an approximate analysis procedure for piled raft foundations

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 17, 849-869 (1993)

A N APPROXIMATE ANALYSIS PROCEDURE FOR PILED RAFT FOUNDATIONS

P. CLANCY AND M. F. RANDOLPH

Department of Civil Engineering, University of Western Australia, Nedlands, Western Australia 6009, Australia

SUMMARY

A piled raft foundation comprises both piles and a pile cap that itself transmits load directly to the ground. The aim of such a foundation is to reduce the number of piles compared with a more conventional piled foundation where the bearing effect of the pile cap, or raft, is ignored. This paper describes a 'hybrid' approach for the analysis of piled raft foundations, based on a load transfer treatment of individual piles, together with elastic interaction between different piles and with the raft. The numerical analysis is used to evaluate a simple approximate method of estimating the overall response of the foundation from the response of the component parts. The method leads to estimates of the overall foundation stiffness, the proportion of load carried by the pile group and the raft, and an initial assessment of differential settlements. Parametric studies are presented showing the effect of factors such as raft stiffness and pile spacing, length and stiffness, and a worked example is included demonstrating the accuracy of the approximate design approach.

INTRODUCTION

It is common practice to cast the cap of a pile group foundation directly on the ground. Where competent soil conditions exist at ground level, a significant proportion of the structural load may then be transmitted directly from the pile cap to the ground. In effect, the foundation behaves as a piled raft, although the performance is usually estimated conservatively by ignoring the bearing effect of the raft. This can lead to an unnecessarily high density of

In order to improve the efficiency of piled raft design, thus minimizing the number of piles, it is necessary to develop simple analytical approaches that are able to quantify the foundation performance, in terms of both overall and differential settlements, and the load sharing between the raft and the piles. However, any simple approach must first be calibrated against a more rigorous numerical analysis.

Approximate analytical methods for piled rafts have been proposed by Pouios and Davis,3 based on the analysis of individual 'pile-raft' units and appropriate interaction factors, and by Randolph4 who combined pile group and raft response through a single interaction factor (also based on an individual pile-raft unit). However, rigorous numerical solutions have been confined to the work of Hain and Lee,' Weisner and Brown6 and Bilotta et aL7 using the boundary element method to analyse piled raft foundations with a relatively small number of piles. Both the boundary element method and the finite element method are limited in the size of problem that can be a d d r e ~ s e d , ~ . ~ owing to the three-dimensional (3-D) nature of the problem that leads to very large stiffness matrices.

The present paper describes a 'hybrid' numerical method which has been developed specificaily to minimize the amount of computation.'' The method combines finite elements and simplified closed form expressions for the diagonal terms of the stiffness matrices, with off-diagonal terms

0363-9061/93/120849-21$15.50 0 1993 by John Wiley & Sons, Ltd.

Received I7 December I992 Revised 29 March I993

850 P. CLANCY AND M. F. RANDOLPH

calculated using Mindlin's' solution and lumped forces. These simplifications, which have been checked for accuracy against more refined approaches, allow problems of practical proportions to be analysed. In addition, a simple design approach for piled rafts is proposed, extending the work of Randolph4 Lo allow the foundation response to be estimated directly from the separate responses of the raft and pile group.

METHOD OF ANALYSIS

The numerical method of analysis employed depends on a hybrid model, combining finite elements to model the structural elements of the foundation with analytical solutions for the soil response. A brief description of the various elements of the analysis is given here, but a fuller description has been presented by Griffiths et cri."

The approach is based on a method of analysis for pile groups presented by C h ~ w ' ~ ' ' ~ and in computer code by Smith and Griffiths.14 This employs one-dimensional (I-D) rod finite elements to represent each axially-loaded pile. coupled with a soil response at each pile node modelled by discrete laod transfer (or r - z ) springs.' ' An analytical method for calculating the gradient of the load transfer springs was proposed by Randolph and Wroth.I6 The soil deformation around the pile shaft was idealized as the shearing of concentric cylinders, and the base response was analysed separately as a rigid punch on the surface of a semi-infinite half space. The two parts of the analysis were combined by forcing displacement compatibility at the base of the pile. This approach has been extended to permit a non-linear (hyperbolic) variation in shear modulus, leading to the derivation of a formula involving the secant shear m o d u l ~ s . ' ~ ~ ' ~ ChowI2 developed this idea further to produce a more efficient formula for computational purposes based on the tangent shear modulus.

Interaction between piles through the soil is calculated using Mindlin's" elastic continuum solution. This allows the displacement at a point in the interior of a semi-infinite linear elastic half space, due to a point load at another point in the half space, to be calculated. Mindlin's solution has been used in the boundary element method.' 9 .20 where numerical integration of the equation over the pile surfaces is required. In the present method no integration is involved since the forces are considered to be lumped at each pile node. This is an acceptable approximation since the use of load transfer springs eliminates the need to calculate interactions between nodes on the same pile.

A separate raft analysis has been developed in a consistent manner, using two-dimensional (2-D) 'thin' plate-bending finite element^'^ to model the raft. An equivalent soil 'spring' response is calculated for each raft node using an analytical solution due to Giroud" for the average displacement under a uniformly loaded rectangular area. The area of raft contributing to each node is calculated by summing the area of each raft eiement to which the node is attached, and by dividing this area by four (since each element has four nodes). The contributing area does not necessarily centre on the node itself. and so the centre of this area is calculated for each node. These centroidal points are then used to determine the interaction between raft nodes through the soil, which again makes use of Mindlin's equation. The adequacy of this approach has been demonstrated by Griffiths et al."

The two analyses are combined by attaching piles to the raft via common nodes at the connecting points. The wrtical freedoms are linked, resulting in only axial load being transmitted to the piles. It is assumed that there is no raft-soil contact at the common nodes. Interaction between pile nodes and raft nodes is calculated using Mindlin's equation.

This has resulted in a relatively rigorous and yet considerably more efficient method of analysis for piled rafts than has previously been available, and allows for variable geometry, pile stiffness,

ANALYSIS PROCEDURE FOR PILED RAFT FOUNDATIONS 85 1

0 One-dimensional pile element @ Ground resistance at each pile node represented by non-linear 'T-Z' springs @ Two-dimensional plaie-bending tinite element ratt mesh @ Ground resistance at each raft node represented by an eqiiiavslerii !qiring @ Pile-soil-pile interaction effects calculated betu-een p:tirs of odes tiring

Mindlin's equation

@ Pile-soil-raft interaction @ Raft-soil-raft iriteraction

Figure I . Numerical representation of piled raft

soil stiffness and raft stiffness. Figure 1 illustrates the various components of the analysis. In the present paper only vertical loading is considered, but i t would be a relatively simple matter to allow for horizontal or inclined loading in a consistent manner (at the expense of increasing the size of the numerical problem). Although only linear elastic soil conditions have been considered to date, non-linear load transfer springs may be used at the pile-soil interface to model local yielding.

PARA MGTERS

In the development stage of the analysis, attention has been restricted to homogeneous soil conditions, and square groups of piles at uniform spacing. The performance of the piled raft is then determined by the material and geometrical properties of the soil, piles and raft, which have been summarized in Table I. These parameters may be grouped in the appropriate non- dimensional ratios summarized in Table 11. The form of the dimensionless group for the raft-soil stiffness ratio (I&) is based on the work of Brown.22


Table I. Parameters for piled raft foundation

Soil Pile Raft ~

Youngs modulus E , Youngs modulus E , Youngs modulus E, Poissons ratio y Length L Poissons ratio v,

Diameter d Thickness t Spacing s Length Lr

Breadth

Table 11. Dimensionless groups for piled raft foundation - ~ ~~

Dimensionless group Definition Practical range

Pile spacing ratio sld 2.5-8 Pile slenderness ratio L l d 10-100 Pile-soil stiffness ratio K, , = E , P S 100-10000

4E,B,t3(1 - v: ) 37cE,LP

Raft-soil stiffness ratio K , , = 0.01-10

ACCURACY OF PRESENT ANALYSIS

A sensitivity study was undertaken in order to explore the accuracy of the numerical analysis, and to assess the level of discretization necessary to yield acceptable results. This was done in a number of stages, the first involving a single pile analysis. A medium length relatively flexible pile was investigated, with the materail and geometrical properties given in Table 111.

The number of rod finite elements was varied from 5 to 25 and the load induced due to an applied unit displacement of the pile head (the overall pile stiffness) was calcualted in each case. The results have been tabulated (Table IV) along with solutions due to Randolph and WrothI6 and Poulos and Davis.3 The present method shows good comparison, even when only five pile elements were used. The results converge on the solution due to Randolph and Wroth because the load transfer approach is used in both cases, whereas the solution due to Poulos and Davis is calculated from a full boundary element analysis.

The effect of mesh refinement on pile-soil-pile interaction was also investigated using the same pile properties. Calculations were performed for two adjacent piles at a spacing of 2.5d, and for a 2 x 2 square pile group with a similar spacing. Again, the results for individual pile stiffnesses were compared with those due to Fleming et aLZ3 and Poulos and Davis3 (Table V).

Table 111. Properties for single pile and pile group mesh refinement analyses

-

Soil Pile Dimensionless groups

E, 280MPa E, 35000MPa LJd 25 vS 0.4 L 20m K,, 125

d 0.8 m

ANALYSIS PROCEDURE FOR PILED RAFT FOUNDATIONS 853

Table IV. Results of mesh refinement analysis for single pile

Number of rod finite elements

Overall pile stiffness (MN/mm)

5 10 15 20 25 Randolph and Wroth Poulos and Davis

1.569 1.548 1.544 1.543 1.542 1.542 1.697

Table V. Results of mesh refinement analysis for pile group

Number of rod Two piles 2 x2 pile group finite elements single pile stiffness single pile stiffness

(MNImm) (MN/mm)

5 10 15 20 25 Fleming et al. Poulos and Davis

1.166 1.151 1.148 1.147 1.147 1.077 1.237

0.797 0.789 0787 0.787 0.787 0,752 0817

Finally, three different meshes were used to analyse the effect of mesh refinement on the raft in a piled raft situation (Figure 2). The piles had similar material and geometrical properties to those used previously, and were each discretized into 15 rod finite elements. A flexible raft was used in the analysis with the properties given in Table, VI.

The results of the piled raft analyses (Table VII) show that even the coarsest raft mesh is sufficiently fine for the calculation of displacements and load distribution. In later analyses, 15 pile elements were used together with a raft mesh in which there were two plate bending finite elements between each pile.

DEVELOPMENT OF A SIMPLIFIED APPROACH

The present method provides a rigorous approach to the piled raft problem, but is limited because of the amount of computing power required to analyse larger pile groups. This arises from the fact that the problem is attacked from a flexibilility approach, requiring inversion of a fully-populated flexibility matrix before the stiffness matrix can be formed. For practical purposes, the analysis is limited to groups of around 50 piles or less.

In order to allow routine design of piled rafts with a larger number of piles, it is necessary to develop approximate methods which allow extrapolation of the rigorous analyses. To this end, an extensive series of analyses have been carried out for smaller piled-raft systems of square geometry and up to 36 piles. The influence of varying a number of parameters (as defined above) was investigated: pile group size, pile spacing ratio, pile slenderness ratio, pile-soil stiffness ratio, and raft-soil stiffness ratio.

HdlOaNVU 'MI aNV A3NV73 .d

8 5 4

P . C L A N C Y A N D M . F . R A N D O L P H

- - - _ - - _ _ _ _ _ _ . . .

O n e w c h c u r r e n t l y a w i l a h l e a p p r o x i m a t e m e t h o d i s t h e a p p r o a c h d e s c r i b e d b y R a n d o l p h '

e m p l o y i n g a ' f l e x i b i l i t y ' m a t r i x m e t h o d t o c o m b i n e t h e i n d i v i d u a l s t i f h e s s e s ( i . e . l o a d - d i s p l a c e -

m e n t r e s p o n s e ) o f p i l e g r o u p a n d r a f t . T w o l a c t o r s a r e i n t r o d u c e d , a p r a n d a , , , w h i c h d e s c r r h c t h e

i n t e r a c t i o n o f t h e t w o f o u n d a t i o n e l e m e n t s w h e n c o m b i n e d .


Table VII. Results of mesh refinement analysis for raft -

Refinement Mesh 1 Mesh 2 Mesh 3

wp (mm) 7.978 797 1 7.967 w , (mm) 8 W Y 8.088 8.083 p, (MN) 29.66 29.21 29.05 p, (MN) 6.34 6.79 6.95 R u n time (min) 5.3 415 404.3

Thus,

where

w, = average displacement of pile grcbup in combined foundation

w, = average displacement of raft in combined foundation

P, = total load carried by pile group in combined foundation

P, = total load carried by raft in combined foundation

k , = overall stiffness (P/w) of pile group in isolation

k , = overall stiffness ( P / w ) of raft in isolation

clrP = interaction factor of pile group on raft

a,, = interaction factor of raft on pile group

For a rigid raft w, = w,, and the overall pileeraft displacement will subsequently be referred to as wpr. This is not generally true for systems with more flexible rafts, since the average displacement of the pile group will usually not be equal to the average displacement of the raft. In order to simplify matters, the difference has been ignored at this stage.

If appropriate work-compatible average displacements of the pile group and the raft are used in equation ( I ) , then the off-diagonal terms of the flexibility matrix must be equal (i.e. npr/kr = zr,/k,). This can easily be demonstrated by considering the trivial example of two unequal piles (Figure 3). The two piles have stiffnesses of k , and k 2 , respectively, and are loaded with P , and P,. According to the liexibility matrix ofequation ( I ) , this causes a total displacement oC(P, /k l + PzgIz;L2) i n Pile 1, and ( P , zL1 . X I i- t? , A 2 } in Pile 2. I f Pilc 1 is 1o;tded before Pile 2. the total work done by the loads is equal to C 1 . l . where

It Pile 2 is loaded before Pile 1 , then the total work done is equal to M >

HI the. prin\iple of superpovticm. M, mu5t be equd to I + - since the cl.istrc encrp) \ t o ted in each case 1s the same, and therefore X , ~ / L , and z 2 , A , must also be equal


I I

Kl Ki I Pile 1 Pile 2

-E I I I

Pile 1 load PI

Pile 2 load 0 disp. PdKi

I I

I I I - I - I I

1

i

11 I Load pile 1: I

Pile 1 load PI I I Pile 2 load 0 I

disp. PdKI

Load pile 2: Pile 1 load 0

Pile 2 load Pz J disp. PdKz disp. PzcLIdKz

Load pile 1, pile 2: Pile 1 load PI

Pile 2 load PZ disp. PI/KI+P~CCI~KI

disp. Qlazi/KI+PyKz

I I

Figure 3. Reciprocal displacements between two adjacent piles

This reciprocal theorem is only valid if work-compatible displacements of the various components are used. For general piled raft systems, the overall settlements of the pile group and raft are calculated by an averaging method which does not satisfy this requirement. In addition, each of the components deforms differently in isolation from their behaviour, as part of a piled raft system. When combined, the influence of the pile group on the raft is more likely to result in an average displacement of the raft similar to the work-compatible value simply because the average displacement of a raft foundation is relatively independent of the raft-soil stiffness ratio. The net result of this is that arp/kp is a more reliable parameter for determining piled-raft behaviour than is upr/kr.

Randolphs4 analytical work was based on a rigid raft, so that wp = w, = wpr. This also meant that the assumption of work-compatible displacements was only a small approximation, and he was able to show that arp/kr z upr/kp. Assuming these two conditions to hold true, the load


distribution between the pile group and the raft can be found in terms of arp by solving equations ( 1 ) simultaneously for Pp and Pr:

The load distribution and overall stiffness can be derived in terms of apr in a similar manner. Randolph4 suggested a method of calculating the two factors arp and apr by considering a single

pile-raft unit. The method involves superposing the displacement fields induced by a single pile and by a circular raft. This results in the following formulae for the calculation of arp and apr:

k r apr = arp -- fiP

L

t c

#-

0 Pile spacing, s 0 Pile diameter, d @ Pile length, L @ Raft length, Lr 0 Raft breadth, Br @ Raft thickness, t @ Radius to give

equivalent raft area per piIe

Figure 4. Piled raft divided into single pile-raft units


where n is the ratio of the circular raft diameter to the pile diameter, rm is a measure of the radius of influence of the pile [rm = 2.5pL(1 - us)] and p is the degree of homogeneity of the For very slender piles, the pile length L is replaced by a limiting effective pile length which may be calculated according to Fleming et dZ3:

L, = 1.5dJ2(1 + v,)E,/E,. In order to calculate the overall stiffness of a piled raft foundation having more than one pile,

the values of arP and up, for a single pile-raft unit are assumed to be directly applicable. The single pile-raft unit is taken to have a raft area equal to the mean raft area per pile in the complete system (Figure 4). The pile group stiffness may be calculated in a manner similar to that for a single pile, allowing for interaction effects between the piles, and the raft stiffness for a uniformly loaded rectangular or circular raft may be found directly using closed form analytical solutions.

The present work tackles the problem from the opposite direction, i.e. to calculate real values of arP and apr from knowledge of the results of full piled raft analyses (i.e. Pp, P, and wpr). The results are compared with the approximate values calculated from Randolphs analytical approach. Values of and apr can be back-calculated, if wpr, Pp and P, are known, by rearranging equations (1).

RESULTS

A comprehensive range of values for arp and apr were calculated for a single pile and circular raft using the analytical method of Randolph: and are plotted in Figure 5. The pile spacing indicated on the abscissae is an indicator of the relative raft size, and is similar to the rectangular pile-pile spacing used in larger pile groups. Thus, for a single pile-raft unit with a raft radius r, the equivalent rectangular pile spacing in a larger piled raft system is given by s/d = & r/d.

As has been mentioned, the analytical method is based on a single pile-raft unit and assumes a rigid raft. Figure 6 shows the numerical results which are directly comparable, i.e. a single pile under a rigid raft (Krs = 10). Since rectangular plate-bending finite elements were used to model the raft, these results were produced for a set of square rafts, each having an area equal to that of the corresponding circular raft. To derive each point on the curves it was necessary to perform three separate analyses, isolating the pile group and the raft in turn to obtain k , and k,, and then performing a full piled raft analysis to obtain kpr, Pp and P,.

It can be seen that there is good agreement between the two sets of results, both in general trends and in numerical values, particularly for arP. This indicates the validity of the approximate method when applied to single pile-raft units, but it is necessary to confirm that the method can be extended to larger pile groups. Observing the trends, arP decreases as the pile spacing is increased, or as the pile stiffness is decreased; apr shows the opposite behaviour. Since arp represents the interaction of the pile group on the raft, these trends are intuitively correct. Increasing the slenderness ratio of the pile merely increases the spread of values at a given pile spacing, resulting from pile stiffness being more important in longer piles.

Figure 7 is a similar set of plots obtained numerically for a 3 x 3 pile group. In all cases the values of arP and apr are higher than for the corresponding single pile-raft unit, and the spread of values is tighter. Thus, the interaction between the separate elements increases as the size of the pile group is increased, leading to a reduction in the overall stiffness of the system per pile-raft


1.0. 0.8 0.6 0.4 - 0.2

yi -0.2

pile 'spacing'. r/d

-0.8 -1.0 Lld = 10

1.0, 08 0 6 0 4

^p 02

Zpopzka s 6 7 b b pile 'spacing'. sld -0.4

-0.6 1 -0.8 -1.0 L/d = 25

1.0. 0.8 0.6 0.4

^a 0.2

pile 'spacing'. sld -0.6 -0.8 -1.0 Lld = 100

1.0

4 5 6 7 8 9 7 -02 pile 'npcmg', sld

-0 8 - I 0 Lld = 10

1.0

pile 'spang', sld

-0.8 -1 0 L/d = 25

1.0,

pile 'spacing'. $Id -0.4 -0.6 1 -0.8 -1.0 Lld = 100

~ apr I

Figure 5. Randolph analytical values for all pile group sizes and raft stiffnesses

unit in addition to the reduction in stiffness of each element (per pile-raft unit). As the size of the foundation is increased, the effects of pile stiffness and pile slenderness on the interaction are reduced. It is worth noting that the values of a,, become negative when the pile group consists of long, stiff piles. The assumption of reciprocity breaks down in these cases because the raft is no longer rigid in comparison to the pile group, allowing the piles to displace by different amounts, and the average pile displacement is no longer a good approximation to the work-compatible value. Numerical errors are likely to occur if the stiffness of the raft is increased, but applying a uniform displacement instead of a uniform load would overcome the problem. However, it was noted earlier that apr was likely to be a less reliable parameter than arp, and for this reason all calculations will use arp.

The trend from the single pile to a 6 x 6 pile group has been generated for piles of slenderness ratio L/d of 25 and is presented in Figure 8. It is clear that convergence of the interaction factors is approached as the size of the pile group is increased. Of particular interest is that the value of arp approaches a value of approximately 0.8 for all pile spacings and across the full range of pile


I 1.0 7 I 0.8 0.6

~ 0.4 ^p 0.2

0.8 0.6

~ a 0.4* 0.2 2 fl

pile 'rpcing', r/d

-1.0 Lld = 25

0.8 06 I a 0 . 4 k 0.2 i3

pile 'spacing'. dd -0.6 -0.8 -1 0 Lld = 100

L. arp- i

1.0 08 06 0.4

c 0.2

% -0.2 4 5 6 7 8 9 2 0.0 , --- ' ,

pile 'rpcing'. r/d

-0.8 - I 0 Lld = 10

I .o 0.8 T

pile 'spang'. dd -0.4 -0.6 -0.8 -1.0 Wd = 25

1 .o 0.8 T

id , __ v - 0.0 4.0.2i 3 4 3 6 7 8 9 pile 'spcmg'. r/d

-0 8 -I 0 Lld = 100

Figure 6. Computed values for single pileeraft unit (Kr3 = 10)

stiffnesses. Figure 9 is a replot of the data shown in Figure 7, this time highlighting that variation of pile slenderness ratio has little effect on the value of arp.

It is known that the average displacement of a raft foundation is approximately constant over the range of raft ~tiffness.'~ The small effect of raft-soil stiffness on the parameters arp and apr is demonstrated in Figure 10, which shows the results for a 3 x 3 pile group of L/d = 25, with raft stiffnesses ranging from very flexible (Krs = 0.01) to very stiff ('rigid' K , , = 10).

The physical significance of clip remaining at a constant value while the geometry and material properties are varied is difficult to extract due to the complex interplay of a number of factors. However, it is possible to determine whether it would be reasonable to accept the validity of such results. Normalizing equation (9) for aIp with respect to the total load ( P ) carried by the piled raft system gives


~

1.0 0.8 0.6 0.4

^e 0.2 g 0 0 2 -0.2 6 7 8 9

-0.4 pile 'spacing', s/d -0.6 1 -0.8 -1.0 Lld = 10

1.0 0.8 0.6 0.4

0.0

-0 4 -0.6

^e 0.2 -

-0.8 f - I 0 L/d = 25

pile 'spacing', d d

Lld = 100

1.0, 0.8 0.6 0.4

pile 'spacing'. r/d

-0.8 -1.0 Lld = 10

10,

: --+.-- 3 4 0.0 =.o.2 3 4 5 6 1 8 9

pile 'apacmg'. r/d

-0.8 -1.0 Lld = 25

1.0 0.8 I 06 0 4 - 00 g 0 2

%O* pile 'spacing'. s/d

-0.8 -1 0 Lld = 100

Figure 7. Computed values for 3 x 3 pile group (KrT = 10)

As an example, Figure 11 analyses in detail the variation with pile spacing of all the factors

1. As s/d is increased, both k , and k, increase, as might be anticipated. 2. k , increases at a smaller rate than k , , so that the proportion of load taken by the pile group

(P,/P) should decrease while the proportion of load taken by the raft (P, /P) should increase, and this is also observed.

involved in the calculation of a,, for a 6 x 6 pile group.

3. Since k , is increasing and Pp/P is decreasing, k p / ( P , / P ) must increase. 4. k , and P, are both increasing, so any variation in ( P , / P ) / k , is dependant on the relative

magnitudes of increase. In fact ( P , / P ) / k , can be seen to decrease slightly with increasing s /d . 5. Since both k , and k , are increasing, it is obvious that wpr/P must decrease, and it does this at

a greater rate than ( P , / P ) / k , . 6. Thus, [(wp,/P) - ( P , / P ) / k , ] decreases while kp/Pp increases, and the combined effect results

in a constant value of a,,.


1.0 0.8 0.6 0 4

^a 0.2

pile 'spacing'. r/d

-0 8 EplEs = 100

1 0 ~~

0.8 0.6 0 4

~ 0 . 4 -0.6 I -0.8 EplEs = 100000

pile 'rpeing', s/d

I L O T I 0.8 I :::

2 0.2

pile 'ipemg'. s/d

-0.8 -1.0 Ep/Es = 100

0.8 0.6 0.4

6 1 8 9

pile 'spang'. sld

EplEs = 100000

-

'1x1' pile group 0 2x2 pile group

0 4x4 pile group A 5x5 pile group

3x3 pile group

6x6 pile group

Figure 8. Computed values: L/d = 25; K,, = 10

1.00 0.8 0.6 0 ,-, 7 0.4

@ % -0.2 7 8 9 pllc 'spacing', sld

-0.8 Ep/Es = 100

pile 'spacing', rld

I

1.0

pile 'spacing', r/d

1.0

pile 'spacing'. d d

-0.8 Ep/Es = 1OOOOO

a P r

Figure 9. Computed values for 3 x 3 pile group (Krs = 10)

ANALYSIS PROCEDURE FOR PILED RAFT FOUNDATIONS

g o z - ~ - 8 0.0- z.o.2:L

-0.4 -0.6 -0 8 .I.o:

863

+---I ' ~ : :

3 4 5 6 7 8 9

pile 'spacing'. sld ~~

~~

EpiEs = 100

1.0 I 08 0.6 0.4

pik 'spacing'. d d

08 Ep/Es = 100000

0.8 0.6 0.4 f.ip . 6 ._ ---. 7 8 . -+ 9

pile 'spacing', d d -0.6 -0.8 .,,,, Ep/Es = 100

_ _ _ _ _ _ _ ~ _ _ _ ~ ~ _. 1.0 n R I _ _ 06 0 4

-z 0 2 4 00

pile 'spcmg'. r/d

-

0 Krs=O.l Krs=1 ~ ~ _ _

Figure 10. Computed values for 3 x 3 pile group (L/d = 25)

l .k+06 8.0e+05 6.Oe+05 4.0e+05 2.0e+05 o.oe+oo

2 3 4 5 6 7 8 9

Pile spacing, s/d ~ _ _ _ _ _

5.k+05

o.oe+oo I- 2 3 4 5 6 1 8 9

Pile spacing, s/d

2.5e-06 T

2 3 4 5 6 7 8 9

Pile spacing, s/d

1 .k+Oo 8.k-01 6.Oe-01 4.0e-01

2 3 4 5 6 7 8 9

Pile spacing, s/d

- WPrP 3.oe-06 2.5e-06 2.k-06 1.5e-06 1.k-06 5.k-07

2 3 4 5 6 7 8 9

Pile spacing, s/d

l.oe+Oo 8.k-01 6.Oe-01 4.k-01 2.k-01

O.k+Oo

alpha(rp) r 1 2 3 4 5 6 1 8 9

Pile spacing, s/d

Figure 11. Investigation of factors contributing to urp, 6 x 6 pile group: L/d = 25; E p / E , = 1ooO; Kro = 10


DIFFERENTIAL SETTLEMENTS

The approximate method outlined above focuses entirely on average settlement and does not address the key problem of differential settlement. A method of predicting the differential settlements of a piled raft system is currently being investigated, and will be dealt with in a separate publication. However, a very simple method that appears to work well for groups of up to 80 piles is to factor the raft displacements by k, /k , , . For convenience of plotting the variation of settlement across foundations of differing size, a normalized co-ordinate is introduced (Figure 12). This has a value of 0 at the corner of the raft, 0.5 at the mid-edge, and 1 at the centre. Figure 13 shows a series of results for a square 3 x 3 pile group across the range of raft stiffnesses as pile stiffness is varied. The displacements have been normalized by a factor of L,E, /P(l - uf). The differential settlements are predicted satisfactorily, with a tendency to overpredict the central displacement and underpredict that at the corner, for all but the most flexible raft. This is to be expected, since the local stiffening due to the piles will be more pronounced under a raft of low stiffness.

A study of the effect of pile spacing, pile slenderness and group size was also made (Figure 14), showing that this method is generally applicable except in the case of very small pile groups. Here the local stiffening effect of the piles again becomes considerable.

WORKED EXAMPLE

A worked example is presented, considering a 9 x 9 square pile group (Figure 15). If five rod finite elements are used to model each pile and the coarsest raft mesh is used, the piled raft problem requires approximately 10 MB of computer memory to run in single precision (4 bytes per real number). This compares with 2 MB for a similar 6 x 6 pile group analysis and 0.2 MB for a 3 x 3 pile group. The analysis takes 15 h to solve on a Sun SparcStation IPC (similar speed to IBM 486), compared with less than 5 min for the 3 x 3 pile group. Obviously, it is advantageous to have a reliable approximate method for the analysis of larger systems.

The material and geometrical properties used in the worked example are given in Table VIII.

0.0 - 0.5 r - r - i I t I

I I I +--+-I

I I I

I I I I c - I c -

L - L - L - L - I - I - I

- + - + - 4 - A - -I

1 1 I I I I I

Figure 12. Normalized co-ordinate applied to 3 x 3 pile group. Raft nodes to be plotted marked by x


normalimd coordinate

0 0 0 5 10 _i , ~ _ _

E 0.2 m B 3 0.4 P 2 0.6 .u

0

0.8 Krs = 0.01

normalised coordinate

0.0 0.5 1 .o 0.0 +-....,-- -

$ 1 B 0.2 M 5

0.4 P a 0.6

U

.-

0.8 Krs = 0.1


0.0 0.5 1 .o 0.0 1- ~ i

$ 1 B 0.2 M 3 0.4 Fi

P 0.6

0

0.8 Krs = 1.0

norrnalised coordinate

0.0 0.5 1 .o 0.0 1--

$ . I 0.2

piled raft, Kps = 1 0 0

0 estimate, Kps = 100

piled raft, Kps = loo0

0 estimate, Kps = loo0

A piled raft, Kps = loo00

A estimate, Kps = loo00

piled raft, Kps = 1OOOOO

0 estimate, Kps = IOOOOO

Figure 13. Comparison between actual piled raft displacements and estimates from raft alone. Variation of pile stiffness and raft stiffness. 3 x 3 pile group: L/d = 25; s/d = 2

The first step is to analyse the pile group alone and calculate its overall stiffness, k,. Next the raft is analysed in isolation to find k, . Finally, a full piled raft analysis is performed to calculate P,, P, and wpr. A uniform load of 1 MPa was applied to the raft, giving a total load of 1296 MN.

k , = 14.31 MN/mm

k, = 13.08 MN/mm

Pp = 867.9 MN

P, = 428.1 MN

wpr = 84.03 mm (w, = 84.11 mm)


normalised coordinate 0.0 0.5 1.0

0.0 I-' -

normalised coordinate 0.0 0.5 1 .o

-i 0.0 1 "

0 .*+ A A

0.8 ' normalised coordinate

0.0 0.5 1 .o 0.0 t--- i

0.8 1

I piled raft, s/d = 2

7 estimate, s/d = 2

piled raft, s/d = 4

> estimate, s/d = 4

piled raft, s/d = 6

1 estimate, s/d = 6

3x3 pile group

Krs = 0.1

Lld = 25 Kps = 10000

piled raft, L/d = 10

0 estimate, L/d = 10

piled raft, Ud = 25

0 estimate, L/d = 25

A piled raft, L/d = 100

_ _ ~~ piled raft, 2x2 group

0 estimate, 2x2 group

piled raft, 4x4 group

0 estimate, 4x4 group

A piled raft, 6x6 group

A estimate, 6x6 group ~

3x3 pile group d d = 2 Kps = 10000 Krs = 0.1

L/d = 25 d d = 2 Kps = 10000 Krs = 0.1

Figure 14. Comparison between actual piled raft displacements and estimates from raft alone. Variation of pile spacing, pile length and size of pile group

These results may then be used to calculate a,, and a,,, using equations (9):

a,, = 0.85

apr = 0.71

In this case a,, is greater than 0.8, as might be expected for a relatively flexible system. The effect this has on the prediction of settlement and load distribution can be investigated by substituting the calculated values of k , and k , into equation (6). Assuming a,, = 0.8 gives k,, = 15.42 MN/mm, allowing wpr under the applied total load of 1296 MN to be calculated. This value of wpr may then be used in equations (4) and (5) to calculate Pp and P,. A comparison of the accurate analysis and approximate results is given in Table IX, showing that there is a very small error in estimating


36 m

1

o Pile ll Raft element Figure 15. 9 x 9 piled raft mesh

Table VIII. Properties for 9 x 9 piled raft worked example

Soil Pile Raft Dimensionless groups

E , 280MPa E p 35000MPa E , 35000MPa sJd 5 v, 0.4 L 20m 0, 0.3 Lld 25

d 0.8 m t 5 m Kps 125 S 4 m L, 36m K S , 0.1 19

B, 36m

Table IX. Comparison of actual and predicted results 9 x 9 piled raft worked example

Full piled raft Predicted Prediction analysis values error (YO)

% 0.85 0.8 - 5.9 Wpr 84 mm 84 mm 0 PP 868MN 778MN - 10 pr 428 MN 518 MN 21

wpr. The estimation of load distribution errs by 90 MN, or 6.9% of the total load, resulting in an underestimation of the load taken by the pile group.

To estimate the differential settlements of the piled raft, the results of the isolated raft analysis were factored by k,/k, , (using the predicted value of k,, gives k,/k, , = 0.889). The accurate and



-1 -0.8

8 -0.6 3 -0.4 3 0

4 0.4 0.6

1

-g -0.2

-g 0.2

s 0.8

. full piled raft 0 estimate from factored raft analysis

Figure 16. Comparison between actual piled raft displacements and estimates from raft alone. 9 x 9 square pile group

approximate displacements have been plotted in Figure 15, using the normalized co-ordinate as per Figure 12, and normalizing the displacements with respect to the range of differential displacements as follows:

where

w = displacement (actual piled raft or factored raft)

wav(pr), wrnax(pr), w ~ ~ ~ ( ~ ~ ) = average, maximum and minimum displacement of piled raft

The approximate displacements underestimate the range of piled raft displacements, with a maximum difference of 0.87 mm, or 1 . 1 % . The corner-centre differential displacement is underestimated by 1.68 mm, or 13%.

In summary, the approximate method requires the following procedure:

(1) calculate the pile group stiffness and raft stiffness in isolation, ( 2 ) combine these stiffnesses using equation (6) and an clrp value of 0.8 to calculate the full piled

(3) the load sharing between the pile group and raft in the combined foundation may now be raft stiffness kpr,

found from equations (4) and (5).

CONCLUSIONS

A rigorous numerical method of piled raft analysis has been developed, based on a hybrid approach of load transfer analysis of individual piles, together with elastic interaction between piles and between the various raft elements and the piles. A comprehensive range of foundation geometries has been analysed, within the practical limitations of the method, where computational times become excessive for groups of more than 100 piles.

The numerical analysis has been used to evaluate an existing approximate method for estimating the overall stiffness of the foundation and the proportion of load carried by the pile


group and by the raft. The approximate method was found to be satisfactory for single pile units but became progressively less accurate as the size of the pile group increased. However, a simple modification to the method has allowed it to be extended to pile groups of practical proportions.

The basis of the method allows the overall foundation response to be estimated from the response of the component parts, through the use of an appropriate interaction factor. While this allows the average settlement of the foundation to be estimated, it gives no information concerning the differential settlements. At present, a simple factoring of the differential settlements that would occur for a raft foundation alone is proposed. This gives satisfactory results for groups of up to 9 x 9 piles, except in the extreme cases of a very flexible raft, or very small pile groups (2 x 2 or less) where the pattern of settlement is rather different from a raft alone. Further refinement of this approach is in hand, and will be addressed in a separate publication.

REFERENCES

1 . R. W. Cooke, D. W. Bryden-Smith, M. N. Gooch and D. F. Sillett, Some observations of the foundation loading and settlement of a multi-storey building on a piled raft foundation in London clay, Proc. lns t . o fC iv . Engnrs. Part 1,

2. C. J . Padfield and M. J. Sharrock, Settlement of structures on clay soils, Special Publication 27, Construction

3. H. G. Poulos and E. H. Davis, Pile Foundation Analysis and Design, Wiley, New York, 1980. 4. M. F. Randolph, Design of piled raft foundations, Proc. Int. Symp. on Recent Developments in Laboratory and Field

Tests and Analysis of Geotechnical Problems, Bangkok, 6-9 December 1983, pp. 525-537. 5. S. J. Hain and I. K. Lee, The analysis of flexible pile-raft systems, Geotechnique, 28 (I), 65-83 (1978). 6. T. J. Weisner and P. T. Brown, Behaviour of piled strip footings subject to concentrated loads, Australian Geomech. J .

7. E. Bilotta, V. Caputo and C. Viggiani, Analysis of soil-structure interaction for piled rafts, Proc. 10th European Conf:

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9. P. Clancy, Computer analysis of pile groups, M.Sc. Thesis, University of Manchester, 1990. 10. D. V. Griffiths, P. Clancy and M. F. Randolph, Piled raft foundation analysis by finite elements, Proc. 7th Int. Con$

1 1 . R. D. Mindlin, Force at a point in the interior of a semi-infinite solid, Phys. 7 , 195-202 (1936). 12. Y. K. Chow, Analysis of vertically-loaded pile groups, Int. J . Numer. Analytic. Meth. Geomech., 10, 59-72 (1986). 13. Y. K. Chow, Three-dimensional analysis of pile groups, ASCE J . Geotech. Eny. Diu., 113(6), 637-651 (1987). 14. 1. M. Smith and D. V. Griffiths, Programming the Finite Element Method, 2nd edn, Wiley, New York, 1988. 15. H. B. Seed and L. C. Reese, The actions of soft clay along friction piles, Trans. ASCE, 122, 731-754 (1957). 16. M. F. Randolph and C. P. Wroth, Analysis of deformation of vertically loaded piles, ASCE J . Geotech. Eny. Diu, 104,

17. M. F. Randolph, A theoretical study of the performance of piles, Ph.D. Thesis, University of Cambridge, 1977. 18. L. M. Kraft, R. P. Ray and T. Kagawa, Theoretical t - z curves, ASCE J . Geotech. Eng. Div., 107, GT11, 1543-1561

19. H. G. Poulos and E. H. Davis, The settlement behaviour of single axially loaded incompressible piles and piers,

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an approximate analysis procedure for piled raft foundations

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