Short-term deformations in clay under a
formwork during the construction of a bridge
A design study
Alexander Berglin
© Alexander Berglin, 2017 Master of Science Thesis
Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering Division of Soil- and Rock Mechanics
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
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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
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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
ix
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 [-]
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]
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
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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)
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).
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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).
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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.
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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.
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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)
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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:
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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)
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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).
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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).
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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.
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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).
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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.
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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 ( )
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)
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)
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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).
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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)
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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.
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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:
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
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
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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)
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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).
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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.
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
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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.
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.
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 [m 2
/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).
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.
