Simplified analysis of piled rafts with irregular geometry

Vrettos, C. Technical University of Kaiserslautern, Germany

Keywords: piled rafts, analysis methods, case studies

ABSTRACT: Piled rafts are increasingly used for the foundation of high-rise buildings on competent ground. During preliminary design, the position and geometry of the piles are optimized in order to minimize differential settlements and consequently the sectional forces in the slabs. In engineering practice, the analysis of this complex soil-structure interaction problem is performed by means of a pseudo-coupled procedure that yields as output the spring stiffness for each pile and the modulus of subgrade reaction for the raft. The paper presents an approximate method for determining these values for piled rafts with irregular geometry. It is based on the linear-elastic analysis method suggested by Randolph with appropriate modifications to include in an approximate manner variable pile distances and pile lengths within the pile group. If further considers the different behavior of edge and central piles and also the influence of the pile load level. The method can be easily implemented into a spreadsheet program. The application is shown by means of case study of a high-rise founded on clay. 1 INTRODUCTION

Piled rafts are a new foundation concept for important high-rise buildings and have been successfully used in Germany since the beginning of the 1990’s (Katzenbach et al. 2000). This foundation type is a viable alternative to conventional pile or raft foundations in competent ground. The combined foundation is able to support the applied axial loading with an appropriate factor of safety at a tolerable level of settlement under working loads. The implementation of this foundation type has led to an abolition of complicated settlement-correction techniques. In recent years, the computational methods available in combination with measurements on real projects allowed the realistic modelling of the complicated bearing behaviour of that composite foundation system.

The overall bearing behaviour is described by means of the piled raft coefficient that defines the proportion of load carried by the piles. Due to the strong nonlinearity of the pile bearing behaviour the piled raft coefficient depends on the stress level and accordingly on the amount of settlement of the piled raft foundation as well. The piles can be loaded up to their ultimate bearing capacity, and are spaced strategically to achieve a more uniform settlement so

as to reduce sectional forces in the raft, giving a more economical solution.

The associated design work consists in estimating the deformation of the composite system and the distribution of the load into its two components, pile group and raft. The available methods may be divided into (i) approximate analytical, (ii) approximate numerical, and (iii) refined numerical, the choice being dictated by the importance of the project. Methods belonging to the first category are those by Randolph (1983; 1994), which is adopted here, Poulos and Davis (1980), and Lutz et al. (2006). Methods of the second category model the structural elements with finite-elements and apply approximate methods of elastic continua for calculating the interaction between the structural elements, Clancy and Randolph (1993), Horikoshi and Randolph (1998), Kitiyodom and Matsumoto (2003), Poulos (1994), Russo (1998), Small et al. (2003), Ta and Small (1996), Yamashita et al. (1998). The third category includes boundary element methods, applied by Butterfield and Banerjee, (1971), EI-Mossallamy and Franke (1997), Hain and Lee (1978), Kuwabara (1989), as well as finite element methods, applied by Arslan et al. (1994), Katzenbach et al. (1998), Reul and Randolph (2004), Smith and Wang (1998). The latter methods

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progressively dominate final design analyses, since they offer the possibility of capturing soil behaviour by appropriate non-linear constitutive models. Overviews are presented by Poulos (2001) and Mandolini (2003).

For the preliminary design, where different foundation alternatives are compared, a flexible, simplified method is required to assess the influence of the pile group configuration and of the soil parameters. The aim of this process is to optimize the position and the geometry of the piles in order to minimize the differential settlements and sectional forces in the raft. The solution of this complex soil-structure interaction problem is obtained by means of a pseudo-coupled procedure that is based on an interaction between the designers of the superstructure and the foundation system, respectively. The interface in this design procedure is defined jointly in terms of the modulus of subgrade reaction for the raft and the spring constant for each pile.

A simplified analytical method based on elastic continuum solutions has been presented by Randolph (1983; 1994). For uniform pile configurations, it leads to simple expressions and diagrams that allow a hand-calculation of the composite foundation system. However in most cases, piled raft foundations of high-rise buildings exhibit an irregular geometry (variable pile distance and length) calling for a modification of the method that is presented in the sequel.

2 METHOD BY RANDOLPH

The method for estimating the load-displacement behavior of piled rafts described by Randolph (1983; 1994) is similar to that by Poulos and Davis (1980) and is based on the solution of a pile-raft unit. The stiffness of a pile-raft unit is estimated through an approximation of the respective elastic continuum solution:

prrp

rprppr K/K

)(KKK 21

21

(1)

where prK is the overall stiffness of the pile group and raft system, pK is the stiffness of the free-standing pile group, rK is the stiffness of the free-standing raft, and rp is the pile-raft interaction factor.

The proportion of load carried by the raft is:

)(KK)(K

PP

rprp

rprr

21

1

(2)

where rP is the load carried by the raft, and P is the total load borne by the piled raft foundation.

The interaction factor rp , which is the essential parameter in the above expressions, is calculated from equation (3) by assuming that the settlement of the surrounding ground is decaying with distance according to a logarithmic law:

)/(ln1

)/(ln)/(ln 0,

0

, rrrr

rr eqr

m

eqrmrp (3)

The parameter , or equivalently mr , is estimated from the following relation:

0/])25.0)1(5.2(25.0[ln rl (4) In the above equations, l is the pile length, 0r the

pile radius, mr the influence radius of a single

free-standing pile, eqrr , the equivalent radius of

the raft area associated with each single pile, the Poisson’s ratio of the linear-elastic soil, the degree of inhomogeneity of the soil defined as the ratio of the average deformation modulus over the pile length to the modulus value at the level of pile base, and is the ratio of the end-bearing for end-bearing piles. Usually for piled rafts, 1 .

The stiffness of the pile group pK is obtained by the weighted superposition of the stiffness of each pile according to the approximate formula suggested by Fleming et al. (1992), assuming that all piles are identical and uniformly arranged within the pile group:

1)1( KNK e

p (5) where N is the number of piles, e an efficiency exponent, and 1K the stiffness of a single pile.

The stiffness of the single pile is taken either from the solutions by Poulos and Davis (1980) or from the following approximate expression derived by Randolph and Wroth (1978) that is an extension of the work by Frank (1974).

454

0

001 )tanh(

)1(41

)tanh(2)1(

4

rl

ll

rl

ll

rGK l

(6)

with

lp GE / (7)

)/(/2 0rll (8) where pE is the modulus of elasticity of the pile material, lG the shear modulus of the soil at the level of the pile base, and the ratio of underream for underreamed piles. Usually, 1 for piled rafts.

The efficiency coefficient e is obtained from a basic value in dependency of the slenderness ratio of the single pile and four correction factors 1c to 4c :

)()()/()()/( 43211 ccdsccdlee (9) where d is the pile diameter and s the average pile spacing within the pile group.

Curves for 43211 ,,,, cccce are given by Fleming et al. (1992). They are approximated here by polynomials using linear regression:

47.0109.4)/(

100.9)/(105.4)/(3

52731

dl

dldle (10)

85.045.0)(log28.0))((log108.3))((log

10

210

23101

c

(11)

32.113.0)/(106.9)/(105.2)/( 3243

2

dsdsdsc (12)

69.056.020.02

3 c (13)

07.120.014.024 c (14)

The above equations are valid for 100/10 dl ,

4)(log2 10 , 15.0 , and 5.00 . We observe that piles with d = 1.20 m and l =

12 m, typically used for piled rafts in sandy soil, are already at the lower limit of the slenderness ratio, reflecting the fact that the above curves were developed primarily for slender piles.

The raft stiffness rK is obtained via elastic continuum theory, taking into account the actual variation of the soil stiffness with depth. Available solutions for rigid rectangular plates with dimensions 2b x 2a (b > a) on homogeneous soil have been presented by various authors, and the solutions take the general form

IaGKr )1(

(15)

where I is a shape factor, and G is the effective shear modulus of the soil. I can be approximated by the following equation (Pais and Kausel, 1988):

75.0)/(1.36.1 abI (16) In order to consider a depth-dependent soil modulus and/or layering, the inhomogeneous soil is replaced by its homogeneous equivalent. The corresponding modulus can be computed, e.g., by adjusting the settlement of a perfectly flexible plate resting on the actual inhomogeneous soil to be equal to that of the same plate on the homogeneous equivalent. The Steinbrenner-approximation may be used for this purpose (cf. Poulos and Davis, 1980).

- Limitations

The application of the method by Randolph has some associated limitations: To guarantee a positive valued overall stiffness prK in equation (1), it is required that

rprp KK (17) For high-rise building foundations this condition is violated in those situations where the reduction of the differential settlements requires only a small number of piles underneath the rigid core of the building. In this case the foundation can be modeled as a pier-supported raft foundation, with the equivalent pier corresponding to a short, thick pile with slenderness ratio 1/ dl . From equation (17) it follows that for small values of the slenderness ratio the Randolph method becomes inaccurate.

Furthermore, the method in its original form determines the overall stiffness, the average settlement, and the proportion of load carried by each system component. Each pile in the group is assumed to have the same stiffness, i.e., it makes no difference if a particular pile is located in the centre,

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at the corner or along the periphery of the group. The approximate formulae have been derived from numerical solutions (finite element or boundary element methods) for large pile groups. With a decreasing ratio between footprint area and perimeter of the pile group, the proportion of peripheral piles within the group increases. The treatment of the peripheral and corner piles is an inherent drawback of any numerical solution based on elasticity theory. The imposed boundary condition of a rigid plate yields unrealistically high stiffness values for the peripheral piles and even higher for the corner piles. The modification of the method presented herein circumvents this problem in an approximate manner, and offers the possibility to compute the stiffness of each pile in dependence of its length and its actual position within the pile group.

3 MODIFIED METHOD

In the modified method, first the method by Randolph (1994) as described above is applied.

For deriving the raft stiffness rK , an irregular raft of footprint area rA is transformed into a rectangle of equal area by selecting an aspect ratio that yields almost the same ratio of footprint area to perimeter as in the original raft. The pile group footprint area *

gA is determined by adding to the area that includes all piles a strip of width equal to 1 to 1.5 times the pile diameter. From this gross area, the average pile distance s is computed.

After selecting appropriate values for the soil parameters for this so-called reference configuration, the interaction factor rp , the raft stiffness rK , the pile group stiffness )(sK p , and the overall stiffness of the piled raft )(sKpr are computed. This can easily be performed by means of a spreadsheet calculation. The proportion of load carried by the raft )(s is obtained from equation (2), and the spring stiffness for each pile within the uniform group is:

NK

sc prPile

)1()(

(18)

Then, the pile group interaction factor )(sRs is computed, that is defined as the ratio between the stiffness of the pile group and the sum of the stiffnesses of N identical single piles:

1

)()(

KNsK

sR ps

(19)

Next, the pile group configuration in its actual geometry is considered. For each individual pile, a representative, average pile distance to its direct neighbors ),..,1( Njs j is defined. These neighbors are selected in such a way that the particular pile lies in the centre of a subgroup. Typically, four piles should be included. In case the pile considered is a peripheral pile, the influence of the free boundary is simulated by means of a fictitious neighboring pile at a distance of mr . Recall that mr in Randolph and Wroth (1978) is the radius of influence of a free-standing single pile. Similarly, for a corner pile two real and two fictitious piles are considered. From the four individual distances, an average distance is computed. This procedure is applied to all piles within the pile group, and the range of values of the pile distance ),( maxmin ss is determined.

Figure 1: Concept for the representative pile spacing of centre, corner, and peripheral piles.

In a next step, a uniform pile group consisting of 44 piles is analyzed using the method by

Randolph for the various values of the pile distance ),1( Njs j and for the prevailing soil conditions.

The respective pile group factors are denoted by )(16, js sR . The pile group factor for the reference

configuration is )(16, sRs . The spring stiffnesses for each of the N piles

within the pile group are then obtained by increasing or decreasing the value of the reference configuration )(scPile according to the following rule:

NjsRsR

scscs

jsPilejPile ,...,1;

)()(

)()(16,

16, (20)

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The selection of a subgroup consisting of 4x4 piles in scaling the individual pile stiffnesses is arbitrary, 3x3 could also be taken.

A plausibility check for the adequacy of this approximation has been made by comparison with the rigorous results given by Hanisch et al. (2001).

Another deviation from the system of identical piles during the optimization of the pile configuration is the variation of pile length aiming at avoiding stress concentrations and large differential displacements in the raft. Usually, peripheral piles are made shorter than piles underneath the building cores. In most cases, the variation of pile length is not too large compared to the pile length, which justifies the application of the following approximation.

First, the procedure outlined above to assess the influence of the actual position of the piles within the group is applied. The pile length in the reference configuration is set to be equal to the average pile length of all piles. It is assumed that the additional pile length contributes solely to a proportional increase of skin friction along the pile shaft, i.e., pile base resistance remains constant. The proportional increase of skin friction under working load conditions is determined either from the results of pile load tests carried out in the frame of the project, or from code recommendations. Thus,

Njlsclsc

lsc

lscc

Pile

jPilejPile

jjPilejPile

,..,1;),(),(

),(

),(,

(21)

With the values rp and rK unchanged, the overall stiffness of the piled raft prK , and the proportion of load carried by the raft are computed from equations (1) and (2), respectively.

To take into account the finite bending stiffness of the raft, the computed stiffness of the rigid raft is reduced by 15%. This represents a reasonable midpoint between a rigid and a perfectly flexible plate.

The next steps in the design process are carried out through an interaction with the designer of the superstructure until an optimum configuration is reached for the load transfer into the ground with capacity utilization of the piles as uniform as possible, and with small differential deformations in the raft. Parameters varied during this optimization procedure are the spring stiffness of the individual piles jPilec , , and the average modulus of subgrade reaction for the raft

r

prRaft A

Kk

(22)

In case some of the piles are found to be subjected to higher loads than others, during this optimisation procedure, their stiffness has to be reduced by adjusting it to the actual load level. For this, the method proposed by Mayne and Schneider (2001) is applied that adopts the approximate non-linear load-deformation relationship suggested by Fahey and Carter (1993),

guqqfGG )/(1/ max (23)

with values f = 1 and g = 0.3, where maxG is the secant shear modulus at small strains, G is the shear modulus corresponding to the load q, and uq is the ultimate bearing resistance of the pile as determined, e.g., from pile load tests or from code specifications. This means, for example, that if the service load of a pile is increased from 50% of the ultimate bearing resistance to 60%, the effective spring stiffness is reduced by 24%.

4 CASE STUDY

The study presented in the sequel refers to the high-rise building “Skyper” recently constructed in Frankfurt. It consists of a tower 153 m high, which is connected to lower buildings. The entire building complex is underlain by a parking garage with three underground levels founded on a continuous raft. Due to the eccentric loading of the building complex and in order to reduce the associated differential settlements a piled raft foundation was selected for the high-rise building section.

The piled raft considered exhibits an irregular pile configuration as shown in Figure 2 with pile lengths varying between 31 m and 35 m. The diameter of the 46 bored piles is 1.5 m. The raft has a thickness of 3.5 m and is placed at a depth of 13.40 m below ground surface.

The soil stratigraphy is typical for Frankfurt: The top layer consists of 7.4 m thick quaternary, gravelly sand deposits with groundwater level at 5.0 m below the ground surface. These deposits are underlain by the Hydrobien layer known as Frankfurt Clay, followed at a depth of 56.4 m by the Inflaten / Frankfurt Limestone layer that is considered incompressible. Hence, the thickness of the compressible layer underneath the raft amounts to 43 m, cf. Figure 3.

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Figure 2: Plan view of the high-rise section of the Skyper building complex.

Figure 3: Foundation cross section with soil profile. The relevant soil parameter for the piled raft

design is the soil stiffness that is given in terms of a depth-dependent constraint modulus as determined from the back calculation of observed building settlements in the area. For the over-consolidated Frankfurt Clay, a two layer profile is usually adopted in design that is described by a constant modulus

value in the top layer, and a linearly increasing one underneath. Assuming a Poisson´s ratio of

,33.0 the profile is expressed in terms of the Young´s modulus E in [MPa] as follows:

)(zE = 62.4 for 6.19z (24a)

zzE 183.26.19)( for 6.19z (24b)

where z in [m] is the depth below the raft. Soil strength is defined by an angle of friction φ´ = 20° and cohesion c´ = 20 kPa.

The settlement inducing load from the superstructure, including the raft’s own weight and the average uplift force, was estimated to be P = 810 MN.

Before proceeding further with the analysis, the adequacy of the modulus depth profile adopted and of the analysis obtained by the method by Randolph were verified by comparing the results with the settlements measured at the piled-raft foundation of the nearby high-rise building “Messeturm”.

Following the procedure outlined above for the Skyper tower, we first determine the representative (fictitious) pile distances for the individual piles

),1( Njs j considering their actual position within the group. These values ranged from 15 to 17 m for the corner piles to 4.5 m for the center piles. Next, we determine from the footprint area of the pile group enlarged by a strip of 2 m around its periphery a gross footprint area of 1414 m2, which yielded an average pile distance of 5.54 m.

The footprint area of the raft is 1900 m2 with a perimeter of 173 m corresponding to an area-to-perimeter ratio of 11. This transformed to an equal-area square with side length 43.6 m that has approximately the same area-to-perimeter ratio as the original raft.

In order to determine the value of the raft stiffness ,rK we first calculate an equivalent modulus using the Steinbrenner approximation for a perfectly flexible raft resting on multi-layered soil, yielding a value of 125 MPa. Entering this value in equation (16) for a rigid raft and reducing the resulting value by 15% to capture the finite rigidity of the raft, we obtain rK = 6100 MN/m.

Next, the reference pile group configuration consisting of the 46 piles of 33 m length with constant spacing of 5.54 m is analysed using the method by Randolph summarized above. The soil profile, equation (24) corresponds to 4.34lG MPa, = 0.75. The analysis yields an interaction factor

rp = 0.644, and an average pile stiffness 4.101Pilec MN/m.

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The influence of the pile spacing on the pile interaction is assessed by means of the pile group factor 16,sR . The results obtained for the soil profile are approximated by the relationship

216, )40/(15.1)40/(14.0)( jjjs sssR with sj

given in [m]. The individual pile stiffnesses are calculated from

equation (20) with values ranging between min,Pilec = 92 MN/m and max,Pilec = 142 MN/m, and

an average of meanPilec , = 114 MN/m. Here, we omit a further correction for the pile length according to equation (21) for the sake of simplicity.

With 6100rK MN/m, meanPilec , = 114 MN/m, and rp = 0.644, we obtain by solving the system of equations (1), (2), and (18): stiffness of pile group

7370pK MN/m, overall stiffness of piled raft 8550prK MN/m, and proportion of load carried

by the raft 0.387. The average settlement of the piled raft then is 810/8550 = 0.095 m. The settlement of a raft

without pile support would be 810/6100 = 0.133 m. The modulus of subgrade reaction for the raft is calculated from equation (22) to Raftk = 1.74 MN/m3. The average pile load is meanPilemeanPile cNPQ ,, /)1( = 10.8 MN with minimum and maximum values min,PileQ = 8.7 MN and max,PileQ = 13.5 MN, respectively.

It should be kept in mind that these values correspond to a uniform loading of the raft. In the detailed final design, the actual load distribution from the superstructure has to be considered.

The piled raft foundation described above has been further analyzed by several other methods. The interested reader may find the results in the summary paper of Richter and Lutz (2010). Results on pile load measurements are not available. The average pile load meanPileQ , as predicted by the other methods varied between 10.3 and 13.9 MN. Two of the methods yielded position-dependent pile loads with values min,PileQ / max,PileQ = 8.5/20.5 MN and 10/20 MN, respectively.

5 CONCLUSIONS

The modification of the method by Randolph outlined above allows the accommodation of the variable pile distance and length as well as the different stiffnesses of central, peripheral, and corner piles. The analysis of a piled raft system is a

deformation problem and so should be treated with the same precision as the settlement prediction of raft foundations. It is therefore justified during the preliminary design to use elastic solutions with soil modulus values that take into account the expected average strain level. For the final design, a nonlinear finite element analysis with an appropriate soil model is recommended, particularly in cases of limited experience with the actual ground conditions.

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