.

In a load controlled test, the load is applied according to a predetermined law, usually a sinusoidal

cyclic load, and the result is the evolution of the induced deformations. The main objective of

these tests is to know the number of cycles necessary to get the failure of the material or a

determined level of deformation for different stress ratios or different confinement pressures.

In an axial deformation controlled test, the deformation is varied in a predetermined way and the

result is the load, or the stress, necessary to obtain the prescribed deformation in each cycle.

The main aim of these tests is to analyze the change in the soil mechanical properties during the

load application.

The deformation modulus is usually determined as the secant modules which is numerically equal

to the slope of the secant of a stress-strain curve, as it can be seen in figure 30a.

The material damping ratio β is obtained according to equation (30). The cyclic shear test allows

to determine the areas ∆W and W directly as the cycle area and the area outlined by the triangle

OAB, respectively.

It is important to remark that, in a load controlled test, permanent deformations can be accu-

mulated which makes that to define the hysteretic cycle the distance between two consecutive

deformation pikes must be smaller than 0.2% of axial deformation. This fact is known as cycle

closure error and can be seen in figure 30b.

Determination of resilient modulus

Traffic loads typically have a very short application time. After removal of the load, part of the

previously induced deformation is canceled but the other part remains. The first is called the


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(a) (b)

Figure 30: (a) Theoretical stress-strain curve for one cycle and (b) stress-strain curve of a cycle

in a load controlled test with cycle closure error.

resilient deformation, the latter is the permanent deformation. The resilient modulus is defined as

the ratio between the applied deviatoric strain and the resilient deformation.

In the following figure, corresponding to one cycle of a test, the permanent and resilient deforma-

tions are indicated.

Figure 31: Permanent and resilient deformations in a cyclic test.

6.5.2 Laboratory equipment

The laboratory equipment necessary to perform dynamic triaxial tests is constituted by (figure 32)

a triaxial cell, transducers to measure the load, displacement and pore water pressure, equipment

for the application of a confinement pressure, equipment for the application of the cyclic loading


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and for the control capable of applying a uniform sinusoidal deformation at a frequency range of

0.1 to 2 Hz, and equipment for data recording, capable to register 40 points per cycle.

Figure 32: Components of triaxial equipment (ASTM D-3999-91).

The measurement of deformations must be done very carefully so that probe and plate defor-

mations, or mistakes due to lack of precision in the probe preparations are not included in the

measured deformation. To avoid such mistakes, the axial and diameter deformation transducers

are placed inside the triaxial cell, as indicated in figure 33.

Figure 33: Position of axial and diameter deformation transducers inside the triaxial cell.


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6.5.3 Test procedure

The procedure to perform a dynamic triaxial test has the following steps:

1. The probes are prepared. The probes can be unaltered or remolded. In case of remoulded

probes, the density and water content of the material must be indicated previously. In all

the cases, the maximum particle size must be less than 1/6 of probe diameter.

2. All the material pores are filled with water so that the pore water pressure during the loading

action can be measured correctly.

3. The material is consolidated under the desired initial effective stress state. This stress

situation can be isotropic or anisotropic.

4. The dynamic loading is imposed in non drained conditions to simulate the real conditions,

for instance of an earthquake, where the soil permeability inhibits water drainage.

If the test is made to obtain the liquefaction potential, a deviatoric sinusoidal load is applied

which can be less than the confinement pressure.

If the aim of the test is to determine the deformation modulus and the camping, the test

can be done controlling the load or the axial deformation. In the case of axial deformation

control, the axial deformation applied is convenient to be smaller than 10−4.

To obtain the variation curve of deformation modulus and camping with the deformation

level, different test with different probes can be done. Another alternative is to use only

one probe and to apply increasing deformation amplitudes during predetermined number

of cycles.


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7. SEISMIC IN SITU TESTS

7.1 Introduction

This section provides an overview of in situ methods that are commonly used to estimate the

small strain dynamic soil properties.

Field tests have some important advantages over laboratory tests. Firstly, they do not require

sampling and consequently do not disturb the soil. Secondly, field tests provide data that are

measured over a large volume of soil. Since soils are typically heterogeneous, this results in the

averaging of soil parameters, in such a way that the test results lead to simulations that approx-

imate the overall soil response. Finally, in situ tests employ strain levels that are comparable to

levels of interest for railway vibrations.

On the other hand, in situ tests also have some disadvantages. Firstly, they do not allow to test

the soil under any other than the present conditions. Also, they do not allow for the controlled

drainage of pore water. These two disadvantages are important when significant alterations to the

current conditions, or very low frequency loads are expected. This is not the case for train induced

vibrations. A third disadvantage is that in situ tests often do not measure properties directly but

use theoretical analysis or empirical relations to derive the desired parameters. These derivations

introduce an extra source of uncertainty. Besides, it is often impossible for in situ measurements

to reach the same level of accuracy as obtained in laboratory tests.

One should always be aware of the epistemic uncertainty of estimated soil properties [47]. Many

sources give rise to this uncertainty, examples are the spatial variability of the properties, the

inherent and induced anisotropy, the nonlinearity of the material, the soil disturbance due to

drilling and sampling, and testing or interpretation errors. While some of these errors can be

limited, others cannot. One should always keep this uncertainty in mind, and if possible quantify

its magnitude.

This section starts with general guidelines for in situ tests in subsection 7.2. The remaining

subsections each focus on a particular test method; subsection 7.3 deals with seismic refraction,

subsection 7.4 with the down-hole method, subsection 7.5 with the up-hole method, subsection

7.6 with the cross-hole method, subsection 7.7 with the seismic cone penetration test, subsection

7.8 with PS suspension logging, subsection 7.9 with the Spectral Analysis of Surface Waves test

and subsection 7.10 with seismic tomography.

7.2 General guidelines for seismic in situ methods

As explained in section 3, soils typically behave nonlinear. This means that measured parameters

depend on strain levels that are reached in the soil. It is therefore advised that tests always aim

to replicate stress-strain conditions and apply anticipated cyclic loading levels. In the scope of

train induced vibrations, these loading levels should induce small strain levels, in the order of

magnitude 10−6 − 10−5.

Seismic geophysical tests are often based on the measurement of body wave velocities, others


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are based on surface wave velocities. The procedures of in situ tests therefore involve the cre-

ation of transient or steady-state elastic waves and the observation of these waves at one or sev-

eral locations. Several sources can be used to generate the waves: sledgehammers, mechanic

hammers, explosions, dropweights, etc. These sources create dilatational, shear and surface

waves. The importance of a particular wave type depends on the type of excitation. Explosions

and vertical impacts are rich in P-wave content while lateral impacts create more S-waves. The

type of excitation also determines the frequency range of the excitation. One should choose the

source that excites the entire frequency range of interest. If there is no such source type, different

types of loading, as to cover the entire frequency range, can be combined.

As explained in subsection 2.3, pressure waves travel faster than shear waves. Consequently, the

pressure waves arrive first at the receiver locations and the detection of their arrival is therefore

fairly straightforward. Due to wave reflection and refraction, P-waves continue to arrive at the

receiver location for some time after the first arrival. When the first S-waves arrive, reflected or

refracted P-waves are usually still recorded. The detection of the S-wave arrival time is therefore

more cumbersome. A common procedure to facilitate the detection of S-waves is to apply two

impulses with opposite direction. Depending on the excitation and the measurement direction,

the S- or P-waves will then have a 180 phase difference.

The strain levels that are reached in in situ tests are rather low. This implies that the recorded

vibration signals will be weak and therefore, low signal to noise ratios (SNR) can be expected.

Due to geometric and material damping, the response will decrease with increasing distance

from the source. Hence, the lowest SNRs will be observed at large distances from the source.

In fact, the SNR of the recorded signals will often be very low. Therefore, techniques to enhance

the signal quality are often used. A possible technique is the stacking of records from different

impulses.

7.3 Seismic refraction

The seismic refraction method is a non-invasive test method, used for the determination of the

P-wave velocity Cp of the different subsurface layers and the thickness of each layer. The use of

the refraction method dates back to 1914 in Germany. Initially it was used for military purposes,

and later on often applied in oil exploration as well.

Physical principle

The method is based on the physical principle of seismic refraction. Seismic refraction is the phe-

nomenon that describes the directional change of a wave when it crosses an interface between

two layers with different mechanical properties. This change in direction is described by Snell’s

law (figure 34):

Cp1 sin θ1 = Cp2 sin θ2 (48)

in which Cp1 and Cp2 represent the phase velocities in the top and bottom layer, respectively. θ1and θ2 represent the angles between the ray path and the vertical in the top and bottom layer,

respectively. The critical angle is the incident wave angle θ1 for which the refracted wave travels


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θ1θ1

θ2

Cp1Cp1

Cp2Cp2

θcθc

(a) (b)

Figure 34: Snell’s law: (a) incident, reflected and transmitted wave and (b) critically refracted

wave.

horizontally (θ2 = 90):

θc = arcsin

(

Cp2

Cp1

)

(49)

An incident wave with angle θc refracts at the interface in such a way that it travels along the

interface. It travels along this interface with a velocity Cp2. This wave creates a disturbance on

the interface and, as a result, an upward wave propagates in the layer above it. According to

Snell’s law, this wave has an angle θc. This situation is explained in figure 35. The impact leads

to body waves, initially only in the top layer (t1 till t2). At a certain moment, the wavefront hits the

interface and waves are refracted. This is happening at t3, but the wave under the critical angle

has not hit the interface yet. At t4, the wave with the critical angle hits the interface, which creates

the wave along the interface. This critically refracted wave travels with a velocity Cp2 > Cp1 and

creates at its wavefront an upward wave, traveling with velocity Cp1 in the top layer. Between t4and t9, the upward waves arrives at points at the surface after the direct body wave did. From

point xc on, the refracted waves arrive first and are followed by the direct wave. This happens

somewhere between t9 and t10.

Test procedure

For the seismic refraction test, seismic energy is generated by an artificial source located on

the surface, usually by an impact on a metal plate. This energy travels through the subsurface

layers as described in figure 35. The waves are detected on the surface using a linear spread of

geophones or accelerometers spaced at regular intervals, and it is the observation of the travel

times of direct and refracted arrivals that gives information about the subsoil velocity layers (figure

36).

The maximum depth reached by this method depends on the distance between the geophones

and the relative distances between the geophones and the different shooting points. The ideal

conditions for data acquisition are maintaining a small spacing between geophones to obtain a

good resolution, and a maximum relative distance of the shooting point large enough to reach the

desired depth (bearing in mind that a greater distance allows to reach a larger depth). In section


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eplacements

t1

t2

t3

t4 t5 t6 t7 t8 t9 t10

t11

t12

xc

θcθc

Cp1

Cp2 > Cp1

impact

direct wave first refracted wave first

Figure 35: Seismic refraction principles. The wavefronts are represented by the dashed lines, the

wave paths with the solid arrows.

Figure 36: Simple two-layer model example. Direct and critically refracted waves and the time

distance diagram showing the first and secondary breaks from these ray paths [49].


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2 it was recommended that soil properties should be known to a depth of about 20 m. For a soil

with one layer with thickness H on top of a halfspace, the point xc can be computed as:

xc = 2H

√

Cp1 + Cp2

Cp2 − Cp1

(50)

If any prior knowledge of the soil stratigraphy is available, this equation can be used to determine

the distance between the furthest receiver and the shooting point. As an example, consider a

soil that has a layer with a P-wave velocity around 250 m/s on top of a half space with a P-

wave velocity of 1000 m/s at a depth of 20 m. The point xc is then located at 52 m from the

source. The number of receivers located farther than 52 m from the source, should be sufficient

to accurately measure the slope of the last line segment in the distance-arrival time diagram. The

other receivers should be evenly distributed between the shooting point and the furthest receiver.

Soil parameter estimation from measurement data

For generating the travel time graphs it is necessary to manually pick up the first arrivals for each

geophone register. The STA/LTA technique may facilitate the detection of this first arrival time

[97]. Travel time versus distance graphs are then plotted with all shots for the same spread. The

different slopes in this graph provide information about the wave velocities in the layers and the

intersect between line segments gives information about the depth of the layers (figure 36). Two

different inversion methods can be employed. The first technique is called “time-term” inversion

or “intercept-time” method, and it employs a combination of linear least squares and delay time

analysis to invert the first arrivals for a velocity section [49]. A more complex inversion method

is the “reciprocal method” or “plus-minus time analysis method” and is based on the travel time

reciprocity and the determination of the crossover point [62]. In this second method it is necessary

to have overlap between two opposite registered shots.

Advantages and disadvantages

The seismic refraction test is a popular test because of its simplicity and because of the low costs

involved. It is a non-invasive test, meaning that no boreholes or cone penetrations are necessary.

This is a significant benefit of the test, since boreholes are expensive and cone penetration tests

may have difficulties penetrating hard layers.

However, there are some disadvantages to the seismic refraction test. Firstly, the test assumes

that the P-wave velocity increases monotonically with depth. Layers that are softer than the layer

above cannot be detected with the seismic refraction test. When the underlying layer is softer,

wave paths will become more vertical and therefore no critically refracted wave can exist. Not

only will the soft layer be impossible to detect, it will also make the underlying layers seem deeper

than is effectively the case. Secondly, the assumed horizontality of the layers can lead to wrong

estimations of the P-wave velocities when the layers are actually inclined, as the assumed wave

path is not valid for inclined layers. This problem is overcome by measuring the arrival times in

two directions from the source. Finally, one should be aware that “blind spots” can be caused by

layers that are too thin or layers that have a too low impedance contrast with the overlying layer.


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An important advantage of the seismic refraction test is that it can be combined with the SASW

test (section 7.9). Both tests use the same test set-up and measurement data. They differ only

in the postprocessing of these data. Combining the postprocessing of the two tests has some

important synergies. This will be elaborated in subsection 7.9.

7.4 Down-hole testing

Seismic down-hole tests measure the shear and dilatational wave velocities by placing vibration

receivers in a borehole and measuring the arrival times of both types of waves.

Test procedure

Figure 37 shows a schematic of the test setup. The test starts with the drilling of a borehole

and the installation of a vibration source next to it. A vibration receiver is then clamped to the

borehole wall at a certain depth z1. This receiver may be a geophone or an accelerometer. Next,

the excitation source is used to generate dilatational and shear waves, of which the arrival is

detected by the receiver in the borehole. Pressure sensors are installed at the source, to serve

as a trigger for the start of the measurements. To reduce the uncertainty due to noise, the test

is repeated a number of times. The results of all tests are stacked to obtain a higher SNR. The

receiver is then lowered into the borehole, with steps ∆z that are typically between 0.5 and 1 m.

The test is repeated for every receiver depth zi.

Figure 37: Down-hole method.

If enough sensors are available, then an alternative to this stepwise procedure is to install a

series of sensors simultaneously in the borehole. This procedure allows for a faster execution of

the down-hole test.


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As for the seismic refraction test, the STA/LTA technique may be used to facilitate the arrival

time detection. The detection of the arrival of S-waves is more cumbersome than for P-waves,

because they arrive later. Reversing the polarization of the waves can again be helpful for the

detection of the S-wave arrival time.

The detected arrival times can be used to plot the arrival time of the P- and S-waves versus the

depth of the receiver. This plot holds the information of the shear and dilatational wave velocities,

which can be calculated for every receiver interval as:

Cpn =zi+1 − zi

∆tp,i+1 −∆tp,i(51)

for the P-wave velocity, in which zi is the depth of receiver i and ∆tp,i is the arrival time of the

P-wave at receiver i. The formula for the S-wave velocity is the same, but uses the arrival times

of the S-waves.

Caution is needed when the distance between the excitation source and the borehole is large.

When the ray path deviates considerably from the vertical, the difference in travelled distance for

two sensors can no longer be approximated by the difference in depth.


The advantage of the down-hole method is that it provides fairly accurate results, with a high

spatial resolution. The disadvantage over the SCPT method is the elevated cost of the method.

7.5 Up-hole testing

Seismic up-hole tests measure the shear and dilatational wave velocities by placing an excitation

source in a borehole and measuring the arrival times of both types of waves at the surface.

Test procedure

The up-hole test is very similar to the down-hole test. Figure 38 shows the test setup. The

excitation source is now located in the borehole and the vibrations are recorded at the surface.

The test thus starts with the drilling of a borehole and the installation of a vibration receiver, a

geophone or an accelerometer, next to it. An excitation source is then lowered into the borehole.

The possibilities for the excitation source are limited due to the fact that it has to be useable in a

confined space. To generate P-waves, a vertical source needs to be used, this can for example

be a falling weight. To generate S-waves, a rotary source may be used.

As for the down-hole method, the test is repeated a number of times for every excitation depth.

The different data records are then stacked to obtain a higher SNR.


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Figure 38: Up-hole method.


The soil parameter estimation procedure is completely analogous to that of the down-hole method.

The travel times are known and so are the traveled distances. Equation (51) is therefore still ap-

plicable, as are the comments concerning the distance between the receiver and the borehole.


The advantage of the up-hole method is that it provides fairly accurate results, with a high spatial

resolution. The disadvantage over the SCPT method is the elevated cost of the method.

7.6 Cross-hole testing

The cross-hole method uses two or more boreholes, with an excitation source in one borehole

and receivers in the others. Using more than two boreholes makes the method more reliable and

makes the method useable for material damping estimations.

Test procedure

The method starts with the drilling of the required number of boreholes. In soft soils, the boreholes

typically need a lining, preferably made of plastic tubing. There are two ways of performing the

cross-hole method. The first method starts with the drilling of the complete boreholes. After the

drilling of the boreholes, the source and receivers are installed at a certain depth and stepwise

lowered into the hole. In the second method, the drilling is also done stepwise. After each drilling

step, the source and receiver are installed at the bottom of the borehole and the experiment is

executed.

As for the up-hole method, one can use a falling weight or a rotary source as excitation. For


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the cross-hole test, the rotary source can be used to create horizontally polarized shear waves

and the falling weight can be used to create vertically polarized shear waves. The combination

of both can identify an anisotropic soil model, with a different vertical and horizontal shear wave

velocity. An explosive source can be used to generate dilatational waves. High frequency signals

are preferred, so that near field effects are avoided.

(a) (b)

Figure 39: Cross-hole method.

The other boreholes are used for the installation of vibration receivers, for which geophones or

accelerometers may be used. The receivers are lowered to the same depth as the excitation

source, which means that horizontally traveling waves are measured. To ensure a good coupling

between soil and receiver, a backfill material may be needed. This is not needed if the drilling is

also done stepwise.

When all receivers and the excitation source are installed at the same depth, the experiment can

start. The excitation source is activated, triggering the data acquisition and the ground motions

are recorded in each of the receiving boreholes. This process is repeated until a sufficiently

high SNR is obtained by stacking the time signals. The averaged time signals are used for the

determination of the arrival time and the motion intensity. The latter is needed for the estimation

of the material damping ratio. The source and receiver are then lowered further in the borehole.

Typical step lengths are between 0.5 and 1m.

When measurements are done to depths over 20m, it might be necessary to measure the in-

clination of the holes. This inclination has an effect on the distance between boreholes, which

becomes significant for large depths. If the inclination is measured, the cross-hole method can

be used up to depths of 50 to 80m.


The wave velocities can be estimated in a very similar way as for the up- and down-hole method.

The traveled distance ∆x equals the distance between the excitation borehole and the receiver

borehole and the travel time ∆t(z) at depth z is measured. The dilatational or shear wave velocity


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then becomes:

Cp/s (z) =∆x

∆t(z)(52)

When multiple receiver boreholes are used, then the difference in arrival time between receiving

boreholes is used. This is more reliable than using the travel time between the source and the

receivers, since it eliminates the errors due to a trigger delay. When multiple boreholes are used,

the method can also be used to estimate the material damping ratio. For this purpose, one

considers the decrease in amplitude of the vibrations with increasing distance from the source.

After correction for the effect of geometric damping, the remaining amplitude decrease is due to

the effect of material damping.


Because multiple boreholes are needed, the method is very expensive. However, results for the

wave velocities are of good to very good quality.

One has to be aware of the risk that refracted waves, traveling through nearby stiffer layers, may

arrive before the direct horizontal wave. This risk increases when layers are thin and when the

receiving boreholes are further from the transmitting borehole. This makes the spacing between

the boreholes a crucial test parameter. The numerical accuracy increases when the boreholes are

well separated, but a large distance between the boreholes increases the likelihood that refracted

waves arrive before the the direct waves.

A disadvantage of the method is the assumption of horizontal layers. When this assumption does

not hold, the results will be incorrect. Another disadvantage is the difficulty of estimating the

damping properties. To obtain a reasonable accuracy, the receivers need to be clamped to the

borehole wall very well.

7.7 Seismic cone penetration test

The Seismic Cone Penetration Test (SCPT) is a variant of the classical cone penetration test

[6, 32, 39]. It is an invasive test and allows for an estimation of the dilatational wave velocity and

the shear wave velocity, with corresponding material damping ratios.

Test procedure

Figure 40 shows the configuration of a typical cone, used in seismic cone penetration tests. The

cone is equipped with two receivers (triaxial geophones or accelerometers), vertically separated

by typically 1 m. This cone is pushed into the soil. A seismic source is installed next to the

penetration point of the cone. Vertical or horizontal hammer impacts on a foundation are typically

used as a source. Dependent on the direction of the hammer impact, P-waves or S-waves prop-

agate through the soil along the shaft. The complete test setup is illustrated in figure 41. The

lateral offset between the source and the seismic cone is typically between 1.0 and 1.2 m. In

order to ensure that the loading beam is coupled to the soil, it is loaded by the weight of the CPT

truck. When a horizontal load is applied, rollers can be placed between the beam and the truck


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to avoid energy leakage to the truck. It has been demonstrated that these rollers lead to shear

wave amplitudes that are three to four times higher than without them.

189 1033 260

392 720 370

1482

60°

Ø 4

3.70

2(A

=15

cm

²)

Ø 3

5.68

2(A

=10

cm

²)

Ø 4

7.87

3(A

=18

cm

²)

Cone tip

Triaxial sensor (bottom part)

[mm]

Ø 3

5.6 8

2(A

=10

cm

²)

Ø 4

3.70

2(A

=1 5

cm

²)

Triaxial sensor (top part)

Figure 40: Typical cone used for Seismic Cone Penetration Tests.

Beam

Static load

Seismic cone penetrometer

Triaxial accelerometers

Lateral offset

Sledge−hammer

Mechanicalhammer

1

x

zz

y

2

Front viewSide view

1

2

Figure 41: Seismic Cone Penetration Test setup.

The test is started by pushing the cone into the soil, to its initial depth. A mechanical hammer or

sledgehammer is then used to generate the impact on the foundation. This triggers the recording

of the vibrations in the two receivers in the cone. The passage of the resulting waves at the

receivers is recorded and stored on a computer. The cone is then pushed one step further into

the soil and the experiment is repeated. The step length is usually 0.5 or 1 m. These SCPTs can

reach as deep as conventional SPTs.


The wave velocity is estimated from the arrival times of the waves at both receivers or from the

phase of the transfer function between both receivers. For the first method, one can choose a

typical point on the time history, such as the first significant upward peak and look for this point

in the two time signals. Other possibilities are the first sudden rise or the first through of the time

function. The latter uses the auto- and cross-correlation spectra to estimate the transfer function.

The material damping ratio of the soil can be estimated from the modulus of the transfer function,

provided that the effect of geometrical attenuation is properly eliminated. In order to estimate the

variation of the soil properties with depth, the test is repeated for various cone penetration depths.


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This technique has the advantage that a high resolution can be obtained. Another advantage

is that it does not require a borehole, nor the difficulties associated with placing sensors in a

borehole. However, being an invasive method, the cone might be unable to penetrate hard layers,

placing a potentially unacceptable limit on the measurement depth.

7.8 Suspension PS logging

The use of a suspended probe in a borehole to obtain the P- and S-wave velocities in an ongoing

way is a relatively new method, dating back to the 1950’s [95]. It is a technique that fulfills the

need of measuring shear wave velocities in deep, uncased boreholes.

Physical principle

The suspension PS logging test is similar to the up-hole test. However, to be able to reach large

depths, the source and receivers of the vibration are both lowered into the borehole. Neither the

receivers, nor the source are coupled to the borehole wall but are suspended in the borehole fluid.

The receivers therefore record the waves (P → PR → P and P → SR → P), in which the subscript

R refers to refracted waves on the walls of the borehole. For this critical refraction phenomenon

to happen, the shear wave velocity Cs in the soil has to be larger than the P-wave velocity Cf

in the fluid. When Cs < Cf there is no wave conversion P → SR → P and it is not possible to

measure the shear wave velocity in the ground by means of this procedure.

Test procedure

In order to move both the receivers and a source down the borehole, a probe (figure 42a) is hung

from a cable into the borehole and suspended in a borehole filled with water (or drilling fluid).

The source of vibration and the sensors are not linked to the borehole walls, but use water (or

drilling fluid) as a coupling with the ground. The method determines directly the average velocity

Cs (or Cp) of the 1 m high soil column, located between the two horizontal geophones (or vertical

accelerometers).

The impulsive source of vibration generates a horizontal point force, normal to the boring walls.

The created wave field (figure 42b) can approximately be considered as produced by a point

force located within an infinite, elastic medium, provided that the wave length λ is much larger

than the borehole diameter d. The predominant radiation of S-waves occurs in the direction of

the borehole axis, whereas that of the P-wave is normal to it (figure 43).

An electromagnetic exciter (figure 44) is used to generate an impulsive force that is transmitted

through the water (or the drilling fluid) to the ground. In order to reach a maximum transmis-

sion efficiency (close to unity), it is necessary that the vibration source has a low mechanical

impedance and an apparent density similar to that of water (or drilling fluid).

The geophones (figure 45) are suspended in the drilling fluid so that, because of the apparent

density of the sensor which equals that of the drilling fluid, neutral buoyancy occurs and the


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(a) (b)

Figure 42: (a) Suspension PS velocity logging system and (b) mode of soil deformation within a

half-space due to a horizontal impulse.

Figure 43: Radiation pattern of a single point force and conceptual displacement features of a

borehole induced by S-waves with wave length λ much higher than the borehole diameter d [43].


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(a) (b)

Figure 44: (a) Electromagnetic exciter for the indirect-excitation type source and (b) principle of

the indirect excitation type source [43].

displacement U of the geophone equals that of the displacement UW of the drilling fluid. On

the other hand, the drilling fluid displacement equals the borehole wall displacement, provided

that the wavelength λ is much larger than the borehole diameter d. Consequently, the geophone

displacement also equals the soil displacement.

Figure 45: Principle of suspension type detector [43].

Figure 46 shows a time signal as recorded by the aforementioned receivers and generated by the

discussed excitation source. In this figure, HN and HR stand for the signals captured by the hori-

zontal geophones when activating the horizontal wave source in the normal (N) and reversed (R)

direction. V in turn represents the signals of the near and far vertical accelerometers, generated

when striking horizontally in the normal direction.


As for all tests that measure the wave velocities by measuring vibrations in different receivers, the

shear wave velocity Cs can be determined from the distance between the horizontal geophones

(1 m) divided by the time difference of the peak values of the S-wave signals captured by each


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Figure 46: Example of measured time history.

receiver. The same procedure is used to obtain Cp values from the signals recorded by the near

and far accelerometers.

By calibrating the two horizontal geophones of the probe it is possible to identify particle velocities

induced by S-waves. Dividing those velocities by their corresponding Cp values, shear strains

can be estimated at each depth. Shear strain levels less than 10−8 were determined at different

depths [93].


The major advantage of the method is the depth to which it can be used, maximum depths up to

700 m have been reported.

The test duration depends on the desired spatial resolution. The conventional test is executed by

lowering the probe one meter at a time. In this case, there is no overlapping of the soil intervals

for which average velocities are measured and one can approximately measure 100 soil intervals

in 2.5 hours. If the resolution is increased, by using a step of half a meter, the test time is almost

doubled. In some tests, and for certain depth intervals, intervals of 0.2 m have been used. In

both cases a SIRT (Simultaneous Iterative Reconstruction Technique) analysis routine can be

used [60] to perform a least-squares inversion of the overlapping average velocities at each one

meter depth interval.

7.9 Spectral Analysis of Surface Waves

The Spectral Analysis of Surface Waves (SASW) test is a non invasive test method to determine

the dynamic shear modulus and the material damping ratio of shallow soil layers [58, 100]. The

SASW method has been used to investigate pavement systems [59], to assess the quality of


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ground improvement [16], to determine the thickness of waste deposits [42], and to identify the

dynamic soil properties for the prediction of ground vibrations [52, 54, 67].

Physical principle

The method relies on the measurement of surface waves, which were discussed in section 2.5.

Surface waves propagate in the horizontal direction and decay exponentially with depth. For low

frequencies, the wavelength of the surface waves is large and the surface waves reach deep soil

layers. These layers are generally stiff and weakly damped, resulting in a high phase velocity

and a low attenuation coefficient. For high frequencies, the wavelength of the surface waves

is smaller and the surface waves travel through shallow soil layers. These layers are generally

softer and more strongly damped, resulting in a lower phase velocity and a higher attenuation

coefficient. As a consequence, the phase velocity and attenuation coefficient of surface waves

vary with the frequency and are determined by the variation of the soil properties with depth.

Test procedure

The SASW method consists of three steps. First, an in situ measurement is performed where sur-

face waves are generated by means of a falling weight device, an impact hammer, or a hydraulic

shaker. The response is measured by means of accelerometers or geophones located at the

soil’s surface. Priority should be given to vertically measuring sensors and a good coupling of the

sensors to the ground is crucial. Second, the experimental dispersion curve CER(ω) and attenua-

tion curve AER(ω) (i.e. the phase velocity and the attenuation factor as a function of the frequency)

are determined from the measurement data. Three different procedures to obtain this dispersion

curve will be discussed further, in order of increasing complexity. In the third step, an inverse

problem is solved in order to find a soil profile corresponding to the experimental dispersion and

attenuation curves. This will be discussed later.

The oldest and most basic method is to apply a steady-state harmonic force with frequency f on

the surface and use a receiver to find the distance to the nearest point that moves in phase with

the excitation source. The distance between the source and the receiver is then assumed to equal

one wavelength λ. The wave velocity CER is calculated as fλ. Performing this experiment for var-

ious frequencies provides the data required to find the experimental dispersion curve. Compared

to the following methods, this first method is inefficient and is therefore not recommended.

A more efficient estimate of the surface wave velocity is obtained by means of Nazarian’s method

[58], where an impulsive source and a line of multiple receivers are used. The wave velocities are

estimated from the phase of the transfer functions between pairs of receivers. The H1 estimator

of the transfer function is used for this purpose [22]. The accuracy of the estimator increases

proportionally to√N , with N the number of events recorded (e.g. hammer impacts). For each

receiver pair, the phase velocity of the surface wave is estimated as:

CER(ω) =

ω∆rijθij(ω)

(53)

where ∆rij is the distance between the receivers and θij(ω) is the unfolded phase of the transfer

function. The results for each receiver pair are withheld for a certain frequency only if the co-

herence between the signals is sufficiently high [76] and the measured wavelength λER is not too


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high or too low compared to the receiver pair distance ∆rij. The lower bound rmin for the ratio

∆rij/λER acts as a high-pass filter that limits the contribution of body waves:

f ≥ rminCER

∆rij(54)

while the upper bound rmax serves as a low-pass filter to remove the high frequency components

contaminated by coherent noise [58]:

f ≤ rmaxCER

∆rij(55)

Because of these thresholds, it is recommended to use different receiver pair distances. A rec-

ommended receiver positioning scheme is to place the first sensors at 2 and 3 m from the source

and then use multiples of these distances for the other sensors (e.g. 2m, 3m, 4m, 6m, 8m,

12m, etc.). The receivers at 2m and 4m, 3m and 6m, 4m and 8m, 6m and 12m, 8m and

16m, 12m and 24m, 16m and 32m, and 24m and 48m from the excitation source can then, for

example, be taken as pairs. The lowest distance determines the highest measurable frequency,

while the largest distance determines the lowest measurable frequency and therefore the depth

to which can be measured. This puts a minimum limit on the distance of the farthest separated

receiver pair. The dispersion curves of the different pairs are finally combined by fitting a high or-

der polynomial to the cloud of points obtained from all pairs. Nazarian’s method can be adjusted

for use in a passive SASW, for which no excitation source is needed and ambient vibrations are

used instead [8]. These passive SASWs are suitable to measure in a low frequency range [63].

The third method, which also uses an impulsive load and multiple receivers, is based on the

transfer function H(r, ω) between the impact force and the vibrations at the receivers at distance

r from the source [64, 65]. Most recent methods are based on a transformation of the transfer

function H(r, ω) to the frequency-wavenumber domain. The resulting frequency-wavenumber

spectrum exhibits peaks corresponding to the occurrence of the surface waves, similar to the

peaks in the transfer function of a finite structure corresponding to the eigenmodes. The positions

of the peaks reveal the dispersion curves, while their width is used to determine the attenuation

curves, using the half-power bandwidth method [9]. Other methods to determine the attenuation

curve exist but are based on the hypothesis that the response of the soil is due to a single

surface mode [25, 48, 71]. Several transformation schemes are available for the computation

of the frequency-wavenumber spectrum. Most of these can be regarded as (approximations

of) a Fourier transformation. While a Fourier transformation leads to a frequency-wavenumber

spectrum that can be used for a reliable estimation of the dispersion curve, it can not be used for

the determination of the attenuation curve. Instead, a Hankel transformation should be applied

[24]: this transformation decomposes the (axisymmetric) wave field in a series of plane waves,

which is a prerequisite for the application of the half-power bandwidth method. In a similar way as

for the previous methods, the inter-receiver distance and the array length determine shortest and

longest measurable wavelength and, consequently, the frequency range where the dispersion

and attenuation curves can be determined. The number of receivers determines the accuracy of

the frequency-wavenumber spectrum. If only the dispersion curve is of interest, it is suggested to

use the same setup as for the method based on receiver-pair transfer functions. This approach

gives a relatively rough approximation of the frequency-wavenumber spectrum, but it is sufficiently

fine to determine the positions of the peaks in an accurate way. If the attenuation curve needs to

be determined as well, a higher number of receivers is required (e.g. 30 instead of 10), so that


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the shape of the peaks is resolved sufficiently well for the application of the half-power bandwidth

method.

Figure 47 shows the frequency-wavenumber spectrum for the site in Lincent. Only a single peak

can be distinguished, which means that the response at the soil’s surface is dominated by a

single (probably the fundamental) surface wave. The corresponding experimental dispersion and

attenuation curves are shown in figure 48.

Figure 47: Experimental f–k spectrum H(kr, ω) for the site in Lincent.

(a)0 20 40 60 80 100

0

50

100

150

200

250

300

Frequency [Hz]

Pha

se v

eloc

ity [m

/s]

(b)0 20 40 60 80 100

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency [Hz]

Atte

nuat

ion

[1/m

]

Figure 48: Experimental (a) dispersion curve CER(ω) and (b) attenuation curve αE

R(ω) for the site

in Lincent.

In summary, a few practical guidelines can be formulated for the two most recent methods. First,

one should be aware that the smallest receiver pair distance determines the maximum measur-

able frequency and the highest distance determines the minimum measurable frequency. The

frequency range that is of interest depends on the soil itself, and can be estimated if historical

data are available. Second, in order to obtain a high coherence over a wide range of frequencies,

different sources should be used. Each source is rich in certain frequencies so when multiple

sources are used, a broad bandwidth is excited. Third, most of the experiment time goes into the

set-up of the receivers. Recording multiple impacts does not cost a lot of extra time. It is there-

fore recommended to record at least 20 times. SASW tests with 100 repetitions are certainly not

uncommon. Finally, it is strongly recommended to measure the input force, also in the second

method. This enables the computation of experimental transfer functions, which can be used for

a validation of the results.


The soil profile is finally determined from the experimental dispersion and attenuation curves

through the solution of an inverse problem. The direct stiffness method [41] or an equivalent


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formulation is used to calculate the theoretical dispersion and attenuation curves of a soil with a

given stiffness and damping profile. The theoretical dispersion curve corresponds to the first (or

fundamental) mode of a layered halfspace or to the effective dispersion curve (a combination of

multiple modes) in the case of inverse layering where still layers are underlain by softer layers

[26, 92]. The profile is iteratively adjusted in order to minimize a misfit function that measures

the distance between the theoretical and the experimental dispersion and attenuation curves.

The minimization problem is usually solved with a gradient based local optimization method [58].

An initial soil profile can be estimated by approximating the experimental CER − λE

R curve with a

stepwise function, using the following rules of thumb:

Cs = 1.1CER (56)

z =λER

3(57)

and similar for the material damping ratio:

β =1

2πλERA

ER (58)

z =λER

3(59)

However, the dispersion and attenuation curves are insensitive to variations of the soil properties

on a small spatial scale or at a large depth. The information on the soil properties provided by

these curves is therefore limited. As a result, the solution of the inverse problem is non-unique:

the soil profile obtained from the inversion procedure is only one of the profiles that fit the exper-

imental data. The non-uniqueness of the solution of the inverse problem in the SASW method

and the resulting impact on the accuracy of ground vibration predictions have been investigated

in reference [79].

The ElastoDynamics Toolbox EDT [78] may be used for the computation of the theoretical disper-

sion and attenuation curves in the inversion analysis.


As the SASW method is based on a non-invasive test, it is an inexpensive method since it does

not require the use of cone penetration or boreholes. The test can also be executed relatively

quickly. It is also an advantage that the test can lead to estimations of both the shear wave

velocity and the material damping ratio.

On the other hand, the resolution of the test is limited in terms of depth and spatial scale: it is

difficult to identify the properties of deep layers (more than 10-15 m) and to detect thin layers.

This may not be an issue, as deep and thin layers may also have little impact in ground vibration

predictions, depending on the frequency range of interest. Furthermore, the depth problem can

be overcome by the use of a passive SASW.

The experimental setup of the SASW method is the same as for the seismic refraction test,

which means that the data collected from an SASW test allow for the simultaneous determination

of the dilatational wave velocity. Combining both inversion analyses into one, gives important

synergies. In every iteration step, one soil profile is generated, which is used to compute the


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theoretical dispersion curve and surface response. The first is compared with the experimental

curve from the SASW method and the latter is compared to experimental surface response data.

A combined residual is now minimized. In this technique, the layer interfaces for the P- and S-

wave velocity will match. Furthermore, one can use information from the S-wave velocity profile

to make assumptions about possible inverted P-wave velocity profiles. This would increase the

reliability of the P-wave velocity estimations.

7.10 Seismic tomography

Traditional geotechnical investigations require drilling through the embankment and taking of sam-

ples and/or performance of in situ tests. In many cases, and particularly for railway embankments,

it is desired to use methods that do not require access to the embankment and do not interfere

with the ongoing traffic. For this purpose, the method of seismic cross-hole tomography can be

applied, obtaining a fairly good picture of wave velocities.

Tomography is a well-known technique in many branches of science to create images of projec-

tions (tomograms) of hidden objects by the use of X-rays, ultrasound or electromagnetic waves

(tomo = slice, graph = picture). During the past few decades the use of tomography has become

more common in the earth sciences, mainly in oil and gas prospectus. There are different kinds of

tomographic measurement techniques and what was used in this project is termed seismic cross

well direct wave travel time tomography. However, it is commonly called cross-hole tomography.

Physical principle

The basic principle of the technique is to estimate a velocity model of the ground by measuring

the time for elastic waves to propagate from a source to a receiver.

Test procedure

To perform seismic cross well tomography measurements it is necessary to have two (or more)

boreholes, figure 49(a). An array of geophones are inserted in one hole and an elastic wave is

generated in the other. A seismograph measures the time it takes for the wave to propagate from

the source point to the geophones. The source is then moved to another position in the hole

and the procedure is repeated. The measurement equipment consists of a summit seismograph,

borehole geophones and a seismic source (combined for both compression and shear waves)

run by an impulse generator (figure 49(b)).

The shear wave source and the geophones are clamped against the borehole wall by use of air

hoses to ensure good contact with the ground. Shear waves display a phase shift when rotating

the source 180. This property can be used to make sure that the correct waves are identified in

the seismograms. Therefore, for the shear wave measurements two data collections should be

made at each depth, one with the source oriented 90 from the location of the geophone (positive

phase) and one with the source located 270 from the direction of the geophone (negative phase).

Plotting the data from the two different source orientations on top of each other facilitates the

detection of the shear wave arrival time. The compression wave measurements should be made

with the source oriented directly towards the geophone, without clamping the source tool to the


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(a)

(b)

Figure 49: (a) Instrument setup and ray paths during a seismic tomography and (b) measurement

equipment.

borehole wall. This increases the possibility of receiving high quality compression wave data.

It is not necessary to rotate the source since the compression waves always arrive first to the

geophone and are thus fairly easy to identify.

The measurements produce numerous arrival times of waves that have crossed the investigated

area. The geophone distance and the wave frequency mainly govern the data resolution; the

shorter the distance and the higher the frequency, the better the resolution. The geometry of the

investigated area, or the ratio between the depth of the boreholes and the distance between the

boreholes is also an important parameter since shallow boreholes and a large distance will lead

to poor ray coverage.


The area between the boreholes is divided into a grid of velocity cells. The size of the cells can

be varied in any particular way, but it is seldom relevant to use a smaller size than the geophone

distance. Each cell is assigned an initial start value for the wave velocity. The program then

calculates the time it takes for different rays to travel through the area between the boreholes. The

calculated times are compared to the measured travel times, and the errors in the calculations are

the differences between these two parameters. Different rays intersect each cell and the best-fit

velocity is estimated by the least squares method. The procedure is repeated for a predetermined

number of iterations or until a chosen limit is reached, the so-called RMS residual. The calculated

velocity model does not provide a unique solution to the inversion problem, but with information

about the geological conditions at the site, it is possible to determine if the established model is


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physically reasonable.

(a)

(b)

Figure 50: (a) Shear wave and (b) compression wave tomograms from beneath an embankment.

Typical results of evaluated shear wave velocity measurements from beneath an embankment are

presented in figure 50a, the results of the evaluated compression wave velocity measurements

are presented in figure 50b. These results were obtained by performing measurements at every

1.0 m depth level, starting at the bottom of one borehole, ending 1 m depth below the ground

surface.


The advantage of seismic tomography is the possibility to obtain a high spatial resolution and

that it can estimate properties in an area that cannot be reached, for example the area under


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train tracks. Its disadvantages are the high cost that is associated with the boreholes and the

non-uniqueness of the solution.


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8. DYNAMIC SOIL CHARACTERISTICS FROM EMPIRICAL

RELATIONS

8.1 Introduction

This section gives empirical relations that can be used to estimate dynamic soil properties from

classical mechanical soil properties. The section is split into two subsections. Subsection 8.2

uses classical soil properties such as the void ratio e, the overconsolidation ratio OCR, the plas-

ticity index Ip, the water content w, the undrained shear strength cu, the liquid limit wl and the

Poisson ratio ν to obtain the small strain dynamic soil properties that are required in numerical

analyses. Subsection 8.3 uses the measurement data from classical in situ tests to estimate the

same small strain dynamic soil properties directly.

The dynamic soil properties are affected by the effective stress level, by the strain level and can

vary with time. The ground water conditions and soil layering can be important as well and need

to be considered. The most important factors, influencing soil behaviour during vibratory loading,

are the strain level, the number of loading cycles and the loading rate.

Empirical relations for the dynamic properties have been elaborated for various soils and in re-

lation with different problems. The relations for the maximum shear modulus are probably best

verified and some of them have been elaborated with particular reference to different sources

described in literature. There exists a large amount of empirical and semi-empirical methods to

estimate dynamic properties of ground materials based on general geotechnical and rock me-

chanical index parameters.

Empirical methods should only be used to obtain a first estimation of the soil properties, which is

then used in planning subsequent in situ tests. Parameters obtained with empirical relations only

are not acceptable for the characterization of test sites in the RIVAS project.

8.2 Dynamic properties from classical soil mechanical properties

8.2.1 Small strain shear modulus

Granular Soils

For coarse- and medium-grained soils, the small strain shear modulus µ0 is estimated with guid-

ance from its grain size distribution, grain shape, void ratio and stress condition. The following

empirical relation is often used to estimate the small strain shear modulus µ0 [kPa]:

µ0 = AF (e) (σ′

0)n

(60)

in which F (e) is a function of the void ratio and σ′

0 is the effective confining stress in kPa. A

summary of these formulae is given in table 6, in which it can be seen that a typical value for n is

0.5.


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Ref. A F (e) n Soil material Test method

Sand

[29]7000 (2.17− e)2/(1 + e) 0.5 Round grained Ottawa sand Resonant column

3300 (2.97− e)2/(1 + e) 0.5 Angular grained crushed quartz Resonant column

[83] 42000 0.67− e/(1 + e) 0.5 Three kinds of clean sand Ultrasonic pulse

[38] 9000 (2.17− e)2/(1 + e) 0.38 Eleven kinds of clean sand Resonant column

[44] 8400 (2.17− e)2/(1 + e) 0.5 Toyoura sand Cyclic triaxial

[99] 7000 (2.17− e)2/(1 + e) 0.5 Three kinds of clean sand Resonant column

Clay

[28] 3300 (2.97− e)2/(1 + e) 0.5 Kaolinite, etc. Resonant column

[53]4500 (2.97− e)2/(1 + e) 0.5 Kaolinite, Ip = 35 Resonant column

450 (4.4− e)2/(1 + e) 0.5 Bentonite, Ip = 60 Resonant column

[101] 2000 ∼ 4000 (2.97− e)2/(1 + e) 0.5 Remolded clay, Ip = 0 ∼ 50 Resonant column

[46] 141 (7.32− e)2/(1 + e) 0.6 Undisturbed clays, Ip = 40 ≈ 85 Cyclic triaxial

Table 6: Constants in proposed empirical relations for the small strain modulus [45].

In fine-grained soils, the effect of overconsolidation on the shear modulus becomes pronounced.

The equation for the small strain shear modulus µ0 [kPa] then changes according to Hardin [27]

to:

µ0 = A F (e) OCRk′ (σ′

0pa)n

(61)

in which OCR is the overconsolidation ratio, A is 625, F (e) is (0.3 + 0.7e2)−1

, n is 0.5, pa is a

reference pressure of 98.1 kPa, σ′

0 is the effective confining stress in kPa and k′ is an adjustment

factor depending on the plasticity index Ip of the soil (figure 51). Equation (61) is mainly used for

low-plastic clays and layered or otherwise inhomogeneous soils, where it can be difficult to obtain

satisfactory and representative values of the undrained shear strength and the consistency limits.

Figure 51: Overconsolidation adjustment factor k′ versus plasticity index Ip [27].

The variation of the shear modulus µ0 in dry sand as a function of depth (confining pressure) is

shown in figure 52. A K0-value of 0.50 has been assumed. It is apparent that the most important


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parameter, which affects the shear modulus, is the void ratio e. The shear modulus increases

with depth from a low value at the ground surface and is thus not a constant value, as is often

assumed. Even a relatively small change in porosity has a large influence on the shear modulus.

Figure 52: Variation of the maximum shear modulus µ0 in dry sand with an assumed coefficient

of lateral earth pressure K0 = 0.5.

Cohesive soils

In soft, normally consolidated clays, the following relationship can be used to estimate the small

strain shear modulus µ0 [kPa]:

µ0 =5150

0.3 + 5.1w2

√

σ′

v (62)

in which w is the water content and σ′

v is the effective vertical stress in kPa.

The small strain shear modulus µ0 [kPa] in normally consolidated or slightly overconsolidated

soils can be estimated as [50]:

µ0 =504 cuwl

(63)

in which cu is the undrained shear strength in kPa and wl is the liquid limit. This relation covers the

whole range of fine-grained soils from low-plastic silty soils to high-plastic clayey organic soils.

An alternative relation, which may yield slightly better estimates for the small strain shear modulus

µ0 [kPa] in high- and medium plastic clays is:

µ0 =

(

208

Ip+ 250

)

cu (64)

in which cu is the undrained shear strength in kPa. In overconsolidated clays, the modulus es-

timated from the undrained shear strength is reduced. Tentatively, this can be done as follows:

µ0(OC) = µ⋆µ0(NC) (65)


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in which µ0(OC) is the small strain shear modulus in overconsolidated soil, µ0(NC) is the small strain

shear modulus for normally consolidated soil, and µ⋆ is the correction factor for overconsolidation.

This correction is based on data presented by Andersen et al. [5]. Atkinson has later presented

data, which indicate a similar correction [7].

The undrained shear strength depends on the plasticity index, Ip and the effective stress. Döringer

developed the following relationship between the shear modulus µ0 [kPa] and the undrained shear

strength cu [kPa] for cohesive soils [17]:

µ0√cupa

= A F (e) OCRk

√

1 + 2K0

3 (0.0029Ip + 0.13)(66)

in which A equals 625, F (e) is (0.3 + 0.7e2)−1

, Ip (%) is the plasticity index, pa is a reference

pressure of 100 kPa and K0 is the coefficient of lateral earth pressure at the rest. The constant

k equals unity for granular soils (sand and gravel). Döringer (1997) also performed a compre-

hensive literature survey of the shear modulus determined from field and laboratory tests [17],

which also included the data reported by Larsson and Mulabdic [50]. Field as well laboratory test

data were used to assess a correlation between the natural water content wn, the plasticity index,

Ip and the undrained shear strength cu as shown in figure 53. The maximum shear modulus at

small strain, µ0 has been calculated for the undrained shear strength cu and a reference stress

pa of 100 kPa.

Figure 53: Relation between the normalize shear modulus and water content for normally con-

solidated clay and silt [17]. The relationship is shown for different values of the plasticity index.

Different investigation methods (field and laboratory) have shown a surprisingly good correlation

for a wide range of soils (with water content ranging from 20%− 180%). The shear modulus data

from laboratory tests are slightly lower and show a larger scatter than results from in situ tests.

An important conclusion from figure 53 is that the shear modulus does not have constant value,

but can vary within wide limits. The water content (and thus the void ratio e) is an important

parameter. The relationship in figure 53 can also be applied to soils with organic content and

peat. However, in soils with a high organic content it is advisable to perform seismic field tests to

check the validity of empirical correlations.


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In peat, it is difficult to obtain relevant values of both undrained shear strength and liquid limit.

The shear modulus µ0 [kPa] in peat may be estimated from [10]:

µ0 = 13800 w−0.67n σ′

0.55

vo (67)

in which wn is the natural water content (%) and σ′

vo is the effective overburden pressure in

kPa. This equation is based on experience from Japanese peat and its relevance for other soil

conditions is uncertain. It should therefore be used with caution.

8.2.2 Estimation of shear modulus degradation curve

The shear stress-strain relations for soils are not linear but follow a hyperbolic of slightly modified

hyperbolic relation. The relation has been found to be different for coarse-grained and fine-

grained soils. The relative decrease of the modulus with strain is generally quicker in coarse-

grained soils. Data from a wide range of sandy and gravelly soils have been compiled by Rollins

et al. [75]. The best hyperbolic relation for all the data points is shown in figure 54. This relation

also more or less coincides with the center of the range found for sands by Seed et al. (1970)

[80]. It approximately follows a curve with equation [75]:

µ

µ0=

1

1.2 + 1600γ (1 + 10−2000γ)(68)

in which γ is the shear strain. This relation also agrees well with empirical studies performed

for more fine-grained non-plastic soils, i.e. Ip = 0. It can thus be used for the whole range of

medium- and coarse-grained soils from coarse silt and upward. In coarser soils, there is an effect

of the stress level in the soil. It is obvious that there is a decrease of modulus which is more rapid

at low effective stresses. The effect is considered to be small and is normally not accounted for.

However, it may be significant at very low stress levels.

Figure 54: Relation between µ/µ0 and shear strain [75].

In fine-grained soils, the stress-strain relations have been found to depend on the plasticity of

the soil. There is a relatively brittle behaviour in low-plastic soils in contrast to ductile behaviour


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and large failure strains in high-plastic soils, particularly in organic soils. Vucetic and Dobry have

presented empirical data for the relative decrease of the shear modulus with strain in fine-grained

soils [96]. The curves in figure 55 show the influence of the plasticity of the soil on this relation.

The corresponding curves for peat fall on or to the right of the curve for Ip = 200.

Figure 55: Influence of shear strain and plasticity on the shear modulus in fine-grained soils [96].

8.2.3 Small strain phase velocities

Indirect method, from shear modulus estimation

According to equation (17), the estimations for the small strain shear modulus lead directly to

estimates for the shear wave velocity if the mass density ρ is known or estimated.

Correspondingly, the compression wave velocity Cp can be calculated from the constrained (oe-

dometer) modulus M0 or from the shear modulus µ0 if the Poisson ratio ν is known, according to

equations (15) and (18), respectively.

Poisson’s ratio ν for saturated soft soils (Sr = 100%) is approximately 0.5. The small-strain

Poisson’s ratio for unsaturated soils (Sr < 99%) is about 0.15 and increases with strain level to

about 0.30. The same relationships can be used to calculate the shear modulus and the confined

modulus if the wave velocities are known. The shear wave velocities for normally consolidated

soils have been calculated and are shown in figure 56.

The shear wave velocity is not constant but increases with depth. The void ratio (and thus in

saturated soils the water content) is an important soil parameter, which affects the shear wave

velocity. In sands overconsolidation does not appear to have a significant influence on the shear

wave velocity. In overconsolidated clays, the overconsolidation effect should be considered.

Direct empirical relations

The shear wave velocity can also be determined based on empirical relationships and experience.

Typical values of the compression wave velocity and the shear wave velocity for different materials

are given in table 7.

For fully saturated soils, the dilatational wave velocity can be estimated according to equation

(25).


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Figure 56: Shear wave velocities in dry sand for different values of the void ratio e.

Soil TypeP-Wave Velocity S-Wave Velocity

[m/s] [m/s]

Ice 3 000 - 3 500 1 500 - 1 600

Water 1480 - 1520 0

Granite 4 500 - 5 500 3 000 - 3 500

Sandstone, Shale 2 300 - 3 800 1 200 - 1 600

Fractured Rock 2 000 - 2 500 800 - 1400

Moraine 1400 - 2000 300 - 600

Saturated Sand and Gravel 1400 - 1800 100 - 300

Dry Sand and Gravel 500 - 800 150 - 350

Clay below GW level 1480 - 1520 40 - 100

Organic soils 1480 - 1520 30 - 50

Table 7: Indicative values of compression wave velocity Cp and shear wave velocity Cs for differ-

ent materials.


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8.2.4 Small strain material damping

During each loading cycle, material damping dissipates a small amount of energy, which is con-

verted into heat. At small strain (< 10−5), the material damping ratio β is low and in the range

of 2 − 4% for most soils. The variation is much less at small strain than in the case of the shear

modulus. Damping values at small strain can be estimated based on experience, figure 57.

Figure 57: Typical values of the material damping ratio for different soils at small strain (< 10−5).

The material damping ratio in the soil can be estimated in a similar way with a division of the soil

into medium- and coarse grained soils and fine-grained soils. However, the picture for the varia-

tion of this parameter is not quite as consistent as for the shear modulus. In general, the damping

ratio increases with increasing strain and is highest for coarser non-plastic soils and lowest for

high-plastic and organic soils. With increasing shear strain, the influence of the plasticity index Ipon the material damping ratio β becomes more pronounced. Material damping is much higher in

granular soils. At a shear strain level of 10−3, the damping ratio is about 10 to 15% in low-plastic

(granular) soils, while damping is much less in cohesive soils, in the order of 3 to 8%.

The average of the data for sand and gravel presented by Seed et al. [80] agrees very well with

the relation proposed for non-plastic silts by Vucetic and Dobry [96]. Seed et al. also proposed

that the damping ratios in sand and gravel are very similar. In contrast, many of the data for

gravelly soils compiled by Rollins et al. show lower damping ratios, which are almost in the same

range as for high plastic clays [75]. The variation in damping ratio with strain for gravelly soils

presented by Rollins et al. is shown in figure 58. The figure also shows the range of data for

sands and gravels presented by Seed et al. [80].

The damping ratio - strain relation is affected by the stress level and the relative density. An in-

crease in stress level results in a lower damping ratio and an increase in relative density results in

a higher damping ratio. It thus appears as if the relations for damping ratio versus strain proposed

for sand and gravel by Seed et al. [80] and by Vucetic and Dobry [96] for non-plastic fine-grained

soils can be used for all non-plastic soils with grain sizes from coarse silt and upwards. In fine-

grained soils a clear influence of the plasticity on the damping ratio has been found. The damping

ratio thus decreases with increasing plasticity and organic content. Empirical guidelines for the

variation of damping ratio with strain and plasticity index have been proposed by Vucetic and

Dobry [96] (figure 58). The curves for damping ratios in peat fall on or below the curve for PI =

200.


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Figure 58: Damping ratio versus strain for coarse-grained soils [75].

Figure 59: Damping ratio versus strain for fine-grained soils [96].


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8.3 Small strain shear wave velocity from CPT and SPT results

Cohesive soils

In [55] the following equation for the estimation of µ0 [kPa] for clays, as a function of qc and the

void ratio e is given:

µ0 = pa99.5

e1.13

(

qcpa

)0.695

(69)

where pa is a reference pressure of 100 kPa and qc is in units of kPa. A correlation with N ,

the result from a SPT test, has been identified [13]. A coefficient of determination r2 = 0, 901indicates a strong correlation. Using equation (69) and the correlation between NSPT and qc given

in [13], table 8 has been produced, relating qc with N and Cs for different void ratios.

e0qc Cs NSPT

[kPa] [m/s] (correlated)

0.5 1000 232 3− 40.5 2500 320 8− 90.5 5000 407 16− 170.5 10000 517 33− 340.5 15000 596 50

1.0 1000 157 3− 41.0 2500 216 8− 91.0 5000 275 16− 171.0 10000 350 33− 341.0 15000 403 50

1.5 1000 125 3− 41.5 2500 172 8− 91.5 5000 219 16− 171.5 10000 278 33− 341.5 15000 320 50

Table 8: Variations for clays of Cs with qc [55] and with NSPT [13].

Cohesionless soils

For cohesionless soils, µ0 [kPa] can be estimated with CPT results:

µ0 = 1634 q1/4c

(

σ′

v

pref

)0.375

(70)

in which σ′

v is the effective vertical stress in kPa, pref is a reference pressure of 1 kPa and qc is in

units of kPa.

For cohesionless soils, SPT results can be used as well to estimate the shear wave velocity. More

than twenty correlations between Cs and N , for different types of soils, as established by different

authors, are identified [51]. Focusing mainly on cohesionless soils, the relationships proposed


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by Imai [36], Sykora and Stokoe [88], and Ohta and Goto [61] have been retained in [51]. The

following relationship proposed by Seed et al. [82] has also been considered:

Cs = 69N0.17D0.2F1F2 (71)

with D the depth from the surface in meter, F1 a factor which is 1 for alluvial deposits and 1.3 for

diluvial deposits and F2 a factor that depends on the soil type as indicated in table 9.

Soil Type F2

Clay 1.00

Fine sand 1.09

Medium sand 1.07

Coarse sand 1.14

Sandy gravel 1.15

Gravel 1.45

Table 9: F2 for different types of soil [80].

Using equation (71) (for D = 5 m and F1F2 = 1.25), together with the abovementioned relation-

ships, calculated values of Cs for increasing N result of SPT test (N = 5, 10, 15, . . . , 30), have

been given in table 10. Values of qc estimated according to [56] have also been incorporated in

the second column of this table.

Meyerhof Imai Sykora and Stokoe Ohta and Goto Seed

SPT (1963) (1977) (1983) (1987) (1986)

Cs = 80.6N0.331 Cs = 330N−0.29 Cs = 85.35N0.348 Cs = 69N0.17j D0.2F1F2

Nqc Cs Cs Cs Cs

(kPa) (m/s) (m/s) (m/s) (m/s)

5 2000 137 160 149 156

10 4000 173 196 190 176

15 6000 198 220 219 189

20 8000 217 239 242 198

25 10000 234 255 262 206

30 12000 248 269 279 212

Table 10: Variations of Cs with NSPT according to different authors in [51] and [56]. For the

formula of Seed, D is taken 5 m and F1F2 = 1.25.


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9. RECOMMENDATION FOR RIVAS TEST PROCEDURE

9.1 Introduction

It is inferred from sections 6, 7 and 8 that many different test methods can be used to obtain the

dynamic soil characteristics that are needed for the experimental validation of numerical predic-

tions in the RIVAS project. This section therefore proposes a recommended procedure.

9.2 Recommended procedure

The first step in any soil investigation campaign involves a study of archive records like geological

maps and results of previous geotechnical investigations such as drillings, samplings, laboratory

tests and in situ tests. An estimation of the soil layering and the dynamic characteristics of each

layer is crucial for the planning of soil sampling and in situ tests and can be obtained from archive

records and empirical relations (section 8) between classical soil mechanics parameters and

dynamic soil characteristics. It should be emphasized that, within RIVAS, estimations of dynamic

soil characteristics based on empirical relations cannot replace their determination by means of

in situ or laboratory tests.

In the second step, (undisturbed) soil samples are taken. These samples are needed for the

determination of the mass density ρ, which is one of the five parameters per soil layer (section

2), and other important parameters such as the void ratio, the degree of saturation, etc. This

enhances the understanding of the soil behaviour; based on these parameters, empirical relations

can provide an update of the estimated dynamic soil characteristics. The required number of

samples depends on the soil layering, as estimated in the first step; at least one sample per soil

layer is recommended. Due to the heterogeneous nature of the soil, it is highly recommended,

however, to take samples at different lateral positions as to obtain spatially averaged values per

soil layer.

In the third step, in situ seismic experiments are planned, testing a representative volume of soil in

natural stress and compaction conditions at small strain levels. A combined surface wave - seis-

mic refraction test (sections 7.3 and 7.9) is proposed as this test is easy to perform at relatively

low cost (as it is non-invasive) and allows to determine all required dynamic soil characteris-

tics; possible disadvantages, however, are lack of penetration depth and resolution. A combined

inversion (with a combined objective function) of the dynamic soil characteristics is highly recom-

mended. It is also strongly recommended to measure the input force, so that the soil’s transfer

functions are available and can be used for validation of the dynamic soil characteristics derived

from the test.

If possible, it is advisable to also employ borehole methods or an SCPT in addition to the com-

bined surface wave - seismic refraction test. These methods are able to provide a better spatial

resolution at depth, as well as results upto larger depths.

Laboratory tests on undisturbed samples, such as the resonant column test and the cyclic tri-

axial test (combined with bender element measurements), may be used to determine the strain

dependency of the shear modulus and material damping ratio.


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Steps 1-3 in this recommended course of action do not depend on budget or local soil conditions,

but are minimum constituents of a reliable soil investigation campaign. The use of borehole

methods is not a minimum requirement, but is nevertheless highly recommended if the budget

is available. Laboratory tests are useful to verify the assumption of linear behaviour and to gain

insight in nonlinear effects close to the source of vibration. They can, however, not replace in situ

testing.

The results of the soil investigation campaigns need to be reported in a detailed manner, con-

taining information about all steps including the study of archive records, the soil sampling, the

classical soil mechanics tests, the seismic in situ tests and the dynamic laboratory tests. Test

reports should contain all information that renders the tests reproducible.

The data from all performed tests should be summarized, providing soil profiles including all

relevant information for each layer: the layer thickness d, the wave velocities Cp and Cs, the

material damping ratios βp and βs and the mass density ρ. It is also advisable to report on the

validation of the test results, in order to provide an indication of the reliability of the obtained

results.


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