Estimativa de Rigidez em Fundação

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AUSTRALIAN GEOMECHANICS VOLUME 56: NO.3 SEPTEMBER 2021 57
 PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION 
Harry G Poulos 
Coffey Services Australia & University of Sydney 
 
ABSTRACT 
This paper makes use of the expressions of Randolph (1994) to derive simple expression for the proportion of raft stiffness 
that can be added to the pile group stiffness in order to estimate the overall stiffness of a piled raft foundation system. 
Simple methods of estimating raft and piled group stiffness values are summarized, and an example illustrating the 
application of the approach is presented. Finally, the approach is used to estimate the settlement of a piled raft foundation 
supporting a high-rise tower in the city of Frankfurt. 
1 INTRODUCTION 
In the preliminary design of a piled raft foundation, when estimating the order of magnitude of the settlement, it is very 
useful to have a simple means of estimating the stiffness of the foundation system. This can then provide a check on the 
final design that may use more complex methods. This note presents a simple approach for estimating the proportion of 
the stiffness of the raft that can be added to the stiffness of the pile group. Use is made of the expressions developed by 
Randolph (1994) for idealized soil profiles. Alternatively, for rafts with few piles, the proportion of pile group stiffness 
that can be added to the raft stiffness can also be derived. 
Some results for the above stiffness proportions are given, and simple methods of estimating the raft and pile group 
stiffness values are presented. The approach presented herein initially assumes elastic soil and pile behavior, but 
allowance for non-linear behavior can be made via the “PDR” approach described by Poulos (2000). 
An example illustrating the application of these methods is presented, together with the calculations for a high-rise tower 
in Frankfurt. 
2 RANDOLPH’S EXPRESSIONS 
Randolph (1994) provided the following very convenient approximate expression for estimating the vertical stiffness of 
a piled raft foundation system: 
KPR = [KP+KR(1-2Dcp)] / [1-Dcp
2KR/KP] (1) 
where KP = stiffness of the group of piles, KR= stiffness of raft, and Dcp = pile-raft interaction factor. 
The pile-raft interaction factor Dcp is given by the following expression: 
Dcp = 1-[ln(rc/r0)]/�] (2) 
where rm = radius at which soil displacements become vanishingly small, 
r0 = radius of pile = 0.5d, where d = pile diameter 
] = ln[rm/r0] (3) 
rc = average radius of pile cap, given by 
rc = [Ar/(S.n)]0.5 (4) 
AR = raft area 
n = number of piles in group 
In some cases, equation (2) can give values of Dcp which lie outside the range 0 to 1.0, and so the values of Dcp used in 
the analysis have been constrained to fall within this range. 
Randolph considered a simplified soil profile, as shown in Figure 1, in which the shear modulus of the soil increased 
linearly with depth, and for the case of an end bearing pile founded on a stiffer stratum. 
The following parameters are defined: 
rm = [0.25+ [(2.5U(1-Q) - 0.25]L (5) 
[ = Gl/Gb (6) 
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PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
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U = Gav/GL (7) 
where L = pile length 
 Q = Poisson’s ratio of soil 
GL = soil shear modulus at level of pile tip 
Gb = soil shear modulus of founding stratum (for end-bearing pile) 
Gav = GL/2 = average shear modulus along pile shaft 
 
Figure 1: Simplified soil profile (Randolph, 1994) 
3 ASSESSING THE PROPORTION OF RAFT STIFFNESS TO ADD TO THE PILE 
GROUP STIFFNESS 
The expression in equation (1) can be modified to express the proportion, KR, of raft stiffness that is added to the pile 
group stiffness to obtain the stiffness of the piled raft system, so that stiffness of the piled raft system is then given as: 
KPR = Kp + KR*KR (8) 
where KR can be calculated as follows: 
KR = {[R+(1-2Dcp)] / [1-Dcp
2R-1]} – R (9) 
and R = KP/KR = ratio of pile group stiffness to raft stiffness. 
Alternatively, the stiffness of a piled raft system can be expressed in terms of the proportion, Kp, of pile stiffness that is 
added to the raft stiffness, so that the piled raft stiffness is then given by 
KPR= KR + KP*KP (10) 
It can be shown readily that KP is related to KR by the following simple expression: 
KP = 1 + (KR-1)/R (11) 
4 VALUES OF PROPORTION OF RAFT STIFFNESS KR 
Equation 9 indicates that the proportion of raft stiffness, KR, is dependent only on the pile-raft interaction factor Dcp and 
the stiffness ratio R = KP/KR. Figure 2 plots KR versus R for various values of Dcp. KR decreases as Dcp increases, and 
tends to decrease as R increases. For values of R in excess of about 2, KR becomes constant, and so limiting values of KR 
can be plotted against Dcp, as shown in Figure 3. 
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PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
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U = Gav/GL (7) 
where L = pile length 
 Q = Poisson’s ratio of soil 
GL = soil shear modulus at level of pile tip 
Gb = soil shear modulus of founding stratum (for end-bearing pile) 
Gav = GL/2 = average shear modulus along pile shaft 
 
Figure 1: Simplified soil profile (Randolph, 1994) 
3 ASSESSING THE PROPORTION OF RAFT STIFFNESS TO ADD TO THE PILE 
GROUP STIFFNESS 
The expression in equation (1) can be modified to express the proportion, KR, of raft stiffness that is added to the pile 
group stiffness to obtain the stiffness of the piled raft system, so that stiffness of the piled raft system is then given as: 
KPR = Kp + KR*KR (8) 
where KR can be calculated as follows: 
KR = {[R+(1-2Dcp)] / [1-Dcp
2R-1]} – R (9) 
and R = KP/KR = ratio of pile group stiffness to raft stiffness. 
Alternatively, the stiffness of a piled raft system can be expressed in terms of the proportion, Kp, of pile stiffness that is 
added to the raft stiffness, so that the piled raft stiffness is then given by 
KPR= KR + KP*KP (10) 
It can be shown readily that KP is related to KR by the following simple expression: 
KP = 1 + (KR-1)/R (11) 
4 VALUES OF PROPORTION OF RAFT STIFFNESS KR 
Equation 9 indicates that the proportion of raft stiffness, KR, is dependent only on the pile-raft interaction factor Dcp and 
the stiffness ratio R = KP/KR. Figure 2 plots KR versus R for various values of Dcp. KR decreases as Dcp increases, and 
tends to decrease as R increases. For values of R in excess of about 2, KR becomes constant, and so limiting values of KR 
can be plotted against Dcp, as shown in Figure 3. 
PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
 
 
 
Figure 2: KR versus R 
 
Figure 3: Limiting KR versus Dcp (for R>2) 
The pile-raft interaction factor Dcp can be evaluated from equation 2 (together with equations 3-7). Figure 4 shows 
computed values of Dcp for piles in a uniform soil, and for values of rc/d ranging between 2 and 10. Both floating piles 
and end bearing piles (with a ratio of moduli of end bearing stratum to overlying soil (Eb/Es) equal to 5) are considered. 
The following points can be noted from Figure 4: 
1. Dcp increases as the length to diameter ratio L/d increases. 
2. Dcp decreases as rc/d increases. 
3. Dcp is smaller for floating piles than for end bearing piles. 
Corresponding values of Dcp for both floating and end bearing piles are shown in Figure 5. For end bearing piles, the ratio 
of the modulus of the bearing stratum to the soil modulus at the level of the pile tip is 5. The characteristics of behaviour 
are similar to those for a uniform soil in Figure 4. However, the Dcp values for the Gibson soil tend to be somewhat smaller 
than the corresponding values for the uniform soil. 
It should be noted that smaller values of Dcp imply a smaller amount of pile-raft interaction, and therefore a larger stiffness 
of the piled raftsystem. 
Figures 2 to 5 provide a convenient means of estimating the proportion of cap stiffness to be added to the stiffness of the 
piles to obtain the stiffness of the piled raft system. The following section will discuss some convenient approaches to 
estimating the stiffness of the raft and the piles. 
2
2 2
K 2
2
2
 K
D
 K 2
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Figure 4:�Dcp values for piles in uniform soil: floating and end bearing (Eb/Es = 5) 
 
Figure 5: Dcp values for piles in Gibson soil: floating and end bearing (Eb/Esl = 5) 
5 INCLUSION OF NON-LINEAR RESPONSE 
The above analyses are for linear pile and soil response, but they can be modified in an approximate way via the so-called 
“PDR” method described by Poulos (2000). This approach is illustrated in Figure 6, which shows a tri-linear load-
settlement relationship for the piled raft. The first part of the curve can be obtained via the piled raft stiffness KPR 
calculated from equations 8 or 10. The behaviour is assumed to remain elastic until the load reaches the value P1 (Point 
A in Figure 6) at which the pile capacity is fully utilised, assuming that this occurs simultaneously for all piles, and that 
the ultimate pile capacity is fully mobilised before the ultimate raft capacity. P1 is given by: 
P1 = Pup/(1 – Ep) (12) 
where Pup = ultimate capacity of the piles 
Ep = proportion of load carried by the piles in the piled raft system 
Beyond Point A, the load-settlement relationship if governed by the stiffness of the raft, KR, until Point B is reached, at 
which point the ultimate capacity, Pu, of the entire piled raft system is mobilised. 
2
2
 
 2
 
 
 
 
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2
2
2
 
 2
 
 
 
 
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Figure 4:�Dcp values for piles in uniform soil: floating and end bearing (Eb/Es = 5) 
 
Figure 5: Dcp values for piles in Gibson soil: floating and end bearing (Eb/Esl = 5) 
5 INCLUSION OF NON-LINEAR RESPONSE 
The above analyses are for linear pile and soil response, but they can be modified in an approximate way via the so-called 
“PDR” method described by Poulos (2000). This approach is illustrated in Figure 6, which shows a tri-linear load-
settlement relationship for the piled raft. The first part of the curve can be obtained via the piled raft stiffness KPR 
calculated from equations 8 or 10. The behaviour is assumed to remain elastic until the load reaches the value P1 (Point 
A in Figure 6) at which the pile capacity is fully utilised, assuming that this occurs simultaneously for all piles, and that 
the ultimate pile capacity is fully mobilised before the ultimate raft capacity. P1 is given by: 
P1 = Pup/(1 – Ep) (12) 
where Pup = ultimate capacity of the piles 
Ep = proportion of load carried by the piles in the piled raft system 
Beyond Point A, the load-settlement relationship if governed by the stiffness of the raft, KR, until Point B is reached, at 
which point the ultimate capacity, Pu, of the entire piled raft system is mobilised. 
2
2
 
 2
 
 
 
 
 2
 
 
 
 
2
2
2
 
 2
 
 
 
 
 2
 
 
 
 
PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
 
 
 
Figure 6: Simplified load-settlement relationship for piled raft 
The proportion of load carried by the piles can be obtained from then following relationship: 
 Ep = 1 - (1-Dcp) /(R + (1-2.Dcp)] (13) 
Figure 7 plots the relationship between Ep, Dcp and R, and shows that once R exceeds 1, the piles take more than half of 
the load. 
 
Figure 7: Proportion of load carried by piles 
The ultimate capacity of the piled raft system, Pu, can be estimated from the following relationship (Mandolini et al, 
2013): 
 Pu = Pup + DuR.PuR (14) 
where Pup = ultimate load capacity of pile group 
 PuR = ultimate capacity of raft 
 DuR = proportion of ultimate raft capacity = 1-3.FF 
 FF = filling factor = (AG/AR)/(s/d) ≥ 0 
 s = average pile spacing 
 d = pile diameter 
 AG = area occupied by pile group 
 AR = area of raft 
The average pile spacing, s, can be calculated as: 
 s = (AR/n)0.5 (14a) 
PuP and PuR can be estimated by conventional methods of pile and raft ultimate load estimation. 
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6 SIMPLE APPROACHES FOR ESTIMATING RAFT AND PILE STIFFNESSES 
To facilitate hand calculation of the stiffness of the raft and the pile group, some solutions that can be employed are 
summarized below. 
6.1 RAFT STIFFNESS 
The average vertical stiffness of a raft on the surface of an elastic layer of finite depth can be obtained from the solutions 
for a rigid circular area in Poulos and Davis (1974). These solutions can be re-expressed as follows: 
 KR= IRK de Esav (15) 
where IRK = stiffness influence factor, plotted in Figure 8, de = equivalent diameter of the raft, Es = average Young’s 
modulus of the soil profile, and nu = Poisson’s ratio of soil. 
 
Figure 8: Raft stiffness factor IRK (nu = Poisson’s ratio) 
The equivalent raft diameter can be obtained as: 
 de = 2(AR/S)0.5 (16) 
where AR = area of raft. 
For a layered soil profile, the average Young’s modulus of the soil below the raft, Esav, can be estimated from the approach 
described by Poulos (1994): 
Esav = 6 [Wihi] / 6 [Wihi/Esi] (17) 
where Wi = weighting factor for a layer i, hi = thickness of layer i, Esi = Young’s modulus of layer i, and the summations 
are carried for the number of layers within the soil profile. 
The weighting factor Wi can be obtained from elastic theory, but convenient linear approximations are shown in Table 1. 
Table 1: Approximations for weighting factor Wi 
Poisson’s ratio = 0.3 
 
Poisson’s ratio = 0.5 
Range of relative depth to 
centerline of layer i, zi/de 
Wi Range of relative depth to 
centerline of layer i, zi/de 
Wi 
0 to 0.2 0.70+1.5zi/de 0 to 0.35 0.3+2zi/de 
0.2 to 1.0 1.0-0.75(zi/de-0.2) 0.3 to 1.0 1.0-0.5(zi/de-0.3) 
1.0 to 3.0 0.4-0.2(zi/de-1) 1.0 to 3.0 0.5-0.25(zi/de-1) 
>3.0 0 >3.0 0 
 
2
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6 SIMPLE APPROACHES FOR ESTIMATING RAFT AND PILE STIFFNESSES 
To facilitate hand calculation of the stiffness of the raft and the pile group, some solutions that can be employed are 
summarized below. 
6.1 RAFT STIFFNESS 
The average vertical stiffness of a raft on the surface of an elastic layer of finite depth can be obtained from the solutions 
for a rigid circular area in Poulos and Davis (1974). These solutions can be re-expressed as follows: 
 KR= IRK de Esav (15) 
where IRK = stiffness influence factor, plotted in Figure 8, de = equivalent diameter of the raft, Es = average Young’s 
modulus of the soil profile, and nu = Poisson’s ratio of soil. 
 
Figure 8: Raft stiffness factor IRK (nu = Poisson’s ratio) 
The equivalent raft diameter can be obtained as: 
 de = 2(AR/S)0.5 (16) 
where AR = area of raft. 
For a layered soil profile, the average Young’s modulus of the soil below the raft, Esav, can be estimated from the approach 
described by Poulos (1994): 
Esav = 6 [Wihi] / 6 [Wihi/Esi] (17) 
where Wi = weighting factor for a layer i, hi = thickness of layer i, Esi = Young’s modulus of layer i, and the summations 
are carried for the number of layers within the soil profile. 
The weighting factor Wi can be obtained from elastic theory,but convenient linear approximations are shown in Table 1. 
Table 1: Approximations for weighting factor Wi 
Poisson’s ratio = 0.3 
 
Poisson’s ratio = 0.5 
Range of relative depth to 
centerline of layer i, zi/de 
Wi Range of relative depth to 
centerline of layer i, zi/de 
Wi 
0 to 0.2 0.70+1.5zi/de 0 to 0.35 0.3+2zi/de 
0.2 to 1.0 1.0-0.75(zi/de-0.2) 0.3 to 1.0 1.0-0.5(zi/de-0.3) 
1.0 to 3.0 0.4-0.2(zi/de-1) 1.0 to 3.0 0.5-0.25(zi/de-1) 
>3.0 0 >3.0 0 
 
2
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PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
 
 
6.2 PILE GROUP STIFFNESS 
The stiffness of a pile group can be related to the stiffness of a single pile, Kp1. Kp1 can be estimated using solutions from 
elastic theory, for example, the closed form solution of Randolph and Wroth (1978), or the charts presented by Poulos 
and Davis (1980). To conveniently allow for both floating and end bearing piles, the solutions presented by Poulos (2001) 
can also be used, whereby Kp1 is given by: 
 Kp1 = dp.Es/IU (18) 
where Es = average Young’s modulus along pile shaft, and IU = influence factor shown in Figure 9, and Eb = Young’s 
modulus of bearing stratum. 
The equivalent soil modulus along the shaft can be approximated as follows: 
 Es = 6 (hi/Esi)/ L (19) 
where hi = thickness of layer i, Esi = Young’s modulus of layer i, and the summation is carried out for the soil layers along 
the pile shaft. 
Young’s modulus at the base of the pile, Eb, can be estimated from equation 14, with the summation being carried out 
from the level of the pile base to a depth of 3 pile base diameters below the pile base. 
The corresponding case of a pile in a Gibson soil is shown in Figure 10. In this case, the single pile stiffness is given by: 
 Kp1 = d.L.Nv/ IU (20) 
where Nv = rate of increase of Young’s modulus with depth and L = length of pile. 
For a group of n piles, group stiffness can be expressed as follows: 
 KP = Kp1.n.RG (21) 
where Kp1 = stiffness of a single pile, n = number of piles, RG = group stiffness reduction factor. 
 
 
Figure 9: Influence factor for axial stiffness of a single pile in a uniform soil (Poulos, 2001) 
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Figure 10: Influence factor for axial stiffness of a single pile in a Gibson soil (Poulos, 2001) 
Values of the group stiffness reduction factor RG can be computed from a pile group analysis program such as DEFPIG 
(Poulos, 1990). For a typical case of piles with length-to-diameter (L/d) ratios of 25 and 50, and for a relative pile to soil 
modulus ratio of 600, Figures 11 and 12 plot values of RG for both floating and end bearing piles, and for various values 
of the average centre-to-centre spacing of the piles. For the end bearing piles, the ratio of Young’s modulus of the bearing 
stratum to that of the soil is 5. To try and obtain more practically relevant values of RG, allowance has been made for the 
greater stiffness of the soil between the piles, using the approximate approach described by Poulos (1988), and assuming 
that the near-pile modulus of the soil is 1/3 of the value between the piles. 
It can be seen that the value of RG for floating and end bearing piles are similar, especially for the wider pile spacings and 
for the relatively longer piles. As the spacing decreases, RG decreases, and the value for floating piles is somewhat less 
than for the corresponding end bearing piles. However, for relatively large groups, there is very little difference between 
the RG values for floating and end bearing piles. 
 
Figure 11: Group stiffness reduction factor for floating and end bearing piles; L/d=25 
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Figure 10: Influence factor for axial stiffness of a single pile in a Gibson soil (Poulos, 2001) 
Values of the group stiffness reduction factor RG can be computed from a pile group analysis program such as DEFPIG 
(Poulos, 1990). For a typical case of piles with length-to-diameter (L/d) ratios of 25 and 50, and for a relative pile to soil 
modulus ratio of 600, Figures 11 and 12 plot values of RG for both floating and end bearing piles, and for various values 
of the average centre-to-centre spacing of the piles. For the end bearing piles, the ratio of Young’s modulus of the bearing 
stratum to that of the soil is 5. To try and obtain more practically relevant values of RG, allowance has been made for the 
greater stiffness of the soil between the piles, using the approximate approach described by Poulos (1988), and assuming 
that the near-pile modulus of the soil is 1/3 of the value between the piles. 
It can be seen that the value of RG for floating and end bearing piles are similar, especially for the wider pile spacings and 
for the relatively longer piles. As the spacing decreases, RG decreases, and the value for floating piles is somewhat less 
than for the corresponding end bearing piles. However, for relatively large groups, there is very little difference between 
the RG values for floating and end bearing piles. 
 
Figure 11: Group stiffness reduction factor for floating and end bearing piles; L/d=25 
2
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2 
2 
 
 
 
 
2
PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
 
 
 
Figure 12: Group stiffness reduction factor for floating and end bearing piles; L/d=50 
6.3 APPROXIMATIONS FOR GROUP STIFFNESS REDUCTION FACTOR 
Using an approximation that has been employed previously (Fleming et al, 1985, Poulos, 1989), the group stiffness 
reduction factor RG can be expressed as follows: 
 RG = n-w (22) 
where n = number of piles in group, and w = exponent that depends on relative pile spacing and other pile-soil parameters. 
Via calibration with the values of RG in Figures 11 and 12, the values of w derived are plotted against the dimensionless 
average spacing s/d in Figure 13, where s is obtained from equation 14a. There is little difference between the values for 
floating and end bearing piles, and thus the following best-fit linear approximation is found for w: 
 w = 0.561 – 0.0421(s/d) (23) 
This approximation applies to the range of L/d (25 to 50) and for the floating and end bearing cases considered herein. 
From equations 16 and 22, the pile group stiffness can therefore be expressed as: 
 KP = Kp1.n.n(-w) = Kp1.n(1-w) (24) 
 
Figure 13: Values of group stiffness reduction factor exponent, w 
It should be noted that there are other approaches that have been developed for estimating the group stiffness reduction 
factor RG, for example, Mandolini et al (2005), Sheil and McCabe (2013). These approaches tend to give smaller values 
of RG than equations 22 and 23. 
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7 SUMMARY OF SIMPLIFIED PROCEDURE 
Table 2 summarizes the procedure set out in this paper. Both the general steps in this procedure, and their implementation 
via the simplified methods described, are included. 
Table 2: Steps in Simplified Procedure 
Step 
No. 
Process 
 
Implementation via simplified procedure 
1 Develop simplified geotechnical model for 
both raft and pile analyses 
For raft, average modulus for raft from equation 17 
and Table 1. For piles, average modulus along pile 
shaft from equation 19, and average modulus at pile 
base from equation 17. 
2 Compute stiffness of raft, KR Equations 15 & 16, & Figure 8 
3 Compute equivalent relativebeen examined subsequently by a number of authors, including 
O’Neill et al (1996), Poulos et al (1997), and Reul and Randolph (2003). The approach that has been developed in this 
paper will be applied to this case. 
The key details are as follows: 
x Raft area = 43 m by 29 m. 
x Raft is founded 15 m below the ground surface, in a deep deposit of Frankfurt clay. 
x Average Young’s modulus of soil (via pressuremeter tests) Es = 62.4 MPa. 
x Piles 30 m long and 1.3 m in diameter. 
x Total applied load = 968 MN. 
The equivalent diameter of the raft, from equation 16, is de = 2(43x29/S)0.5 = 39.8 m. 
From Figure 8, for a deep layer with Poisson’s ratio = 0.3, the raft stiffness factor IRK = 1.1. 
Thus the stiffness of the raft, from equation 15, is found to be KR = 1.1x39.8x62.4 = 2714 MN/m. 
Assuming a pile modulus of EP = 30,000 MPa, the pile stiffness factor is 30,000/62.4 = 481. Also, the pile length to 
diameter ratio is 30/1.3 = 23.1. Interpolating from Figure 9, the single pile influence factor is about 0.092. Thus, the single 
pile stiffness, from equation 20, is Kp1 = 1.3x62.4/0.092 = 882 MN/m. 
Considering now the pile group, the average spacing of the piles, from equation 14a, is s = (43x29/40)0.5 = 5.6 m, so that 
s/d = 5.6/1.3 = 4.3. From equation 23, the exponent w = 0.561 – 0.0421x4.3 = 0.38. 
From equation 24, the pile group stiffness is therefore KP = 882x40 (1-0.38) = 8678 MN/m. 
The ratio R = KP/KR is then 8679/2714 = 3.31. 
The average pile cap radius rc is obtained from equation 4 as (43x29/Sx40)0.5 = 3.15 m, so that rc/d = 3.15/1.3 = 2.42. 
From Figure 4, the pile-raft interaction factor Dcp is found to be about 0.71. The proportion of raft stiffness, KR can then 
be obtained from Figure 3, since R > 2, and is found to be 0.10. 
Thus the stiffness of the piled raft, from equation 8, is 8678 + 0.1x2714 = 8949 MN/m. 
The corresponding average settlement is then 968/8949 = 0.108 m = 108 mm. This compares well with computed values, 
from various methods, of between about 105 mm and 170 mm, and a measured value of 110 mm, as reported by Poulos 
et al (1997). 
10 CONCLUSIONS 
A simplified approach to estimating the vertical stiffness of a piled raft foundation system is set out in this paper. 
Expressions derived from Randolph’s equations are given for the proportion, KR, of raft stiffness to be added to the pile 
group stiffness, or the proportion, KP, of pile stiffness to be added to the raft stiffness. Both of these proportions are 
dependent on the pile-raft interaction factor Dcp, which is in turn dependent on the ratio of pile group to raft stiffness, R, 
and it is found that a limiting value of Dcp is reached once R exceeds about 2. 
Simplified expressions are presented for estimating the raft and pile group stiffness values without the need to implement 
computer programs. An example illustrating the application of the proposed approach is presented. The results are found 
to be similar to, and in this case somewhat more conservative than, those from two other computer programs. The 
approach also gives a reasonable estimate of the settlement of a high-rise tower in Frankfurt. 
11 REFERENCES 
Fleming, W.G.K., Weltman, A.J., Randolph, M.F. & Elson, W.K. (1985). “Piling engineering”. Surrey University Press, 
Halsted Press. 
Franke, E., Lutz, B. & El- Mossallamy, Y. (1994). “Measurements and numerical modelling of high-rise building 
foundations on Frankfurt clay”. Vert and Horiz. deformation of foundations and embankments, ASCE Geot. Spec. 
Pub. No. 40. 2: 1325-1336. 
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9 APPLICATION TO A CASE HISTORY 
Franke et al (1994) have documented the case history of the Westend Tower in Frankfurt. This is a 208 m tall tower which 
is supported on a piled raft with 40 bored piles, and has been examined subsequently by a number of authors, including 
O’Neill et al (1996), Poulos et al (1997), and Reul and Randolph (2003). The approach that has been developed in this 
paper will be applied to this case. 
The key details are as follows: 
x Raft area = 43 m by 29 m. 
x Raft is founded 15 m below the ground surface, in a deep deposit of Frankfurt clay. 
x Average Young’s modulus of soil (via pressuremeter tests) Es = 62.4 MPa. 
x Piles 30 m long and 1.3 m in diameter. 
x Total applied load = 968 MN. 
The equivalent diameter of the raft, from equation 16, is de = 2(43x29/S)0.5 = 39.8 m. 
From Figure 8, for a deep layer with Poisson’s ratio = 0.3, the raft stiffness factor IRK = 1.1. 
Thus the stiffness of the raft, from equation 15, is found to be KR = 1.1x39.8x62.4 = 2714 MN/m. 
Assuming a pile modulus of EP = 30,000 MPa, the pile stiffness factor is 30,000/62.4 = 481. Also, the pile length to 
diameter ratio is 30/1.3 = 23.1. Interpolating from Figure 9, the single pile influence factor is about 0.092. Thus, the single 
pile stiffness, from equation 20, is Kp1 = 1.3x62.4/0.092 = 882 MN/m. 
Considering now the pile group, the average spacing of the piles, from equation 14a, is s = (43x29/40)0.5 = 5.6 m, so that 
s/d = 5.6/1.3 = 4.3. From equation 23, the exponent w = 0.561 – 0.0421x4.3 = 0.38. 
From equation 24, the pile group stiffness is therefore KP = 882x40 (1-0.38) = 8678 MN/m. 
The ratio R = KP/KR is then 8679/2714 = 3.31. 
The average pile cap radius rc is obtained from equation 4 as (43x29/Sx40)0.5 = 3.15 m, so that rc/d = 3.15/1.3 = 2.42. 
From Figure 4, the pile-raft interaction factor Dcp is found to be about 0.71. The proportion of raft stiffness, KR can then 
be obtained from Figure 3, since R > 2, and is found to be 0.10. 
Thus the stiffness of the piled raft, from equation 8, is 8678 + 0.1x2714 = 8949 MN/m. 
The corresponding average settlement is then 968/8949 = 0.108 m = 108 mm. This compares well with computed values, 
from various methods, of between about 105 mm and 170 mm, and a measured value of 110 mm, as reported by Poulos 
et al (1997). 
10 CONCLUSIONS 
A simplified approach to estimating the vertical stiffness of a piled raft foundation system is set out in this paper. 
Expressions derived from Randolph’s equations are given for the proportion, KR, of raft stiffness to be added to the pile 
group stiffness, or the proportion, KP, of pile stiffness to be added to the raft stiffness. Both of these proportions are 
dependent on the pile-raft interaction factor Dcp, which is in turn dependent on the ratio of pile group to raft stiffness, R, 
and it is found that a limiting value of Dcp is reached once R exceeds about 2. 
Simplified expressions are presented for estimating the raft and pile group stiffness values without the need to implement 
computer programs. An example illustrating the application of the proposed approach is presented. The results are found 
to be similar to, and in this case somewhat more conservative than, those from two other computer programs. The 
approach also gives a reasonable estimate of the settlement of a high-rise tower in Frankfurt. 
11 REFERENCES 
Fleming, W.G.K., Weltman, A.J., Randolph, M.F. & Elson, W.K. (1985). “Piling engineering”. Surrey University Press, 
Halsted Press. 
Franke, E., Lutz, B. & El- Mossallamy, Y. (1994). “Measurements and numerical modelling of high-rise building 
foundations on Frankfurt clay”. Vert and Horiz. deformation of foundations and embankments, ASCE Geot. Spec. 
Pub. No. 40. 2: 1325-1336. 
PRACTICAL APPROACH FOR PILED RAFT STIFFNESS ESTIMATION POULOS 
 
 
Mandolini, A., Russo, G. and Viggiani, C.(2005). “Pile foundations: experimental investigations, analysis and design”. 
Proc. 16th Int. Conf. Soil Mechs. Geot. Eng., Osaka, IOS Press, 177-213. 
Mandolini, A., Di Laora, R, and Mascaruddi, Y. (2013). “Rational design of piled raft”. Procedia Engineering, 57: 45-52. 
O’Neill, M.W., Caputo, V., De Cock, F., Hartikainen, J. and Mets, M. (1996). “Case histories of pile-supported rafts”. 
Report of ISSMFE Technical CommitteeTC18, 116 pp. 
Poulos, H.G. (1988). “Modified calculation of pile-group interaction”. Jnl. Geot. Eng., ASCE, Vol. 114, No. 6, pp. 697-
706. 
Poulos, H.G. (1989). “Pile behaviour - theory and application”. 29th Rankine Lecture. Géotechnique, 39(3): 365-415 
Poulos, H.G. (1990). “DEFPIG user’s manual”. Centre for Geot. Research, Univ. of Sydney. 
Poulos, H.G. (1994). “Settlement prediction for driven piles and pile groups”. Spec. Tech. Pub. 40, ASCE, 2: 1629-1649. 
Poulos, H.G. (2001). “Pile foundations”. Ch. 10 of Geotechnical and Geoenvironmental Engineering Handbook, Ed. R.K. 
Rowe, Kluwer Academic Press, Boston, pp. 261-304. 
Poulos, H.G. and Davis, E.H. (1974). “Elastic solutions for soil and rock mechanics”. John Wiley, New York. 
Poulos, H.G. and Davis, E.H. (1980). “Pile foundation analysis and design”. John Wiley, New York. 
Poulos, H.G., Small, J.C., Ta, L.D., Sinha, J. and Chen, L. (1997). “Comparison of some methods for analysis of piled 
rafts”. Proc. 14th Int. Conf. Soil Mechs. Foundn. Eng. Hamburg, Balkema, Rotterdam, Vol. 2, 1119-1124. 
Randolph, M.F. (1994). “Design methods for pile groups and piled rafts”. State of the Art Rep., Proc., 13th ICSMFE, 
New Delhi, Vol. 5, 61–82. 
Randolph, M.F. and Wroth, C.P. (1978). “Analysis of deformation of vertically loaded piles”. Jnl. Geot. Eng. Div., ASCE, 
104(GT12): 1465-1488. 
Reul, O. and Randolph, M.F. (2003). “Piled rafts in overconsolidated clay: comparison of in situ measurements and 
numerical analyses”. Geotechnique, 53(3): 301-315. 
Sheil, B.B. and McCabe, B.A. (2014). “A finite element-based approach for predictions of rigid pile group stiffness 
efficiency in clays”. Acta Geotechnica, 9:469-484. 
Small, J.C. and Poulos, H.G. (2007). “A method of analysis of piled rafts”. Proc. 10th ANZ Conf. Geomechanics, Brisbane, 
Australian Geomechanics Society, Vol. 2, pp.602-607.TC18, 116 pp. 
Poulos, H.G. (1988). “Modified calculation of pile-group interaction”. Jnl. Geot. Eng., ASCE, Vol. 114, No. 6, pp. 697-
706. 
Poulos, H.G. (1989). “Pile behaviour - theory and application”. 29th Rankine Lecture. Géotechnique, 39(3): 365-415 
Poulos, H.G. (1990). “DEFPIG user’s manual”. Centre for Geot. Research, Univ. of Sydney. 
Poulos, H.G. (1994). “Settlement prediction for driven piles and pile groups”. Spec. Tech. Pub. 40, ASCE, 2: 1629-1649. 
Poulos, H.G. (2001). “Pile foundations”. Ch. 10 of Geotechnical and Geoenvironmental Engineering Handbook, Ed. R.K. 
Rowe, Kluwer Academic Press, Boston, pp. 261-304. 
Poulos, H.G. and Davis, E.H. (1974). “Elastic solutions for soil and rock mechanics”. John Wiley, New York. 
Poulos, H.G. and Davis, E.H. (1980). “Pile foundation analysis and design”. John Wiley, New York. 
Poulos, H.G., Small, J.C., Ta, L.D., Sinha, J. and Chen, L. (1997). “Comparison of some methods for analysis of piled 
rafts”. Proc. 14th Int. Conf. Soil Mechs. Foundn. Eng. Hamburg, Balkema, Rotterdam, Vol. 2, 1119-1124. 
Randolph, M.F. (1994). “Design methods for pile groups and piled rafts”. State of the Art Rep., Proc., 13th ICSMFE, 
New Delhi, Vol. 5, 61–82. 
Randolph, M.F. and Wroth, C.P. (1978). “Analysis of deformation of vertically loaded piles”. Jnl. Geot. Eng. Div., ASCE, 
104(GT12): 1465-1488. 
Reul, O. and Randolph, M.F. (2003). “Piled rafts in overconsolidated clay: comparison of in situ measurements and 
numerical analyses”. Geotechnique, 53(3): 301-315. 
Sheil, B.B. and McCabe, B.A. (2014). “A finite element-based approach for predictions of rigid pile group stiffness 
efficiency in clays”. Acta Geotechnica, 9:469-484. 
Small, J.C. and Poulos, H.G. (2007). “A method of analysis of piled rafts”. Proc. 10th ANZ Conf. Geomechanics, Brisbane, 
Australian Geomechanics Society, Vol. 2, pp.602-607.