ANALYSIS OF LATERALLY LOADED PILES IN WEAK ROCK
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By Lymon C. Reese, l Honorary Member, ASCE
ABSTRACT: The p-y method for the analysis of piles under lateral loading is extended here to the analysis of
single piles in rock. Rational equations are presented for developing a solution, but the method is termed
"interim" principally because of the meager amount of experimental data available to validate the equations.
Nonlinearity, both in the p-y curves and in the bending stiffness of the pile, must be considered in solving ~or
the loading that will cause a failure in bending, deflection, or buckling under combined loading. Two case studies
are presented whereby the analytical method is shown to agree well with results from experiments. However,
loading tests of full-sized piles are recommended at any site where a sizable number of piles are needed, to
further improve the analytical method presented herein.
INTRODUCTION
Rock has received little attention by authors of papers on
the lateral loading of piles. If rock is encountered in installing
a deep foundation, overburden usually exists of sufficient
thickness, that the computed deflection of a pile at the rock is
so small that the resistance of the rock may be neglected, regardless of the stiffness of the rock. However, the combination
of rock near or at the surface with a significant magnitude of
lateral loading does occur, and lateral loading may dictate the
penetration of the pile even though the axial load is substantial.
The theory of elasticity has been used by Kulhawy and his
coworkers (Carter and Kulhawy 1987, 1992), with useful results, and their ideas have influenced the development presented herein. A serious problem with regard to applying any
analytical method to the response of rock is the dominant role
played by the secondary structure of rock. The Canadian
Foundation engineering manual (1978) has addressed secondary structure by basing the behavior of rock on the spacing
and thickness of soil-filled cracks and joints, in addition to the
compressive strength of intact specimens.
SUBSURFACE INVESTIGATION
The writer served on a panel to consider the foundations for
a bridge at Northumberland Strait Crossing, Canada. The panel
members included a number of experts on rock mechanics.
The water depth was moderate, but rock existed at the floor
of the strait. One of the schemes under consideration was to
install piles into predrilled sockets. During winter, the columns, which were to be extensions of the piles, would be
subjected to large lateral loads from moving sheets of ice. One
of the panel members opined that each of the piles should be
proof tested under lateral load because a soil-filled joint could
exist near the surface of the rock. The weak joint would allow
a mass of the rock to slide away from the pile with very low
lateral resistance. The validity of the comment was apparent
if, in fact, soil-filled joints existed at the site. The question was
not resolved because the planning was abandoned for other
reasons. (The bridge was under construction during 1996 with
a different type of foundation.)
For the design of piles under lateral loading in rock, special
emphasis is necessary in the coring of the rock. Experience
has shown that careful attention is required to establish procedures and specifications for field work. The values for the
'Professor Emeritus, The Univ. of Texas at Austin, TX; Prine., Lymon
C. Reese & Associates, Inc.
Note. Discussion open until April I, 1998. To extend the closing date
one month, a written request must be filed with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review and
possible publication on September 30, 1996. This paper is part of the
Journal of Geotechnical and Geoenvironmental Engineering, Vol. 123,
No. 11, November, 1997. ©ASCE, ISSN 1090-0241197/0011-10101017/$4.00 + $.50 per page. Paper No. 14234.
Rock Quality Designation (RQD), percent of recovery, and
compressive strength can probably be more seriously in error
from improper procedures than are the corresponding properties of soil. The procedures that follow are based on results
from field tests of piles in rock with differing characteristics.
In neither case, however, did soil-filled joints influence the
response of the piles. Methods of investigation should reveal
detailed information, and designers must address the potential
behavior of the rock in a site-specific manner. Therefore, the
judgment of the geotechnical engineer is critical with respect
to characterizing the rock and applying the technique shown
herein.
p-y METHOD OF ANALYSIS
A model describing the method of analysis is shown in Fig.
1. An elevation view of a pile is shown in Fig. l(a), with a
lateral load PIt an axial load Px , and a moment M applied at
the pile head. The pile is shown as an elastic line "in Fig. l(b)
in a coordinate system with deflection y and length x along
the pile. The rock (soil, usually) is modeled according to the
Winkler concept with a number of nonlinear, discrete mechanisms. The mechanisms, shown in the first quadrant for convenience, are characterized by a spring and sliding block
merely to indicate nonlinearity, and they are described by the
p-y curves in Fig. l(c), where p is the resistance of the rock
and y is the local deflection. The parameter p refers to the line
load from the rock resistance and is the integral of the unit
stresses acting around the circumference of the pile. A number
of authors have made recommendations for predicting p-y
curves for different soils [for example, Matlock (1970); Welch
and Reese (1972); O'Neill and Murchison (1983)].
The continuum is not modeled faithfully by the Winkler-
P,
M,
pb
P~
Pk=Y
p1==y
W
x
pC
~
y
~
FIG. 1. Model of Laterally Loaded Pile: (a) Elevation View; (b)
As Elastic Line; (c) p-yCurves
1010/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / NOVEMBER 1997
J. Geotech. Geoenviron. Eng. 1997.123:1010-1017.
type curves; however, the recommendations for formulating Py curves are based strongly on results from field-load tests, as
done herein, where the continuum effect is fully satisfied. The
method is described in the technical literature [for example,
McClelland and Focht (1956); Matlock and Reese (1960);
Reese (1984)], and computer codes implementing the method
are in use in many offices in the United States and abroad.
The nonlinear differential equation to be solved for deflection, rotation, bending moment, shear, and soil resistance along
the pile is
d (d
2
2y
d
2
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y
)
-dx 2 EI+P• -Pdx 2
dx 2
w-o
-
(1)
where p. = axial load on pile, (F); y = lateral deflection of
pile at point x, (L); P = soil resistance per unit length along
pile, (F/L); EI = flexural rigidity (bending stiffness), (F - L 2);
and W = distributed load along pile, (F/L).
The equation is the standard beam-column equation where
the value of EI may change along the length of the pile and
may also be a function of the bending moment. The equation
(1) allows a distributed load to be placed along the upper portion of a pile; (2) can be used to investigate the axial load at
which a pile will buckle; and (3) can deal with a layered profile of soil or rock.
Solutions may be developed readily using difference-equation techniques and Gaussian elimination. The usual method
of developing a solution is to increment the loading, employing nonlinear p-y curves and a nonlinear EI curve, to find the
failure loading from excessive deflection, a plastic hinge, or
axial buckling. Analytical methods are available for computing
the values of EI as a function of bending moment and axial
load, and for computing the moment at which a plastic hinge
will develop (Reese and Wang 1994). After finding the loading
that will cause the governing type of failure, the loads may be
factored to find the design loading. With input data at hand,
various parameters may be investigated, and an acceptable solution can be found with computer codes that are relatively
straightforward and easy to use.
3. The initial slope K lr of the p-y curves must be predicted
because small lateral deflections of piles in rock can result in resistances of large magnitudes. For a given value
of compressive strength, Klr is assumed to increase with
depth below the ground surface.
4. The modulus of the rock Elro for correlation with Klro may
be taken from the initial slope of a pressuremeter curve.
Alternatively, the results of compressive tests of intact
specimens may be used to obtain values of E lr • The data
in Fig. 2 (Horvath and Kenny 1979; Peck 1976; Deere
1968) may be useful, but, as may be seen, the E lr values
for samples of the same type of rock may vary by several
orders of magnitude. Therefore, Fig. 2 can be expected
to yield only approximate correlations between compressive strength and modulus. Fig. 3 (Bieniawski 1984)
shows a correlation between Emus/Ecore and RQD. Values
of Emus may be estimated if tests have been performed
of cored specimens (Ems.. and E ir are assumed to be
equivalent). Again, scatter is significant. The modulus for
the mass of rock is assumed to be implemented in the
expressions that follow.
5. The ultimate resistance Pur for the p-y curves will rarely,
if ever, be developed in practice, but the prediction of
Pur is necessary in order to reflect nonlinear behavior.
6. The component of the strength of rock from unit weight
is considered to be small in comparison to that from
compressive strength quro therefore, unit weight is ignored.
7. The compressive strength of the intact rock qur for computing a value of Pur may be obtained from tests of intact
specimens.
8. The assumption is made that fracturing will occur at the
surface of the rock under small deflections; therefore, the
compressive strength of intact specimens is reduced by
multiplication by a r to account for the fracturing. The
value of a r is assumed to be 1/3 for RQD of 100 and to
increase linearly to unity at RQD of zero. If RQD is zero,
0.1
FIELD TESTS
Results from two programs of testing of full-scale, bored
piles (drilled shafts) in rock are available for analysis (Nyman
1980; D. Speer, unpublished report, 1992). In both cases, data
were available on the geometry of the piles, magnitude and
point of application of loads, and characteristics of the rock.
Curves showing deflection versus lateral load were reported
for both of the programs. Comparisons of the results from
analyses, using the procedures described herein, and results
from the experiments are presented later.
INTERIM RECOMMENDATIONS FOR COMPUTING
p-yCURVES FOR ROCK
Concepts
An analysis of the results from the tests noted earlier, and
a study of other information, formed the basis for the recommendations given here. The recommendations are termed "interim" for a number of reasons, and comments on their appropriate use in analysis and design are given.
The following concepts and procedures establish the framework for the recommendations:
1. The secondary structure of rock, related to joints, cracks,
inclusions, fractures, and any other zones of weakness,
can strongly influence the behavior of the rock.
2. The p-y curves for rock and the bending stiffness EI for
the pile must both reflect nonlinear behavior in order to
predict loadings at failure.
10
100
I. 1x104
•I
I,i
0.1
1
10
100
UNIAXIAL COMPRE88IVE STRENGTH -
1000
M'"
FIG. 2. Engineering Properties for Intact Rocks [after Horvath
and Kenney (1979); Peck (1976); Deere (1968)]
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1 NOVEMBER 1997/1011
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p
1.0
(FILl
Pur
Jlj
f
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o
0.6
I
!l
0.8
YA Yrm
0.4
I
o
o
FIG. 4.
o _
0.2
-
-
-_--D -..... ...
-1-
_1-
_1- -
-
o
n.
"C;,lI
00
0
O.OL-....:...L_...L-_L-......L_...L-_L-......L_-'---IL--:t
o
60
80
100
Rock Qu.1l1y D••llln.tlon, RQD, %
FIG. 3. Modulu8 Reduction Ratio a8 Function of RQD [after
Blenlaw8kl (1984)]
the compressive strength may be obtained directly from
a pressuremeter curve, or approximately from Fig. 2, by
entering the value of the pressuremeter modulus.
Ultimate Resistance of Rock
The following expression for the ultimate resistance Pu, for
rock is based on limit equilibrium and reflects the influence
of the surface of the rock:
Pu,
= cx,qw,b
Pur
(1 + 1.4;); 0~
= 5.2cx,qw,b;
Xr ~
x,
~ 3b
3b
(2)
Slope of Initial Portion of p-yCurves
If one were to consider a strip from a beam resting on an
elastic, homogeneous, and isotropic solid, the initial modulus
K I (PI divided by YI) may be shown to have the following value
(using the symbols for rock):
(4)
where E 1, = initial modulus of rock; and ki , = dimensionless
constant. Eqs. (5) and (6) for ki , are derived from experiment
and reflect the assumption that the presence of the rock 'surface
will have a similar effect on kin as was shown for Pu, for
ultimate resistance.
o ~ x, ~
= 500;
Xr
> 3b
Sketch of p-yCurve for Rock
lower values of the K i , in relation to the modulus of rock or
soil; however, the modulus of the rock at San Francisco, used
principally in developing the correlations, was obtained by
pressuremeter. The details employed in the pressuremeter test
could well reveal a much lower value of the initial modulus
of rock in comparison with that exhibited by a pile of large
diameter deflected against the rock. Third, the increase in K 1,
with depth in (5) is consistent with results obtained from the
lateral loading of piles in overconsolidated clays.
Formulas for Family of p-y Curves
With guidelines for computing Pu, and Kin the equations for
the three branches of the family of p-y curves for rock can be
presented. The characteristic shape of the p-y curves is shown
in Fig. 4. The equation for the straight-line, initial portion of
the curves is given by (7) and, for the other branches, (8)(10)
p
=KI,y;
Pu, Y
P=- ( 2 Yrm
(3)
where qu, = compressive strength of rock, usually lower-bound
as function of depth; cx, = strength reduction factor; b = diameter of pile; and x, = depth below rock surface.
kl,
r(l)
3b
(5)
(6)
Eqs. (5) and (6), developed from experimental data, show
that the initial portions of the P-Y curves are very stiff in order
to model the relatively very low deflections observed during
beginning loads. Some further comments are in order about
the equations. First, the equations have no influence on solutions beyond the value of Y" and probably will have no influence on designs based on the ultimate bending moment of a
pile. Second, available theory, while incomplete, shows much
)
(7)
y ~ y"
O.2~
; Y ~y,,;
p
~
(8)
Pur
P =pw,
Yrm
= krmb
(9)
(10)
where krm = constant, ranging from 0.0005 to 0.00005 (see
case studies to follow), that serves to establish overall stiffness
of curves.
The value of y" is found by solving for the intersection of
(7) and (8), and is shown by (11)
Pu,
] 1.333
[
y" = 2(Yrml~KI'
(11)
COMMENTS ON EQUATIONS FOR PREDICTING p-y
CURVES FOR ROCK
The equations predict with reasonable accuracy the behavior
of single piles under lateral loading for which experimental
data are available. Because of the meager amount of data, the
equations should be used with caution. An adequate factor of
safety should be employed in all cases; preferably, field tests
should be undertaken with full-sized piles, with appropriate
instrumentation.
If the rock contains joints that are filled with weak soil, the
selection of properties of strength and stiffness must be sitespecific and will require a comprehensive geotechnical investigation. In those cases, the application of the method presented herein should proceed with even more caution than
normal.
The equations are based on the assumption that p is a linear
function of y, an idea that appears to be valid if loading is
static and if resistance is due only to lateral stresses. However,
O'Neill points out that "in large-diameter drilled shafts, mo-
1012/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I NOVEMBER 1997
J. Geotech. Geoenviron. Eng. 1997.123:1010-1017.
ment is resisted in the push-pull resistance produced by the
axial shears caused by the rotation of the pile. In rocks, this
effect could be significant, especially for small deflections, if
the diameter of the pile is large" (M. W. O'Neill, personal
communication, 1996).
COMPUTATION OF MUir AND EI AS FUNCTION OF
BENDING MOMENT AND AXIAL LOAD
The aim of the case studies that follow is to employ the
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p-y curves for rock for computing the pile-head deflection for
comparison with experimental values. Experience and preliminary computations show the necessity of considering the nonlinear bending stiffness EI of a pile in order to make meaningful comparisons with experimental data. Methods of
computing the value of EI as a function of the applied moment
and axial load, as well as the ultimate bending moment Mol"
are summarized here. The methods are implemented in the
studies that follow.
FIG. 5.
Streaa-Straln Curve for Concrete
Analytical Procedure
Equations for the behavior of a slice from a beam or from
a beam-column under bending and axial load are formulated.
A reinforced-concrete section is assumed in the presentation,
but the concepts can be applied to a steel shape. The EI of the
concrete member will experience a significant change when
cracking occurs. In the procedure described herein, the assumption is made that the tensile strength of concrete is relatively small and that cracks will be closely spaced when they
appear. Actually, such cracks will initially be spaced at some
distance apart, and the change in the EI will not be so drastic
as computed. Therefore, the EI for a reinforced-concrete pile
will change more gradually in practice than in the computations by the suggested analytical method.
Because the nonlinear stress-strain curves for steel and concrete do not indicate a condition for collapse, values of the
ultimate strain of these materials are selected to reflect their
failure. For concrete, the ultimate value of strain is 0.003; for
reinforcing steel, the ultimate value of strain is 0.015. These
values appear to be consistent with those frequently used in
practice.
The curve for the deformational characteristics of concrete
implemented in the procedure is shown in Fig. 5. The values
of f~, the compressive strength, and E e , the modulus of elasticity, are found from standard tests of cylinders or from other
appropriate tests. The following equations are for concrete of
normal weight, and apply to the branches of the curve:
fc
= Eee;
0::5
fc
fc
=f~ [2 (t) - (t)1
fc
=fe" ( 1 -
::5 J,
(12)
O::5fc
0.15£0) ; fe,,~
0.0038
<? Je
<?
::5f~
0.8 5f"e
(13)
-----Jr--:L-----------'E
FIG. 6.
Streu.straln Curve for Steel
from the American Concrete Institute (ACI) (1989) and is for
concrete with normal weight.
The stress-strain curve for steel is shown in Fig. 6, and there
is no limit to the amount of plastic deformation. The curves
for tension and compression are identical. The yield strength
of the steel h is selected according to the material being used,
and E is the initial modulus of the steel. The following equations apply:
(19)
E = 200,000 MPa
(20)
The derivation adopts the concept that plane sections in a
beam or beam-column remain plane after loading. Thus, an
axial load and a moment can be applied to a section with the
result that the neutral axis will be displaced from the center
of gravity of a symmetrical section.
The equations to be solved are as follows:
h
(14)
b
f
'
a dy
= p.
(21)
ay dy
=M
(22)
-h,
where
h
f~
= 0.85f:
(15)
(16)
J,= 19.7Vif
(17)
The approximate value of E e , in the absence of experimental
data, may be taken from the following equation:
Ee
= 151,oooVif
(18)
The terms in (17) and (18) have units of kPa. Eq. (18) is
b
f
'
-h,
where b = depth of section; hit h2 = distance from neutral axis
to extreme fibers; a = normal stress; y = distance from neutral
axis; p. = axial load; and M = moment on section. The steps
in a convenient procedure of computation are as follows: (1)
Select the angle of rotation e for a section; (2) Estimate the
position of the neutral axis; (3) Compute the strain across the
section; (4) Use numerical methods to solve for the distribution of stresses across the cross section; (5) Compute the magnitude of the axial load by summing the forces across the
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1 NOVEMBER 1997/1013
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section, as indicated in (20); (6) Modify the position of the
neutral axis if the computed value of axial load does not agree
with the applied load; (7) Repeat the computations until convergence is achieved; (8) Solve for the bending moment by
numerical methods, implementing (22); and (9) Obtain the
bending stiffness by use of (23) where p is equal to the radius
of curvature of de divided by the length of the element dx.
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M
EI
1
(23)
P
A computer code has been written to solve the relatively
straightforward equations because of the iteration due to the
nonlinear stress-strain characteristics of concrete and steel. The
computations continue until the maximum strain selected for
failure is reached. Then, the ultimate moment MUll can be
found.
Approximate Method
The analytical procedure results in a sharp decrease in the
value of EI because the mechanics predicts continuous cracking at a given tensile strain of the concrete. Observations of
the behavior of reinforced-concrete sections have yielded an
empirical equation that gives values of bending stiffness that
reduce more gradually, as a function of the applied bending
moment, than do the values from mechanics (ACI 1989).
I.
= (~:r Ig +
[I - (~:r]
Ier
(24)
where
Mer
=g
(25)
Ye
f,
= 19.7Vjf (for normal-weight concrete)
(26)
and where I. = effective moment of inertia for computation of
deflection; I g = moment of inertia of gross concrete section
about centroidal axis, neglecting reinforcement; Ye = distance
from centroidal axis of gross section, neglecting reinforcement, to extreme fibers in tension; I er = moment of inertia of
cracked section; and M a = maximum moment in pile.
The value of I er may be computed by the analytical method,
using standard mechanics. In computing bending stiffness, the
value of E is assumed to remain constant.
The absence of a term for axial load in (24) means that the
approximate method or ACI method is limited in scope. However, a comparison of results from (24) with those from the
analytical method with no axial load and with a specified axial
load will reveal a trend that should prove useful in solving a
practical problem.
In applying the ACI method, the difference between the response of a beam and a pile to lateral loading may be great.
For many practical cases, the computed bending-moment
curve for a pile changes rapidly with depth, and the region of
the maximum bending moment may be a small fraction of the
length of the pile. Therefore, the engineer may wish to apply
(24)-(26) point-by-point along the length of the pile rather
than to the entire length.
CASE STUDIES
Islamorada
The test was performed under sponsorship of the Florida
Department of Transportation and was carried out in the Florida Keys (Nyman 1980). The rock was a brittle, vuggy, coral
limestone, allowing a steel rod to be driven into the rock to
considerable depths, apparently because the limestone would
fracture and the debris would fall into the vugs. Cavities in
the order of a third of a meter in the largest dimension existed
in the limestone in some regions, but only the vugs were encountered at the test site. Two specimens were obtained for
compressive tests. The small discontinuities at the outside surface of the specimens were filled with gypsum cement to minimize stress concentrations. The ends of the specimens were
cut with a rock saw and lapped flat and parallel. The compressive strengths were found to be 3.34 and 2.60 MPa. The
axial deformation was measured during testing, and the average value of the initial modulus of the rock was found to be
7,240 MPa. In the absence of additional data, the value from
the cores is assumed to be equal to the modulus of the mass.
The rock at the site was also investigated by in-situ-groutplug tests under the direction of Dr. John Schmertmann (unpublished report, 1977). A 140-mm-diameter hole was drilled
into the limestone, a high-strength-steel bar was placed to the
bottom of the hole, and a grout plug was cast over the lower
end of the bar. The bar was pulled to failure and the hardened
grout was examined to ensure that failure occurred at the interface of the plug and the limestone. Tests were performed at
three locations, and the results are shown in Table I. Nyman
(1980) studied all of the data, and a compressive strength of
3.45 MPa was selected as representative of the rock in the
zone near the rock surface where the deflection of the pile was
most significant. Values of RQD were not obtained. However,
in view of the difficulty in obtaining intact specimens, RQD
was assumed to be close to zero and a r was taken as unity.
The bored pile was 1.22 m in diameter and penetrated 13.3
m into the limestone. A layer of sand over the rock was retained by a steel casing, and the lateral load was applied at
3.51 m above the surface of the rock. A maximum lateral load
of 667 kN was applied, and the resulting curve of load versus
deflection was nonlinear.
Nyman (1980) recommended p-y curves for the vuggy rock
at Islamorada. A key feature of the recommendations was that
the rock was assumed to fracture and lose all strength after a
small amount of deflection. Such a failure was not observed
-only postulated. In the absence of other recommendations,
Nyman's suggestions were used for other kinds of rock (Reese
and Wang 1989).
In the absence of details on the strengths of the concrete
and steel and on the amount and placement of the rebars, the
bending stiffness of the gross section was used for the initial
solutions. The following values were used in the equations for
p-y curves; qur = 3.45 MPa; a r = 1.0, ETI = 7,240 MPa; krm =
0.0005; b = 1.22 m; L = 15.2 m; and EI = 3.73 X 106 kN·
m 2•
The comparison of pile-head deflection for results from experiment and from analysis is shown in Fig. 7. The figure
shows excellent agreement between the two methods up to
about 350 kN, using unmodified values of the bending stiffness. A sharp change in the load-deflection curve occurs at a
lateral load of about 350 kN.
TABLE 1. Results of Grout-Plug Tests by Schmertmann (Unpublished Report, 19n)
Depth range
(m)
(1 )
Ultimate resistance
(MPa)
(2)
0.76-1.52
0.76-1.52
0.76-1.52
2.44-3.005
2.44-3.005
2.44-3.005
5.49-6.10
5.49-6.10
2.28
1.31
1.15
1.74
2.08
2.54
1.31
1.013
10141 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1NOVEMBER 1997
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Computed curves giving deflection and bending moment as
a function of depth are presented in Fig. 8 for a lateral load
of 334 kN, one-half of the ultimate lateral load. Only depths
of about 4.5 m for deflection and about 6 m for bending moment are plotted. The values to the full length are too small
to plot. The stiffness of the rock, compared to the stiffness of
the pile, is reflected by a total of 13 points of zero deflection
over the length of the pile of 15.2 m. With an increase in the
lateral loads, the deflections will increase and fewer points of
zero deflection will result. However, for the data employed
here, the pile will behave as a long pile through the full range
of loading. With the use of nonlinear p-y curves, there can be
no a priori decision about the classification of a pile under
lateral loading as short, intermediate, or long. Such a decision
can be made only after solution of (1), using nonlinear mechanics.
As shown in Fig. 7, values of EI were reduced gradually in
zones of large bending moments to find deflections that would
agree fairly well with values from experiment. The plot of
bending moment in Fig. 8 shows that the largest moment occurs in the zone of about 2.5 -4.5 m. The following combinations of values of load and bending stiffness were used in
the analyses in units of kN and kN 'm2 , respectively: 350 and
below 3.73 X 106 ; 400, 1.24 X 106 ; 467, 9.33 X 105 ; 534,
7.46 X lOS; 601, 6.23 X 105 ; and 667, 5.36 X 105 • In each
case, the computed bending-moment curve was examined and
the reductions were only made in the zone where the bending
stiffness was expected to be in the nonlinear range. The lowest
700
-r--+----.,.------,.--..---.....
600
600
J
1
)
400
- - - Araysls wi elastic EI
Analysis wi reduced EI
o Experimental
300
200
100
o
5
10
15
Oroundllne deflection, mm
20
FIG. 7. Comparison of Experimental and Computed Values of
Pile-Head Deflection, Islamorada Test
Bending moment, kN·m
o
-400
San Francisco
The California Department of Transportation performed lateral-load tests of two bored piles near San Francisco (Speer,
unpublished report, 1992), and the results of the tests, while
unpublished, have been provided courtesy of Caltrans.
Two borings were made into the rock, and the following
statements describe the experimental techniques. "After bedrock was encountered, sampling was continued using a NWD4
core barrel in a 4-inch [102 mm] diameter cased hole. A three
and seven eighth inch [98 mm] tricone rock bit was used to
advance the casing and clean the borehole." The sandstone
was found to be medium-to-fine grained (with grain sizes from
0.1 to 0.5 mm), well sorted, and thinly bedded (25 - 75 mm
thick). Except in a few cases, recovery was 100%. Twenty
values of RQD were reported, ranging from zero to 80, with
an average of 45. In most of the corings, the sandstone was
described as very intensely to moderately fractured with bedding joints, joints, and fracture zones. With respect to the selection of a value of an it was assumed that there was little
chance of brittle fracture and a r was taken as unity.
Pressuremeter tests were performed at the site, and the results, as might be expected, were scattered. The plotted results
of the values obtained for the moduli of the rock are shown
in Fig. 9. The averages that were used for analysis are shown
as a function of depth by the dashed lines. The following values were estimated for the compressive strength of the rock:
0-3.9 m, 1.86 MPa; 3.9-8.8 m, 6.45 MPa; and below 8.8 m,
16.0 MPa. The right-hand curve in Fig. 2 was employed in
developing the correlation between E jr from Fig. 9 and qur'
Two piles, 2.25 m in diameter, with penetrations of 12.5 m
and 13.8 m, were tested simultaneously. Lateral loading was
accomplished by hollow-core rams, acting on high-strength
steel bars, that were passed through tubes, transverse and perpendicular to the axes of the piles. The load was measured by
load cells, and the piles were instrumented with slope indicators and strain bars. Deflection was measured by transducers,
and slope and deflection of the tops of the piles were obtained
by readings from the slope indicators.
The load was applied in increments at 1.41 m above the
ground line for pile A and 1.24 m for pile B. The pile-head
deflection was measured at slightly different points above the
1200
BOO
400
value of EI that was used is believed to be roughly equal to
that for the fully cracked section. The assumption that the decrease in slope of the curve of Y, versus P, at Islamorada can
be explained by reduction in values of EI is reasonable. However, the Islamorada example gives little guidance to the designer of piles in rock, except for early loads. The example
from San Francisco that follows is more instructive.
O+-................-l,,......-'-.........l.-l,.........--..-'-"-T.......-j
EJr,MPe
o
BOO
1200
1600
2000
o+-...,.........i--'-.........'-!-..............................."J...............-1
400
I
01 0
2
,
'
°186 MFa
,
ooV----:-----, _
cJ
4-1-_°"':_
3 .9
......--r_ _
o
0:
5
8
6
- - - -: - --
7
-0.5
0
0.5
1.5
2
2.5
....
3
Deflection, mm
10
_0:'
:
I
m:
Q
v6~5~F~
.- -b"--:-
,
_ .,_
--;-S.ifril-
_:__ -0- __
~_1:
00 MFa
12...L..------------'1'---...J
FIG. 8. Plot of Computed Curves of Deflection and Bending
Moment versus Depth, Islamorada Test, Lateral Load of 334 kN
FIG. 9. Initial Moduli of Rock from Pressuremeter. San Francisco Test
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / NOVEMBER 1997/1015
J. Geotech. Geoenviron. Eng. 1997.123:1010-1017.
1‫סס‬oo
40,...--.,..---,.--......--....,..--...,
30
......
25
.... -:'.
.. .- ..... _ .. ,
--- .. ~ .. -- .. --- ------- ~ . ----
35
8000
:G
1'!
~
___ . L • • • • • _ .
,
,
8000
- .. - -
~
- . - . i .. - . - - -
,
j
,
- - - - - -
Z
.¥
........
- l' --G__
4000
2000
1:.
lC
Unmodlfled EI
Anelytical
ACI
Expartmental
20
'
._~~._
.....
, ..
..... ,,. ..
,
,
....
ACI
_
Experimental
_
Analytical
.
,
,
-. -
": 15
r
e
. . .;
......
.............. ~
iii
;.
.;.-
,
........ ;.
10
,..
...... L .. ~
Downloaded from ascelibrary.org by MARRIOTT LIB-UNIV OF UT on 11/28/14. Copyright ASCE. For personal use only; all rights reserved.
..
~
• ••
- .
.
,
L
..
0
0
10
20
30
40
o-t-T""'T""T"'t"""........"T"",.............,...,.""T".,...,,.....,.."T"".......-1
o
4000
8000
12000 16000 20000
50
Qroundllne denectlon, mm
FIG. 10. Comparison of Experimental and Computed Values
of Pile-Head Deflection for Different Values of EI, San Francisco
Test
M, kN·m
FIG. 11.
Values of Elfor Three Methods, San Francisco Test
1‫סס‬oo
rock line, but the results were adjusted slightly to yield equivalent values for each of the piles.
In addition to the previously noted values, the following
values were used in the analyses: k rm =0.‫סס‬OO5 and EI =35.15
6
2
6
4
X 10 kN· m (E = 28.05 X 10 kPa; I = 1.253 m ) for the
beginning loads. The ultimate bending moment MUll was computed to be 17,740 m·kN. The values of EI and MUll were
computed from the following properties of the cross section:
compressive strength of the concrete was 34.5 MPa, tensile
strength of the rebars was 496 MPa, there were 40 bars with
a diameter of 43 mm, and cover thickness was 0.18 m.
The curve for pile B (see Fig. 10) exhibited a large increase
in pile-head deflection at the largest load, which suggests that
a plastic hinge developed. Therefore, the assumption was made
that the ultimate bending moment, 17,740 m . kN, was reached.
Analysis of the previously reviewed Islamorada test showed
that the bending stiffness of a reinforced-concrete member decreases significantly with increased bending moment; however,
previous work showed that the computation of bending moment is not strongly dependent on the specific value of EI.
Therefore, the beginning computations for the solution of (1)
were based on the initial values of EI. Values of qu" from
results from the pressuremeter (see Fig. 2), were found to predict MUll with reasonable accuracy.
Then, attention was given to the probable reduction in the
values of EI with increasing load, and three methods were used
to predict the reduced values. The methods were the analytical
method, the approximate method (or ACI method, used since
no axial load was applied during the testing), and the experimental method. The three plots of the values of EI as a function of M are shown in Fig. 11.
The experimental method employed the average of the observed deflections, the applied loading, and iteration to find
the values of EI and the corresponding values of maximum
bending moment that fitted the results. In these computations,
and for those that follow, the value of EI was changed for the
entire length of the pile for ease in computations. Errors in
using constant values of EI in the regions of low values of M
are thought to be small.
All three curves exhibit a sharp decrease in the value of EI
with increase in bending moment, but the analytical method
yielded a precipitous drop, for the reason noted earlier. All of
the values of EI start from 35.15 X 106 kN· m 2 , (the value of
EI from the analytical method was slightly higher because of
the presence of the steel). The values of I from ACI were
multiplied by a constant value of E of 28.05 X 106 kPa to get
the values of EI. Surprisingly, the values of I from the experiment and from ACI fall below the value from analysis for
over half of the range of values. The concrete may be cracking
8000
~
,
,
..
,
,
,,,
,,
.............. -:
,
,
,
,
,
~
6000
··.·.·· ...,-------·r·.-··
e
~
4000
...
2000
,,
,,
'._._-
.•••.
~
..
,
,
,,
..,
..
"fi
~
:,
- ~ •• ~ • - •••' • • • • - • - • & • - • • • • • •
- - Unmodilled EI
- K - Analytical
:
-e- ACI
: .. , -lil- Experimental
o.................-r-...............-r-..............-r-..............-r-...............-I
4000
6000
12000
18000
20000
o
Maximum bending moment, kN-m
FIG. 12. Comparison of Experimental and Computed Values
of Maximum Bending Moments for Different Values of EI, San
Francisco Test
at a smaller strain than 0.003, there may be some crushing of
the concrete in compression, or the analytical theory may not
faithfully reflect the real behavior of the reinforced-concrete
section in some other ways. The next step was to investigate
the influence of the values of I (and El) on the computations
of deflection and maximum bending moment.
The computed pile-head deflections, using the values of I
for the three cases in Fig. 11, are shown in Fig. 10. The experimental values agree well with computations, of course,
because of the fitting noted earlier. The computations with the
ACI equations fit the experimental values better than do the
computations with the analytical method. However, if load factors of 2.0 and greater are selected, the computed deflections,
taking into account the methods for reducing the value of EI,
would be about 2 or 3 mm, with the experiment showing about
4 mm. The differences are probably not very important in the
range of the service loading.
Also shown in Fig. 10 is a curve showing deflection as a
function of lateral load with no reduction in the values of EI.
The necessity of employing a reduced value of EI is clear.
The values of I in Fig. 11 were used to compute the maximum bending moment as a function of the applied load. The
curves are given in Fig. 12, and the close agreement among
all three methods is striking. Also shown in Fig. 12 is the plot
of maximum bending moment using the gross EI. The curve
is reasonably close to the curves from adjusted values of EI,
indicating that the computation of bending moment is not very
1016/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / NOVEMBER 1997
J. Geotech. Geoenviron. Eng. 1997.123:1010-1017.
sensitive to the selected values of bending stiffness. The value
of the ultimate bending moment Mult , computed by the analytical method, is shown in Fig. 12. Assuming Muir to be correct, all of the methods predict failure in bending with good
accuracy.
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CONCLUSIONS
1. The p-y method for the analysis of piles under lateral
loading, in wide use for the analysis and design for piles
in soil, can be used for the analysis of piles installed in
rock.
2. The subsurface investigation for the characteristics of
rock at a particular site is critical because secondary
structure can dictate the behavior of a pile under lateral
loading. Special geotechnical studies are necessary for
cases whereby joints are filled with weak soil.
3. The analytical method for computing the value of Muir is
essential to the analysis of reinforced-concrete piles under lateral loading in order to compute the loading at
which failure will occur in bending.
4. The interim method of computing p-y curves, employed
in the solution of (1), using unadjusted values of EI, can
be used to determine the combined loading that will develop the ultimate bending moment Muir in the pile.
5. Adjusted values of bending stiffness are necessary in
computing pile-head deflection. The ACI equations are
recommended for the computation of adjusted values of
EI.
6. The computation of maximum bending moment is much
less affected by the selection of appropriate values of EI
than is the computation of deflection.
7. The deflection under service (unfactored) loads of the
pile head for piles in rock, even for relatively soft rock,
will be relatively small.
8. Field-load tests with instrumented piles in rock are
strongly desirable for improving the design at a particular
site, especially if a large number of piles are to be used,
and to add to the experimental-data base. The method
presented herein may be used in interpreting the results
from the particular experiment.
ACKNOWLEDGMENTS
The writer wishes to acknowledge the contributions of the Florida
Department of Transportation and the California Department of Transportation in performing the experiments that aIlowed for the development
of the method presented herein. In particular, Tom PoIlack, acting chief,
Division of Structures, Caltrans, is thanked for providing a copy of the
first draft of an unpublished report (1996), entitled "Shaft lateral load
test terminal separation." K. J. Nyman is acknowledged for the data
acquisition and the preliminary analysis of the results of the experiment
at Islamorada. Dr. Michael W. O'Neill, University of Houston, and Paul
D. Passe, Florida Department of Transportation, read an early draft and
made valuable suggestions. The writer's coIleagues, Dr. Shin-Tower
Wang and Jos6 Arrel1aga, made significant contributions. Both read the
drafts and offered useful suggestions about the method of analysis. Dr.
Wang coded some of the techniques for solution by computer, and JostS
Arr611aga prepared the drawings.
APPENDIX I.
Carter, J. P., and Kulhawy, F. H. (1992). "Analysis of lateral1y loaded
shafts in rock." J. Geotech. Engrg., ASCE, 118(6), 839-855.
Deere, D. V. (1968). "Chapter 1: geological considerations. Rock mechanics in engineering practice, K. G. Stagg and O. C. Zienkiewicz,
eds., John Wiley & Sons, Inc., New York, N.Y., 1-20.
Horvath, R. G., and Kenney, T. C. (1979). "Shaft resistance of rocksocketed drilled piers." Proc., Symp. on Deep Found., ASCE, New
York, N.Y., 182-184.
Matlock, H. (1970). "Correlations for design of laterally loaded piles in
soft clay." Proc., 2nd Annu. Offshore Technol. Con/., Paper no. OTC
1204, 1,577-594.
Matlock, H., and Reese, L. C. (1960). "Generalized solutions for laterally
loaded piles.' J. Soil Mech. and Found. Div., ASCE, 86(5), 63-91.
McClel1and, B., and Focht, J. A. Jr. (1956). "Soil modulus for laterally
loaded piles." J. Soil Mech. and Found. Div., ASCE, 82(4).
Nyman, K. J. (1980). "Field load tests of instrumented dril1ed shafts in
coral limestone," MS thesis, Grad. School, The Univ. of Texas at Austin, Tex.
O'Neill, M. W., and Murchison, J. M. (1983). "An evaluation of p-y
relationships in sands." Rep. PRAC 82-41-1 Prepared for American
Petroleum Institute, Univ. of Houston, University Park, Houston, Tex.
Peck, R. B. (1976). "Rock foundations for structures." Proc., Spec. Con/.
on Rock Engrg. for Found. and Slopes, ASCE, New York, N.Y.
Reese, L. C. (1984). Handbook on design ofpiles and drilled shafts under
lateral load. FHWA-1P-84-11, Fed. Hwy. Admin., U.S. Dept. of
Transp., Washington, D.C.
Reese, L. C., and Wang, S.-T. (1989). "Documentation of computer program LPILE." Ensoft, Inc.
Reese, L. C., and Wang, S.-T. (1994). "Analysis of piles under lateral
loading with nonlinear flexural rigidity." Proc., U.S. FHWA Int. Con/.
on Des. and Constr. of Deep Found., Fed. Hwy. Admin., U.S. Dept. of
Transp., Washington, D.C.
Welch, R. C., and Reese, L. C. (1972). "Laterally loaded behavior of
dril1ed shafts." Res. Rep. no. 3-5-65-89, Center for Highway Research.
APPENDIX II.
NOTATION
The following symbols are used in this paper:
Symbols Relating to p-yCurves
b
EI
E tT
kiT
k rm
L
Pr
Px
p
qUT
= diameter of pile, (L);
= flexural rigidity (bending stiffness), (F = initial modulus of rock, (FIL
= dimensionless constant;
XT
y
lXT
L 2 );
);
= dimensionless constant, ranging from 0.0005-0.0005,
that serves to establish overall stiffness of p-y curves;
= length of pile (L);
= lateral load on pile, (F);
= axial load on pile, (F);
= soil resistance per unit length along pile, (FIL);
= compressive strength of rock, usually lower bound, as
function of depth, (FIL
= distributed load along pile, (FIL);
= axial coordinate along pile, (L);
= depth below rock surface, (L);
= lateral deflection of pile at point x, (L); and
= strength reduction factor.
2
W
x
2
);
Symbols Relating to Bending Stiffness
b
f,
hh h2
leT
I.
= depth of section, (L);
= modulus of rupture of concrete, (F/L
= distance from neutral axis to extreme fibers, (L);
= moment of inertia of cracked section, (L4);
= effective moment of inertia for computation of deflec2
);
tion, (L4 );
REFERENCES
American Concrete Institute. (1989). "Building code requirements for
reinforced concrete." ACI 318-89, Detroit, Mich.
Bieniawski, Z. T. (1984). Rock mechanics design in mining and tunneling.
A. A. Balkema, Rotterdam, The Netherlands.
Canadian Foundation engineering manual. part 3. deep foundations.
(1978). Can. Geoteeh. Soc., Montreal, Quebec, Canada.
Carter, J. P., and Kukhawy, F. H. (1987). "Analysis and design of dril1ed
shafts socketed into rock." Res. Rep. 1493-4, Geotech. Engrg. Group,
Cornell Univ., Ithaca, N.Y.
I,
M
M.
MeT
Px
y
Ye
(J'
= moment of inertia of gross concrete section, (L
4
);
= moment on section, (F - L);
= applied maximum bending moment, (F - L);
= cracking bending moment, (F - L);
= axial load, (F);
= distance from neutral axis, (L);
= distance from centroidal axis of gross section, neglecting
reinforcement, to extreme fibers in tension, (L); and
= normal stress, (F/L
2
).
JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1 NOVEMBER 1997/1017
J. Geotech. Geoenviron. Eng. 1997.123:1010-1017.