The 12th International Conference of
International Association for Computer Methods and Advances in Geomechanics (IACMAG)
1-6 October, 2008
Goa, India
Ultimate Lateral Load of a Pile in Soft Clay under Cyclic Loading
D. M. Dewaikar
Professor, Department of Civil Engineering, Indian Institute of Technology, Bombay, Mumbai, Maharashtra, India.
S. V. Padmavathi
Former Post-Graduate Student, Geotechnical Engineering, Indian Institute of Technology, Bombay, Mumbai,
Maharashtra, India.
R.S.Salimath
Post Graduate Student, Geotechnical Engineering, Indian Institute of Technology, Bombay, Mumbai, Maharashtra,
India.
Keywords: Piles, load, deflection,
ABSTRACT: In this paper, the ultimate lateral resistance of a long, flexible, unrestrained vertical pile in soft clay is
computed under cyclic loading condition using p-y curves. A new method for the construction of p-y curves is
proposed. The comparison of calculated results of the proposed method with the field test results reported by Matlock
(1970) shows a good agreement. An iterative analysis is employed with secant modulus approach using a matrix
method known as moment area method developed by Sawant and Dewaikar (1994). The load-ground line
displacement relationship is obtained for different load eccentricities, number of cycles and pile diameters using the
proposed p-y curves. Ultimate lateral load is computed from the log-log plot of ground line displacement and applied
load and the effect of number of load cycles on the ultimate lateral load resistance is investigated for various pile
diameters and load eccentricities.
1
Introduction
Piles in the offshore environment are frequently large and in quite a few situations, driven into soft saturated clay,
which exhibits highly non-linear stress-strain behavior, even at low levels of applied loads. Cyclic lateral loading is an
aspect of the problem that introduces additional complexity and is encountered in offshore foundations as well as
other applications.
Amongst limited research works in this field, the major contributions are made by Matlock (1970), Idriss et al. (1978),
Poulos (1982), Vucetic et al. (1988), Grashuis et al. (1990) and Georgiadis et al. (1992). A brief review of these
works indicates that the degradation of the pile soil system under lateral cyclic loading mainly depends upon the
following parameters: i) reduction in soil modulus and undrained shear strength, ii) number of cycles, iii) cyclic load
level (ratio of cyclic load to the lateral static ultimate pile capacity, in case of load-controlled mode) and iv) cyclic
displacement level (ratio of cyclic displacement applied at the pile head to the external pile diameter, in the case of
displacement controlled mode).
As pointed out by Poulos (1982), the degradation is mainly due to
1) Partial to zero dissipation of excess pore water pressure generated during cyclic loading in progress,
2) Destruction of inter-particle bond with particle realignment and
3) Rearrangement and gradual accumulation of permanent plastic deformation around pile.
Over the past several years, considerable advances have been made in understanding the behavior of soils during
cyclic loading. In order to predict the behavior in a reliable manner, a new degradation model for one-way cyclic
lateral loading on piles in soft clay is proposed and is incorporated with the hyperbolic p-y curve for static loading
3498
proposed by Dewaikar and Patil (2005).
An iterative procedure is adopted to perform a parametric sensitivity analysis and the effect of cyclic lateral load on
deflection is studied. Based on the lateral deflection at the end of first cycle (static load), empirical relations were
established for strength degradation and gap effect, degradation model is assumed and the p-y curve is modified.
The cyclic load analysis is carried out using the static analysis program idealizing the soil by modified p-y curve, which
considers the effect of number of cycles, degradation in soil strength and magnitude of cyclic lateral load. The
allowable lateral deflection of pile top is taken as 20% of the pile diameter and load values corresponding to this
deflection level are computed for both static and cyclic conditions. Response of pile under one-way cyclic lateral
loading is studied using these load values for varying number of load cycles, load eccentricities, pile diameters and pile
embedment lengths.
2
Proposed method of analysis
For computing moment, rotation and deflection in respect to a long, flexible and unrestrained pile in cohesive soil, a
non-linear analysis is performed. The proposed method of analysis is briefly described as follows.
2.1
Initial Tangent Modulus
In the analysis, the initial soil modulus, Es, is taken as increasing with depth according to the expression suggested
by Poulos and Davis (1980).
Es=ηhz
(1)
Where, z represents depth below ground surface and ηh represents coefficient of soil modulus variation with depth. It
is taken as 1800kN/m3, average of 160-3450kN/m3 suggested by Reese and Matlock (1956) for soft normally
consolidated clays.
2.2
Development of p-y curves
For the validation purpose, analysis is performed using cyclic p-y curves developed by Matlock (1970), as shown in
Fig. 1.
Fig 1. p-y curve for soft clay under cyclic loading conditions, Matlock (1970)
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Where,
pu = ultimate soil resistance
x = depth at which the p-y curve is constructed
p= soil resistance
y = deflection
y50=deflection at one half the ultimate soil resistance
xr= transition depth
2.3
Method of Analysis
The moment area method developed by Sawant et al. (1994) is used here for analysis. In this matrix method, the pile
is discretized into small segments (150 nos.) of equal length and at the center of each segment, a node is
considered. For each node, a p-y curve is developed. Over each segment the soil reactive pressure is assumed to be
uniformly distributed. The problem then becomes identical to that of a vertical beam subjected to a load at its top and
reactions at the nodes, which correspond to the soil pressures. Using conventional beam theory, equations are
formulated for the evaluation of the unknown nodal displacements including the displacement and rotation at the pile
head. One advantage of the moment area method is that, no assumption is required to be made regarding soil
modulus variation with depth. For the analysis, the secant modulus approach is used in which the applied
predetermined load is analyzed repeatedly.
An iterative scheme is employed in the nonlinear analysis with secant modulus approach, using the moment area
method. First, the analysis is carried out with an assumed value of the soil modulus, usually the initial tangent
modulus values as suggested by Poulos and Davis (1980). In the next analysis, the assumed value of soil modulus
Es is revised (Es=p/y) to become consistent with the evaluated deflection. The procedure is repeated until the
calculated deflections between two successive analyses vary within a prescribed tolerance limit (10-7m).
3
Validation of proposed method of analysis
A program is developed using C language, to facilitate the validation. Results of proposed method of analysis are
compared with theoretical results given by Matlock (1970) as shown in Fig. 2, which are in close agreement.
Maximum difference is of the order of 1% to 2%. So this method of analysis is used for further computations.
Moment (kNm)
Depth of embedment (m)
-10
0
0
10
20
30
40
50
60
70
1
2
3
4
5
6
7
8
9
10
Sabine field test data (1970)
Matlock theoritical(1970)
Computed using Matlock's(1970)
cyclic p-y curves
Figure 2: Comparison of moments along the
pile length (for load =35.6kN) for cyclic loads
Fig 2. Comparison of moments along the pile length (for load =35.6kN) for cyclic loads
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4
Matlock’s (1970) field test data
Matlock carried out field tests on a long, flexible, unrestrained vertical pile of 0.324m diameter with 12.802m length,
embedded in soft clay at Sabine in Texas, with undrained shear strength 14kPa and submerged unit weight of
7.85kN/m3.
5
A New degradation model
Degradation of soft clay such as that found in many marine deposits during cyclic loading is well documented in
numerous studies (Vucetic, 1988; Matasovic and Vucetic, 1992). The cyclic strength degradation of soil is
implemented by using an exponential function that models the decreasing soil strength as a function of number of
cycles. Experimental evidence for this type of strength degradation has been reported by Heerama (1979) and Smith
(1987). Based on these finding, a degradation model is proposed as a function of number of load cycles, ratio of
ground line deflection at mud line to 20% of the pile diameter and 10% of the ratio of elasticity modulus of soil to the
undrained shear strength. The proposed expression is as per Equation 2.a.
Δ
E
⎡
− B×
× ×0.1 ⎞ ⎤
⎛
puN = pu ⎢1 − A⎜1 − N 0.2×b cu ⎟ ⎥
⎟⎥
⎜
⎢⎣
⎠⎦
⎝
ηhN = ηh × N − B×0.5
(2.a)
(2.b)
Where,
ƒ
ƒ
ƒ
ƒ
ƒ
puN is the soil resistance after N number of cycles
pu is the soil resistance for first cycle of load
Parameter A defines the residual soil strength level after an infinite number of cycles and is taken as 0.7
since the residual soil strength after an infinite number of cycles results in at least 30% of the initial strength
value (Grashuis 1990)
Δ is the ground line deflection under the applied load
b is the external diameter of the pile
ƒ
E
cu
ƒ
ƒ
ƒ
ƒ
y is the pile deflection at each spring level
ηh is the coefficient of subgrade reaction for first cycle of load
ηhN is the coefficient of subgrade reaction at N cycles of load
B is empirical constant derived after various trials and is taken as 0.0273 (Little and Briaud 1988)
is to be selected from Table 1, based on the soil conditions (Skempton 1951)
5.1
Table 1. Typical soil parameters for soft clay (Skempton, 1951)
Consistency of clay
•
•
6
ε50
E/cu
Soft
0.020
50
Medium
0.010
100
Stiff
0.005
200
E – elasticity modulus of soil
cu – undrained shear strength
Development of p-y curve
The following static p-y curve proposed by Dewaikar and Poonam (2005) was found to fit remarkably well with the
field test data points given by Matlock (1970).
3501
p=
y
(3)
y
1
+
E s pu
Where,
p = soil pressure (kN/m) at a displacement, y
pu = ultimate lateral soil resistance (kN/m) as defined by Matlock (1970); it is a function of soil un-drained cohesion,
effective soil unit weight, pile diameter and the desired depth
Es = initial stiffness of p-y curves which increases with depth (Poulos and Davis, 1980) (Refer Eq.1)
Equation 3 is modified by incorporating the degradation model (Equations 2.a & 3) for cyclic loading conditions. The
cyclic p-y curve which fulfills the features of cyclic behavior is as follows:
p=
6.1
y
(4)
y
1
+
E sN p uN
Validation of the proposed model
Analyses are performed as explained above (Section 2) using the proposed p-y curve (Eq. 4). Results are compared
with field test results reported by Matlock (1970), as shown in Fig. 3.
M o m e n t (k N m )
-1 0
0
10
20
30
40
50
60
70
Depth of embedment (m)
0
1
2
3
4
5
6
7
8
9
10
11
12
S a b in e f ie ld t e s t d a t a ( 1 9 7 0 )
C o m p u t e d u s in g s t a t ic p - y c u r v e s p r o p o s e d b y
D e w a ik a r & P a t il ( 2 0 0 5 ) in c o r p o r a t e d w it h
N e w d e g r a d a t i o n m o d e l f o r c y c li c l o a d i n g c o n d i t i o n s
C o m p u t e d u s i n g G e o r g i a d is e t a l . ( 1 9 9 2 )
c y c lic p - y c u r v e s
Fig 3 Comparison of moments along the pile length (for load =35.6kN) for cyclic loads
Results are in very good agreement with the field test data, the difference is only of the order of 2% to 3% where as
the difference between computed values using the Georgiadis (1992) cyclic p-y curves with field test data is of the
order of 30 to 50%. The proposed model is used for further computations, to calculate the ultimate lateral load.
7
Method of calculating ultimate lateral load
From the analysis, pile top deflections are obtained for various cyclic loads applied at an eccentricity, e from the
ground surface. Computations are carried out for various load eccentricities, pile diameters and number of cycles.
The relationships between applied cyclic loads and pile top deflections are obtained and plotted on log-log scale for
various cases as shown in Figs. 4 to 20. Then, two straight lines of different slopes passing through most of the
points are identified. The intersection of these two lines gives the lateral load capacity, Pu (Indian Standard Code of
Practice in Soil Engineering, Method of Load Test on Soils - IS 1888: 1988). As seen from these figures, points of
higher loads lie on a straight line. But for points corresponding to lower loads, the best-fit line is drawn.
3502
100
1000
Pu =39.66kN
Pu =148.97kN
100
Load (kN)
Load (kN)
10
10
1
1
0.1
1E-4
1E-3
0.01
0.1
1
0.1
1E-4
10
1E-3
0.01
Displacement (m)
0.1
1
10
Displacement (m)
Fig 4 D=0.3m
Fig 5 D=0.6m
1000
1000
Pu =543.97kN
Pu =314.80kN
100
Load (kN)
Load (kN)
100
10
10
1
1
0.1
1E-4
1E-3
0.01
0.1
1
0.1
1E-4
10
1E-3
0.01
0.1
1
10
Displacement (m)
Displacement (m)
Fig 6 D=0.9m
Fig 7 D=1.2m
Pu =845.34kN
1000
Load (kN)
100
10
1
0.1
1E-4
1E-3
0.01
0.1
1
10
Displacement (m)
Fig 8 D=1.5m
Figs 4 to 9 Computation of ultimate lateral load capacity of a pile with load eccentricity 0.0 m and embedment length
30.0m in soft clay at 100 number of load cycles with change in pile diameter
1000
1000
1000
Pu =145.93kN
Pu =83.15kN
100
Load (kN)
10
10
1
1
0.1
1E-4
Pu =68.35kN
100
Load (kN)
Load (kN)
100
1E-3
0.01
0.1
Displacement (m)
Fig 9 e=0
1
10
0.1
1E-4
10
1
1E-3
0.01
0.1
Displacement (m)
Fig 10 e=5
3503
1
10
0.1
1E-4
1E-3
0.01
0.1
1
Displacement (m)
Fig 11 e=7.5
10
1000
1000
Load (kN)
Load (kN)
100
Pu =56.76kN
100
10
Pu =49.38kN
10
1
1
0.1
1E-4
1E-3
0.01
0.1
1
0.1
1E-4
10
1E-3
0.01
0.1
1
10
Displacement (m)
Displacement (m)
Fig 12 e=10.0
Fig 13 e=12.5
Figs 9 to 13 Computation of ultimate lateral load capacity of a pile of diameter 0.6m and embedment length 30.0m in
soft clay at 100 number of load cycles with change in load eccentricity
1000
1000
1000
Pu =184.07kN
Pu =170.12kN
100
Pu =157.37kN
100
10
10
1
0.1
1E-4
Load (kN)
Load (kN)
Load (kN)
100
1
1E-3
0.01
0.1
1
0.1
1E-4
10
10
1
1E-3
0.01
0.1
1
Displacement (m)
Displacement (m)
Fig 14 N=1
Fig 15 N=5
10
0.1
1E-4
1E-3
0.01
0.1
1
10
Displacement (m)
Fig 16 N=25
1000
1000
1000
Pu =152.70kN
Pu =143.51kN
Pu =148.57kN
100
100
10
1
0.1
1E-4
Load (kN)
Load (kN)
Load (kN)
100
10
1
1
1E-3
0.01
0.1
1
0.1
1E-4
10
10
1E-3
0.01
0.1
Displacement (m)
Displacement (m)
Fig 17 N=50
Fig 18 N=100
1
0.1
1E-4
10
1E-3
0.01
0.1
1
10
Displacement (m)
Fig 19 N=200
Figs 14 to 19 Computation of ultimate lateral load capacity of a pile of diameter 0.6m and embedment length 30.0m
in soft clay at varying number of load cycles
Table 2 Variation of lateral load capacity of pile with pile diameter
1
Pile Diameter
(m)
0.3
39.96
Pu0.2D
(kN)
44.83
2
0.6
148.97
166.30
0.89
3
0.9
314.80
356.10
0.88
4
1.2
543.97
611.30
0.89
5
1.5
845.34
959.10
0.88
SR.No.
Pu(kN)
3504
Pu/ Pu0.2D
0.89
Table 3 Variation of lateral load capacity of pile with load eccentricity
1
Load eccentricity
(m)
0.0
145.93
Pu0.2D
(kN)
166.30
2
5.0
83.15
93.10
3
7.5
68.35
75.95
0.89
4
10.0
56.76
64.06
0.88
5
12.5
49.38
55.36
0.89
SR.No.
Pu(kN)
Pu/ Pu0.2D
0.88
0.89
Table 4 Variation of lateral load capacity of pile with number of load cycles
1
Number of
cycles
1
184.07
Pu0.2D
(kN)
207.50
2
5
170.12
190.97
0.89
3
25
157.37
176.83
0.89
SR.No.
Pu(kN)
Pu/ Pu0.2D
0.88
4
50
152.70
171.31
0.89
5
100
148.57
166.30
0.89
6
200
143.51
161.25
0.89
200
1000
Ultimate lateral load, Pu(kN)
Ultimate lateral load, Pu(kN)
180
800
600
400
200
160
140
120
100
80
60
40
20
0
0
0.0
0.3
0.6
0.9
1.2
1.5
-1
1.8
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15
Load eccentricity (m)
Diameter of the pile (m)
Fig 20 Variation of ultimate lateral load of a pile of embedment
length 30.0m at 100 number of load cycles with change
in pile diameter (e=0.0m)
Fig 21 Variation of ultimate lateral load of a pile of
diameter 0.6m and embedment length 30m
at 100 number of load cycles with change in
load eccentricity.
Ultimate lateral load, Pu(kN)
200
180
160
140
120
0
20
40
60
80
100
120
140
160
180
200
Number of cycles
Fig. 22: Degradation of ultimate lateral load of a pile of diameter 0.6 m and embedment length 30.0m in soft clay with
increasing number of cycles (e=0.0m)
8
Results and discussion
The computation of ultimate lateral load capacity of flexible unrestrained piles in cohesive soils is performed
considering pile with embedment length as 30.0m and with diameters 0.3, 0.6, 0.9, 1.2 and 1.5m. Load eccentricity is
taken as 0.0, 5.0, 7.5, 10.0 and 12.5. Analysis is carried out with number of load cycles as 1, 5, 25, 50, 100 and 200.
The results are tabulated in Tables 2 to 5.
In the first case, ultimate load capacity is calculated with embedment length as 30.0m and with varying diameters.
Load eccentricity is taken as zero in this case. The analysis is carried out for 100 number of load cycles. Results are
tabulated in Table 2. A significant variation in ultimate lateral load capacity with change in pile diameter is observed;
3505
when the diameter is increased to 5 times, increase in lateral capacity is 20 times the initial value. Computations are
as shown in Figs. 5 to 9 and the variation is as shown in Fig.21.
In the second case, ultimate load capacity is calculated with embedment length as 30.0m and with diameter as 0.6m.
Load eccentricity is taken as varying from 0 to 12.5m. The analysis is carried out for 100 number of load cycles.
Results are tabulated in Table 3. Decrease in ultimate lateral load capacity by changing the load eccentricity from 0.0
to 12.5m is 66%. Computations are as shown in Figs. 10 to 14 and the variation is as shown in Fig.22.
Computation of ultimate lateral load capacity of a pile of diameter 0.6m and embedment length 30.3m number of
cycles increasing from 1 to 200 is carried out as shown in Figs. 15 to 20. Degradation of ultimate lateral load with
increasing number of cycles is as shown in Fig. 22.Initially degradation is very high for a first few cycles. From one to
five cycles the degradation is of the order of 8% and for the next 20 cycles, it is of the order of 7%. For the next 25
cycles it is 3%. After that, degradation rate is very less compared to the initial degradation rate. For the next 50 cycles
it is 6% and then reaching equilibrium after a large number of load cycles. From 100 to 200 cycles, degradation is
only of 3%, which is very less compared to initial degradation rate.
As no other method or field test data is available in literature to validate this method, the results are compared with
the load values corresponding to 0.2D deflection at pile top. The ratio of ultimate load to the load corresponding to
0.2D deflection (Pu/ Pu0.2D) is calculated. It is clearly seen from the tabulated results (Tables 2 to 4) that, this ratio is
almost constant in all the cases considered. From this it can be concluded that, it is possible to compute the ultimate
load under cyclic loading conditions using this new technique as the trend of variation of ultimate load with diameter,
load eccentricity and number of cycles is same as that of the variation of load corresponding to 0.2D deflection. The
results obtained using this technique is on lower side as compared to the load corresponding to 0.2D deflection.
Therefore, the method can be safely used for the ultimate load computations under cyclic loading conditions.
9
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th
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