draft...weak rocks. these investigators the measured related peak side resistance (f sp) of rock...

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Evaluation of the peak side resistance for rock socketed shafts in weak sedimentary rock from an extensive

database of published field load tests: a limit state approach

Journal: Canadian Geotechnical Journal

Manuscript ID cgj-2018-0590.R1

Manuscript Type: Article

Date Submitted by the Author: 02-Oct-2018

Complete List of Authors: Asem, Pouyan; University of Illinois at Urbana-Champaign, Department of Civil and Environmental EngineeringGardoni, Paolo; University of Illinois at Urbana-Champaign

Keyword: Peak side resistance, weak rock mass, load test database, resistance factor, rock socket

Is the invited manuscript for consideration in a Special

Issue? :Not applicable (regular submission)

https://mc06.manuscriptcentral.com/cgj-pubs

Canadian Geotechnical Journal

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Evaluation of the peak side resistance for rock socketed shafts in weak sedimentary rock

from an extensive database of published field load tests: a limit state approach

Pouyan Asem, Ph.D., A.M. ASCE, M. JGS Adjunct lecturer in Civil and Environmental Engineering

University of Illinois at Urbana-Champaign 205 N. Mathews Ave.

Urbana, IL 61801 [email protected]

and

Paolo Gardoni, Ph.D.

Professor of Civil and Environmental Engineering University of Illinois at Urbana-Champaign

205 N. Mathews Ave. Urbana, IL 61801

[email protected]

Corresponding author: Pouyan Asem

January 19, 2019

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ABSTRACT

This paper presents the analyses of the measured peak side resistance of rock sockets constructed

in weak claystone, shale, limestone, siltstone, and sandstone. The peak side resistance is obtained

from the in situ axial load tests on drilled shafts, anchors, and plugs. The parameters that affect

the development of the peak side resistance are determined using the in situ load test data. It is

found that the peak side resistance increases with the unconfined compressive strength and the

deformation modulus of the weak rock, and decreases with increase in the length of the shear

surface along the rock socket sidewalls. The increase in the socket diameter also slightly

decreases the peak side resistance. Additionally, it is found that the initial normal stresses do not

significantly affect the measured peak side resistance in the in situ load tests. The in situ load test

data are used to develop an empirical design equation for the determination of the peak side

resistance. The proposed model for the peak side resistance, and the reliability analysis are used

to determine the corresponding resistance factors for use in the load and resistance factor design

framework for the assessment of the strength limit state.

KEYWORDS

Peak side resistance, weak rock mass, load test database, resistance factor, rock socket.

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INTRODUCTION

The near-surface soils usually have low shear strength, are compressible, and usually do not offer

the required bearing capacity to carry the loads of the heavy structures. In such conditions, a

shallow footing where the ratio of the embedment depth (measured from the ground surface DGS)

to the foundation diameter (B) is less than 2 (Terzaghi 1943) may not be a feasible foundation

solution. Accordingly, a deep foundation (DGS/B > 4, after Peck et al. 1974; Coduto 2001) is

typically used to transfer the load of the heavy structures to the more competent strata, which are

often found at greater depths. Among the available deep foundation options, drilled shafts are

frequently used, and are typically extended through the upper incompetent and compressible

residual soils into the underlying weak rocks (Rosenberg and Journeaux 1976; Horvath and

Kenney 1979; Rowe and Armitage 1987; Kulhawy and Phoon 1993; Hassan and O’Neill 1997;

Seidel and Collingwood 2001; Vu 2013; Asem 2018) to support the heavy loads. Weak rocks are

commonly encountered in the zone between the near-surface residual soils, and the more

competent and unweathered bedrock at greater depths. Socketing drilled shafts into weak rocks

has been used more frequently in the recent years (Horvath and Kenney 1979; Horvath and Chae

1989; Hassan and O’Neill 1997) because weak rocks often provide the necessary bearing layer to

support the heavy loads, and at the same time, result in more economical and effective

construction compared to unweathered rocks found at greater depths.

The base resistance, the side resistance, or a combination of both contribute to the overall axial

resistance of the drilled shafts that are embedded (socketed) in weak rock formations (Horvath

and Kenney 1979; Zhang and Einstein 1998; Lo and Hefny 2001; Seidel and Collingwood 2001).

The in situ load test measurements, and the analyses based on the theory of elasticity, however,

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show that a substantial portion of the axial service loads on the rock sockets is carried using the

side resistance (Osterberg and Gill 1973; Horvath and Kenney 1979; Goodman 1980) because

the side resistance is mobilized at significantly smaller axial displacements compared to the base

resistance (Carter and Kulhawy 1988; Seidel and Collingwood 2001). Therefore, when designing

a rock socketed drilled shaft in weak rock, it is necessary to accurately determine the load

transfer to the rock socket sidewalls. The maximum shear stress mobilized along the rock socket

sidewalls is represented by the peak side resistance (fsp). The peak side resistance (fsp) may be

measured using the in situ axial load tests (Rosenberg and Journeaux 1976; Horvath and Kenney

1979; Williams 1980; Rowe and Armitage 1987; Abu-Hejleh et al. 2003; Miller 2003; Kulhawy

et al. 2005; Vu 2013). An in situ load test, however, is costly and time consuming. Therefore,

predictive models, which are commonly developed based on the results of in situ load tests,

laboratory interface tests, or numerical models, have been used to estimate the peak side

resistance (fsp) (Horvath and Kenney 1979; Rowe and Armitage 1987; Kulhawy and Phoon

1993; Hassan and O’Neill 1997; Seidel and Collingwood 2001; Vu 2013).

The work reported by Williams (1980), Williams and Pells (1981), and Seidel and Collingwood

(2001) showed that in addition to the unconfined compressive strength (qu), parameters such as

the rock mass deformation modulus (Em) may also be needed to obtain a more accurate estimate

of the values of peak side resistance (fsp) for rock sockets embedded in weak rocks. In the

following sections, we first develop a database consisting of axial load tests on drilled shafts,

anchors, and plugs that were constructed in weak rocks. In the drilled shaft and some of the plug

load tests, the components of side and base resistances were separated using strain gauges that

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were placed along the rock socket sidewalls, or using a load cell at the base of the drilled shafts

or the plugs. Occasionally, a compressible base or a void was created near the base of the drilled

shafts or plugs to eliminate the base resistance to directly measure the side resistance. In the

anchor load tests, no component of the base resistance was present, and thus the side resistance

was measured directly. The load testing methods will be discussed in more detail in the

subsequent sections of the paper. These load test data are then used to investigate the parameters

that affect the peak side resistance (fsp) in the rock sockets that are constructed in weak rocks.

Next, the method of Gardoni et al. (2002), and the in situ load test data are used to develop a

predictive model for the peak side resistance (fsp) of the rock sockets in weak rocks. A reliability

analysis is then used to determine the corresponding resistance factors (φ) for use in the Load

and Resistance Factor Design (LRFD) framework.

DEFINITION OF THE WEAK ROCK

The stress relief in the rock masses that follows the removal of the overburden geomaterials by

the glacial movements and other similar geologic events causes the opening of the structural

discontinuities such as joints and fissures that may be originally closed (Terzaghi 1936), and

subsequently results in increase of the water content (w) of the rock materials exposed on the

face of open joints and fissures by absorbing water (Terzaghi 1936; Skempton 1948; Skempton

and Hutchinson 1969; Mesri and Shahien 2003). The increase in the water content (w) leads to

the swelling of the rock materials under zero pressure along the exposed walls of the open cracks

(Terzaghi 1936), and it results in the softening and deterioration of the rock blocks that are

separated by the said structural discontinuities. The resulting softening causes additional

movements in the rock formation, and more fissures and joints will open (Terzaghi 1936;

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Skempton 1948), which further reduces the shear strength and increases the compressibility of

the rock mass.

It follows from the discussion above that a weak rock mass is an assemblage of the relatively

weathered and softened rock blocks, which are separated by the individual structural

discontinuities (Hoek 1983; Singh and Rao 2005). The weathered rock block refers to the

“unfractured blocks which occur between structural discontinuities in a typical rock mass” (Hoek

1983). The structural discontinuities include “fold structures, joints, faults, shears and

slickensided surfaces” (Hoek 1983; Goodman 1980; Goodman 1993).

Because a weak rock mass consists of relatively weathered and softened rock blocks and

structural discontinuities that are also subjected to the weathering and softening effect, and both

components affect the response of the weak rock to external loads, the definition of weak rock

should reflect how weathering and softening affects the rock blocks and the discontinuity

surfaces. Accordingly, the values of qu that are obtained from the representative samples are

used to characterize the effect of the mineralogy and the weathering state of the rock blocks, and

the Geological Strength Index (GSI) is used to reflect the blockiness of the rock mass, and the

degree of the alteration of the natural structural discontinuities such as joints and fissures. The

review of the technical literature suggests that for weak rocks, the values of the unconfined

compressive strength (qu) ranges from 0.5 MPa to slightly greater than 30 MPa (Deere and

Miller 1966; Barton et al. 1978; Rowe and Armitage 1987; Cepeda-Diaz 1987; Kanji 2014), and

the geological strength index (GSI) is commonly less than 70 (Hoek and Brown 1997).

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METHODS FOR PREDICTING PEAK SIDE RESISTANCE IN ROCK SOCKETS

Methods based on in situ load tests

Rosenberg and Journeaux (1976), Horvath and Kenney (1979), Williams (1980), Rowe and

Armitage (1984; 1987), Carter and Kulhawy (1988), Abu-Hejleh et al. (2003), Miller (2003),

Kulhawy et al. (2005), Vu (2013), and the Canadian Foundation Engineering Manual (2006)

(which recommend the methods of Horvath et al. 1983; Rowe and Armitage 1984; Carter and

Kulhawy 1988) introduced predictive models for the peak side resistance (fsp) of rock sockets in

weak rocks. These investigators related the measured peak side resistance (fsp) of rock socket

sidewalls in the in situ axial load tests on drilled shafts, anchors, and plugs to the unconfined

compressive strength (qu) of the rock. These models use the following general mathematical

form

(1)

where α and n are obtained by fitting Equation (1) to the load test data. Table 1 summarizes the

empirical parameters α and n for some of the existing models (α and n in Table 1 are obtained

from O’Neill et al. 1996; Seidel and Collingwood 2001; Canadian Foundation Engineering

Manual 2006; Asem 2018). The following observations are noteworthy in relation to the models

based on the in situ load tests

1. Some of the existing models were developed based on a limited number of sites with

similar rock formations and rock socket properties, which covered different ranges of

rock weathering conditions. As a result, such models may be considered as site- or rock

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type-specific. Examples include methods of Williams (1980) that used the data from six

sites (i.e., Stanley Avenue, Middleborough Road, Westgate Freeway, Eastern Freeway,

Flinders Street, and Johnson Street) in Melbourne Siltstone, Abu-Hejleh et al. (2003) that

used the data from four sites (i.e., I-225, County Line, Franklin, and Broadway) in

Colorado claystone, sandstone, and siltstone, Miller (2003) that used the data from three

sites (i.e., Lexington, Grandview Triangle, and Waverly) in Missouri shale, and Vu

(2013) that used the data from two sites (i.e., Frankford, and Warrensburg) in Missouri

shale. Due to their site-specific nature, these methods may need to be re-calibrated using

additional data before they can be used to predict the peak side resistance (fsp) of the rock

sockets sidewalls that are constructed in other rock formations with different rock mass

characteristics.

2. Some design methods relate the values of fsp to only qu of the weak rocks (Rosenberg

and Journeaux 1976; Abu-Hejleh et al. 2003; Miller 2003; Kulhawy et al. 2005; Stark et

al. 2013; Vu 2013) do not account for the effects of (i) the secondary structure of the rock

mass such as joins and slickensided surfaces, and (ii) the rock socket geometry such as

diameter (B) and the length of the shear surfaces (L) that form along the rock socket

sidewalls, on the mobilized fsp. Williams and Pells (1981) and Seidel and Collingwood

(2001) showed that the peak side resistance of the rock socket sidewalls is also affected

by other parameters such as rock socket sidewall roughness and rock mass fracturing.

3. The existing load test data for the peak side resistance (fsp) of the rock socket sidewalls in

weak rocks (e.g., data presented in Fig. 6.1 of Stark et al. 2013) show that the relationship

between the values of fsp and qu for the weak rocks could best be described using a linear

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function where the empirical parameter n = 1. Yet, some of the existing methods use an

empirical parameter n that is less than unity (i.e., n < 1).

Methods based on laboratory tests or numerical models

In this approach (Hassan 1994; Hassan and O’Neill 1997; Seidel and Collingwood 2001;

Wainshtein et al. 2008), the design models were developed based on (i) the laboratory constant

normal stress or constant normal stiffness direct shear tests on natural- or artificial-rock/concrete

interfaces (e.g., Seidel 1993; Seidel and Collingwood 2001), or (ii) the numerical models (e.g.,

Hassan and O’Neill 1997). These methods commonly relate the shear strength of the interface to

(i) the frictional properties of the interface such as interface friction angle (φint), and the initial

average roughness height (∆r) (e.g., Seidel 1993; Seidel and Collingwood 2001), (ii) initial

normal stress (σno) acting on the interface (e.g., Hassan and O’Neill 1997), and (iii) the rock

mass deformation modulus (Em) (e.g., Seidel and Collingwood 2001; Hassan and O’Neill 1997).

The following should be noted in relation to the models based on laboratory tests or numerical

models

1. Some of the existing predictive methods were developed based on the constant normal

stiffness (CNS) direct shear tests on rock/concrete interfaces. Some aspects of these tests

are noteworthy. For examples use of a CNS direct shear test method by Seidel (1993) is

more realistic than previously used traditional constant normal stress direct shear tests,

and it appears that it can properly model the dilation of the interface and the resulting

changes in the normal stresses on the mobilized shear surface. Recent studies by the

Author shows that the fsp measurements from constant normal stiffness direct shear tests

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reported by Seidel (1993) are in reasonable agreement with fsp measurements in the

drilled shaft, plug, and anchor load tests in weak rocks.

2. Seidel and Collingwood (2001) proposed a method for prediction of fsp that accounts for

the average rock/concrete interface roughness height (∆r) in rock sockets. Seidel and

Collingwood (2001) also proposed an approach for prediction of the ∆r that is shown in

Fig. 1 as upper- and lower-bound solid lines. Figure 1 also presents measured ∆r and qu

data that are reported in Collingwood (2000). While the proposed model of Seidel and

Collingwood (2001) is in good agreement with measured rock socket sidewall roughness

data, a wide range of ∆r is possible for any given value of qu.

3. The model of Hassan and O’Neill (1997) relates the normalized peak side resistance (i.e.,

fsp/qu) to (i) the corresponding values of the unconfined compressive strength (qu) of the

weak (or “soft” according to Hassan and O’Neill 1997) rock, (ii) to the initial normal

stress (σno) on the rock socket sidewalls, and (iii) to the interface friction angle (φint).

While the values of σno in each drilled shaft, plug, and anchor may be reasonably

estimated using the hydrostatic pressure of fresh concrete, Vesic (1963) and, the data

presented in this paper (see Fig. 9) suggest that the values of the σno and fsp are not

strongly related. This is because the normal stresses (acting on the rock socket sidewalls)

at the time of mobilization of fsp are not necessarily the same as the values of the σno

because the rock socket often (with some exceptions such as carbonate rocks) dilates

against the normal stiffness (Kn, Equation 4) of the weak rock mass.

IN SITU LOAD TEST DATA

Existing databases

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The existing load test databases (e.g., Horvath and Kenney 1979; Williams 1980; Rowe and

Armitage 1984; Hassan 1994; Collingwood 2000; Stark et al. 2013) were used in the

development of some of the relationships for the prediction of the peak side resistance (fsp) in the

rock sockets. The predictive models developed based on these databases, with the exception of

methods such as Williams and Pells (1981) and Seidel and Collingwood (2001), often relate the

measured peak side resistance (fsp) of the rock socket sidewalls to the unconfined compressive

strength (qu) of rock measured using the specimens obtained from the adjacent boreholes. The

databases of Horvath and Kenney (1979), Williams (1980), Rowe and Armitage (1984), Hassan

(1994), and Collingwood (2000) are summarized in Table 2. One or more of the following

parameters were not reported in the existing databases: (i) the rock mass properties such as the

deformation modulus (Em), and (ii) the rock socket characteristics, such as the depth of

embedment from the ground surface and top of rock (DGS or DTOR, respectively) and the length

of the shear surface (L) mobilized along the rock socket sidewalls. Therefore, the effect of these

missing parameters on the peak side resistance (fsp) cannot be readily examined using the such

databases.

Updated load test database

The updated database consists of the in situ axial load tests on rock sockets in weak rocks, which

are found in the literature. The rock sockets analyzed are a part of the in situ load tests on drilled

shafts, anchors, and plugs in weak rock masses. The database was first reported by Asem (2018),

and is summarized in the Supplemental Data section. In addition to the length of the shear

surface formed along the rock socket and the rock socket diameter (L and B, respectively), the

database summarizes the load test and rock socket excavation methods used at each load test site,

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the estimated deformation modulus of the rock mass (Em), the initial shear stiffness (Ksi)

mobilized along the shear surfaces formed on the rock socket sidewalls, the unconfined

compressive strength (qu) of the rock, the measured peak side resistance (fsp), and the

corresponding vertical socket displacement (δp). The database is briefly discussed blow.

1. Deformation modulus of weak rock mass (Em): the value of Em is estimated using the

slope of the tangent line drawn to the initial part of the relationship between the shear

stress (fs) and axial rock socket displacement (δ) (Fig. 2) that were measured for each

load test. The initial slope of the fs-δ relationship is represented by the initial shear

stiffness (Ksi). Equation (2), which is based on the theory of elasticity, may be used to

relate the values of Ksi to Em that will eventually be used to estimate the values of Em

(2)

In Equation (2), P is the load on the rock socket, and I is an embedment influence factor

that is obtained from Pells and Turner (1979). Dividing both sides of Equation (2) by the

circumferential area of the shear plane ( ), we obtain Equation (3) for the

estimation of Em based on the in situ load test measurements

(3)

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Figure 3 shows the estimated values of Em using Equation (3) that are plotted on the

available data from literature that include (i) the estimated Em values from the base

pressure-base displacement relationships in the rock sockets that are reported by Asem

(2018), and (ii) the results of the plate load tests on weak rocks that are reported by Chern

et al. (2004). Figure 3 shows that the values of Em estimated using Equation (3) follow a

similar trend as the values of Em reported in the published literature for weak rocks. It is

noted that the use of the method of Pells and Turner (1979) (that uses the theory of

elasticity) has resulted in meaningful estimates of Em (see Fig. 3) because the values of

Ksi, which are used to estimate the corresponding values of Em are obtained at the early

stages of the load testing where the rock socket behaves elastically, and the relative

displacements between the rock mass and the rock socket concrete are small (Lo and

Hefny 2001).

2. Peak side resistance (fsp) and the corresponding displacement (δp): the values of the

peak side resistance (fsp), and the corresponding displacement (δp) are obtained from the

measured fs-δ relationships (e.g., Fig. 2) that are reported by Asem (2018). The

definitions of fsp and δp are shown in Fig. 2. The fsp is the shear stress where the slope of

the tangent line to the shear stress and displacement relationship for the sidewalls of the

rock sockets first approaches zero, i.e., , and the reported values of δp are the

corresponding displacements.

3. Rock socket geometry: the diameter for each rock socket (B), the depth of embedment

from the ground surface (DGS) and from the top of rock formation (DTOR) to the center of

rock socket, and the length (L) of the shear surface formed along the rock socket

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sidewalls for each load test are summarized in the database. The quantity L is equivalent

to the length of the rock socket for most cases. When strain gauges are used to measure

the load transfer from the shaft to the surrounding weak rock mass, L represents the

distance between the strain gauges.

4. Load test approach: Figure 4 shows the methods of load testing used in the load test

database. Figures 4(a), (b) and (c) represent the top-down (compression) load test method

where the loads are applied to the drilled shaft butt, Fig. 4(d) is an Osterberg load test

method where the loads are applied bi-directionally using a load cell embedded in the test

shaft at a desired depth, and Fig. 4(e) is a tension plug or anchor load test method. In Fig.

4(a), a load cell is placed at the base of the drilled shaft to separate the side and base

resistances. In Figs. 4(b) and (d), the values of the side and base resistances are separated

using strain gauges. In Fig. 4(c) a void or a compressible base is provided at the base of

the drilled shaft to eliminate the base resistance to directly measure the side resistance.

Lastly, in Fig. 4(e), the loads are applied in tension in the plug or anchor load tests.

The load test data are first used to examine the factors that affect the peak side resistance (fsp).

The load test data are then used for the development of a predictive model for the peak side

resistance (fsp) of rock sockets in weak rocks, and for the development of the corresponding

LRFD resistance factors (φ).

PARAMETERS AFFECTING PEAK SIDE RESISTANCE

Rock type

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The variation of the measured peak side resistance (fsp) in the drilled shaft, anchor, and plug load

tests with different rock mass and rock socket properties are shown in Figs. 5 to 8. Different rock

types are shown with different symbols. It appears that the rock type does not significantly affect

the measured values of fsp and may be of secondary importance for load tests in weak

sedimentary rocks. However, due to the significant scatter in the load test data, a definitive

conclusion on the effect of rock type requires additional high quality data that are not available at

the present time, and this conclusion may need to be updated as more data become available.

Unconfined compressive strength

Figure 5 shows that the peak side resistance (fsp) and the unconfined compressive strength (qu)

are related, which is in agreement with Rosenberg and Journeaux (1976), Horvath and Kenney

(1979), Williams (1980), Rowe and Armitage (1984; 1987), Carter and Kulhawy (1988), Abu-

Hejleh et al. (2003), Miller (2003), Kulhawy et al. (2005), Vu (2013), Stark et al. (2013), and

Asem (2018). The relationship between the values of fsp and qu may be explained as follows

1. A jointed rock mass typically consists of weathered rock blocks, which are separated by

several joint sets (Hoek 1983). The shear strength of the weathered rock blocks that

affects the rock mass shear strength and deformational properties, may be represented by

qu. Williams (1980), Williams and Pells (1981), and Hassan and O’Neill (1997)

suggested that the shear surface mobilized on the rock socket sidewalls is commonly

formed within the adjacent rock mass and not entirely at the rock/concrete interface.

Therefore, the measured values of fsp are expected to be affected by the shear strength of

the rock mass and therefore its qu.

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2. As the drilled shaft, anchor, or plug displaces with respect to the adjacent rock mass, it

commonly dilates against the normal stiffness (Kn) of the rock mass due the roughness of

the mobilized shear surface. It should be noted, however, that in some occasions it has

been observed by previous investigators (e.g., Seidel 1993) that the interface can show a

contractive behavior. However, for most rocks in the database presented in this paper, a

dilative behavior is expected. The normal stresses that act on the shear surface are

proportional to the dilation times the normal stiffness (Kn) of the rock mass and increase

with axial displacement of the rock socket. As the compressive strength of the shear

surface increases, (i) the magnitude of the dilation will increase because the rock socket

sidewall asperity crushing decreases because the sidewall has a greater compressive

strength due to larger qu, and (ii) the stiffness of the rock mass increases because the

values of qu and Em are related (see Fig. 3), and Em governs Kn (see Equation 4).

Therefore larger normal stresses are generated at failure that leads to larger values of fsp.

Consistently, Fig. 5 shows that fsp increases with qu. The scatter in the fsp-qu data in Fig. 5 can

be explained as follows

1. The construction method of rock sockets, the load testing methods used in the drilled

shaft, anchor, and plug load tests, the sampling approach of the rock mass to obtain rock

core for unconfined compression tests, and the rate of shearing of rock specimens in the

laboratory for measurement of the unconfined compressive strength (qu) are not similar

for all cases in the database and will contribute to the scatter of fsp-qu data in Fig. 5.

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2. As will be shown in the subsequent parts of this section, the peak side resistance (fsp) of

the rock socket sidewalls is not only a function of the unconfined compressive strength

(qu). Other parameters such as the deformation modulus (Em) and the length of the shear

surface (L) also affect the measured peak side resistance (fsp). These missing parameters

in Fig. 5 contribute to the data scatter.

Deformation modulus of rock mass

As the rock socket displaces in the axial direction, the shear surface formed along the rock socket

sidewalls dilates against the normal stiffness (Kn) of the adjacent weak rock mass, and the

normal stresses on the shear surface increase accordingly. The value of the peak side resistance

(fsp) for rock socket sidewalls in weak rocks is affected by the change in the normal stresses that

act on the mobilized shear surface on the rock socket sidewalls. The mobilized normal stresses

are affected by the normal stiffness (Kn) of the rock socket sidewalls, which can be written as

(4)

where ν is the Poisson’s Ratio of the weak rock. Equation (4) (Seidel and Collingwood 2001)

shows that Em and Kn are dependent. As Em increases, Kn also increases, and larger normal

stresses are generated as the rock socket displaces axially and dilates with respect to the adjacent

rock mass. Therefore, fsp is greater for a rock mass with larger Em as shown in Fig. 6 that is also

in agreement with previous published work by Williams (1980), Williams and Pells (1981), and

Seidel and Collingwood (2001).

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Rock socket geometry

Figure 7 shows that the measured values of the peak side resistance fsp decrease with the rock

socket diameter (B) (Fig. 7a) (Horvath and Kenney 1979), and the length of the shear surface (L)

(Fig. 7b). The shear length (L) is the distance over which an average fsp is calculated based on

the load test results. The value of L may correspond to the overall length of the rock socket, or

the distance between the strain gauges if they are used in the load test. Bandis (1980) and Bandis

et al. (1983) also showed that the shear strength of rock joints decreases with increase in their

length, which is in agreement with the trends observed for the drilled shafts, anchors, and plugs

socketed in weak rocks (Fig. 7b). The decrease in the measured values of the peak side resistance

(fsp) with B and L is explained below

1. As B and L in a rock socket increase, the zone of the influence of the rock socket also

increases (Lo and Hefny 2001), and the number of discontinuity sets that interact with the

rock socket increases accordingly. Therefore, the stiffness and shear strength of the rock

socket sidewalls decrease because the mobilized shear strength and stiffness of weak rock

are scale dependent parameters (Bieniawski and Van Heerden 1975; Goodman 1980;

Hoek 1983; Lo and Hefny 2001). As Em decreases, Kn decreases and fsp decreases

because smaller normal stresses are generated on the rock socket sidewalls. Data

presented in Fig. 7, however, show that the effect of L on fsp is more significant than the

effect of B on the mobilized values of fsp.

2. The non-uniform stress distribution along the rock socket sidewall can result in

progressive failure along the rock socket sidewalls as loads on the rock socket increase.

This will result in the overall decrease in the measured side resistance as L increases.

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Initial state of stress

The embedment depth (DGS and DTOR) may be used to estimate the lateral pressures that are

exerted by the concrete, which may be related to the initial normal stresses on the shear surfaces

that are formed on the rock socket sidewalls (Hassan and O’Neill 1997). The data presented in

Fig. 8 show that the embedment depth (DGS and DTOR) does not significantly affect the

mobilized values of the peak side resistance (fsp). Therefore, the initial normal stresses (σno) and

the peak side resistance (fsp) are not strongly related. The peak side resistance (fsp), however, is

affected by the final normal stresses that are mobilized on the shear surface at the time of failure

(Seidel 1993). These final normal stresses are different from the initial normal stresses (Vesic

1963, Seidel 1993) due to the dilation of the rock socket against the normal stiffness (Kn) of the

rock socket side walls (Williams and Pells 1981; Seidel and Collingwood 2001).

Fig. 9 shows a plot of the normalized peak side resistance (fsp/qu) and qu. In Fig. 9, the fsp/qu

data are separated based on the estimated normalized initial normal stress (σno/σp). The trend

lines representing the relationship between fsp/qu, qu and σno/σp, which are reproduced based on

the method of Hassan and O’Neill (1997) are also plotted on the load test data in Fig. 9. In the

normalized initial normal stress (σno/σp), σp is the atmospheric pressure, and its value is equal to

0.101 MPa, and σno is estimated based on the hydrostatic pressure of fresh concrete at the time of

pour by Equation (5)

(5)

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where γc and hc are, respectively, the unit weight of concrete, and the head of concrete above the

depth where the corresponding values of the σno were calculated. Equation (5) results in

reasonable estimates of σno because high slump concrete was used in the construction of most of

the rock sockets reported in this study. Bernal and Reese (1983) showed that for high slump

concrete, the lateral pressure exerted by the concrete approaches hydrostatic condition. The

scatter in Fig. 9 is significant, however, this figure shows that the values of fsp/qu and σno/σp are

not strongly related.

Figure 10(a) shows the variation of the rock mass deformation modulus (Em) with the

embedment depth from the ground surface (DGS) for drilled shaft load tests in weathered

Melbourne Siltstone (after Williams 1980), and Fig. 10(b) shows the variation of fsp with DGS,

for the same load test site. Figure 10(a) shows that Em increases with DGS, which may be

explained by the decrease in the weathering with the embedment depth at this particular load test

site. Larger values of Em results in a larger rock socket sidewall Kn (see Equation 4) and causes

the mobilization of larger normal stresses at the time of failure, which leads to mobilization of

greater fsp values as shown in Fig. 10(b). It is noted that increase in the initial normal stresses

due to increase in the values of DGS may also be contributing to some extent, however, it is clear

that increase in Em with DGS is also a contributing factor for increase in fsp with DGS. Based on

the discussion presented above, it may thus be concluded that the increase in fsp with embedment

depth could be more related to decrease in the weathering, and the subsequent increase in Em

(and the corresponding values of Kn), and not necessarily due to increase in initial normal

stresses (σno) acting on the shear surface.

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If, on the other hand, the weathering increases with DGS, the values of Em and thus Kn will

decrease with depth accordingly, and fsp will also decease. An example of such scenario is

shown in Fig. 11 that is obtained based on the load tests in weathered Melbourne Siltstone,

performed by Williams (1980). The variation of fsp with DGS shown in Fig. 11 may not be

explained, nor could it be predicted using the method of Hassan and O’Neill (1997), which

suggests that fsp should tacitly increase with DGS, with no exception.

Vertical socket displacement

The vertical displacement that is required to mobilize the values of fsp in rock sockets in weak

rocks (δp) is plotted versus the corresponding values of the rock socket diameter (B) in Fig. 12.

Figure 12 shows that, on average, a vertical displacement of 10 mm is required to mobilize fsp.

This observation is in agreement with Brown et al. (2010). Figure 12 also shows that the

displacement required to mobilize fsp is not a discernible function of B.

Roughness

The importance of the shear surface roughness on the measured values of the peak side

resistance (fsp) has been emphasized by Williams (1980), Williams and Pells (1981), and Seidel

(1993) and many others. Johnston and Lam (1989), Kodikara (1989), Seidel (1993), and Seidel

and Collingwood (2001) developed design models that relate the shear stress to the pre-concrete

pour average roughness height (∆r) of the rock/concrete interfaces. The roughness of the

mobilized shear surface along the rock socket sidewalls cannot be accurately predicted, and was

not included in the development of the proposed design model for fsp for the following reasons

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1. The method of Seidel and Collingwood (2001) for the prediction of ∆r does not consider

the following effects: (i) formation of the smeared and disturbed materials (Williams

1980) on the rock socket sidewalls, and (ii) increase in the water content of the rock

socket sidewalls beyond its initial condition due to migration of the moisture from the

concrete to the adjacent rock (Meyerhof and Murdock 1953). The formation of the

disturbed materials and the change in the water content of the interface can result in post-

construction changes (i.e., softening) in the characteristics of the socket wall roughness

and can decrease its resistance to wear and crushing which in turn can affect the dilation

of the shear surface.

2. Actual rock socket sidewall roughness measurements are not made in most of the load

tests in the database that is compiled herein.

3. The available roughness measurements are those of rock/concrete interface and provide

little information on the roughness of the actual shear surface which is mobilized.

PROPOSED PREDICTIVE EQUATION FOR PEAK SIDE RESISTANCE

The method of Gardoni et al. (2002) is used to develop a predictive model for the peak side

resistance (fsp). The following general equation is adopted for the proposed model

(6)

where fsp(x, Θ) represents a probabilistic model for peak side resistance, γ (x,θ) is a set of terms

that are developed based on the properties of (i) the rock socket and (ii) rock mass, which control

the development of the peak side resistance, Θ = (θ,σ) is a set of model parameters used to fit the

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proposed model, of the form shown in Equation (6), to the observed data, σε is an additive

model error (additivity assumption) where ε is assumed to be a normal random variable

(normality assumption) with zero mean and unit variance, and σ denotes the standard deviation

of the model error that is assumed to be constant (homoskedasticity assumption). The validity of

these assumptions can be assessed using diagnostics plots following Rao and Toutenburg (1997)

and Gardoni et al. (2002).

Following Gardoni et al. (2002), the γ (x,θ) term can be expressed using Equation (7)

(7)

The term hj(x) is the jth explanatory function used to include all parameters (represented by the

vector x) that affect the development of fsp in the design model, and θj is the jth model parameter

that is used to fit the proposed model to the observed data.

The following procedure is used to estimate the model parameters Θ = (θ,σ), which are obtained

by fitting Equation (6) to the in situ load test data

1. The load test data that are used in the analysis correspond mostly to rock sockets with

relatively clean and rough rock socket sidewalls. Quantitative measure of roughness of

shear surface was not available in most cases. The proposed methods herein should be

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applied with caution to the design of rock sockets where drilling method is anticipated to

result in production of smooth or smeared rock socket sidewalls.

2. The quantities of interest (fsp, qu, L, B, and Em) are first transformed using a natural

logarithm. This is to justify the following assumptions: (i) the model variance (σ2) is

independent of x (homoskedasticity assumption), and (ii) ε has a normal distribution

(normality assumption) (Gardoni et al. 2002).

3. Once all parameters of interest are transformed using the proper transformation technique

(in this case natural logarithm), the Maximum Likelihood (Ang and Tang 2007) method

is used to estimate the model parameters, Θ = (θ,σ).

4. Once the model parameters Θ = (θ,σ) are estimated, both sides of the model are

transformed back into the original space using an exponential transformation, e(.).

MODEL CALIBRATION

The review of the load test data shows that the peak side resistance (fsp) depends on the values of

qu, L, B, and Em. The effect of such parameters has been emphasized in the published literature

(Pells et al. 1980; Williams 1980; Williams and Pells 1981; O’Neill et al. 1996).

In this study, we first develop a “Base Model” that relates the measured values of the peak side

resistance (fsp) to the corresponding values of the unconfined compressive strength (qu) of the

weak rock along the rock socket walls. Additional terms, including the rock socket and rock

mass properties (i.e., explanatory functions including L, B, and Em), are then added to this “Base

Model” to investigate any potential improvement in the model accuracy. The “Base Model” is

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calibrated using the load test data and the method of Gardoni et al. (2002). Equation (8) shows

the calibrated Base Model

(8)

where fsp and qu are in units of MPa. The standard deviation (σ) of the model error for Equation

(8) is 0.825 MPa. The statistics of the model error and model parameters are summarized in

Table 3.

Analysis of the in situ load test data further shows that the rock socket geometry affects the

mobilized values of fsp. Therefore, we investigate the contribution of additional information on L

and B to the overall accuracy of the “Base Model” by updating the “Base Model” using the

values of B and L. The updated models are shown in Equations (9) and (10)

(9)

and

(10)

where the values of B and L are in units of meter. The estimated values of σ for the model error

in Equation (9) is 0.766 MPa, and in Equation (10) is 0.768 MPa. These results indicate that

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updating the “Base Model” using information on the values of L and B improves the model

accuracy as indicated by the reduced values of σ in Equations (9) and (10) compared to the

value of σ in Equation (8). Analyses, however, show that the effect of B on σ is insignificant

compared to the effect of L on σ. This conclusion may be reached by comparing the values of σ

in Equations (9) and (10). The statistics of the model error and model parameters for Equations

(9) and (10) are, respectively, summarized in Tables 4 and 5.

Analysis of load test data also shows that fsp and the estimated values of rock mass Em are

related. To investigate possible contributions of Em to the accuracy of the “Base Model,” the

“Base Model” is now updated using the additional information on the values of Em and L. The

updated models are summarized in Equations (11) and (12)

(11)

and

(12)

In Equation (11), Em is added to Equation (8) and in Equation (12), L and Em are added to

Equation (8). The estimated value of σ for the model error for Equation (11) is 0.799 MPa, and

value of σ for Equation (12) is 0.645 MPa. The statistics of the model error and model

parameters for Equations (11) and (12) are, respectively, summarized in Tables 6 and 7.

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Comparison of the values of σ in Equations (8) to (12) indicates that Equation (12) is the most

accurate model for the prediction of fsp. The parameters required to evaluate Equation (12) may

be estimated for design purposes. The proposed model for fsp includes qu, L, and Em (Equation

12). The following are noted in relation to this model

1. Equation (12) states that fsp increases with qu and Em but decreases with L. These are in

agreement with the in situ load test data presented in Figs. 5 to 7 and published literature.

2. Load test data shows that B does not significantly affect fsp (Fig. 7a), particularly for

large diameter rock sockets (i.e., B > 400 mm).

3. Initial state of stress is not considered in the updating process presented above because

load test data (Fig. 8 which shows the variation of the fsp with DGS and DTOR that are

assumed to crudely represent the initial state of stress in the rock mass adjacent to the

rock socket walls) do not show a significant relationship between these parameters and

fsp. This is because fsp is related to normal stresses at the time of failure. DTOR and DGS

only crudely represent the initial normal stresses on the rock socket walls.

4. Because the roughness of the actual shear surface for the load tests reported in this study

are not available, the roughness is not included in the formulation of the proposed model.

The existing models are evaluated using the load test database, and the results are summarized in

Table 8. The models are compared in terms of the coefficient of variation of their bias (δλ). The

bias is defined as the ratio of measured peak side resistance (fsp,m) to predicted side resistance

(fsp,p), and is denoted by λ. Standard formulation (e.g., Ang and Tang 2007) may be used to

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evaluate the mean (µλ), and the standard deviation (σλ) of the bias, λ = fsp,m/fsp,p. The coefficient

of variation (δλ) for bias (i.e., fsp,m/fsp,p) may then be defined using Equation (13)

(13)

Coefficient of variation for bias (δλ) is a measure of the “dispersion” or “variability” of fsp,m/fsp,p

relative to its mean value (Ang and Tang 2007), and may be used as a measure of the accuracy of

the predictive model. The statistics for predictive model in Equation (12) are summarized in

Table 9. The results presented in Table 8 show that δλ for all of the existing models are greater

than the proposed model (i.e., Equation 12). The results presented herein indicate that addition of

L and Em to the “Base Model” has resulted in improved performance of the model.

CALIBRATION OF RESISTANCE FACTOR

Traditionally, designers have used a factor of safety (FS) in the Allowable Stress Design (ASD)

approach to account for the uncertainties both in resistances and loads. The FS is selected

subjectively, and does not properly quantify the design uncertainties (Kulhawy and Phoon 2006;

Roberts and Misra 2009; Paikowsky et al. 2010). As an alternative to the ASD, designers have

used the Load and Resistance Factor Design (LRFD) framework to ensure the design is safe

against each limit state (e.g., bearing capacity failure). The LRFD accounts for the uncertainty in

loads and resistances separately. Additionally, the LRFD framework quantifies the design

uncertainties more rigorously using the theory of probability.

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In the design of foundations, two limit states should be recognized (Meyerhof 1951; Allen 1975;

Phoon et al. 2000). The foundation should be safe against the bearing capacity failure, and the

foundation settlements should be less than the tolerable values. The evaluation of the strength

limit state is discussed in the following sections to ensure a bearing capacity failure will not

materialize. In the context of the LRFD framework, the strength limit state is determined using

Equation (14)

(14)

where φs and φb, respectively, are the LRFD resistance factors for the side and base resistances,

Rs and Rb are, respectively, the total side and base resistances, and the values of γi are the load

factors for the ith axial load effects (Qi) at the top of drilled shaft, anchor, or plug.

To design a rock socket using the LRFD framework, φs and φb should be calibrated to produce a

design that is consistent with the acceptable level of failure probability in the current practice. In

this section, the predictive model for fsp (Equation 12) is used to determine φs using the First-

Order Reliability Method (FORM) following the procedure proposed by Briaud et al. (2013).

First-Order reliability method (FORM)

The limit state function (g), which is required for the FORM analysis (Briaud et al. 2013;

Gardoni 2017) is introduced in Equation (15)

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(15)

where λR is the bias of the predictive model for fsp (i.e., Equation 12), λDL and λLL, respectively,

are the bias for dead and live loads, and DL and LL are, respectively, the dead and live loads.

Following the recommendations of Paikowsky et al. (2004), Paikowsky et al. (2010), and Briaud

et al. (2013), an HS-20 design truck is used to represent the LL for typical bridge piers (i.e., LL =

445 kN). The calibration of the LRFD resistance factors also requires an estimate of the dead

load to live load ratio (DL/LL). The actual value of the DL/LL ratio is a function of the span

length (l) of the structure. For example, McVay et al. (2000) provided the following values of

DL/LL based on the values of l: DL/LL = 0.52, 1.06, 1.58, 2.12, 2.64, 3.00, and 3.53,

respectively, for l values of 9, 18, 27, 36, 45, 50, and 60 m. These values are used in the

calibration process. A lognormal distribution is used to represent the variation of the bias for fsp

predictive model (i.e., Equation 12) (i.e., λR). Following Briaud et al. (2013), the statistics for

λDL and λLL are also assumed to follow a lognormal distribution, and are reported in Table 9.

The values reported in Table 9 are obtained from the structural engineering literature to be

consistent with the existing practice for the design of the superstructure.

Equation (15) is now used in the FORM analysis to iterate about Rs until the calculated

reliability index (β) (Ditlevsen and Madsen 1996; Haldar and Mahadevan 2000; Gardoni 2017)

approaches the target reliability index (βT) that is consistent with the acceptable level of failure

probability that is commonly used for design of the superstructure. Based on a review of

literature, βT is selected as follows

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1. ISO (2015) evaluated βT for different “consequences of failure” and “relative cost of

safety measure.” The consequences of failure included minor, moderate, and large. The

relative cost of safety measure included large, normal, and small. The recommended βT

ranges from 3.1 (for large cost of safety measure and large consequence of failure) to 4.7

(for small cost of safety measure and large consequence of failure).

2. The resistance factors recommended by Brown et al. (2010) (i.e., FHWA-NHI-10-016)

for the design of drilled shafts in rocks correspond to a βT of 3.0.

3. The redundancy and the possibility of load redistribution (i.e., load sharing among

foundation components) in the event of failure of one of the load bearing components in a

system is an important criterion in determination of βT (Allen et al. 2005). Liu et al.

(2001) defined a redundant foundation system as one in which the calculated β value for

the overall system is 0.5 higher than the reliability of the individual foundation members

within the group. Barker et al. (1991) determined that βT for drilled shafts ranges from

2.0 to 3.7 (from MVFOSM analysis), and 2.0 to 4.3 (from more advanced analysis).

Barker et al. determined that βT is 3.5 for non-redundant systems, 2.5 to 3.0 for drilled

shafts, and 2.0 to 2.5 for highly redundant systems. Barker et al. (1991) relied heavily on

the implied level of safety by previous design practice to provide their recommendations

for the values of βT and the resulting resistance factors. Allen et al. (2005) proposed that

the ability of the soil to redistribute the load allows for the foundation system to be

designed for a lower reliability index compared to the superstructure. The only exception

to this rule is the case of a single drilled shaft that supports the entire bridge pier where a

βT value of 3.5 is desirable (Allen et al. 2005).

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4. Paikowsky et al. (2004) recommended a βT of 3.0 when the drilled shaft group contains

less than 5 drilled shafts, and recommended a βT of 2.3 for the cases where the number of

shafts in a shaft group exceeds five. Therefore, the βT for which individual drilled shafts

are designed is heavily affected by the redundancy of the drilled shaft foundation system.

This conclusion is in agreement with the work of other researchers (e.g., Liu et al. 2001;

Allen et al. 2005).

5. Briaud et al. (2013) used a βT of 3.5 for the calibration of resistance factors for the

assessment of scour of the bridge foundations.

6. Phoon et al. (2000) recommended a βT of 3.2 for the assessment of the strength limit state

in drilled shafts. Phoon et al. (2000) reported that the actual annual probabilities of failure

for foundations is between 0.1% to 1%. These values of probability of failure may be

used to back-calculate βT of 2.5 to 3.0 for these foundations using the method of

Rosenblueth and Esteva (1972) (i.e., ).

7. Phoon and Kulhawy (2005) determined the reliability of drilled shafts in different soils

and rocks, and stated that the value of βT varied from 2.6 to 3.4 for all drilled shafts in

their database that were subjected to an undrained compression loading.

8. Kulhawy and Phoon (2006) adopted the recommendations of the Canadian Building

Code and the AASHTO bridge specifications. The Canadian Building Code recommends

a βT of 3.5 for the superstructure and the foundation and the AASHTO specifications

recommends a βT of 3.5 for the superstructure and a βT of 2.0 to 3.5 for the foundation.

9. Nowak (1995) used a βT of 3.5 for the superstructure. Allen et al. (2005) cited a βT of 3.5

for the superstructure and a βT of 3.0 for the foundation design from past practice

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although Allen et al. (2005) indicates that it is desirable to “maintain a consistent level of

safety across all limit states of a given type (e.g., strength limit state).”

10. Ellingwood and Galambos (1982) recommended βT values of 2.5 to 4.0 for structural

steel elements, and 2.3 to 3.6 for reinforced concrete elements in the building structures.

The βT values for foundations should be selected so as to maintain a degree of

consistency between level of safety of the foundation and the superstructure.

11. The typical range of βT for different load combinations is reported by Allen (1975).

According to Allen (1975), βT often ranges from 3.0 to 4.0.

Therefore, the quantity Rs is calculated for βT values of 2.0, 2.5, and 3.0 that fall within the rage

suggested in the published literature. Once Rs is calculated from the FORM analyses, Equation

(16) may be used to calculate the corresponding φs

(16)

Equation (16) and the FORM analyses are used to determine the values of φs for fsp that is given

by Equation (12), and for a range of l and βT, which are discussed in the previous sections.

The resistance factors obtained from the FORM analysis are summarized in Fig. 13. The

calculated range of the resistance factors is in agreement with the recommended resistance

factors by other investigators (e.g., Paikowsky et al. 2010). Figure 13 shows that the resistance

factor increases with decrease in βT, and slightly decreases with increase in l.

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CONCLUSIONS

In situ drilled shaft, anchor, and plug load tests are used to develop a framework for the

prediction of peak side resistance (fsp) of rock socketed drilled shafts in weak rocks. A

comprehensive database is compiled. The effects of rock mass, and rock socket geometry on

peak side resistance (fsp) are evaluated. A “Base Model” is developed that relates the measured

values of peak side resistance (fsp) to the unconfined compressive strength (qu) of the weak rock.

The method of Gardoni et al. (2002) is used to update the “Base Model” using additional data

gathered and reported in the new database introduced in the Supplemental Data section.

Analysis results shows that a model that includes the unconfined compressive strength (qu), the

rock mass deformation modulus (Em), and the length of the mobilized shear surfaces (L)

provides a more accurate estimate of fsp than all other methods reviewed herein. The values of qu

represents the shear strength of the intact blocks that make the rock mass. The values of Em and

L reflect the size effect and the number of discontinuity surfaces that interact with the rock

socket.

The proposed model is not a function of the initial normal stress (σno) on the rock socket walls

because analysis showed that the σno does not affect the fsp. The estimated values of σno are

related to the embedment depth. The proposed model for fsp is also not a function of ∆r because

the average roughness height was not measured and are not available for the drilled shaft, plug,

and anchor load tests reported in herein. Additionally, the proposed model should be used when

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drilling method leads to development of relatively rough rock socket sidewalls that are free from

smeared materials.

The First-Order Reliability Method (FORM) is then used to develop the corresponding Load and

Resistance Factor Design (LRFD) resistance factors (φ). The recommended resistance factors are

functions of structure span length (l), and the target reliability index (βT). It is shown that the

calculated values of resistance factor are not sensitive to the span length of structure (l).

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ACKNOWLEDGEMENTS

The Authors would like to thank and acknowledge the University of Illinois at Urbana-

Champaign for providing access to the technical and computational resources that were required

for completion of this work. The Authors thank Professor Emeritus James H. Long for his

contributions to this paper. The Authors also thank Dr. Armin Tabandeh for his assistance with

MATLAB code used for reliability analysis performed in this study.

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FIGURE CAPTIONS

Fig. 1. Relationship between the measured initial rock socket concrete/rock interface

average roughness height (∆r) with the unconfined compressive strength (qu) of

weak rock (after Collingwood 2000; Asem 2018). Solid lines represent the

method of Seidel and Collingwood (2001) for the prediction of ∆r based on qu.

Fig. 2. Definition of the peak side resistance (fsp) and initial shear stiffness (Ksi) using

measured shear stress (fs) and displacement (δ) relationships.

Fig. 3. Relationship between modulus of deformation of rock mass (Em) from base

resistance database, plate load tests, and side resistance database and the weak

rock unconfined compressive strength (qu) (after Asem 2018).

Fig. 4. Load testing methods used in the load test database (after Asem 2018).

Fig. 5. Variation of peak side resistance (fsp) with unconfined compressive strength (qu)

of weak rock (data from Asem 2018).

Fig. 6. Variation of peak side resistance (fsp) with deformation modulus of weak rock

mass (Em) (data from Asem 2018).

Fig. 7. Variation of peak side resistance (fsp) with rock socket diameter (B) and rock

socket length (L) (data from Asem 2018).

Fig. 8. Variation of peak side resistance (fsp) with rock socket embedment depth from the

ground surface, and top of rock (DGS and DTOR, respectively) (data from Asem

2018).

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Fig. 9. Variation of the normalized peak side resistance (α = fsp/qu) with normalized

initial normal stress on rock socket walls (σno/σp), and unconfined compressive

strength (qu) with method of Hassan and O’Neill (1997) (data from Asem 2018).

Fig. 10. Variation of modulus of deformation of weak rock mass (Em) and peak side

resistance (fsp) with depth of embedment from the ground surface (DGS) for

drilled shaft load tests in weathered Melbourne Siltstone (data from Williams

1980).

Fig. 11. Variation of modulus of deformation of weak rock mass (Em) and peak side

resistance (fsp) with depth of embedment from ground surface (DGS) for drilled

shaft load tests in weathered Melbourne Siltstone (data from Williams 1980).

Data present an example of a case where fsp decreases with embedment depth.

Fig. 12. Variation of the displacement required to mobilize the peak side resistance (δp)

with the rock socket diameter (B) (data from Asem 2018).

Fig. 13. Variation of the calculated LRFD resistance factors with structure span length and

target reliability index.

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FIGURES

Figure 1

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2

Figure 2

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3

Figure 3

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Figure 4

Top of rock

Groundsurface

B

Base loadcell

Test shaftReaction

shaftsDGS DTOR

Reaction frame

Top loadcell

Top of rock

Ground surface

B

Test shaft Reaction shaftsDGS DTOR

Reaction frame

Top load cell

Straingauge

Top of rock

Ground surface

B

Void or compressible

base

Test shaftReaction

shaftsDGS DTOR

Reaction frame

Top load cell

Top of rock

Ground surface

B

O-cell

Test shaftDGS DTOR

Strain gauge

Top of rock

Ground surface

B

Plug or anchor

Reactionshafts

DGS DTOR

Reaction frameT

Method (a) Method (b) Method (c)

Method (d) Method (e)

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Figure 5

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Figure 6

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Figure 7

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Figure 8

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Figure 9

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Figure 10

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Figure 11

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Figure 12

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Figure 13

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TABLES

Table 1 Existing empirical coefficients ( and n) for peak side resistance (fsp) of rock

sockets in weak rocks.

Design method nRosenberg and Journeaux (1976) 0.362 0.52Horvath and Kenney (1979) 0.20 0.50Pells et al. (1979) 0.20 1.00Williams (1980) 0.43 0.37Horvath et al. (1983) 0.20-0.30 0.50Rowe and Armitage (1984) 0.45 0.50Carter and Kulhawy (1988) 0.20 0.50Stark et al. (2013) 0.30 1.00Stark et al. (2017) 0.31 1.00

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Table 2 Summary of the existing drilled shaft load test database (after Asem 2018).

Database Number of load test

Rock type Rock socket

diameter, B (mm)

Unconfined compressive

strength, qu (MPa)

Horvath and Kenney (1979)

76 (in situ load tests on small- and large-diameter rock sockets)

Shale, claystone, mudstone, siltstone, sandstone, limestone and chalk

13 to 1220 0.35 to 110 for shale and mudstone, 7 to 24 for sandstone and 1 to 7 for limestone and chalk

Williams (1980) (in situ load tests, embedment depth, DGS < 2000 mm)

Weathered

draft...weak rocks. these investigators the measured related peak side resistance (f sp) of rock...

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