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Approximate pile-to-pile interaction factors between two dissimilar piles
Computers and Geotechnics 32 (2005) 613–618 www.elsevier.com/locate/compgeo
Technical communication
Approximate pile-to-pile interaction factors between two dissimilar piles Sii Chung Wong a, Harry G. Poulos a
b,*
Chee Lian Development Construction Sdn. Bhd., Sarawak, Malaysia Coffey Geosciences, P.O. Box 125, North Ryde, Sydney, Australia
b
Received 22 June 2005; received in revised form 1 November 2005; accepted 11 November 2005 Available online 6 January 2006
Abstract This paper develops approximations for the settlement interaction factors between two dissimilar piles. Via an extensive parametric study using the computer program GEPAN, approximations are developed for interaction factors for piles having dissimilar diameters but equal lengths, piles having dissimilar lengths, and for piles having dissimilar ground conditions at the pile tips. Correction factors are then given to allow for the effects of piles stiffness and length-to-diameter ratio. The approximations may be employed in analyses for group settlements to allow for the case where not all piles in the group are identical. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In predicting the response characteristics of pile groups, and piled rafts, the application of interaction factors based on the theory of elasticity that employs the equations of Mindlin [1] has been widely adopted. The interaction factor, denoted by aij, was defined as the additional displacement at the top of pile i due to a loaded adjacent pile j, divided by the settlement of pile j under its own load. Poulos [2,3], Ta [4], and Zhang [5] have successfully employed the interaction factor into their analysis for predicting the response characteristics of pile groups and piled raft foundation systems. However, the analyses carried out by the above-mentioned investigators were for foundations in which the piles had identical dimensions and properties. More rigorous numerical analyses such as GEPAN [7], are required to provide a more accurate response characteristic for piled foundations with dissimilar piles. However, GEPAN demands more computing resources and computing time, especially for cases involving large pile groups. In order to provide a rapid prediction of the response characteristics of piled foundations with dissimilar piles, Hewitt *
Corresponding author. Tel.: +61 2 9911 1000; fax: +61 2 9911 1001. E-mail address: harry_poulos@coffey.com.au (H.G. Poulos).
0266-352X/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2005.11.001
[6] proposed approximate relationships for dissimilar piles to determine the pile-to-pile interaction factor between two dissimilar piles. However, there are some shortcomings in these relationships, which will be discussed later. This paper presents a parametric study carried out using GEPAN [7] to examine in detail the interaction characteristics between two dissimilar piles. Subsequently, attempts are made to provide an approximate relationship to enable the interaction factors between two dissimilar piles to be obtained rapidly for more complicated combinations of two dissimilar piles, namely for dissimilar pile lengths, dissimilar pile tip conditions, and for dissimilar pile diameters. Assessments of the potential for the approximate relationship to be used as a design tool are also made. 2. Analysis method The program GEPAN is described by Xu and Poulos [7], and has been used in the present study. GEPAN employs a boundary element analysis in which the soil is modelled as an elastic continuum and each pile is discretised via a series of elements along and around the pile, which is assumed to be circular in cross-section and perfectly elastic. Use is made of Mindlin’s equations of elasticity to compute the vertical and lateral soil move-
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
Lj = varies Dj = 0.3m Ep = 30.0GPa
Es = 4.0MPa νs = 0.35 Fig. 1. Piles with dissimilar lengths.
ð1Þ
3. Piles with dissimilar diameters Hewitt [6] postulated approximate relationships to determine the pile-to-pile interaction factor between two piles with dissimilar diameters as shown in the following equation:
0.70
4. Piles with dissimilar length (incorporating dissimilar pile tip conditions) In this study, two piles of equal diameter, (Di = Dj) but with different lengths, Li and Lj have been analysed (see Fig. 1). As in the previous section, both piles are installed in a deep homogeneous elastic soil layer. The influence of the loaded pile on the unloaded pile is shown in Figs. 2 and 3. Both figures show the variation of the interaction factors for pile i due to pile j, aij, with the dimensionless ratio of the pile spacing to the diameter of the loaded pile j, s/Dj. Fig. 2 shows the aij obtained for cases whereby pile i has a constant length (Li = 50 m), while the length of the loaded pile j, Lj, changes from 15 m to 45 m. The interaction factors plotted in Fig. 3 are for cases where pile i has varying length, Li, while the length of the loaded pile j remains constant.
ij
0.50 0.40 0.30 0.20 0.10
ð2Þ
0.00 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
s/D j
Fig. 2. Interaction factors aij for two piles with different lengths (Li > Lj).
0.80 0.70
L j :L i 50m:45m (GEPAN)
0.60
50m:35m (GEPAN)
ij
i.e. the interaction factor for pile i due to pile j, aij, is approximately the same as the interaction factor between two identical piles j, ajj. An extensive parametric study carried out in this present study using GEPAN has also shown that the approximate relationship proposed by Hewitt [6] in Eq. (2) can be applied in estimating the interaction factors between two piles with dissimilar pile diameters. Subsequently, no further consideration needs to be given to this case.
L i :L j 50m:45m (GEPAN) 50m:35m (GEPAN) 50m:25m (GEPAN) 50m:15m (GEPAN) 50m:45m (Hewitt,1988) 50m:35m (Hewitt,1988) 50m:25m (Hewitt,1988) 50m:15m (Hewitt,1988)
0.60
Interaction factor,
where Pi is the load on pile i, Ki is the axial head stiffness of pile i, aij is the interaction factor for effect of pile j (influencing loaded pile) on pile i (influenced pile), Pj is the load on pile j and Kj is the axial head stiffness of pile j.
aij ajj
Li = varies Di = 0.3m Ep = 30.0GPa
Interaction factor,
S i ¼ P i =K i þ aij P j =K j
Pile i
ments due to the stresses developed at the pile–soil interface. Compatibility of all components of displacement is imposed at the pile–soil interface in order to obtain the pile–soil stresses and the pile displacements. Allowance can also be made for specifying limiting values of the pile–soil stresses so that non-linear response of the pile can be simulated. The program can consider groups of piles with different pile dimensions and properties, and thus is a very suitable tool for the present parametric study. In this case, consideration is confined to purely elastic pile and soil behaviour and to the axial response of two dissimilar piles. Once the interaction factor has been obtained, the settlement Si of a pile i due to its own load and due to an adjacent loaded pile j, can be expressed as follows:
Pile j
614
50m:25m (GEPAN) 0.50
50m:15m (GEPAN) 50m:50m (Hewitt, 1988)
0.40 0.30 0.20 0.10 0.00 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
s /D j
Fig. 3. Interaction factors aij for two piles with different lengths (Li < Lj).
Hewitt [6] attempted to obtain approximate relationships to determine interaction factors for two friction piles with different pile lengths and the approximate relationships postulated by Hewitt [6] are reproduced in the following equations:
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
For Li > Lj: aii þ ajj aij 2
fls ¼ ð3Þ
ð4Þ
where, aii and ajj represents the interaction factor for two identical piles having the dimensions of pile i and pile j, respectively. The approximate interaction factors, aij, between two piles with dissimilar pile lengths obtained using Hewitt’s [6] recommendations have also been plotted in both Figs. 2 and 3. Hewitt’s [6] approximate relationship in Eq. (3) is seen to overestimate the interaction of the shorter pile on the longer pile when the difference in pile length (h) between the two piles is greater than 30% of the longer pile (see Fig. 2), while. Eq. (4) tends to underestimate the interaction of the longer pile on the shorter pile for a length difference (h) greater than about 30% of the longer pile (see Fig. 3). It has been found by the authors that a more accurate approximate relationship is given in the following equations: For Li > Lj: aii þ ajj aij fls For Li < Lj: ajj aij fsl
A lnðDj =sÞ þ B 1 105
ð7Þ
C lnðs=Dj Þ2 þ D lnðs=Dj Þ þ E 1 106
ð8Þ
and,
For Li < Lj: aij ajj
fsl ¼
where parameters A, B, C, D and E can be expressed as functions of Es/Eb and h/Ll. The approximate expression for parameters A, B, C, D and E are detailed in Appendix I, and have been obtained via curve fitting. 5. Incorporation of the effects of pile stiffness and length-todiameter ratio A parametric study has been carried out using GEPAN to assess the effects on the interaction factors of the pile stiffness factor (Kp = Ep/Es) and the length to diameter ratio (Lj/Dj) of the longer pile, when it is end bearing on a stiffer stratum. Corrections can be introduced for the effects of Kp and Lj/Dj on the ‘‘base’’ values given in Eqs. (5) and (6). The resulting interaction factors, a0ij can be expressed as follows:
ð5Þ
For Li > Lj: aij a0ij ¼ K L Rls Rls
ð6Þ
For Li < Lj: aij a0ij ¼ K L Rsl Rsl
where, fls and fsl represent factors that vary depending on the dimensionless ratios of pile spacing to pile diameter, s/Dj, the ratio of difference in pile length to the length of the longer pile, h/Ll, and the ratio of Young’s modulus of soil layer to the underlying bearing stratum, Es/Eb. A parametric study has been carried out based on the hypothetical case illustrated in Fig. 4, and it has been found that fls and fsl can be expressed as shown in the following equations:
615
ð9Þ
ð10Þ
where, RKls and RKsl are the correction factors for Kp, and RLls and RLsl are the correction factors for Lj/Dj, and aij is the ‘‘base’’ interaction factor given by Eqs. (5) and (6). The correction factors can be approximated by the expressions detailed in Appendix II, which have been obtained by curve fitting. Figs. 5 and 6 show that the above approximations provide relatively good agreement with the interaction factors computed directly by GEPAN.
Fig. 4. Typical cases adopted for derivation of approximate method to determine the interaction factors between two dissimilar piles.
616
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618 L i :L j = 50m:45m
Interaction factors, ij
0.5 0.4 0.3 0.2
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.6 0.5
ij
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.6
L i :L j = 50m:15m
0.7
Interaction factors,
0.7
0.4 0.3 0.2
0.1
0.1
0.0
0.0 0.0
10.0
20.0
30.0 s /D j
40.0
50.0
60.0
0.0
10.0
20.0 30.0 s /D j
40.0
50.0
60.0
Fig. 5. Computed (GEPAN) and estimated (approximate method) interaction factors, aij (Li > Lj).
L i :L j = 45m:50m
ij
Interaction factors,
0.6 0.5 0.4 0.3 0.2
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.7
ij
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.7
L i :L j = 15m:50m
0.8
Interaction factors,
0.8
0.1
0.6 0.5 0.4 0.3 0.2 0.1
0.0
0.0
0.0
20.0
40.0
60.0
0.0
20.0
40.0
60.0
s /D j
s /D j
Fig. 6. Computed (GEPAN) and estimated (approximate method) interaction factors, aij (Li < Lj).
The application of the approximate expressions for two piles with dissimilar pile length in a two-layer soil profile is restricted to the condition where the length of the longer pile is equal to the depth of the upper soil layer. Such a limitation is applicable only when a problem with different pile lengths is analysed. In addition, the approximate expressions should only be applied in cases within the range of parameters analysed in this study (see Table 1). 6. Statistical verification To evaluate the reliability of the approximate expressions, a statistical analysis based on the linear regression analysis was carried out. Statistical samples used in this study consist of pre-existing predictions used for deriving the formulations and some additional random cases. The
Table 1 Range of parameters adopted in the approximate expressions Parameter
Range
S/Dj Eb/Es h/Ll Kp = Ep/Es Lj/Dj
2–60 1–100 0.1–0.7 500–10,000a 50–300b
a b
Only for two-layer soil profile with Eb/Es > 1. Referring to the longer pile in two layers soil profile with Eb/Es > 1.
agreement factor, fr is defined herein as the ratio of the estimated interaction factor to the computed (GEPAN) factor. Hence, an ‘‘ideal agreement’’ between the estimated interaction factor and that (GEPAN) will have fr equal to unity. Within the 90% confidence level, fr is very close to unity for all cases analysed (see Figs. 7 and 8). This demonstrates that the interaction factors estimated via the approximate
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
Appendix I
1.0
Approximate expressions
617
Ideal agreement line
Note: 296 total samples.
0.8
1
The definition of parameters A and B to estimate fls are as follows:
1
A lnðDj =sÞ þ B 1 105
0.6
fls ¼
0.4
where
0.2
A1 ðEs =Eb Þ þ A2 ðEs =Eb Þ þ A3 ðEs =Eb Þ þ A4 A ¼ exp 1 104
Approximate vs. GEPAN
(
3
ðAI:1Þ
2
ðAI:2Þ
Regression predictions 0.0
2
0.0
0.2
0.4
0.6
0.8
1.0
GEPAN
A1 ¼ 5517ðh=Ll Þ 128817ðh=Ll Þ þ 55215
ðAI:3Þ
2
ðAI:4Þ
2
A3 ¼ 25882ðh=Ll Þ 69907ðh=Ll Þ þ 26987
ðAI:5Þ
2
ðAI:6Þ
A2 ¼ 136431ðh=Ll Þ þ 266509ðh=Ll Þ 108536
Fig. 7. Approximate expressions versus GEPAN predictions of aij with Eb/Es P 1 (Li > Lj).
A4 ¼ 21419ðh=Ll Þ þ 4437:5ðh=Ll Þ þ 93903 2
B ¼ B1 ðEs =Eb Þ þ B2 ðEs =Eb Þ þ B3
Approximate expressions
1.0
0.8
ðAI:7Þ
2
B1 ¼ 109113ðh=Ll Þ þ 46376ðh=Ll Þ þ 167406
Ideal agreement line
Note: 296 total samples.
)
1
2
B2 ¼ 42487ðh=Ll Þ 179552ðh=Ll Þ 366847
1
2
B3 ¼ 660219ðh=Ll Þ þ 51276ðh=Ll Þ þ 413296 0.6
ðAI:8Þ ðAI:9Þ ðAI:10Þ
The definition of parameters C, D and E to estimate fsl are as follows:
0.4
2
fsl ¼ 0.2
where
Approximate vs. GEPAN Regression predictions 0.0 0.0
0.2
0.4
0.6
0.8
C lnðs=Dj Þ þ D lnðs=Dj Þ þ E 1 106
1.0
GEPAN
(
2
C 1 ðEs =Eb Þ þ C 2 ðEs =Eb Þ þ C 3 C ¼ exp 1 104
ðAI:11Þ ) ðAI:12Þ
Fig. 8. Approximate expressions versus GEPAN predictions of aij with Eb/Es P 1 (Li < Lj).
C 1 ¼ 4610:7 lnðh=Ll Þ 23868 C 2 ¼ 2776:8 lnðh=Ll Þ þ 22090
ðAI:13Þ ðAI:14Þ ðAI:15Þ
expressions in Eqs. (5)–(10) are in good agreement with those computed using GEPAN.
C 3 ¼ 398:01 lnðh=Ll Þ þ 101847 ( ) D1 ðEs =Eb Þ2 þ D2 ðEs =Eb Þ þ D3 D ¼ exp 1 104
7. Conclusion
D1 ¼ 1864:6 lnðh=Ll Þ 17566
ðAI:17Þ
D2 ¼ 5127:9 lnðh=Ll Þ þ 12117 D3 ¼ 91:189 lnðh=Ll Þ þ 118549
ðAI:18Þ ðAI:19Þ
E ¼ E1 ðEs =Eb Þ2 þ E2 ðEs =Eb Þ þ E3
ðAI:20Þ
E1 ¼ 61156 lnðh=Ll Þ 611439 E2 ¼ 100166 lnðh=Ll Þ þ 1099260
ðAI:21Þ ðAI:22Þ
E3 ¼ 72270 lnðh=Ll Þ þ 431828
ðAI:23Þ
A simplified approach for approximating the settlement interaction factors for two dissimilar piles has been presented in this paper. This approach can provide a convenient means of estimating the interaction factors for piles that have dissimilar diameters, lengths, and pile tip conditions. To provide a rapid method of prediction for a practical problem, the simplified approach presented herein can easily be adopted in the existing methods of pile group and piled raft analysis that employ interaction factors. The approximations set out in Eqs. (5)–(10) are the basis of the approximations. As always, the use of a simplified approach requires sound engineering judgement in idealizing the real situation, developing the geotechnical model, and assessing the relevant geotechnical parameters.
ðAI:16Þ
Appendix II The correction factors were derived from a series of GEPAN analyses with fixed values of h/Ll equal to 0.1 and s/Dj equal to 5. It was assumed that the correction factors are applicable for a wider range of s/Dj and h/Ll.
618
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
For Li > Lj: RKls ¼
2 AKls ðK p Þ
where BKls ðK p Þ 5
þ 1 10
þ
C Kls
2
ðAII:1Þ
where 2
C Kls
2
BKsl ¼ 506 ðEs =Eb Þ 166 ðEs =Eb Þ 4:7 C Ksl
AKls ¼ 0:026 ðEs =Eb Þ 0:006 ðEs =Eb Þ 0:0002 BKls
AKsl ¼ 0:028 ðEs =Eb Þ 0:01 ðEs =Eb Þ 0:00005 ðAII:15Þ
2
¼ 213:86 ðEs =Eb Þ þ 24:8 ðEs =Eb Þ þ 8:89
ðAII:3Þ
2
¼ 34988 ðEs =Eb Þ þ 182174 ðEs =Eb Þ þ 44702 ðAII:4Þ
RLls ¼ ALls ðDj =Lj Þ3 þ BLls ðDj =Lj Þ2 þ C Lls ðDj =Lj Þ þ DLls
¼ 2140959 ðEs =Eb Þ
ðAII:2Þ
ðAII:5Þ
þ 667228 ðEs =Eb Þ þ 133632 RLsl ¼
2 ALsl ðLp =Dp Þ
BLsl ðLp =Dp Þ 6
þ 1 10
ðAII:16Þ
2
þ
C Lsl
ðAII:17Þ ðAII:18Þ
where ALsl ¼ 313 ðEs =Eb Þ2 77 ðEs =Eb Þ 4
ðAII:19Þ
2
where
BLsl
¼ 194892 ðEs =Eb Þ þ 52583 ðEs =Eb Þ þ 1678 ðAII:20Þ
For Es/Eb P 0.1:
C Lsl
¼ 15685521 ðEs =Eb Þ 4252896 ðEs =Eb Þ þ 874788
ALls ¼ 320596 ðEs =Eb Þ þ 215675
ðAII:6Þ
BLls ¼ 4366 ðEs =Eb Þ 10732
ðAII:7Þ
C Lls DLls
¼ 186 ðEs =Eb Þ þ 190
ðAII:8Þ
¼ 2:04 ðEs =Eb Þ 0:05
ðAII:9Þ
For Es/Eb < 0.1: ALls ¼ 2953632 ðEs =Eb Þ 47629
ðAII:10Þ
BLls C Lls DLls
¼ 133883 ðEs =Eb Þ þ 2219
ðAII:11Þ
¼ 1123 ðEs =Eb Þ þ 60
ðAII:12Þ
¼ 0:7 ðEs =Eb Þ þ 0:3
ðAII:13Þ
For Li < Lj:
RKsl ¼
AKsl ðK p Þ2 þ BKsl ðK p Þ þ C Ksl 1 105
ðAII:14Þ
2
ðAII:21Þ
References [1] Mindlin RD. Force at a point in the interior of a semi-infinite solid. Physics 1936;7:195. [2] Poulos HG. Analysis of the settlement of pile group. Geotechnique 1968;18:449–71. [3] Poulos HG. An approximate numerical analysis of pile–raft interaction. Int J Numer Anal Method Geomech 1994;18:73–92. [4] Ta LD. A finite layer method for analysis of pile groups, rafts and piled raft foundations in layered soils. Ph.D thesis, School of Civil and Mining Engineering, University of Sydney, Australia; 1996. [5] Zhang HH. Finite layer method for analysis of piled raft foundations. Ph.D thesis, Department of Civil Engineering, University of Sydney, Australia; 2000. [6] Hewitt CM. Cyclic response of offshore pile groups. Ph.D thesis, School of Civil and Mining Engineering, The University of Sydney; 1988. [7] Xu KJ, Poulos HG. General elastic analysis of piles and pile groups. Int J Numer Anal Method Geomech 2000;24:1109–38.
Technical communication
Approximate pile-to-pile interaction factors between two dissimilar piles Sii Chung Wong a, Harry G. Poulos a
b,*
Chee Lian Development Construction Sdn. Bhd., Sarawak, Malaysia Coffey Geosciences, P.O. Box 125, North Ryde, Sydney, Australia
b
Received 22 June 2005; received in revised form 1 November 2005; accepted 11 November 2005 Available online 6 January 2006
Abstract This paper develops approximations for the settlement interaction factors between two dissimilar piles. Via an extensive parametric study using the computer program GEPAN, approximations are developed for interaction factors for piles having dissimilar diameters but equal lengths, piles having dissimilar lengths, and for piles having dissimilar ground conditions at the pile tips. Correction factors are then given to allow for the effects of piles stiffness and length-to-diameter ratio. The approximations may be employed in analyses for group settlements to allow for the case where not all piles in the group are identical. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction In predicting the response characteristics of pile groups, and piled rafts, the application of interaction factors based on the theory of elasticity that employs the equations of Mindlin [1] has been widely adopted. The interaction factor, denoted by aij, was defined as the additional displacement at the top of pile i due to a loaded adjacent pile j, divided by the settlement of pile j under its own load. Poulos [2,3], Ta [4], and Zhang [5] have successfully employed the interaction factor into their analysis for predicting the response characteristics of pile groups and piled raft foundation systems. However, the analyses carried out by the above-mentioned investigators were for foundations in which the piles had identical dimensions and properties. More rigorous numerical analyses such as GEPAN [7], are required to provide a more accurate response characteristic for piled foundations with dissimilar piles. However, GEPAN demands more computing resources and computing time, especially for cases involving large pile groups. In order to provide a rapid prediction of the response characteristics of piled foundations with dissimilar piles, Hewitt *
Corresponding author. Tel.: +61 2 9911 1000; fax: +61 2 9911 1001. E-mail address: harry_poulos@coffey.com.au (H.G. Poulos).
0266-352X/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2005.11.001
[6] proposed approximate relationships for dissimilar piles to determine the pile-to-pile interaction factor between two dissimilar piles. However, there are some shortcomings in these relationships, which will be discussed later. This paper presents a parametric study carried out using GEPAN [7] to examine in detail the interaction characteristics between two dissimilar piles. Subsequently, attempts are made to provide an approximate relationship to enable the interaction factors between two dissimilar piles to be obtained rapidly for more complicated combinations of two dissimilar piles, namely for dissimilar pile lengths, dissimilar pile tip conditions, and for dissimilar pile diameters. Assessments of the potential for the approximate relationship to be used as a design tool are also made. 2. Analysis method The program GEPAN is described by Xu and Poulos [7], and has been used in the present study. GEPAN employs a boundary element analysis in which the soil is modelled as an elastic continuum and each pile is discretised via a series of elements along and around the pile, which is assumed to be circular in cross-section and perfectly elastic. Use is made of Mindlin’s equations of elasticity to compute the vertical and lateral soil move-
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
Lj = varies Dj = 0.3m Ep = 30.0GPa
Es = 4.0MPa νs = 0.35 Fig. 1. Piles with dissimilar lengths.
ð1Þ
3. Piles with dissimilar diameters Hewitt [6] postulated approximate relationships to determine the pile-to-pile interaction factor between two piles with dissimilar diameters as shown in the following equation:
0.70
4. Piles with dissimilar length (incorporating dissimilar pile tip conditions) In this study, two piles of equal diameter, (Di = Dj) but with different lengths, Li and Lj have been analysed (see Fig. 1). As in the previous section, both piles are installed in a deep homogeneous elastic soil layer. The influence of the loaded pile on the unloaded pile is shown in Figs. 2 and 3. Both figures show the variation of the interaction factors for pile i due to pile j, aij, with the dimensionless ratio of the pile spacing to the diameter of the loaded pile j, s/Dj. Fig. 2 shows the aij obtained for cases whereby pile i has a constant length (Li = 50 m), while the length of the loaded pile j, Lj, changes from 15 m to 45 m. The interaction factors plotted in Fig. 3 are for cases where pile i has varying length, Li, while the length of the loaded pile j remains constant.
ij
0.50 0.40 0.30 0.20 0.10
ð2Þ
0.00 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
s/D j
Fig. 2. Interaction factors aij for two piles with different lengths (Li > Lj).
0.80 0.70
L j :L i 50m:45m (GEPAN)
0.60
50m:35m (GEPAN)
ij
i.e. the interaction factor for pile i due to pile j, aij, is approximately the same as the interaction factor between two identical piles j, ajj. An extensive parametric study carried out in this present study using GEPAN has also shown that the approximate relationship proposed by Hewitt [6] in Eq. (2) can be applied in estimating the interaction factors between two piles with dissimilar pile diameters. Subsequently, no further consideration needs to be given to this case.
L i :L j 50m:45m (GEPAN) 50m:35m (GEPAN) 50m:25m (GEPAN) 50m:15m (GEPAN) 50m:45m (Hewitt,1988) 50m:35m (Hewitt,1988) 50m:25m (Hewitt,1988) 50m:15m (Hewitt,1988)
0.60
Interaction factor,
where Pi is the load on pile i, Ki is the axial head stiffness of pile i, aij is the interaction factor for effect of pile j (influencing loaded pile) on pile i (influenced pile), Pj is the load on pile j and Kj is the axial head stiffness of pile j.
aij ajj
Li = varies Di = 0.3m Ep = 30.0GPa
Interaction factor,
S i ¼ P i =K i þ aij P j =K j
Pile i
ments due to the stresses developed at the pile–soil interface. Compatibility of all components of displacement is imposed at the pile–soil interface in order to obtain the pile–soil stresses and the pile displacements. Allowance can also be made for specifying limiting values of the pile–soil stresses so that non-linear response of the pile can be simulated. The program can consider groups of piles with different pile dimensions and properties, and thus is a very suitable tool for the present parametric study. In this case, consideration is confined to purely elastic pile and soil behaviour and to the axial response of two dissimilar piles. Once the interaction factor has been obtained, the settlement Si of a pile i due to its own load and due to an adjacent loaded pile j, can be expressed as follows:
Pile j
614
50m:25m (GEPAN) 0.50
50m:15m (GEPAN) 50m:50m (Hewitt, 1988)
0.40 0.30 0.20 0.10 0.00 0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
s /D j
Fig. 3. Interaction factors aij for two piles with different lengths (Li < Lj).
Hewitt [6] attempted to obtain approximate relationships to determine interaction factors for two friction piles with different pile lengths and the approximate relationships postulated by Hewitt [6] are reproduced in the following equations:
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
For Li > Lj: aii þ ajj aij 2
fls ¼ ð3Þ
ð4Þ
where, aii and ajj represents the interaction factor for two identical piles having the dimensions of pile i and pile j, respectively. The approximate interaction factors, aij, between two piles with dissimilar pile lengths obtained using Hewitt’s [6] recommendations have also been plotted in both Figs. 2 and 3. Hewitt’s [6] approximate relationship in Eq. (3) is seen to overestimate the interaction of the shorter pile on the longer pile when the difference in pile length (h) between the two piles is greater than 30% of the longer pile (see Fig. 2), while. Eq. (4) tends to underestimate the interaction of the longer pile on the shorter pile for a length difference (h) greater than about 30% of the longer pile (see Fig. 3). It has been found by the authors that a more accurate approximate relationship is given in the following equations: For Li > Lj: aii þ ajj aij fls For Li < Lj: ajj aij fsl
A lnðDj =sÞ þ B 1 105
ð7Þ
C lnðs=Dj Þ2 þ D lnðs=Dj Þ þ E 1 106
ð8Þ
and,
For Li < Lj: aij ajj
fsl ¼
where parameters A, B, C, D and E can be expressed as functions of Es/Eb and h/Ll. The approximate expression for parameters A, B, C, D and E are detailed in Appendix I, and have been obtained via curve fitting. 5. Incorporation of the effects of pile stiffness and length-todiameter ratio A parametric study has been carried out using GEPAN to assess the effects on the interaction factors of the pile stiffness factor (Kp = Ep/Es) and the length to diameter ratio (Lj/Dj) of the longer pile, when it is end bearing on a stiffer stratum. Corrections can be introduced for the effects of Kp and Lj/Dj on the ‘‘base’’ values given in Eqs. (5) and (6). The resulting interaction factors, a0ij can be expressed as follows:
ð5Þ
For Li > Lj: aij a0ij ¼ K L Rls Rls
ð6Þ
For Li < Lj: aij a0ij ¼ K L Rsl Rsl
where, fls and fsl represent factors that vary depending on the dimensionless ratios of pile spacing to pile diameter, s/Dj, the ratio of difference in pile length to the length of the longer pile, h/Ll, and the ratio of Young’s modulus of soil layer to the underlying bearing stratum, Es/Eb. A parametric study has been carried out based on the hypothetical case illustrated in Fig. 4, and it has been found that fls and fsl can be expressed as shown in the following equations:
615
ð9Þ
ð10Þ
where, RKls and RKsl are the correction factors for Kp, and RLls and RLsl are the correction factors for Lj/Dj, and aij is the ‘‘base’’ interaction factor given by Eqs. (5) and (6). The correction factors can be approximated by the expressions detailed in Appendix II, which have been obtained by curve fitting. Figs. 5 and 6 show that the above approximations provide relatively good agreement with the interaction factors computed directly by GEPAN.
Fig. 4. Typical cases adopted for derivation of approximate method to determine the interaction factors between two dissimilar piles.
616
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618 L i :L j = 50m:45m
Interaction factors, ij
0.5 0.4 0.3 0.2
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.6 0.5
ij
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.6
L i :L j = 50m:15m
0.7
Interaction factors,
0.7
0.4 0.3 0.2
0.1
0.1
0.0
0.0 0.0
10.0
20.0
30.0 s /D j
40.0
50.0
60.0
0.0
10.0
20.0 30.0 s /D j
40.0
50.0
60.0
Fig. 5. Computed (GEPAN) and estimated (approximate method) interaction factors, aij (Li > Lj).
L i :L j = 45m:50m
ij
Interaction factors,
0.6 0.5 0.4 0.3 0.2
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.7
ij
Rb = 1 Rb = 2 Rb = 5 Rb = 10 Rb = 100 Rb=1 (approx.) Rb=2 (approx.) Rb=5 (approx.) Rb=10 (approx.) Rb=100 (approx.)
0.7
L i :L j = 15m:50m
0.8
Interaction factors,
0.8
0.1
0.6 0.5 0.4 0.3 0.2 0.1
0.0
0.0
0.0
20.0
40.0
60.0
0.0
20.0
40.0
60.0
s /D j
s /D j
Fig. 6. Computed (GEPAN) and estimated (approximate method) interaction factors, aij (Li < Lj).
The application of the approximate expressions for two piles with dissimilar pile length in a two-layer soil profile is restricted to the condition where the length of the longer pile is equal to the depth of the upper soil layer. Such a limitation is applicable only when a problem with different pile lengths is analysed. In addition, the approximate expressions should only be applied in cases within the range of parameters analysed in this study (see Table 1). 6. Statistical verification To evaluate the reliability of the approximate expressions, a statistical analysis based on the linear regression analysis was carried out. Statistical samples used in this study consist of pre-existing predictions used for deriving the formulations and some additional random cases. The
Table 1 Range of parameters adopted in the approximate expressions Parameter
Range
S/Dj Eb/Es h/Ll Kp = Ep/Es Lj/Dj
2–60 1–100 0.1–0.7 500–10,000a 50–300b
a b
Only for two-layer soil profile with Eb/Es > 1. Referring to the longer pile in two layers soil profile with Eb/Es > 1.
agreement factor, fr is defined herein as the ratio of the estimated interaction factor to the computed (GEPAN) factor. Hence, an ‘‘ideal agreement’’ between the estimated interaction factor and that (GEPAN) will have fr equal to unity. Within the 90% confidence level, fr is very close to unity for all cases analysed (see Figs. 7 and 8). This demonstrates that the interaction factors estimated via the approximate
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
Appendix I
1.0
Approximate expressions
617
Ideal agreement line
Note: 296 total samples.
0.8
1
The definition of parameters A and B to estimate fls are as follows:
1
A lnðDj =sÞ þ B 1 105
0.6
fls ¼
0.4
where
0.2
A1 ðEs =Eb Þ þ A2 ðEs =Eb Þ þ A3 ðEs =Eb Þ þ A4 A ¼ exp 1 104
Approximate vs. GEPAN
(
3
ðAI:1Þ
2
ðAI:2Þ
Regression predictions 0.0
2
0.0
0.2
0.4
0.6
0.8
1.0
GEPAN
A1 ¼ 5517ðh=Ll Þ 128817ðh=Ll Þ þ 55215
ðAI:3Þ
2
ðAI:4Þ
2
A3 ¼ 25882ðh=Ll Þ 69907ðh=Ll Þ þ 26987
ðAI:5Þ
2
ðAI:6Þ
A2 ¼ 136431ðh=Ll Þ þ 266509ðh=Ll Þ 108536
Fig. 7. Approximate expressions versus GEPAN predictions of aij with Eb/Es P 1 (Li > Lj).
A4 ¼ 21419ðh=Ll Þ þ 4437:5ðh=Ll Þ þ 93903 2
B ¼ B1 ðEs =Eb Þ þ B2 ðEs =Eb Þ þ B3
Approximate expressions
1.0
0.8
ðAI:7Þ
2
B1 ¼ 109113ðh=Ll Þ þ 46376ðh=Ll Þ þ 167406
Ideal agreement line
Note: 296 total samples.
)
1
2
B2 ¼ 42487ðh=Ll Þ 179552ðh=Ll Þ 366847
1
2
B3 ¼ 660219ðh=Ll Þ þ 51276ðh=Ll Þ þ 413296 0.6
ðAI:8Þ ðAI:9Þ ðAI:10Þ
The definition of parameters C, D and E to estimate fsl are as follows:
0.4
2
fsl ¼ 0.2
where
Approximate vs. GEPAN Regression predictions 0.0 0.0
0.2
0.4
0.6
0.8
C lnðs=Dj Þ þ D lnðs=Dj Þ þ E 1 106
1.0
GEPAN
(
2
C 1 ðEs =Eb Þ þ C 2 ðEs =Eb Þ þ C 3 C ¼ exp 1 104
ðAI:11Þ ) ðAI:12Þ
Fig. 8. Approximate expressions versus GEPAN predictions of aij with Eb/Es P 1 (Li < Lj).
C 1 ¼ 4610:7 lnðh=Ll Þ 23868 C 2 ¼ 2776:8 lnðh=Ll Þ þ 22090
ðAI:13Þ ðAI:14Þ ðAI:15Þ
expressions in Eqs. (5)–(10) are in good agreement with those computed using GEPAN.
C 3 ¼ 398:01 lnðh=Ll Þ þ 101847 ( ) D1 ðEs =Eb Þ2 þ D2 ðEs =Eb Þ þ D3 D ¼ exp 1 104
7. Conclusion
D1 ¼ 1864:6 lnðh=Ll Þ 17566
ðAI:17Þ
D2 ¼ 5127:9 lnðh=Ll Þ þ 12117 D3 ¼ 91:189 lnðh=Ll Þ þ 118549
ðAI:18Þ ðAI:19Þ
E ¼ E1 ðEs =Eb Þ2 þ E2 ðEs =Eb Þ þ E3
ðAI:20Þ
E1 ¼ 61156 lnðh=Ll Þ 611439 E2 ¼ 100166 lnðh=Ll Þ þ 1099260
ðAI:21Þ ðAI:22Þ
E3 ¼ 72270 lnðh=Ll Þ þ 431828
ðAI:23Þ
A simplified approach for approximating the settlement interaction factors for two dissimilar piles has been presented in this paper. This approach can provide a convenient means of estimating the interaction factors for piles that have dissimilar diameters, lengths, and pile tip conditions. To provide a rapid method of prediction for a practical problem, the simplified approach presented herein can easily be adopted in the existing methods of pile group and piled raft analysis that employ interaction factors. The approximations set out in Eqs. (5)–(10) are the basis of the approximations. As always, the use of a simplified approach requires sound engineering judgement in idealizing the real situation, developing the geotechnical model, and assessing the relevant geotechnical parameters.
ðAI:16Þ
Appendix II The correction factors were derived from a series of GEPAN analyses with fixed values of h/Ll equal to 0.1 and s/Dj equal to 5. It was assumed that the correction factors are applicable for a wider range of s/Dj and h/Ll.
618
S.C. Wong, H.G. Poulos / Computers and Geotechnics 32 (2005) 613–618
For Li > Lj: RKls ¼
2 AKls ðK p Þ
where BKls ðK p Þ 5
þ 1 10
þ
C Kls
2
ðAII:1Þ
where 2
C Kls
2
BKsl ¼ 506 ðEs =Eb Þ 166 ðEs =Eb Þ 4:7 C Ksl
AKls ¼ 0:026 ðEs =Eb Þ 0:006 ðEs =Eb Þ 0:0002 BKls
AKsl ¼ 0:028 ðEs =Eb Þ 0:01 ðEs =Eb Þ 0:00005 ðAII:15Þ
2
¼ 213:86 ðEs =Eb Þ þ 24:8 ðEs =Eb Þ þ 8:89
ðAII:3Þ
2
¼ 34988 ðEs =Eb Þ þ 182174 ðEs =Eb Þ þ 44702 ðAII:4Þ
RLls ¼ ALls ðDj =Lj Þ3 þ BLls ðDj =Lj Þ2 þ C Lls ðDj =Lj Þ þ DLls
¼ 2140959 ðEs =Eb Þ
ðAII:2Þ
ðAII:5Þ
þ 667228 ðEs =Eb Þ þ 133632 RLsl ¼
2 ALsl ðLp =Dp Þ
BLsl ðLp =Dp Þ 6
þ 1 10
ðAII:16Þ
2
þ
C Lsl
ðAII:17Þ ðAII:18Þ
where ALsl ¼ 313 ðEs =Eb Þ2 77 ðEs =Eb Þ 4
ðAII:19Þ
2
where
BLsl
¼ 194892 ðEs =Eb Þ þ 52583 ðEs =Eb Þ þ 1678 ðAII:20Þ
For Es/Eb P 0.1:
C Lsl
¼ 15685521 ðEs =Eb Þ 4252896 ðEs =Eb Þ þ 874788
ALls ¼ 320596 ðEs =Eb Þ þ 215675
ðAII:6Þ
BLls ¼ 4366 ðEs =Eb Þ 10732
ðAII:7Þ
C Lls DLls
¼ 186 ðEs =Eb Þ þ 190
ðAII:8Þ
¼ 2:04 ðEs =Eb Þ 0:05
ðAII:9Þ
For Es/Eb < 0.1: ALls ¼ 2953632 ðEs =Eb Þ 47629
ðAII:10Þ
BLls C Lls DLls
¼ 133883 ðEs =Eb Þ þ 2219
ðAII:11Þ
¼ 1123 ðEs =Eb Þ þ 60
ðAII:12Þ
¼ 0:7 ðEs =Eb Þ þ 0:3
ðAII:13Þ
For Li < Lj:
RKsl ¼
AKsl ðK p Þ2 þ BKsl ðK p Þ þ C Ksl 1 105
ðAII:14Þ
2
ðAII:21Þ
References [1] Mindlin RD. Force at a point in the interior of a semi-infinite solid. Physics 1936;7:195. [2] Poulos HG. Analysis of the settlement of pile group. Geotechnique 1968;18:449–71. [3] Poulos HG. An approximate numerical analysis of pile–raft interaction. Int J Numer Anal Method Geomech 1994;18:73–92. [4] Ta LD. A finite layer method for analysis of pile groups, rafts and piled raft foundations in layered soils. Ph.D thesis, School of Civil and Mining Engineering, University of Sydney, Australia; 1996. [5] Zhang HH. Finite layer method for analysis of piled raft foundations. Ph.D thesis, Department of Civil Engineering, University of Sydney, Australia; 2000. [6] Hewitt CM. Cyclic response of offshore pile groups. Ph.D thesis, School of Civil and Mining Engineering, The University of Sydney; 1988. [7] Xu KJ, Poulos HG. General elastic analysis of piles and pile groups. Int J Numer Anal Method Geomech 2000;24:1109–38.