Cavity Expansion Analysis to Predict Side Shear Set-Up in Clayey Soils

In this study, fifty driven piles located in the Santos Coastal Plain (“Baixada Santista”), Brazil, have been dynamically tested at various times after installation, indicating gain of capacity over time. The average side shear resistance, related to a low-OCR clayey layer – known as SFL soil , was evaluated over time, indicating side shear set-up within a relatively narrow range of values (between 2.0 and 2.5) approximately 20 days after installation. The cavity expansion theory, which considers an ideal cohesive soil, has been used to predict the effective radial stress over time, considering two main parameters: undrained shear strength and horizontal coefficient of consolidation. Finally, the average of measured pile shaft capacities were compared to the unit shear resistances predicted by the Method (effective stress method). The theoretical values of side shear set-up are fairly similar to the results of the analyzed load tests, which seems to support the viability of the presented method.


Introduction
In the design of deep foundations, there are still some poorly understood phenomena, which are not therefore properly considered when calculating foundations, such as time-dependent effects that alter the load capacity of driven piles over time.In some cases, it is possible to observe a decrease in pile resistance after driving, known as relaxation, and in other cases the opposite effect is noticed, causing an increase in soil resistance over time, known as set-up or freeze.Its incorporation into pile design can offer substantial benefits, resulting in significant economic gain.
A reliable prediction of the set-up is only possible when pile tests are carried out beforehand in order to verify capacity gain over time.The tests must be performed for many days, while set-up occurs, generating an increase in terms of both cost and time for the construction.Consequently, a reliable and cost-effective method of predicting the long-term set-up magnitude based on numerical analyses would be very advantageous.The purpose of this paper is to evaluate the ability of the Cavity Expansion Analysis to predict the side shear set-up of driven piles in clayey soils.
This paper presents a brief summary of pile set-up and some methods for predicting the set-up phenomenon.In addition, it describes a method for predicting pile set-up using the Cavity Expansion Analysis and presents a Brazilian case, comparing load tests conducted on 50 piles and the results of numerical analyses.
Studies by Komurka et al. (2003) indicate that set-up is directly related to shaft resistance, while Chow et al. (1998) indicate that the loss of pile capacity (relaxation) is more influenced by toe resistance.Therefore, it can be concluded that friction piles have greater set-up than endbearing piles.
Two concepts are used to define set-up, as described below: • set-up: obtained by the ratio between total load capacity after some time and load capacity at the end of driving; • side shear set-up: obtained by the ratio between shaft resistance after some time and shaft resistance at the end of driving.
Set-up values can vary widely due to the characteristics of each soil and pile material.However, it is observed that values above 2 are relatively common and maximum values are around 10. Long et al. (1999) analyzed the results of 80 load tests conducted on piles comprised of several materials driven in different soil types (sand, silt and clay).It was observed that set-up can occur in all cases, although this phenomenon is more pronounced in clayey soils.Bilfinger (2010) states that set-up mechanisms can be divided into two main groups, i.e., the mechanisms associated with pore pressure variations and those that are related to different phenomena, such as aging and creep.
In the case of clayey soils, as a pile is driven, the soil displacement along the shaft is predominantly radial (Komurka et al., 2003).Randolph. & Wroth (1979) state that, in clay, the soil around the shaft is remolded up to 20 radius from the pile axis and significant excess pore pressure is generated, thus causing a reduction in the effective stress and, consequently, facilitating the driving process.At the end of driving, the excess pore water pressure begins to dissipate, allowing the soil around the pile to reconsolidate.During reconsolidation, the soil undergoes a gradual increase in resistance, hence pile load capacity increases.
For soils with low OCR, at the end of driving, the excess pore pressure generated in the soil-pile interface is between 3 and 6 times the undrained shear resistance of the soil (Randolph. & Wroth, 1979).However, in the case of soils with high OCR, the excess pore pressure can be null or negative (Coop & Wroth, 1989;Bond & Jardine, 1995).The time to dissipate the excess pore pressure is proportional to the square of the radius of the pile and inversely proportional to the horizontal coefficient of consolidation of the soil (Soderberg, 1961).Consequently, piles with a larger diameter take longer to set up than the ones with a smaller diameter (Long et al., 1999) and excess pore water pressure dissipates slower for a group of piles than for a single pile (Camp & Parmar, 1999).
At the end of driving, total radial stress around the pile is higher than the initial horizontal stress, having been observed values between 6 and 11 times the undrained shear resistance (Earth Technology Corporation, 1986;Lehane & Jardine, 1994;Paikowsky & Hart, 2000).These studies also show reductions in total radial stress during consolidation, indicating that the total radial stress after consolidation is about 40% lower than the total stress observed at the end of driving.
The most popular methods to predict set-up in driven piles are actually based on empirical methods, correlations with in-situ tests or previous studies on test piles.

Empirical methods to predict pile set-up
Most of the studies regarding set-up are based on the results of load tests used to formulate methods for predicting the gain of load capacity as a function of time.Typical set-up values for different soil types were presented by Rausche et al. (1996).For instance, for clays these authors recommends set-up of 2.0, for silty 1.5 and for sand 1.0.
The most popular relation between load capacity and time was initially presented by Skov & Denver (1988), according to whom the increase in the total resistance of the pile is considered proportional to the log of time, as shown in Eq. 1.
where Q t = axial capacity at time t after driving, Q 0 = axial capacity at time t 0 , A = constant, depending on the soil type and t 0 = an empirical initial time value.Komurka et al. (2003) presented pile load capacity as a function of time divided in 3 phases.During phase 1, because of the highly disturbed state of the soil, the rate of variation in load capacity is not constant with the logarithm of time, taking place a short period after driving (see Fig. 2).Phase 2 begins when the dissipation rate of excess pore water pressure becomes constant (linear) with the log of time, occurring until the total dissipation of excess pore pressure.Phase 3 is set after primary consolidation ends.In this phase, set-up rate is independent of effective stress when load capacity gain occurs due to aging.During this phase, the increase in resistance remains linear with the logarithm of time, but with lower rates than the ones observed in phase 2. It is important to note that t 0 refers to the time when set-up becomes linear with the logarithm of time.Komurka et al. (2003) state that the duration of the logarithmically nonlinear dissipation rate of excess pore water pressure depends on the properties of both the soil (type, permeability and sensitivity) and the pile (type, permeability and size).
The t 0 values recommended in several studies are presented according to soil type and pile characteristics, varying between 0.01 day (Long et al., 1999;Svinkin & Skov, 2000), 1.0 day (Axelsson, 1988;Bullock, 1999;McVay et al., 1999) and 2.0 days (Camp & Parmar, 1999).Studies indicate that the larger the diameter of the pile, the greater the value of t 0 (Camp & Parmar, 1999).Paikowsky et al. (1996), when evaluating the results of dynamic load tests in different piles, concluded that parameters "A" and t 0 must be obtained for each situation, considering that both depend on the characteristics of the soil as well as those of the pile.In practice, parameter "A" either can be predicted using field tests or can be obtained empirically.Table 1 presents some values for parameter "A" and the corresponding t 0 , obtained in several studies.

Methods for predicting set-up with in-situ tests
For an exploration-phase field test to be valuable for evaluating set-up, the test must have a significant side shear component, as well as the ability to separate side shear from end bearing (Bullock, 1999).Several studies have been conducted looking for correlations between geotechnical in-situ tests and set-up, including torque measurement in SPT tests (Bullock & Schmertmann, 2003), analyses of torque tests on driven rods (Axelsson, 1998;Axelsson & Westin, 2000) and uplift measurement in SPT tests (Rausche et al., 1996), among others.Bullock & Schmertmann (2003) evaluated several results of standard penetration test with torque tests (SPT-T test) performed in different soil types, divided into two groups (clay and sand).During the test, both torque and rotation angle were recorded.The tests can measure both peak and residual torque and can be performed at various times after driving.The results show that in clayey soils the maximum torque tends to increase over time, while in sandy soils the maximum torque does not present considerable variations over time.
Similar studies conducted with the SPT sampler were developed by Axelsson (1998) and Axelsson & Westin (2000) in noncohesive soils.During the studies, torque measures applied to several small-diameter rods in different periods of time after driving were recorded.The tests with small-diameter rods have indicated an increase in resistance of approximately 30%, while static load tests performed on piles embedded into the same site indicated an increase in resistance of approximately 40% in comparison with the initial resistance.Axelsson (2002) found that with staged testing the increase in peak torque after some time is considerably higher than the increase in residual torque.Rausche et al. (1996) evaluated SPT-T and uplift measurement in SPT tests in order to predict "damping" and "quake" parameters for dynamic analyses on piles.Uplift tests carried out 10, 25 and 70 min after the end of pile driving indicate that the development of uplift strength is linear with the logarithm of time, as well as with pile shaft resistance.The values obtained for the uplift tests were equivalent to approximately 80% of the maximum torque.

Test piles to predict set-up
The results of dynamic and/or static load tests conducted in different periods after the end of driving on test piles are currently the most effective way to predict set-up due to the uncertainties surrounding theoretical and empirical methods.
The procedure proposed by Bullock (2008) suggests performing dynamic load tests at the end of pile driving and some time after that, in order to calibrate resistance variation over time.By considering this method, it is important to determine the exact time when the tests are carried out, as it is known that in the early hours after driving variation in pile capacity is more significant.According to Bullock (2008), it is possible to predict the set-up factor by performing dynamic load tests 15 min, 60 min and 1 day after the end of driving, with the exception of sandy soils, in which larger periods are required to observe the effects of aging.However, the same author recommends performing load

Numerical Simulation Procedure
In the present study, set-up is evaluated by considering the cylindrical cavity expansion theory, since field studies indicate that soil displacements around the pile during pile driving can be considered exclusively radial (Komurka et al., 2003).
Isolated piles are usually evaluated using axisymmetric models, according to which structures are circular with symmetrical cross-section and uniform loads around the central axis (y-axis), where deformation and stress are equal in all radial directions.Fig. 3 presents a schematic example of an axisymmetric model, in which the x-axis represents the radial direction and the y-axis corresponds to the axial line of symmetry.
In applications such as driven pile modeling, the driving process is usually simulated using the cylindrical cavity expansion theory with the initial radius equal to zero.In practice, the numerical analysis must begin with a cylindrical cavity with radius different from zero to avoid infinite stress that would appear in case an initial cavity with null radius was considered.Carter et al. (1979) defined some relations between the radius of the cylindrical cavity (model) and that of the pile, leading to satisfactory results when comparing field data to numerical analyses.The study suggests that to simulate pile driving it is necessary to consider a cavity expansion model with initial radius (a 0 ) defined by: a r where a 0 : initial radius of the cylindrical cavity and r 0 : pile radius.
The final radius (a f ) is equivalent to 2 times a 0 .This relationship is indicated in Fig. 4, which schematically presents the cavity radius considered in the model, comparing it to the pile radius (r 0 ).It is important to note that the initial and final cavity radii are equivalent to 0.58 and 1.15 time r 0 , respectively.
Figure 5 shows the cylindrical cavity expansion model adopted in the numerical analysis developed on the Plaxis software.It is important to highlight that the analyses were performed with a unit height model, in order to speed up calculations.The model parameters were thus defined considering a "slice" from the center of the soil layer analyzed; therefore, the results are average values.
Using the numerical analysis, it is possible to predict the excess pore pressure and radial stress around the pile shaft over time.By considering the stress obtained in the numerical analysis, the variation in unit shaft resistance (t l ) can be predicted with the b method (Burland, 1973), also known as effective stress method, defined by: where t l : unit shaft resistance, s' r : radial effective stress on the pile shaft and d: friction angle between pile and soil.
It is important to note that effective radial stress, s' r , is proportional to the initial effective vertical stress, s' vo , defined by the equation: It is also important to highlight that coefficient K, representing the ratio between radial stress and the initial vertical stress, is different from K 0 , which represents the in situ horizontal stress ratio, especially in the case of driven piles whose horizontal stress acting on the pile shaft is usually higher than the initial horizontal stress of the soil.

Subsurface conditions
The Santos Coastal Plain is characterized by a succession of sedimentary layers extending between the mountains and the ocean.These sediments were deposited during the last 100,000 years at different sedimentation cycles and with occasional erosion.Studies indicate that during periods of glaciation, there was significant drawdown in the water level (about 130 m), influencing the origin and history of the stress of the soil deposited in this area, causing an increase in effective stress and drying the soil layers closer to the surface.After the end of glaciation, sea level rose again, experiencing new sedimentation cycles (Massad, 2009).Another important phenomenon that influenced soil characteristics in the region are temporary depositions of high sand dunes.
Soil investigation consisted in several SPTs, 2 CPTUs and 7 Vane Tests.The main layers of sedimentary soil, composed predominantly of clayey materials and deposited in different periods, are described below: • Mangrove (0 to -5 m): surface layer, with null N SPT values, which is still in the formative phase.• Fluvio-Lagoon Sediments (SFL) (-5 to -25 m): deposited after the sea level rose again, with small OCR (between 1.0 and 3.0) and N SPT values around 0 to 5 blows/30 cm.• Transitional Clays (AT) (-25 to -40 m): materials of geological origin characterized by high OCR due to the increase in effective stress as a consequence of sea level drawdown, with N SPT values ranging from 5 to 25 blows/30 cm.Interspersed with the argillaceous materials there are sandy lenses and, under these horizons, it is usual to find residual soil, up to 10 m thick, resting on the rock mass.
Figure 6 presents the undrained shear resistance (s u ) vs. depth, obtained in 2 CPTUs and 7 Vane Tests performed in the area.CPTU values were obtained by the known ex-pression derived from the theory of cylindrical cavity expansion: where N KT is an empirical factor usually varying in the range of 10-13 in the Santos Coastal Plain (Massad, 2009).In this paper, a value of 11.5 was adopted in consonance with VT results (see Fig. 6).
Six CPTU pore pressure dissipation tests, shown in Fig. 7, were carried out at different depths, allowing to predict the horizontal coefficient of consolidation (c h ).Considering the method by Houlsby and Teh (1988), the values obtained for c h were between 5.10 -3 and 50.10 -3 cm 2 /s.The results indicate a value of excess pore water pressure (Du) generated on the cone shaft at the end of driving between approximately 4 and 6 times s u .
The cavity expansion analysis was carried out considering exclusively the SFL layer (-5 to -25 m), with Mohr-Coulomb model (linear elastic perfectly plastic behavior) and Hardening Soil Model, that accounts for stress-dependency of stiffness moduli.The average geotechnical parameters, around the -15.0 m level, are presented in Table 2 and Table 3.

Load tests
To evaluate set-up, 62 dynamic load tests with increasing energy were carried out in 50 piles, shown in Fig. 8, with an 80 cm diameter and driven lengths ranging from 28 to 57 m.All load tests were performed with a Juntan HHK 16A hammer, weighing 160 kN, with increasing falling heights ranging from 0.20 to 1.20 m.
Figure 9 presents the maximum load capacities, obtained through CAPWAP analyses considering the highest energy blow of each pile and performed in some cases at the end of driving (EOD) and in other cases after some time elapsed (1 to 87 days).
The dynamic load tests, performed at different intervals of time from the end of driving until up to 87 days, resulted in total resistance ranging from 3,570 to 11,030 kN.A tendency of resistance gaining over time can be noticed.However, the relation between resistance and time is not well defined.
Figure 10 shows the average unit shaft resistance (t l ) of the SFL layer found in the CAPWAP analysis between -5.0 and -25.0 m deep.It is possible to note that the SFL material is the main focus of the analysis, since this layer is composed predominantly of clayey material with low OCR values, which means that its set-up should be more significant than that of sandy layers (sand lenses and residual soil) or clayey soils ("AT" soil) with high OCR.
In this case, the average unit shaft resistance at the end of driving resulted in a range of values from 0 to 12 kPa, while results obtained after some time reached up to 42 kPa.It is important to note that the results from Pier 2 indicated lower values than the results from other areas, despite the fact that the geotechnical investigations did not point to any significant variation in the soil in this location.It is thus possible to assume that some variation may have occurred during driving or even when processing the CAPWAP analysis, resulting in lower capacities than expected.Therefore, for theoretical analyses, data obtained in this area (Pier 2) will be disregarded.

Numerical Simulation Results
Numerical simulation analyses were carried out, considering Mohr-Coulomb and Hardening-Soil models, to predict variation in pore water pressure and radial stress around an 80 cm prestressed pile over time.It is important Table 2 -Geotechnical parameters of the SFL soil (-15.0 m) -Moh-Coulomb.to say that the results obtained with Hardening-soil model will not be presented, as they were almost the same as the results considering the Mohr-Coulomb model.The cylindrical cavity expansion model is shown in Fig. 5.

Excess pore pressure (Du)
Variations in pore water pressure, resulting from cylindrical cavity expansion adopting Mohr-Coulomb model, are presented as a function of time in Figs.11 and 12.It is worth remembering that numerical analyses were performed considering s u values between 40 kPa (Fig. 11) and 60 kPa (Fig. 12) and c h values between 5.10 -3 and 50.10 -3 cm/s 2 .
The results indicate maximum values of excess pore water pressure (Du) generated in the soil-pile interface around 180 and 240 kPa for soils with undrained shear resistance (s u ) of 40 and 60 kPa, respectively.These results are equivalent to approximately 4.0 to 4.5 times s u , in consonance with the CPTU dissipation tests (see Fig. 7).

Total radial stress (s r )
As previously mentioned, pile driving generates excess in both pore water pressure and total radial stress around the pile.Variations in total radial stress around the pile, due to cavity expansion, are presented in Figs. 13 (s u = 40 kPa) and 14 (s u = 60 kPa).
The results indicate an increase in total radial stress (s r ) between 220 and 300 kPa at the end of driving.After full consolidation, total radial stress is reduced to about 75% of the radial stress at the end of driving, ranging from 170 to 240 kPa.Both ranges are for soils with undrained shear resistance (s u ) equal to 40 and 60 kPa, respectively.
It is worth noting that at the end of driving, predicted s r is between 2.0 and 2.3 times the initial radial stress (s ro ), and after the total stress reduction predicted s r varies between 1.8 and 2.1 times s ro .

Effective radial stress (s' r )
Figures 15 and 16 present the effective radial stress acting on the pile shaft over time for undrained shear resistance of 40 kPa (Fig. 15) and 60 kPa (Fig. 16).
During the analyzed period, between the end of driving and after full consolidation, an increase in effective radial stress of about 120 and 170 kPa was observed for undrained shear resistance of 40 and 60 kPa, respectively.
The ratio between effective radial stress after cavity expansion (s' r ) and effective vertical stress at rest (s' vo ) is defined by:    The values of K shortly after driving vary between 1.6 and 2.0, and after full consolidation these values are between 3.2 and 4.1.For the numerical analysis, an average value of 3.6 was considered.In these computations a value of K 0 = 0,86 was assumed, as shown in Table 2.

Evaluation of Side Shear Set-Up
To evaluate the ultimate value of the unit shaft resistance (t l,ult ), the following expression was used: According to Burland (1973) the proportionality between s' r and s' v0 is a simplifying assumption and, in his words, "it represents a simple and logical starting point".
Parameter d, representing the friction angle between the pile and the soil, depends on the characteristics of both the soil and the pile material and can be set through laboratory tests, although most of the studies on this topic are related to sandy soils, so little information on pile driving in clay is available.
The product K.tan(d) was defined by Burland (1973) as coefficient b, resulting in: Several studies indicate values of b ranging from 0.25 to 0.30 for driven piles in clays with low OCR.Thus, considering K = 3.6 and b = 0.3 leads to tan(d) equals approximately to 0.08.Note that this figure implies a value of d roughly equals to 5º, which may seem low.However, Atkinson (1993) cites that d lies generally between the residual friction angle (f' r ) and the peak friction angle (f' p ). Laboratory tests performed on SFL clays in the overconsolidated range of stress lead to f' p from 6°to 8° ( Massad, 2016), with a non zero cohesion.The same material, when tested above preconsolidation stress, results in f' p around 15°and 25°with zero cohesion (Massad, 2009).In this context and taking into account the sensitivity of the SFL clays, ranging from 3 to 5, indicating a low value for f' r , it can be assumed that d = 5º is within the range of expected values.
Taking into account effective radial stress as a function of time (Figs.15 and 16), obtained using numerical modeling, and considering tan(d) equals to 0.08 in Eq. 7, the values predicted for unit shaft resistance as a function of time are shown in Fig. 17.
The average unit shaft resistance acting on the SFL layer varies between 10.7 and 13.1 kPa at the end of driving and 23.1 and 31.2 kPa after full consolidation, considering s u equal to 40 and 60 kPa, respectively.These results are consistent with static load tests performed on floating piles around this area, according to data presented by Massad (2009).
Despite of the scattering of t l , there is a trend of results within a relatively narrow range of values.In addition,    side shear set-up ranges from 2.0 to 2.5 approximately 20 days after driving.
It is important to note that for the 80 cm prestressed piles, the time for 90% of the total set-up is approximately 6 and 70 days, related to horizontal coefficients of consolidation (c h ) of 50.10 -3 and 5.10 -3 cm 2 /s, respectively.The results of the numerical analyses indicate that after 1 day of pile driving, about 50% to 70% of the total pile capacity gain predicted was observed.
Finally, it is possible to conclude that the methodology proposed to predict side shear set-up by considering cavity expansion analyses and b method is in consonance with field results obtained from CAPWAP analyses.

Conclusions
Literature review indicated that: • increases in pile load capacity over time can occur in any kind of soil or pile material, although pile set-up is more pronounced in normally consolidated clays; • set-up is more influenced by shaft capacity than by toe resistance; and • changes in pore water pressure and total radial stress after the end of driving are the main mechanisms responsible for set-up in clayey soils.
The unit shear resistance of the case history SFL layer obtained from the dynamic load tests ranged from 0 to 12 kPa at the end of driving (t = 0 day) and a maximum of 42 kPa a certain time after driving.
A method based on the cavity expansion theory has been presented to predict side shear set-up of driven piles.Based on the comparisons between observed and predicted results for side shear set-up in low-OCR clay soils of the case history, known as SFL clay, the following conclusions can be drawn: Predictions of unit shaft friction using the cavity expansion theory and application of effective stress analyses (b method) indicate that: • the average unit side friction acting on the SFL layer varies between 10.7 and 13.1 kPa at the end of driving and between 23.1 and 31.2 kPa after full consolidation.The latter figures are in consonance with the results of dy-namic load tests and consistent with static load tests performed on floating piles in Santos Coastal Plain; • the predicted side shear set-up found in cavity expansion analyses varies between 2.0 and 2.5 for the SFL layer, in consonance with the results of dynamic load tests; • the time required for 90% of the set-up to take place is approximately equal to 6 and 70 days for c h between 5.10 -3 and 50.10 -3 cm 2 /s, respectively; and • the numerical analysis indicated that one day after the end of driving, about 50% and 70% of the total set-up predicted occurred, for values of c h between 5.10 -3 and 50.10 -3 cm 2 /s, respectively.
The applied method proved a valuable tool for estimating pile set-up as it is cost-and time-efficient.Moreover, the results showed good correlation between the side shear set-up evaluated through cavity expansion analyses and load tests on driven piles.
Figure 2 -Set-up development with the logarithm of time (Komurka et al., 2003).

Figure 4 -
Figure 4 -Cylindrical cavity expansion model for evaluating driven piles.
The test site is located in the Brazilian Southeast coast, specifically in the Santos Coastal Plain, around 80 km east of the city of São Paulo.The tests were conducted during the construction of a port terminal, comprising a pier around 1,100 m long, supported by approximately 2,300 prestressed concrete piles, with external diameter of 80 cm and wall thickness of 15 cm.

Figure 5 -
Figure5-Numerical model adopted to simulate cavity expansion in an axisymmetric one-dimensional model.

Figure 7
Figure 7 -CPTU pore pressure dissipation tests, normalized by s u .

Figure 13 -
Figure 13 -Total radial stress (s r ) from cavity expansion analyses s u = 40 kPa.

Figure 14 -
Figure 14 -Total radial stress (s r ) from cavity expansion analyses s u = 60 kPa.

Figure 17 -
Figure 17 -Unit shaft resistance for the SFL layer over time from CAPWAP and numerical analyses.

Table 1 -
Summary of time t 0 and parameter "A" obtained in several studies.
Thorburn, S. & Rigden, W.J.(1980).A practical study of pile behavior.Proc.12thAnnualOffshore Technology Conf.,Houston.Titi, H.H. & Wathugala, G.W. (1999).Numerical procedure for predicting pile capacity -setup/freeze.Transportation Research Record 1663, paper n. 99-0942, pp.25-32.Yang, N.C.(1956).Redriving characteristics of piles.Journal of the Soil Mechanics and Foundations Division, v. 82, paper n. 1026, SM 3, ASCE.Constant, depending on the soil type a o : Initial radius of the cylindrical cavity c h : Horizontal coefficient of consolidation d: Friction angle between pile and soil f' r : Soil residual friction angle f' p : Soil peak friction angle Du: Excess porewater pressure E: Young's modulus g: Unit weight K: Ratio between radial stress and the initial vertical stress K 0 : In situ horizontal stress ratio N KT : Factor relating corrected cone end resistance and undrained shear strength n: Poisson's ratio q t : Cone resistance Q 0 : Axial capacity at time t 0 Q initial : Axial capacity at the end of driving Q t : Axial capacity at time t after driving r 0 : Pile radius s' r : Effective radial stress on the pile shaft s r : Total radial stress s' ro : Effective radial stress at rest, before driving s ro : Total radial stress at rest, before driving s u : Undrained shear resistance s v0 : Initial total vertical stress s' vo : Initial effective vertical stress t 0 : Empirical initial time value t l,ult : Ultimate unit shaft resistance t A: l : Unit shaft resistance