Short-term deformations in clay under a

formwork during the construction of a bridge

A design study

Alexander Berglin

Master of science thesis 2017

© Alexander Berglin, 2017

Master of Science Thesis

Royal Institute of Technology (KTH)

Department of Civil and Architectural Engineering

Division of Soil- and Rock Mechanics

ISSN 1652-599X 17:05

i

Abstract

During the casting of a concrete bridge deck, the temporary formwork is causing the

underlying ground to deform if a shallow foundation solution is used. There are often

demands on the maximum deformation of the superstructure when designing the foundation

for the formwork. To keep the deformations within the desired limits, several ground

improvement methods like deep mixing columns or deep foundation methods like piling

can be used. Permanent ground improvement methods are however expensive, and far from

always needed. To reduce the need for unnecessary ground improvements, it is crucial to

calculate the predicted deformations accurately during the design phase.

The purpose of this thesis was to investigate how short-term deformations in clay under a

formwork during bridge construction should be calculated more generally in future

projects.

Three different calculation models have here been used to calculate the ground

deformations caused by the temporary formwork. A simple analytical calculation and two

numerical calculations based on the Mohr Coulomb and Hardening Soil-Small constitutive

models. The three calculation models were chosen based on their complexity. The

analytical calculation model was the most idealised and the Hardening Soil-Small to be the

most complex and most realistic model.

Results show that the numerical calculation model Mohr Coulomb and the analytical

calculation model gives the best results compared to the measured deformation. One of the

most probable reasons for the result is that both of the models require a few input

parameters that can easily be determined by well-known methods, such as triaxial-, routine-

and CRS-tests. The more advanced Hardening soil small model requires many parameters

to fully describe the behaviour of soil. Many of the parameters are hard to determine or

seldom measured. Due to the larger uncertainties in the parameter selection compared with

the other two models, the calculated deformation also contains larger uncertainties.

Key words: Small-Strain stiffness, Plaxis, Short-term deformations, Elasticity modulus,

Correlations

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Sammanfattning

Vid gjutning av betongbrodäck kommer den underliggande marken att deformeras av den

temporära formställningen, som tar upp lasterna medan betongen härdar. Det finns oftast

krav på hur stora markdeformationerna maximalt får vara. För att hålla deformationerna

inom gränserna kan diverse markförstärkningsmetoder, så som kalkcementpelare eller

pålar, användas. Permanenta markförstärkningar är oftast väldigt dyra och inte alltid

nödvändiga. Ett alternativ till att använda dyra markförstärkningar skulle kunna vara att

beräkna den förutspådda deformationen med stor exakthet i projekteringsstadiet.

Syftet med det här arbetet var att undersöka hur korttidsstätningar i lera vid en

bronybyggnation ska beräknas mer generellt i framtida projekt.

I detta arbete har tre beräkningsmodeller använts för att beräkna markdeformationerna från

den temporära formställningen. En enklare analytisk modell samt två numeriska

beräkningsmodeller som baseras på Mohr Coulomb och Hardening Soil Small teorierna. De

tre beräkningsmodellerna valdes utifrån deras komplexitet. Den analytiska beräkningen

ansågs vara den mest förenklade modellen medan Hardening Soil-Small var den mest

komplexa och realistiska modellen.

Resultatet visar att trots sin enkelhet så ger den numeriska beräkningsmodellen Mohr

Coulomb och den analytiska beräkningen bäst resultat jämfört med de uppmätta

deformationerna. En möjlig anledning till det goda resultatet är att modellerna endast kräver

ett fåtal ingångsparametrar som kan bestämmas med hjälp av välkända fält- och

laboratoriemetoder så som triaxialförsök, rutinlaboratorieförsök och CRS-försök. Den mer

komplexa modellen Hardening Soil Small kräver flera ingångsparametrar för att kunna

modellera jordens beteende. Många av parametrarna är svåra att bestämma då mätdata

oftast saknas. Osäkerheterna i valet av ingångsparametrar för den mer komplexa hardening

soil small modellen är större än de två andra studerade modellerna, vilekt även ger upphov

till större osäkerheter i dem beräknade deformationerna.

Nyckelord: Small-Strain Stiffness, Plaxis, Korttidssättningar, Elasticitetsmodulen,

Korrelationer

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Preface

This thesis concludes my studies in Civil Engineering at KTH. The idea for the thesis was

provided by ELU Konsult where the thesis was written. I would like to thank the extremely

skilled and kind people at the geotechnical division at ELU for their help when it was

needed. A special thanks to my two supervisors at ELU Konsult; Sebastian Addensten and

Anders Beijer-Lundberg for your knowledge, guidance and encouragement!

I would also like to thank Martin Holmén at SGI for providing me with data and expertise

regarding triaxial tests and Dr. Johan Spross at KTH for your valuable comments about the

thesis.

Furthermore I would like to thank Professor Stefan Larsson at KTH. Your knowledge and

enthusiasm about geotechnical engineering has inspired me.

Last but not least, thank you to all the other people who have been supporting me during

my time at KTH.

Stockholm, June 2017

Alexander Berglin

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Table of Contents

1 Introduction........................................................................................... 1

2 Literature study ..................................................................................... 3

2.1 Brief introduction to consolidation theory ....................................................................... 3

2.2 Brief introduction into elastic theory ............................................................................... 3

2.2.1 Background ................................................................................................................. 3

2.2.2 Theory ......................................................................................................................... 3

2.2.3 Formulation ................................................................................................................. 5

2.3 Elasticity modulus............................................................................................................ 7

2.3.1 Internal factors ............................................................................................................. 7

2.3.2 External factors ............................................................................................................ 7

2.4 Small-strain stiffness........................................................................................................ 8

2.4.1 The influence of diagenesis ........................................................................................... 9

2.4.2 The influence of confining stress ................................................................................. 10

2.4.3 The influence of void ratio .......................................................................................... 11

2.5 Measuring small-strain stiffness .................................................................................... 12

2.5.1 In-Situ tests................................................................................................................ 12

2.5.2 Laboratory tests.......................................................................................................... 14

2.6 Ground investigation methods ....................................................................................... 16

2.6.1 Oedometer tests.......................................................................................................... 16

2.6.2 Triaxial tests .............................................................................................................. 16

2.7 Empirical correlations for determining the soil stiffness ............................................... 21

3 Soil modeling ....................................................................................... 24

3.1 Introduction to numerical modelling ............................................................................. 24

3.2 Mohr coulomb (MC) ...................................................................................................... 24

3.3 Hardening Soil (HS) ...................................................................................................... 26

3.4 Hardening Soil Small (HSS) .......................................................................................... 29

4 Case study: Bridge over Ulvsundavägen ............................................. 31

4.1 Introduction .................................................................................................................... 31

4.2 Studied section ............................................................................................................... 33

viii

4.2.1 Ground conditions ...................................................................................................... 34

4.3 Ground deformation measurements ............................................................................... 36

5 Calculation procedure ......................................................................... 37

5.1 Formwork geometry....................................................................................................... 37

5.2 Empirical correlation study ............................................................................................ 38

5.3 Analytical Calculation.................................................................................................... 39

5.4 2D Numerical simulation ............................................................................................... 41

5.4.1 Assumptions .............................................................................................................. 42

5.4.2 Input parameters......................................................................................................... 42

5.5 3D Numerical simulation ............................................................................................... 45

5.6 Parameter sensitivity analysis ........................................................................................ 46

6 Results ................................................................................................. 47

6.1 Empirical correlation study ............................................................................................ 47

6.2 Calculated deformations ................................................................................................ 50

6.2.1 Analytical calculations ................................................................................................ 50

6.2.2 2D Mohr Coulomb ..................................................................................................... 51

6.2.3 2D HSS Model........................................................................................................... 53

6.2.4 Comparison between 2D and 3D numerical calculations ............................................... 55

6.3 Sensitivity analysis of the HSS model ........................................................................... 56

7 Analysis and discussion ....................................................................... 57

7.1 Empirical correlation for the elasticity modulus............................................................ 57

7.2 Plaxis parameter optimisation function ......................................................................... 57

7.3 Calculated deformations ................................................................................................ 57

7.3.1 Analytical vs measured deformations ........................................................................... 57

7.3.2 MC-calculations ......................................................................................................... 58

7.3.3 HSS-Calculations ....................................................................................................... 58

7.3.4 2D vs 3D ................................................................................................................... 59

7.4 Sensitivity analysis......................................................................................................... 59

7.5 Conclusions and recommendations for practical design ................................................ 60

8 Bibliography ........................................................................................ 61

Appendix A Soil Data................................................................................. 64

Appendix B Numerical input parameters .................................................. 66

ix

Appendix C Performed triaxial tests.......................................................... 70

x

Notations

Abbreviations Explanation

OCR Over-consolidation ratio

CRS Constant rate of strain

NC Normally consolidated

OC Overconsolidated

HS Hardening soil

HSS Hardening soil-small

MC Mohr Coulomb

Roman letters Explanation Unit

Help paramter [kPa]

b Width of the fictive plate [m]

Cohesion, shear strength [Pa]

Corrected shear strength [Pa]

Coefficient of consolidation [m2/s]

Void ratio [-]

Young’s modulus [Pa]

Initial Young´s modulus in the elastic range [Pa]

Undrained young´s modulus [Pa]

Secant modulus (50% of peak strength) [Pa]

Secant stiffness in drained triaxial tests [Pa]

Plastic modulus [Pa]

Tangent stiffness for oedometer loading [Pa]

Unloading modulus [Pa]

Unloading/reloading stiffness [Pa]

Young´s modulus in the vertical direction [ ]

Young´s modulus in the horizontal direction [ ]

Yield function [-]

Shear stiffness [Pa]

Specific gravity [-]

Initial Shear modulus for small strains [ ]

Shear modulus in the vertical plane [ ]

Shear modulus in the horizontal plane [ ]

Thickness of soil layers [m]

Drainage distance [m]

Density index [-]

Permeability [m/s

xi

Coefficient of lateral earth pressure at rest [-]

Bulk Modulus [Pa]

Elastic modulus [Pa]

Plastic modulus [Pa]

Pore-water pressure [Pa]

Deviatoric stress [Pa]

Q Distributed load [kN/m2]

Reference stress [ ]

Force [N]

Deviatoric stress [Pa]

Failure ratio [-]

Sensitivity [-]

Time [s]

Time factor [-]

Velocity of P-wave propagations [m/s]

Velocity of S-wave propagations [m/s]

Wave propagation velocity in soil [m/s]

Plastic limit [%]

Liquid limit [%]

Water content [%]

Greek Symbols Explanation Unit

Deformation [m]

Strain [%]

Friction angle [°]

Lamé constant [-]

Shear strain [Pa]

Shear strain at 30% degradation of small-strain stiffness [Pa]

Dilatancy [°]

Major principal stress [Pa]

Minor principal stress [Pa]

ρ Bulk density [kg/m3]

Poissons ratio [-]

Uncorrected undrained shear strength [Pa]

Shear stress at failure [Pa]

1

1 Introduction During the casting of a concrete bridge deck, the temporary formwork is causing the

underlying ground to deform if a shallow foundation solution is used. The soil deformations

occur within in the first few days, before the load from the bridge deck can be transferred

through the supports of the bridge when the concrete structure cures. There are often demands

on the maximum deformation of the superstructure when designing the foundation for the

formworks. To keep the deformations within the desired limits, several ground improvement

methods like deep mixing columns or deep foundation methods like piling can be used.

Ground improvement is a possible way to strengthen the soil and therefore reducing the

deformations. Ground improvements are however expensive and sometimes superfluous. To

reduce the need for unnecessary ground improvements it is crucial to be able to predict the

ground deformation accurately during the design phase and adjust the height of the formwork

accordingly.

The deformation response of soil is dependent on many different parameters. These include

the elasticity modulus and the small strain-stiffness of the soil. The modulus of elasticity is

hard to decide in geotechnical engineering, due to the highly non-linear behaviour of soil.

Despite that, the elasticity theory has shown that the calculated deformation of a soil

corresponds well to the measured deformations, if the elasticity modulus is chosen carefully.

Triaxial tests are generally the most suitable and easily available method for investigating the

strength and deformation properties of soil (SGF, 2012). The parameters obtained from the

triaxial tests are used for idealized analytical or more advanced numerical models in order to

calculate the deformation of the soil. However, triaxial tests are far from always performed in

geotechnical projects. An alternative method for the estimation of the elasticity modulus is by

using empirical correlations that are based on parameters that can be obtained from in-situ or

routine laboratory tests.

Recent studies by Benz (2007), Clayton (2011) and Wood (2016) have shown the importance

of small-strain stiffness in soils in the serviceability limit state. At very small strains, the soil

behaves elastically, but with increasing strain the stiffness decays non-linearly. Despite this,

small-strain stiffness is not too common in design for the serviceability limit state in

geotechnical projects.

This thesis is based on a study of the ground deformations during the construction of the

bridge over Ulvsundavägen in Stockholm. Calculations of the predicted ground deformations

were done in the design phase of the bridge in order to determine if piles had to be installed

under the temporary formworks. In-situ tests as well as triaxial and CRS-tests were carried out

to determine the soil profile and soil parameters needed to calculate the deformations

accurately.

2

The purpose of this thesis was to investigate how short-term deformations in clay under a

formwork during bridge construction should be calculated more generally in future projects.

Three different calculation models were chosen, a simple analytical and two numerical based

on the Mohr Coulomb and Hardening soil small constitutive model. The three models were

chosen based on their complexity, where the analytical method was considered to be the most

idealised and the hardening soil-small to be the most complex and realistic model.

3

2 Literature study This chapter discusses the mechanical response of soils during stress and deformation change,

in order to relate the later chapter on laboratory tests and numerical models to the scientific

literature.

2.1 Brief introduction to consolidation theory The deformation of soil is a process that involves three stages: The Elastic stage, followed by

primary- and secondary consolidation (Lambe & Whitman, 1979). Elastic deformations occur

instantly when the soil gets exposed to a load. Elastic deformations mainly occur in friction

material such as sand and gravel. Primary consolidation is when the pore water is being

squeezed out from the soil skeleton. Primary consolidation occurs over a longer time span in

cohesive soils. Secondary consolidation is when the soil skeleton gets deformed, a process

occurring over a long period of time (Larsson, 2008).

2.2 Brief introduction into elastic theory

2.2.1 Background When a body is subjected to changing forces, it will to some extent change its shape or

volume, hence deformations will occur. The body is said to be elastic if the shape goes back

to its original state when the forces are removed. The phenomenon that the deformation

(strain) is related to the force (stress) was formulated by Robert Hooke in the 1676

(Timoshenko, 1983). Today this phenomenon is known as the generalized Hooke´s law

(Davis & Selvadurai, 1996).

Hooke´s law is an example of a constitutive relation (Timoshenko, 1983). A constitutive

relation is an equation that relates the cause and effect. The constitutive equations all involve

at least one parameter which takes different values for different materials.

Hooke´s law works well for isotropic mechanical behaviour, i.e. same properties in all

directions. Soil is very complex and the behaviour normally not considered to be isotropic.

However, to simplify the behaviour of soil, assumptions of soil being isotropic is done in

many geotechnical areas (Davis & Selvadurai, 1996).

2.2.2 Theory When an elastic bar gets subjected to uniaxial tension stress it elongates in the direction of

the applied stress, resulting in an extensional strain , (Timoshenko, 1940). According to

the generalized hook’s law, is dependent on , Eq. (1).

(1)

When the bar gets elongated in the stress direction it also gives rise to lateral contraction,

causing the bar to become skinnier. The lateral contraction leads to more strains, and

4

in the lateral directions. If the material is assumed to be linear elastic the relationship between

the strains are:

(2)

The results can be generalized by studying a cube subjected to uniform normal stress in all

directions, Figure 2.1. will be dependent on the stresses in all directions and can be

formulated as Eq.(3).

(3)

Equation (3) can be rewritten and by taking and into account the following equations

will be:

[ ( )]

[ ( )]

[ ( )]

(4)

The shear modulus relates the shear stress at any given point in a body to the shear strain

that occurs at that point, Eq. (5)

(5)

is related to the Young´s modulus by the following relationship:

( ) (6)

Figure 2.1 Cube subjected to normal stresses in all directions

5

By combining equations (4) - (6) the generalized hook´s law can be expressed in a matrix

form in a six-dimensional stress-strain vector space, Eq.(7).

[

]

[

]

[

]

(7)

There are two more elastic constants, the bulk modulus and the Lamé constant , (Davis &

Selvadurai, 1996). These two constants are related to , and . relates the sum of the

normal stresses to the volumetric strain and can be obtained from Eq.(8).

( )

(8)

can be expressed as Eq.(9)

( )( ) (9)

2.2.3 Formulation

2.2.3.1 Isotropic elasticity The response of an isotropic material is independent of the orientation (Davis & Selvadurai,

1996). Isotropic materials can be fully described using two of the five elastic constants, and

, and , Eq.(10) and (11).

( )

(10)

( )

(11)

2.2.3.2 Anisotropic elasticity Anisotropic elasticity is here presented to provide a more realistic description of real soil

behaviour.

The elastic properties in an anisotropic material are dependent on the orientation of the

sample, (Muir Wood & Arroyo, 2004). To fully describe the anisotropic elasticity of a

material a total number of 21 independent parameters are needed (Jamiolkowski et al., 1996).

This can be compared to the isotropic behaviour, where only two parameters are needed.

However, many materials show a more limited version of anisotropy. One example of this is

6

transverse isotropy or cross-anisotropy. The cross-anisotropy have the same elastic parameters

in the horizontal direction, but different parameters in the vertical direction (Piriyakul, 2006).

The cross-anisotropic elasticity can be described by the following matrix, Eq. (12)

[

]

[

( )

]

[

]

(12)

Where:

= Poisson´s ratio for horizontal strain due to horizontal strain at right angles

= Poisson´s ratio for vertical strain due to horizontal stress

= Poisson´s ratio for horizontal strain due to vertical stress

= Young´s modulus in the vertical direction

= Young´s modulus in the horizontal direction

= Shear modulus in the vertical plane

= Shear modulus in the horizontal plane

2.2.3.3 Incompressible elasticity In Soil mechanics, incompressibility is relevant when the response of a fully saturated soil in

undrained conditions needs to be analysed (Atkinson, 2000). Undrained condition is when the

pore fluid cannot move freely inside the soil particles. An incompressible body is

characterized by an infinite bulk modulus, , which implies the following relationships:

, and (Davis & Selvadurai, 1996; Lambe & Whiteman, 2008).

7

2.3 Elasticity modulus E is a measurement of the soils stiffness and a crucial input when performing calculations to

predict ground deformations. However due to its complexity, E is a difficult parameter to

determine. Both external and internal factors, such as, water content, stress history,

cementation, particle organization along with loading factors all influence the elasticity

modulus (Briaud, 2001).

2.3.1 Internal factors If the particles in the soil are close to each other, the modulus tends to be higher than if the

particles are more spaced. How close the particles are to one and another can be obtained by

measuring the dry density of the soil. The higher the dry density, the closer the particles are to

each other. How the soil particles are organized is another important factor that affects the

modulus. Depending on the internal structure, two samples with the same dry density may

have different elasticity moduli.

The water content in soil is one of the most important factors affecting the modulus. At low

water content the water binds the soil particles and increases the effective stress between

them, leading to higher moduli. However, if the water content is too low the modulus will be

lower.

The previous loading history of the soil also influences the moduli. If the soil previously has

been exposed to stresses it is called overconsolidated. Overconsolidated soils often have a

higher elasticity modulus than soils that have not been exposed to previous stresses, normally

consolidated soils (Briaud, 2001).

2.3.2 External factors Stresses that are induced by the loading process of soil can be normal stresses, shear stresses

or a combination of them. At any arbitrary spatial point, there will be a set of three principal

normal stresses in the soil. The mean value of these stresses will influence the modulus of the

soil. The phenomena were the mean stresses influences the modulus is known as the

confinement effect. The higher the confinement is, the higher the modulus of the soil will be,

Figure 2.2a.

Stresses are induced when loading a soil, due to the non-linearity of soils, the secant modulus

will depend on the strain level. The secant modulus will generally decrease as the strain

increases, this due to the shape of the stress-strain curve, Figure 2.2b.

The rate at which the soil is being loaded also affects the modulus. Soil is a viscous material,

which means that the faster the soil is loaded the higher the modulus will become. The

exponent b in Figure 2.2c is dependent on the soil type. In clays the exponent b often varies

between 0.02 for stiff clays to 0.1 for very soft clays (Briaud, 2001).

The number of times the soil is being loaded also influences the modulus. The more loading

cycles the soil experiences, the lower the modulus will become, Figure 2.2d. The exponent c

varies, but a value of -0.1 to -0.3 is commonly used (Briaud, 2001).

8

Figure 2.2 Loading factors affecting the modulus: a) Mean stress level. b) Strain level in the soil. c) Strain rate. d)

Number of cycles experienced by the soil (Briaud, 2001).

2.4 Small-strain stiffness Small-strain stiffness refers to how soil behaves at small strains ( ). At small strains

the soil behaviour is considered to be truly elastic (Atkinson, 2000; Benz, 2007). The range of

strain at which the soil behaves truly elastic is dependent on the material composition and the

stress-strain history of the soil (Wood, 2016). The small-strain stiffness and its degradation

with increasing strains is often described with (Seed & Idriss, 1969). The modulus

describes both the drained and undrained conditions of the soil. However, the degradation of

the stiffness with increasing strains can also be described with (Thiers & Seed, 1968).

Another common way to present the stiffness degradation is in terms of , where is

the initial stiffness at small strain, Figure 2.3 (Benz, 2007).

Several studies have been made regarding the stiffness at small strains and its degradation at

increasing strain, e.g. Benz (2007); Burland, (1989); Jardine, (1986). The studies have shown

that the parameter is affected by several factors, Table 2.1.

9

Figure 2.3 Stiffness-strain behaviour of soil (Benz, 2007).

Table 2.1 Factors affecting the stiffness at small strains (Benz, 2007; Burland, 1989)

Parameter Importance to for Cohesive

soils

Strain amplitude Very Important

Confining stress Very Important

Void ratio Very Important

Plasticity Index Very Important

Overconsolidation ratio Very Important

Diagenesis Less Important

Strain History Relatively Unimportant

Strain rate Relatively Unimportant

Effective material strength Less Important

Grain Characteristics (size, shape) Less Important

Degree of saturation Very Important

Dilatancy Relatively Unimportant

2.4.1 The influence of diagenesis Diagenesis is a process involving seawater, subsurface brines or meteoric water that alters the

sediments up to the point of metamorphism. The diagenesis process alters the interparticle

structure and therefor also alters the stiffness of the soil with time (Benz, 2007).

The diagenesis process that have a large influence are cementation and aging, which

according to (Terzaghi K, 1996;Mitchell & Soga, 2005) are defined as change in various

mechanical properties resulting from a secondary compression under an external load.

10

2.4.2 The influence of confining stress (Hardin & Richart, 1963) proposed a relationship between and the effective confining

stress :

( ) (13) Where is a factor accounting for the type of soil.

For cohesive soils the exponent was previously set to 0.5. However, the value is very

sensitive and dependent on the liquid limit and the plasticity index . Figure 2.4 shows a

compilation of the exponent as a function of and for different clays at very small

strains (Benz, 2007).

Figure 2.5 shows how the stiffness decays with decreasing .

Figure 2.4 exponent m as a function of plasticity index and liquid limit (Benz, 2007)

Figure 2.5 Influence of the confining stress on the decay of small -strain stiffness (Benz, 2007)

11

2.4.3 The influence of void ratio (Hardin & Richart, 1963) proposed another relationship between the propagation velocity

and void ratio for Ottawa sand:

( ) (14)

Where and are material constants.

Based on equation (14) (Hardin & Richart, 1963) derived a formula for how is dependent

on the :

( )

(15)

Equation (15) has proven to correspond reasonably well for clays with low surface activity.

For higher surface activity the coefficient 2.97 is replaced by a higher one (Benz, 2007).

Other relationships between and that are frequently used is, (Benz, 2007; Burland,

1989):

(16)

Where for sand and clay and in the range of for various clays.

Figure 2.6 shows how the stiffness decays with the soil relative density for tests performed at

a confining stress of 80 kPa. The soil relative density is expressed by the density index:

Figure 2.6 Influence of soil relative density on the decay on small-strain stiffness (Benz, 2007)

12

2.5 Measuring small-strain stiffness

2.5.1 In-Situ tests In-situ tests are not measuring the small-strain stiffness directly but rather the elastic soil

mechanical propagation properties, e.g. (Mayne et al., 1999). Instead other parameters are

measured and related to the stiffness by mathematical relationships (Jardine et al., 1986). One

of the most common, indirect, methods of measuring the small-strain stiffness in the elastic

domain is by using wave propagation velocities.

The wave propagation velocity is dependent on the stiffness and density of the material that

the waves are propagating through. The higher the stiffness of a material is, the higher the

velocity will become (Kramer, 1996).

√

(17)

√

(18)

Where and are the wave propagation velocities for Primary (compression) and

Secondary (shear) waves, Figure 2.7.

( )

(19)

( )( )

(20)

By combining equation (17),(18),(19) and (20) the following relationship can be derived:

(21)

(

)

(

) (22)

13

Figure 2.7 Primary and secondary waves (Kramer, 1996)

2.5.1.1 Cross hole seismic To perform a cross hole seismic survey at least two vertically drilled bore holes are needed. In

one of the boreholes an energy source is lowered to the desired depth. In the other holes

receivers are placed at the same depth, Figure 2.8. The distance between the energy source

and the receivers must be known exactly, which often demands inclinometer reading in each

hole. By knowing the distance and by measuring the time it takes for the signal to reach the

receive it is possible to calculate the wave propagation velocities. By using the measured

velocity, can be obtained by Eq.(21) (Kramer, 1996).

The cross hole seismic survey is the most expensive testing method for in-situ small strain

stiffness, however it is also the most reliable (Benz, 2007).

Figure 2.8 Cross hole seismic survey (Clayton, 2011)

14

2.5.1.2 Continuous surface wave A vibrator sends out single-frequency sinusoidal force at the ground surface. The waves travel

through the ground and are measured by geophones at a certain distance from the vibrator,

Figure 2.9. By using different frequencies, a profile of phase velocity against wavelength is

obtained. From the phase velocity – wavelength profile it is possible to calculate a stiffness-

depth profile (Clayton, 2011).

2.5.2 Laboratory tests Several laboratory testing methods have been developed to determine the static and dynamic

small-strain stiffness. Laboratory testing tends to give a lower small strain-stiffness value than

field tests (Jardine et al., 1986). The two main reasons for this are explained by sample

disturbances and errors related to interpretation, such as assumptions or idealisations. For the

laboratory results to be closer to the in-situ values, great care must be taken to both of the

possible error sources (Wood, 2016).

2.5.2.1 Bender elements The bender element method was first introduced at the end of the 1970s (Viggiani &

Atkinson, 1995). The bender element method has become more popular over the past decade,

mostly due to its perceived simplicity. The bender element consists of two thin piezo-ceramic

plates that are bonded together, with the soil sample in-between them, Figure 2.10.

By applying a voltage at one of the plates, it will contract or extend, generating seismic waves

in the soil. The element on the opposite side will register the incoming waves. By knowing

the distance between the elements and the time obtained for a wave to travel through the soil,

between the plates, the velocity of the waves can be calculated (Clayton, 2011).With the

velocities known, can be obtained by using Eq.(21).

Figure 2.9 Continuous surface wave survey (Clayton, 2011)

15

Figure 2.10 Sketch of a bender element (Clayton, 2011)

2.5.2.2 Resonant column testing The resonant column testing has been used for more than 40 years. It is a method for

determining and at very small strains. The method also estimates the rate of stiffness

degradation with increasing strain (Clayton, 2011). The method works by vibrating the soil

specimen with a certain frequency. The frequency is increased until the first-mode of

vibration for the specimen is reached. At this frequency, the resonance frequency and

amplitude is measured. By knowing the geometry of the specimen, the measured data is used

to calculate the wave propagation velocity. The velocity is then used to calculate , Eq.(21).

Figure 2.11 shows a schematic drawing of the resonant column testing apparatus.

Figure 2.11 Schematic drawing of a resonant column apparatus (Clayton, 2011)

16

2.6 Ground investigation methods Ground investigations are crucial to obtain knowledge about the soil and groundwater

conditions but also to determine the soils properties, (Lambe & Whitman, 1979) Ground

investigations can be divided into two sections, in-situ tests or laboratory tests.

The advantage with laboratory testing is that there is a high degree of control over the

conditions compared to the field tests (Wood, 2016). The disadvantage with laboratory tests is

that the soil sample is being taken out of its original conditions, which can lead to different

results. Great care must be taken when soil samples are being extracted from the field.

Experience has shown that many soil types, including clay, are very sensitive to sampling

disturbance. If the sampling disturbance is significant it may lead to the laboratory test being

almost worthless (Davis & Selvadurai, 1996).

In this section, the focus will be on the laboratory test method triaxial testing.

2.6.1 Oedometer tests The oedometer test is a common method of determining the soils deformation properties

(Larsson, 2008). The oedometer test can be performed in two variants, incremental load steps

or constant rate of strain (CRS). In Sweden, the CRS test is the more common of the two

methods. The CRS method determines several parameters such as, the pre-consolidation

pressure , coefficient of consolidation , permeability and the modulus .

2.6.2 Triaxial tests

The triaxial test is one of the most common and versatile performed geotechnical laboratory

tests for determining the shear strength and stiffness of soil (SGF, 2012). In the triaxial test a

soil specimen gets loaded in its axial and horizontal direction (Lambe & Whitman, 1979). The

deformations and strains due to the loading are measured and evaluated to determine the

parameters of the soil. The primary parameters that are determined from the test are: Angle of

shearing resistance ϕ΄, cohesion c and the undrained shear strength , depending if the

shearing is carried out in a drained in an undrained way. Other parameters such as the

compression index , and can also be determined (GDS, 2013). The advantage of the

triaxial test compared to other methods is the ability to simulate the original stresses and pore

water pressures in the soil (SGF, 2012).

Figure 2.12 shows an illustration of a triaxial cell.

17

Figure 2.12 Illustration of a triaxial cell (GDS, 2013)

2.6.2.1 Standard triaxial tests

A triaxial test usually consists of four stages: specimen and system preparation, saturation,

consolidation and shearing (Jardine, Symes, & Burland, 1984). In the first stage the soil

sample taken from the field is prepared and put into the triaxial cell. It is important to keep the

disturbance of the sample to a minimum during the preparation since sample disturbances can

affect the results. The purpose of the saturation stage is that all voids within the test sample

are filled with water. Before the specimen gets sheared, the specimen is brought up to the

desired effective stress. In the fourth and last stage of the test, the specimen is sheared by

applying an axial strain at a constant rate. The rate at which the specimen is being sheared is

dependent on which triaxial test that is conducted.

Figure 2.13 Stress state during triaxial compression (GDS, 2013)

18

There are three types of standard triaxial tests that can be performed (Lambe & Whitman, 1979):

Unconsolidated Undrained test (UU)

Consolidated Undrained test (CU)

Consolidated Drained test (CD)

The Unconsolidated undrained test is the fastest and simplest test procedure used for

evaluating the short-term soil stability. During loading of the specimen, the total stresses are

measured which allows the undrained shear strength to be determined.

The consolidated drained test describes the long-term response of the soil and is used for

determining the angle of shearing resistance and the cohesion. It is a more time-consuming

process for testing cohesive soils than the unconsolidated undrained test. The reason for this is

that the rate of strain must be slower to allow for small pore water pressure changes.

The consolidated undrained test is the most common triaxial test (Larsson, 2008). It

determines the same parameters as the consolidated drained test, but at a shorter time. During

the consolidated drained test the change in the excess pore pressure can be measured within

the specimen as shearing takes place, leading to the ability of using a higher rate of strain

(GDS, 2013).

The standard triaxial tests can be performed as active or passive tests. During active triaxial

tests the specimen is being loaded by a higher axial than horizontal load. During passive tests,

the horizontal load applied is higher than the axial. The purpose of the active and passive tests

is to simulate real stress behaviours that would occur in the field.

Figure 2.13 illustrates the stress state during triaxial compression.

2.6.2.2 Triaxial test presentation Performed triaxial tests are often presented graphically through several plots. The most

common ones are:

Stress against axial strain (

)

Change in pore pressure against axial strain ( )

Stress path with effective stress ( )

Figure 2.14 illustrates results from an active undrained triaxial test performed on clay.

19

Figure 2.14 Presentation of an active undrained triaxial test: a)

b) c) (SGF, 2012)

2.6.2.3 Evaluation of the elasticity modulus in triaxial tests The elasticity modulus of a material can be determined by the slope of a line in the stress-

strain curve, Figure 2.15. However, the elasticity modulus will vary depending on how the

line is defined. The line can be drawn as a tangent or as a secant to the strain-stiffness curve.

The line can be defined in the beginning of the stress-strain curve, when the strains are small

or at the end of the curve, when the stresses have decreased. There is no rule of thumb for

deciding where this line should be placed in order to get a realistic value of the elasticity

modulus (Briaud, 2001).

The elasticity modulus can be decided through several equations, following the chapter 2.2.2:

[ ( )]

( ( ))

[ ( )]

( ( ))

[ ( )]

( ( ))

(23)

(24)

( )

(25)

20

(26)

Depending on where the line is drawn, several elasticity moduli can be determined (Schanz,

Vermeer, & Bonnier, 1999). , Unloading modulus , secant modulus at 50% of the shear

strength reloading modulus and cyclic modulus are examples of moduli that can be

determined, Figure 2.16.

Figure 2.15 Determination of the elasticity modulus (Briaud, 2001)

Figure 2.16 Definition of different slopes used to evaluate the corresponding elasticity modulus (Briaud, 2001)

21

2.7 Empirical correlations for determining the soil stiffness In the absence of laboratory tests for determining the elasticity modulus empirical correlations

can be used (Duncan & Bursey, 2013). The correlations are derived from parameters that can

be obtained from in-situ, CRS or routine laboratory tests, , or . The results from the

correlations are often more useful and effective than direct measurements of the elasticity

modulus (Duncan & Bursey, 2013).

The correlation seen in Eq. (28) below is dependent on the plasticity index, , Eq. (27). In

Sweden, the plastic limit is not determined in routine laboratory tests, however Table 2.2

shows correlations between the and (IEG, 2011)

(27)

Table 2.2 Correlation between and for clay

Explanation Liquid limit [%] Plasticity index

Low plasticity clay 15-30

Medium plasticity clay 30-50 10-25

High plasticity clay 50-80 25-50

Very high plasticity clay

(28)

(29)

Eq. (29) is based on an assumption often used in practical design (Trafikverket, n.d.)

Where is calculated according to (Trafikverket, 2014)

(

)

(30)

The small-strain stiffness is measured, as mentioned earlier, through dynamical tests

performed in a laboratory or in the field, (Atkinson, 2000; Burland, 1989). However, the

small-strain stiffness is not too often used in routine design and therefore far from always

measured. In Sweden, it is common to relate to the undrained shear strength through

several empirical correlations. Table 2.3 below is originally found in the thesis by Wood,

(2016).

In the same thesis by Wood, (2016), the correlations were compared with field measurements

of for two different geological deposits at Gothenburg and Uppsala, site 1 and site 10

22

respectively. The spheres and rings in Figure 2.17 and Figure 2.18 represent the degree of

sample disturbance in the laboratory test and the quality of the field test respectively. Large

spheres indicates large sample disturbances and large rings represents very good quality of the

field test (Wood, 2016).

Figure 2.17 and Figure 2.18 illustrates the accuracy of the correlations given in Table 2.3 for

determining .

Table 2.3 Empirical correlations for the small-strain shear modulus (Wood, 2016).

Source Soils Correlation

Andréasson (1979)

(uncorrected shear vane) High plasticity post glacial soft

clays (Gothenburg Area)

Stokoe (1980) (average Su from CAUC, CAUE & DSS) Clays (

)

Larsson & Mulabdic (1991)

(corrected shear vane and at some sites SuDSS)

High- low plasticity soft clays

(Western and central Sweden

and Norway) (

)

Larsson & Mulabdic (1991)

(corrected shear vane and at

some sites SuDSS)

Low plastic and varved or

otherwise inhomogeneous soils

& organic clays (

)

Bråten et al. (1991)

(SuDSS)

Medium and low plasticity soft

clays (Norway) (

)

Long et al. (2013)

(Su from CAUC tests)

Medium plasticity firm clays

(Ireland)

23

Figure 2.17 Comparison between the empirical correlation for and field measurements from site 1, Gothenburg.

(Wood, 2016).

Figure 2.18 Comparison between the empirical correlation for and field measurements from site 10, Uppsala

(Wood, 2016).

24

3 Soil modeling

3.1 Introduction to numerical modelling During the past 40 years, the fast development of computers and numerical modelling

software has made it possible for sophisticated geotechnical problems to be analysed by most

engineering practices. The numerical software offers a variety of constitutive models that the

engineers can choose from. The constitutive models are often based on different theories and

require different input parameters (Brinkgreve et al., 2015).

Plaxis is an advanced numerical simulation program that was created during the 1970’s at the

University of Delft, in the Netherlands (Brinkgreve et al., 2015). It was originally used for

elastic-plastic calculations for axi-symmetrical problems based on higher-order elements.

Today Plaxis offers a wide range of advanced soil models and simulations in both 2D and 3D

(Brinkgreve et al., 2015).

In this thesis, the constitutive models Mohr Coulomb and Hardening Soil Small have been

studied in both 2D and 3D models of the case study presented later. The Mohr-Coulomb

model is generally used for drained analysis in geotechnical engineering, and the hardening

soil model is similar to other types of small strain model used in numerical analysis, e.g. in

Jardine et al., (1986)

3.2 Mohr coulomb (MC) The Mohr Coulomb model is one of the most generally used plasticity constitutive models,

(Lambe & Whitman, 1979). It assumes the soil to be elastic perfectly-plastic, Figure 3.1, and

normally non-associated plastic strain is assumed (Vermeer & de Borst, 1984). The Mohr

Coulomb model is a straight-forward constitutive model with few parameters method that can

be used as a first analysis of the problem. The model only includes a small number of features

that the soil behaviour shows in reality. The model does not take stress, stress-path nor strain

dependency of stiffness or anisotropic stiffness into account (Brinkgreve et al., 2015).

The first part of the Mohr Coulomb constitutive model, the linear elastic part, is based on

Hooke’s law of isotropic stiffness, Eq.(7). The second part, the perfectly-plastic part, is based

on the Mohr Coulomb failure criterion. The Mohr Coulomb Failure Criterion can be

expressed as Eq.(31) and illustrated in Figure 3.2.

(31)

Where is the shear stress and is the normal effective stress on the failure plane.

25

Figure 3.1 Illustration of the elastic perfectly-plastic model (Brinkgreve et al., 2015)

Figure 3.2 Mohr Coulomb failure criterion for undrained case (Brinkgreve et al., 2015)

Since the plastic state is an important part of the Mohr Coulomb, it is important to know when

plastic yield occurs. For plastic yield to occur, development of irreversible strains needs to

take place. To evaluate if irreversible strains have taken place a yield function is introduced

as a function of the stress and strain. Eq. (32) shows the yield function as formulated in

terms of principal stresses (Brinkgreve et al., 2015).

The Mohr Coulomb yield criterion is represented by a hexagonal cone in the principal stress

space, Figure 3.3.

(

) ( ) ( ) (32)

The Mohr Coulomb model requires five input parameters that can be obtained from basic soil

testing, Table 3.1.

26

Figure 3.3 Mohr Coulomb yield surface in the principal stress space (c=0) (Brinkgreve et al., 2015)

Table 3.1 Basic Input parameters for the Mohr Coulomb soil model

Parameter Description

Internal friction angle

Cohesion

Dilatancy

Young’s modulus

Poisson’s ratio

3.3 Hardening Soil (HS) The hardening soil model is more advanced than the Mohr Coulomb model for simulating the

behaviour of soil, (Schanz et al., 1999), and is primarily based on the double hardening

model presented by Vermeer in 1978, (Vermeer, 1978). The total strains are calculated using

a stress-dependent stiffness. The stress-dependent stiffness is different for both loading and

unloading or reloading (Surarak et al., 2012).

In the HS model, the stress-strain relationship due to primary loading is assumed to be a

hyperbolic curve, Figure 3.4. The hyperbolic function for the undrained triaxial test was stated

by (Kondner, 1963) and can be formulated as:

(33)

Where is the deviatoric stress and is the axial strain.

is the ultimate deviatoric stress at failure and derived from the Mohr Coulomb failure

criterion involving the strength parameters and and is defined as:

( ) (34)

27

Figure 3.4 Hyperbolic Stress-strain relationship in primary loading for a standard drained triaxial test (Brinkgreve et

al., 2015)

And is defined as:

(35)

Where is the failure ratio, which in PLAXIS has the standard value of 0.9.

in Figure 3.4, is the initial stiffness, and is due to the highly non-linearity of soil the is

used instead of for calculations. is the confining stress dependent stiffness modulus, at

50 % of the shear strength, for primary loading and given by Eq.(36).

(

)

(36)

Where

is the reference stiffness modulus corresponding to the reference stress, which in

PLAXIS is set to as a default value.

Equation (36) shows that the actual stiffness for the HS model depends on the minor principle

stress , which is the effective confining pressure in a triaxial test (Brinkgreve et al.,

2015). The amount of stress dependency is given by the power-law coefficient (Surarak,

2010). For clay soils the value of is in the range of 0.7-1.0 (Benz, 2007).

The stress dependent stiffness modulus for unloading and reloading stress paths can be

calculated by Eq.(37).

(

)

(37)

is the reference modulus for unloading and reloading corresponding to the reference

pressure . In PLAXIS is set equal to

due to practical

reasons.

28

In the HS model the shear yielding function is defined as:

(38)

Where is given by Eq. (39) and is given by Eq. (40).

{

( )}

( )

(39)

(40)

By looking at Eq. (38)- (40) it can be seen that the parameters,

, obtained

from triaxial test controls the shear hardening yield surface.

Another important stiffness parameter to control the magnitude of the plastic strains is the

reference oedometer modulus

. Similar to the Unloading modulus and secant

modulus , the oedometer modulus can be calculated by Eq. (41).

(

)

(41)

The HS model needs the following input parameters, Table 3.2.

Table 3.2 Input parameters for the Hardening soil model

Parameter Description

Internal friction angle

Cohesion

Dilatancy

Secant stiffness in standard drained triaxial tests

Unloading/reloading stiffness

Tangent stiffness for primary oedometer loading

Exponent for stress-level dependency

Poisson’s ratio for unloading-reloading

Reference stress for stiffness

value for normal consolidation

Failure ratio

29

3.4 Hardening Soil Small (HSS) The Hardening Soil Small Model is an addition to the Hardening Soil model, with the

capacity of modelling the soil behaviour at very small strains, and is similar to other types of

soil models presented in, e.g. Jardine et al, (1986). The HSS model uses the same input

parameters as the HS model, Table 3.2, except for two additional parameters (Benz, 2007):

which is the initial or very small-strain shear modulus

which is the shear strain level at which the secant shear modulus is reduced to

about 70% of

is, similarly to , dependent on the confining pressure, Eq. (42)

(Brinkgreve et al., 2015).

(

)

(42)

The parameter defines the level of shear strain where has been reduced to 70% of its

initial value, Figure 3.5. One of the most used models in soil dynamics is the Hardin-Drnevich

relationship, the hyperbolic law for larger strains, Eq. (43).

|

| (43)

Where is the threshold shear strain and is quantified as:

(44)

Where is the shear stress at failure.

Eq.(43) relates to large strain behaviour of soil and has been modified in a study by

(Santos & Gomes Correia, 2001) in order to fit small strain behaviour of soil, Eq.(45)

|

| (45)

Where is a factor set to 0.385.

can be approximated with Eq.(46).

[ ( ( )) ( ) ( )

(46)

Where is negative.

30

Figure 3.5 Illustration of the parameter (Brinkgreve et al., 2015)

31

4 Case study: Bridge over

Ulvsundavägen

4.1 Introduction The project studied in this thesis is a new construction of an 85 meters long and 16 meters

wide bridge for public transport, cyclist and pedestrians located between Bromma airport and

Ulvsundavägen in Stockholm, Sweden, Figure 4.1.

During the casting of the bridge decks temporary formworks were used as supports, Figure

5.1. The formwork transfers the entire load to the ground before the bridge deck has hardened

and the loads can be taken up by the supports of the bridge. During this time, the temporary

formwork is causing the underlying soil to deform.

The whole bridge rests on ten supports, creating nine spans in between them. A study was

carried out during the design phase of the bridge. The purpose of the study was to investigate

if the loads from the formworks could be transferred directly to the ground or if piling was

necessary to keep the deformations within desired limits. The purpose was also to calculate

the deformations caused by the temporary formworks, so that the height of the formworks

could be adjusted accordingly.

Several ground investigations have been made in the area. The investigations consisted of

both field test and laboratory tests. The purpose of the investigations was to determine the

ground conditions and soil parameters in order to calculate the predicted ground deformations

under the temporary formwork.

Piston sampling, Soil-rock probing, weight sounding and groundwater measurements were

carried out during the site investigation. Clay samples that were extracted from the field by

the piston sampler were sent to a laboratory for routine, triaxial and CRS-tests to determine

the soil properties.

The results from field and laboratory test for a studied section will be presented in Appendix

A and used for analytical and numerical calculations in chapter 5.

Material parameters that could not be determined by field or laboratory tests were obtained

from TK Geo 13 (Trafikverket, 2014).

32

Figure 4.1 Satellite picture of the location of the bridge. The highlighted area shows the studied section in this thesis.

Picture taken from (www.eniro.se 2014-04-20)

33

4.2 Studied section The ground deformations in one of the nine spans were chosen to be investigated in this

thesis, namely the deformations between supports 6-7.

The span between supports 6-7 was chosen due to two main reasons:

The ground deformations were measured prior to and after the casting of the concrete

bridge deck at two locations close to support seven.

Triaxial- as well as CRS-tests have been performed on clay samples taken from

borehole 15E03, Figure 4.2.

Figure 4.2 Studied span and location of borehole 15E03

34

4.2.1 Ground conditions The soil profile between supports 6-7 was determined through weight soundings, soil-rock

probing, CRS-, triaxial- and routine laboratory tests. Table 4.1 show the results from the

routine laboratory and CRS-tests. Figure 4.3 shows the result from a sounding in borehole

15E03.

The ground water level was measured at two locations, support 4 and support 10, Table 4.2.

Based on the measurements the ground water level was assumed to be 0.9 meters below the

ground surface at point 15E03.

The soil between supports 6-7 initially consisted of an approximately 1 m thick layer of fill

material, mostly silt, sand and gravel. Underneath the fill layer there is a 0.6 m thick layer of

stiffer clay. The stiffer clay is resting on softer clay with the thickness of approximately 10

meters. The softer clay rests on an up to two-meter-thick layer of moraine resting on solid bed

rock. During the construction of the bridge the initial 1-2 meters were excavated and replaced

by crushed rock for increased bearing capacity. In the studied section an assumption was

made that the upper two meters were excavated and replaced by crushed rock.

Figure 4.4 shows the interpreted soil profile that have been used for the analytical and

numerical calculations in chapter 5.

The depths in Figure 4.4 are based on the distance from the ground surface +0.0

Table 4.1 Results from the routine laboratory and CRS -tests taken from borehole 15E03

Depth [m]

[

] [ [ [ [ [ [ [ M´

k [m/s]

1.5 1.67 69 67 17 19 15.56 54 531 88 12.4 3.7E-10

2.5 1.58 80 63 22 12 10.11 40 266 55 12.2 5.9E-10

3.5 1.54 93 72 26 12 9.52 44 234 57 13.2 6.9E-10

5.5 1.64 65 61 10 12 10.25

Table 4.2 Conducted ground water measurements in the area

Measuring

point

Measuring period Ground water level Depth below ground surface (m)

09R187GV

(+4.7)

26/7-2009 – 30/1-2015

+2.9 - +1.9 1.8-2.8

13W018G

+(4.0)

8/28-2013 – 30/1-

2015

+3.3 - +2.9 0.7-1.1

35

Figure 4.3 Result from sounding in borehole 15E03

Figure 4.4 Interpreted original soil profile(left) and soil profile after replacing the upper two meters during

construction(right)

36

4.3 Ground deformation measurements Ground deformation measurements were conducted at a total of 12 points along the bridge.

The measurements were done directly and 21 days after the casting of the concrete bridge

deck.

Figure 4.5 shows where deformations have been measured between support 6-7.

The measured deformations for the studied section are presented in Table 4.2 and are used as

a reference to the calculated deformations in chapter 5. The accuracy of the measured

deformations is around .

Figure 4.5 Conducted ground deformation measurements between support 6-7

Table 4.3 Measured ground deformations between supports 6-7

Point Load group Directly after

casting [mm]

21 days after

casting [mm]

Total

deformation

[mm]

11 1 4 9 13 ( ) 12 2 5 9 14 ( )

37

5 Calculation procedure

5.1 Formwork geometry The formwork supporting the bridge deck during construction consists of 153 steel rods

placed on double rectangular wooden plates of sizes 0.1 m*0.4*0.4 m, Figure 5.1. The

spacing between the steel rods and the rod forces can be seen in Figure 5.2.

Figure 5.1 Temporary formwork for casting the bridge deck

Figure 5.2 Geometry and load distribution in the steel rods

1 2

38

5.2 Empirical correlation study Empirical correlations are very important in geotechnical engineering. One of the most

important correlation used in the design phase is the correlation for the elasticity modulus.

Many correlations exist for determining the elasticity modulus, as shown in section 2.7. The

correlations were compared to the initial elasticity modulus and the secant modulus at 50%

of the peak strength ,evaluated from triaxal test from several sites around Sweden, Figure

5.3, to study the accuracy of the correlations.

Figure 5.3 Approximate location of the performed triaxial and laboratory tests used in the correlation study.

39

5.3 Analytical Calculation A simplified analytical calculation was made for the span between supports 6-7 as a reference

to the measured deformations and the calculated deformations using more advanced

numerical soil models. Deformations due to consolidation in the clay have been assumed to be

occurring during the first seven days. The assumption is based on that the concrete bridge

deck would have cured enough for the loads to be transferred to the ground through the

permanent supports rather than through the temporary formwork.

The analytical calculation was based on Eq. (47)-(49).

∫

(47)

∫ (

)

(48)

∫

(

) (49)

Where:

= vertical stress increase caused by the form work.

was calculated using the 2-1 method, which assumed the loads to decrease with increased

depth, due to load spread, Eq. (50). The load spread was assumed only to be occurring in the

fill material.

( )

(50)

Where:

b = Width of a fictive plate, z =Depth below the ground surface, Q = Distributed load from

the form work and h= Thickness of the layer

The distributed loads were calculated by summing the forces in the steel rods and dividing

them by an area, which is illustrated by the two larger blue rectangles 1 and 2 in Figure 5.2.

For simplicity an average distributed load of 29 kN/m2 was used for both areas.

40

Equations (47)-(49) calculates the deformations after an infinite period of time. To be able to

calculate the deformations after seven days, as assumed previously, a time factor was

implemented, Eq. (51).

(51)

Where:

= Time factor connected to the degree of consolidation for the clay

= Coefficient of consolidation (evaluated from the performed CRS-test, Appendix A -

Figure 1)

k = Permeability

= Drainage distance in the clay

= Time in second

= Unit weight of water

The modulus used in Eq. (47)-(49) has for simplicity been set to , as previously seen in

Eq.(29).

The deformation in the clay after seven days were calculated as ∑

Where: √

The clay was divided into a single layer and the deformation properties obtained from CRS-

test at 3.5 m depth was used for the calculation of the deformation, Table 5.1.

Table 5.2 shows the evaluated stress state in the soil.

Table 5.1 Results from the CRS -test performed at 3.5 m depth

Depth [m]

[

] [ [ [ [ [ [ M´ k [m/s] cv [m2/s]

3.5 1.54 93 72 9.52 44 234 57 13.2 6.9E-10

Table 5.2 Evaluated stress state in the soil

Depth

[m] [ [kPa] [kPa] [kPa] [kPa]

0 0 0 0 39 40

0.9 18 0 18 24.6 42.6

2.5 50.9 16 34.9 21.4 56.3

3.5 66.3 26 40.3 21.4 61.7

The deformations occurring in the fill material were calculated using theory of elasticity, Eq.

(52).

(52)

41

5.4 2D Numerical simulation The 2D numerical calculations have been performed using two constitutive models, the

simpler Mohr Coulomb- and the more refined Hardening soil small model both with the

Undrained-A condition in Plaxis.

Figure 5.4 shows the geometry used in the 2D numerical simulations.

Figure 5.4 2D geometry used in the MC and HSS simulations

42

5.4.1 Assumptions The loads had to be simplified due to the 2D geometry. The point loads from each steel rod

were transformed into line loads by dividing the forces with the distances between the rods.

The two load cases that were used in calculations, will be referred to as load group 1 and load

group 2, see the orange rectangular in Figure 5.2.

Where:

In the numerical simulations, the settlements have been assumed to be occurring during the

first seven days, see section 5.3.

5.4.2 Input parameters The input parameters for the MC and HSS-model can be chosen from empirical correlations,

field data or customized to fit laboratory test results by the help of the Plaxis “SoilTest”

program. The SoilTest program is based on a single point algorithm and allows the user to

simulate soil lab tests quickly. The input parameters can be manipulated manually in the

SoilTest program until the simulated test corresponds well to the actual performed test.

However, Plaxis also has a built-in function, “parameter optimisation”, that allows the user to

choose which and how much the parameters should get manipulated for the simulated test to

match the performed laboratory test.

Two ways of obtaining the input parameters for the Mohr Coulomb calculations have been

investigated:

Strictly empirical correlations

Parameters obtained from triaxial test combined with correlations for the other

parameters

Three ways of obtaining the input parameters for the Hardening soil small calculations have

been investigated:

Using the Plaxis Parameter Optimisation function - Allowing Plaxis to automatically

match the fictive stress-strain curve obtained from Plaxis “SoilTest” program to the

real curve obtained from the triaxial test.

Strict empirical correlations – All input parameters are based on empirical

correlations.

Customized input – Parameters obtained from the triaxial test were mixed with

parameters obtained from the Plaxis parameter optimisation function and from

empirical correlations.

43

The Plaxis parameter optimisation requires a laboratory test data to be used. The strict

empirical correlation method can be used when simpler ground investigation methods

have been performed, e.g. routine laboratory tests.

5.4.2.1 Stiffness moduli The elasticity moduli used in the Mohr Coulomb model are the same as previously stated in

section 5.3.1.

The Hardening soil small model requires three different elasticity moduli as input parameters,

.

was assumed to have no effect on the actual deformation since no unloading or reloading

occurred. Therefor the value of

was set to a standard value of

. The value for

varies depending on the method that was used to obtain the input parameters.

was set to for the customized input and automatically determined by Plaxis for the

parameter optimisation case. For the strict empirical method

was set to

. The value for

was automatically determined in Plaxis for the parameter optimisation

and the customized input methods. For the strict empirical correlation method, the value for

was set to

.

5.4.2.2

and are two important parameters that can be determined by correlations or by

performing triaxial test. Equations (53)-(55) were used for determining empirically

(Trafikverket, n.d.). Eq.(56) was used for evaluating from the performed triaxial test. Due

to the OCR of 1.37 the correlations for both normally consolidated and overconsolidated clay

was studied.

(53)

( ) (54)

(55)

(56)

44

5.4.2.3 and are two important parameters in the HSS model. As mentioned earlier, is not

frequently measured in Sweden.

In this thesis, has been determined by the parameter optimisation function in Plaxis and by

the correlation (

) .

Wood (2016) showed that the correlation is very conservative compared to the measured field

values, Figure 2.17 and Figure 2.18. To make the correlation less conservative and closer to a

realistic value a factor 3 was used with the correlation for , Eq.(57).

[(

) ]

(57)

The factor 3 was determined based on the results from field measurements of at a site in

Uppsala, Sweden, in the thesis by (Wood, 2016).

The correlation for is dependent on the plasticity index. For the determination of the

plasticity index was set to 30 %, based on the average liquid limit obtained from routine

laboratory tests and the correlation, Table 2.2, resulting in a value of .

The parameter was determined in three different ways:

Automatically by the Plaxis optimisation function

Approximated by Eq.(58) for the customized parameter method

Assumed value for the strict empirical method.

[ ( ( )) ( ) ( )

(58)

45

5.5 3D Numerical simulation The 3D numerical calculations have been performed using the same two constitutive models

as for the 2D case. The calculations done in the 3D are based on the same parameters as in the

2D case and will therefore not be discussed here. Just like the 2D model, the 3D model takes

consolidation settlements into account. The settlements have been assumed to be occurring

during the seven first days, see section 5.3.

The permeability of the clay, obtained from the CRS-tests, was assumed to be the same in all

directions. The assumption that the permeability is the same in all direction is probably not

correct, but there are no measurements in the horizontal direction. The geometry that was

used in the 3D numerical simulation was based on the soil profiles stated in section 4.2.1.

Figure 5.5 shows the initial geometry and loads used for the 3D simulation.

Figure 5.5 Initial geometry for the 3D numerical simulation

46

5.6 Parameter sensitivity analysis Due to the complexity of the Hardening Soil Small constitutive model and due to the

uncertainties of determining some of the input parameters, e.g. a simplified

sensitivity analysis was performed. The purpose of the sensitivity analysis was to investigate

how large effect each of the input parameters would have on the calculated deformations. The

result of the simplified sensitivity analysis would give an indication of which of the

parameters that needed to be determined with greater caution.

The customized parameters were used to calculate a deformation which constituted as a

reference deformation in the sensitivity analysis. Most of the input parameters were varied by

increasing or decreasing the reference value by 5,10 and 15% and thereafter calculating a new

deformation. However, some parameters could not be varied as much due to regulations in

Plaxis.

Table 5.3 shows which parameters that were studied.

Table 5.3 Parameter variation

Parameter

m

47

6 Results

6.1 Empirical correlation study The results from the comparison show that the correlations for determining the elasticity

modulus show large variations. The standard correlation used for high plasticity clays in

Sweden, gives a very low elasticity modulus compared to the evaluated, for

the area where the bridge was built, as seen in Figure 6.1-6.3. The standard correlation

is also underestimating the elasticity modulus from the Norrköping test.

However, the correlation seems to be reasonable for the Uppsala test, Figure 6.6.

In general, the less commonly used correlation in Sweden

seems to give the

best result compared to the triaxial test, Figure 6.1-6.3.

Triaxial tests have only been performed at one depth for the Stockholm, Ulvsunda area. The

moduli are considered constant at the increasing depth, as indicated by the black lines.

Figure 6.1 Comparison between correlations and triaxial results from borehole 15E01 taken from the construction

site.

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0 2000 4000 6000 8000 10000 12000

De

pth

be

low

gro

un

d s

urf

ace

[m

]

Elasticity modulus [kPa]

Stockholm, Ulvsunda 15E01

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Correlation 45% IP

Correlation 50% IP

EI

E50

E50 EI

48

Figure 6.2 Comparison between correlations and triaxial results from borehole 15E09 taken from the construction

site.

Figure 6.3 Comparison between correlations and triaxial results from borehole 15E03 taken from the construction

site.

2

2.5

3

3.5

4

4.5

5

5.5

0 5000 10000 15000

De

pth

be

low

gro

un

d s

urf

ace

[m

]

Elasticity modulus [kPa]

Stockholm, Ulvsunda 15E09

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Correlation 45% IP

Correlation 50% IP

EI

E50

E50 EI

1

1.5

2

2.5

3

3.5

4

0 5000 10000 15000

De

pth

be

low

gro

un

d s

urf

ace

[m

]

Elasticity Modulus [kPa] Stockholm, Ulvsunda 15E03

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Correlation 45% IP

Correlation 50% IP

EI

E50

E50 EI

49

Figure 6.4 Comparison between correlations and triaxial results from the Norrköping Area

Figure 6.5 Comparison between correlations and triaxial results from the Norrköping Area

4

6

8

10

12

14

16

0 5000 10000 15000 20000 25000

De

pth

be

low

gro

un

d s

urf

ace

[m

]

Elasticity Modulus [kPa]

Norrköping, Location 1

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Correlation 45% IP

Correlation 50% IP

EI

E50

4

6

8

10

12

14

16

0 5000 10000 15000 20000 25000

De

pth

be

low

th

e g

rou

nd

su

rfac

e [

m]

Elasticity modulus [kPa]

Norrköping, Location 2

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Correlation 45% IP

Correlation 50% IP

EI

E50

50

Figure 6.6 Comparison between correlations and triaxial results from the Uppsala Area

6.2 Calculated deformations The results from the calculations have been normalized, i.e.

. Where are the

calculated deformations and are the measured deformations. It is important to have in

mind that the measured deformations are very small (13-14 mm). Due to the small

deformations, even a small change of will lead to deformation changes of .

The change represents the resolution of the measurements.

6.2.1 Analytical calculations The analytical calculations give good result compared to the measured deformation. The

difference between them is approximately 5-6 mm, which is considered to be very good due

to the simplicity of the analytical calculation.

Table 6.1 Results from the analytical calculation

Point Total

deformation in

fill layer

Total

deformation

in clay

Degree of

consolidation

Deformation after

seven days

Normalized

result

11 1.6 mm 690 mm 2.56 % 19.23 mm 147 %

12 1.6 mm 690 mm 2.56 % 19.23 mm 137 %

0

10

20

30

40

50

60

0 5000 10000 15000 20000 25000

De

pth

be

low

gro

un

d s

urf

ace

[m

]

Elasticity Modulus [kPa]

Uppsala

250*cu

Correlation 30% IP

Correlation 35% IP

Correlation 40% IP

Corelation 45% IP

EI

E50

51

6.2.2 2D Mohr Coulomb The Mohr Coulomb calculations in 2D show that the results are strongly dependent on the

choice of soil modulus and the magnitude of the loads. For load group 1 the deformation

ranges between 140-500% where the initial elasticity modulus gives the most accurate results,

Figure 6.7. The correlation that is commonly used in Sweden for high plasticity clays,

overestimates the deformations by a factor of approximately 5.

As the load decreases for load group 2, the correlations show better accuracy. For load group

2 the deformation ranges between 109-340%, Figure 6.8. The standard correlation

overestimates the deformation by a factor of approximately 3.4 for load group 2.

Figure 6.9 and Figure 6.10 shows the deformations and strain levels in the soil using 2D MC

with the initial elasticity modulus obtained from triaxial test.

Figure 6.7 Influence of the elasticity modulus for load group 1

Figure 6.8 Influence of the elasticity modulus for load group 2

100% 143% 174%

267% 303%

338% 374%

408%

486%

0%

100%

200%

300%

400%

500%

600%

1

No

rma

lize

d D

efo

rma

tio

ns

[%]

2D MC Normalized deformations. Load group 1

Measured deformation EI E50

Correlation 30% IP Correlation 35% IP Correlation 40% IP

Correlation 45% IP Correlation 50% IP Correlation 250*Cu

100% 109% 133% 189% 213% 218%

264% 287% 335%

0%50%

100%150%200%250%300%350%400%

1

No

rma

lize

d d

efo

rma

tio

ns

[%]

2D MC Normalized deformations. Load group 2

Measured deformation EI E50

Correlation 30% IP Correlation 35% IP Correlation 40% IP

Correlation 45% IP Correlation 50% IP Correlation 250*Cu

52

Figure 6.9 Calculated deformations using 2D MC model, Load group 1, with the initial elasticity modulus obtained

from triaxial test

Figure 6.10 Strain level in the soil profile for load group 1 using 2D MC model with initial elasticity modulus obtained

from triaxial test

53

6.2.3 2D HSS Model The results from the 2D HSS simulation show that the deformation varies depending on the

method of determining the parameters, Figure 6.11 and Figure 6.12 The most accurate result

is obtained when the Plaxis Parameter optimisation was used for determining the input

parameters based on matching the fictive to the real triaxial test curve. The 2D HSS model

overestimates the deformations for all cases except for the Plaxis parameter optimisation in

load group 2. Figure 6.13 and Figure 6.14 shows the deformations and strain levels in the soil

using 2D HSS model with the customized parameter input method.

Figure 6.11 2D HSS Results for load group 1

Figure 6.12 2D HSS Results for load group 2

100%

143% 174%

125%

293%

220%

0%

50%

100%

150%

200%

250%

300%

350%

1

No

rma

lize

d d

efo

rma

tio

ns

[%]

2D HSS Model. Load group 1

Measured deformation 2D MC (EI)

2D HSS Customized Parameters 2D HSS Plaxis Parameter Optimisation

2D HSS Correlation NC-Clay 2D HSS Correlation OC-Clay

100% 109%

154%

98%

212%

157%

0%

50%

100%

150%

200%

250%

1

No

rma

lize

d d

efo

rma

tio

ns

[%]

2D HSS Results. Load group 2

Measured deformations 2D MC (EI)

2D HSS Customized Parameters 2D HSS Plaxis Parameter Optimisation

2D HSS Correlation NC-Clay 2D HSS Correlation OC-CLay

54

Figure 6.13 Calculated deformations for load group 1 using 2D HSS model with the customized parameters method

Figure 6.14 Strain level in the soil profile for load group 1 using 2D HSS model with the customized parameter

method

55

6.2.4 Comparison between 2D and 3D numerical calculations Both the 2D and 3D numerical calculations overestimate the deformations in most of the

simulations. However, the results from the 2D and 3D numerical calculations are relatively

similar to each other, with the 3D giving slightly lower results for most of the simulations for

both load group 1 and 2, Figure 6.15and Figure 6.16 respectively.

Figure 6.15 2D vs 3D Simulations for load group 1

Figure 6.16 2D vs 3D Simulations for load group 2

100%

143% 141% 174% 160%

125% 124%

293% 293%

220% 195%

0%

50%

100%

150%

200%

250%

300%

350%

1

No

rma

lize

d d

efo

rma

tio

ns

[%]

Comparison between 2D and 3D Calculations

Measured deformations 2D MC

3D MC 2D HSS Customized Parameters

3D HSS Customized Parameters 2D HSS Plaxis Parameter Optimisation

3D HSS Plaxis Parameter Optimisation 2D HSS Correlation NC Clay

3D HSS Correlation NC Clay 2D HSS Correlation OC Clay

3D HSS OC Clay

100% 109% 122%

154% 130%

98% 101%

212% 210%

157% 157%

0%

50%

100%

150%

200%

250%

1No

rma

lize

d D

efo

rma

tio

ns

[%]

2D vs 3D Load group 2

Measured deformation 2D MC 3D MC

2D HSS Customized parameters 3D HSS Customized parameters 2D HSS Parameter optimisation

3D HSS Parameter Optimisation 2D HSS Correlation NC-Clay 3D HSS Correlation NC-Clay

2D HSS Correlation OC-Clay 3D HSS Correlation OC-Clay

56

6.3 Sensitivity analysis of the HSS model The sensitivity analysis for the 2D HSS model shows that many of the studied parameters had

little effect on the calculated deformation. The stiffness moduli and the shear strength show

little influence on the deformations. A 15% decrease of the modulus gave rise to an

approximate deformation increase of 2%. The shear strength of the clay showed an even less

influence, the 15% change only influenced the deformations by approximately 0.5%.

The parameters showing the largest influence on the deformations were the lateral earth

pressure coefficient , the power law coefficient and the initial small strain stiffness

By decreasing by 15%, the deformations increased by approximately 26%. The parameter

also had a large influence on the deformations, by decreasing the reference value by 15%,

the deformations decreased by approximately 12%.

Figure 6.17 Results from the sensitivity analysis

80.00%

85.00%

90.00%

95.00%

100.00%

105.00%

110.00%

115.00%

120.00%

125.00%

130.00%

-20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00%

NO

RM

ALI

ZED

DEF

ORM

ATI

ON

S [%

]

PARAMETER VARIATION [%]

SENSI TI VI TY ANALYSIS RESULTS

E50ref Eoed m K0nc c´ref Psi γ0,7 G0 K0

57

7 Analysis and discussion

7.1 Empirical correlation for the elasticity modulus The comparison between performed triaxial tests and existing correlations give varying results

depending on the geographical location. Another problem with determining the stiffness

based on the correlations is that they do not take the strain level in the soil into account. The

stiffness, as mentioned earlier, is dependent on the strain level in the soil. The smaller the

strains, the higher the stiffness. To obtain realistic values of the elasticity modulus, triaxial

tests needs to be performed with minimal disturbances.

7.2 Plaxis parameter optimisation function The Plaxis parameter optimisation function shows good similarities with the actual performed

triaxial test, see Appendix B - Figure 1 and Appendix B - Figure 2. However, by slightly

changing the parameter range in the optimisation function a totally different result can be

obtained. This means that great care must be taken when using the parameter optimisation

function. The Plaxis parameter optimisation function uses the Plaxis program SoilTest, which

simulates real life soil tests. The SoilTest program is based on a single point algorithm. Soil

behaviour is very complex and it is highly unlikely that a simple mathematical algorithm can

reproduce the actual behaviour of soil.

7.3 Calculated deformations In geotechnical engineering many simplifications are often required due to the complexity of

soil. Soil is often simplified by assuming homogeneity and linear elasticity, while in reality

soil is not a homogenous material and also far from linear elastic. Due to the required

simplifications, a natural deviation between measured deformation and calculated is common.

Another important factor that needs to be discussed when it comes to deformations in clay is

time. Deformations increase over time, due to consolidation. In the three calculation models

the deformations were assumed to be occurring during the first seven day, before the concrete

had hardened. The concrete may very well harden after three days, which would result in

slightly lower deformations. The uncertainties in the time it takes for the concrete to harden

also gives rise to uncertainties in the calculated deformation.

7.3.1 Analytical vs measured deformations The analytical calculation, which was the simplest of the three studied models, was based on

well-known formulas and easily determined parameters obtained from the commonly used

CRS-test. Despite its simplicity the analytical calculation produces a really good result

compared to the measured deformation, with a difference of approximately 5 mm.

58

7.3.2 MC-calculations The Mohr Coulomb constitutive model is an idealization of the soil response and does not

take the stress nor the strain dependency into account. Due to the simplifications, the method

requires fewer input parameters. Most of the parameters that affect the deformations, such as

the modulus of elasticity and the shear strength of the soil can easily be measured in the field

or in a laboratory.

The results from the MC calculation show that the calculated deformations vary between

109 % to 500 % depending on the load group and choice of correlation for the elasticity

modulus. The best result is obtained when the elasticity modulus is evaluated from a triaxial

test and the worst result is when using the approximate correlation . The result

from the MC-model is heavily dependent on the choice of elasticity modulus. By evaluating

the triaxial test and choosing an elasticity modulus based on the predicted strain level in the

soil, the results match the measured deformations well.

7.3.3 HSS-Calculations The more advanced HSS-model takes the stress and strain dependency of the soil into

account. Due to its complexity, the model requires more input parameters to fully capture the

true behaviour of soil.

The calculated deformations vary depending on how the input parameters were determined.

The best result was obtained when the Plaxis parameter optimisation function was used to

match a fictive triaxial test to the actual performed test. By using optimised parameters, the

results were almost identical to the measured deformations. The optimisation function is

however believed to be wrongly determining some parameters, such as the stiffness

degradation parameter . The parameter obtained from the optimisation function is

approximately a factor 100 times higher than the value obtained from the correlation. The

larger value means that the soil is a lot stiffer even at higher strains, which in turn gives rise to

smaller deformations, which is observed in the results.

The customized parameter method which was a mix between correlations, Plaxis parameter

optimisation and data based from triaxial test overestimated the deformations with

approximately 150-180 % compared to the measured deformations.

The HSS calculations based on strict empirics show varying results depending on if the clay is

considered to be normally consolidated or overconsolidated. The OCR influences the lateral

earth pressure coefficient , which has a large effect on the calculated deformations, Figure

6.17. Clay is considered to be normally consolidated if the and overconsolidated if

the . With an OCR of 1.37 the clay can be considered to be slightly

overconsolidated. By assuming slightly overconsolidated clay, the results from the strict

empirical correlation overestimates the deformations by a factor 2 for load group 1,Figure

6.11 and approximately a factor of 1.5 for load group 2, Figure 6.12.

Overall the three different methods of determining the input parameters studied in this thesis

have uncertainties. To decrease the uncertainties and thereby increasing the accuracy of the

59

calculated ground deformations more advanced ground investigation methods, such as cross

hole seismic tests or resonant column tests, needs to be conducted to asses the small-strain

stiffness properties.

7.3.4 2D vs 3D The results obtained from the 2D and 3D numerical simulations are very similar. The small

difference that exists may come from different mesh sizes and the required load simplification

in the 2D geometry. The 2D numerical simulations use very a fine mesh while the 3D

simulations are using a coarse mesh. The main reason for the mesh difference is the

computation time, where both converged during the model testing. A 2D simulation with

very fine mesh took about 10-15 min for the simple geometry meanwhile the 3D simulation

with coarse mesh took approximately 6-7 hours. The 2D geometry in Plaxis is also faster to

set up than the 3D geometry.

7.4 Sensitivity analysis Many parameters in the HSS model are intercorrelated, for example the stiffness degradation

parameter is affected by the small strain stiffness. By varying only one of the parameters

while keeping the other parameter constant the sensitivity analysis fails to show the true

importance of some parameters. Therefor it is important to keep in mind that the results from

the simplified sensitivity only gives an indication of the importance of some parameters.

The simplified sensitivity analysis shows that many of the parameters studied had a negligible

effect on the deformation. The elasticity modulus that is important in the MC model has

almost no effect on the deformations in the HSS model according to the analysis. The main

reason for this is believed to be that the elasticity modulus is correlated to the level of strains.

At lower strain levels, the elasticity modulus increases, therefor the starting input stiffness has

less of an affect compared to the MC model.

The three parameters that had the largest effect on the deformations were .

The power law coefficient , had a great impact on the deformations, however, determining

the coefficient is not easy. (Surarak et al., 2012) mentions that m should be set to 0.9-1.0 for

clay soils while (Benz, 2007) recommends a value between 0.7-1.0. The coefficient is also

dependent on the liquid limit and the plasticity index on the clay (Benz, 2007).

The lateral earth pressure coefficient also had a large influence on the deformations. A

small decrease of 5% gave rise to an 8% deformation increase. Therefor it is important to

determine accurately.

One of the more difficult and yet most important parameters to determine is the small-strain

stiffness . The parameter is dependent on a lot of variables and seldom measured for

practical design cases. Due to its dependency of many variables the span for the small-strain

stiffness is large. The value of in this thesis is assumed to be in the range of 15 000-50 000

kPa. The chosen value of used as a reference value in the sensitivity

analysis lies in the mid of the large span. Based on the obtained deformations the true value of

60

the small-strain stiffness may however be in the range of 40 000 – 45 000 kPa from back

analysis of the deformations.

7.5 Conclusions and recommendations for practical design Soil is a complex material and calculating the predicted deformations is tricky due to the

many simplifications that need to be done during the process. There are many models for

calculating the predicted ground deformations when the soil is subjected to external loads.

The different models often require different parameters obtained from different types of field

or laboratory tests.

The result from this thesis show that despite its simplicity the analytical calculation based on

parameters obtained from CRS-test manages to predict the ground deformation with great

accuracy.

The numerical Mohr Coulomb constitutive model managed to predict the ground deformation

with greater accuracy than the analytical calculation, however the MC model is heavily

dependent on having a realistic value on and E. In order to determine the elasticity

modulus realistically, triaxial test must be performed and evaluated correctly. Triaxial tests

are however seldom performed in smaller projects due to the cost.

The Hardening Soil Small constitutive model, which simulates the soil behavior more

accurately than the simpler MC model, failed to give more accurate deformations than the MC

model. The main reason is believed to be the difficulty of determining its parameters.

In order to use the HSS-model with greater accuracy many of the parameters, such as and

needs to be measured by dynamic tests in a laboratory or out in the field.

The difference between 2D and 3D numerical calculations showed minor differences. The

difference that occurred may have been caused by different mesh qualities in Plaxis.

For geometries similar to those in the thesis, the 2D model is recommended due to its

simplicity and faster computation time. The 3D model can be used for complex problems

where the simplifications into 2D may have a significant effect on the result. However,

modeling with 3D often requires other types of simplifications.

For future practical design it is recommended to skip the most advanced model, hardening soil

small, due to the lack of quality input data. This thesis has shown that the analytical

calculation model works very well for calculating the predicted short-term ground

deformations. The method is fast and can be done by parameters obtained from routine- and

CRS-tests, which often are performed even in smaller projects, which makes it very important

in future practical design.

It is also recommended to perform some kind of numerical simulation as a complement to the

analytical calculation. The numerical simulation must however be based on quality input data

obtained from e.g. correctly performed and evaluated triaxial tests.

61

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64

Appendix A Soil Data Appendix A presents data obtained from in-situ and laboratory tests performed between

supports 6-7.

Appendix A - Table 1 Results from the routine and CRS laboratory tests from borehole 15E03

Depth [m]

[

] [ [ [ [ [ [ [ M´

k [m/s]

1.5 1.67 69 67 17 19 15.56 54 531 88 12.4 3.7E-10

2.5 1.58 80 63 22 12 10.11 40 266 55 12.2 5.9E-10

3.5 1.54 93 72 26 12 9.52 44 234 57 13.2 6.9E-10

5.5 1.64 65 61 10 12 10.25

Appendix A - Table 2 Consolidation stage from triaxial test performed on samples from Borehole 15E03

Stage [kPa] [kPa]

1 6 6

2 34 21

3 25 19.5

Appendix A - Table 3 Shearing from triaxial test performed on samples from Borehole 15E03

Back pressure 300 kPa

Dry density 1.59 g/cm3

Liquid limit before test 74 %

Liquid limit after test 68 %

Consolidation strain 2.70 %

Consolidation strain 3.52 %

Shear strength 11.6 kPa

Axial strain failure 1.40 %

Axial failure stress 34 kPa

Radial failure stress 11 kPa

65

Appendix A - Figure 1 Consolidation coefficient cv evaluated from the CRS -test performed at 3.5 m depth

66

Appendix B Numerical input parameters Appendix B presents the input parameters used for the numerical calculation models as well

as the result from matching the fictive triaxial test to the performed test to obtain input

parameters for the numerical calculations.

Appendix B - Table 1 Input parameters for the numerical Mohr Coulomb model

Varies kN/m2

0.33 -

1 kPa

30 °

0 °

0.78 -

m/day

Appendix B - Table 2 The different elasticity moduli that were studied in the numerical MC model

Method of obtaining the stiffness Elasticity modulus [kPa]

– Triaxial test 10470

– Triaxial test 8149

5053

4330

3790

3369

3032

2526

67

Appendix B - Table 3 Input parameters for the Fill material

20 kN/m3

23 kN/m3

50000 kN/m2

0.2265 -

0.1 kPa

45 °

15 °

0.2929 -

Appendix B - Table 4 Input parameters for the HSS calculations

Where .

Parameters not mentioned above were set to standard values according to Plaxis.

Parameter Customized

parameters

Parameter

optimisation

Correlations

NC-Clay

Correlations

OC-Clay

Units

10470 10230 5053 5053 kN/m2

5282 5427 4043 4043 kN/m2

31410 29780 31410 31410 kN/m2

0.9016 0.9057 1 1 -

0.2 0.2 0.2 0.2 -

0.5130 0.5062 0.5 0.5 -

0.78 0.78 0.632 0.751 -

100 100 100 100 kN/m2

28300 24320 28300 28300 kN/m2

0.04270 -

1 1.476 1 1 kN/m2

30 30 30 30 °

-3.829 -2.930 0 0 °

0.9 0.9 0.9 0.9 -

16.10 16.10 16.10 16.10 kN/m3

16.10 16.10 16.10 16.10 kN/m3

2.020 2.020 2.020 2.020 -

68

Appendix B - Figure 1 Results from the Plaxis parameter optimisation compared with triaxial test from borehole

15E03

Appendix B - Figure 2 Results from the Plaxis parameter optimisation compared with triaxial test from borehole

15E03

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

She

ar s

tre

ss [

kPa]

Axial strain [%]

Stress-Strain diagram

Triaxial test

Plaxis parameteroptimisation

0

2

4

6

8

10

12

14

0 5 10 15 20 25 30

τ [k

Pa]

s´ [kPa]

Stress-Path diagram

Triaxial test

Plaxis paramteroptimisation

69

Appendix B - Table 5 Parameter range for the automatically determined parameters in the customized method

Appendix B - Table 6 Plaxis parameter optimisation parameter range

Parameter Min Value Optimal value Max Value Units

- 10470 - kN/m2

4000 5282 10000 kN/m2

- 31410 - kN/m2

0.7 0.9016 1.0

- 0.2 -

0.5 0.513 0.62

- 100 - kN/m2

- 28300 - kN/m2

- - -

- 1 - kN/m2

- 30 -

-7 -3.829 -2 °

- 0.9 °

- 16.10 - kN/m3

- 16.10 - kN/m3

Parameter Min Value Optimal value Max Value Units

8000 10230 10470 kN/m2

4000 5427 10000 kN/m2

- 29780 - kN/m2

0.8 0.9057 1.0

- 0.2 -

0.5 0.5062 0.62

- 100 - kN/m2

10000 24320 30000 kN/m2

0.01 -

1 1.476 2 kN/m2

- 30 -

-7 -2,930 -3 °

- 0.9 °

- 16.10 - kN/m3

- 16.10 - kN/m3

70

Appendix C Performed triaxial tests Appendix C presents data from different triaxial-, routine- and CRS-tests are here presented

for clay samples taken from Stockholm, Norrköping and Uppsala, Figure 5.3.

Appendix C - Figure 1 Results from the triaxial test from borehole 15E01, Ulvsunda, Stockholm

Appendix C - Figure 2 Results from the triaxial test from borehole 15E03, Ulvsunda, Stockholm

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14

She

ar s

tre

ss[k

Pa]

Axial strain [%]

Stockholm, Ulvsunda. Borehole 15E01

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

She

ar s

tre

ss[k

Pa]

Axial strain [%]

Stockholm, Ulvsunda. Borehole 15E03

71

Appendix C - Figure 3 Results from the triaxial test from borehole 15E09, Ulvsunda, Stockholm

Appendix C - Figure 4 Results from triaxial test from Norrköping, Location 1 Clay. 5m Depth

0

2

4

6

8

10

12

14

0 2 4 6 8 10 12 14

She

ar s

tre

ss [

kPa]

Axial Strain [kPa]

Stockholm, Ulvsunda. Borehole 15E09

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20

She

ar S

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Loaction 1. Depth 5m

72

Appendix C - Figure 5 Results from triaxial test from Norrköping, Location 1 Clay. 8m Depth

Appendix C - Figure 6 Results from triaxial test from Norrköping, Location 1. 13m Depth

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18 20

She

ar s

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Location 1. Depth 8m

0

5

10

15

20

25

30

35

40

45

0 2 4 6 8 10 12 14 16 18 20

She

ar s

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Location 1. Depth 13 m

73

Appendix C - Table 1 Results from routine and CRS -test for Norrköping, Location 1 Clay

Depth

[m] ρ [t/m3] [ [ [ [ [

4 1.61 72.2 82 15 40 29.9 -

5 1.56 76.4 83 14 31 23.1 97

6 1.59 83.9 93 14 33 23.3 98

8 1.57 78.3 82 18 32 23.9 110

9 1.56 81.9 86 18 33 24.2 128

13 1.55 80.4 84 15 28 20.7 116

14 1.56 78.9 83 16 29 21.5 136

Appendix C - Figure 7 Results from triaxial test from Norrköping, Location 2. 5m Depth

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20

She

ar s

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Location 2. Depth 5m

74

Appendix C - Figure 8 Results from triaxial test on Norrköping, Location 2. 8m Depth

Appendix C - Figure 9 Results from triaxial test on Norrköping, Location 2. 13m Depth

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20

She

ar s

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Location 2. Depth 8m

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20

She

ar s

tre

ss [

kPa]

Axial Strain [%]

Norrköping, Location 2. Depth 13m

75

Appendix C - Table 2 Results from routine and CRS -test for Norrköping, Location 1 Clay

Depth

[m] ρ [t/m3] [ [ [ [ [

4 1.48 92 89 15 19 13.7 -

5 1.57 86.9 84 15 18 13.3 62

6 1.53 83.9 77 18 20 15.4 64

8 1.52 78.3 93 14 27 19.1 74

9 1.51 81.9 94 15 20 14.1 68

13 1.61 80.4 87 17 19 13.8 88

14 1.55 78.9 86 11 19 13.9 93

Appendix C - Figure 10 Results from triaxial test from Uppsala 3m Depth

0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20

She

ar S

tre

ss [

kPa]

Axial Strain [%]

Uppsala. Depth 3m

76

Appendix C- Table 3 Results from routine and CRS -test for Uppsala

Depth [m]

ρ (t/m3) [ [ τ [kPa] cu [kPa] [

3 1,6 64 70 12 26 20,88046 62

5 1,54 80 93 8,6 25 17,66784 67

9 1,53 79 93 9,3 37 26,1484 90

12 1,51 80 94 7,9 46 32,35274 103

14 1,56 75 88 7,8 44 31,87838 109

22 1,64 61 71 7,1 41 32,71737 144

28 1,66 66 74 7,7 31 24,28109 160

35 1,81 46 53 6,7 32 29,12639 185

42 1,83 48 56 7,1 40 35,51699 248

55 1,89 38 45 6,2 46 45,06849 294