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Numerical Study for Modulus of Subgrade Reaction of Laterally-Loaded
Piles in Elastic Soils
Xin Zhou, Liu Yang, Ming-xin Li, Dr. Yun-gang Zhan*
School of Naval Architecture and Ocean Engineering, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, China * Corresponding author e-mail: [email protected]
ABSTRACT Several sophisticated methods, such as finite element method, boundary element method, and p-y method can be used to predict the response of laterally loaded pile. The elastic foundation beam method (Winkler method) proposed in 1867, however, is still used in engineering design for its simplicity. Many studies have been made to determine the coefficient of subgrade reaction of elastic soil for Winkler method, but a generally accepted equation has not yet developed. Combining the three-dimensional solid finite element analysis with analytical solution of finite-length beam on elastic foundations, the coefficients of subgrade reaction of laterally loaded pile/elastic soil systems were ascertained and the effects of concerning factors on them were examined. An equation to estimate the coefficients of subgrade reaction was recommended based on this study.
KEYWORDS: coefficients of subgrade reaction, Winkler method, laterally loaded piles, three-dimensional finite element.
INTRODUCTION Piles serving as foundations of offshore structures are often subjected to lateral loads or
overturning moments, which lead to lateral and rotational displacements at pile heads. From the perspective of safety of superstructures, more attention has been recently focused on the displacement of pile foundations, rather than the limit bearing capacity. Several methods have been developed to model the response of piles under lateral loads. Based on the solutions of horizontal displacement caused by a horizontal point load within the interior of semi-infinite, elastic mass (Mindlin, 1936), the boundary element method has been used to provide the numerical solutions for piles in homogeneous soils (Poulos, 1971; Davies & Budhu, 1986) or soil with linearly increasing soil modulus with depth (Budhu & Davies,1988). Finite element method (FE method) is a powerful and versatile tool to estimate the deformation of piles in soils with different strength profiles (Desai & Appel, 1976; Randolph, 1981). However, this method was predominantly used for the purpose of research, because it requires elaborate analysis techniques and needs high computation time, especially for three-dimensional (3-D) analyses. Currently, the finite element (FE) method is gradually used as a supplementary tool for design of pile foundations for important structures, with the development of computation techniques (Kellezi &
Vol. 18 [2013], Bund. O 3186 Hansen, 2003). The Winkler method or elastic foundation beam method was extensively used to analyze the laterally loaded pile problems (Hetenyi, 1946; Davisson & Gill, 1963) after Winkler proposed it for analyzing beams on soils. The coefficient of subgrade reaction, sk used in Winkler method represents the ratio of contact pressure, p and displacement, y at a given point of the interface of piles and soils. Several equations have been developed to estimate sk for elastic soils based on tests and theoretical analyses, but they do not agree well with each other (Sadrekarimi & Akbarzad, 2009). A more widely accepted relationship between sk and material properties of elastic soils, suggested by Vesic (1961), is
1/12
4
2
0.65
(1 )s s
sp ps
E E Bk
E IB
(1)
where Es and s is the modulus of elasticity and Poisson’s ratio of soils, respectively; B is the width of footings (or replaced by piles diameter, D); and EpIp is the flexural rigidity of footings (or piles).
The constant coefficient of subgrade reaction with the displacement, y was extended to a variable depending on y to account for the nonlinearity of the soils, which is called p-y method (McClelland & Focht, 1958). The p-y relationship for a laterally loaded pile in a specific construction site is usually established through back analysis of a full-scale lateral load test, which lumps many factors affecting the behavior of the laterally loaded pile/soil system. Several p-y relationships have been developed for different soil types (Reese et al., 1975; O’Neill & Murchison, 1983) and were incorporated into analyzing programs for laterally loaded pile (Reese et al., 2000) or were used in engineering designs (Chen et al., 2010)). Though the p-y method is an improvement of Winkler method, Winkler method is still in use in research and engineering practice for its simplicity (Pranjoto & Pender, 2003; Dan Tappel, 2010).
In present study, the coefficients of subgrade reaction for laterally loaded piles in elastic soils were studied by using 3-D solid finite element method and analytical solution for finite-length beams on elastic foundations. Based on it, an equation consulting the one proposed by Vesic (1961) was recommended.
ANALYTICAL AND FE METHOD SOLUTIONS FOR A PILE IN AN ELASTIC FOUNDATION
Hetenyi (1946) provided analytical solutions for a variety of infinite beams on a Winkler foundation. For a beam of finite length on an elastic foundation with point load at one end, the analytical solution can easily be deduced based on Winkler method. These analytical solutions can be used to analyze the response of laterally loaded piles with constant subgrade reactions. General-purpose finite element package ABAQUS provides an option of “FOUNDATION” to model structures on elastic foundations by prescribing the foundations stiffness per unit area.
The modulus of subgrade reaction along per unit length of circular piles or rectangular beams used in analytical methods is ks×D or ks×B, which are the same if D equated to B, no matter what the shape of cross section of the piles or beams is. This also means the coefficient of subgrade reaction for circular piles is defined on its longitudinal section. However, the stiffness of elastic foundation used in FE method provided by software ABAQUS is defined per unit area in normal direction of pile/elastic foundation interface. This leads to a smaller modulus of subgrade reaction
Vol. 18 [2013], Bund. O 3187 along per length of circular piles than rectangular piles with the same width (diameter), if the same ks defined. Fig. 1 shows a semi-circular pile resting on an elastic foundation with stiffness of ks per unit area. The vertical displacement at the longitudinal axis of an infinitesimal segment of the pile ( l ) is y. Then the normal displacement of a point in the interface between the pile and the elastic foundation is ( siny ), where is the central angle showed in Fig. 1(a). The vertical
resistance of elastic foundation applied on the area element ( d / 2D l ) is yp (=2
s sin d / 2D lk y ), as shown in Fig. 1(b). Integrating yp from zero to , a total vertical
resistance ( s / 4lDk y ) can be obtained. According to the above explanation, the foundation
stiffness in FE elastic foundation analysis should be raised to ( 4 / ) times of that used in analytical solutions to obtain a same modulus of subgrade reaction, Dks.
Figure 1: Subgrade reaction of semi-circular footing on elastic foundation
An example of a circular pile/elastic foundation system was analyzed by using these three methods, to provide a benchmark solution for 3-D solid FE analysis. The pile is 25.0m long and its diameter is 1.0m. The Young’s modulus of the pile is 30GPa with Poisson's ratio of 0.2. The coefficient of subgrade reaction is 10000kN/m3 used for analytical solution, while 12732.4kN/m3 for the FE analysis. The pile is categorized as a flexible beam according to the elastic foundation beam theory. Therefore, the infinite beam solution should be the same as that of finite-length beam. A point load of 200kN was applied at the top of the pile. Fig. 2 presents the distributions of displacements and internal forces of the pile varying with the pile depth. The good agreement of results of these three methods indicate that the finite-length beam analytical solution used in this study is correct, and treating the pile as structural beam member analyzed by the analytical method or solid member analyzed by the FE method has no difference in responses of laterally load pile in elastic foundation. The analytical method can be used to calibrate the coefficient of subgrade reaction according to the responses of laterally loaded piles predicted by 3-D solid FE analyses.
0 2 4 6 8 10
25
20
15
10
5
0
Dep
th /
m
Lateral displacement / mm
Infinite beam solution FEM solution Finite-length beam solution
(a) Displacements
semi-circular footing
Elastic foundation
d
pyp
xp
Elastic foundation
y
yny ty
/ 2D
semi-circular footing
Elastic foundation
d
pyp
xp
Elastic foundation
y
yny ty
/ 2D
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-50 0 50 100 150 200 250 300 350
25
20
15
10
5
0
Dep
th /
m
Moment / kN.m
Infinite beam solution FEM solution Finite-length beam solution
(b) Moments
-50 0 50 100 150 200 250
25
20
15
10
5
0
Dep
th /
m
Shear force / kN
Infinite beam solution FEM solution Finite-length beam solution
(c) Shear forces
Figure 2: Distribution of displacements, moments, and shear forces along the depth of the pile
DETERMINING COEFFICIENT OF SUBGRADE REACTION USING 3-D SOLID FINITE ELEMENT
MODEL The coefficient of subgrade reaction of a pile/elastic soil system is not only a function of the
mechanical behaviors of the pile and soil, but also affected by the size of pile. In this section, the 3-D solid FE analysis in conjunction with analytical method was employed to determine coefficient subgrade reaction of circular pile/elastic soil systems.
Numerical pile/elastic soil models The FE analyses were performed with half models due to the symmetry of the problems. A
generalized numerical model is shown in Fig. 3. The circular pile length is L and the pile diameter is D. The size of the surrounding soil is 10.5D from the central line of the pile in radial direction and the depth is (L+ 10D). The pile was simulated as linear elastic material with elastic modulus, Ep = 30GPa and Poisson's ratio, p 0.2 . A variable modulus Es and Poisson's ratio s were used
to describe the linear elastic soil, for the purpose of parametric studies. A fixed condition was
Vol. 18 [2013], Bund. O 3189 adopted at the bottom of the soil and the horizontal displacement was constrained along the side of the soil. The displacement in the normal direction of the symmetry plane was not allowed. Constraint of "Tie" was used in pile/soil interface, which do not allow slip and detachment between the pile and the soil. Solid element C3D20R was used to mesh pile/soil model.
(a) Model size (b) Finite element mesh
Figure 3: Numerical Pile/soil model
Determining subgrade reaction of elastic soils The pile/soil model with L = 25m and D = 1m exemplified in the above section was re-
analyzed by 3-D solid FE method. The results of 3-D FE analysis with different soil elastic moduli and constant Poisson's ratio, s 0.4 are presented in Fig. 4, along with the analytical solution (finite-length beam solution) obtained in previous section. It can be seen that the fundamental behaviors of laterally loaded pile predicted by these two methods are consistent, but cannot agree quite well no matter which value of modulus of elasticity of soil was adopt in 3-D solid FE analysis. The distributions of shear forces in transverse section of pile along pile shaft are affected slightly by the elastic modulus of soil and are more close to the analytical solution than other behaviors. The curves of the distribution of moments and lateral displacements, varying with elastic modulus of the soil, show two opposite tendencies along the depth of the pile, approaching the analytical solution with the increase of elastic modulus of soil before the first turning point of distribution curve arising but deviating from analytical solution after that. For this example, the elastic modulus of soil of 5.5MPa should be a good match with the coefficient of subgrade reaction, 10000kN/m3, considering all the behaviors shown in Fig. 4. Lateral displacement at the top of piles, however, is more interested in engineering design. The elastic modulus of soil of 6.0MPa should be selected if it was taken as a reference, according to Fig. 4(c). In the following studies, lateral displacement at the top of piles was used to determine the coefficient of subgrade reaction, ks of pile/elastic soil systems.
21D
L
10D
D
L
Pile
Soil
21D
L
10D
D
L
Pile
Soil
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-50 0 50 100 150 200 250
25
20
15
10
5
0
Dep
th /
m
Shear force / kN
Analytical solution k
s=10000kN/m3
FEA Es=6.0MPa
FEA Es=5.5MPa
FEA Es=5.0MPa
FEA Es=4.5MPa
(a) Shear forces
0 50 100 150 200 250 300 350
25
20
15
10
5
0
Dep
th /
m
Moment / kN.m
Analytical solution k
s=10000kN/m3
FEA Es=6.0MPa
FEA Es=5.5MPa
FEA Es=5.0MPa
FEA Es=4.5MPa
(b) Moments
0 2 4 6 8 10
25
20
15
10
5
0
Dep
th /
m
Lateral displacement / mm
Analytical solution k
s=10000kN/m3
FEA Es=6.0MPa
FEA Es=5.5MPa
FEA Es=5.0MPa
FEA Es=4.5MPa
(c) Displacements
Figure 4: Responses of pile under lateral load
The relationship of ks and Es Based on the above analysis, a series of 3-D solid FE analyses for pile/soil models with
s 0.4 and different Es were performed to determine the coefficient of subgrade reaction, ks. For
clarity of comparing the effects of concerning factors on subgrade reaction, a parameter was defined referring to Equation (1) as follows,
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1/12
4
p ps
s s
E IDk
E E D
(2)
The values of varying with the elastic modulus of soils for three pile/soil models with same pile diameter D =1.0m but different pile length L =10m, 25m, and 40m were presented in Fig. 5. The pile with 10m in length belongs to semi-rigid elastic foundation beam and the other two belong to flexible one, according to elastic foundation beam theory. As can be seen, the value of for semi-rigid beam is higher than flexible beam and the difference decreases gradually with the increase of soil modulus. For flexible beams, obviously, the pile length has no effect on the parameter and these parameters are mostly around 2.53 except that for elastic modulus equated to 1.0MPa and 5.0MPa, which are 2.82 and 2.61, respectively. This study was limited to flexible elastic foundation beam, and these values of pile with D =1.0m and L=40m were set as references for piles with different diameters in the following analyses.
0 10 20 30 402.0
2.5
3.0
Es / MPa
L=25m L=40m L=10m
Figure 5: Parameter for piles with different length
The effect of Poisson's ratio of elastic soils, s on were examined by a pile/soil model with
pile diameter D =1.0m and pile length L =40m, shown in Fig. 6. It is seen that the value of is almost the same when s ranging from 0.2 to 0.4. However the value of for 0.49s is bigger
than that for 0.2s by 4.0% or so. If ks is estimated by Equation (1), varies with s distinctly, and the increase of 4.0% changes to 26.3% in this case. Based on this study, the effect of Poisson's ratio of soil could be neglected when developing an equation to estimate the coefficient of subgrade reaction, ks of elastic soils, and this merely underestimate ks slightly.
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0 10 20 30 402.0
2.5
3.0
Es / MPa
s
s
s
s
Figure 6: The effect of Poisson's ratio s on parameter
The effect of pile diameter on ks was investigated using pile/soil models with pile length, L= 40m and pile diameter, D = 0.6m, 1.0m, 1.6m, 2.0m, 2.6m, 3.0m, and 4.0m, respectively. The flexural rigidity of piles, EpIp were kept constant with the pile diameter by changing the elastic modulus of piles accordingly in FE analysis. The elastic soils were modeled with 0.4s and Es
= 20MPa and 30MPa, respectively. The values of for pile with D = 1.0m and L= 40m was set as normalization parameters. Fig. 7 presents the relationship of normalized parameter and pile diameter along with the fitting curve, in which the Dref is 1.0m and ref
D is 2.53. It can be seen
that the fitting curve gives a slightly conservative estimation of ref/D D .
Figure 7: The relationship of parameter and pile diameter
By virtue of tension stress could not be sustained in the interface of piles and soils, the reference value of ref
D should be halved to 1.265 to account for the tension stress against pile
displacement introduced by "Tie" constraint in the 3-D solid FE analyses. Combining the fitting curve, equation (2), and the value of ref
D , one can obtain the following equation to estimate
the coefficient of elastic soils for laterally loaded piles,
0 1 2 3 40
1
2
3
4
5
6Fitting curve
Da
/ (D)
ref
D/Dref
Es=20MPa
Es=30MPa
1.3
ref/ 0.868
ref
DD D
D
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1/120.3 4s s
sref p p
1.098E E DDk
D D E I
(3)
CONCLUSIONS In this study, analytical solutions for beams on elastic foundations and FE method of
structures on elastic foundations were used to analyze a circular pile on an elastic foundation firstly, which provided a benchmark solution for 3-D solid FE analyses for laterally loaded pile/soil systems. Lateral displacement at piles top obtained by 3-D solid FE analysis was selected as a reference to calibrate the coefficient of subgrade reaction through analytical solution of finite-length beam. Based on a series of FE analyses, the following results are obtained:
The pile length has an effect on the coefficient of subgrade reaction of pile/elastic soil systems, but the effect vanishes if the pile belongs to a flexible beam in terms of elastic foundation beam theory.
The Poisson's ratio of elastic soil has a little effect on the coefficient of subgrade reaction when it ranging from 0.2 to 0.4. Even it is 0.49, the ks value increase merely by 4.0%. This is different from that of previous study, such as Vesic's (1961).
The correlation between the coefficient of subgrade reaction (ks) and the modulus of elastic soil for a given pile/soil system can be described by Equation (3), which includes the effect of pile diameter D with D=1m as a reference.
ACKNOWLEDGEMENT This study was sponsored by the Undergraduate Innovation Project of Jiangsu University of Science and Technology.
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8. Kellezi, L. and Hansen, P. B. (2003) "Static and dynamic analysis of an offshore mono-pile windmill foundation," Offshore mono-pile, Lyngby, Denmark, GEO - Danish Geotechnical Institute.
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