Soil Dynamics and Earthquake Engineering 97 (2017) 478–481
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Soil Dynamics and Earthquake Engineering
journal homepage: www.elsevier.com/locate/soildyn
A practical method for construction of p-y curves for liquefiable soils
a
b
c
MARK
b,⁎
Suresh Dash , Mehdi Rouholamin , Domenico Lombardi , Subhamoy Bhattacharya
a
b
c
Indian Institute of Technology Bhubaneswar, India
University of Surrey, UK
The University of Manchester, UK
A R T I C L E I N F O
A BS T RAC T
Keywords:
p-y curves
Liquefied soil
Pile-soil interaction
BNWF model
In practice, the analysis of laterally loaded piles is often carried out using a “Beam on Non-linear Winkler
Foundation method” whereby the lateral pile-soil interaction is modelled as a set of non-linear springs (also
known as p-y curves). During seismic liquefaction, the saturated sandy soil changes its state from a solid to a
viscous fluid like material, which in turn alters the shape of the p-y curve. Typically, p-y curves for non-liquefied
soil looks like a convex curve with an initial stiff slope that reduces with increasing pile-soil relative
displacement (y) i.e., elasto-plastic softening response. However, recent research conclusively showed that py curve for liquefied soil has a different shape, i.e., upward concave with practically-zero initial stiffness (due to
the loss of particle to particle contact) up to a certain displacement, beyond which the stiffness increases due to
reengaging of the sand particles. This paper presents a practical method for construction of the newly proposed
p-y curves along with an example.
1. Introduction
A reasonable representation of pile-soil interaction is important in
the evaluation of response of pile foundation to earthquake shaking.
Winkler approach (also known as Beams on Non-Linear Winkler
Foundation, BNWF or p-y approach) for analysis of laterally loaded
pile is widely used in the practice due to its ease of nonlinearity
modelling and mathematical and computational efficiency. Fig. 1(a)
schematically demonstrates the BNWF model where the lateral, axial
and end bearing soil-pile interactions are modelled by lateral springs
(p-y spring), axial springs (t-z spring) and end-bearing spring (q-z
spring), respectively. For evaluating the lateral capacity of pile foundation, p-y springs play a vital role, and the backbone force deformation
behaviour defined for this spring is known as p-y curve. In a p-y curve,
p is the soil reaction per unit length of the pile and y is the
corresponding relative pile-soil displacement. For liquefied soil, the
p-y curve used in current practice is a factored value from its nonliquefied state, which is found to be in disagreement with the p-y curve
observed from full scale, centrifuge and shaking table tests, see for
example [1,4]. Fig. 1(b) and (c) shows the shape of p-y curves in pre
and post-liquefaction stage. Further details of the shape of the p-y
curves for liquefied and non-liquefied soils can be found in [1–7]. This
paper presents a step-by-step procedure that can be used for constructing p-y curves for liquefiable soils from a typical field bore log
data.
⁎
Corresponding author.
E-mail address: S.Bhattacharya@surrey.ac.uk (S. Bhattacharya).
http://dx.doi.org/10.1016/j.soildyn.2017.03.002
Received 2 March 2017; Accepted 3 March 2017
Available online 05 April 2017
0267-7261/ © 2017 Elsevier Ltd. All rights reserved.
2. Major steps in the construction of p-y curves for
liquefiable soils
The construction of p-y curves for liquefiable soils involves four
steps, as follows.
1)
2)
3)
4)
Evaluation of soil parameters from bore-log data
Consideration of dimensions of pile foundation
Construction of simplified stress-strain curve for liquefied soil
Generation of p-y curve for liquefied soil from stress-strain curve
obtained in Step 3 above.
2.1. Evaluation of soil parameters from bore-log data
Normally the bore-log data from a site provides the variation of STP
(N) values with depth. However, some advanced soil properties are
necessary if the soil is liquefiable to construct its p-y curve. These
necessary soil parameters are: corrected SPT value (N1), relative
density (Dr), maximum shear modulus (Gmax), and residual shear
strength of liquefied soil (Sr). These parameters can be estimated from
element testing or through standard correlations available in the
literature. This paper provides some standard correlations available
in the literature which can be used in the absence of experimental
results obtained from representative soil samples. However, the
limitations and applicability must be checked for the actual site
Soil Dynamics and Earthquake Engineering 97 (2017) 478–481
S. Dash et al.
Fig. 1. (a) BNWF model of pile-soil interaction, (b) pre liquefaction, and (c) post liquefaction p-y curves.
condition in hand for the adopted correlations. Some typical correlations as per available literature are given below for ready reference.
(a) For sandy soil, the field SPT value can be corrected for overburden
pressure as
N1 = (N / (σv′ /98) )
(1)
where σ′v is the initial effective overburden pressure in kPa.
(b) The relative density of the soil can be estimated using correlations
available in the literature from SPT N1 value. The relationship of
N1 and Dr suggested by [8] as follows may be used.
Dr =
N1/ CD
(2)
1.7
where CD = (9/(emax − emin )) . CD can be taken as 20 for sand with
fines, 41 for clean sand and 70 for gravelly sand
(c) The maximum shear modulus of the soil can be determined either
from field test or using any suitable correlation. For the calculations in this paper, the correlation of Gmax (in kPa) with SPT value
(N) as suggested by Imai and Tonouchi [16] given in Eq. (3) as
Gmax = 14400N 0.68
Fig. 2. Residual shear strength of liquefied soil (after [11]).
whereas steel pile may be considered as smooth.
2.3. Construction of simplified stress-strain curve for liquefied soil
(3)
In absence of results from element tests, the following simplified
model can be used for liquefied soil. This simplified model is defined
using four parameters: (a) take off strain (γto ), initial shear modulus
(G1), critical state shear modulus (G2) and maximum shear strength
(τmax ). Following equations (Eqs. 4–7) can be used to estimate these
parameters of simplified stress-strain model (see Fig. 3 a) for liquefied
soil [4].
(d) The residual shear strength of liquefied soil (Sr ) can be defined as
the shear strength mobilized at large deformation after liquefaction is triggered. There are many correlations available in the
literature to specify the residual shear strength of liquefied soil,
either based on SPT or CPT, initial effective overburden, etc. The
upper and/or lower bound values may be used to capture the worst
possible response while analyzing lateral pile-soil interaction.
Fig. 2 can be referred for Sr based on corrected SPT value.
(a)
Take off strain=γto = 74.34 − 17.71 ln(Dr ) [in %]
(4)
(b)
Initial Shear Modulus = G1 = 1/ γto [in kPa]
(5)
(c)
Critical state shear modulus = G2 =
2.2. Consideration of pile foundation data
The diameter of a pile is necessary for p-y curve estimation. The
friction between pile and soil will influence the soil resistance with
rough interface resulting in higher resistance than smooth interface. Typically, concrete pile interface is considered as rough,
(d)
479
Ultimate shear strength = τmax = sr
Gmax
5 pini ′
[in kpa]
(6)
(7)
Soil Dynamics and Earthquake Engineering 97 (2017) 478–481
S. Dash et al.
Fig. 3. Schematic representation of obtaining p-y curve. (a) Simplified stress-strain curve for liquefied soil, (b) linearly scaled p-y curve model for stress-strain model, and (c) smoothed
p-y curve model.
Fig. 4. (a) Soil profile with SPT N-value at the site along with (b) pile details and, (c) the p-y curve for liquefied soil layer at 5 m depth.
to represent the behaviour in a pragmatic way, the p-y curve obtained
in the above step is smoothened at its transition using weighing factors.
The smooth p-y relationship can be obtained by using the following
expression.
2.4. Preparation of p-y curve for liquefied soil from stress-strain
curve
Once the simplified stress-strain model of liquefied soil is formulated, the corresponding p-y curve parameters are calculated by using
scaling factors, Ms and Ns (see [1,5] and [4] for the derivation of these
factors). Fig. 3 schematically represents the process involved in
preparing stress-strain model to the p-y curve of liquefied soil.
The initial and ultimate lateral resistance and displacement can be
calculated using the below equations.
The scaling factor Ns = 9.2 for smooth interface and 11.94 for rough interface.
(8)
The strain scaling factor Ms = 1.87 for fully liquefied soil
(9)
p=ω
ω=
The ultimate lateral resistance pu = NsτmaxD
⎡
⎛ 6π ⎛
4y + yu ⎞⎞⎤
1⎢
1 − tanh⎜⎜ ⎜y − 1
⎟⎟⎟⎥
2 ⎢⎣
y
6 ⎠⎠⎥⎦
⎝
⎝ u
(15)
3. Example for calculating p-y curves for liquefied soil from a
typical ground profile
(10)
Typically for onshore practice, SPT N value are recorded through
geotechnical investigation. Fig. 4(a) shows the ground profile along
with a pile for which p-y curves are to be constructed under liquefied
conditions. Stepwise description is given below to obtain p-y curves for
liquefied soil at a depth of 5 m.
Steps to obtain p-y curve for liquefied soil from ground profile:
Step – 1: The soil considered at a depth of 5 m with SPT value N=5
and unit weight=17 kN/m3
1.25γtoD
Ms
⎡p + p
⎛
p − p1
y + y1 ⎞⎤
2π
1
y + A(1 − ω)⎢ u
+ u
tanh
⎟⎥
⎜y − u
⎢⎣ 2
2
3(yu − y1) ⎝
2 ⎠⎥⎦
(14)
The initial lateral displacement corresponding to take−off strain = y1
=
y1
where A=0 for y=0, and A=1 for y‡0 and ω is a weight function given by
Eq. (15).
The initial lateral resistance corresponding to take−off strain = p1
= Ns1.25γtoG1D
p1
(11)
(12)
⎛
τmax − (G11.25γto ) ⎞
The ultimate lateral displacement = yu = ⎜1.25γto +
⎟
G2
⎝
⎠
D
×
Ms
(13)
(a) The SPT value corrected for overburden =N1=8, using Eq. (1).
(b) The effective initial overburden=σ′v = 5 × (17 − 9.81) = 36 kPa
To make the p-y curve model effective in the numerical analysis and
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Soil Dynamics and Earthquake Engineering 97 (2017) 478–481
S. Dash et al.
Novelty of submission
(c) The effective confining stress = pini ′= 2/3 x 36 = 24kPa
(d) The relative density of soil=Dr =0.45=45%, by using Eq. (2),
considering it as clean sand
(e) For N1=8, the residual strength=su =1.8 kPa (lower bound) and
20 kPa (upper bound) using Fig. 2
1. Mechanics based shape of p-y curves
2. Proposed p-y curves for liquefiable soils.
3. Understanding the parameters required for construction of p-y
curves.
Step-2: The pile diameter considered as 0.6 m, and the pile material
is taken as steel
Step – 3: For the considered soil and pile properties
(a)
(b)
(c)
(d)
References
[1] Dash SR. Lateral pile soil interaction in liquefiable soils [Ph.D. Thesis]. UK:
University of Oxford; 2010.
[2] Bhattacharya S, Adhikari S, Alexander NA. Simplified method for unified buckling
and dynamic analysis of pile supported structures in seismically liquefiable soils. Soil
Dyn Earthq Eng 2009;29:1220–35. http://dx.doi.org/10.1016/jsoildyn.2009.01.006.
[3] Lombardi D, Bhattacharya S. Evaluation of seismic performance of pile-supported
models in liquefiable soil. Earthq Eng Struct Dyn 2016;45:1019–38.
[4] Lombardi D, Dash SR, Bhattacharya S, Ibraim E, Wood DM, Taylor CA. Construction
of simplified design p-y curves for liquefied soils. Geotechnique 2017 http://dx.doi.
org/10.1680/igeot.15.P.116.
[5] Bouzid DJ, Bhattacharya S, Dash SR. Winkler Springs (p-y curves) for pile design
from stress-strain of soils: FE assessment of scaling coefficients using the Mobilized
Strength design concept. Geomech Eng 2013;5(5):379–99.
[6] Lombardi D, Bhattacharya S, Hyodo M, Kaneko T. Undrained behaviour of two silica
sands and practical implications for modelling SSI in liquefiable soils. Soil Dyn
Earthq Eng 2014;66:293–304.
[7] Rouholamin M, Bhattacharya S, Orense R. Effect of initial relative density on the
post-liquefaction of sand. Soil Dynamics and Earthquake Engineering. (Accepted
with corrections); 2017.
[8] Cubrinovski M, Ishihara K. Empirical correlation between SPT N value and relative
density for sandy soils. Soil Found, Jpn Geotech Soc 1999;39(5):61–71.
[11] Cubrinovski M, Bradley B. Assessment of seismic performance of soil–structure
systems. In: Proceedings of the 18th New Zealand Geotechnical Society 2008
Symposium, Auckland, New Zealand; 2008, p. 111–127.
[16] Imai T, Tonouchi K. Correlation of N-value with S-wave velocity and shear
modulus. In: Proceedings of 2nd European Symposium on Penetration Testing,
Amsterdam; 1982, p. 57–72.
Take off strain=γto = 6.97% =0.0697 for Dr = 45% using Eq. (4)
Initial shear modulus=G1=14 kPa using Eq. (5)
Critical state shear modulus=G2 =1758 kPa using Eq. (6)
The maximum shear strength of liquefied soil =τmax = Sr = 1.8 kPa
(lower bound) and = 20 kPa (upper bound).
Step-4: From the estimated simplified stress-strain curve, the p-y
curve parameters are calculated as:
(a) The scaling factor for stress Ns =9.2 (considering smooth interface
for steel pile)
(b) The scaling factor for strain Ms is 1.87 for fully liquefied soil.
(c) p-y curve parameters using Eqs. (11–14) are:
p1 =6.9 kPa and y1 =0.0279 m, pu =9.94 kN/m and
yu =0.028 m (Lower Bound)
p1 =6.9 kPa and y1 =0.0279 m, pu =110.4 kN/m and
yu =0.0314 (Upper Bound)
(d) The smooth p-y curve is obtained using Eq. (14).
The above calculation enables to plot the p-y curve at 5 m depth, for
both lower bound and upper bound condition as shown in Fig. 4(b).
Similar calculation can be also done for various depths.
Further reading
[1] Bowles JE. Foundation analysis and design, Fifth ed.. New York: The McGraw-Hill
Companies, Inc.; 1996.
[2] American Petroleum Institute. Recommended practice for planning, designing and
constructing fixed offshore platforms-Load and resistance factor design, API-RP-2A,
21st Ed.; 2010.
[3] Vaid Y, Thomas J. Liquefaction and post liquefaction behaviour of sand. J Geotech
Eng 1995;121(2):163–73.
[4] Sivathayalan . Static cyclic and post liquefaction simple shear response of sands
[M.Sc. Thesis]. University of British Columbia; 1994.
[5] Yasuda S, Masuda T, Yoshida N, Nagase H, Kiku H, Itafuji S, Mine K, Sato K.
Torsional shear and triaxial compression tests on deformation characters of sands
before and after liquefaction. In: Proceedings of the 5th US-Japan workshop on
earthquake resistant design of lifelines and countermeasures against soil
liquefaction; 1994, p. 249–265.
[6] Hua Pan, Guoxing Chen, Hanlong Liu, Binghui Wang. Behavior of large postliquefaction deformation in saturated Nanjing fine sand. Earthq Eng Eng Vib
2011;10(2):187–93.
4. Conclusion
In this paper a practical method for construction of p-y curves for
liquefiable soils is presented. A step by step calculation procedure has
been provided with an example considering a standard borelog data.
The calculation uses basic soil properties such as relative density, SPT
profile, basic pile geometry and material. Other required soil properties
are estimated based on proposed empirical relationships. Once the
stress-strain behaviour of post-liquefied soil is defined, the p-y curve
can be constructed using two scaling factors for stress and strain,
respectively. The proposed p-y curve captures the actual behaviour of
the liquefied soil when sheared under undrained conditions, i.e., strainhardening behaviour with practically-zero stiffness at low strain.
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