Analysis of pile behaviour in liquefying sloping ground

Computers and Geotechnics 37 (2010) 115–124

Contents lists available at ScienceDirect

Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo

Analysis of pile behaviour in liquefying sloping ground D.S. Liyanapathirana a,*, H.G. Poulos b a b

School of Engineering, University of Western Sydney, Locked Bag 1797, Penrith South DC, NSW 1797, Australia Coffey Geotechnics, 8/12 Mars Road, Lane Cove West, NSW 2066, Australia

a r t i c l e

i n f o

Article history: Received 11 September 2008 Received in revised form 23 July 2009 Accepted 3 August 2009 Available online 29 August 2009 Keywords: Pile foundations Soil liquefaction Earthquake loading Lateral spreading Sloping ground

a b s t r a c t In this paper, a numerical procedure based on the finite element method is outlined to investigate pile behaviour in sloping ground, which involves two main steps. First a free-field ground response analysis is carried out using an effective stress based stress path model to obtain the ground displacements, and the degraded soil stiffness and strength over the depth of the soil deposit. Next a dynamic analysis is carried out for the pile. The interaction coefficients and ultimate lateral pressure of soil at the pile–soil interface are calculated using degraded soil stiffness and strength due to build-up of pore pressures, and the soil in the far field is represented by the displacements calculated from the free-field ground response analysis. Pore pressure generation and liquefaction strength of the soil predicted by the stress path model used in the free-field ground response analysis are compared with a series of simple shear tests performed on loose sand with and without an initial static shear stress simulating sloping and level ground conditions, respectively. Also the numerical procedure utilised for the analysis of pile behaviour has been verified using centrifuge data, where soil liquefaction has been observed in laterally spreading sloping ground. It is demonstrated that the new method gives good estimate of pile behaviour, despite its relative simplicity. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction In sloping ground, before the occurrence of seismic loading, the ground is subjected to static shear stresses due to the weight of the soil and the slope of the ground. These stresses will act as driving forces and may cause very large ground deformations, known as lateral spreading, even before the onset of soil liquefaction, which in turn may cause severe damages to foundations. Reliable prediction of ground deformations due to lateral spreading is essential in predicting pile behaviour in liquefying sloping ground. Numerical simulation of pile behaviour in liquefying sloping ground under earthquake loading is a complex problem. The loss of soil stiffness and strength due to excess pore pressure generation during earthquake loading may develop large bending moments and shear forces in piles, leading to pile damage. The major earthquakes that have occurred during past years such as the 1964 Niigata, 1989 Loma-Prieta and 1995 Hyogoken-Nambu have clearly demonstrated the significance of soil liquefaction-related damage to pile foundations. Still there are uncertainties in the mechanisms involved in pile–soil-structure interaction in liquefying soil. Hence, there is a great demand for validated numerical procedures to predict pile behaviour in liquefying soil.

* Corresponding author. Tel.: +61 2 4736 0653. E-mail address: [email protected] (D.S. Liyanapathirana). 0266-352X/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2009.08.001

There are two main approaches available to evaluate the amount of pore pressure generation in saturated ground during an earthquake, effective stress methods and total stress methods. The deficiency of a total stress analysis is that it calculates pore pressures based on the total stresses developed in the ground during the earthquake loading without directly considering the effective stresses. There is no way to evaluate the progressive degradation of soil stiffness and strength during a total stress analysis. In general, the amount of resistance provided by the soil against deformation at any point in the soil deposit depends on the effective stress level at that point. Therefore, it is important to evaluate pore pressures using an effective stress based analysis. Subsequently ground deformations and pile behaviour can be evaluated using the degraded soil stiffness and strength due to increase in pore pressures during earthquake loading. The numerical procedure adopted in this paper is based on an effective stress approach and involves two main steps. In the first step a free-field ground response analysis is carried out in order to obtain degraded soil stiffness and strength due to pore pressure generation, and the ground deformations using the stress path model proposed by Ishihara and Towhata [13]. In the second step a dynamic finite element analysis has been used to investigate the pile behaviour. On level ground, effective stresses will reduce to near zero at the onset of liquefaction and any residual displacements will be small in the absence of driving stresses acting on the ground.

116

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

However, in sloping ground, the situation is different. A soil element near the surface of a slope is subjected to a driving static shear stress. Hence, with the increase in pore pressure, ground deformations can be very large due to these driving shear stresses but the effective stress may not reach near zero at the onset of liquefaction. The application of the stress path model for the sloping ground has been verified using the cyclic simple shear tests carried out at the University of British Columbia, Canada [21]. The dynamic finite element analysis used for the prediction of pile behaviour utilises a one-dimensional Winkler model, where the pile is modelled as a beam and the lateral soil pressure acting on the pile is modelled using a non-linear spring-dashpot model. For the seismic analysis of piles in liquefying soil, Winkler type models have been developed by Kagawa [17], Yao and Nogami [29], Fuji et al. [10]. Degraded p–y curves may be used in Winkler models to define the behaviour of non-linear springs at any depth, where p is the soil resistance per unit length of the pile and y is the pile lateral displacement. These p–y curves should be back-figured from field or model test data. However, for liquefying soil, available data are limited to back calculate p–y curves. Therefore in this paper, a method based on the Mindlin’s equation is presented to determine non-linear spring constants of the Winkler model [19,20]. Depending on the amount of pore pressure development, the spring coefficients in the spring-dashpot model are degraded while the effect of radiation damping is taken into account separately. The validity of the computer program developed by the authors is examined with reference to the pile behaviour observed during centrifuge tests carried out by Brandenberg et al. [3] in liquefying sloping ground subjected to lateral spreading. It will be demonstrated that the proposed method is accurate enough to predict the pile behaviour in liquefying soil for design purposes, despite its relative simplicity. 2. Numerical procedure The two steps involved in the one-dimensional numerical model developed for the analysis of pile foundations in liquefying sloping ground subjected to lateral spreading is described in this section. 2.1. Free-field ground response analysis The numerical model used for the free-field ground response analysis is one-dimensional and it is based on the finite element method. The soil deposit is divided into a number of layers and converted to a lumped mass system by concentrating one-half of the mass of each layer at the layer boundaries as shown in Fig. 1. The constitutive behaviour of the non-linear springs, which represent the soil, is modelled using a hyperbolic stress–strain relationship. They reflect the non-linear, strain-dependent and hysteretic behaviour of the soil. In addition to the hysteretic damping, 20% of critical damping is added to take into account the viscous damping of the soil [26]. Pore pressure generation during cyclic loading is evaluated using the stress path model proposed by Ishihara and Towhata [13]. Fig. 2 shows the stress path obtained from the numerical model for a typical case of sand subjected to a constant amplitude cyclic shear stress. The stress path from points 1–2 is assumed to be a parabola given by,

r0m ¼ m 

B0p 2 s m

ð1Þ

where m is a parameter to locate the parabolic stress path and B0p is a soil constant representing the pore pressure generation characteristics of the soil. For stress paths 1–2, m is the initial effective ver-

1 1 2

P2

2 3 i i

j

j

Pj

j+1

n n n+1 Fig. 1. Lumped mass representation of the discretised system.

tical stress. A stress path beyond the yield loci (e.g., 3–4, 5–6 and 7– 8) is also defined by the same equation but m should be computed each time the stress path crosses the yield loci (e.g., points 5 and 7). Within the current yield surface (e.g., 2–3, 4–5 and 6–7), the stress path is given by:

Dr0m ¼ B0u



s r0mo

 rs0m



mo

r0m r0mo

  j Ds

Dr0m ¼ 0

r0m P jr0mo r0m < jr0mo

ð2Þ

where B0u is a soil parameter representing pore pressure generation characteristics of the soil and j is a parameter which takes into account the fact that pore pressure build up ceases when the r0m decreases to jr0mo . At point ‘A’ the stress path crosses the phase transformation line, which has a gradient of 0:625 tan u0l , and beyond point ‘A’ for both loading and reloading, the stress path is given by:



r0m m

2

 

2

s

m tan /0l

¼1

ð3Þ

where /0l is the friction angle at very small effective confining stress. Eq. (3) approaches the failure line asymptotically as shown in Fig. 2. For unloading beyond point ‘A’, the stress path follows a straight line, which is tangential to the gradient of Eq. (3) at the point of stress reversal. During the analysis, pore pressure generation as well as pore pressure dissipation within the soil due to vertical drainage has been considered [25]. Based on the effective stress level of the soil, soil stiffness and strength are degraded. Hence the shear modulus and shear strength of the soil are time-dependent parameters. The shear modulus at any time is given by the following equation [15]:

ðGs Þt¼t ¼ ðGs Þt¼0

"



r0m t¼t r0v o

#0:4 ð4Þ

where Gs is the shear modulus of the soil and r0m is the effective stress at any time t. Subscripts t = 0 and t = t denote the initial values and values at any time t, respectively. For the results presented in this paper, soil is assumed to follow the Mohr–Coulomb failure criterion. Therefore the degraded shear strength of the soil is given by the following equation:

ðsf Þt¼t ¼



r0m

 t¼t

tan /0 þ C 0

ð5Þ 0

where sf is the shear strength of the soil, / is the friction angle and C0 is the cohesion of the soil. According to Eq. (4), shear modulus of the soil, Gs, is a function of r0m and theoretically, it is not possible to carry out an analysis

117

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

Phase transformation line φ’l θ’s

2

6

Yield loci

Shear stress

5

3

1

Effective vertical stress

Yield loci 8

Failure line

7

4

Fig. 2. Stress path computed for a cyclic simple shear test.

with zero vertical effective stress. Therefore, in our analysis, when pore pressure has reached 97% of the initial effective overburden pressure, pore pressure build-up is arrested. This assumption is reasonable because in normal ground the shear stress application is multi-directional. At the onset of liquefaction, there will be a shear stress applied on the soil in the direction perpendicular to the direction of shear stress causing soil liquefaction [13]. This is clearly demonstrated during the rotational simple shear tests performed by Ishihara and Yamasaki [14]. The last hyperbolic curve given by Eq. (3) intersects the r0m axis when r0m ¼ 0:03r0mo . Since there is no further decrease in r0m , the stress path is made to follow the same hyperbolic curve back and forth [13]. 2.2. Pile analysis Pile analysis is carried out based on the method proposed by Liyanapathirana and Poulos [19,20], where the interaction between the soil and the pile is modelled using the analysis method for a dynamically loaded beam on a non-linear Winkler foundation. In Winkler models, the lateral soil pressure acting on the pile is modelled using a spring-dashpot model as shown in Fig. 3 and the pile is modelled as a beam. The partial differential equation for a beam on Winkler foundation is given by,

EP I p

@4 Up @z4

! þ Mp

@2 Up @t2

!

  @U ff @U p ¼ KxðU ff  U p Þ þ C x  @t @t

U x¼0 ¼

P f ðc1 ; c2 ; b; y; z; mÞ G

ð6Þ

U ij ¼ P j F ij

Pile

Fig. 3. Beam on Winkler foundation model for dynamic pile analysis.

ð8Þ

where Fij is the influence coefficient determined from Eq. (6). Since the analysis is based on the finite element method, nodal displacements should be calculated along the pile. The displacement at node i due to loading over the pile is given by:

ð9Þ

This can be written in matrix form as:

fUg ¼ ½FfPg

Superstructure

ð7Þ

where P is the uniformly distributed load over the rectangular area, G is the shear modulus of the soil and m is the Poisson’s ratio of the soil. Here y extends from d/2 to +d/2 where d is the diameter of the pile. If Uij represents the displacement at the centroid of the ith rectangle due to a uniform pressure Pj distributed over the jth rectangle as shown in Fig. 1, then we can write:

U i ¼ P1 F i1 þ P2 F i2 þ    þ Pj F ij þ    þ Pn F in

where Ep is the Young’s modulus of the pile material, Ip is the inertia of the pile, Up is the pile displacement, Uff is the free-field lateral soil displacement, Mp is the mass of the pile, and Kx and Cx are the spring and dashpot coefficients of the Winkler model. A solution to the

Ground displacement profile from the site response analysis (at each time step)

problem can be obtained by solving Eq. (6) using the finite element method. Time integration of Eq. (6) is performed using the constant average acceleration method. To obtain the spring coefficients of the Winkler model, the integration of the Mindlin’s equation over a rectangular area in the yz plane extending from y = b/2 to +b/2 and z = c2 to c1 given by Douglas and Davis [9] is used. Then the displacement at any point in the rectangular area, b(c2  c1), is given by

ð10Þ

where F is the flexibility matrix of influence coefficients. F is calculated at y = d/2, 0 and +d/2, and the average value is obtained at the centre of each loaded area, i.e. at each pile node. By inverting F, spring coefficients of the Winkler model can be obtained. During the pile analysis, F is computed using the degraded shear modulus obtained from the free-field ground response analysis. Therefore, whenever the pore pressure level in the soil changes, F should be computed and inverted to obtain the stiffness matrix, which represents the interaction between the pile and the soil away from the pile. The Mindlin hypothesis does not include the soil radiation damping and this should be incorporated into the analysis separately. Kagawa and Kraft [18] modelled pile radiation damping using the expression given by Berger et al. [1]. They assumed that a horizontally moving pile cross section generates one-dimensional dilation waves in the direction of shaking and shear waves in the direction perpendicular to shaking, and derived the following expression to compute the radiation damping per unit length of the pile:

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

  Vp cr ¼ 2dqs V s 1 þ Vs

ð11Þ

where d is the pile diameter, qs is the density, Vs is the shear wave velocity and Vp is the dilation wave velocity of the soil. If Vp is replaced with Vs, Eq. (11) gives 4qsVsd as the radiation damping, which is used in the pile computer program PAR [22]. Gazetas and Dobry [11] derived an expression for radiation damping coefficient applicable to a circular cross section with diameter d as follows:

"

cr ¼ 1þ 2dqs V s



3:4

4=5 # 3=4 p

pð1  mÞ

4

a1=4 o

ð12Þ

where ao is the non-dimensional frequency of the vibration and m is the Poisson’s ratio of the soil. Boulanger et al. [2] used the above expression to represent radiation damping in a Winkler model, where soil liquefaction effects are not taken into account. The disadvantage of Eq. (12) is that it is frequency dependent. Tabesh and Poulos [27] used a value of 5qsVs for the dashpot coefficient. According to Tabesh and Poulos [28], this value tends to underestimate the actual damping. All previous values of radiation damping are derived assuming that soil layers are elastic, horizontally infinite and nonliquefiable. The amount of radiation damping, which occurs in non-linear liquefiable soil is still unknown. Several researchers have shown that it is far less than the value obtained from the elastic assumption. Using centrifuge tests, Chacko [7] showed that the elastic formulation of radiation damping is valid only during small amplitude shaking. During the comparison of numerical results with centrifuge data it has been found that 5qsVs is extremely high for laterally spreading sloping ground conditions. Therefore, a reduced value of qsVs is used here for the dashpot coefficient, assuming that this dashpot has the ability to absorb shear waves travelling away from the pile, along each sloping layer of soil. During the analysis, dashpot coefficients have not been changed and Vs is calculated based on the initial shear modulus of the soil [20]. At each time step of integration, the lateral pressure at the soilpile interface is monitored. Since the amount of radiation damping during yielding is unknown, an iterative procedure has been used to keep only the contribution from the spring below the limiting value. This will be further discussed in Section 4.3. This method has the potential of predicting p–y behaviour, where p is the lateral soil resistance against the pile at a given depth and y is the lateral displacement of the pile relative to the far field ground displacement at the same depth. Due to the change in stiffness of the soil as a result of pore pressure increase, the p–y curve at a particular depth also changes with time and it is not possible to provide a generalised set of p–y curves as in the case of static load analysis of laterally loaded piles.

3. Numerical simulation of cyclic simple shear test data The ability of the numerical model in simulating soil liquefaction in sloping ground has been evaluated using a series of constant volume, simple shear tests performed on Fraser River sand at UBC (University of British Columbia, Canada). These test data are available at http://www.civil.ubc.ca/liquefaction. Numerical simulations based on the plastic constitutive model UBCSAND [6,24] are described by Park and Byrne [21]. When applying the stress path model presented here to simulate the constitutive behaviour of liquefying soil, four model parameters (h0s ; j; B0p and B0u ) need to be determined prior to the analysis. In this study, the phase transformation angle, h0s , and angle of internal friction, /0l , at very small effective confining stress,

existing in the sand at the stage near the onset of liquefaction, are related to the friction angle of the soil, /, as follows [13]:

tan /0l ¼ 1:4 tan / 5 tan h0s ¼ tan /0l 8

ð13aÞ ð13bÞ

The soil constant, j, is determined based on the cyclic shear strength curve and given by

j¼

sa 1 r0mo tan /0

ð14Þ

where sa =r0mo is the minimum cyclic stress ratio below which liquefaction does not occur. For all analyses carried out here for Dr = 44% Fraser River sand, j = 0.01 has been used. The third soil constant B0p is determined using Eq. (1). The effective stress level is determined from the pore pressure increase recorded in the first cycle during virgin loading in compression. The value of B0p should be a constant for a particular soil type with a particular relative density. In this study B0p of 8.2 has been used, based on the experimentally observed pore pressure increase in the first cycle during virgin loading in compression. The fourth parameter B0u governs the amount of pore pressure generation during unloading and reloading inside the current yield surface as given by Eq. (2). A value for B0u has been selected using the chart provided by Ishihara and Towhata [13] where B0p and B0u are related to the Cyclic Stress Ratio (CSR) required to obtain liquefaction after application of 20 stress cycles to the soil. Figs. 4 and 5 show, respectively, the variation of shear stress with effective stress level predicted by the numerical model and that observed during the experiments for a soil sample subjected to a CSR of 0.08 without any static shear stress. Fig. 6 shows the excess pore pressure generation predicted from the numerical model. When the effective stress of the soil is below 75 kPa, the experimental results show rapid softening of the soil. According to the numerical model, soil softens rapidly when the effective stress of the soil is below 50 kPa. Pore pressure generation predicted using the stress path method is slightly high after the 10th cycle. Fig. 7 shows the stress–strain behaviour of the soil sample during cyclic loading. The predicted shear strains are symmetrical around the y-axis but the experimental results are not symmetrical about the y-axis. Numerical predictions given by Park and Byrne [21] based on the UBCSAND constitutive model also shows symmetrical strains about the y-axis when there is no static shear stress acting on the soil and theoretically they should be symmetrical. However, the overall agreement between the predicted and measured

20

Shear stress (kPa)

118

10

0

0

50

100

150

200

-10

-20

Effective vertical stress (kPa) Fig. 4. Stress path predicted by numerical model (CSR = 0.08, a = 0.0).

119

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

Cyclic Stress Ratio (CSR)

20

Shear stress (kPa)

10

0

0

50

100

150

200

0.2

0.15

0.1

0.05

Prediction Experiment

0 1

-10

10

100

Number of cycles to liquefaction Fig. 8. Predicted and measured liquefaction response of Fraser River sand (a = 0.0).

-20

Effective vertical stress (kPa) 40

1.0

Shear stress (kPa)

Pore pressure ratio

Fig. 5. Stress path from the cyclic simple shear test (CSR = 0.08, a = 0.0).

0.8 0.6 0.4

Prediction

0.2

30

20

10

Experiment

0.0 0

5

10

15

20

25

30

35

Number of cycles

0 0

Fig. 6. Pore pressure increase during cyclic loading (CSR = 0.08, a = 0.0).

50

100

150

200

Effective vertical stress (kPa) Fig. 9. Stress path predicted by numerical model (CSR = 0.08, a = 0.1).

Shear Stress (kPa)

20

10

0 -2

-1

0 -10

1

2

Prediction Experiment

-20 Shear strain % Fig. 7. Stress–strain behaviour of soil (CSR = 0.08, a = 0.0).

stress–strain behaviour is good. The onset of soil liquefaction is predicted by the numerical model to occur after 35 cycles, and this agrees well with the simple shear test data. Fig. 8 shows the number of cycles required for liquefaction at different CSRs, when the static shear stress acting on the soil is zero. At cyclic stress ratios higher than 0.1, numerical prediction shows a slightly higher resistance to liquefaction compared to that observed during the cyclic simple shear test. The largest difference observed between the numerical model and the experiment is within ±10% and this is acceptable for design purposes. In sloping ground, before application of cyclic loads leading to pore pressure generation and subsequent soil liquefaction, the ground is subjected to static shear stress due to the weight of the soil. Park and Byrne [21] simulated this condition in cyclic simple shear tests, where tests were carried out with a static shear stress,

ss. The static shear stress ratio, a, is defined as the driving static shear stress to initial effective vertical stress, r0mo . Figs. 9 and 10 show the stress paths obtained from the numerical model and the experiment for CSR = 0.08 and a = 0.1. In both cases, liquefaction is initiated after four stress cycles. When this result is compared with Figs. 4 and 5, the influence of initial static shear stress on the soil liquefaction is clear. When there is no static shear stress, soil could sustain 35 stress cycles with CSR = 0.08 before liquefaction but when there is a static stress of 0:1  r0mo , soil can sustain only four stress cycles before liquefaction. Fig. 11 shows the predicted pore pressure generation and Fig. 12 shows the stress–strain behaviour of the soil. Figs. 13–16 show the stress paths, excess pore pressure generation and stress–strain behaviour for CSR = 0.1 and a = 0.1. The predicted variation of pore pressure ratios during the cyclic loading closely agrees with the experimental results. Experiment shows a rapid increase in shear strains with the onset of soil liquefaction but the numerical model shows a gradual increase in shear strains with the increasing pore pressures. When there is a static shear stress, there is no stress reversal and hence the accumulated shear strain became larger and larger irrespective of the pattern of the cyclic stress. Results given in this section clearly show that the stress path model has the ability to simulate the soil behaviour in both level and sloping ground. In the following sections, application of this method to simulate a centrifuge test will be described. 4. Comparison with centrifuge test results Here the validation of the numerical model is carried out using the centrifuge tests carried out by Brandenberg et al. [3] for pile groups founded on laterally spreading sloping ground. The soil

120

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

50

40

Shear stress (kPa)

Shear stress (kPa)

40 30

20

10

30

20

10

0

0 0

50

100

150

200

0

50

1.0

200

Pore pressure ratio

1.0

0.8 0.6 0.4

Prediction 0.2

0.8 0.6 0.4 Prediction

0.2

Experiment

Experiment 0.0

0.0 0

1

2

3

4

5

0

6

1

2

Fig. 11. Pore pressure increase during cyclic loading (CSR = 0.08, a = 0.1).

3

4

Number of cycles

Number of cycles

Fig. 15. Pore pressure increase during cyclic loading (CSR = 0.1, a = 0.1).

50

50 40

40

Shear Stress (kPa)

Shear Stress (kPa)

150

Fig. 14. Stress path predicted by numerical model (CSR = 0.1, a = 0.1).

Fig. 10. Stress path from the cyclic simple shear test (CSR = 0.08, a = 0.1).

Pore pressure ratio

100

Effective vertical stress (kPa)

Effective vertical stress (kPa)

30 20

Prediction 10

Experiment 0

30

20

Prediction 10

0

5

10

15

20

Experiment

Shear Strain % 0 Fig. 12. Stress–strain behaviour of soil (CSR = 0.08, a = 0.1).

0

5

10

15

20

Shear Strain % Fig. 16. Stress–strain behaviour of soil (CSR = 0.1, a = 0.1).

50

profile in the centrifuge model consisted of a 1.4 m thick layer of Monterey sand, overlying a 2.8 m thick deposit of heavily overconsolidated Bay Mud, overlying a 5.6 m thick layer of loose Nevada sand (Dr = 35%), overlying 18.2 m thick layer of dense Nevada sand (Dr = 75%) in prototype scale. All soil layers were built to a slope of 3°. Properties for Bay Mud and Nevada sand are given in Table 1.

Shear stress (kPa)

40

30

20

4.1. Calibration of the stress path model

10

0 0

50

100

150

200

Effective vertical stress (kPa) Fig. 13. Stress path predicted by numerical model (CSR = 0.1, a = 0.1).

During the centrifuge test, the model was subjected to the Santa Cruz motion scaled to a maximum acceleration level of 0.67 g. Fig. 17 shows the input acceleration record. The stress Path model has been calibrated using the cyclic shear strength data given by Popescu and Prevost [23] for loose Nevada sand. Fig. 18 shows

121

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124 Table 1 Soil parameter values used for the analysis. Monterey sand Soil constant Soil constant

Bay Mud

B0u B0p

Parameter j in pore pressure model Earth pressure coefficient at rest, Ko Void ratio, e Density, qs (kg/m3) Go (N/m2) Friction angle, /0 Undrained shear strength, cu (N/m2)

0.53 0.8 1950 3.2  105 37 0.0

0.53 1.53 1580 2.2  105 28 1.7  104

0.8

Loose Nevada sand

Dense Nevada sand

2.0 1.0

1.0 0.4

0.06 0.5 0.755 1936.8 2.2  105 35 0.0

0.06 0.4 0.614 2018.35 2.2  105 37 0.0

80

0.4 0.2 0 - 0.2 0

5

10

15

20

Time (s )

- 0.4

Pore pressure (kN/m2)

Acceleration (g)

0.6 60 40

Numerical model Centrifuge data

20

- 0.6 0

- 0.8

0

5

10

15

20

Time (s)

Fig. 17. Santa Cruz motion scaled to 0.67 g.

Fig. 19. Pore pressure generation at the middle of the loose Nevada sand layer.

Cyclic stress ratio

0.4

Loose Nevada Sand 0.3 0.2 0.1 0 1

10

100

Number of cycles to liquefaction Simple shear test (VELACS) Triaxial test (VELACS) Element test - Simple shear (Popescu and Prevost, 1993) Element test - Triaxial (Popescu and Prevost, 1993) Stress Path model simulation Fig. 18. Cyclic shear strength simulated from the stress path method.

the cyclic shear strength curve for the the loose Nevada sand obtained from the numerical model, one element simulations by Popescu and Prevost [23] and experimental data (VELACS). The Stress Path model agrees well with the other data.

test for dense and loose Nevada sand layers. Pore pressure generation has been considered only for the loose and dense Nevada sand layers. In addition to the hysteretic damping, viscous damping of the soil has also been taken into account. According to Seed and Idriss [26], under the amplitude of motions likely to develop during earthquakes, viscous damping should be on the order of 20% of the critical damping. Figs. 19 and 20 show the pore pressure generation at the centre of loose and dense Nevada sand layers. As observed during the centrifuge test, the loose sand layer has liquefied about 10 s after application of the earthquake load. The numerical model predicts higher pore pressure development during the early stages of loading than that observed during the centrifuge test. In the dense Nevada sand layer, large pore pressure reductions have been observed during the earthquake loading due to dilation of the soil. Although the numerical model does not have the ability to model dilation of the soil, the overall agreement between the measured and observed pore pressure distributions in both loose and dense Nevada sand layers is reasonable. Fig. 21 shows the ground displacement at the surface obtained from the numerical model is in close agreement with the centrifuge data.

4.2. Ground response analysis for the sloping ground

G ¼ Go

 0:4 ð2:17  eÞ2 1 þ 2K o 0 rm 1þe 3

kN=m2

ð15Þ

90 60

Numerical model Centrifuge data

30 0 -30

0

5

10

15 Time (s)

20

2

where Go and rm are in kN/m , e is the void ratio of soil and Ko is the earth pressure coefficient at rest. The constant Go is determined by matching the shear wave velocities measured during the centrifuge 0

120

Pore pressure (kN/m2)

Before the application of earthquake load, sloping ground is subjected to a static shear stress equivalent to the component of effective overburden stress parallel to the slope of the ground. In this case, the ground slope is 3°. Therefore an initial static shear stress of r0mo sinð3Þ has been applied to the ground. In sloping ground, displacements can become very large before the onset of soil liquefaction due to this static shear stress acting on the ground. Shear modulus of the soil is represented by [13]:

-60 Fig. 20. Pore pressure generation at the middle of the dense Nevada sand layer.

122

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

The dotted line in Fig. 23 shows the pressure, p, acting on the pile at 6.7 m below the ground surface, which is in the liquefiable sand, calculated from the recorded moment distribution along the pile using the beam theory according to the equation [4]:

ground surface (m)

Displacement at the

0.2 0.0 -0.2 0

5

10

15

20 Time (s)

-0.4

2

pðzÞ ¼

-0.6

Numerical model -0.8

Centrifuge data

-1.0 Fig. 21. Displacement at the ground surface.

4.3. Pile analysis The pile group considered for the analysis consisted of six pipe piles. In the prototype scale, the piles had an outer diameter of 1.0 m, a wall thickness of 0.05 m, Young’s modulus of 200 GN/m2 and a density of 7500 kg/m3. The cap mass was shared between the six piles. Hence each pile in the group carried a capmass of 100 t. The embedded length of the pile was 25.2 m. Since the pile cap height was nearly equal to the thickness of the layer of Monterey sand at the top of the model container, pile analysis was carried out only for the pile section below the Monterey sand layer. Fig. 22 shows the bending moment distributions at z = 9.1 m for three different piles in the group. It can be seen that they have nearly the same bending moment. Usually in the design of pile groups in liquefying soil, it is assumed that all piles behave in the same way [12]. The experimental results validate that this assumption is reasonable. Therefore, in the pile analysis, although the pile group has six piles, only one of them was considered. The degraded soil shear modulus recorded during the each time step of the free-field ground response analysis has been used to compute spring-dashpot coefficients at the pile–soil interface. A lateral force equivalent to capmass  ground surface acceleration has been applied at the pile head during the analysis. This force is time dependent as it varies with the ground surface acceleration during the analysis.

Bending moment (MNm)

2

Pile at SE corner Pile at NE corner Centre pile (West)

1 .5 1 0 .5 0 - 0 .5

0

5

10

15

20

Time (s)

-1 - 1 .5

Pressure acting on pile 6.7 m below ground surface (kN/m)

Fig. 22. Pile bending moment close to the bottom of the loose sand layer (z = 9.1 m) observed for three piles in the group.

d

2

dz

MðzÞ

ð16Þ

where M is the bending moment and z is the depth along the pile. According to Fig. 19, about 10 s after the application of the earthquake loading, liquefaction is commenced in the loose sand layer. According to Japanese Road Association (2002), the pressure acting on pile should be 30% of the total overburden stress, which corresponds to p = 29 kN/m for a 1.0 m diameter pile. Dobry et al. [8] suggests that p in the liquefiable sand can be neglected when there is a nonliquefiable crust above the liquefiable sand. However, according to the pressure distribution given in Fig. 23, maximum value of p reaches 235 kN/m in liquefying soil contradicting the above two recommendations. Therefore, in this study, the lateral pressure at the pile–soil interface was monitored and an iterative procedure was used to keep only the static part of the lateral pressure represented by the spring at or below the ultimate lateral pressure, Py. When the static pressure at the pile–soil interface reaches the ultimate value, soil yielding occurs. For piles subjected to static loads in clay, Broms [5] has suggested that the ultimate lateral pressure varies from 2cu to 9cu within top four diameters of the pile length and stays at 9cu below that level. Therefore, in this analysis, only static part of the lateral pressure is limited using these values for the pile–soil interface in the Bay Mud layer. For the liquefiable sand layer, Py ¼ 0:3r0m has been used [16] to limit the static part of the lateral pressure. For the nonliquefiable dense sand layer, the values suggested by Broms [5] has been used as follows:

Py ¼ Np Pp

ð17Þ

where Np = factor ranging between 3 and 5, and Pp = Rankine passive pressure given by:

Pp ¼ r0m tan2 ð45 þ /0 =2Þ

ð18Þ

Fig. 23 shows the pressure acting on the pile predicted from the numerical model. Although the experimental and numerical values do not agree well before the onset of soil liquefaction, after soil liquefaction, the range of p obtained from the numerical simulation agrees with the centrifuge test. Fig. 24 shows the pile head displacement during the earthquake loading. Similar to the pressure distribution given in Fig. 23, the agreement between measured and computed pile head displacement agrees well after the commencement of liquefaction. The lower pile displacements obtained from the numerical model before the onset of liquefaction, compared to the centrifuge test, is due to the higher pressure at the pile–soil interface simulated by the numerical model. Fig. 25 shows the bending moment distribution at z = 9.1 m obtained from the numerical model, and Fig. 26 shows measured and computed maximum positive and negative bending moment distributions along the pile. Overall, the agreement between the centrifuge data and the simpli-

400

Numerical model

200

Centrifuge data

0 -200 0

5

10

15

-400 -600

Time (S)

Fig. 23. Pressure acting on pile 6.7 m below the ground.

20

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

Pile head displacement (m)

0.1

Time (s)

0.0 -0.1 0

5

10

15

20

-0.2 -0.3

Numerical model

-0.4

Centrifuge data

-0.5

Data files of centrifuge tests carried out at the University of California, Davis, used in this study have been provided by Dr. Daniel W. Wilson and Dr. Scott Brandenberg. Their support is gratefully acknowledged.

2

Bending moment (MNm)

subjected to a static shear stress prior to application of cyclic shear stresses. The spring coefficients of the Winkler model used for the pile analysis are derived by integrating the Mindlin’s equation. Pile group performance observed during centrifuge tests show that all piles in the group have similar bending moment distributions. Therefore, it is reasonable to analyse a single pile in the design process. The response of a pile in the group analysed from the method outlined in this paper agrees well with the behaviour observed during the centrifuge test. Acknowledgements

Fig. 24. Displacement at the pile head.

1.5 1 0.5

References

0 -0.5

123

0

5

10

15

20

Time (s)

-1

Pile at SE corner

-1.5

Numerical model

-2 Fig. 25. Pile bending moment close to the bottom of the loose sand layer (z = 9.1 m).

Bending moment (MNm) -6

-4

-2

0

2

4

0 -5

Numerical model

-10

Centrifuge data

-15

Depth (m)

-20 -25 -30 Fig. 26. Maximum positive and negative bending moment distributions along the pile.

fied analysis method presented in this paper is good. Hence the simplified method has the ability to predict pile behaviour with sufficient accuracy for design purposes.

5. Conclusions This paper has described a numerical procedure, which may be used to compute pile behaviour in liquefying sloping ground subjected to lateral spreading. An effective stress based free-field ground response analysis is first carried out and the resulting ground displacements and degraded soil stiffness are used to obtain the pile performance. Application of the stress path model to simulate soil liquefaction in sloping ground has been verified using simple shear tests and results clearly show that the model has the ability to simulate liquefaction in sloping ground, where the soil is

[1] Berger E, Mahin SA, Pyke R. Simplified method for evaluating soil–pile structure interaction effects. In: Proceedings of the 9th offshore technology conference, Houston (TX); 1977. p. 589–98. [2] Boulanger RW, Curras CJ, Kutter BL, Wilson DW, Abghari A. Seismic soil–pile structure interaction experiments and analysis. J Geotech Geoenviron Eng, ASCE 1999;125(9):750–9. [3] Brandenberg SJ, Chang D, Boulanger RW, Kutter BL. Behaviour of piles in laterally spreading ground during earthquakes – centrifuge data report for SJB03. Report No. UCD/CGMDR-03/03, Centre for Geotechnical Modelling, Department of Civil Engineering, University of California, Davis, CA; 2003. [4] Brandenberg SJ, Boulanger RW, Kutter BL, Chang D. Behaviour of pile foundations in laterally spreading ground during centrifuge tests. J Geotech Geoenviron Eng, ASCE 2005;131(11):1378–91. [5] Broms BB. Lateral resistance of piles in cohesive soils. J Soil Mech Found Eng Div, ASCE 1964;90(SM2):27–63. [6] Byrne PM, Roy D, Campanella RG, Hughes J. Predicting liquefaction response of granular soils from pressuremeter tests. ASCE National Convention, San Diego, geotechnical special publication 56; 1995. p. 122–35. [7] Chacko MJ. Analysis of dynamic soil–pile structure interaction. Masters Thesis, Department of Civil and Environmental Engineering, University of California Davis; 1995. [8] Dobry R, Abdoun T, O’Rourke TD, Goh SH. Single piles in lateral spreads. J Geotech Geoenviron Eng, ASCE 2003;129(10):879–89. [9] Douglas DJ, Davis EH. The movement of burried footings due to moment and horizontal load and the movement of anchor plates. Geotechnique 1964;14:115–32. [10] Fuji S, Cubrinovski M, Tokimatsu K, Hayashi T. Analyses of damaged and undamaged pile foundations in liquefied soils during the 1995 Kobe Earthquake. In: Proceedings of the conference on geotechnical earthquake engineering and soil dynamics III, Seattle (Washington), vol. 2; 1998. p. 1187–98. [11] Gazetas G, Dobry R. Horizontal response of piles in layered soils. J Geotech Eng, ASCE 1984;110(1):20–40. [12] Ishihara K, Cubrinowski M. Performance of large-diameter piles subjected to lateral spreading of liquefied deposits. In: Thirteenth Southeast Asian geotechnical conference, Taipei, Taiwan; 1998. [13] Ishihara K, Towhata I. Dynamic response analysis of level ground based on the effective stress method. In: Pande GN, Zienkiewicz OC, editors. Soil mechanics – transient and cyclic loads. New York: John Wiley & Sons Ltd.; 1982. p. 133–72. [14] Ishihara K, Yamasaki F. Cyclic simple shear tests on saturated sand in multidirectional loading. Soils Found 1980;20(1):45–59. [15] Iwasaki T, Tatsuoka R. Effects of grain size and grading on dynamic shear moduli of sands. Soils Found 1977;17(3):19–35. [16] Japan Road Association. Specifications for highway bridges. Public Works Research Institute and Civil Engineering Research Laboratory, Japan; 2002. [17] Kagawa T. Lateral pile response in liquefying sand. In: Proceedings of 10th world conference on earthquake engineering, Madrid (Spain), paper no. 1771; 1992. [18] Kagawa T, Kraft LM. Lateral load deflection relationships of piles subjected to dynamic loading. Soils Found 1980;20(4):19–36. [19] Liyanapathirana DS, Poulos HG. Seismic lateral response of piles in liquefying soil. In: Proceedings, Pacific conference on earthquake engineering, Christchurch (New Zealand), paper no 21; 2003. [20] Liyanapathirana DS, Poulos HG. Seismic lateral response of piles in liquefying soil. J Geotech Geoenviron Eng, ASCE 2005;131(12):1466–79. [21] Park SS, Byrne PM. Numerical modelling of soil liquefaction at slope site. In: Proceedings of international conference on cyclic behaviour of soils and liquefaction, Bochum (Germany); 2004. p. 571–80. [22] PMB, Eng. Inc. PAR, Pile Analysis Program. Users manual, San Francisco (CA, USA); 1994.

124

D.S. Liyanapathirana, H.G. Poulos / Computers and Geotechnics 37 (2010) 115–124

[23] Popescu R, Prevost JH. Centrifuge validation of a numerical model for dynamic soil liquefaction. Soil Dyn Earthq Eng 1993;12:73–90. [24] Puebla H, Byrne PM, Phillips R. Analysis of CANLEX liquefaction embankments: prototype and centrifuge models. Can Geotech J 1997;34(5):641–57. [25] Seed HB, Idriss IM. On the importance of dissipation effects in evaluating pore pressure changes due to cyclic loading. In: Pande GN, Zienkiewicz OC, editors. Soil mechanics – transient and cyclic loads. New York: John Wiley & Sons Ltd.; 1982 [chapter 4]. [26] Seed HB, Idriss IM. Analysis of soil liquefaction: Niigata earthquake. J Soil Mech Found Div, ASCE 1967;93(SM3):83–108.

[27] Tabesh A, Poulos HG. A simple method for the seismic analysis of piles and its comparison with the results of centrifuge tests. In: Proceedings of the 12th world conference on earthquake engineering, Auckland (New Zealand), paper no. 1203; 2000. [28] Tabesh A, Poulos HG. The effect of soil yielding on seismic response of single piles. Soils Found 2001;41(3):1–16. [29] Yao S, Nogami T. Lateral cyclic response of piles in viscoelastic Winkler subgrade. J Eng Mech, ASCE 1994;120(4):758–75.