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A simplified nonlinear analysis method for piled raft foundation in layered soils under vertical loading
Computers and Geotechnics 38 (2011) 875–882
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
A simplified nonlinear analysis method for piled raft foundation in layered soils under vertical loading Maosong Huang ⇑, Fayun Liang, Jie Jiang Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, PR China Department of Geotechnical Engineering, Tongji University, Shanghai 200092, PR China
a r t i c l e
i n f o
Article history: Received 13 June 2010 Received in revised form 6 June 2011 Accepted 7 June 2011 Available online 13 July 2011 Keywords: Piled raft Layered soil Nonlinear analysis Interaction Stiffness matrix
a b s t r a c t This paper presents a simplified nonlinear solution for piled raft foundations in layered soils under vertical loading. Based on the elastic–plastic analysis of a single pile in a layered soil, the shielding effect between a receiver pile and the soil is taken into account to modify the conventional interaction factor between two piles. An approximate approach with the concept of the interaction factor is employed to study the nonlinear behavior of pile groups with a rigid cap. Considering the variation of soil properties, the solution to multilayered elastic materials is used to calculate the settlement of the soil. The interactions between pile–soil–raft are taken into account to determine the stiffness matrix of the piled raft. By solving the stiffness matrix equations, the settlement and the load shared by the piles and raft could be obtained. Compared with results of the available published literatures, the proposed solution provides reasonable results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In the past few years, there has been an increasing recognition that the raft has significant contribution to the performance of a piled raft foundation. In order to analyze this complex threedimensional problem, the foundation is usually assumed to be in an elastic state under working load conditions. A boundary element solution was reported by Kuwabara [9] to investigate the behavior of piled rigid raft foundations. A three-dimensional finite element method was conducted by Ottaviani [15] and Liang et al. [10] to analyze the piles–cap–soil interaction. Typically, it needs a large amount of computer time for analysis with these numerical methods. In order to improve the efficiency, Chow et al. [4] and Liang and Chen [11] used a variational approach to analyze the piled rafts. Liang et al. [12] proposed an integral equation method to analyze the piled rafts in an elastic state. It is noted that these analytical methods are limited to analysis of elastic problems. They cannot be used to simulate the behaviors of piled raft foundations in a plastic state. Besides the above methods, Ta and Small [19] and Small and Zhang [18] have done a systematic research on the piled raft in layered soils with a finite layer method. Compared with finite element methods or boundary element methods, it is more ⇑ Corresponding author at: Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, PR China. Tel./fax: +86 21 6598 3980. E-mail addresses: [email protected] (M. Huang), [email protected] (F. Liang), [email protected] (J. Jiang). 0266-352X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2011.06.002
effective. But compared with some simplified analytical methods, it is still a little complicated for the engineering calculation. In practice, more and more designs have been done based on a settlement criterion, under which it is tolerable to have a certain number of yielding piles [16]. In this paper, a nonlinear solution is proposed to analyze the response of a vertically loaded piled raft in a layered soil. Based on the Winkler model, the load transfer matrices of a single pile in the elastic–plastic state in a layered soil are used to obtain the settlement and internal forces of a single pile. As presented by Mylonakis and Gazetas [13], pile–soil–pile interaction factors depend not only on the displacement field arising from the settlement of a loaded (‘source’) pile, but also on the interplay between the adjacent (‘receiver’) pile and the soil subjected to the displacement field. The shielding effect between receiver pile and soil was taken into account to modify the conventional interaction factor between two piles. An approximate approach is proposed by extending the Mylonakis and Gazetas [13] approach to study the nonlinear behavior of pile groups in layered soils. For the variation of layered soil properties, the proposed method utilized the nonlinear layered Winkler model to simulate the pile to pile interaction and the surface to pile interaction, and the multilayered elastic half-space model to simulate the surface to surface interaction. Furthermore, the approach is extended to analyze the nonlinear response of a vertically loaded piled raft. Based on the layered soil model, the pile–soil–raft interactions are taken into account to obtain the nonlinear analysis of piled raft foundation. Compared to rigorous approaches, the proposed method is fairly simple in formulation and efficient in computation.
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
Thus it is feasible to perform the analysis for a large piled raft foundation. The proposed approach is validated against case studies. The reasonably accurate prediction results suggest the potential practical use of this proposed method.
2. Formulation of the problem
are assumed to be nonlinear and layered. The soil in contact with the raft is divided into a number of approximate square discrete elements and the pile head fits within one soil element, as illustrated in Fig. 1b. For piled rigid raft foundation, the stiffness matrix formulations of a piled raft system can be written as the following [4,11].
½K sp fwsp g ¼ fPsp g
Fig. 1a shows a piled raft system under vertical loads. The raft is rectangular and rigid. The raft–soil interface is assumed to be smooth and continuous. Therefore, the connection between the raft and the pile is assumed to be a simple connection, only the vertical forces transmitted from the raft to the head of piles. With these two assumptions there will be only vertical contact stress, and the interface displacement can be represented as either the raft deflection or the pile group-soil interface deformation [6]. Piles are assumed to be isotropic and linearly elastic. However, the soils
(a) Pile raft system in layered soils
where {wsp} is the displacement vector of the raft, {Psp} is the external force vector. [Ksp] is the stiffness matrix of the pile group-soil system. The vertical soil deformations at the center of soil elements and the displacements at the head of piles are determined using the principle of superposition. [Ksp] can be calculated from the inversion of the flexibility matrix.
½K sp ¼ ½F sp 1
ð2Þ
(b) Discretization of the interface betweensoiland raft
Fig. 1. Model of the problem analyzed.
Ppi f
Ps pp
f ps ai
L
L
s
s
(a) Pile to pile interaction effect
(b) Surface to pile interaction effect
Ppi f
Ps sp
f ss ai
L
ð1Þ
s
s
(c) Pile to surface interaction effect
(d) Surface to surface interaction effect
Fig. 2. Pile and/or surface interaction.
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
where [Fsp], which has the order of (ns + np) (ns + np), is the flexibility matrix of pile group-soil system, and can be rewritten in submatrix form of:
½F sp ¼
f ss
f sp
f ps
f pp
1 ½ð1 2mÞ shnz nz chnz U22 2ð1 mÞ 1 ¼ nz shnz þ chnz 2ð1 mÞ
U21 ¼
ð3Þ
1þm 1 nz shnz U24 ð1 mÞE 2n 1þm 1 ½ð3 4mÞshnz nz chnz ¼ ð1 mÞE 2n
in which the submatrices [fss], [fsp], [fps], [fpp] represent the interaction of surface to surface, surface to pile, pile to surface and pile to pile, respectively. Hain and Lee [6] defined these submatrices as shown in Fig. 2. np is the number of piles, ns is the number of soil element. A simplified approach to analyze the nonlinear response of pile groups is proposed to determine the flexibility coefficients in these submatrices in Eq. (3).
U23 ¼
3. Proposed solution for the stiffness matrix
U33 ¼
As Chow [3] described, high strain and slippage may occur at the pile–soil interfaces with the increment of load acting on piles. Nonlinearity due to slippage is confined only to a narrow zone of soil adjacent to the pile, whereas most of the soil between the piles is subjected to relatively low strain levels, and hence remains essentially elastic. Thus, at this stage, the nonlinear response of the group is dominated by the nonlinear behavior of the individual piles whereas interaction effects remain elastic. The flexibility coefficients will be determined for submatrices.
U31 ¼
E E n ½shnz þ nz chnz U32 ¼ n nz sh 2ð1 m2 Þ 2ð1 m2 Þ
1 nz shnz þ chnz U34 2ð1 mÞ 1 ¼ ðð1 2mÞ shnz nz chnzÞ 2ð1 mÞ E n nz shnz U42 2ð1 m2 Þ E ¼ n ðshnz nz chnzÞ 2ð1 m2 Þ
U41 ¼
1 ðsh nz þ nz chnzÞ U44 2ð1 mÞ 1 ¼ nz shnz þ chnzÞ 2ð1 mÞ
U43 ¼ 3.1. Surface to surface interaction in a layered soil Considering variation of each layered soil properties, the Burmister solution of multilayered elastic materials [2] is used to calculate the settlement of soil. In the calculations, the raft elements are approximately treated as the circular loads with equivalent areas. [fss], the flexibility matrix of soil, representing the interaction between the raft and the layered soil can be determined by Wang et al. [20] as follows.
Z 1 1 F 33 F 24 F 34 F 23 fii ¼ n 2abp 0 F 33 F 44 F 34 F 43 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 Z Z a2 b A J 0 @n x þ y dxdydn 2 2 X
fij ¼
1 2p
Z
1
n 0
F 33 F 24 F 34 F 23 J ðnrÞdn F 33 F 44 F 34 F 43 0
ð5Þ
ð6Þ
where U(n, z) is the 4 4 size transfer matrix and can be written as:
1 nz shnz þ chnz U12 2ð1 mÞ 1 ¼ ½ð1 2mÞ shnz þ nz chnz 2ð1 mÞ
U11 ¼
1þm 1 ½ð3 4mÞ shnz þ nz chnz U14 ð1 mÞE 2n 1þm 1 ¼ ðnz shnzÞ ð1 mÞE 2n
U13 ¼
3.2. Interaction of pile to pile in layered soils Though the present work is essentially a modification of the previous researches by Randolph and Wroth [17] and Mylonakis and Gazetas [13], the main analysis procedures are outlined briefly for completeness in the following.
ð4Þ
where the flexibility coefficients fii is the displacement of soil element i due to unit load acting on itself; fij is the surface displacement of soil element i due to unit load acting on soil element j; a and b are the length and width of the soil element i respectively; r is the distance between center of soil element i and soil element j; J is the Bessel function of the first kind; X is the area of soil element i; Fij is the coefficient of transfer matrix [F], which is given by
½F ¼ ½Uðn; h1 Þ ½Uðn; h2 Þ ½Uðn; hn Þ
where E and v are the Young’s modulus and the Poisson ratio of soil, respectively.
3.2.1. Analysis of single pile in layered soil The method by Randolph and Wroth [17] assumed that the soil surrounding a pile be represented by distributed Winkler springs. The governing equation for pile–soil interaction is given below by Randolph and Wroth [17]. 2
d WðzÞ 2
dz
k2 WðzÞ ¼ 0
ð7Þ
where W(z) is the axial pile deformation, and the variable k is
sffiffiffiffiffiffiffiffiffiffi Kz k¼ E p Ap
ð8Þ
where Ep and Ap are the Young’s modulus and cross-sectional area of an equivalent solid cylinder pile, respectively; K z ¼ lnð2rpm Þ Gs , Gs is r0
the soil shear modulus; r0 is the pile radius; and rm represents the maximum radius of influence of the pile beyond which the shear stress becomes negligible. In the general case of an inhomogeneous soil, rm is given by Randolph and Wroth [17]. For n soil layers, the displacement along the pile W11(z) was determined using Eq. (9) by Mylonakis and Gazetas [13].
W 11 ðzÞ ¼ A11 expðkzÞ þ B11 expðkzÞ
ð9Þ
where A11 and B11 are integration constants and can be obtained using the method proposed by Mylonakis and Gazetas [13].
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
According to Mylonakis and Gazetas [13], for each soil layer, by imposing the continuity of forces and displacements at the interface, we can have
W 11 ðLÞ
P11 ðLÞ
W 11 ð0Þ ¼ ½T P11 ð0Þ 1
ð10Þ
n
where ½T 1 ¼ P ð½t 1 i Þ, the transfer matrices ½t 1 i is
"
½t 1 i ¼
i¼1
coshðki hi Þ
Ep Ap ki sinhðki hi Þ
#
ðEp Ap ki Þ1 sinhðki hi Þ coshðki hi Þ
K b T 111 T 121
ð12Þ
T 122 K b T 112
dEb d Kb ¼ 1 þ 0:65 hb 1 v 2b
ð13Þ
where the factor in the parentheses on the right-hand side of Eq. (13) accounts for the presence of a rigid bedrock at depth equal to hb below the pile tip [5]; evidently, with no bedrock present, hb ? 1 and the term in the parentheses of Eq. (13) equals to 1. Eb and vb are the Young’s modulus and the Poisson ratio of the soil at the pile base level, respectively; and d is the diameter of the pile. Moreover, for a given force atop the pile, the pile response at the bottom of any layer j can be calculated as follows by Mylonakis and Gazetas [13].
W 11 ðzÞ
P11 ðzÞ
j
( ) 1 K
1
¼ P ð½t i Þ j
1
i¼1
P11 ð0Þ
d W i ðzÞ 2
dz
¼
2
d w21 ðzÞ 2
dz
K z ½W 21 ðzÞ W s ðs; zÞ ¼ 0
ð18Þ
The solution to Eq. (18) is
k W 21 ðzÞ ¼ A21 ekz þ B21 ekz þ wðsÞz½A11 ekz þ B11 ekz 2
ð19Þ
where A21 and B21 are integration constants, which can be determined by applying the method proposed by Mylonakis and Gazetas [13]. According to Mylonakis and Gazetas [13], for two interacting piles, the interaction factor a is readily obtained by imposing the continuity of forces and displacements at each interface.
a¼
W 21 ð0Þ W 11 ð0Þ
ð20Þ
The flexibility coefficients in submatrices [fpp] are given by approximate analytical solutions of the interaction factor determined by Eq. (20).
fiipp ¼ 1=K i
ði ¼ 1; 2; . . . ; npÞ
ð21Þ
fijpp ¼ aij =K
ði; j ¼ 1; 2; . . . ; np; i–jÞ
ð22Þ
where fiipp is the displacement at the head of pile i due to unit load acting on itself; fijpp is the displacement at the head of pile i due to unit load acting on the head of pile j; K is the elastic stiffness of every identical pile; Ki is the nonlinear stiffness of pile i; and aij is the interaction factor between the piles i and j.
ð14Þ
As the pile-head load increases, the friction force of pile element i reaches the soil shear strength sfi and is assumed to be kept constant. This means that the pile element is in a state of perfect plasticity. In this study, the analysis method is extended to allow for non-linear behavior of the piles on real soil by using the ‘‘cut-off load’’ method. The governing equation for the pile element i can be written as 2
ð17Þ
The presence of the ‘‘receiver’’ pile modifies (usually reduces) the displacement estimated above. The relative displacement is W21(z) Ws(s, z) as proposed by Mylonakis and Gazetas [13]. The vertical equilibrium of an element of the receiver pile gives:
Ep Ap
where the parameter Kb, represents the stiffness of soil at the pile base. As Randolph and Wroth [17] proposed, it is reasonable to assume that the pile base (‘tip’) acts as a rigid circular disk on the surface of a homogeneous elastic stratum. The corresponding force–displacement relationship can be written as
W s ðs; zÞ ¼ wðsÞW 11 ðzÞ
ð11Þ
where hi is the thickness of ith soil layer. Enforcing the boundary conditions W11(0) = 1 and P11(L) = KbW11(L), the stiffness of a single pile is easily obtained from Eq. (10).
K¼
their neighbors. At the location of the unloaded ‘receiver’ pile, if this pile were not present, the soil displacement would be
pd sfi Ep Ap
V Case 1
Case 2
Case 3
0.3L
Es
Es
4Es
0.4L
Es
2Es
2Es
Es
4Es
Es
h 0.3L D s
ð15Þ
h = 2 L, L / D = 25,ν s = 0.3, Ep / Es = 1000
The Eq. (15) can be resolved in terms of an increment as follows.
DW i DP i
¼ ½t f i b
DW i DP i
ð16Þ t
where the transfer matrix [tf]i is: ½t f i ¼ 1 0
ðEP AP Þ1 hi . 1
3.2.2. Analysis of piles group in layered soil Generally, pile groups are analyzed by the superposition principle. To determine the interaction factor between two piles, one may start by calculating the displacement field around a single loaded (‘source’) pile. The plane-strain approximation yields a logarithmic variation of vertical soil displacement Ws with radial distance s from the pile and thereby to an attenuation function w(s) expressed as Randolph and Wroth [17]. In reality, however, piles do not follow exactly the free-field displacement generated by
0.9m
3m
1.5m
10m 1.5m
3m
0.4m
Fig. 3. Configuration of foundation and soil profiles.
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
3.3. Interaction of surface to pile in layered soils The flexibility coefficients in submatrices [fsp] can be obtained from the load–displacement at the pile shaft approximately when the interaction of pile base to soil surface is weak [4,11].
fijsp ¼
lnðrm =s00ij Þ lnðr m =r0 Þ
fjjpp
ði ¼ 1; 2; . . . ; ns; j ¼ 1; 2; . . . ; npÞ
ð23Þ
where fijsp is the surface displacement of the soil element i due to unit load acting on the head of pile j; and s00ij is the distance between the center of soil element i and pile j. As to the interaction of pile to soil, the flexibility coefficients in submatrices [fps] can be given by the Maxwell’s reciprocal theorem by Hain and Lee [6].
fijps ¼ fjisp
Thus the stiffness matrix of pile group-soil systems defined in Eq. (1) can be obtained from the inversion of the flexibility matrix in Eq. (2), which is expressed as [Ksp] = [Fsp]1. For a given force atop the ‘source’ pile, the ‘receiver’ pile response at the bottom of any layer j can be calculated as
ði ¼ 1; 2; . . . ; np; j ¼ 1; 2; . . . ; nsÞ
ð24Þ
W 21 ðzÞ
P21 ðzÞ
( ¼ ½T 1 j
j
aP i K
0
( )
) þ ½T 2 j
1 K
1
Pi
ð25Þ
For the usual case of a group of m identical piles on a rigid cap, the ‘source’ pile can be analyzed with nonlinear method while the ‘receiver’ pile can be considered as linearly elastic, one can solve for the cap settlement WG in terms of the resultant force PG.
W G ¼ W i ¼ ai1 P1 =K þ ai2 P 2 =K þ þ aii Pi =K i þ þ aim Pm =K
PG ¼
where fijps is the displacement at the head of pile i due to unit load acting on the center of soil element j.
m X
Pi
i¼1
(a) Case 1
(b) Case 2
(d) Case 3 Fig. 4. Comparison of calculated solutions for piled raft.
ð26Þ
ð27Þ
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
where Wi and Pi are the head displacement and load at the top of the pile i, respectively, K is the elastic stiffness of every identical pile and Ki is the nonlinear stiffness of pile i. For a rigid raft, the settlement of the raft, the displacement of the soil at the surface, and the displacement of the pile heads are compatible. The proposed method is also suitable for analyzing the flexible raft combining with the finite element method. 4. Validation of the proposed solution 4.1. Finite element method The analyses using FEM by Kitiyodom and Matsumoto [8] and proposed method were performed for a piled raft foundations with a square raft supported by four piles as shown in Fig. 3. Three types of soil profiles considered are also presented in Fig. 3. The pile and the raft as well as the soil were modeled by linear elastic materials. The geometrical and mechanical properties of the raft, the piles and the soil are shown in terms of dimensionless parameters in Fig. 3. In order to model a rigid raft, the raft–soil stiffness ratio, Krs, which can be calculated using Eq. (28) by following Brown [1] was set equal to 10.
K rs ¼
4ER BR ð1 v 2s Þt3R
ð28Þ
3pEs L4R
Fig. 4 shows the calculated displacements and load distribution of the piled raft embedded in various soil profiles. The calculated results are shown in terms of dimensionless parameters IwV defined as Eq. (29) for the settlement and CaV defined as Eq. (30) for the axial force along the pile.
IwV ¼ C aV ¼
finite element approach. Thus, it is concluded that this method can be used with some confidence in the preliminary design of axially loaded piled raft foundation embedded in layered soils for the determination of the raft size, the number of piles, the pile spacing and the pile length. 4.2. A field pile group load test O’Neill [14] described a full-scale loading test conducted by the American Railway Engineering Association on a nine-pile group in a soft to medium-stiff clay. The layout of the foundation is shown in Fig. 5. Specifically, the piles were tapered steel pipes with external diameters of 419 and 203 mm at the head and tip of the pile, respectively, wall thickness of 4.6 mm, and embedment length of 18.45 m. They were installed by driving at a spacing of 1.22 m. The pile cap was not in contact with the soil. The soil was modeled by O’Neill as a two-layered soil system of saturated clay, with the upper layer extending to a depth of 12.2 m. Poisson’s ratio has been assumed to be very close to 0.5. The soil elastic moduli for the upper and lower layers are 43 MPa and 74.8 MPa, respectively. The soil shear strength for the upper and lower layer is 21.5 and 37.4 kPa, respectively. The computed and measured load–displacement curve of the pile group is shown in Fig. 6. It can be seen that the computed displacement of the pile groups matches the measured values well. A comparison of the computed and measured loads carried by the corner pile, edge pile and center pile is shown in Fig. 7. It can be seen that the most of load is carried by the corner pile. With the load applied to the cap increasing, the corner pile will transfer a proportion of load to center pile. 4.3. A load test on piled raft in layered soil
Es Dw V
ð29Þ
N V
ð30Þ
The accuracy of the nonlinear method for analysis of piled raft foundation in a layered soil in this paper has been verified by comparing the results with those from a full-scale test on a five pile
where w and N are the displacement of raft and axial force along the pile, respectively. If the foundation is assumed to be in elastic state, there are reasonably good agreements between the calculated results from the proposed method and those from the more rigorous
1
2
1
1.22 m
2
3
2
1.22 m
1
2
1 Fig. 6. Load–displacement behavior for the pile group.
3.66 m
1.22 m
12.2 m
6.25 m Fig. 5. Layout of the pile group.
Fig. 7. The loads carried by piles with the load applied to the cap increasing.
M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
881
4.4. Some remarks
0.4m
9.1m
11.6m
6.8m 6.8m 4.3m 4.3m
26m
Fig. 8. Layout of the piled raft.
Load / MN
Displacement / mm
0
0
2
4
6
8
10
5 10
5. Conclusions
15 Measured Nonlinear method Elastic method
20 25 30
Fig. 9. Load-Settlement behavior for the piled raft.
4
Load carried by piles / MN
In some other situations, engineers need to know the working behaviors of the piled rafts in the different stress states with the increasing of acting loads. If a full load–displacement curve of the piled rafts can be obtained, it is easy to determine whether the soil behavior is in elastic stage. Furthermore, the ultimate bearing capacities of the piled rafts may be determined. It will obviously help the engineers to understand the behavior of the piled rafts better. The main purpose of this paper is to propose a simplified solution to analyze the nonlinear behaviors of the piled rafts. The proposed method may be also suitable for analyzing such situations mentioned by Liang et al. [12]. Liang et al. [12] proposed an integral equation method to investigate the behavior of piled rafts with dissimilar piles in an elastic state. In realty, however, some of the piles (especially short piles) may yield before other piles. It should be pointed out that the calculated load on each pile (especially a short pile) should be checked against its capacity to ensure it is within an elastic state.
Measured Nonlinear method Elastic method
3
Acknowledgements
2
1
0
0
1
2
3
4
A nonlinear analytical method has been developed for the analysis of the deformation and the load distribution of axially loaded piled raft foundation embedded in a layered soil. The proposed method was verified through comparisons with the results from published field load test data and a rigorous finite element approach. These comparisons suggest that the proposed method is capable of predicting reasonably well the deformation and the load distribution of single piles, pile groups and piled rafts in a layered soil. The proposed method has the potential practical use for piled raft foundations.
5
6
7
8
Total load / MN Fig. 10. Load carried by piles.
group with raft in contact with the ground by Kakurai et al. [7]. The geometrical and mechanical properties of the raft, the piles and the soil are shown in Fig. 8. The raft can be assumed to be rigid. The soil is divided into four layers and the Young’s modulus from surface to base are 8.78 MPa, 10.84 MPa, 16.03 MPa and 30.28 MPa, respectively. Shear strength of the clay from the top to the tip of piles are 22 kPa, 26 kPa, 35 kPa, respectively. The Poisson ratio of the soil is 0.45. As to the fourth layers of soil blow the pile tip, the stiffness of soil can be determined by Eq. (13). The computed elastic and elastic–plastic load–displacement behavior of the piled raft is shown Fig. 9, together with the field measurements. It can be seen that the displacements of the piled raft calculated by the nonlinear method agrees closely with the measured displacement. A comparison of the computed and measured load carried by the piles is shown in Fig. 10. In the stage of elasticity, the computed results agree reasonably well with the measured field test results. It is noted that in the stage of plasticity, the proposed method tends to slightly underestimate the load carried by the raft.
The work reported herein was supported by the National Funds for Distinguished Young Scientists of China (Grant No. 50825803), the National Natural Science Foundation of China (Grant No. 50708078), and the Shanghai Rising-Star Program from Science and Technology Commission of Shanghai (Grant No. 10QA1407000). The authors wish to express their gratitude for the above financial supports. Special acknowledgements are given to Dr. Ke Yang of CH2M HILL, USA for his suggestions and comments. The anonymous reviewers’ comments have improved the quality of this paper and are also greatly acknowledged. References [1] Brown PT. Strip footing with concentrated loads on deep elastic foundations. Geotech Eng 1975;6(1):1–13. [2] Burmister DM. The general theory of stresses and displacements in layered soil systems. J Appl Phys 1945;6(2):89–96 [6(3):126–127, 6(5):296–302]. [3] Chow YK. Analysis of vertically loaded pile groups. Int J Numer Anal Methods Geomech 1986;10(1):59–72. [4] Chow YK, Yong KY, Shen WY. Analysis of piled raft foundations using a variational approach. Int J Geomech 2001;1(2):129–47. [5] Gazetas G. Analysis of machine foundation vibration: state of the art. Soil Dynam Earthqua Eng 1983;2(1):2–41. [6] Hain SJ, Lee IK. The analysis of flexible raft–pile systems. Geotechnique 1978;28(1):65–83. [7] Kakurai M, Yamashita K, Tomono M. Settlement behaviour of piled raft foundation on soft ground. In: Proceedings of 8th Asian regional conference on SMFE. vol. 1; 1987. p. 373–6. [8] Kitiyodom P, Matsumoto T. A simplified analysis method for piled raft foundations in non-homogeneous soils. Int J Numer Anal Method Geomech 2003;27(2):85–109. [9] Kuwabara F. An elastic analysis of piled raft foundations in a homogeneous soil. Soils Found 1989;29(1):82–92. [10] Liang FY, Chen LZ, Shi XG. Numerical analysis of composite piled raft with cushion subjected to vertical load. Comput Geotech 2003;30(6):443–53.
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[16] Poulos HG. Pile raft foundations: design and applications. Geotechnique 2001;51(2):95–113. [17] Randolph MF, Wroth CP. Analysis of deformation of vertically loaded piles. J Geotech Eng Div ASCE 1978;104(12):1465–88. [18] Small JC, Zhang HH. Behaviour of piled raft foundations under lateral and vertical loading. Int J Geomech 2002;2(1):29–45. [19] Ta LD, Small JC. Analysis of piled raft systems in layered soils. Int J Numer Anal Methods Geomech 1996;20(1):57–72. [20] Wang YH, Tham LG, Tsui Y, Yue ZQ. Plate on layered foundation analyzed by a semi-analytical and semi-numerical method. Comput Geotech 2003; 30(5):409–18.
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A simplified nonlinear analysis method for piled raft foundation in layered soils under vertical loading Maosong Huang ⇑, Fayun Liang, Jie Jiang Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, PR China Department of Geotechnical Engineering, Tongji University, Shanghai 200092, PR China
a r t i c l e
i n f o
Article history: Received 13 June 2010 Received in revised form 6 June 2011 Accepted 7 June 2011 Available online 13 July 2011 Keywords: Piled raft Layered soil Nonlinear analysis Interaction Stiffness matrix
a b s t r a c t This paper presents a simplified nonlinear solution for piled raft foundations in layered soils under vertical loading. Based on the elastic–plastic analysis of a single pile in a layered soil, the shielding effect between a receiver pile and the soil is taken into account to modify the conventional interaction factor between two piles. An approximate approach with the concept of the interaction factor is employed to study the nonlinear behavior of pile groups with a rigid cap. Considering the variation of soil properties, the solution to multilayered elastic materials is used to calculate the settlement of the soil. The interactions between pile–soil–raft are taken into account to determine the stiffness matrix of the piled raft. By solving the stiffness matrix equations, the settlement and the load shared by the piles and raft could be obtained. Compared with results of the available published literatures, the proposed solution provides reasonable results. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction In the past few years, there has been an increasing recognition that the raft has significant contribution to the performance of a piled raft foundation. In order to analyze this complex threedimensional problem, the foundation is usually assumed to be in an elastic state under working load conditions. A boundary element solution was reported by Kuwabara [9] to investigate the behavior of piled rigid raft foundations. A three-dimensional finite element method was conducted by Ottaviani [15] and Liang et al. [10] to analyze the piles–cap–soil interaction. Typically, it needs a large amount of computer time for analysis with these numerical methods. In order to improve the efficiency, Chow et al. [4] and Liang and Chen [11] used a variational approach to analyze the piled rafts. Liang et al. [12] proposed an integral equation method to analyze the piled rafts in an elastic state. It is noted that these analytical methods are limited to analysis of elastic problems. They cannot be used to simulate the behaviors of piled raft foundations in a plastic state. Besides the above methods, Ta and Small [19] and Small and Zhang [18] have done a systematic research on the piled raft in layered soils with a finite layer method. Compared with finite element methods or boundary element methods, it is more ⇑ Corresponding author at: Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, PR China. Tel./fax: +86 21 6598 3980. E-mail addresses: [email protected] (M. Huang), [email protected] (F. Liang), [email protected] (J. Jiang). 0266-352X/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2011.06.002
effective. But compared with some simplified analytical methods, it is still a little complicated for the engineering calculation. In practice, more and more designs have been done based on a settlement criterion, under which it is tolerable to have a certain number of yielding piles [16]. In this paper, a nonlinear solution is proposed to analyze the response of a vertically loaded piled raft in a layered soil. Based on the Winkler model, the load transfer matrices of a single pile in the elastic–plastic state in a layered soil are used to obtain the settlement and internal forces of a single pile. As presented by Mylonakis and Gazetas [13], pile–soil–pile interaction factors depend not only on the displacement field arising from the settlement of a loaded (‘source’) pile, but also on the interplay between the adjacent (‘receiver’) pile and the soil subjected to the displacement field. The shielding effect between receiver pile and soil was taken into account to modify the conventional interaction factor between two piles. An approximate approach is proposed by extending the Mylonakis and Gazetas [13] approach to study the nonlinear behavior of pile groups in layered soils. For the variation of layered soil properties, the proposed method utilized the nonlinear layered Winkler model to simulate the pile to pile interaction and the surface to pile interaction, and the multilayered elastic half-space model to simulate the surface to surface interaction. Furthermore, the approach is extended to analyze the nonlinear response of a vertically loaded piled raft. Based on the layered soil model, the pile–soil–raft interactions are taken into account to obtain the nonlinear analysis of piled raft foundation. Compared to rigorous approaches, the proposed method is fairly simple in formulation and efficient in computation.
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Thus it is feasible to perform the analysis for a large piled raft foundation. The proposed approach is validated against case studies. The reasonably accurate prediction results suggest the potential practical use of this proposed method.
2. Formulation of the problem
are assumed to be nonlinear and layered. The soil in contact with the raft is divided into a number of approximate square discrete elements and the pile head fits within one soil element, as illustrated in Fig. 1b. For piled rigid raft foundation, the stiffness matrix formulations of a piled raft system can be written as the following [4,11].
½K sp fwsp g ¼ fPsp g
Fig. 1a shows a piled raft system under vertical loads. The raft is rectangular and rigid. The raft–soil interface is assumed to be smooth and continuous. Therefore, the connection between the raft and the pile is assumed to be a simple connection, only the vertical forces transmitted from the raft to the head of piles. With these two assumptions there will be only vertical contact stress, and the interface displacement can be represented as either the raft deflection or the pile group-soil interface deformation [6]. Piles are assumed to be isotropic and linearly elastic. However, the soils
(a) Pile raft system in layered soils
where {wsp} is the displacement vector of the raft, {Psp} is the external force vector. [Ksp] is the stiffness matrix of the pile group-soil system. The vertical soil deformations at the center of soil elements and the displacements at the head of piles are determined using the principle of superposition. [Ksp] can be calculated from the inversion of the flexibility matrix.
½K sp ¼ ½F sp 1
ð2Þ
(b) Discretization of the interface betweensoiland raft
Fig. 1. Model of the problem analyzed.
Ppi f
Ps pp
f ps ai
L
L
s
s
(a) Pile to pile interaction effect
(b) Surface to pile interaction effect
Ppi f
Ps sp
f ss ai
L
ð1Þ
s
s
(c) Pile to surface interaction effect
(d) Surface to surface interaction effect
Fig. 2. Pile and/or surface interaction.
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where [Fsp], which has the order of (ns + np) (ns + np), is the flexibility matrix of pile group-soil system, and can be rewritten in submatrix form of:
½F sp ¼
f ss
f sp
f ps
f pp
1 ½ð1 2mÞ shnz nz chnz U22 2ð1 mÞ 1 ¼ nz shnz þ chnz 2ð1 mÞ
U21 ¼
ð3Þ
1þm 1 nz shnz U24 ð1 mÞE 2n 1þm 1 ½ð3 4mÞshnz nz chnz ¼ ð1 mÞE 2n
in which the submatrices [fss], [fsp], [fps], [fpp] represent the interaction of surface to surface, surface to pile, pile to surface and pile to pile, respectively. Hain and Lee [6] defined these submatrices as shown in Fig. 2. np is the number of piles, ns is the number of soil element. A simplified approach to analyze the nonlinear response of pile groups is proposed to determine the flexibility coefficients in these submatrices in Eq. (3).
U23 ¼
3. Proposed solution for the stiffness matrix
U33 ¼
As Chow [3] described, high strain and slippage may occur at the pile–soil interfaces with the increment of load acting on piles. Nonlinearity due to slippage is confined only to a narrow zone of soil adjacent to the pile, whereas most of the soil between the piles is subjected to relatively low strain levels, and hence remains essentially elastic. Thus, at this stage, the nonlinear response of the group is dominated by the nonlinear behavior of the individual piles whereas interaction effects remain elastic. The flexibility coefficients will be determined for submatrices.
U31 ¼
E E n ½shnz þ nz chnz U32 ¼ n nz sh 2ð1 m2 Þ 2ð1 m2 Þ
1 nz shnz þ chnz U34 2ð1 mÞ 1 ¼ ðð1 2mÞ shnz nz chnzÞ 2ð1 mÞ E n nz shnz U42 2ð1 m2 Þ E ¼ n ðshnz nz chnzÞ 2ð1 m2 Þ
U41 ¼
1 ðsh nz þ nz chnzÞ U44 2ð1 mÞ 1 ¼ nz shnz þ chnzÞ 2ð1 mÞ
U43 ¼ 3.1. Surface to surface interaction in a layered soil Considering variation of each layered soil properties, the Burmister solution of multilayered elastic materials [2] is used to calculate the settlement of soil. In the calculations, the raft elements are approximately treated as the circular loads with equivalent areas. [fss], the flexibility matrix of soil, representing the interaction between the raft and the layered soil can be determined by Wang et al. [20] as follows.
Z 1 1 F 33 F 24 F 34 F 23 fii ¼ n 2abp 0 F 33 F 44 F 34 F 43 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 Z Z a2 b A J 0 @n x þ y dxdydn 2 2 X
fij ¼
1 2p
Z
1
n 0
F 33 F 24 F 34 F 23 J ðnrÞdn F 33 F 44 F 34 F 43 0
ð5Þ
ð6Þ
where U(n, z) is the 4 4 size transfer matrix and can be written as:
1 nz shnz þ chnz U12 2ð1 mÞ 1 ¼ ½ð1 2mÞ shnz þ nz chnz 2ð1 mÞ
U11 ¼
1þm 1 ½ð3 4mÞ shnz þ nz chnz U14 ð1 mÞE 2n 1þm 1 ¼ ðnz shnzÞ ð1 mÞE 2n
U13 ¼
3.2. Interaction of pile to pile in layered soils Though the present work is essentially a modification of the previous researches by Randolph and Wroth [17] and Mylonakis and Gazetas [13], the main analysis procedures are outlined briefly for completeness in the following.
ð4Þ
where the flexibility coefficients fii is the displacement of soil element i due to unit load acting on itself; fij is the surface displacement of soil element i due to unit load acting on soil element j; a and b are the length and width of the soil element i respectively; r is the distance between center of soil element i and soil element j; J is the Bessel function of the first kind; X is the area of soil element i; Fij is the coefficient of transfer matrix [F], which is given by
½F ¼ ½Uðn; h1 Þ ½Uðn; h2 Þ ½Uðn; hn Þ
where E and v are the Young’s modulus and the Poisson ratio of soil, respectively.
3.2.1. Analysis of single pile in layered soil The method by Randolph and Wroth [17] assumed that the soil surrounding a pile be represented by distributed Winkler springs. The governing equation for pile–soil interaction is given below by Randolph and Wroth [17]. 2
d WðzÞ 2
dz
k2 WðzÞ ¼ 0
ð7Þ
where W(z) is the axial pile deformation, and the variable k is
sffiffiffiffiffiffiffiffiffiffi Kz k¼ E p Ap
ð8Þ
where Ep and Ap are the Young’s modulus and cross-sectional area of an equivalent solid cylinder pile, respectively; K z ¼ lnð2rpm Þ Gs , Gs is r0
the soil shear modulus; r0 is the pile radius; and rm represents the maximum radius of influence of the pile beyond which the shear stress becomes negligible. In the general case of an inhomogeneous soil, rm is given by Randolph and Wroth [17]. For n soil layers, the displacement along the pile W11(z) was determined using Eq. (9) by Mylonakis and Gazetas [13].
W 11 ðzÞ ¼ A11 expðkzÞ þ B11 expðkzÞ
ð9Þ
where A11 and B11 are integration constants and can be obtained using the method proposed by Mylonakis and Gazetas [13].
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According to Mylonakis and Gazetas [13], for each soil layer, by imposing the continuity of forces and displacements at the interface, we can have
W 11 ðLÞ
P11 ðLÞ
W 11 ð0Þ ¼ ½T P11 ð0Þ 1
ð10Þ
n
where ½T 1 ¼ P ð½t 1 i Þ, the transfer matrices ½t 1 i is
"
½t 1 i ¼
i¼1
coshðki hi Þ
Ep Ap ki sinhðki hi Þ
#
ðEp Ap ki Þ1 sinhðki hi Þ coshðki hi Þ
K b T 111 T 121
ð12Þ
T 122 K b T 112
dEb d Kb ¼ 1 þ 0:65 hb 1 v 2b
ð13Þ
where the factor in the parentheses on the right-hand side of Eq. (13) accounts for the presence of a rigid bedrock at depth equal to hb below the pile tip [5]; evidently, with no bedrock present, hb ? 1 and the term in the parentheses of Eq. (13) equals to 1. Eb and vb are the Young’s modulus and the Poisson ratio of the soil at the pile base level, respectively; and d is the diameter of the pile. Moreover, for a given force atop the pile, the pile response at the bottom of any layer j can be calculated as follows by Mylonakis and Gazetas [13].
W 11 ðzÞ
P11 ðzÞ
j
( ) 1 K
1
¼ P ð½t i Þ j
1
i¼1
P11 ð0Þ
d W i ðzÞ 2
dz
¼
2
d w21 ðzÞ 2
dz
K z ½W 21 ðzÞ W s ðs; zÞ ¼ 0
ð18Þ
The solution to Eq. (18) is
k W 21 ðzÞ ¼ A21 ekz þ B21 ekz þ wðsÞz½A11 ekz þ B11 ekz 2
ð19Þ
where A21 and B21 are integration constants, which can be determined by applying the method proposed by Mylonakis and Gazetas [13]. According to Mylonakis and Gazetas [13], for two interacting piles, the interaction factor a is readily obtained by imposing the continuity of forces and displacements at each interface.
a¼
W 21 ð0Þ W 11 ð0Þ
ð20Þ
The flexibility coefficients in submatrices [fpp] are given by approximate analytical solutions of the interaction factor determined by Eq. (20).
fiipp ¼ 1=K i
ði ¼ 1; 2; . . . ; npÞ
ð21Þ
fijpp ¼ aij =K
ði; j ¼ 1; 2; . . . ; np; i–jÞ
ð22Þ
where fiipp is the displacement at the head of pile i due to unit load acting on itself; fijpp is the displacement at the head of pile i due to unit load acting on the head of pile j; K is the elastic stiffness of every identical pile; Ki is the nonlinear stiffness of pile i; and aij is the interaction factor between the piles i and j.
ð14Þ
As the pile-head load increases, the friction force of pile element i reaches the soil shear strength sfi and is assumed to be kept constant. This means that the pile element is in a state of perfect plasticity. In this study, the analysis method is extended to allow for non-linear behavior of the piles on real soil by using the ‘‘cut-off load’’ method. The governing equation for the pile element i can be written as 2
ð17Þ
The presence of the ‘‘receiver’’ pile modifies (usually reduces) the displacement estimated above. The relative displacement is W21(z) Ws(s, z) as proposed by Mylonakis and Gazetas [13]. The vertical equilibrium of an element of the receiver pile gives:
Ep Ap
where the parameter Kb, represents the stiffness of soil at the pile base. As Randolph and Wroth [17] proposed, it is reasonable to assume that the pile base (‘tip’) acts as a rigid circular disk on the surface of a homogeneous elastic stratum. The corresponding force–displacement relationship can be written as
W s ðs; zÞ ¼ wðsÞW 11 ðzÞ
ð11Þ
where hi is the thickness of ith soil layer. Enforcing the boundary conditions W11(0) = 1 and P11(L) = KbW11(L), the stiffness of a single pile is easily obtained from Eq. (10).
K¼
their neighbors. At the location of the unloaded ‘receiver’ pile, if this pile were not present, the soil displacement would be
pd sfi Ep Ap
V Case 1
Case 2
Case 3
0.3L
Es
Es
4Es
0.4L
Es
2Es
2Es
Es
4Es
Es
h 0.3L D s
ð15Þ
h = 2 L, L / D = 25,ν s = 0.3, Ep / Es = 1000
The Eq. (15) can be resolved in terms of an increment as follows.
DW i DP i
¼ ½t f i b
DW i DP i
ð16Þ t
where the transfer matrix [tf]i is: ½t f i ¼ 1 0
ðEP AP Þ1 hi . 1
3.2.2. Analysis of piles group in layered soil Generally, pile groups are analyzed by the superposition principle. To determine the interaction factor between two piles, one may start by calculating the displacement field around a single loaded (‘source’) pile. The plane-strain approximation yields a logarithmic variation of vertical soil displacement Ws with radial distance s from the pile and thereby to an attenuation function w(s) expressed as Randolph and Wroth [17]. In reality, however, piles do not follow exactly the free-field displacement generated by
0.9m
3m
1.5m
10m 1.5m
3m
0.4m
Fig. 3. Configuration of foundation and soil profiles.
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3.3. Interaction of surface to pile in layered soils The flexibility coefficients in submatrices [fsp] can be obtained from the load–displacement at the pile shaft approximately when the interaction of pile base to soil surface is weak [4,11].
fijsp ¼
lnðrm =s00ij Þ lnðr m =r0 Þ
fjjpp
ði ¼ 1; 2; . . . ; ns; j ¼ 1; 2; . . . ; npÞ
ð23Þ
where fijsp is the surface displacement of the soil element i due to unit load acting on the head of pile j; and s00ij is the distance between the center of soil element i and pile j. As to the interaction of pile to soil, the flexibility coefficients in submatrices [fps] can be given by the Maxwell’s reciprocal theorem by Hain and Lee [6].
fijps ¼ fjisp
Thus the stiffness matrix of pile group-soil systems defined in Eq. (1) can be obtained from the inversion of the flexibility matrix in Eq. (2), which is expressed as [Ksp] = [Fsp]1. For a given force atop the ‘source’ pile, the ‘receiver’ pile response at the bottom of any layer j can be calculated as
ði ¼ 1; 2; . . . ; np; j ¼ 1; 2; . . . ; nsÞ
ð24Þ
W 21 ðzÞ
P21 ðzÞ
( ¼ ½T 1 j
j
aP i K
0
( )
) þ ½T 2 j
1 K
1
Pi
ð25Þ
For the usual case of a group of m identical piles on a rigid cap, the ‘source’ pile can be analyzed with nonlinear method while the ‘receiver’ pile can be considered as linearly elastic, one can solve for the cap settlement WG in terms of the resultant force PG.
W G ¼ W i ¼ ai1 P1 =K þ ai2 P 2 =K þ þ aii Pi =K i þ þ aim Pm =K
PG ¼
where fijps is the displacement at the head of pile i due to unit load acting on the center of soil element j.
m X
Pi
i¼1
(a) Case 1
(b) Case 2
(d) Case 3 Fig. 4. Comparison of calculated solutions for piled raft.
ð26Þ
ð27Þ
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M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
where Wi and Pi are the head displacement and load at the top of the pile i, respectively, K is the elastic stiffness of every identical pile and Ki is the nonlinear stiffness of pile i. For a rigid raft, the settlement of the raft, the displacement of the soil at the surface, and the displacement of the pile heads are compatible. The proposed method is also suitable for analyzing the flexible raft combining with the finite element method. 4. Validation of the proposed solution 4.1. Finite element method The analyses using FEM by Kitiyodom and Matsumoto [8] and proposed method were performed for a piled raft foundations with a square raft supported by four piles as shown in Fig. 3. Three types of soil profiles considered are also presented in Fig. 3. The pile and the raft as well as the soil were modeled by linear elastic materials. The geometrical and mechanical properties of the raft, the piles and the soil are shown in terms of dimensionless parameters in Fig. 3. In order to model a rigid raft, the raft–soil stiffness ratio, Krs, which can be calculated using Eq. (28) by following Brown [1] was set equal to 10.
K rs ¼
4ER BR ð1 v 2s Þt3R
ð28Þ
3pEs L4R
Fig. 4 shows the calculated displacements and load distribution of the piled raft embedded in various soil profiles. The calculated results are shown in terms of dimensionless parameters IwV defined as Eq. (29) for the settlement and CaV defined as Eq. (30) for the axial force along the pile.
IwV ¼ C aV ¼
finite element approach. Thus, it is concluded that this method can be used with some confidence in the preliminary design of axially loaded piled raft foundation embedded in layered soils for the determination of the raft size, the number of piles, the pile spacing and the pile length. 4.2. A field pile group load test O’Neill [14] described a full-scale loading test conducted by the American Railway Engineering Association on a nine-pile group in a soft to medium-stiff clay. The layout of the foundation is shown in Fig. 5. Specifically, the piles were tapered steel pipes with external diameters of 419 and 203 mm at the head and tip of the pile, respectively, wall thickness of 4.6 mm, and embedment length of 18.45 m. They were installed by driving at a spacing of 1.22 m. The pile cap was not in contact with the soil. The soil was modeled by O’Neill as a two-layered soil system of saturated clay, with the upper layer extending to a depth of 12.2 m. Poisson’s ratio has been assumed to be very close to 0.5. The soil elastic moduli for the upper and lower layers are 43 MPa and 74.8 MPa, respectively. The soil shear strength for the upper and lower layer is 21.5 and 37.4 kPa, respectively. The computed and measured load–displacement curve of the pile group is shown in Fig. 6. It can be seen that the computed displacement of the pile groups matches the measured values well. A comparison of the computed and measured loads carried by the corner pile, edge pile and center pile is shown in Fig. 7. It can be seen that the most of load is carried by the corner pile. With the load applied to the cap increasing, the corner pile will transfer a proportion of load to center pile. 4.3. A load test on piled raft in layered soil
Es Dw V
ð29Þ
N V
ð30Þ
The accuracy of the nonlinear method for analysis of piled raft foundation in a layered soil in this paper has been verified by comparing the results with those from a full-scale test on a five pile
where w and N are the displacement of raft and axial force along the pile, respectively. If the foundation is assumed to be in elastic state, there are reasonably good agreements between the calculated results from the proposed method and those from the more rigorous
1
2
1
1.22 m
2
3
2
1.22 m
1
2
1 Fig. 6. Load–displacement behavior for the pile group.
3.66 m
1.22 m
12.2 m
6.25 m Fig. 5. Layout of the pile group.
Fig. 7. The loads carried by piles with the load applied to the cap increasing.
M. Huang et al. / Computers and Geotechnics 38 (2011) 875–882
881
4.4. Some remarks
0.4m
9.1m
11.6m
6.8m 6.8m 4.3m 4.3m
26m
Fig. 8. Layout of the piled raft.
Load / MN
Displacement / mm
0
0
2
4
6
8
10
5 10
5. Conclusions
15 Measured Nonlinear method Elastic method
20 25 30
Fig. 9. Load-Settlement behavior for the piled raft.
4
Load carried by piles / MN
In some other situations, engineers need to know the working behaviors of the piled rafts in the different stress states with the increasing of acting loads. If a full load–displacement curve of the piled rafts can be obtained, it is easy to determine whether the soil behavior is in elastic stage. Furthermore, the ultimate bearing capacities of the piled rafts may be determined. It will obviously help the engineers to understand the behavior of the piled rafts better. The main purpose of this paper is to propose a simplified solution to analyze the nonlinear behaviors of the piled rafts. The proposed method may be also suitable for analyzing such situations mentioned by Liang et al. [12]. Liang et al. [12] proposed an integral equation method to investigate the behavior of piled rafts with dissimilar piles in an elastic state. In realty, however, some of the piles (especially short piles) may yield before other piles. It should be pointed out that the calculated load on each pile (especially a short pile) should be checked against its capacity to ensure it is within an elastic state.
Measured Nonlinear method Elastic method
3
Acknowledgements
2
1
0
0
1
2
3
4
A nonlinear analytical method has been developed for the analysis of the deformation and the load distribution of axially loaded piled raft foundation embedded in a layered soil. The proposed method was verified through comparisons with the results from published field load test data and a rigorous finite element approach. These comparisons suggest that the proposed method is capable of predicting reasonably well the deformation and the load distribution of single piles, pile groups and piled rafts in a layered soil. The proposed method has the potential practical use for piled raft foundations.
5
6
7
8
Total load / MN Fig. 10. Load carried by piles.
group with raft in contact with the ground by Kakurai et al. [7]. The geometrical and mechanical properties of the raft, the piles and the soil are shown in Fig. 8. The raft can be assumed to be rigid. The soil is divided into four layers and the Young’s modulus from surface to base are 8.78 MPa, 10.84 MPa, 16.03 MPa and 30.28 MPa, respectively. Shear strength of the clay from the top to the tip of piles are 22 kPa, 26 kPa, 35 kPa, respectively. The Poisson ratio of the soil is 0.45. As to the fourth layers of soil blow the pile tip, the stiffness of soil can be determined by Eq. (13). The computed elastic and elastic–plastic load–displacement behavior of the piled raft is shown Fig. 9, together with the field measurements. It can be seen that the displacements of the piled raft calculated by the nonlinear method agrees closely with the measured displacement. A comparison of the computed and measured load carried by the piles is shown in Fig. 10. In the stage of elasticity, the computed results agree reasonably well with the measured field test results. It is noted that in the stage of plasticity, the proposed method tends to slightly underestimate the load carried by the raft.
The work reported herein was supported by the National Funds for Distinguished Young Scientists of China (Grant No. 50825803), the National Natural Science Foundation of China (Grant No. 50708078), and the Shanghai Rising-Star Program from Science and Technology Commission of Shanghai (Grant No. 10QA1407000). The authors wish to express their gratitude for the above financial supports. Special acknowledgements are given to Dr. Ke Yang of CH2M HILL, USA for his suggestions and comments. The anonymous reviewers’ comments have improved the quality of this paper and are also greatly acknowledged. References [1] Brown PT. Strip footing with concentrated loads on deep elastic foundations. Geotech Eng 1975;6(1):1–13. [2] Burmister DM. The general theory of stresses and displacements in layered soil systems. J Appl Phys 1945;6(2):89–96 [6(3):126–127, 6(5):296–302]. [3] Chow YK. Analysis of vertically loaded pile groups. Int J Numer Anal Methods Geomech 1986;10(1):59–72. [4] Chow YK, Yong KY, Shen WY. Analysis of piled raft foundations using a variational approach. Int J Geomech 2001;1(2):129–47. [5] Gazetas G. Analysis of machine foundation vibration: state of the art. Soil Dynam Earthqua Eng 1983;2(1):2–41. [6] Hain SJ, Lee IK. The analysis of flexible raft–pile systems. Geotechnique 1978;28(1):65–83. [7] Kakurai M, Yamashita K, Tomono M. Settlement behaviour of piled raft foundation on soft ground. In: Proceedings of 8th Asian regional conference on SMFE. vol. 1; 1987. p. 373–6. [8] Kitiyodom P, Matsumoto T. A simplified analysis method for piled raft foundations in non-homogeneous soils. Int J Numer Anal Method Geomech 2003;27(2):85–109. [9] Kuwabara F. An elastic analysis of piled raft foundations in a homogeneous soil. Soils Found 1989;29(1):82–92. [10] Liang FY, Chen LZ, Shi XG. Numerical analysis of composite piled raft with cushion subjected to vertical load. Comput Geotech 2003;30(6):443–53.
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