Design method of piled-raft foundations under vertical ...

Computers and Geotechnics 47 (2013) 16–27

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Design method of piled-raft foundations under vertical load considering interaction effects Dang Dinh Chung Nguyen ⇑, Seong-Bae Jo, Dong-Soo Kim Department of Civil and Environmental Engineering, KAIST, 291 Daehakro, Yuseong-gu, Daejeon 305-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 9 February 2012 Received in revised form 29 May 2012 Accepted 27 June 2012 Available online 3 August 2012 Keywords: Piled-raft Centrifuge modelling Individual pile Soil–structure interactions Nonlinear behaviour Settlement

a b s t r a c t The paper has proposed a design method considering interaction effects for a piled raft foundation. In this method, the raft is considered as a plate supported by a group of piles and soil. The ultimate load capacity of the pile group is taken into account in calculating the settlement when the foundation is subjected to a large vertical external load. In addition, this method supports estimation of the nonlinear behaviour of the piled raft foundation by considering the nonlinear behaviour of the piles. A step-by-step procedure to apply the proposed method to calculate the settlement and distribution of the bending moment of the piled raft foundation is introduced. To verify the reliability of the proposed method, models of a 16-pile raft and a 9-pile raft with different pile lengths embedded in homogeneous silica sand were tested in a centrifuge and comparisons were made between the results of the proposed method, the results of centrifuge tests, and those of Plaxis 3D. Good agreement between centrifuge modelling and the proposed method is demonstrated, thus showing the potential of the proposed method. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The piled raft foundation system has recently been widely used for many structures, especially high rise buildings. In this foundation, the piles play an important role in settlement and differential settlement reduction, and thus can lead to economical design without compromising the safety of the structure. In several design cases, the piles are allowed to yield under the design load. Although the load capacity of the pile is exceeded, the piled raft foundation can hold additional loads with controllable settlement. Thus, accurately determining the settlement of the foundation is critical and for this the designers must consider the role of the raft and the role of piles in combination, as well as the interactions between the foundation’s components. In efforts to solve this problem, Poulos and Davis [4], Burland [5] and Randolph [6] have proposed simplified methods involving a number of simplifications in relation to the soil profile and load on the raft. A second group of methods is the approximate methods proposed by Poulos [1] and Clancy and Randolph [2]. These methods employ a ‘‘strip on springs’’ or ‘‘plate on springs’’, where the raft is represented by a strip or a plate and the piles as springs. Other methods include more sophisticated computer-based methods, combining boundary elements for piles and a finite element analysis for the raft (e.g. [3,7,8,12,16]). Commercial programs such as FLAC [19], AMPS (developed by AMPS Technologies, 2005),

Plaxis 3D Foundation (developed by Delft University of Technology & Plaxis bv) and others provide good options for a numerical analysis. Nevertheless, the simplified methods provide less accurate calculated results, the other groups of methods are not easily applied to engineering work, and not everyone has access to the commercial programs. This paper proposes a design method that effectively simplifies the calculation procedure considering the nonlinear behaviour of the raft for application to engineering work. This method considers the foundation as a plate and employs a finite element analysis to solve the stress distribution of the raft. The plate is supported by springs and subjected to vertical loads. The interactions between the piles, raft, and soil are considered by means of the interactions between these springs. Consequently, this method can provide reasonable results for the settlement and bending moment of the raft. To verify the reliability of the proposed method, centrifuge tests with two piled raft models, sixteen piles with 15 m pile length and nine piles with 9 m pile length, were performed with high resolution instruments. The centrifuge test results are compared with the results obtained by the proposed method and by the commercial program Plaxis 3D Foundation.

2. Background of analysis 2.1. Interactions in piled raft foundation

⇑ Corresponding author. Tel.: +82 42 350 7219; fax: +82 42 350 7200. E-mail addresses: [email protected] (D.D.C. Nguyen), siderique@kaist. ac.kr (S.-B. Jo), [email protected] (D.-S. Kim). 0266-352X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compgeo.2012.06.007

The behaviour of a piled raft foundation is influenced by the interactions between the piles, raft and soil, and consequently

D.D.C. Nguyen et al. / Computers and Geotechnics 47 (2013) 16–27

interaction factors have been widely adopted for the prediction of the response of a piled raft. In reality, there are two basic interactions, pile–soil–pile interaction and pile–soil–raft interaction, as shown in Fig. 1. The pile–soil–pile interaction is defined as the additional settlement of a pile caused by an adjacent loaded pile, and the pile–soil–raft interaction is defined as superposing the displacement fields of a raft caused by a pile supporting the raft. The pile–soil–pile interaction is an important consideration in the analysis of pile groups and piled rafts, and the pile–soil–raft interaction is necessary for analysing piled rafts. Several approaches for determination of these two interaction factors are tabulated in Table 1. The approach of Poulos and Davis [4] for obtaining the pile–pile interaction factor considers a pair of vertical piles spaced at (S) and embedded in a horizontally layered soil. The formulation is based on the additional settlement of a pile under the interaction of the other pile. Poulos [9] proposed another approach that can be used for calculating the additional settlement of a pile caused by a pile group surrounding it by superposing additional settlement caused by each pile. The difference between these two approaches is that in the later method the additional settlement of a pile is a function of the forces of other piles in the pile group. In the approach developed by Clancy and Randolph [2], the interaction factor of a pile to a raft (arp) is calculated based on the additional settlement of a circular rigid raft caused by its supporting pile. This formulation, however, does not consider the change in soil stiffness along the pile, and therefore Randolph [6] proposed a modified version of his earlier formulation by considering the stiffness of soil at the pile head and the pile tip and along the pile shaft. Nevertheless, neither of Randolph’s approaches considers strength characteristics of soil (e.g., friction angle, cohesion) or the flexibility of the raft. The approach of Clancy and Randolph [2] is used to calculate arp when the settlement of the piled raft and the load transmitted to the piles and to the raft are known. The advantage of this method is that it can determine the interaction of the pile group to the raft. The difficulty of estimating the load transmitted to the pile group and the raft, however, hinders practical use of this formulation.

2.2. Review of design methods for piled raft foundation 2.2.1. Simplified method – Randolph [6] method This method is based on calculation of the total stiffness of the piled raft by means of the stiffness of the pile group and the stiffness of an unpiled raft in isolation and the interaction between one pile with the region of the raft surrounding the pile. Thus, the

1. Pile-soil-pile interaction 2. Pile-soil-raft interaction Fig. 1. The interactions in a piled raft foundation system.

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settlement of the foundation and the ratio of transmitted load to the raft can be calculated. This method can obtain the behaviour of the piled raft in the form of a tri-linear load–settlement curve [10]). Nevertheless, this method only considers the interaction between the piles and the raft and not the interaction between piles in the pile group. The application, however, is quiet easy for hand calculation, as the method is fairly straightforward. 2.2.2. Approximate method – plate on springs approach [1] This method is based on the elastic theory and interactions between the components of the piled raft foundation. Poulos [1] modelled a piled raft in the form of a plate supported by springs representing piles. This method is implemented via the program GARP (Geotechnical Analysis of Raft with Piles) which allows consideration of the layered soil profile to failure behaviour and the effect of piles reaching their ultimate capacity. Four interactions were considered in this program, interaction between elements of the raft, interaction between piles, influence of the raft on the piles, and influence of the piles on the raft. The remarkable advantage of this method is that it can obtain the distribution of the stress inside the raft and can consider the ultimate capacity of piles. Nevertheless, the behaviour of piles under the transmitted load depends on the soil model, with many parameters required when using the GARP program. This can cause that the behaviour of piles deviate from the real behaviour if the soil is not modelled reliably. Consequently, the obtained settlement of the foundation will inevitably include some errors. Moreover, the based on the elastic theory of the analysis is another limitation, and its complexity also prohibits its application for design (it can be only applied by using the GARP program). 2.2.3. More sophisticated computer-based methods 2.2.3.1. Hain and Lee [3] method. Hain and Lee [3] analysed two components, soil and piles, to solve the problem of a piled raft, where the soil elements and the pile elements were arranged compatibly with the raft. In the model, the soil surface was meshed into a number of square or rectangular 4-node finite elements. The nodes located at the piles are called pile nodes and the remaining nodes are referred to as soil nodes. The pile–pile interaction and pile–soil interaction are used to calculate the vertical settlement of each node. The total supporting soil–pile group stiffness can then be obtained and the settlement and stress on the raft are estimated. This analysis rigorously considers the interaction between soil and piles and solves many problems such as flexibility of the raft, ultimate capacity of piles. Nevertheless, this method also has several limitations. First, if the raft consists of a series of bending plates, the number of soil nodes will be very high, and the calculation cost is consequently large. Conversely, if the soil nodes are reduced, the number of bending plates is decreased, and the calculation results, especially the internal stress of the raft, will be less accurate. In addition, because the effect of the raft in the interactions is not considered and the pile stiffness is only correlated with Young’s modulus of the pile (Ep) and Young’s modulus of the soil mass (Es) (Kp = Ep/Es), the accuracy of the results will be further worsened. 2.2.3.2. Reul and Randolph [16] method. This method bases totally on the finite element method. The authors used Abaqus program to simulate piled rafts and proposed that modelling the soil and foundation by finite elements can allow the most rigorous treatment of the soil–structure interaction. This method has several remarkable advantages. First, the soil can be modelled as a multiphase medium which consists of three components solid phase (grains), liquid phase (pore water) and gaseous phase (pore air), so the geotechnical characteristics of soil can be considered effectively. Second, the nonlinear material behaviour of the soil can be taken in to account with

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Table 1 Approaches for determining interaction factors. Method

Type of interaction

Equation

Comment

Poulos and Davis [4]

Pile–pile interaction ( a)

a ¼ DWW

– Considering the additional settlement of a pile (DW) caused by an adjacent pile

Poulos [9]

Pile–pile interaction ( a)

Dwk ¼ x1

Randolph [11]

Pile–raft interaction

– The contribution of the adjacent pile was not presented clearly in the equation

arp ¼ 1

Pn

j¼i Q j akj

ln ln

rr rp

2rm rp


ln


rr

– Not considering the change in soil stiffness along the pile and the flexibility of the raft. The parameter for considering soil stiffness is q which is the degree of homogeneity of the soil – Considering the additional settlement of a circular rigid raft caused by a pile

rp arp ¼ 1

ln

Clancy and Randolph [2]

– Considering the additional settlement of a circular rigid raft caused by a pile

rm = 2.5qL(1 ts) Randolph [6]

– Considering the additional settlement of a pile caused by adjacent piles in the term of axial loads of adjacent piles Qj

rm rp

rm = 0.25 + Lf[2.5q(1 t) 0.25] f = Esl/Esb q = Esav/Esl arp ¼ Pkpp wpr Pkrr

– Considering the soil stiffness along the pile (Esav), at pile tip (Esb) and pile head (Esl) – Not considering the flexibility of the raft – The settlement of piled raft and load transmitted for piles and for raft are needed – Can consider the flexibility of the raft

Nomenclature: Dwk = additional of pile k caused by other piles; x1 = displacement due to unit load of pile k and pile j; Qj = the load on pile j; akj = interaction factor for pile k due to any other pile j within the group; arp = interaction factor between a pile and a raft; rr = the diameter of the raft; rp = the diameter of the pile; q = the degree of homogeneity of the soil; L = the length of the pile; ts = Poisson ratio of soil; Esl = soil Young’s modulus at level of pile tip; Esb = soil Young’s modulus of bearing stratum below pile tip; Esav = average soil Young’s modulus along pile shaft; Pp = total load carried by pile group in combined foundation; Pr = total load carried by raft in combined foundation; kp = overall stiffness of pile group in isolation; kr = overall stiffness of raft in isolation; wpr = settlement of a piled raft foundation.

the elastoplastic cap model. Third, the plastic behaviour of soil can be considered by the nonassociated flow potential (Gs) of the shear surface and the associated flow potential (Gc) of the cap. Fourth, the contact between structure and soil and the various types of applied load can be simulated. Nevertheless, this method still has some problems. In modelling pile–soil interaction, the interface element was not used so the method could not consider the relative motion between the pile elements and soil elements. Moreover, the calculating time for obtaining a solution is long and Abaqus program is not easy for practicing engineers to use. In general, the existing methods can be employed to solve the piled raft problem fairly completely. However, there exist some limitations as mentioned above, especially with application to engineering work, because they are quite complex (i.e., the approximate methods and more rigorous computer-based methods). This paper hence proposes an analysis method to solve several problems, including: Simplification for ease of application for practicing engineers. Solving piled raft problem without any sophisticated finite element model for soil and helping practicing engineers control well the mechanism of piled rafts. Using a combination of the single pile and unpiled raft behaviours to estimate the behaviour of the piled raft fairly exactly. Obtaining nonlinear behaviour of a piled raft foundation by using the nonlinear behaviour of the single pile. Estimation of the settlement of the foundation and the distribution of the bending moment in the raft with reasonable accuracy. 3. Proposed design method 3.1. Modelling of piled raft foundations In this study, the raft is modelled as a series of bending plates, each pile is modelled as a pile spring at the pile’s position, and the

relative raft–soil stiffness is modelled by means of raft springs with the quantity and the position decided by the designer, as shown in Fig. 2. However, in order to solve the stiffness of the pile springs, it is convenient to assume that vertical forces are only transmitted from the raft to the head of a pile [3]. This assumption involves neglecting the lateral pile head force and the lateral pile movement. In general, the total vertical loads are considerably greater than the total lateral loads, so the lateral movements of the raft are small. In the case of large lateral loads subjected to the raft, the batter piles will be used and can be modelled as lateral pile springs. However, this problem is not considered in the scope of this paper. Then, the vertical displacement of a pile is given by: n1 X

wpK ¼

d1J P pJ aKJ þ d1K PpK

ð1Þ

J¼1;J–K

where wpK is the vertical displacement of the pile K; d1J, d1K are the displacement due to the unit load of the piles J and K, respectively, which can be derived from the load–settlement curve of a single pile having the same size; PpJ is the load on pile J; aKJ is the pile–soil–pile interaction factor of pile J on pile K; PpK is the load on pile K; and n is the number of piles. The stiffness of pile spring K is given by:

K p ¼ K pK ¼

PpK wpK

ð2Þ

where KpK is the stiffness of pile spring K To solve the stiffness of raft springs, the lateral force and lateral movement are also neglected. The vertical displacement of a raft spring is given by:

wrpM ¼

n X

ðq1M Q M bKM Þ þ wrM

ð3Þ

K¼1

where wrpM is the vertical displacement of the raft spring M in consideration of pile interactions; wrM is the vertical displacement of the raft spring M without pile interactions; q1M is the displacement

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Fig. 2. Model of piled raft foundation. (a) A piled raft foundation. (b) Modelling for proposed design method.

of the raft spring M due to the unit load, which can be calculated from elastic theory or derived from the load–settlement curve of an unpiled raft having the same size as the raft of piled raft; QM is the load on the raft spring M; and bKM is the pile–soil–raft interaction factor of pile K for the raft spring M. The stiffness of the raft spring M is given by:

K r ¼ K rM ¼

QM wrpM

ð4Þ

where KrM is the stiffness of the raft spring M 3.2. Pile–soil–pile interaction factor The pile–soil–pile interaction factor, a, is used to calculate the additional settlement of a pile caused by adjacent piles. In the case of two piles K and J, additional settlement for pile K can be written as follows:

DwK ¼ x1J Q J aKJ

ð5Þ

Table 2 Plaxis 3D input parameters. Parameter

Input value

Soil material L/d = 25 and Dr = 70% Dry density (cd) Secant Young’s modulus (E50) Friction angle (/) Confinement pressure Poisson’s ratio (t)

15.13 kN/m3 50 MPa 43° 150 kPa 0.25

L/d = 25 and Dr = 40% Dry density (cd) Secant Young’s modulus (E50) Friction angle (/) Confinement pressure Poisson’s ratio (t)

13.7 kN/m3 25 MPa 40° 150 kPa 0.25

L/d = 16.7 and Dr = 70% Dry density (cd) Secant Young’s modulus (E50) Friction angle (/) Confinement pressure Poisson’s ratio (t)

15.13 kN/m3 28.5 MPa 43° 100 kPa 0.25

Pile material Young’s modulus (E) Density (c) Poisson’s ratio (t)

2.82E+04 MPa 15 kN/m3 0.16

Then,

aKJ ¼

DwK x1J Q J

ð6Þ

where aKJ is the interaction factor of pile J on pile K; Dwk is the additional settlement of pile K caused by pile J; and x1J is the settlement due to a unit load of pile J; QJ is the load on pile J The interaction factor between pile K and pile J can be defined as follows:

aKJ ¼

additional displacement of pile K caused by unit load on pile J displacement of pile J due to unit load ð7Þ

In this paper, the finite element method (FEM) via Plaxis 3D Foundation program is employed to obtain this factor. The input parameters are presented in Table 2. The soil used is silica sand with two different relative densities, 70% and 40%. The parameters of soil are chosen at the level of depth about two-third of the pile length from the pile head. The characteristics of the tested soil taken from the drained triaxial tests are presented in detail in the tested soil section. In the FEM, the soil is considered as a hardening model of the case of hyperbolic relationship for standard drained triaxial test and the dilatancy of the sand is also considered [15]). The piles are represented by embedded pile elements. The embedded pile beam

can be placed arbitrarily in a soil volume element, and at the position of the beam element nodes virtual nodes are created in the soil volume element from the element shape functions. Then, the special interface forms a connection between the beam element nodes and these virtual nodes, and thus with all nodes of the soil volume element. The interaction with soil at the pile skin and at the pile foot is described by means of embedded interface elements. These interface elements are based on 3-node line element with a pairs of nodes instead of single nodes. One node of each pair belongs to the beam element, whereas the other (virtual) node is a point in the 15-node wedge element belonging to soil element. The skin interaction is taken into account by the development of skin traction and the foot interaction is considered by the development of the foot force. First, the model of a single pile subjected to load F was simulated to obtain the settlement. The modelling of two separate piles subjected to the same load F with the given spacing (S) was then

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performed to obtain the settlement of each pile. The additional settlement was calculated by the difference in the settlements between the case of a single pile and the case of two piles. Using Eq. (7), the pile–soil–pile interaction for the given pile spacing can be obtained. In this paper, the number of pile spacing (up to S/d = 16) was selected corresponding with the piled raft models of centrifuge tests. Thus, a large value of pile spacing is not necessary for this study. The pile–soil–pile interaction curve was constructed as shown in Fig. 3. Fig. 3a shows a comparison between the FEM results obtained for silica sand (Dr = 70% and t = 0.25) and the results of Poulos and Davis [4] (overall stiffness K = 1000 and Poisson’s ratio t = 0.5) in the case of a length–diameter ratio L/d = 25. General agreement in the trend that the interaction factor decreases with increasing pile spacing between the two results is observed. The difference in the values originates from the different Poisson’s ratio, type of soil, and the analysis method, but is still acceptable. Poulos and Davis [4] suggested the pile–soil–pile interaction curves for various types of soil and pile lengths based on an elastic continuum analysis. These curves can also be used to obtain the pile–soil–pile interaction factor in an alternative way if the FEM is unavailable. Fig. 3b and c shows pile–soil–pile interaction factors, obtained by the finite element method, as a function of the pile spacing to diameter ratio (S/d). When this ratio increases, the interaction factor decreases. This means that when the distance (S) is small, the additional settlement of a pile caused by the other pile is large but when (S) is large, the additional settlement becomes small and the behaviour of a pile more closely approximates the behaviour of a single pile. The interaction factors are also dependent on the density of soil and the ratio between the pile length and the diameter of the pile (L/d). It simply explains in prose the form of the interaction equation that the coefficient a depends on the displacement of pile J due to unit load so when the relative density of soil decreases the settlement due to unit load of pile J increases and thus it makes the value of a reduces. In the case of increasing pile length, when the length of pile J increases the capacity of this pile develops making the settlement due to unit load of this pile lessens and thus the value of a increases. Fig. 3 presents a comparison of pile–pile interaction curves for two states of silica sand, dense and loose states. 3.3. Pile–soil–raft interaction factor The FEM was employed to construct the pile–soil–raft interaction curve. The soil same with the soil used to determine the pile–soil–pile interaction was represented. For the given pile spacing, to derive the pile soil raft interaction factor, b, it was necessary to prepare two separate models. The first model was a piled raft (four piles with the given pile spacing) and the second model was an unpiled raft having raft with the same size as that considered in the piled raft model as shown in Fig. 4. The additional settlement (DW) of the raft caused by piles was obtained. The applied load for the unpiled raft model equals the transmitted load to the raft in the piled raft model. The additional settlement equals the difference in the settlement values for the raft between the two models. The additional settlement caused by a pile for the raft equals DW divided by the number of piles. The pile–soil–raft interaction coefficient, b, is then determined as follows:

b¼

additional displacement of the raft caused by a pile displacement of the unpiled raft

ð8Þ

The pile–soil–raft interaction curves were constructed as shown in Fig. 5. Fig. 5a provides a comparison of pile–soil–raft interaction factors between the results obtained by the FEM and by the

Fig. 3. Pile–soil–pile interaction factors for homogeneous soil layer. (a) Comparison of interaction factors between FEM result obtained for Silica sand Dr = 70%, t = 0.25 with the result of Poulos and Davis [4] K = 1000, t = 0.5. (b) Comparison of interaction factors between dense and loose sands in same L/D ratio. (c) Comparison of interaction factors between two different L/D ratios in dense sand.

equation from Randolph [6]. The curve constructed by Randolph’s approach was calculated with a Young’s modulus of soil at the pile head Esl = 8.61 MPa, Young’s modulus at the bearing stratum below the pile tip Esb = 74.75 MPa, the average Young’s modulus along pile shaft Esav = 46.63 MPa, and Poisson’s ratio t = 0.25 (these parameters are derived from silica sand in a dense state as used for the study in this paper). It can be seen that there is good agreement between the two set of results, both in general trends and in numerical values. This indicates that the factor can also be determined by the method of Randolph [6] in an alternative way if the FEM is unavailable. Fig. 5b and c shows the pile–raft interaction factor, obtained from finite element method, as a function of the ratio S/d. When the distance (S) increases, the interaction factors decrease. The interaction curves tend to converge to a value of about 10% when the distance (S) becomes large. In comparison, the interaction curve of L/d = 25 is higher than that of L/d = 16.7. This illustrates that as the length of the pile and the soil stiffness respectively increase, the interaction effect between the pile and one point of the raft becomes accordingly higher. The reason is that the coefficient b depends on the displacement of the unpiled raft (according to Eq. (8)). When the relative density of soil increases the capacity of the unpiled raft develops so its settlement reduces, and thus the value of b increases. In the case of increasing pile length, the interaction


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length of piles to the raft increases raising the pile–soil–raft interaction up. 3.4. Analysis procedure using SAP 2000

Fig. 4. Model scheme for obtaining pile–soil–raft interaction.

Fig. 5. Pile–soil–raft interaction factors for a square raft in homogeneous soil. (a) Comparison of interaction factors between FEM and Randolph [6] in dense sand. (b) Comparison of interaction factors between dense and loose sands in same L/D ratio. (c) Comparison of interaction factors between two different L/D ratios in dense sand.

The proposed method is used to estimate the settlement and bending moment induced in the raft of two piled raft models which are performed in centrifuge tests. To conduct this method in a civil engineering analysis package and instruct the practicing engineer reproducing the method, the SAP 2000 structural commercial program, which is being used by many civil design companies at present, is used. This program cannot simulate any soil model but with the help of the proposed method, it can solve the piled raft problem well. The analysis requires the following input data: the single pile behaviour, the unpiled raft behaviour, and the pile–soil–pile and pile–soil–raft interaction factors. The pile–soil–pile interaction factor plays a role of connecting pile springs working as a pile group while the pile–soil–raft interaction factor helps the plate models operate as a real raft in the piled raft foundation. As final outputs, the settlement of the piled raft and the distribution of the bending moment and shearing force of the raft can be obtained. Fig. 6 shows a flow chart of the analysis procedure. When the assumed load transmitted to the piles in the piled raft in the first calculation is larger than the piles’ capacity, the ‘‘load cut-off’’ procedure [3]) is applied, where the piles’ capacity is taken to calculate the settlement of pile springs. The remaining load is transmitted to the raft to calculate the settlement of the raft springs. The analysis procedure using SAP 2000 is as follows: 1. Model the piled raft foundation with SAP 2000, where the raft is modelled by a bending plate having the same size as the raft, piles are modelled by pile springs and relative raft–soil stiffness is modelled by raft springs. The plate is meshed into a series of small plates to solve the bending moment and stress of the raft. 2. Determine the pile–soil–pile interaction factors for each pile spring and pile–soil–raft interaction factors for each raft spring based on the interaction curves. 3. From the total applied load, assume the load for piles and the load for the raft (about 80% and 20% of the total applied load, respectively), and assume the axial force transmitted for each pile spring and each raft spring (usually at the first calculation, assume that all axial forces of all pile springs are equal and all the axial forces of all raft springs are equal). 4. Calculate the vertical settlement for each pile spring by Eq. (1) and each raft spring by Eq. (3). The displacements of the pile due to unit load in Eq. (1) (d1J and d1K respectively) are derived from the load–settlement curve of a single pile, and the displacement due to the unit load for the raft spring (q1M) is calculated from the load–settlement curve of an unpiled raft or the solution of Boussinesq for shallow foundations [17]. 5. Calculate the stiffness of each pile spring by Eq. (2) and the stiffness of each raft spring by Eq. (4). 6. Assign all the calculated stiffness for all springs of the piled raft model in SAP 2000. This step establishes the boundary conditions for the plate model. 7. Solve the system of equations to obtain the preliminary settlement of the piled raft and axial forces transmitted for each spring. 8. Calculate the difference in the axial forces of pile springs with a tolerance of about 5–7%. If the differential values are larger than 7%, repeat the calculation from step 4 to step 8 to estimate the stiffness of all springs and the settlement of the foundation a second time. This iterative process is terminated when the

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strong model box (900 mm in a diameter) until the thickness of the sand layer was approximately 400 mm (corresponding with 20 m at 50 g and 24 m at 60 g in the prototype scale). 4.2. Test program and models In order to verify the applicability of the proposed method, two cases of piled raft model tests were evaluated in the centrifuge tests for comparison; the two piled rafts have different numbers of piles as well as different pile length and pile spacing. The first case is sixteen piles (15 m pile length and 0.6 m in diameter in prototype) and 4d pile spacing, and the second case is nine piles (10 m pile length and 0.6 m in diameter in prototype) and 3d pile spacing. The piled raft models were loaded by loading equipment fixed to the rigid frame of the box. A load cell was installed in the equipment to measure the amount of total applied load. The raft settlement was measured by two linear displacement transducers (LVDT) fixed on a frame that was connected to the frame supporting loading equipment. The tips of the core of LVDTs rested on the two opposite corners of the raft. The average values of data measured by the two LVDTs were taken as the settlement of the foundation. The test model set-up is shown in Fig. 7b. The target of

Fig. 6. Schematic of flow chart for SAP 2000 analysis.

differential axial forces of the pile springs are in a range of 5–7%. This tolerance is based on experience so it may be chosen at the designer discretion. 4. Centrifuge testing program Centrifuge tests were performed using the 240 g ton geotechnical centrifuge equipment at KAIST (Korea Advanced Institute of Science and Technology) in Korea. The maximum capacity of the KAIST beam centrifuge, with a 5 m radius, is 2400 kg for up to 100 g of centrifugal acceleration and 1300 kg at 130 g of maximum centrifugal acceleration. The detailed specifications of the centrifuge equipment can be found in Kim et al. [14]. Fig. 7a shows the centrifuge equipment with the testing system developed in this study. The tests were carried out at two centrifugal acceleration levels, 50 g and 60 g. Two centrifugal accelerations were adopted for making a difference in the pile length between two piled raft models. 4.1. Tested soil Silica sand, with particle mean diameter D50 = 0.22 mm, a uniformity coefficient CU = 1.96 and classified as SP type (according to the Unified Soil Classification System), was used for all centrifuge tests. Triaxial drained tests were performed to obtain characteristics of the tested soils, with relative densities, DR 70% and DR 40%. The test results are presented in Table 3. A homogeneous dry soil model was prepared by means of pluvial deposition to relative densities DR of 70% and 40% (dense state and loose state, respectively) using a travelling sand spreader, which controls the fall height and travel speed of the deposition curtain. The spreader was passed repeatedly over the circular

Fig. 7. KOCED geotechnical centrifuge with testing system and test model set-up. (a) KOCED geotechnical centrifuge with testing system. (b) Test model set-up.

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D.D.C. Nguyen et al. / Computers and Geotechnics 47 (2013) 16–27 Table 3 Silica sand parameters. Relative density

Confinement pressure (kPa)

Depth (m)

E (MPa) e > 0.2%

Peak friction angle (/)

Critical state friction angle (/cr)

Dense state Dr = 70% (cd = 1.49 t/m3)

50 100 200

3.4 6.8 13.6

8.61 28.42 74.75

43°

33.5°

Loose state Dr = 40% (cd = 1.37 t/m3)

50 100 200

3.8 7.6 15.2

8.47 13.33 36.84

40°

these tests is obtaining the load–settlement curves of the two piled raft models in model scale. Then, these curves are converted to prototype scale using the laws of similitude as in Table 4 [18]) to compare with the load–settlement curves of the two piled raft calculated by the proposed method and Plaxis 3D Foundation. The material and thickness of the pile model and the raft model are selected in considering the equivalent stiffness between the model scale and the prototype scale. In the case of the pile model, the equilibrium of axial strain is considered, and in the case of the raft model the equilibrium of deflection is considered. The number of centrifuge tests performed is 10. The test programme are summarised in Table 5. Two tests were performed at 50 g on the single pile model (pile length 220 mm) and a piled raft model having nine piles (pile length 200 mm and three piles were equipped with load cells to measure transmitted axial forces). In the piled raft tests, the rafts need to be completely contacted with the soil surface, whereas in the pile group tests the rafts (or the caps) do not contact with the soil surface. In order to make the same penetrated pile length in both piled raft tests and pile group tests, the pile length model of pile group tests is longer than the one of piled raft tests about 20 mm (1.2 m at 60 g in the prototype scale). The models were made of aluminium alloy (E = 7E+04 MPa). Details of all models are summarised in Table 6, where the prototype and model scale dimensions are reported. 4.3. Test procedures All models are decided to install at 1 g because if models are installed at 50 g or 60 g the pile group capacity will be increased by 502 or 602 times. Thus, it is very difficult to penetrate the models into the soil with the loading equipment ability used. The following test procedures have been adopted: At 1 g: a homogeneous soil model was prepared by pluvial deposition into the circular box. The box was then placed into the centrifuge basket and the model was then installed to the loading equipment and penetrated the soil. Single pile tests: the embedded lengths are 250 mm for the pile models having 270 mm length and 200 mm for the pile model having 220 mm length. Piled group tests: the embedded length is 250 mm. Unpiled raft tests: the unpiled raft model was placed on the surface of the soil. Piled raft tests: the piled raft model was penetrated into the soil until the raft reached the soil surface.

Table 4 Scaling factor for centrifuge modelling [18]. Parameter

Scaling factor

Parameter

Scaling factor

Acceleration Stress Mass Stiffness

N 1 1/N3 1/N

Length Strain Force Time (diffusion)

1/N 1 1/N2 1/N2

Table 5 Test program. UR = unpiled raft; SP = isolated single pile; PG = pile group; PR = piled raft. Scheme

Test

No. of piles

Pile spacing

Dr of soil (%)

Acceleration N (g)

UR UR SP SP SP PG PG PR PR PR

T1 T2 T3 T4 T5 T6 T7 T8 T9 T10

– – 1 1 1 16 16 16 16 9

– – – – – 4D 4D 4D 4D 3D

70 40 70 40 70 70 40 70 40 70

60 60 60 60 50 60 60 60 60 50

At 50 g or 60 g: After the soil surface settles down completely, the models were penetrated to the soil about 23 mm to ensure the embedded length of piles still equals to the length of piles in the piled raft and/or for the raft perfectly contacted with the soil surface (2–3 mm is the settlement of the soil surface during the increase of the centrifugal acceleration from 1 g to 50 g or 60 g). The loading tests were then performed. For all tests, loading equipment penetrated the models at a rate of 0.04 mm per second (rate of penetration) until a relative displacement w/dp 30% was reached, where w is the measured settlement.

5. Results and discussion 5.1. Single pile tests Fig. 8 shows the load–settlement curves (Psp–wsp curves) of three single piles. It is observed that there are three stages in behaviour when a pile is subjected to a load. In the first stage, the wsp increases almost linearly when the Psp increases up to the critical load point. In the second stage, the Psp–wsp response becomes more curved, because the wsp increases more than the first stage when the load increases. This stage can be called the critical stage of the pile. In the third stage, the path of the pile’s behaviour becomes nearly linear again, but the wsp develops quickly although the Psp does not increase substantially. Fig. 8 also presents a comparison of the behaviour of a single pile among three cases of centrifuge tests. The bearing capacity of a pile includes shaft resistance and end bearing resistance. The end bearing depends on the strength of the soil below the pile tip and the shaft resistance is mainly based on the length of a pile. In the cases of two single pile tests having the same pile length L = 15 m in different soil densities, the pile in the dense state shows stiffer behaviour than that in the loose state. Nevertheless, the single pile of L = 15 m in loose sand is still stiffer than the single pile of L = 9 m in dense sand. This indicates that when the pile length increases the stress state increases corresponding with the increase of embedded depth. This helps to increase considerably the pile capacity.

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Table 6 Dimensions of centrifuge models. Dm-in = inner diameter of pile; Dm-out = outer diameter of pile; L = length of pile; Br = width of raft; tr = thickness of raft. Scheme

UR UR SP SP SP PG PG PR PR PR

Model scale

a/g

Dm-in (mm)

Dm-out (mm)

Lm (mm)

Brm (mm)

trm (mm)

– – 8 8 10 8 8 8 8 10

– – 10 10 12 10 10 10 10 12

– – 270 270 220 270 270 250 250 200

150 150 – – – – – 150 150 180

15 15 – – – – – 15 15 18

60 60 60 60 50 60 60 60 60 50

Prototype scale Dm-in (m)

Dm-out (m)

Lm (m)

Brm (m)

trm (m)

– – 0.2 0.2 0.3 0.2 0.2 0.2 0.2 0.3

– – 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

– – 16.2 16.2 11 16.2 16.2 15 15 10

9 9 – – – – – 9 9 9

1.22 1.22 – – – – – 1.22 1.22 1.22

Fig. 9. Centrifuge raft tests’ result in prototype scale.

Fig. 8. Centrifuge single pile load tests’ result in prototype scale.

The Psp–wsp curve of the single pile is an important input value in the proposed method. The settlement of a pile due to the unit load (d1) (in Eq. (1)) can be obtained by normalising the settlement of the pile with the corresponding load. It is important to note that, in the third stage, the piles reach their limit capacity and the wsp increases considerably, and as a result the settlement of piles due to the unit load cannot be obtained from this stage. 5.2. Raft tests The test results are presented in Fig. 9. It can be seen that the load–settlement behaviour of an unpiled raft is almost linear and the stiffness of the raft on loose sand is much lower than that on dense sand. To calculate the stiffness of the raft springs, the settlement due to the unit load (q1) (in Eq. (7)) is needed and this value can be obtained from the load–settlement behaviour of the unpiled raft. It is noticed that in the case of silica sand this behaviour is almost linear, and thus the unpiled raft stiffness can be estimated via elastic theory as an alternative approach. 5.3. Piled raft tests The results in Fig. 10 show that in the case of this study the piled raft’s behaviour has two stages, where the first stage is curved and the second stage is linear. In the first stage, the piled raft’s behaviour is governed by the piles’ behaviour when they are still in working ability. Under the application of external load, the piled raft settles down and this settlement provides the main contribution to pile settlement. In this stage, the raft supports a small amount of the total load (about 10–20% of total load; [10])

and reduces the settlement of the piled raft relative to the case of the pile group. Nevertheless, when the external load increases to a very large value the piled raft’s behaviour becomes linear (it is assumed that at this time the piles reached their limit capacity). This indicates that in this stage the piled raft’s behaviour is governed by the raft’s behaviour. Initially, the subjected load is transmitted to the piles until reaching their critical capacity. The load is thereafter transmitted to the raft, and the settlement of the piled raft at this time increases corresponding with the increase of settlement of the unpiled raft. Additionally, the results of the centrifuge test also illustrate that the settlement of the piled raft is smaller than that of the unpiled raft due to the contribution of the piles. Besides, in Fig. 10a, with dense sand state the difference in load–displacement (P–w) response between the piled raft and the pile group is small when the total load is less than 50,000 kN. The P–w path of the piled raft just deviates from the one of the pile group when the total load passes 50,000 kN and the difference becomes larger when the load continues to increase. The reason is that the settlement of the piled raft in large applied load is smaller than the pile group because of the contribution of the raft. However, Fig. 10b shows that in loose sand state the P–w path of the piled raft starts to deviate from the one of the pile group when the load is just about 15,000 kN. The piled raft shows much stiffer behaviour than the pile group when comparing with the dense sand state case. It can be concluded that a piled raft can give an increased benefit over a pile group in the looser sand.

5.4. Comparison of piled raft behaviour between centrifuge test, proposed method and Plaxis 3D analysis Two piled rafts are simulated by means of three-dimensional finite element via Plaxis 3D Foundation program in the prototype


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Fig. 10. Piled raft centrifuge testing results. (a) Piled raft with 16 piles (D = 0.6 m; L = 15 m) in dense sand. (b) Piled raft with 16 piles (D = 0.6 m; L = 15 m) in loose sand.

scale to obtain the load–settlement curves. The input parameters are presented in Table 2. In this analysis, the soil is modelled in dry state using a hardening model of the case of hyperbolic relationship for standard drained triaxial test and the dilatancy of the sand is also considered (similar with the simulation of soil in determining the pile–soil–pile and pile–soil–raft interaction factors section). The soil parameters are selected at the level of two-third of the pile length from the pile head. The size of the soil block consists of 54 m length and 24 m height for the case of the piled raft having pile length 15 m (corresponding with the size of the soil model scale using for centrifuge tests 900 400 mm). In the case of the piled raft having pile length 10 m, the size of the soil block is 45 20 m. Bottom and all four sides of soil block are fixed. The raft is represented by floor elements which are structural objects used to model thin horizontal (two-dimensional) structures in the ground with a significant flexural rigidity (bending stiffness). The piles are represented by embedded pile elements and the pile–soil interaction is considered by means of skin and foot interactions. Fig. 11 shows a comparison of total load–average settlement curve obtained by centrifuge tests, the proposed method using SAP 2000, and analysis results obtained with the Plaxis 3D program. It is seen that the results of Plaxis 3D do not match well with the results of the centrifuge test, meanwhile the results of SAP 2000 match well with the results of the centrifuge test. The closer agreement of the proposed method with the centrifuge test results from using the load–settlement curves of the single pile and an unpiled raft having the same size piles and raft as the piled raft as the input data for the proposed method. In addition, the ‘‘load cut-off’’ procedure, i.e., when the piles reach the limit capacity, the additional applied load is transmitted to the raft, helps to estimate

Fig. 11. Piled raft’s behaviour comparison between centrifuge test, proposed method analysis and Plaxis 3D foundation analysis. (a) Piled raft with 16 piles (D = 0.6 m; L = 15 m) in dense sand. (b) Piled raft with 16 piles (D = 0.6 m; L = 15 m) in loose sand. (c) Piled raft with 9 piles (D = 0.6 m; L = 9 m) in dense sand.

the limit point of the piled raft capacity at which the load–settlement curve changes from nonlinear to linear. The deviation of Plaxis 3D results may relate to the nonstrain-softening constitutive model assumed and the use of embedded pile elements. Piled raft modelling is a difficult job since there are many parameters affecting to the piled raft behaviour. The embedded piles can be used and this can give a good evaluation in settlement when a piled raft has not reached the failure yet. The embedded piles also help to reduce the required number of elements to model a piled raft and consider the pile–soil interaction well. However, the group effect and installation effect are not considered. Even if the soil is perfectly modelled, deviations from actual behaviour occur due to pile installation. All these factors can affect to the piled raft behaviour in the modelling of Plaxis 3D. The experience with the piled raft

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Fig. 12. Distribution of total bending moment. (a) Plaxis 3D. (b) SAP 2000.

model should be gathered with time and shared among the Plaxis users. In comparison with the case of Fig. 11a and c shows that the first stage of the piled raft behaviour is more linear. The reason for this result is maybe that the piles were arranged in the centre region of the raft and the pile length is shorter. It helps the raft gives more contribution to the pile raft capacity and makes the piled raft behaviour more linear. However, this phenomenon needs more experiments to give the more accurate conclusion. In general, the behaviour of a pile raft is complex and based on many factors such as pile length, number of piles, arrangement of piles and state of soil, but the general trend is that the first stage of the piled raft behaviour is controlled by piles and the second stage is controlled by the raft. 5.5. Comparison of bending moment between proposed method and Plaxis 3D analysis In the practical design of a piled raft foundation, obtaining the distribution of the bending moment and its maximum value in the raft is crucial. Based on those values, the quantity and distribution of reinforcing steel used for the raft are determined. To verify the validity of the results obtained by the proposed method, the moment distributions obtained by the proposed method are compared with those given by Plaxis 3D, which is frequently used in practice. Fig. 12 shows the distribution of total bending moment of the raft (piled raft with sixteen piles in dense sand) at the total load of about 30,000 kN. The two methods show the same trend that the maximum bending moment is distributed in the central region of the raft and the moment reduces from the centre to the edges of the raft. Fig. 13 presents a comparison in difference of the maximum bending moment. The maximum bending moment of SAP 2000 is greater than that of Plaxis 3D for the cases of 16-pile raft. In fact, this is beneficial for the raft design, as it is more conservative.

Fig. 13. Comparison in difference of maximum bending moment of the raft between SAP 2000 and Plaxis 3D.

However, in the case of 9-pile raft the difference of maximum bending moment between SAP 2000 and Plaxis 3D is fairly small. It can be proved that the maximum bending moment is related to the arrangement of piles, and the results obtained from the proposed method are reliable. 5.6. Comparison of individual piles behaviour between centrifuge test, proposed method and Plaxis 3D analysis In order to determine the behaviour of an individual pile in the piled raft foundation, load cells were installed at the piles to measure the transmitted axial load [13]. A comparison of the behaviours of individual piles 1, 2, and 5 is shown in Fig. 14. According to the results of the centrifuge test, the behaviour of the individual piles in the piled raft shows a similar trend to the single pile behaviour. Initially, the load transmitted to each pile increases nearly linearly with


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6. Conclusions An analytical method for calculating characteristics of a piled raft foundation based on consideration of the interactions between the components of a piled raft foundation, such as the piles, raft, and soil, is proposed in this paper. A comparison of the results obtained by the proposed method with those provided by Plaxis 3D and centrifuge tests shows that the method can estimate the settlement of a piled raft foundation quite accurately and the distribution of stress in the raft reasonably well. The suggested method offers a number of other advantages. First, using the behaviour of a single pile from test data allows the development of the ultimate load of the piles, and also makes it possible to consider a nonlinear response of the foundation system. Second, the combination of pile springs and raft springs allows convenient consideration of the roles of piles and raft. Thus, taking into consideration the pile ‘‘load-cut off’’ procedure, within piles’ capacity, the piles are dominant in terms of the piled raft behaviour, and beyond the piles’ capacity, the raft is dominant. Third, it allows for the use of any structural commercial program to solve the piled raft foundation problem. It can be concluded that the piled raft foundation problem can be solved effectively through a combination of structural responses and geotechnical characteristics without a complex model of the soil and foundation. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-0000154). References

Fig. 14. Comparison of behaviour of pile 1, pile 2 and pile 5 between the result of Centrifuge test, SAP 2000 and Plaxis 3D (prototype scale).

an increase of settlement. After reaching the critical load, the settlement increases with the faster rate than the initial stage when the transmitted load continues to increase, resulting in a curved load– settlement path. The settlement then develops steeply, even though the load is only increased slightly. In this stage, the behaviour of the pile is almost linear. Meanwhile, the calculation by SAP 2000 provides better agreement with the results of centrifuge test than the Plaxis 3D calculation results. The behaviour of piles according to Plaxis 3D is almost linear, whereas the behaviour obtained by the centrifuge test and SAP 2000 are nonlinear. The load–settlement curves of the piles obtained by SAP 2000 and the proposed method also shows three stages of behaviour, as in the centrifuge test, because the input data for calculating the pile spring stiffness were derived from the single pile behaviour.

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