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55
ANALYTICAL METHODS actual soil/rock reaction. In practice, a great deal of effort and
research has been aimed at developing methods for selecting
A laterally loaded deep foundation is the classic example of a appropriate p-y curves. All of the proposed methods are em-
soilstructure interaction problem. The soil or rock reaction pirical and there is no independent test to determine the rel-
depends on the foundation displacement, whereas displace- evant p-y curve. The principal limitations of the p-y method
ment is dependent on the soil or rock response and flexural normally cited are that: (1) theoretically, the interaction of
rigidity of the foundation. In most methods of analysis, the soil or rock between adjacent springs is not taken into ac-
foundation is treated as an elastic beam or elastic beam- count (no continuity), and (2) the p-y curves are not related
column. The primary difference in analytical methods used to directly to any measurable material properties of the soil or
date lies in the approach used to model the soil and/or rock rock mass or of the foundation. Nevertheless, full-scale load
mass response. Methods of analysis fall into two general cate- tests and theory have led to recommendations for estab-
gories: (1) subgrade reaction and (2) elastic continuum theory. lishing p-y curves for a variety of soil types (Reese 1984).
The method is attractive for design purposes because of the
Subgrade reaction methods treat the deep foundation an- following:
alytically as a beam on elastic foundation. The governing dif-
ferential equation (Hetenyi 1946) is given by · Ability to simulate nonlinear behavior of the soil or
rock;
d4y d2y · Ability to follow the subsurface stratigraphy (layering)
EI 4
+ Pz 2 - p - w = 0 (97)
dz dz closely;
· Can account for the nonlinear flexural rigidity (EI) of a
in which EI = flexural rigidity of the deep foundation, y = reinforced-concrete shaft;
lateral deflection of the foundation at a point z along its length, · Incorporates realistic boundary conditions at the top of
Pz = axial load on the foundation, p = lateral soil/rock reac- the foundation;
tion per unit length of foundation, and w = distributed load · Solution provides deflection, slope, shear, and moment
along the length of the shaft (if any). In the most commonly as functions of depth;
used form of subgrade reaction method, the soil reaction- · Solution provides information needed for structural de-
displacement response is represented by a series of indepen- sign (shear and moment); and
dent nonlinear springs described in terms of p-y curves · Computer solutions are readily available.
(Reese 1984). A model showing the concept is provided in
Figure 36. The soil or rock is replaced by a series of discrete Boundary conditions that can be applied at the top of the
mechanisms (nonlinear springs) so that at each depth z the foundation include: (1) degree of fixity against rotation or
soil or rock reaction p is a nonlinear function of lateral de- translation and (2) applied loads (moment, shear, and axial).
flection y. Ideally, each p-y curve would represent the With a given set of boundary conditions and a specified
stressstrain and strength behavior of the soil or rock, effects family of p-y curves, Eq. 97 is solved numerically, typically
of confining stress, foundation diameter and depth, position using a finite-difference scheme. An iterative solution is
of the water table, and any other factors that determine the required to incorporate the nonlinear p-y curves as well as
the nonlinear moment versus EI relationship (material and
geometric nonlinearities) for reinforced-concrete shafts.
The elastic continuum approach for laterally loaded deep
foundations was developed by Poulos (1971), initially for
analysis of a single pile under lateral and moment loading at the
pile head. The numerical solution is based on the boundary
element method, with the pile modeled as a thin elastic strip
and the soil modeled as a homogeneous, isotropic elastic ma-
terial. This approach was used to approximate socketed piles
by Poulos (1972) by considering two boundary conditions at
the tip of the pile: (1) the pile is completely fixed against ro-
tation and displacement at the tip (rock surface) and (2) the
pile is free to rotate but fixed against translation (pinned) at
the tip. The fixed pile tip condition was intended to model a
socketed deep foundation, whereas the pinned tip was in-
tended to model a pile bearing on, but not embedded into,
rock. Although these tip conditions do not adequately model
the behavior of many rock-socketed shafts, the analyses
FIGURE 36 Subgrade reaction model based on p-y curves served to demonstrate some important aspects of socketed
(Reese 1997). deep foundations. For relatively stiff foundations, which

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applies to many drilled shafts, considerable reduction in dis- have been conducted and published to date. This lack of
placement at the pile head can be achieved by socketing, verification can be viewed as a limitation on use of the p-y
especially if the effect of the socket is to approximate a method for rock-socketed drilled shafts. A single study by
"fixed" condition at the soil/rock interface. Reese (1997) presents the only published criteria for selec-
tion of p-y curves in rock. A few state DOTs have developed
The elastic continuum approach was further developed by in-house correlations for p-y curves in rock.
Randolph (1981) through use of the finite-element method
(FEM). Solutions presented by Randolph cover a wide range
of conditions for flexible piles and the results are presented in Reese (1997)
the form of charts as well as convenient closed-form solutions
for a limited range of parameters. The solutions do not ade- Reese proposed interim criteria for p-y curves used for
quately cover the full range of parameters applicable to rock- analysis of drilled shafts in rock. Reese cautions that the
socketed shafts used in practice. Extension of this approach recommendations should be considered as preliminary be-
by Carter and Kulhawy (1992) to rigid shafts and shafts of cause of the meager amount of load test data on which they
intermediate flexibility, as described subsequently, has led to are based. The criteria are summarized as follows. For
practical analytical tools based on the continuum approach. "weak rock," defined as rock with unconfined compressive
strength between 0.5 MPa and 5 MPa, the shape of the p-y
Sun (1994) applied elastic continuum theory to deep foun- curve, as shown in Figure 37, can be described by the
dations using variational calculus to obtain the governing dif- following equations. For the initial linear portion of the
ferential equations of the soil and pile system, based on the curve
Vlasov model for a beam on elastic foundation. This approach
was extended to rock-socketed shafts by Zhang et al. (2000), p = Kiry for y yA (98)
and is also described in this chapter.
For the transitional, nonlinear portion
0.25
pur y for y yA, p pur
p-y Method for Rock Sockets p= (99)
2 yrm
The p-y method of analysis, as implemented in various com- yrm = krmB (100)
puter codes, is the single most widely used method for
design of drilled shafts in rock. Responses to the survey ques- and when the ultimate resistance is reached
tionnaire for this study showed that 28 U.S. state transportation
agencies (of 32 responding) use this method. The analytical p = pur (101)
procedure is dependent on being able to represent the re-
sponse of soil and rock by an appropriate family of p-y where
curves. The only reliable way to verify p-y curves is through Kir = initial slope of the curve,
instrumented full-scale load tests. The approach that forms pur = the rock mass ultimate resistance,
the basis for most of the published recommendations for p-y B = shaft diameter, and
curves in soil is to instrument deep foundations with strain krm is a constant ranging from 0.0005 to 0.00005 that serves
gages to determine the distribution of bending moment over to establish the overall stiffness of the curve.
the length of the foundation during a load test. Assuming that
the bending moment can be determined reliably from strain
gage measurements, the moment as a function of depth can
be differentiated twice to obtain p and integrated twice to
obtain y. Measured displacements at the foundation head
provide a boundary condition at that location. The p-y curves
resulting from analysis of field load tests have then been cor-
related empirically to soil strength and stressstrain proper-
ties determined from laboratory and in situ tests.
An alternative approach for deducing p-y curves from load
tests is to measure the shape of the deformed foundation; for
example, using slope inclinometer measurements and fitting
p-y curves to obtain agreement with the measured displace-
ments. This approach is described by Brown et al. (1994).
Very few lateral load tests on drilled shafts in rock, with
the instrumentation necessary to back-calculate p-y curves, FIGURE 37 Proposed p-y curve for weak rock (Reese 1997).

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The value of yA corresponding to the upper limit of the initial weak rock. The first load test was located at Islamorada,
linear portion of the curve is obtained by setting Eq. 98 equal Florida. A drilled shaft, 1.2 m in diameter and 15.2 m long,
to Eq. 99, yielding was socketed 13.3 m into a brittle, vuggy coral limestone. A
1.333 layer of sand over the rock was retained by a steel casing and
pur lateral load was applied 3.51 m above the rock surface. The
yA = (102)
2 ( yrm ) K ir
0.25
following values were used in the equations for calculating
the p-y curves: qu = 3.45 MPa, r = 1.0, Eir = 7,240 MPa, krm =
The following expression is recommended for calculating 0.0005, B = 1.22 m, L = 15.2 m, and EI = 3.73 × 106 kN-m2.
the rock mass ultimate resistance: Comparison of pile head deflections measured during the
load test and from p-y analyses are shown in Figure 38. With
pur = r qu B 1 + 1.4 r
x a constant value of EI as given above, the analytical results
for 0 xr 3B (103)
B show close agreement with the measured displacements up
pur = 5.2rquB for xr 3B (104) to a lateral load of about 350 kN. By reducing the values of
flexural rigidity in portions of the shaft subject to high mo-
in which qu = uniaxial compressive strength of intact rock, ments, the p-y analysis was adjusted to yield deflections that
r = strength reduction factor, and xr = depth below rock agreed with the measured values at loads higher than 350 kN.
surface. Selection of r is based on the assumption that frac- The value of krm = 0.0005 was also determined on the basis
turing will occur at the surface of the rock under small deflec- of establishing agreement between the measured and pre-
tions, thus reducing the rock mass compressive strength. The dicted displacements.
value of r is assumed to be one-third for RQD of 100 and to
increase linearly to unity at RQD of zero. The underlying as- The second case analyzed by Reese (1997) is a lateral load
sumption is that, if the rock mass is already highly fractured, test conducted on a drilled shaft socketed into sandstone at a
then no additional fracturing with accompanying strength site near San Francisco. The shaft was 2.25 m in diameter
loss will occur. However, this approach appears to have a with a socket length of 13.8 m. Rock mass strength and mod-
fundamental shortcoming in that it relies on the compressive ulus values were estimated from PMT results. Three zones of
strength of the intact rock and not the strength of the rock rock were identified and average values of strength and mod-
mass. For a highly fractured rock mass (low RQD) with a ulus were assigned to each zone. The sandstone is described
high-intact rock strength, it seems that the rock mass strength as "medium-to-fine-grained, well sorted, thinly bedded, very
could be overestimated. intensely to moderately fractured." Twenty values of RQD
were reported, ranging from zero to 80, with an average of
The initial slope of the p-y curve, Kir, is related to the ini- 45. For calculating p-y curves, the strength reduction factor
tial elastic modulus of the rock mass as follows: r was taken as unity, on the assumption that there was "lit-
tle chance of brittle fracture." Values of the other parameters
K ir kir Eir (105)
where Eir = rock mass initial elastic modulus and kir = di-
mensionless constant given by
400 xr
kir = 100 + for 0 xr 3B (106)
3B
kir = 500 for xr > 3B (107)
The expressions for kir were determined by fitting a p-y
analysis to the results of a field load test (back-fitting) in
which the initial rock mass modulus value was determined
from PMTs. The method recommended by Reese (1997) is
to establish Eir from the initial slope of a pressuremeter curve.
Alternatively, Reese suggests the correlation given by Bieni-
awski (1978) between rock mass modulus, modulus of intact
rock core, and RQD, given as expression 2 in Table 12 (chap-
ter two) of this report. According to Reese (1997), rock mass
modulus EM determined this way is assumed to be equivalent
to Eir in Eq. 105.
Results of load tests at two sites are used by Reese (1997) FIGURE 38 Measured and analytical deflection curves for shaft
to fit p-y curves according to the criteria given previously for in vuggy limestone (Reese 1997).

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used for p-y curve development were: krm = 0.00005; qu = range of 0.55 MPa. The user assigns a value to krm. The doc-
1.86 MPa for depth of 03.9 m, 6.45 MPa for depth of umentation (Ensoft, Inc. 2004) recommends to:
3.98.8 m, and 16.0 MPa for depth of more than 8.8 m; Eir =
10qu (MPa) for each layer, B = 2.25 m, and EI = 35.15 x 103 . . . examine the stressstrain curve of the rock sample. Typi-
MN-m2. The value of krm was adjusted to provide agreement cally, the krm is taken as the strain at 50% of the maximum
between displacements given by the p-y method of analysis strength of the core sample. Because limited experimental data
are available for weak rock during the derivation of the p-y
and measured displacements from the load test. criteria, the krm from a particular site may not be in the range
between 0.0005 and 0.00005. For such cases, you may use the
Figure 39 shows a comparison of the measured load- upper bound value (0.0005) to get a larger value of yrm, which in
displacement curve with results produced by the p-y turn will provide a more conservative result.
method of analysis, for various methods of computing the
flexural rigidity (EI) of the test shaft. Methods that account The criteria recommended for p-y curves in the LPILEPLUS
for the nonlinear relationship between bending moment and users manual (Ensoft Inc. 2004) for "strong rock" is illus-
EI provide a better fit than p-y analysis with a constant vale trated in Figure 40. Strong rock is defined by a uniaxial
of EI. The curve labeled "Analytical" in Figure 39 was ob- strength of intact rock qu 6.9 MPa. In Figure 40, su is
tained using an analytical procedure described by Reese to defined as one-half of qu and b is the shaft diameter. The p-y
incorporate the nonlinear momentEI relationships directly curve is bilinear, with the break in slope occurring at a
into the numerical solution of Eq. 97, whereas the curve la- deflection y corresponding to 0.04% of the shaft diameter.
beled "ACI" incorporates recommendations by the Ameri- Resistance (p) is a function of intact rock strength for both
can Concrete Institute for treating the nonlinear momentEI portions of the curve. The criterion does not account explic-
behavior. itly for rock mass properties, which would appear to limit its
applicability to massive rock. The authors recommend veri-
Fitting of p-y curves to the results of the two load tests as fication by load testing if deflections exceed 0.04% of the
described previously forms the basis for recommendations shaft diameter, which would exceed service limit state crite-
that have been incorporated into the most widely used com- ria in most practical situations. Brittle fracture of the rock is
puter programs being used by state DOTs for analysis of lat- assumed if the resistance p becomes greater than the shaft
erally loaded rock-socketed foundations. The program diameter times one-half of the uniaxial compressive strength
COM624 (Wang and Reese 1991) and its commercial version, of the rock. The deflection y corresponding to brittle fracture
LPILE (Ensoft, Inc. 2004), allow the user to assign a limited can be determined from the diagram as 0.0024 times the shaft
number of soil or rock types to each subsurface layer. One of diameter. This level of displacement would be exceeded in
the options is "weak rock." If this geomaterial selection is many practical situations. It is concluded that the recom-
made, additional required input parameters are unit weight, mended criteria applies only for very small lateral deflections
modulus, uniaxial compressive strength, RQD, and krm. The and is not valid for jointed rock masses. Some practitioners
program then generates p-y curves using Eqs. 98107. The apply the weak rock criteria, regardless of material strength,
program documentation recommends assigning "weak rock" to avoid the limitations cited earlier. The authors state that
to geomaterials with uniaxial compressive strengths in the the p-y curve shown in Figure 40 "should be employed with
FIGURE 39 Measured and analytical deflection curves, socket FIGURE 40 Recommended p-y curve for strong rock (Ensoft,
in sandstone (Reese 1997). Inc. 2004).

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caution because of the limited amount of experimental data
and because of the great variability in rock." p ult
The survey questionnaire for this study found that 28
agencies use either COM624 or LPILEPLUS for analysis of
rock-socketed shafts under lateral loading.
p
The following observations are based on a review of the
literature:
· Existing published criteria for p-y curves in rock are
based on a very limited number (two) of full-scale field kh
load tests,
· Recommendations for selecting values of input param-
y
eters required by the published criteria are vague and
unsubstantiated by broad experience, and FIGURE 41 Hyperbolic p-y curve.
· The p-y method of analysis is being used extensively
despite these sources of uncertainty.
It is therefore concluded that research is needed and should be model. Tests were performed on shafts in Piedmont weath-
undertaken with the objective of developing improved criteria ered profiles of sandstone, mica schist, and crystalline rock.
for p-y curves in rock. The research should include full-scale Finite-element modeling was used to calibrate a p-y curve
field load tests on instrumented shafts, much in the same way model incorporating subgrade modulus as determined from
that earlier studies focused on the same purpose for deep foun- PMT readings and providing close agreement with strains
dations in soil. The p-y curve parameters should be related to and deflections measured in the load tests. The model was
rock mass engineering properties that can be determined by then used to make forward predictions of lateral load re-
state transportation agencies using available site and material sponse for subsequent load tests on socketed shafts at two
characterization methods, as described in chapter two. locations in weathered rock profiles different than those used
to develop the model.
The procedure for establishing values of subgrade modulus
Current Research by State Agencies
Kh involves determination of the rock mass modulus (EM) from
The literature review and the survey identified two state trans- PMT measurements. The coefficient of subgrade reaction is
portation agencies (North Carolina and Ohio) with research in then given by:
progress aimed at improving the methodology for construct- 1
0.65E M E M B 4 12
F
ing p-y curves for weathered rock. The North Carolina study kh = units : 3 (109)
is described in a draft report by Gabr et al. (2002) and the B (1 - vr2 ) Es I s
l
Ohio study is summarized in a paper that was under review
at the time of this writing, by Liang and Yang (2006). Both in which B = shaft diameter, EM = rock mass modulus, vr =
studies present recommendations for p-y curves based on a Poisson's ratio of the rock, and Es and Is are modulus and mo-
hyperbolic function. Two parameters are required to charac- ment of inertia of the shaft, respectively. A procedure is
terize a hyperbola, the initial tangent slope and the asymptote. given by Gabr et al. (2002) for establishing the point of rota-
For the proposed hyperbolic p-y models, these correspond to tion of the shaft. For p-y curves above the point of rotation,
the subgrade modulus (Kh) and the ultimate resistance (pult), subgrade modulus is equal to the coefficient of subgrade re-
as shown in Figure 41. The hyperbolic p-y relationship is then action times the shaft diameter or
given as
Kh = kh B (110)
y
p=
1 y (108) For depths below the point of rotation, a stiffer lateral sub-
+
K h pult grade reaction is assigned and the reader is referred to Gabr
et al. (2002) for the equations. An alternative procedure is
A summary of the two studies, including recommenda- presented for cases where rock mass modulus is determined
tions for selection of the required parameters (Kh and pult), using the empirical correlation given by Hoek and Brown
is presented. (1997) and presented previously as expression 7 in Table 12
of chapter two. In that expression, rock mass modulus is cor-
In the North Carolina study, results of six full-scale field related with GSI and uniaxial compressive strength of intact
load tests, at three different sites, were used to develop the rock (qu).

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The second required hyperbolic curve parameter is the ior of the concrete shaft, which reduces the predicted deflec-
asymptote of the p-y curve, which is the ultimate resistance tions more significantly than the p-y criteria. One of the lim-
pult. The proposed expression is given by itations of the p-y criterion proposed by Gabr et al. (2002) is
that it is based on analyses in which EI is taken as a constant.
pult = (pL + max)B (111) For proper analysis of soilrockstructure interaction during
lateral loading, the nonlinear momentEI relationship should
where pL = limit normal stress and max = shearing resistance be modeled correctly.
along the side of the shaft. Gabr et al. adopted the following
recommendation of Zhang et al. (2000) for unit side resistance: The North Carolina DOT also reports using the program
LTBASE, which analyzes the lateral load-displacement
max = 0.20 qu ( MPa) (112) response of deep foundations as described by Gabr and
Borden (1988). The analysis is based on the p-y method, but
The limit normal stress is estimated on the basis of Hoek also accounts for base resistance by including a vertical
Brown strength parameters as determined through correla- resistance component mobilized by shaft rotation and hori-
tions with RMR and GSI, and is given by zontal shear resistance, as illustrated in Figure 43. Base re-
a
sistance becomes significant as the relative rigidity of the
'z shaft increases and as the slenderness ratio decreases. For
pL = 'z + qu mb + s (113)
qu relatively rigid rock sockets, mobilization of vertical and
shear resistance at the tip could increase overall lateral ca-
in which ' = effective unit weight of the rock mass, z = depth pacity significantly, and base resistance effects should be
from the rock mass surface, and the coefficients mb, s, and a considered. Gabr et al. (2002) stated that the hyperbolic WR
are the HoekBrown coefficients given by Eqs. 1215 in p-y curve model is now incorporated into LTBASE, but no
chapter two. results were given.
Results of one of the field load tests conducted for the pur- In the Ohio DOT study, Liang and Yang (2006) also pro-
pose of evaluating the predictive capability of the proposed pose a hyperbolic p-y curve criterion. The derivation is based
weak rock (WR) model is shown in Figure 42. The analyses on theoretical considerations and finite-element analyses.
were carried out using the program LPILE. Analyses were also Results of two full-scale, fully instrumented field load tests
conducted using p-y curves as proposed by Reese (1997), de- are compared with predictions based on the proposed p-y
scribed previously, as well as several other p-y curve recom- curve criterion. The initial slope of the hyperbolic p-y curve
mendations. The proposed model based on hyperbolic p-y is given by the following semi-empirical equation:
curves derived from PMT measurements (labeled dilatometer 0.284
in Figure 42) shows good agreement with the test results. The B -2 v Es I s
K h = EM e r (114)
authors (Gabr et al. 2002) attributed the underpredicted dis- Bref EM B4
placements obtained using the Reese criteria to the large values
of the factor kir predicted by Eqs. 106 and 107. However, the
analysis did not incorporate the nonlinear momentEI behav-
FIGURE 42 Measured lateral load deflection versus predicted FIGURE 43 Base deformation as a function of shaft rotation
(Gabr et al. 2002). (Gabr and Borden 1988).

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61
in which Bref = a reference diameter of 0.305 m (1 ft) and all analysis of full-scale field load tests on instrumented shafts.
other terms are as defined above for Eq. 109. Liang and Yang However, the proposed equations for generating p-y curves
(2006) recommend modulus values EM from PMTs for use in differ between the two proposed criteria and both models will
Eq. 114, but in the absence of PMT measurements they pre- result in different load-displacement curves. It is not clear if
sent the following correlation equation relating EM to modu- either model is applicable to rock sockets other than those
lus of intact rock and GSI: used in its development. Both sets of load tests add to the
database of documented load tests now available to re-
GSI
Er 21 searchers. A useful exercise would be to evaluate the North
EM = e .7 (115) Carolina proposed criteria against the Ohio load test results
100
and vice versa.
where Er = elastic modulus of intact rock obtained during
uniaxal compression testing of core samples. Eq. 115 is also Florida Pier
expression 8 of Table 12 in chapter two. Liang and Yang
(2006) present two equations for evaluating pult. The first cor- Several states reported using other computer programs that
responds to a wedge failure mode, which applies to rock are based on the p-y method of analysis. Seven agencies re-
mass near the ground surface. The second applies to rock port using the Florida Bridge Pier Analysis Program
mass at depth and is given by (FBPIER) for analysis of rock-socketed shafts. Of those
seven, six also report using COM624 and/or LPILE. The
2 FBPIER, described by Hoit et al. (1997), is a nonlinear, finite-
pult = pL + max - pA B (116)
4 3 element analysis program designed for analyzing bridge pier
substructures composed of pier columns and a pier cap sup-
where pL = limit normal stress, max = shearing resistance along ported on a pile cap and piles or shafts including the soil (or
the side of the shaft, and pA = horizontal active pressure. rock). FBPIER was developed to provide an analytical tool
Eq. 116 is similar to Eq. 111 (Gabr et al. 2002), but accounts allowing the entire pier structure of a bridge to be analyzed
for active earth pressure acting on the shaft. Both methods in- at one time, instead of multiple iterations between foundation
corporate the HoekBrown strength criterion for rock mass to analysis programs (e.g., COM624) and structural analysis
evaluate the limit normal stress pL, and both rely on correla- programs. Basically, the structural elements (pier column,
tions with GSI to determine the required HoekBrown cap, pile cap, and piles) are modeled using standard structural
strength parameters. In the Liang and Yang (2006) approach, finite-element analysis, including nonlinear capabilities
pult at each depth is taken as the smaller of the two values (nonlinear MEI behavior), whereas the soil response is
obtained from the wedge analysis or by Eq. 116. modeled by nonlinear springs (Figure 44). Axial soil re-
sponse is modeled in terms of t-z curves, whereas lateral
A source of uncertainty in all of the proposed p-y criteria response is modeled in terms of p-y curves. The program has
derives from the choice of method for selecting rock mass built-in criteria for p-y curves in soil, based on published
modulus when more than one option is available. For exam- recommendations and essentially similar to those employed
ple, using the pressuremeter and GSI data reported by Gabr in LPILE. User-defined p-y curves can also be specified. To
et al. (2002) significantly different values of modulus are ob- simulate rock, users currently apply the criteria for either soft
tained for the same site. In some cases, the measured shaft clay (Matlock 1970) or stiff clay (Reese and Welch 1975)
load-displacement response (from load testing) shows better but with strength and stiffness properties of the rock, or user-
agreement with p-y curves developed from PMT modulus, defined curves are input. Research is underway to incorpo-
whereas another load test shows better agreement with p-y rate improved p-y curve criteria into FBPIER, specifically for
curves developed from GSI-derived modulus. Proper selec-
tion of rock mass modulus for foundation design is one of
the challenges for design of rock-socketed shafts, as pointed
out in chapter two. This issue becomes most important when
p-y curves for lateral load analysis are based on rock mass
modulus. Both the Reese (1997) criteria and the hyperbolic
criteria require rock mass modulus to determine the slope of
p-y curves.
The North Carolina and Ohio programs provide examples
of state DOT efforts to advance the state of practice in design
of rock-socketed foundations. The programs incorporate
careful site investigations using available methods for char-
acterizing rock mass engineering properties (RMR, GSI) as FIGURE 44 Florida pier model for structure and foundation
well as in situ testing (PMT). Both programs are based on elements (Hoit et al. 1997).

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Florida limestone, as described by McVay and Niraula (2004). Continuum Models for Laterally Loaded Sockets
Centrifuge tests were conducted in which instrumented model
shafts embedded in a synthetic rock (to simulate Florida Carter and Kulhawy (1992)
limestone) were subjected to lateral loading. Strain gage
measurements were used to back-calculate p-y curves, Carter and Kulhawy (1988, 1992) studied the behavior of flex-
which are presented in normalized form, with p normalized ible and rigid shafts socketed into rock and subjected to lateral
by shaft diameter and rock compressive strength (p/Bqu) and loads and moments. Solutions for the load-displacement rela-
y normalized by shaft diameter (y/B). There is no analytical tions were first generated using finite-element analyses. The
expression recommended for new p-y curve criteria and the finite-element analyses followed the approach of Randolph
report recommends that field testing be undertaken on full- (1981) for flexible piles under lateral loading. Based on the
size drilled shafts to validate the derived p-y curves estab- FEM solutions, approximate closed-form equations were
lished from the centrifuge tests before they are employed in developed to describe the response for a range of rock-socket
practice. parameters typically encountered in practice. The results pro-
vide a first-order approximation of horizontal groundline dis-
placements and rotations and can incorporate an overlying soil
layer. The method is summarized as follows.
Strain Wedge Model
The strain wedge (SW) model has been applied to laterally Initially, consider the case where the top of the shaft cor-
loaded piles in soil, as described by Ashour et al. (1998). The responds to the top of the rock layer (Figure 45). The shaft is
2006 Interim AASHTO LRFD Bridge Design Specifications idealized as a cylindrical elastic inclusion with an effective
identify the SW model as an acceptable method for lateral Young's modulus (Ee), Poisson's ratio (vc), depth (D), and di-
load analysis of deep foundations. The 3-D soilpile interac- ameter (B), subjected to a known lateral force (H), and an
tion behavior is modeled by considering the lateral resistance overturning moment (M). For a reinforced-concrete shaft
that develops in front of a mobilized passive wedge of soil at having an actual flexural rigidity equal to (EI)c, the effective
each depth. Based on the soil stressstrain and strength prop- Young's modulus is given by
erties, as determined from laboratory triaxial tests, the hori-
zontal soil strain () in the developing passive wedge in front ( EI )c
Ee =
of the pile is related to the deflection pattern (y) versus depth. B 4 (117)
The horizontal stress change (H) in the developing passive 64
wedge is related to the soilpile reaction (p), and the nonlin-
ear soil modulus is related to the nonlinear modulus of sub- It is assumed that the elastic shaft is embedded in a ho-
grade reaction, which is the slope of the p-y curve. The SW mogeneous, isotropic elastic rock mass, with properties Er
model can be used to develop p-y curves for soil that show and vr. Effects of variations in the Poisson's ratio of the rock
good agreement with load test results (Ashour and Norris mass (vr), are represented approximately by an equivalent
2000). Theoretically, the SW model overcomes some of the shear modulus of the rock mass (G*), defined as:
limitations of strictly empirically derived p-y curves because
the soil reaction (p) at any given depth depends on the re- 3v
G = Gr 1 + r (118)
sponse of the neighboring soil layers (continuity) and prop- 4
erties of the pile (shape, stiffness, and head conditions).
Ashour et al. (2001) proposed new criteria for p-y curves in in which Gr = shear modulus of the elastic rock mass. For an
weathered rock for use with the SW model. The criteria are isotropic rock mass, the shear modulus is related to Er and vr by
described by the authors as being based on the weak rock
criteria of Reese (1997) as given by Eqs. 98-104, but modi-
fied to account for the nonlinear rock mass modulus and the
strength of the rock mass in terms of MohrCoulomb
strength parameters c and . Ashour et al. (2001) reported
good agreement between the SW analysis and a field load
test reported by Brown (1994). One state DOT (Washington)
reports using the computer program (S-Shaft) based on the
SW model that incorporates the p-y curve criteria for weath-
ered rock. However, the program has not yet been used for
design of a socketed shaft (J. Cuthbertson, personal com-
munication, Sep. 30, 2005). The SW model and proposed
p-y criteria of Ashour et al. (2001) warrant further consider-
ation and should be evaluated against additional field load
test results (e.g., the tests reported by Gabr et al. 2002 and FIGURE 45 Lateral loading of rock-socketed shaft (Carter and
Liang and Yang 2006). Kulhawy 1992).

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63
Er The accuracy of Eqs. 125 and 126 has been verified for the
Gr = (119)
2 (1 + vr ) following ranges of parameters: 1 D/B 10 and Ee/Er 1.
Based on a parametric study using finite-element analysis, Shafts can be described as having intermediate stiffness
it was found that closed-form expressions could be obtained whenever the slenderness ratio is bounded approximately as
to provide reasonably accurate predictions of horizontal dis- follows:
placement (u) and rotation () at the head of the shaft for two 1 2
E 2 D E 7
limiting cases. The two cases correspond to flexible shafts and 0.05 e < < e (127)
rigid shafts. The criterion for a flexible shaft is G B G
2/7
D Ee For the intermediate case, Carter and Kulhawy suggested
(120)
B G that the displacements be taken as 1.25 times the maximum
of either (1) the predicted displacement of a rigid shaft with
the same slenderness ratio (D/B) as the actual shaft or (2) the
For shafts satisfying Eq. 120, the response depends only
predicted displacement of a flexible shaft with the same mod-
on the modulus ratio (Ee/G*) and Poisson's ratio of the rock
ulus ratio (Ee/G*) as the actual shaft. Values calculated in this
mass (vr) and is effectively independent of D/B. The follow-
way should, in most cases, be slightly larger than those given
ing closed-form expressions, suggested by Randolph (1981),
by the more rigorous finite-element analysis for a shaft of in-
provide accurate approximations for the deformations of termediate stiffness.
flexible shafts:
-1 -3 Carter and Kulhawy next considered a layer of soil of
H E 7
M E 7
thickness Ds overlying rock, as shown in Figure 46. The
u = 0.50 e + 1.08 2 e (121)
G B G G B G analysis is approached by structural decomposition of the
-3 -5
shaft and its loading, as shown in Figure 46b. It was assumed
H E 7
M E 7
that the magnitude of applied lateral loading is sufficient to
= 1.08 2 e + 6.40 3 e (122)
G B G G B G cause yielding within the soil from the ground surface to the
top of the rock mass. The portion of the shaft within the soil
in which u = groundline deflection and = groundline rota- is then analyzed as a determinant beam subjected to known
tion of the shaft. loading. The displacement and rotation of point A relative to
point O can be determined by established techniques of struc-
Carter and Kulhawy (1992) reported that the accuracy of tural analysis. The horizontal shear force (Ho) and bending
the above equations is verified for the following ranges of pa- moment (Mo) acting in the shaft at the rock surface level can
rameters: 1 Ee/Er 106 and D/B 1. be computed from statics, and the displacement and rotation
at this level can be computed by the methods described pre-
The criterion for a rigid shaft is viously. The overall groundline displacements can then be
calculated by superposition of the appropriate parts.
1
D E 2
0.05 e (123)
B G Determination of the limiting soil reactions is recom-
mended for the two limiting cases of cohesive soil in
and undrained loading ( = 0) and frictional soil (c = 0) in drained
loading. Ultimate resistance for shafts in cohesive soils is
Ee based on the method of Broms (1964a), in which the undrained
G 100
(124)
( )
2
B
2D
When Eqs. 123 and 124 are satisfied, the displacements of
the shaft will be independent of the modulus ratio (Ee/Er) and
will depend only on the slenderness ratio (D/B) and Poisson's
ratio of the rock mass (vr). The following closed-form expres-
sions give reasonably accurate displacements for rigid shafts:
1 7
2D - 2D -
u = 0.4 + 0.3 2
H 3 M 8
(125)
G B B G B B
-7 -5
H 2 D 8 M 2 D 3
= 0.3 2 + 0.8 3 (126) FIGURE 46 Rock-socketed shaft with overlying
G B B G B B layer (Carter and Kulhawy 1992).

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64
soil resistance ranges from zero at the ground surface to a soil and/or rock mass behaves linearly elastically at small
depth of 1.5B and has a constant value of 9su below this depth, strain levels and yields when the soil and/or rock mass reac-
where su = soil undrained shear strength. For socketed shafts tion force p (force/length) exceeds the ultimate resistance pult
extending through a cohesionless soil layer, the following lim- (force/length).
iting pressure suggested by Broms (1964b) is assumed:
Analysis of the loaded shaft as an elastic continuum is ac-
pu = 3K p 'v (128) complished using the method developed by Sun (1994). The
numerical solution is by a finite-difference scheme and in-
1 + sin '
Kp = (129) corporates the linear variation in soil modulus and linear
1 - sin ' variation in rock mass modulus above the base of the shaft.
Solutions obtained for purely elastic responses are compared
in which v ' = vertical effective stress and ' = effective with those of Poulos (1971) and finite-element solutions by
stress friction angle of the soil. For both cases (undrained and Verruijt and Kooijman (1989) and Randolph (1981). Rea-
drained) solutions are given by Carter and Kulhawy (1992) sonable agreement with those published solutions is offered
for the displacement, rotation, shear, and moment at point O as verification of the theory, for elastic response.
of Figure 46. The contribution to groundline displacement
and rotation from the loading transmitted to the rock mass The method is extended to nonlinear response by account-
(Ho and Mo) is determined based on Eqs. 121 and 122 or ing for local yielding of the soil and rock mass. The soil and
Eqs. 125 and 126 and added to the calculated displacement rock mass are modeled as elastic, perfectly plastic materials,
and rotation at the top of the socket to determine overall and the analysis consists of the following steps:
groundline response.
1. For the applied lateral load H and moment M, the shaft
Application of the proposed theory is described by Carter
is analyzed by assuming the soil and rock mass are
and Kulhawy (1992) through back-analysis of a single case in-
elastic, and the lateral reaction force p of the soil and
volving field loading of a pair of rock-socketed shafts. The
rock mass along the shaft is determined by solution of
method has not been evaluated against a sufficient database
the governing differential equation and boundary con-
of field performance, and further research is needed to assess
ditions at the head of the shaft.
its reliability. The analysis was developed primarily for ap-
plication to electrical transmission line foundations in rock, 2. The computed lateral reaction force p is compared
although the concepts are not limited to foundations support- with the ultimate resistance pult. If p > pult, the depth of
ing a specific type of structure. The approach is attractive for yield zy in the soil and/or rock mass is determined.
design purposes, because the closed-form equations can be 3. The portion of the shaft in the unyielded soil and/or
executed by hand or on a spreadsheet. rock mass (zy z L) is considered to be a new shaft
and analyzed by ignoring the effect of the soil and/or
Carter and Kulhawy (1992) stated that the assumption of rock mass above the level z = zy. The lateral load and
yield everywhere in the soil layer may represent an oversim- moment at the new shaft head are given by:
plification, but that the resulting predictions of groundline zy
displacements will overestimate the true displacements, giv- Ho = H - pult dz (130)
0
ing a conservative approximation. However, the assumption
pult ( z y - z ) dz
zy
that the limit soil reaction is always fully mobilized may lead M o = M + Hz y - (131)
0
to erroneous results by overestimating the load carried by the
soil and thus underestimating the load transmitted to the 4. Steps 2 and 3 are repeated and the iteration is continued
socket. Furthermore, groundline displacements may be un- until no further yielding of soil or rock mass occurs.
derestimated because actual soil resistance may be smaller 5. The final results are obtained by decomposition of
than the limiting values assumed in the analysis. the shaft into two parts, which are analyzed sepa-
rately, as illustrated previously in Figure 46. The sec-
tion of the shaft in the zone of yielded soil and/or
Zhang et al. (2000) rock mass is analyzed as a beam subjected to a dis-
tributed load of magnitude pult. The length of shaft in
Zhang et al. (2000) extended the continuum approach to pre- the unyielded zone of soil and/or rock mass is ana-
dict the nonlinear lateral load-displacement response of rock- lyzed as a shaft with the soil and/or rock mass behaving
socketed shafts. The method considers subsurface profiles elastically.
consisting of a soil layer overlying a rock layer. The defor-
mation modulus of the soil is assumed to vary linearly with Ultimate resistance developed in the overlying soil layer
depth, whereas the deformation modulus of the rock mass is is evaluated for the two conditions of undrained loading
assumed to vary linearly with depth and then to stay constant ( = 0) and fully drained loading (c = 0). For fine-grained
below the shaft tip. Effects of soil and/or rock mass yielding soils (clay), undrained loading conditions are assumed and
on response of the shaft are considered by assuming that the the limit pressure is given by

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65
Discussion of Analytical Models for Laterally
pult = N p cu B (132) Loaded Sockets
' J
Np = 3 + z+ z9 (133) Each of the analytical methods described above has advan-
cu 2R
tages and disadvantages for use in the design of rock-socketed
shafts for highway bridge structures. The greatest need for
in which cu = undrained shear strength, B = shaft diameter, ' = further development of all available methods is a more thor-
average effective unit weight of soil above depth z, J = a coef- ough database of load test results against which existing
ficient ranging from 0.25 to 0.5, and R = shaft radius. For theory can be evaluated, modified, and calibrated.
shafts in sand, a method attributed to Fleming et al. (1992)
is given as follows: The simple closed-form expressions given by Carter and
Kulhawy (1992) represent a convenient, first-order approxi-
pult = K p
2
'zB (134) mation of displacements and rotations of rock-socketed
shafts. Advantages include the following:
where Kp = Rankine coefficient of passive earth pressure de-
fined by Eq. 129. Ultimate resistance of the rock mass is · Predicts lateral displacements under working load
given by conditions,
· Requires a single material parameter (rock mass
pult = ( pL + max ) B (135) modulus),
· Provides reasonable agreement with theoretically
where max = maximum shearing resistance along the sides of rigorous finite-element analysis, and
the shaft (e.g., Eq. 30 of chapter three) and pL = normal limit · Is the easiest method to apply by practicing design
resistance. The limit normal stress pL is evaluated using the engineers.
HoekBrown strength criterion with the strength parameters
determined on the basis of correlations to GSI. The resulting Limitations include:
expression was given previously as Eq. 113.
· Does not predict the complete lateral load-displacement
According to Zhang et al. (2000), a computer program was
curve,
written to execute this procedure. Predictions using the pro-
· Elastic solution does not provide shear and moment dis-
posed method are compared with results of field load tests re-
tribution for structural design,
ported by Frantzen and Stratten (1987) for shafts socketed into
· Does not account for more than one rock mass layer,
sandy shale and sandstone. Computed pile head deflections
· Does not account directly for nonlinear MEI behavior
show reasonable agreement with the load test results. The
of reinforced-concrete shaft, and
method appears to have potential as a useful tool for founda-
· Does not account for interaction between axial and lateral
tions designers. Availability of the computer program is un-
loading and its effects on structural behavior of the shaft.
known. Programming the method using a finite-difference
scheme as described by Zhang et al. (2000) is also possible.
The method can be best used for preliminary design; for
example, establishing the initial trial depth and diameter of
Discontinuum Models rock-socketed shafts under lateral and moment loading. For
some situations, no further analysis may be necessary. Final
A potential mode of failure for a laterally loaded shaft in rock design should be verified by field load testing.
is by shear failure along joint surfaces. To et al. (2003) pro-
posed a method to evaluate the ultimate lateral-load capacity The method of Zhang et al. (2000) provides a more rigorous
of shafts in rock masses with two or three sets of intersecting continuum-based analysis than that of Carter and Kulhawy.
joints. The analysis consists of two parts. In the first part, the The tradeoff is that more material parameters are required as
block theory of Goodman and Shi (1985) is used to deter- input. Variation of rock mass modulus with depth is required.
mine if possible combinations of removable blocks exist that To fully utilize the nonlinear capabilities, the HoekBrown
would represent a kinematically feasible mode of failure. yield criterion parameters are required, and these are based on
In the second part, the stability of potentially removable establishing the RMR and/or GSI. The method is best applied
combinations of blocks or wedges is analyzed by limit equi- when a more refined analysis is required and the agency is will-
librium. Both steps in the analysis require careful evaluation ing to invest in proper determination of the required material
of the joint sets, in terms of their geometry and strength prop- properties. Advantages include:
erties. Although the method is based on some idealized
assumptions, such as equal joint spacing, and it has not been · Predicts the full, nonlinear, lateral load-deformation re-
evaluated against field or laboratory load tests, it provides a sponse;
theoretically based discontinuum analysis of stability in · Accounts for partial yield in either the rock mass or the
cases where this mode of failure requires evaluation. overlying soil (more realistic);