CE5107 - Lecture 4
CE5107 - Lecture 4
CE5107 - Lecture 4
Dr. Taeseo Ku
1
Brief Review – Lecture 3
2
Total Stress Analysis (α-Method) vs.
Effective Stress Analysis (β-Method)
Brief Overview
Undrained Drained
3
Conceptual Load Transfer
Tip resistance = 0
(not mobilized)
4
Negative Skin Friction (NSF)
Qc Qsn Qc
Neutral plane
Qsp
Stiff soil (non-
consolidating)
Qb Qsp Qb
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Axial Pile Deflection
6
AXIAL LOAD-DISPLACEMENT PREDICTION
OF DEEP FOUNDATIONS
The evaluation of the load-displacement response of driven piles and
drilled shafts subjected to axial compression loading is important in
foundation design. The relative proportions of load carried by the shaft and
base at working loads are also desired. Consequently, elastic continuum
theory is utilized to describe the load-transfer distribution and axial load-
displacement response at the top of the foundation.
The procedures have been developed using boundary element
formulations (Poulos and Davis, 1980), finite elements (Poulos, 1989;
Randolph, 1989) and approximate closed-form solutions by Randolph and
Wroth (1978, 1979).
The generalized method characterizes the soil stiffness by an equivalent
elastic modulus (Es) and Poisson's ratio (νs).
The soil modulus may be taken either uniform with depth (constant Es) or a
Gibson-profile (linearly increasing Es with depth). 7
AXIAL LOAD-DISPLACEMENT PREDICTION
OF DEEP FOUNDATIONS
9
Single Pile Axial Deflection
The elastic continuum solution for the vertical displacement (wt or ρ) at
the top of a pile foundation subjected to axial compression loading is
expressed by (Poulos & Davis 1980):
where Pt = applied axial load at the top of the shaft, EsL = soil modulus at
the foundation base, d = foundation diameter, and Iρ = influence factor.
The values of Iρ for a homogeneous soil (i.e., constant Es = EsL with depth)
are given by Poulos (1972) and Poulos & Davis (1968, 1980) in chart form
according to:
Iρ = Io∙RK∙Rh∙Rν∙Rb
10
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
Io = settlement influence factor for an incompressible pile in a semi-infinite soil mass
with ν = 0.5;
RK = correction factor for pile compressibility;
Rh = correction factor for finite depth to rigid incompressible layer beneath pile tip;
Rν = correction factor for Poisson effect;
Rb = correction factor for relative stiffness of tip bearing stratum.
Values of these correction factors are given in the following figures in terms of the pile
slenderness ratio (L/d, or D/B) and pile stiffness factor (K = RAEp/Es),
RA = pile area ratio = Ap/(πd2/4), note that RA = 1 for a solid pile.
Ap is the area of the pile section
Ep = pile modulus
Es = soil modulus
Default values are given such that a choice of one of the following cases must be made:
(1) floating pile (Rb= 1), or (2) end-bearing pile (Rh = 1) with pile resting on a stiffer
stratum.
11
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
dB = dia. of base
12
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
13
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
14
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
15
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
16
Single Pile Axial Deflection
Iρ = Io∙RK∙Rh∙Rν∙Rb
The value of Io distinguishes between straight shaft piles (db/d = 1) and piles
with underreamed bases, commonly referred to as belled piers (db/d > 1).
For a floating pile, the depth correction factor (Rh) is for a special case
where supporting medium is of finite thickness, and thus depends upon the
depth (h) to a rigid boundary (i.e. rock), expressed as a dimensionless ratio
of either (h/L), or alternate (L/h).
For an end bearing pile, the base modulus correction (Rb) depends upon the
pile-soil stiffness (K), slenderness ratio (L/d), and the relative stiffness of the
bearing stratum (Eb) to soil modulus along the pile shaft (EsL = Es).
17
Single Pile Axial Deflection
Assumption (Chart Solutions, Poulos and Davis 1980):
1) Rigid pile in infinite half-space
2) Homogeneous (Es constant with depth)
3) Undrained, ν = 0.5
18
Cross-sectional Issues
βLT = βo∙CK∙Cb∙Cν
20
Load Transfer
βLT = βo∙CK∙Cb∙Cν (After Poulos & Davis, 1980)
21
Load Transfer
βLT = βo∙CK∙Cb∙Cν (After Poulos & Davis, 1980)
22
Load Transfer
βLT = βo∙CK∙Cb∙Cν (After Poulos & Davis, 1980)
23
Assumed Variation of Soil Shear Modulus
24
Closed Form Solution For Iρ
General Compressible Pile Solution
• Alternatively, an approximate closed-form for the influence
factor (Iρ) has been derived (Randolph and Wroth, 1978, 1979;
Poulos, 1989):
The solution given above is quite general and can accommodate soil
models with constant Es (homogeneous soils) or soils having a linearly-
varying Es with depth (Gibson-type soils). The pile can be either a floating
type pile (EsL = Eb) or end-bearing type where the pile base rests on a
stiffer stratum (Eb > EsL).
25
Closed Form Solution For Iρ
General Compressible Pile Solution
• νs = Poisson's ratio of soil.
• d = pile diameter.
• L = pile length.
• η = eta factor (usually η = 1, or 0 = db/d for belled piers, with db = diameter of base).
- Alternatively, this factor can be used to account for a finite layer thickness beneath the
pile tip, such as IG from the shallow foundation displacement influence factors.
• ξ = EsL/Eb = xi factor (Note: ξ = 1 for floating pile).
• ρE = Esm/EsL = rho factor (ρE = 1 for uniform soil; ρE = 0.5 for simple Gibson soil).
- Note in shallow foundations, Es = Eso +kE∙z, so that ρE = 1 - [2Eso/(kE∙L)+2]-1
• λ = 2(1+νs)Ep/EsL = Ep/GsL =lamda factor.
• ς = ln{[0.25 + (2.5ρ(1- νs) - 0.25)ξ] (2L/d)} = zeta factor.
• μL = 2(2/ ς λ)0.5(L/d) = mu factor.
• Ep = pile modulus (concrete including percentage of reinforcing steel).
• EsL = soil modulus value along pile shaft at level of foundation base (pile tip).
• Esm = soil modulus value at mid-depth of pile shaft.
• Eb = soil modulus below foundation base (Note: Eb = EsL for floating pile).
26
Closed Form Solution – Load Transfer
General Compressible Pile Solution
For the approximate elastic continuum solution of Randolph & Wroth
(1978), the distribution of axial load transfer with depth is also closed
form. The ratio of the displacement at the foundation head (wt) to the
foundation base (wb) is given simply by:
wt/wb = cosh(μL)
wb = Pb(1-νs)(1+νs)η/(Ebd)
27
Closed Form Solution – Load Transfer
General Compressible Pile Solution
The expression for fraction of the total load transferred to the pile base is
given by Randolph (1994):
Pb
32
Prof. Paul Mayne
Pb
33
Prof. Paul Mayne
Summary
Pb
34
Prof. Paul Mayne
Pt =
35
Prof. Paul Mayne
36
Prof. Paul Mayne
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Prof. Paul Mayne
Case Study – Axial Pile Deflections
Drilled shaft case study – Fairfax Hospital, Northern Virginia
A drilled shaft with d = 3 feet and L = 65 feet was constructed in Piedmont residual sandy silts that
grade into decomposed schist for the Fairfax Hospital in Fairfax County, Virginia. Flat plate
dilatometer tests indicated an elastic E=35 MPa (mean of 64 tests), assuming a homogenous
profile of modulus. Based on correlations with DMT and SPT (Mayne and Frost, TRR 1279, 1988),
the estimated modulus of the decomposed rock beneath the pile base was Eb=102 MPa. The
concrete shaft can be assumed to have an equivalent modulus of approx. 27.5 GPa. These
conditions give a pile slenderness ratio, L/d = 65/3 =21.7 and pile stiffness factor, KF = Ep/Es = 764.
38
Case Study – Axial Pile Deflections
Drilled shaft case study – Fairfax Hospital, Northern Virginia
39
Case Study – Linear Elastic Axial Pile Response
Drilled shaft case study – Fairfax Hospital, Northern Virginia
40
Case Study – Linear Elastic Axial Pile Response
Drilled shaft case study – Fairfax Hospital, Northern Virginia
41
Case Study – Linear Elastic Axial Pile Response
Texas Highway Drilled Shaft
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Sample Question
43
Sample Question
1.2m
Es (bar) = 445
L = 24.5 m
0.34×0.34m2 (square)
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Load-Settlement Ratio
Closed-Form Elastic Pile Solutions
VERY LONG PILES: when L/ro > 3 (Ep/GsL)0.5 then load vanishes before length of
pile reached. Then, for this case, tanh(μL) approaches 1 and the axial load-
displacement relationship is:
L = pile length, ro = pile radius, rb = radius of pile base (i.e., for belled shafts), η = rb/ro;
Ep = pile modulus, GsL = soil shear modulus along side at z = L;
ρE = GsM/GsL = Gibson parameter;
GsM = soil shear modulus at mid-shaft;
λ = Ep/GsL = soil-pile stiffness ratio,
ξ = GsL/Gsb; Gsb = soil stiffness below pile tip/toe/base;
ζ = ln(rm/ro) = measure of soil zone affected by pile influence;
rm = L{0.25 + ξ [2.5 ρE(1-ν) – 0.25]} = “magic radius”
For long piles, the ratio of load to displacement is independent of the pile length.
The soil shear modulus GsL must then be interpreted as that value along the active
zone, corresponding to the depth z = 3 ro (Ep/GsL)0.5 45
END BEARING vs FRICTION PILE
SETTLEMENTS
46
Case Study – Long Pile at Sandpoint, Idaho
A steel pipe pile with d = 404 mm was driven closed-ended to an embedded
length L = 45.8 m into soft glacial lake sediments in northwest Idaho.
Seismic piezocone sounding:
47
Case Study – Long Pile at Sandpoint, Idaho
The shear wave velocity data can be used to determine a profile of the initial
small-strain shear modulus with depth
48
Case Study – Long Pile at Sandpoint, Idaho
Plunging failure
No increase of shaft
resistance?
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Case Study – Long Pile at Sandpoint, Idaho
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Case Study – Long Pile at Sandpoint, Idaho
51
Methods of Pile Group Settlement Analysis
(Poulos 2011)
It is now well recognized that the settlement of a pile group can differ significantly
from that of a single pile at the same average load level. There are a number of
approaches commonly adopted for the estimation of the settlement of pile groups:
− Methods which employ the concept of interaction factors and the principle of
superposition (e.g. Poulos and Davis, 1980);
− Methods which involve the modification of a single pile load-settlement curve, to
take account of group interaction effects;
− The settlement ratio method, in which the settlement of a single pile at the
average load level is multiplied by a group settlement ratio Rs, which reflects the
effects of group interaction;
− The equivalent raft method, in which the pile group is represented by an
equivalent raft acting at some characteristic depth along the piles;
− Numerical methods such as the finite element method and the finite difference
method (such as FLAC). While earlier work employed two-dimensional analyses, it
is now less uncommon for full three-dimensional analyses to be employed (e.g.,
Katzenbach et al., 1998).
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Pile Group Interaction
For floating piles, the interaction factor (αs) between piles is given as a
function of the center-to-center spacing distance between piles (s), the
“magic radius” (rm), and influence zone size given by ζ = ln(rm/ro). Thus, per
Fleming et al. (1992):
53
Pile Group Interaction
Piles in groups are normally installed with spacing (s) to
diameter (d) ratios of between 3 < s/d < 10. Closer than this may
result in damage during driving, or disruption or movement of
already-placed pilings. Spacing greater than 10 diameters results
in large pile caps that are expensive and impractical.
55
Pile Group
Interaction
56
Pile Group Interaction
Interaction factor
d
Pile 1 Pile 2 57
Pile Group Interaction (Poulos 2011)
One of the common means of analyzing pile group behaviour is via the interaction
factor method described by Poulos and Davis (1980). In this method, the settlement
wi of a pile i within a group of n piles is given as follows:
where Pav = average load on a pile within the group; S1 = settlement of a single pile under unit
load (i.e., the pile flexibility); αij = interaction factor for pile i due to any other pile (j) within
the group, corresponding to the spacing sij between piles i and j.
Eq. 1 can be written for each pile in the group, thus giving a total of n equations,
which together with the equilibrium equation, can be solved for two simple cases:
1. Known load on each pile, in which case the settlement of each pile can be
computed directly. In this case, there will usually be differential settlements among
the piles in the group.
2. A rigid (non-rotating) pile cap, in which case all piles settle equally. In this case,
there will be a uniform settlement but a non-uniform distribution of load in the
piles. 58
Interaction Factors -Characteristics
α
• Decreases as s/d increases
• Decreases as K decreases
• Decreases as L/D decreases
• Less for end bearing than friction piles
• Less for non-homogeneous than homogeneous
profiles
• Increases as νs increases
• Increases if base enlarged
• Decreases if soil between piles is stiffer
59
Interaction Factors - Issues
• Generally, will tend to overestimate interaction within a
relatively large group, due to effects of:
Greater stiffness of soil between piles
• Should make allowances for stiffer soil & more rapid decay
of α
60
Settlement Ratio Method
▪ Settlement Ratio RS
RS = Average group settlement / Settlement of
single pile at same average load
n > RS > 1
61
Settlement Ratio Method
SG = RS·Pav·S1
RS from:
• Tabulated values (Poulos, 1979)
• Randolph’s approximation
▫ RS ~ nw
▫ Values of w from theoretical analysis
▫ As first approximations:
w ~ 0.5 for floating groups in clay
w ~ 0.33 for floating groups in sand
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Theoretical Solutions for Exponent w
63
Stacked Pile Segments (Layered Soils)
For a rigid pile in a three-layer soil profile, the following components are
evaluated:
64
Randolph-Wroth Solution
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References – Analysis & Prediction
• Banerjee, P.K. and Davies, T.G. (1978), "The behavior of axially and laterally loaded single piles embedded in non-
homogeneous soils", Geotechnique 28 (3), 309-326.
• Fleming, W.G.K., Weltman, A.J., Randolph, M.F. and Elson, W.K (1992). Piling Engineering, Second Edition,
Blackie/Halsted Press, John Wiley & Sons, New York, pp. 122-128.
• Gardner, W.S. (1987), "Design of drilled shafts in the Atlantic Piedmont", Foundations and Excavations in Decomposed
Rock of the Piedmont Province (GSP 9), ASCE, New York, 1-15.
• Poulos, H.G. and Davis, E.H. (1968), "The settlement behavior of single axially-loaded incompressible piles and piers",
Geotechnique 18 (3), 351-371.
• Poulos, H.G. (1972), "Load-settlement prediction for piles and piers", Journal of the Soil Mechanics and Foundations
Division, ASCE, 98 (SM9), 879-897.
• Poulos, H.G. (1979), "Settlement of single piles in non-homogeneous soils", Journal of the Geotechnical Engineering
Division 105 (GT5), 627-641.
• Poulos, H.G. (1989), "Pile Behavior: Theory and Application", 29th Rankine Lecture, Geotechnique, Vol. 39, No. 3,
September, pp. 363-416.
• Poulos, H.G. and Davis, E.H. (1980), Pile Foundation Analysis and Design, John Wiley and Sons, New York, 397 p. (now
published by University of Sydney Press, Australia).
• Randolph, M.F. (1989), PIGLET: Analysis and Design of Pile Groups, Internal Report, Dept. of Civil Engineering,
University of Western Australia, Nedlands, 35 p.
• Randolph, M.F. (1994), "Solution for axially loaded pile in elastic soil", fax communication to P.W. Mayne from
University of Western Australia (April 13), 4 pgs.
• Randolph, M.F. and Wroth, C.P. (1978), "Analysis of Deformation of Vertically Loaded Piles", Journal of the
Geotechnical Engineering Division, ASCE, Vol. 104, No. GT12, Dec., pp. 1465-1488.
• Randolph, M.F. and Wroth, C.P. (1979), "A Simple Approach to Pile Design and the Evaluation of Pile Tests", Behavior
of Deep Foundations, STP 670, ASTM, Philadelphia, PA, pp. 484-499.
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References – Pile Group
• Banerjee, P.K. and Davies, T.G. (1977), "Analysis of Pile Groups Embedded in Gibson Soil",
Proceedings, 9th International Conference on Soil Mechanics and Foundation Engineering,
Vol. 1, Tokyo, 381-386.
• Fleming, W.G.K., Welman, A.J., Randolph, M.F., and Elson, W.K. (1992). Piling Engineering.
Second Edition, Blackie/Halsted Press/John Wiley & Sons, London/New York.
• O’Neill, M.W., Hawkins, R.A., and Audibert, J.M.E. (1982), "Installation of Pile Group in
Overconsolidated Clay", Journal of the Geotechnical Engineering Division, ASCE, 108
(GT11), 1369-1386.
• O’Neill, M.W. (1983), "Group Action in Offshore Piles", Geotechnical Practice in Offshore
Engineering, ASCE Conference, Univ. of Texas, Austin, 25-64.
• Poulos, H.G. (1971), "Behavior of Laterally-Loaded Piles II: Pile Groups", Journal of the Soil
Mechanics and Foundations Division, ASCE 97 (SM5), 733-751.
• Poulos, H.G. and Mattes, N.S. (1971), "Settlement and Load Distribution Analysis of Pile
Groups", Australian Geomechanics Journal, Vol. G1, No. 1, 18-28.
• Poulos, H.G. and Davis, E.H. (1980), Pile Foundation Analysis and Design, Wiley & Sons,
New York (Reprinted 1990 by University of Sydney, Australia).
• Poulos, H.G. (1989), "Pile Behavior: Theory and Application", 29th Rankine Lecture,
Geotechnique, Vol. 39 (3), 365- 415.
• Poulos, H.G. (1988), "Modified Calculation of Pile Group Settlement Interaction", Journal of
Geotechnical Engineering 114 (6), 697-706. 67
Computer Programs Available for Analysis of Deep Foundations
DEFPIG - Analysis of piles and pile groups in an elastic medium based on boundary element formulations.
Poulos, H.G. (1978). University of Sydney, Dept. of Civil Engineering, NSW 2006, Sydney, Australia, 77 p.
PIGLET - Analysis of piles & pile groups in an elastic medium based on finite element and approximate
analytical solutions. Randolph, M.F. (1994). Univ. of Western Australia - Geomechanics, Nedlands 6009,
Australia, 26 p.
COM624 - Laterally-loaded piles and shafts based on p-y method. Reese, L.C. (1977). Laterally-loaded piles:
program documentation", Journal of Geotechnical Engineering Division 103 (GT4), ASCE, 287-305.
LTBASE - Laterally-loaded piers using p-y method. Borden, R.H. and Gabr, M.A. (1987), Laterally-Loaded Pier
Analysis Including Base & Slope Effects. Research Project HPR 86-5, NC State Univ., Raleigh, 48 p.
MFAD - Moment Foundation Analysis and Design for drilled shafts using p-y approach. EPRI Report EL-6420,
Prepared by GAI Consultants, Pittsburgh for Electric Power Research Institute, Palo Alto, CA.
CUFAD - Compression-Uplift Foundation Analysis and Design. EPRI Report EL-6420, Vol. 16, and Report No.
EL-6583-CCML, Prepared by Cornell University for Electric Power Research Institute, Palo Alto, CA.
LPILE - Laterally-loaded flexible piles using p-y approach. ENSOFT, Inc., P.O. Box 180348, Austin, TX 78717.
APILE1 and APILE2 - Ultimate capacity of driven piles and load-displacement curves for piles using t-z
approach. ENSOFT Inc., Box 180348, Austin, TX 78717.
UNIPILE - (for windows). Analysis of axial pile capacity and driveability based on dynamics. Unisoft Ltd., 735
Ludgate Court, Ottawa, Ontario K1J 8K8.
CEMSOLVE and CEMSET - Analysis of axial pile capacity for separate base & side components and
consideration of time effects. Cementation Piling & Foundations Ltd., Maple Cross House, UK.
CAPWAP and GRLWEAP - dynamic analysis of axial pile capacity and pile driveability. Pile Dynamics, Inc.,
Cleveland, OH 44128.
Florida-Pier Program (LPGSTAN) - nonlinear finite element program for integrated bridge structure/piles
under axial & lateral loading; pile groups, for FHWA and FL DOT. Public domain. Download from Univ. FL
website. 68
WEBSITES
There is a selection of software for beam on elastic foundations, sheet piles, axial
piles, & lateral piles available for free from Delft University (and slope stability):
http://dutcgeo.ct.tudelft.nl/software/software_e.htm
Several downloadable programs and links to geotech software sites given by the
Geotech Virtual Library (GVL) that is tied with the Electronic Journal of
Geotechnical Engineering:
http://geotech.civen.okstate.edu/wwwvl/soft-gvl.htm
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Additional References
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