2 VIBRATORY DRIVING TECHNIQUE AND INDUCED

Transcription

1 2 VIBRATORY DRIVING TECHNIQUE AND INDUCED SOIL BEHAVIOUR CHAPTER 2 Vibratory Driving Technique and Induced Soil Behaviour I INTRODUCTION Three basic driving techniques can be used to install piles or sheet-piles in the soil: impact driving, jacking and vibratory driving. Vibratory driving seems to be the preferred alternative to drive piles or sheet-piles in soft soils because it allows a fast installation with little environmental disturbance. Impact driving can generate high energy under difficult soil conditions, but generates noise and high vibrations in the soil. Therefore, impact driving cannot be used near other constructions, and its use is more and more restricted. Jacking does not induce environmental disturbance but it is expensive. This method is only used where sensitive environmental conditions are encountered. The objective of this chapter is to position the present research relatively to the main research efforts investigating the vibratory driving technique and the associated soil behaviour around the pile. The chapter is divided in two sections. The first section discusses the principles of the vibratory driving technique and the equipment generally used on working sites. This section discusses also the parameters influencing the performance of the vibratory driving and summarises the different methods presented in the literature to predict the penetration speed and the refusal depth that can be expected during a vibratory driving. The second section describes the different types of soil behaviour observed under cyclic loading conditions and the different laboratory tests traditionally used to characterise these type of behaviour. 2-1

2 2-2 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour The following review does not pretend to be exhaustive. The ambition is to make the reader familiar with the technique of vibratory driving and the expected soil behaviour around the pile during its application. II VIBRATORY DRIVING TECHNIQUE II.1 Introduction The objective of the present section is to present the principles of the vibratory driving technique and the equipment required on the working site. An overview of the different models available in the literature (Holeyman, 2000; Viking, 1997) to predict the soil and pile behaviours during the penetration is also presented. II.2 Principles of the vibratory driving technique Vibratory driving is a method frequently used to install pile or sheet-piles into granular soils and less frequently in cohesive soils. A vibrator induces a cyclic load on the pile with two consequences. The first effect is to apply on the pile a driving force greater than the static soil resistance. The second effect is to lead to a significant reduction of soil resistance. Indeed, the cyclic movement induces a degradation of soil strength and, in saturated media, the build-up of the pore pressure. The ultimate buildup of pore pressure in saturated sands is referred to as liquefaction, expressing a complete loss of strength. Some consider the pile is not actually installed into the ground primarily by the vibrating force, rather by sinking the pile under gravity forces into degraded material. The vibratory driving technique is a method frequently used to drive sheet-piles because of the short installation time, the low disturbances to the surrounding area, and the low damages to the sheet-pile. This technique also allows the driving of H-pile, tubes for cast-in-steel-shell piles, precast prestressed concrete piles, The technique is particularly efficient under soft saturated, cohesionless soil conditions. Indeed, this type of soil is easily subjected to liquefaction when a cyclic load is applied. On the other hand, it seems the vibratory driving technique becomes more difficult to apply in moderately stiff saturated clays, very dense sands or gravels, because the degradation of the shear resistance is less pronounced for these categories of soils.

3 II. Vibratory driving technique 2-3 II.3 Vibratory equipment The cyclic load applied on the pile is induced by a vibrator placed at the pile top (Fig. 2-1). Four elements can be identified on the vibrator:! Bias mass! Elastomer dampers! Exciter block! Clamping device The vibratory action is generated by the exciter block constituted of an even number of eccentric masses disposed symmetrically around the vibrator axis. Each pair of masses is turning at the same frequency in opposite direction. With this configuration, the horizontal component of the centrifugal forces induced by the rotation are eliminated whereas the vertical component are added. As a result, a sinusoidal vertical load, which amplitude is a function of the sum of the eccentric moments of each masses me and the angular frequency ω, are applied on the pile: ( 2..Freq) 2 F v(t) = me.ω² sin ( ωt) = me. π sin ( 2. π.freq. t) (Eq. 2-1) where me is the eccentric moment of the vibrator ω is the angular frequency of the vibrator Freq is the frequency of the vibrator The vibratory action will be balanced by reactive inertial effects of masses undergoing the vibratory movement and by soil reactions opposing the profile movement. Provided the centre of gravity of the rotating masses coincides at all times with the profile neutral axis, the exciter block is assumed to exert a purely longitudinal force onto the profile. The exciter block is connected to the profile via a clamping device. The purpose of this mechanism is to transfer the vibrations to the profile without damaging it and to handle easily and rapidly the different type of profiles. Fig. 2-1: Mechanical action of a vibrator (Holeyman-2000)

4 2-4 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour The vibrator is generally suspended to a carrier. In order to avoid to transfer the vibration to the carrier, the suspension device is constituted of a vibration isolator mechanism. It consists of a quasi-stationary heavy mass directly suspended to the suspension block (suppressor housing) and an intervening spring, generally consisting of elastomer pads (elastomer dampers). The weight of the suspension device is generally called bias mass or static mass. During the driving operation, the pile penetration can be slowed down by controlling the retaining force T C applied by the carrier. The composite system of the pile and the vibrator is divided into a static and a vibrating part. The bias mass, isolated from the vibrating movement, stays quasi static. This mass increases the static load applied on the pile. The vibrating part is constituted of the pile, the clamping device and the exciter block. The total load applied on the soil during a vibratory pile driving includes a cyclic term and a static term (Eq 2-2). The cyclic load F v is generated by the vibrator, whereas the static term is the sum of the weight of the pile, the clamping device, the exciter block, and the bias mass. This total load is sometimes reduced by the retaining force T C applied by the carrier. F(t) = (M where P + M C + M v 2 + M ).g + me.ω S sin ( ωt) T F is the net load applied to the soil me is the eccentric moment of the vibrator ω is angular frequency of the vibrator M P is the mass of the pile M C is the mass of the clamping device M V is the mass of the exciter block M S is the mass of the static weight T C is the retaining force C (Eq. 2-2) Two main types of vibrators are commercially available: the electrical and the hydraulic vibrators. For both types, the motors are located in the exciter block. These motors are powered by a separated diesel-hydraulic or diesel-electric power generator connected to the vibrator by a transmission line (Fig. 2-1). Due to their light weight and ease of adjustment of the operating frequency, hydraulic vibrators are frequently preferred to electrical vibrators. Based on the frequency and the eccentric moment, the different vibrators are generally separated in five classes as described in Table 2-1.

5 II. Vibratory driving technique 2-5 "Standard Type frequency" Frequency range [rpm] Eccentric moment [kg.m] Maximum centrifugal force [kn] Free hanging double amplitude [mm] up to 230 up to 4,600 up to 30 High frequency to to 2, to 22 Variable eccentricity to to to 17 Excavator accessory 1800 to to to to 20 Resonant driver ,000 (in theory) Self destructing Table 2-1: Different classes of vibrators The free hanging double amplitude S 0 corresponds to the double amplitude of movement for a free hanging vibrator whose vibrating mass is formed of the exciter block and the clamping device (Eq 2-3). The parameter S 0 depends on the eccentric moment me and the vibrating mass of the vibrator (M C +M V ), and is independent from the operational frequency Freq. This amplitude is reduced when the pile is connected to the vibrator due to the increase of the vibrating mass. During the vibratory driving operation, the soil resistance further reduces the value of this amplitude. S 0 2. me = M + M C v (Eq. 2-3) When a vibratory driving is planned, the vibrators are generally chosen based on experience and field tests. The vibrator main characteristics that must be identified, are the eccentric moment me, the frequency Freq and the static surcharge M S. Rodger and Littlejohn (1980) proposed a table helping the selection of the vibrator characteristics based on the soil conditions and the type of pile (Table 2-2). Cohesive soils Dense cohesionless soils Loose cohesionless soils All cases Low point resistance High point resistance Heavy piles Light piles High acceleration Low displacement amplitude High acceleration Low frequency. Large displacement amplitude High acceleration Predominant side resistance Predominant side resistance. Predominant end resistance. Predominant side resistance. Requires high acceleration for either shearing or thixotropic transformation Freq > 40 Hz a: 6-20 g s 0 : 1-10 mm Requires high acceleration for fluidisation Freq : Hz a : 5-15 g s 0 : 1-10 mm Requires high displacement amplitude and low frequency for maximum impact to permit elasto-plastic penetration Recommended parameters Freq : 4-16 Hz a: 3-14 g s 0 : 9-20 mm Requires high acceleration for fluidisation Freq : Hz a: 5-15 g s 0 : 1-10 mm Freq = vibrator frequency; a = the maximum transmitted acceleration (= me.ω²); s 0 = free hanging double amplitude. Table 2-2: Helping procedure to choose the appropriated vibrator (Rodger and Littlejohn, 1980).

6 2-6 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour II.4 Description of some vibratory driving models The soil behaviour and the interaction between the pile and the soil during vibratory driving have been widely investigated for the last decades. The objective is to be able to predict the abilities of a vibrator to drive a pile, to evaluate the average penetration speed, to determine the refusal depth and, finally, to estimate the environmental disturbance. Different models have been proposed and can be classified, according to Viking (1997) in four groups based on the mechanical engineering principles considered: Models based on force equilibrium Models based on momentum conservation Models based on energy conservation Models based on integration of the equation of laws of motion The following paragraphs summarise briefly these different types of models (Viking, ). II.4.1 Models based on force equilibrium The models based on force equilibrium try to determine wether or not the vibrator can overcome the resisting forces applied to the pile during the vibratory driving. These models do not provide any estimation of the penetration speed. They only verify whether the maximum driving force applied by the vibrator is greater than the resistance of the soil. Jonker (1987) proposed the beta formula (Eq 2-4). This equation was deduced from the author s experiment in vibratory driving of large pipe piles in offshore works. The value proposed for parameter β are summarised in Table 2-3. Parameters β are introduced into the equation to take into account the soil strength reduction induced by the cyclic loading. F C + F + F > β. R where F C i F i S F S R S0 β 0 R si β i R t β t 0 so + β. R i si + β R t t (Eq. 2-4) is the maximum force generated by the vibrator (= me.ω²) is the inertial forces of dynamic mass (= (M V +M C +M P ).a) where a is the maximum acceleration during the cycle is the constant surcharge force (= (M V +M C +M P +M S ).g) is the soil friction outside pile shaft is the empirical reduction factor of shaft friction outside pipe pile is the soil friction inside pile shaft is the empirical reduction factor of shaft friction inside pipe pile is the soil resistance at pile toe is the empirical reduction factor of toe resistance

7 II. Vibratory driving technique 2-7 Type of soil Value for β Round coarse sand 0.10 Soft loamed/marl,soft loess, stiff cliff 0.12 Round medium sand, round gravel 0.15 Fine angular gravel, angular loam, angular loess 0.18 Round fine sand 0.20 Angular sand, coarse gravel 0.25 Angular/dry fine sand 0.35 Marl, stiff/very stiff clay 0.40 Table 2-3: Proposed values of parameters β (β 0, β i & β t ) in the beta formula (Jonker-1987) Warrington (1989) developed an other type of formula to determine whether or not the vibrator will be able to drive the pile (Eq 2-5). This formula was widely used by the Tunker company in USA during vibratory driving where the double amplitude of displacement was greater than 2.38mm. The correlation between the soil resistance τ s considered in Eq. 2-5 and the result of SPT test are presented in Table 2-4. FC > τs.a where F C τ s A is the maximum force generated by the vibrator (=me.ω²) is the stress resistance along the pile shaft is the area of the pile shaft (Eq. 2-5) SPT value [ blows/30 cm] Cohesionless soil Cohesive soil Soil resistance τ s [kpa] Table 2-4: Correlation between SPT test and soil resistance for the Tunker formula.

8 2-8 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour II.4.2 Models based on momentum conservation The models based on the momentum conservation (Schmid 1969) suppose the momentum of the total weight of the vibrator, the bias mass and the pile over the period of one vibration cycle, is equal to soil resistance impulse (Eq 2-6). T ( ) c + M + M + M.g T= R.dt= M (Eq. 2-6) s v C p α.r t.tc o where M S is the mass of the static weight M V is the mass of the exciter block M C is the mass of the clamping device M P is the mass of the pile T is the period of the cycles T C is the time of contact between pile toe and the soil during one cycle R t is the soil resistance at the pile toe. α is an empirical coefficient between 0.5 and 1, generally equal to 2/3 The evaluation of time of contact between the pile toe and the soil during one cycle T c requires to determine experimentally the minimum pile acceleration a min. This acceleration is the greatest peak acceleration induced by the vibrator for which the inertial force is cancelled by the soil resistance (i.e. no displacement occurs). The contact time T c is evaluated using the following expression: T C = where 2.v a a P min.t T C is the time of contact between pile toe and the soil during one cycle v P is the average penetration speed a is the peak acceleration during the considered cycle a min is the minimum peak acceleration (Eq. 2-7) Based on Eq 2-6 and 2-7, it is possible to calculate the average penetration speed of the pile (Eq. 2-8). For an acceleration lower than minimum acceleration a min, no penetration occurs, whereas for a greater acceleration, the pile penetrates into the ground with a penetration speed increasing linearly with the acceleration. The key parameter of this method is the minimum acceleration a min that has to be carefully determined. V p = 2 ( a a ).T ( Ms + M v + MC + M p ).g 2 min R t.α (Eq. 2-8)

9 II. Vibratory driving technique 2-9 II.4.3 Models based on energy conservation The principles of the models based on the energy conservation assume that the energy provided by the power supply of the vibrator is equal to the energy dissipated by the friction between the soil and the pile. These models have the following form: R.v = β. W + ( F + F ). v P t t i s P (Eq. 2-9) where R is the soil resistance v P is the average penetration speed W t is the theoretical power transmitted to the system β t is the empirical reduction factor of power transmission F i is the inertial forces of dynamic mass (= (M V +M C +M P ).a) where a is the peak acceleration during the cycle is the constant surcharge force (= (M V +M C +M P +M S ).g) F S This expression allows to calculate directly the penetration speed (Eq. 2-10) v P = β t W ( R F F ) i t s (Eq. 2-10) The determination of the empirical reduction factor β t was based on Davidson s formula (1970) estimating the bearing capacity for a Bodine Resonant Driver and can be calculated using the following equation: R β t = 1 vp se..wt (Eq. 2-11) 1000 where s e is an empirically determined set [mm/cycle] representing all energy losses. As an alternative, Warrington (1989) suggested to use a value of 0.1 for the reduction factor β t. II.4.4 Models based on integration of the equation of laws of motion The different models presented in the previous paragraphs considered only the maximum forces and accelerations during the cycles. Furthermore, they simulated the soil/pile interaction and the soil degradation by means of empirical reduction factors. In recent years, thanks to new computing capacities, new types of models were proposed to simulate more accurately the soil behaviour around the pile, and the interaction between the pile and the soil. The following paragraphs present some of these models. The first paragraph describes models that integrate the equation of motion of the pile based on the forces applied by the vibrator, the pile toe resistance and the pile shaft resistance. The other paragraph discusses about more elaborated models where the soil and the pile are represented by discretising the media in a set of elements.

10 2-10 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour A. Single degree of freedom model The single degree of freedom models are based on the integration of the motion equation considering the pile as a rigid body. The general form of the equation to integrate is the following: 2 u a(t) = = 2 t where ( M + M + M + M ) s v C p 2.g + me.ω.sin( ω.t)! R M + M v C + M p s R a(t)is the instantaneous pile acceleration at time t u is the pile displacement M S is the mass of the static weight M V is the mass of the exciter block M C is the mass of the clamping device M P is the mass of the pile R s is the soil resistance at pile shaft R t is the soil resistance at pile toe t (Eq. 2-12) This kind of model is very flexible and able to integrate the desired phenomenon. For example, it allows a large flexibility in the choice of constitutive laws used to model the soil behaviour. It can also take into account the differences of behaviour between the shaft resistance and the toe resistance. During vibratory driving, the soil resistance applied on the pile shaft and the pile base behave differently as a function of the pile displacement. Indeed, the shaft resistance R s is reversible and is applied in the opposite direction of the pile displacement. On the other hand, the toe resistance R t can be mobilised only when the pile moves downward. Other parameters can be introduced in the motion equation as the friction in the clutches between two sheet-piles, the friction inside an hollow pipe pile, the retaining force of the carrier,... Based on this concept, Holeyman (1993-a) proposed a semi-empirical equation allowing the evaluation of the average penetration speed based on the peak acceleration of the free hanging vibrator/pile system. This equation assumes the soil behaviour is perfectly plastic, and the net acceleration during a cycle can be deduced from the difference between the sinusoidal driving force and the opposing soil resistance. The equation considers also the difference of behaviour between the base and toe resistance. Attention is also paid to the clutch friction which is combined with the shaft resistance. The average penetration is calculated by intuitively integrating the net downward and upward acceleration over a complete cycle. The soil resistance is evaluated based on the results of CPT tests reduced as a function of the friction ratio and the acceleration, to take into account the degradation of the soil strength. This model has been verified and calibrated based on different results of full-scale tests (BBRI, 1994) Gonin (1998) proposed a similar procedure where the penetration speed is deduced from the analytical integration of the excess force.

11 II. Vibratory driving technique 2-11 Vanden Berghe (1997) also proposed a model based on the numerical integration of equation of motion. The soil resistance is assumed to be directly mobilised when the direction of the displacement changes (i.e. perfectly plastic). A distinction is made between soil resistance at pile toe and along the pile shaft. The average penetration speed is deduced from the calculated pile penetration during one cycle. A similar numerical method has been proposed by Diersen (1994) considering more elaborated models to simulate the soil behaviour during the different phases of the cycles. Fig. 2-2 presents the shape of the evolution of the shaft and toe resistance as a function of the pile displacement. Based on experimental measurements, Diersen observed a separation between the pile and the soil at the pile base. Fig. 2-2: Evolution of (a) the shear stress along the pile shaft and (b) the toe resistance during vibratory pile driving (after Dierssen, 1994)

12 2-12 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour B. Radial 1D model Fig. 2-3: Radial 1D model (Holeyman, 1993b) Holeyman (1993-b) suggested to simulate the pile penetration and the wave propagation around the pile by modelling the soil with a set of concentric rings (Fig. 2-3) possessing individual mass and transmitting forces to their neighbouring ones. The pile and the soil elements are assumed to be rigid bodies. To calculate the displacement of the pile as well as the wave propagation around the vibrating profile, the model integrates numerically the equation of motion of each cylinder based on their dynamic shear equilibrium in the vertical direction. Different constitutive laws can be used to calculate the force between the different rings. Holeyman (1993b) proposed to simulate the shear stress along the pile shaft and between soil elements by constructing the hysteretic soil behaviour, which enforces parameters decrease as a function of the cycle number to take into account the soil degradation. C. Longitudinal 1D model The principal weaknesses of the models presented in the previous paragraphs is their limitations to model the pile behaviour in a layered soil. Indeed, each model considers a global shaft resistance. When the profile crosses different layers of soil, some averaging procedures are proposed. The longitudinal 1D model offers the solution for this kind of soil configurations (Gardner 1981; Chua 1981; Middendorp 1988; Ligterink 1990). The 1D longitudinal model is adapted from the Smith s classic lumped parameters model (Fig. 2-4). The pile is divided in a series of elements that are interconnected with springs which stiffness depends on the pile characteristics. The first element represents the static mass and is connected by a soft spring, whereas the second element simulates the exciter block and is subjected to a sinusoidal force. The soil behaviour is modelled using a spring-slider-dashpot system.

13 II. Vibratory driving technique 2-13 Fig. 2-4: longitudinal 1D model (Gardner 1997) The use of this model for vibratory driving of pipe piles (Middendorp 1988) pointed out the need of a constitutive equation of the soil able to simulate the degradation of the soil resistance as a function of the oscillation history. The authors remarked that the soil parameters may depend on the chosen driving frequency and the pile displacement amplitudes. II.5 Concluding remark The previous paragraphs presented the principles and the equipment generally used during pile vibratory driving. However, it must be considered that each contractor and vibrator manufacturer brings his own know-how to increase the efficiency of his equipment. Therefore, it is not surprising to find on working site equipments, accessories and procedures different from the general description presented here. The description of some models developed over the last years to predict the penetration speed and the wave propagation during the vibratory driving of piles or sheet-piles, tried to provide the reader with a general overview of the existing models, with their advantages and their limitations. This analysis does not pretend to be complete. Many other models were developed and proposed in the literature. The purpose was to situate the present research among all the different types of models.

14 2-14 III Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour CYCLIC SOIL PROPERTIES III.1 Introduction Before describing the phenomena at play during cyclic loading of the soil, the first part of this section introduces the usual representation of the stress state in the soil and the different types of behaviour observed on sand during monotonic loading. The next part of the section describes the general behaviour of soil observed during cyclic loading. It introduces the parameters traditionally used to represent that behaviour, and it shows how these parameters are influenced by the soil and loading conditions. The last part of the section describes briefly the different types of laboratory tests used to measure these cyclic soil properties. III.2 Representation of the stress states in the soil A soil consists of an assemblage of individual granular particles (Fig. 2-5-a). This assemblage is characterised by its density that is quantified by the void ratio e (i.e. volume of voids divided by volume of grain). Each particle is in contact with a number of neighbouring particles. The external load applied on the soil produces contact forces between the particles (Fig. 2-5-b). These forces hold individual particles in place and give the soil its strength and stiffness. The quantification of these contact forces uses the concept of effective stress. The effective stresses are a representation of the average values of the stresses between all grains of the soil structure. (a) (b) Fig. 2-5: (a) grains assemblage (b) grain contact forces (from

15 III. Cyclic soil properties 2-15 If the soil is saturated, the external loads applied on the soil are balanced by the contact forces and the water pressure u. The stress state in the soil is then described by the total stress state corresponding to the external load applied on the soil, the pore pressure and the effective stress state. At each point of the soil and for each stress direction, the following relationships links these stresses: σ = σ' + u τ = τ ' where σ is the total normal stress τ is the total shear stress u is the pore pressure σ is the effective normal stress τ is the effective shear stress (Eq. 2-13) For each point of the soil, the effective and the total stress state can be represented by the three circles of Mohr (Fig. 2-6). For a constant total stress state, the effect of pore pressure is to shift horizontally the circles corresponding to the effective stress state. An increase of the pore pressure tends to decrease the effective stresses which can even vanish when the pore pressure becomes equal to the total stress. On the other hand, a decrease of the pore pressure increases the effective stresses. 400 Shear Stress [kpa] Effective stress state (P',q/2) Pore pressure (P,q/2) Total stress state σ3' σ2' σ1' σ3 σ2 σ Normal Stress [kpa] Fig. 2-6: Representation of the soil stress states with Mohr s circles Generally, the Mohr s circles representing the stress states of the soil are characterised by the mean stresses (P and P) and the deviator q. In the present research, these parameters were defined using the Cambridge s definition (Eq 4-14, 4-15 and 4-16). The mean stress represent the normal stress related to the mean value of the normal stresses corresponding to the centres of the three circles. The deviator is a representation of the mean value of the diameters of the Mohr s circles. These parameters are represented on Fig. 2-6.

16 2-16 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour σ ' σ2 ' = + ' 3 σ σ2 = P + σ 3' 1 P + σ3 q= ( σ' σ' ) + ( σ' σ' ) + ( σ' σ' ) where P is the effective mean stress P is the total mean stress q is the deviator σ 1, σ 2, σ 3 are the effective principal stresses σ 1, σ 2, σ 3 are the total principal stresses (Eq. 2-14) (Eq. 2-15) (Eq. 2-16) III.3 Sand behaviour during monotonic loading III.3.1 Critical state When a soil is loaded under monotonic conditions, its effective stress state and density evolves based on initial conditions, the strain state (e.g. triaxial compression, simple shear, ) and the boundary conditions (e.g. drained or undrained conditions). This evolution tends systematically toward a critical state, defined as the state in which the soil flows continuously under constant deviator and constant effective mean stress at constant volume ( q, P and e =0 for non zero strain state). The critical state can be described by the critical mean stress P cr, the critical deviator q cr and the critical void ratio e c. The set of all critical states (P cr, q cr, e cr ) define a surface called the critical state surface. In a diagram of the deviator q versus the effective mean stress P (e=cst) (Fig. 2-7-a), the critical state surface is a straight line. The slope of this line, usually referenced by M, depends principally of the friction angle ϕ of the soil and the type of strain path followed to reach the critical state. In the diagram of the void ratio e versus the mean stress P (Fig. 2-7-a), Castro (1977) showed experimentally that the critical void ratio e c decreases when the critical effective mean stress increases.

17 III. Cyclic soil properties Critical state line (P cr ', q cr ) Critical state line (P cr ', e c ) Deviator q [kpa] M Void Ratio e [-] e = cst Effective Mean Stress P' [kpa] (a) (b) Fig. 2-7: Critical state line (a) in the (P, q) diagram and (b) in (P, e) diagram q = cst Effective Mean Stress P' [kpa] III.3.2 Monotonic loading Based on the initial state of the soil, three types of behaviour can be observed during monotonic loading (Castro, 1977). During a monotonic loading on a dense sand (such as specimen A on Fig. 2-8) starting from an isotropic stress state (q=0), the specimen exhibits a contractive behaviour. The grains are moving relatively to each other, and try to reduce the volume of the soil. For a certain deformation, the soil structure cannot be deformed anymore by reducing the volume any further. Therefore, to follow the imposed further deformation, the grains have to roll on each other. This phenomenon induces a tendency of the soil to dilate. If the shearing is undrained (i.e. no volume change), the tendency of volume change induces a variation of the pore pressure (contraction = increase of the pore pressure and dilation = decrease of the pore pressure). The typical stress path (P,q) of a undrained shearing is drawn on Fig The mean stress P decreases during the contraction phase, until the dilation state where the specimen behaviour changes from contraction to dilation because the specimen can not be deformed further without trying to increase the volume. During the dilation phase, the stress path is aligned to a straight line (called critical state line) until the specimen reaches the critical state.

18 2-18 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour Deviator q [kpa-] Critical state line (P cr ', q cr ) C (P cr ', q cr ) B (P cr ', q cr ) Effective Mean Stress P' [kpa] 1 A M ea > ec > eb Dilative Behaviour Contractive Behaviour Maximu shear stress max [kpa] B A C Flow liquefaction Dilative Behaviour ea > ec > eb Limited liquefaction Shear Strain γ [% ] Fig. 2-8: Types of soil behaviour during undrained monotonic loading (e = cst). Very loose specimen (such as specimen B on Fig. 2-8) exhibit only a contractive behaviour. During the beginning of the shearing, the deviator q increases, but for a threshold value the grain structure becomes unstable and all the structure collapses. It results a rapid flow to large strains at low effective mean stress. This phenomenon is called flow liquefaction. At intermediate densities (such as specimen C on Fig. 2-8), the soil starts the shearing with a contractive behaviour. The deviator presents a peak value followed by a limited period of flow, which ends with an onset of dilative behaviour at the end of the shearing. This type of behaviour is called limited liquefaction. III.4 Cyclic behaviour of soil III.4.1 Grain behaviour during cyclic loading When a dense sand is subjected to large amplitude cyclic loading, the forward and backward movements modify cyclically the amplitude and the direction of the contact force between grains. It results in a reorganisation of the grain skeleton during which the grains become more free and tend toward a more stable state by trying to reduce the volume of the specimen. If the soil is saturated and if the boundary condition prevent any drainage, this tendency results in an increase of the pore pressure. As explained in the previous paragraph, the increase of the pore pressure induces a reduction of the effective stresses and the shear resistance. The continuous increase of the pore pressure consecutive to the successive grain reorganisations leads progressively the soil to a structure where the grains can move freely without being in contact. No more stresses between grains are observed and no more resistance can be obtained. The specimen is liquefied. This phenomena is illustrated on Fig. 2-9 where an initial grain structure with low water pressure and high contact forces is compared

19 III. Cyclic soil properties 2-19 with the grain structure at liquefaction characterised by the high water pressure, the low values of existing contact force and broken grain contacts. (b) (a) Fig. 2-9: Grain structure during undrained cyclic loading: (a) initial state; (b) liquefied state. (from The term liquefaction, introduced by Mogami and Kubu (1953), is not limited to ultimate state during cyclic loading. Paragraph III.3.2 presented an other kind of behaviour inducing liquefaction. In fact, the term liquefaction is related to all phenomena leading to an increase of the pore pressure of sands under undrained loading conditions. These phenomena are systematically consecutive to a grain structure reorganisation trying to reduce the soil volume. The induced pore pressure build-up decreases the effective stresses to a value that may be close to zero. Under this condition, the sand does not possess any shear strength resistance and behaves like a viscous liquid. Hence the name of liquefaction. The authors generally distinguish two main types of phenomena inducing liquefaction:! Flow liquefaction! Cyclic mobility As explained in paragraph III.3.2, flow liquefaction occurs when an undrained saturated loose sand is sheared in a such way that the grain skeleton is deformed in an unstable configuration. It results in a rapid collapse of the grain structure and a tendency of the soil to compact. Since any volume change is prevented by the pores water, this tendency increases the pore pressure inducing a reduction of the effective normal stresses and of the shear resistance. Flow liquefaction failures are characterised by the sudden nature of their origin, the speed with which they develop, and the large deformation observed during the flow. The occurrence of a flow liquefaction failure requires a disturbance strong enough to bring the grain structure in an unstable position and when the flow is engaged, it can not be stopped.

20 2-20 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour Cyclic mobility occurs when a dense undrained saturated sand is subjected to cyclic loading. If the amplitude is large enough, the grain structure subject to cyclic loading is reorganised and tends to compact and decrease in volume. Since drainage conditions are undrained, the pore pressure increases, leading to a reduction of the effective normal stresses and the shear resistance. In contrast to the flow liquefaction, the soil strength degradation induced by cyclic mobility stops if the cyclic loading stops As showed by Castro and Poulos (1977), all sands are not subjected to flow liquefaction. This phenomena can only occur with sands which initial conditions are situated above the critical void ratio line, i.e. for high void ratio and/or high effective stresses. On the other hand, cyclic mobility occurs only for dense sands for which initial conditions are situated below the critical void ratio line. This phenomenon is illustrated on Fig For example, a sample starting from point C, when subjected to a cyclic or monotonic loading, fails by flow liquefaction: a large amount of positive pore pressure is generated and the sample ends up at point A on the steady state line, where the sample has no further tendency to change volume. On the other hand, a dense soil which initial conditions are situated on point D below the steady state line moves, when subjected to a cyclic loading, fails by cyclic mobility. The soil conditions move toward the point B, a condition of zero effective stresses. If the same specimen is sheared by a monotonic loading, the soil condition would move from point D to the right direction until the sample conditions would be situated on the critical void ratio line. (P cr ', q cr ) C CONTRACTIVE SOILS (Loose) A Flow liquefaction Void Ratio e B Cyclic mobility D Monotonic loading (P cr ', q cr ) Critical state line DILATIVE SOILS (Dense) Effective Mean Stress P' Fig. 2-10: Sate diagram showing liquefaction potential based on undrained tests of saturated sands (Castro and Poulos, 1977)

21 III. Cyclic soil properties 2-21 III.4.2 Stress-strain relationship during cyclic shearing When a soil is subjected to symmetric cyclic deformations of amplitude γ a, the relationship between the shear stress and the shear strain is generally non linear and presents an hysteresis loop as shown on Fig This hysteresis loop can be described in two ways: by the actual path of the loop itself, or by parameters describing its shape. The shape of the loop is generally characterised by the slope and the surface. The global slope of the loop is generally characterised by the secant shear modulus Gs defined as the slope between the two extremities of the loop (Eq. 2-17). τ Gsn= γ where n max n max τ γ n min n min Gs is the secant shear modulus τ max is the maximum shear stress τ min is the minimum shear stress γ max is the maximum shear strain amplitude γ min is the minimum shear strain amplitude. (Eq. 2-17) The surface of the loop which is proportional to the energy dissipated during the cycle, is conveniently described by the damping ratio λ: λ d=. 2.π 1 γ W.τ max max where λ d is the damping ratio W is the surface of the hysteresis loop γ max is the shear strain amplitude τ max is the maximum shear stress (Eq. 2-18)

22 2-22 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour Fig. 2-11: Characteristics of hysteresis loops during cyclic shearing The secant shear modulus Gs and the damping ratio λ d are not intrinsic characteristics of the soil, and are dependent of the strain amplitude γ a, the void ratio e, the mean stress P, the plasticity index Pi, the overconsolidation ratio OCR and the number of the cycle N. A method frequently used in earthquake engineering to construct hysteresis loops consecutive to cyclic motion is based on the Masing rules (Masing, 1926). This method uses the curve obtained from monotonic loading, called a backbone curve (curve OA on Fig. 2-12). The unloading branch of the loop (curve ADB) is obtained by two-fold stretching of the backbone curve about B, translating its origin O to the point of stress reversal A. In the same way, the reloading curve (curve BCA) is obtained by enlarging the backbone curve by a factor of two about A, shifting its origin O to the point of stress reversal B. τ+ τa γ+ γa = f 2 2 Shear Stress, τ -γ a C B τ max 0 τ min D A γ a τ= f Backbone curve Shear Strain, γ () γ τ τ a γ γ a = f 2 2 Fig. 2-12: Construction of hysteresis loop using Masing s loop

23 III. Cyclic soil properties 2-23 The backbone curve used traditionally to build the hysteresis loops with Masing s rules is expressed by the hyperbolic equation of Eq (Kondner, 1963). The maximum shear modulus G max is equal to the tangent shear modulus at the origin, and the limit shear stress τ c is equal to the shear stress at failure during monotonic loading: τ = 1+ G. max Gmax τ c γ. γ where γ is the shear strain τ is the shear stress τ c is the limit shear stress G max is the maximum shear modulus (Eq. 2-19) The hyperbolic expression of the backbone curve limits the use of the model to simulating loops with a concave shape (as on Fig. 2-12). However, experimental results performed on dilative soil presented hysteresis loops with a S-shape (Fig. 2-13). The Kondner s equation is unable to represent this kind of shapes. Vanden Berghe (2001) proposed to extend the backbone curve by introducing an inflexion point on the loading curve, and a convex part when the soil has a dilative behaviour. The Masing s rules were used to build successfully the hysteresis loops (Fig. 2-13). Shear Sress [kpa] Model Test Result Shear Sress, [kpa] Model Test Result Shear Strain, γ [% ] Shear Strain, γ [% ] Fig. 2-13: Construction of hysteresis loops using Masing s rules with a modified backbone curve (Vanden Berghe, 2001)

24 2-24 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour III.4.3 Strain amplitude influence A. Small strain amplitude When the shear strain amplitude of a cyclic loading decreases to very low values (around 10-4 %), the soil behaviour becomes linear elastic and the hysteresis loops are reduced to a straight line. The corresponding secant shear modulus Gs measuring the slope of this line is maximum and is equal to the elastic shear modulus of soil. For this particular case, the secant shear modulus Gs is called the maximum shear modulus G max. Since the surface of the loop is equal to zero, the damping ratio, corresponding to this particular case, vanishes. Because the soil behaves as an elastic linear material, the maximum shear modulus G max can be evaluated using the wave theory in elasticity based on the soil density ρ and the shear wave velocity v s : G 2 max= ρ.v S (Eq. 2-20) where G max is the maximum shear modulus ρ is the unit weight V s is the shear wave velocity The maximum shear modulus G max is not an independent property of the soil. The literature identifies it depends principally of the effective mean stress P, the void ratio e, the overconsolidation ratio OCR (i.e. the ratio between the highest stress that was applied on the soil and the current stress), the plasticity index PI, the soil cementation and the geologic age t g. On the other hand, this modulus is independent of the shear strain amplitude and the number of cycle as long as the soil behaviour remains in the linear elastic domain. Table 2-5 summarises the influence of each parameter. Increasing factor Effective mean stress P Void Ratio e Geologic age t g Cementation c Overconsolidation ratio OCR Plasticity index PI Strain rate Maximum shear modulus G max Increase with P Decrease with e Increase with t g Increase with c Increase with OCR Increase with PI if OCR>1 Constant if OCR=1 No affected for non plastic soil Table 2-5: Effect of environmental and loading conditions on maximum shear modulus of normally consolidated and moderately overconsolidated soils (Dobry-1987) Many empirical equations were developed to determine the value of the maximum shear modulus depending of the soil conditions. Generally, these equations are expressed by the following formulation:

25 III. Cyclic soil properties 2-25 G max= A. F(e). P' n. OCR k (Eq. 2-21) where G max is the maximum shear modulus e is the void ratio P is the mean effective stress OCR is the overconsolidation ratio A, n and k are empirical parameters Table 2-6 presents the values of the different empirical constants proposed by different authors and summarised by Kokusho (1987). In this table, the overconsolidation factor is assumed to be equal to 1. Parameters for G max assessment (OCR=1) Sand clay A F(e) n Soil material (2.17-e)²/(1+e) 0.5 Round grain Ottawa sand (2.97-e)²/(1+e) 0.5 Angular grained crushed quartz e/(1+e) kinds of clean sands (2.17-e)²/(1+e) kinds of clean sand (2.17-e)²/(1+e) 0.5 Toyoura sand (2.17-e)²/(1+e) kinds of clean sand (2.97-e)²/(1+e) (2.97-e)²/(1+e) (4.4-e)²/(1+e) (2.97-e)²/(1+e) (7.32-e)²/(1+e) Kaolinite Kaolinite PI = 35 Bentonite PI = 60 Remoulded clay PI = 0-50 Undisturbed clay, PI=40-85 P in kpa, G max in kpa PI= plasticity index Table 2-6: Proposed empirical constant for the calculation of the maximum shear modulus G max (Kokusho-1987) Parameter k depends principally on the plasticity index PI. It is equal to zero for PI equal to 0 (= sand) and increases with the plasticity index PI. The values suggested by Hardin (1972) are summarised in Table 2-7. Parameter k for G max determination Plasticity Index PI >100 Parameter k Table 2-7: Proposed empirical constants of parameter k as a function of the plasticity index PI for the calculation of the maximum shear modulus G max (Hardin-1972)

26 2-26 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour B. Large strain amplitude The shape of the hysteresis loops observed during cyclic loadings is strongly influenced by the shear strain amplitude due to the different induced soil behaviour. Globally, when the shear strain amplitude increases, the secant shear modulus decreases, and the damping ratio increases (Fig. 2-14). According to several authors (e.g. Vucetic 1993; Kokusho ), three types of soil behaviour can be identified as a function of the strain amplitude:! Linear elastic behaviour! Elasto-plastic behaviour without degradation! Elasto-plastic behaviour with degradation Linear elastic domain Non linear elastic domain Elasto-plastic domain with degradation Normalised Secant Shear Modulus Gs/Gmax [-] Elastic Threshold Secant shear modulus ratio Damping ratio Degradation Threshold Shear Strain Amplitude γ a [% ] N = 1 N = 10 N = 100 N = Damping Ratio λ d [-] Fig. 2-14: Typical relationship between secant shear modulus ratio Gs/G max and damping ratio λ d as a function of the shear strain amplitude γ a. As shown in the previous paragraph, the soil behaviour during small strain amplitudes cyclic shearing is virtually linear elastic, and the hysteresis loops are reduced to a straight line. This kind of behaviour is characterised by a normalised secant shear modulus equal to 1 and a low value of the damping ratio. This damping ratio does not correspond to the hysteretic damping defined above (Eq 2-18) but to an intrinsic viscous damping. When the strain amplitude increases, the soil behaviour becomes non linear elastic and hysteresis loops appear. The secant shear modulus Gs characterising the loops decreases when the strain amplitude increases, whereas the damping ratio λ d

27 III. Cyclic soil properties 2-27 increases. The shear strain amplitude corresponding to the transition from the linear elastic behaviour to the non linear elastic behaviour is called the elastic threshold or threshold of non linearity (e.g. Vucetic, 1993) (Fig. 2-14). During larger strain amplitude cycles, the grain structure continuously deteriorates, leading to a reduction of the soil resistance during each cycle. It results in a reduction of the secant shear modulus and of the damping ratio as a function of the number of cycles. The shear strain amplitude corresponding to the transition from the non degrading behaviour to the degrading behaviour is called the degradation threshold or threshold of degradation (Fig. 2-14). Fig. 2-15: Comparison of intergranular behaviours

28 2-28 Chap. 2. Vibratory Driving Technique and Induced Soil Behaviour The differences between the soil behaviours can be explained at the granular scale as illustrated on Fig When the shear strain amplitude is very low, the linear elastic behaviour results of an elastic deformation at the grain interfaces. No plastic zone and no permanent slipping are observed between the grains. When the shear strain amplitude increases (higher than the elastic threshold but smaller than the degradation threshold, i.e. non linear elastic domain), some plastic zones are generated around the centre of the grain contact area but no permanent slipping between the axis of the grains is observed when the soil is unloaded. For strain amplitudes higher than the degradation threshold (elasto-plastic domain), the deformations induce a plastification of all the grain contact area resulting in a permanent slipping of the grains relatively to each other. Previously, the reduction of the mobilised secant shear modulus Gs/G max was treated separately for coarse and fine grained soil (see Seed and Idriss 1970). However, many researches (Zen 1978, Kokusho 1982, Dobry and Vucetic 1987, Sun 1988) comparing different types of soil, observed a gradual transition between the modulus reduction of non plastic coarse grained soil and plastic fine grained soil. It was shown that the shape of the modulus reduction is similar to the shape presented on Fig for each types of soil, and that the position of the shape is influenced more by the plasticity index PI (Fig. 2-16) and the effective mean stress P (Fig. 2-17), than by the void ratio e. The curve of modulus reduction curve corresponding to a plasticity index PI equal to 0 (PI=0) was also found similar to the average curve proposed by Seed and Idriss (1970) for sands. (a) (b) Fig. 2-16: Influence of plasticity index PI on (a) the modulus reduction curve and (b) the damping ratio (Vucetic and Dobry 1991) The analysis of Fig and Fig shows the main parameter influencing the secant shear modulus reduction is the plasticity index PI. For sandy soil, the reduction occurs for smaller strain amplitudes than for high plastic clayed soil. The elastic threshold and the degradation threshold are also increasing with the plasticity