Bayesian estimation of the normal and shear stiffness for rock sockets in weak sedimentary rocks

International Journal of Rock Mechanics & Mining Sciences 124 (2019) 104129

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Bayesian estimation of the normal and shear stiffness for rock sockets in weak sedimentary rocks Pouyan Asem a, *, Paolo Gardoni b a b

Department of Civil, Environmental and Geo-Engineering University of Minnesota Twin-Cities, USA University of Illinois at Urbana-Champaign, USA

A R T I C L E I N F O

A B S T R A C T

Keywords: Bayesian estimation Weak sedimentary rock mass Shear stiffness Normal stiffness

The design of socketed foundations in weak sedimentary rocks is often governed by settlement. The shear stiffness of the rock socket sidewalls, and the normal stiffness of the rock socket base are the main parameters that affect the settlement of rock sockets at service load conditions and are needed for settlement calculation in load-transfer method or using the theory of elasticity. In this paper, in situ load test data for socketed foundations in weak rocks are collected. The shear stiffness of the rock socket sidewalls and the normal stiffness for the rock socket base are, respectively, estimated based on the measured side shear stress and shear displacement, and base contact pressure and normal displacement relationships. The parameters affecting shear and normal stiffnesses are discussed. A Bayesian approach is used to develop probabilistic models for the rock socket shear and normal stiffnesses, which are functions of both rock properties and the rock socket size.

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1. Introduction

δðzÞ ¼ β1 e

The design of foundations in weak rocks of sedimentary origin is often governed by tolerable settlements1,2,3 which are regularly dictated by the allowable deformations of the superstructure. A weak rock is defined as that having an unconfined compressive strength (qu), 0.5 MPa < qu < 30 MPa4,5,6,7,8,9, and a Geological Strength Index (GSI) less than 7010 or a Rock Mass Rating (RMR) less than 40 (Poor to Very Poor rocks11). The settlement of rock socketed foundations at service load level may be estimated using the in situ load tests on similar foundation units. However, in situ load tests are expensive, time consuming, and can rarely cover the whole range of rock and environ­ mental conditions relevant to a foundation1 and are, therefore, not routinely conducted for small size projects. As an alternative approach to in situ load testing for determination of foundation settlement, solutions based on the theory of elasticity12,13,1 may be used for preliminary analysis when settlements at the service load levels are calculated. The theory of elasticity is applicable because the foundation behavior is commonly in the elastic range at low loads observed in the service-load regime (e.g., see load tests reported by Asem14). For instance, the solutions for beams on elastic foundation13 may be used to derive the following equation for calculation of settle­ ment of socketed foundations in weak rocks:

where δ(z) is the settlement calculated at depth z, β1 and β2 are obtained by application of the boundary conditions for a complete rock socket where loads are carried by both the rock socket sidewalls and rock

λz

þ β2 eλz

socket base, z is the depth measured from the top of rock socket and λ ¼ qffiffiffiffiffiffi Ksi C AEc is a constant that depends on the initial shear stiffness of the rock socket sidewalls (Ksi), the rock socket circumference (C), the crosssectional area of the rock socket (A), and the Young’s modulus of the rock socket concrete (Ec). The constant β1 may be obtained from: 2 3 6 7 6 7 6 7 6 7 7 Q 6 1 61 þ 7 � � β1 ¼ 7 AEc λ 6 6 7 λE e2λL 1þ Knc 7 6 6 7 � � 1 þ 4 5 1

λEc Kn

and the constant β2 may be obtained from:

* Corresponding author. Minneapolis, MN, 55455, USA. E-mail addresses: [email protected] (P. Asem), [email protected] (P. Gardoni). https://doi.org/10.1016/j.ijrmms.2019.104129 Received 1 January 2019; Received in revised form 23 September 2019; Accepted 12 October 2019 Available online 24 October 2019 1365-1609/Published by Elsevier Ltd.

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P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

2

3

6 7 6 7 6 7 6 7 7 Q 6 1 6 7 � � β2 ¼ 7 AEc λ 6 6 7 λE c e2λL 1þ Kn 7 6 61 þ � 7 � 4 5 1

Table 1 Summary of side resistance database (after Asem14).

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Database component

Description

Type of soft rock

Sandstone and siltstone Shale, mudstone, and claystone Limestone 0.5 MPa < qu < 30 MPa

Unconfined compressive strength (qu) Rock quality designation (RQD) Geological strength index (GSI) Rock mass friction angle (φm) Back-analyzed modulus of deformation (Em) Socket geometry

λEc Kn

where L is the rock socket length and Kn is the normal stiffness of the rock socket base. The shear stiffness of the rock socket sidewalls (Ksi), and the normal stiffness of the rock socket base (Kn) are required for settlement calcu­ lation in load-transfer approach where the interaction of the rock socket with the surrounding rock mass is modeled using load-transfer functions15,16,17,18,19,20,21,22 or using the solutions based on the theory of elasticity such as those shown in Eqs. (1)-(3).13 In this paper, we analyze a comprehensive database of in situ load tests on (i) drilled shafts, anchors and plugs where side resistance is measured, and (ii) drilled shafts and plates where base resistance is measured. The load tests, the rock formations and their properties are discussed and are documented in the Supplemental Data section (after Asem14). The measured shear stress (fs) and shear displacement (δ) relationship for rock socket sidewalls, and the normal stress (q) and displacement (δ) relationship for rock socket base are used to estimate the shear stiffness of the rock socket sidewalls (Ksi), and the normal stiffness of the rock socket base (Kn). The effect of different parameters on Ksi and Kn are evaluated using the load test results. A Bayesian approach23,24,25,26,27 is used to develop probabilistic models for estimation of Ksi and Kn using the load test database compiled in this paper.

Test shaft concrete Load test method

0 < RQD < 65% 31 < GSI < 70 12 < φm < 42� , φm,avg ¼ 25.4� 10 MPa < Em < 19 GPa 46 < B < 2440 mm 0.25 < DGS < 49 m 0 < DTOR < 33 m 21 < f’c < 55 MPa 22 < Ec < 45 GPa Slump: 0–229 mm Osterberg tests, conventional top loaded tests, anchor load tests and plug load tests

Table 2 Summary of base resistance database (after Asem14). Database component

Description

Type of soft rock

Sandstone and siltstone Shale, mudstone and claystone Limestone and chalk 0.5 < qu < 30 MPa

Unconfined compressive strength (qu) Rock quality designation (RQD) Geological strength index (GSI) Rock mass friction angle (φm) Back analyzed modulus of deformation (Em) Socket geometry

2. Geological history of weak sedimentary rocks The geology of weak sedimentary rock generally consists of three distinct phases.2,28 In phase one, parent materials are eroded, carried and accumulated into rivers, lakes, and oceans, and consolidated or compacted under the weight of overburden. Phase two consists of the removal of the overburden, over-consolidation of young deposits and formation of fissures and joints at shallow depths. Phase three consists of an increase in the water content of materials along the exposed walls of the open cracks, and subsequent softening of the rock blocks that are separated by discontinuity surfaces. Therefore, a weak rock mass is an assemblage of relatively weathered rock blocks, which are separated by the structural discontinuity sets.29,30 Because a weak rock mass consists of relatively weathered rock blocks and structural discontinuities that are also subjected to weath­ ering, and both components affect the response of the rock to external loads, the definition of a weak rock should reflect how weathering af­ fects the rock blocks and the discontinuity surfaces. Accordingly, the values of unconfined compressive strength (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 alteration of the discontinuities such as joints and fissures. The review of the technical literature suggests that for weak rocks, the unconfined compressive strength (qu) ranges from 0.5 to 30 MPa4,5,6,7,9 and the Geological Strength Index (GSI) is commonly less than 70.10 This definition is consistently used in the se­ lection of the load tests in the following section.

Test shaft concrete Load test method

20 < RQD < 100% 20 < GSI < 70 14 < φm < 45� 4 MPa < Em < 14 GPa 100 < B < 2500 mm 0 < DGS < 53 m 0 < DTOR < 48 m 20 < f’c < 55 MPa 10 < Ec < 42 GPa Slump: 25–229 mm Osterberg tests, conventional top loaded tests and plate load test

for each load test are included in the Supplemental Data section in Tables T1, T2, S1 and S2. The detailed information on rock types, rock formations, load tests (e.g., fs-δ and q-δ relationships), and the statistics for rock properties and rock sockets are discussed by Asem.14 The fs-δ and q-δ relationships are, respectively, used to estimate the shear stiff­ ness of the rock socket sidewalls (Ksi), and the normal stiffness of the rock socket base (Kn). 3.1. Rock formations Table 3 provides a summary of the rock formations that includes the rock formation name, information on the secondary structure and the in situ weathering condition of each rock mass. The load tests were con­ ducted in Australia, Canada, Singapore and the United States in twentyfour different rock formations. The database mainly includes weak sedimentary rocks that include claystone, siltstone, sandstone, shale and limestone.

3. Databases

3.2. Initial shear stiffness (Ksi) and initial normal stiffness (Kn)

The published data on load tests on drilled shafts, plugs, anchors and plates in different rock masses were reviewed and compiled in two da­ tabases that are summarized in Tables 1 and 2. The detailed rock socket information, load test information and results and rock mass properties

Fig. 1 is an example of the side shear stress (fs) and displacement (δ) relationship for the rock socket sidewalls that is obtained from an instrumented Osterberg load test LT8718-214 in Graneros shale 2

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Table 3 Summary rock formations included in the load test database (from Asem14). Rock formation

Location

Geomaterial type

In situ features

Dargile and Anderson Creek

Melbourne, Australia

Interbedded layers of claystone, siltstone and sandstone

Well-defined beddings and includes minor and major faults, joint sets are approximately orthogonal, rock discontinuities (i.e., joints) are often tight and clean Poorly laminated and weathered

Graneros shale

Maquoketa shale Carlile shale Oread limestone

Central regions of Kansas, east of Colorado, Nebraska and South Dakota, United States Iowa, United States

Non-calcareous shale which ranges from silty to fine sandy shale

Kansas, United States Kansas, United States

Shale and limey shale Alternating layers of limestone and shale Interbedded relatively thin layers of claystone, siltstone and sandstone Claystone and shale with occasional layers of siltstone and sandstone Fine-grained carbonate mud

Severy shale

Kansas, United States

Denver, Arapahoe and the Lower Dawson Austin Chalk

Colorado, United States

Cherokee and Marmaton

Missouri, United States

Caddo

Oklahoma, United States

Austin, United States

Shale

Sandstone, siltstone, underclay, limestone and coal beds Shale interbedded with limestone

Caddo

Oklahoma, United States

Shale interbedded with limestone

Queenston

Ontario, Canada

Shale

Eagle Ford

Dallas, United States

Gray to brown sandy clay with intermittent limestone layers and shale Shale

Pierre shale Ocala limestone Fort Thompson Belle Fourche

North and South Dakota, United States Florida, United States Florida, United States

Limestone Limestone and calcareous sandstone Marine shale

Table 3 (continued ) Rock formation

Location

Geomaterial type

In situ features

Siltstone

Weathered, horizontally bedded and joints spacing is 100–150 mm Characterized by folding and faulting due to the past tectonic activities with heterogeneous weathering profile Characterized by massive structure

Scoresby siltstone

South Dakota, United States Victoria, Australia

Jurong

Singapore

Conglomerate, sandstone, shale, mudstone, limestone and dolomite

Hawkesbury sandstone

Sydney, Australia

Sandstone

formation at US 36 over Republican River test site in Kansas, United States. This figure represents the mobilization of the shear stresses on the rock socket sidewalls with shear displacement. The initial shear stiffness (Ksi) is defined as the slope of the tangent line to the initial portion of the fs-δ relationship as shown in Fig. 1. Fig. 2 is an example of the contact pressure (q) and displacement (δ) relationship for the rock socket base that is obtained from an Osterberg load test LT885414 in Pennsylvanian age shale at I 235 over Des Moines River, Iowa, United States and represents a typical q-δ relationship for the base of rock sockets in weak rocks. This figure represents the mobilization of the contact pressure at the rock socket base with vertical displacement. The initial normal stiffness (Kn) is defined as the slope of the “initial tangent” line to the initial part of q-δ relationship as shown in Fig. 2. The following should be noted in relation to the initial part of the fs-δ and q-δ relationships: (i) the initial part of the fs-δ and q-δ relationships is not concave upward, which is commonly true for most of the rock sockets in the database. This indicates that the rock socket base and sidewalls were normally cleaned, and (ii) the variation of the shear stresses (fs) on the rock socket sidewalls and contact pressures (q) at the rock socket base with displacement at the early stages of loading is linear and may be represented by the slope of the tangent lines that are shown in Figs. 1 and 2 (Ksi and Kn, respectively) and therefore the theory of elasticity may be used to analyze the initial portion of the fs-δ and q-δ relationships.

Beddings are cut by two systems of joints which intersect at oblique angles Weathered Irregularly bedded

Porosity of the rock mass formations ranges from 6 to 30% with regional fractures Horizontally bedded and dip slightly in a northwesterly direction in the Missouri area Caddo formation contain numerous iron stains resulting from chemical weathering Caddo formation contain numerous iron stains resulting from chemical weathering RQD ranges from 29 to 88% with horizontal to oblique iron stained joints

3.3. Rock mass properties The unconfined compressive strength (qu) of weak rock at each load test site was obtained from the laboratory unconfined compression tests. Information on weak rock specimen size used in the laboratory deter­ mination of qu is not available, and thus a size correction31,10 cannot be performed. The deformation modulus of weak rock mass (Em) is esti­ mated based on fs-δ and q-δ relationships in each load test using the method of Pells and Turner.12 For side resistance data, Em may be ob­ tained from: Em ¼ 2πLKsi I

(4)

where L is the rock socket length and Ksi is the initial shear stiffness of the rock socket sidewalls. For the base resistance data, Em may be ob­ tained from:

Thinly bedded with iron stained fractures and joints Poorly to well indurated

Em ¼ πðB = 2ÞKn I

(5)

where B is the foundation diameter and Kn is the normal stiffness of the rock socket base. The values of the embedment reduction factor (I) in Eqs. (4) and (5) are obtained from Pells and Turner.12 The estimated Em values are shown in Fig. 3. Fig. 3 also includes data from the in situ plate load tests reported by Chern et al.32 for weak rocks in China and Taiwan. 3

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 1. Shear stress (fs) and shear displacement (δ) relationship and definition of the initial shear stiffness (Ksi) (data from Loadtest, Inc., LT8718-214).

Fig. 2. Contact pressure (q) and vertical displacement (δ) relationship and definition of the initial normal stiffness (Kn) (data from Loadtest, Inc., LT885414).

The following should be noted in relation to the estimated Em: (i) an elastic approach is used to estimate Em based on the initial part of the fs-δ and q-δ relationships for the load test database of Asem.14 This approach is reasonable because load test data show that the initial part of most fs-δ and q-δ relationships is linear-elastic at small displacements. Addition­ ally, elastic methods have been used by other investigators to estimate Em,1 (ii) Fig. 3 shows that the values of Em estimated using the method of Pells and Turner12 are in general agreement with the available data on the deformation modulus of weak rocks obtained from Chern et al.32 for weak rocks in China and Taiwan, and (iii) Fig. 3 shows that a wide range of Em values is possible for any given qu. This wide range is expected because Em is not only a function of qu. Other factors that affect Em include the weathering condition of each rock mass, the testing methods used to measure the stress and displacement relationships, and the mode of failure for rock mass in each load test. These parameters are not the same for all cases in Fig. 3 and thus contribute to the wide ranges that are observed for estimated values of Em. The Hoek and Diederichs33 relationship: Em ¼

100; 000 1 þ eð75 GSIÞ=11

estimate GSI for the rock formations in the load test database (by solving Eq. (6) for GSI). The rock masses in the database are assumed to be undisturbed, and thus a disturbance factor D ¼ 0 is used to derive Eq. (6).33 The estimated values of GSI, along with the definition of weak rock discussed earlier, are used to consistently determine whether a rock formation can be considered as a weak rock for inclusion in the load test database. 3.4. Rock socket geometry and load test methods The diameter for each rock socket (B), the length of the shear profile (L) (i.e., length of the rock socket or distance between strain gauges when used along the rock socket sidewalls), and the depth of embed­ ment (DGS) from the ground surface to the center of the shear profile or to the base of rock socket are obtained from the reviewed load tests. The methods of load testing used in the load test databases are shown in Fig. 4. In Fig. 4(a), a load cell is placed at the base of the rock socket to separate side and base resistances. In Fig. 4(b) and (d), the side and base contributions are separated using strain gauges. In Fig. 4(c), a void or compressible base is provided at the base of the rock socket to eliminate base resistance and to directly measure side resistance. In Fig. 4(e), loads are applied in tension. Finally, in Fig. 4(f), loads are directly applied to the rock mass for the measurement of the base contact pressures in plate

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between the deformation modulus of rock mass (Em) and the Geological Strength Index (GSI), and the estimated Em in each load test is used to 4

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 3. Variation of the back-calculated rock mass deformation modulus (Em) with unconfined compressive strength (data from literature32,14).

Fig. 4. Summary of the load test methods used in the database (after Asem14).

load tests.

predictive models for Ksi and Kn as follows:

4. Model formulation

Qðx; ΘÞ ¼ γðx; θÞ þ σε

(7)

where Q is the quantity of interest or its suitable transformation into a new space to satisfy the additivity, homoskedasticity and normality

Following the model formulation proposed by Gardoni et al.,24 a probabilistic model can be formulated for the development of the 5

P. Asem and P. Gardoni

assumptions, γðx; θÞ ¼

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129 n P j¼1

Yn �1 hrj ðθÞi� ϕ LðΘÞ∝ j¼1

θj hj ðxÞ is a function expressed in terms of

σ

observed variable, x, and unknown parameter θ ¼ (θ1, θ2,…., θn), and σε is the model error that captures the model inexactness that may result from missing model parameters or inappropriate selected model form. In the model error term, σ is the standard deviation of the model error, and ε is a random variable with zero mean and unit variance. A suitable transformation of Q (Ksi or Kn) should be used to justify the assumption that σ is constant and independent of the variable x (homoskedasticity assumption), and that ε follows a normal distribution (normality assumption). These assumptions are used in the estimation of the model parameters Θ ¼ (θ, σ). In this paper, a natural logarithmic transformation of Ksi and Kn is used to satisfy these assumptions. The validity of this transformation may be checked using diagnostic plots as described by Rao and Toutenburg.34

5.3. Effect of various covariates on estimated Θ ¼ (θ, σ ) Different covariates such as rock type, test location and load test method could affect the estimated Θ ¼ (θ, σ ). The following are considered in the analyses in the subsequent sections and should be noted: (i) the effect of rock type and location – the rock sockets are constructed at various locations, but in similar fine-grained rocks with similar degrees of weathering. The data are separated based on rock types in Figs. 5–8. These figures show that different rock types exhibit similar trends. Therefore, in the analysis it was assumed that the effect of rock type and location may be insignificant as far as the proposed equations are applied to weak, fine-grained rock as defined earlier in the manuscript, and (ii) load test method – Brown et al.35 and Asem14 showed that the load test method does not introduce much variability as far as the load test database uses similar load test methods conducted in a consistent manner. In this study, we have collected data from well-documented tests where most tests follow similar procedures, i.e., rate of load application and construction method and therefore it is assumed these variables do not exert much influence on the predicted values of Θ ¼ (θ, σ ). It would be ideal, however, to subdivide the database into different load test methods and rock types and to use these smaller databases to calibrate the model parameters using a Bayesian framework and to ac­ count for between-group variation.36,37 If the original database has limited number of observations, however, subdividing based on the load test and rock type would result in databases with small number of ob­ servations, which affects the accuracy of the calibrated model parame­ ters.24 We have therefore, used a “lumped” approach where it has been assumed that separating the database into sub-groups based on rock type, location and test method will not significantly change the esti­ mated Θ ¼ (θ, σ) and have proposed similar Θ ¼ (θ, σ ) for various rock types, locations and test methods.

A Bayesian approach is used to estimate the vector of unknown model parameters Θ ¼ (θ, σ ) using a Markov Chain Monte Carlo (MCMC) approach. A Bayesian approach allows us to include different types of information in the estimation of the unknown model parameters including field data and subjective engineering judgment. In a Bayesian approach, the prior probability distribution density function (PDF) for the model parameters Θ is updated according to the following updating rule23: (8)

where f’ðΘÞ is the prior PDF reflecting our knowledge on Θ before observed data become available, and f’’ðΘÞ is the posterior PDF that represents the updated knowledge about Θ. The posterior PDF is ob­ tained by combining the information from the prior PDF and informa­ tion on Θ from the new observed data and observations that are represented by the likelihood function, LðΘÞ. The parameter k ¼ 1 R is a normalizing factor and ensures that f’’ðΘÞ is a proper LðΘÞf’ðΘÞdðΘÞ

PDF.

5.1. Prior distribution

6. Parameters influencing Ksi and Kn

The prior distribution PDF f’ðΘÞ, represents the state of our knowl­ edge about the model parameters Θ before observed data become available from new measurements or observations. This prior informa­ tion may have been obtained from laboratory test data, field tests and measurements, and subjective engineering judgment. The effect of f’ðΘÞ on the posterior PDF becomes small as the number of new measurements and observations increases. Box and Tiao23 suggested that when no previous information on Θ is available, a noninformative prior PDF may be used for the estimation of the model parameters Θ. Gardoni et al.24 showed that for a univariate model that is linear in parameter θ, the noninformative prior PDF may be represented as follows: f ’ðΘÞ∝

1

σ

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where ϕ [.] is an standard normal PDF and rj(θ) is the residual term that may be given by rj(θ) ¼ Qj- γ(xj, θ).

5. Bayesian assessment of the model parameters

f ’’ðΘÞ ¼ kLðΘÞf ’ðΘÞ

σ

The load test database is used to study the parameters that affect the values of Ksi and Kn. The parameters included in the analysis are the depth of embedment (DGS), the rock socket diameter (B), the length of shear profile along the rock socket sidewalls (L), the rock unconfined compressive strength (qu) and the rock mass deformation modulus (Em). The observations in this section are used in the development of γðx; θÞ ¼ n P j¼1

θj hj ðxÞ term in Eq. (7).

6.1. Initial shear stiffness (Ksi) of rock socket sidewalls

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At small displacements, the initial shear stiffness (Ksi) defines the rate of increase in the side shear stress (fs) with settlement (δ). The variation of Ksi with the unconfined compressive strength (qu), the estimated modulus of deformation (Em) of rock mass, rock socket diameter (B) and length of shear profile (L) are discussed.

5.2. Likelihood function The information on n available observations that is given by the vector of measured quantities Qj ¼ (Q1, Q2, …, Qn) is included in the Bayesian estimation of the model parameters Θ using the likelihood function, L(Θ). The likelihood function is equal to the probability of the observation for the given values of Θ. The formulation of the likelihood function depends on the type of data available. When the available observations are statistically independent, the likelihood function, L(Θ), may be written as:

6.1.1. Rock mass properties Different rock types are shown with different symbols in Figs. 5 and 6. Because most load tests in this study are conducted in fine-grained rocks and based on a visual assessment of the data presented in Figs. 5 and 6, it is assumed that there are no differences in behavior (i.e., measured Ksi) due to rock type. 6

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 5. Variation of the estimated initial shear stiffness (Ksi) with unconfined compressive strength (qu) and the back-calculated rock mass deformation modulus (Em) (data from database of Asem14).

related to the shear stresses that are generated on the shear surface41,1, .42 Additionally, increase in the normal stresses, although small at early stages of loading due to small dilations, will increase the confinement on the rock joints within the rock mass adjacent to the socket sidewalls, and thus will increase the shear strength of rock mass in the vicinity of the rock socket.1 Because Ksi represents the rate of generation of shear stresses on the shear planes with axial displacement of the rock socket, and Em controls the normal stresses on the rock socket sidewalls and indirectly affects the shear strength of the rock mass, Em and Ksi are related as shown in Fig. 5(b).

A rock mass consists of relatively intact blocks that are separated by individual discontinuity surfaces.29 The unconfined compressive strength (qu) represents the shear strength of the intact rock blocks that make up the rock mass. Therefore, as qu increases, the shear strength of the rock mass also increases. The measured values of Ksi represent the rate of mobilization of the shear strength along the rock socket sidewalls with the shear displacement along the shear surfaces that are commonly formed inside the rock socket sidewalls.38 Therefore, Ksi is greater for a rock mass that has a greater qu. The increase in Ksi with qu is shown in Fig. 5(a). The measured values of Em have often been related to different rock mass classification systems such as the Rock Mass Rating (RMR)39 or the Geological Strength Index (GSI).33 Therefore, as Em increases, it is generally expected that the degree of weathering of the joints and fis­ sures decreases, and their spacing increases.40,10 Consequently, it may be argued that - to some extent - Em represents the quality and hence the normal stiffness of the rock socket sidewalls (Kn,s), which controls the development of the normal stresses (confinement) on the mobilized shear planes that are parallel to the rock sock sidewalls:

σ n ¼ σ ni þ Kn;s δn ¼ σni þ

Em

� �δn ð1 þ νÞ B2

6.1.2. Rock socket size Fig. 6 illustrates the effect of rock socket size on the measured values of Ksi. Although the scatter in the data presented in Fig. 6 is significant, this figure can provide some tentative insight on the effect of rock socket size (i.e., diameter, B, and shear profile length, L) on Ksi. These con­ clusions can be updated as additional data and in situ observations become available. Fig. 6(a) shows that Ksi generally decreases with increase in the length of the mobilized shear plane along the rock socket sidewalls (L) and later remains more or less constant as L increases beyond a threshold value of 1 m. A possible explanation is that as L increases, the volume of the surrounding rock mass that is affected by the rock socket will also increase. Therefore, a greater number of discontinuity surfaces such as shears, joints and fissures interact with the rock socket, which results in decrease in Ksi.43,1 However, from visual assessment of the data, it ap­ pears that after a threshold L value of approximately 1 m is reached, the

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where σ n is the normal stress at a given vertical displacement, σni is the initial normal stress on the rock socket sidewalls, δn is the dilation of the shear surface and ν is the Poisson’s ratio. These normal stresses are 7

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 6. Variation of the estimated initial shear stiffness (Ksi) with the length of the shear surface along rock socket sidewalls (L) and the rock socket diameter (B) (data from database of Asem14).

and simple regression analyses (i) for Fig. 7a: Kn ¼ 0.902q0.17 u R2 ¼ 0.0142, and (ii) for Fig. 7(b): Kn ¼ 0.057E0.51 and R2 ¼ 0.3087, m where R2 is the goodness of fit. These results show that Kn increases monotonically with both qu and Em but the relationship between Kn and Em is stronger as indicated by a greater R2 value of 0.3087. It was mentioned that as qu increases, the shear strength of the rock mass also increases. The initial normal stiffness (Kn) defines the rate of increase in the contact pressure (q) with settlement (δ)50 which are closely related to the shear strength of rock mass. Therefore, as qu increases, Kn also increases as shown in Fig. 7(a) but the correlation is weak as shown by the low value of R2 ¼ 0.0142. The modulus of deformation (Em) is an indicator of the stiffness of the rock mass and controls the mobilization of the confining pressures around the fractured zone (failure mechanism), which forms under the rock socket base. Ladanyi52 developed an expression for the prediction of the radial pressures (qr) on the walls of spherical cavities: 2 3a pffiffiffiffiffiffi � � Em 4 K 6 pr Vio 7 qu 7 qr ¼ 6 (12) 43�pffi2ffi 0:5ð1 νÞ� 1 Vi 5 qu

volumetric density of the joints and fissures will no longer increase with L, and thus on average, Ksi will not decrease any further. Fig. 6(b) shows that the rock socket diameter (B) have a similar effect on the measured values of Ksi. This is because as B increase, the overall contact area along the rock socket sidewalls will increase and thus the rock socket will interact with a greater number of discontinuities. The size dependency of rock mass behavior has been recognized, and documented in the technical literature44,45,1,46,47,10,48,49 and is in agreement with the findings summarized in Fig. 6. 6.2. Initial normal stiffness (Kn) of rock socket base At small displacements, the initial normal stiffness (Kn) defines the rate of increase in the contact pressure (q) with settlement (δ).50 The variation of Kn with the unconfined compressive strength (qu), the estimated modulus of deformation (Em) of rock mass, the rock socket diameter (B), and the embedment depth from the ground surface (DGS) are discussed. 6.2.1. Rock mass properties Different rock types are shown with different symbols in Figs. 7 and 8. Based on a visual assessment of the data presented in these figures, it is assumed herein that there are no differences in the behavior due to the rock type that is in agreement with Zhang and Einstein.51 Fig. 7 shows that Kn is related to qu and Em, however, a stronger relationship is observed between Kn and Em as indicated by the results of

which may be used to represent the confining pressure on the bound­ aries of the failure mechanism under the rock socket base due to the increase in the axial foundation loads and the subsequent increase in the volume of the fractured rock mass under the rock socket base. In Eq.

m (12), Kpr ¼ 1þsinφ 1 sinφ is the passive earth pressure coefficient that is a m

8

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 7. Variation of the estimated initial normal stiffness (Kn) with the unconfined compressive strength (qu) and the back-calculated rock mass deformation modulus (Em) (data from database of Asem14).

function of the rock mass friction angle (φm), ν is the weak rock Poisson’s ratio, Vio/Vi represents the change in the volume of the failure mecha­ nism and a is a function of the friction angle of the rock mass (φm). Equation (12) shows that, at early stages of loading, confining pressures (qr) are directly related to the deformation modulus of the rock mass (Em). The mobilized confining pressures result in crack and joint closure, which will lead to increase in the shear strength of rock mass by increasing the rock joint wall interlock.53,54,55,1 Therefore, a rock mass with greater Em mobilizes a greater shear strength and higher contact pressures, which is in agreement with the data shown in Fig. 7(b) as well as the results of the simple regression analysis discussed above.

greater for the small diameter rock sockets (small diameter definition, B < ~0.5 m, after Horvath and Kenney58) and on average remains con­ stant after a diameter of approximately B ¼ ~0.5 m is reached. A possible explanation of this behavior is that for small diameter rock sockets (B < ~0.5 m), the rock socket base is mainly interacting with the intact rock and thus slightly greater Kn values are mobilized compared to the case when large diameter rock sockets are tested. In the case of larger diameter rock sockets, the rock socket base interacts with a larger vol­ ume of rock mass discontinuities, thereby mobilizing a slightly smaller Kn. After a diameter of approximately B ¼ ~0.5 m is reached, visual assessment of the data indicates no additional noticeable decrease in Kn. Similar size effects were observed by Bieniawski and Van Heerden45 and Goodman1 where shear strength of blocks of Cedar City quartz diorite decreased with specimen length until a threshold specimen length was reached after which the shear strength remained more or less constant. This particular size effect may be attributed to the fact that as the size of the affected zone in the rock mass increases, the volumetric intensity of the discontinuities will also increase. However, after a threshold size is reached, the volumetric intensity of discontinuities will not increase significantly with additional increases in rock socket diameter (B) and thus Kn will essentially and on average remain a constant. The small dependence of initial normal stiffness on rock socket diameter may be attributed to weathering of weak rocks. The shear strength and the deformational properties of the two components of the

6.2.2. Rock socket geometry It is expected based on the well-known size effect44,45,46,47,1,48,56,49 that as the rock socket diameter (B) increases, the zone of influence increases57,8,3 and the number of discontinuities within the failure mechanism will proportionally increase.1,3 Therefore, the shear strength and stiffness of the rock socket base will decrease until a threshold diameter (B) is reached after which these rock mass properties will remain more or less constant. The size effect becomes especially important in stiff rocks where the shear strength and deformational properties of the intact rock blocks and the discontinuity surfaces are noticeably different. Visual assessment of the data in Fig. 8(a) shows that Kn is on average 9

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Fig. 8. Variation of the estimated initial normal stiffness (Kn) with depth of embedment from the ground surface (DGS) and the rock socket diameter (B) (data from database of Asem14).

rock mass (i.e., intact blocks and discontinuities) are similar due to excessive degrees of physical and chemical weathering. The rock mass natural discontinuities are also narrow or closed as suggested by the concave downward shape of the q-δ relationships. Therefore, the pres­ ence of discontinuities does not significantly reduce the initial normal stiffness of rock mass. This may be observed by comparing Kn for small (B < ~0.5 m) and large (B > ~0.5 m) diameter rock sockets in Fig. 8(a). Lee et al.,22 using numerical models, suggested that the initial normal stiffness (Kn) of the rock socket base should decrease indefinitely with increase in the rock socket base diameter. Lee et al.22 conclusion is in contradiction with the in situ load test results shown in Fig. 8 and therefore may result in conservative estimates for Kn if used for rock sockets in weak rocks. Visual assessment of the data in Fig. 8(b) indicates that the initial normal stiffness (Kn) does not increase with increase in the embedment depth. A simple regression analysis has been performed which resulted in Kn(DGS) � 0.8 which indicates that Kn is more or less independent of DGS. Some possible explanations for this observed behavior include (i) at the early stages of load testing when Kn is measured, a well-defined failure mechanism has not been mobilized yet and the failed rock mass is confined to regions below the base of the rock socket, and does not penetrate the overburden soil or rock and thus the shear strength of the overburden will not significantly contribute to the rate of

mobilization of the contact pressures at small displacements that is represented by Kn, and (ii) the depth of embedment represents the initial state of stress around the failure mechanism, which is not necessarily the same as that corresponding to the failure state55 which governs the contact pressures53,54,.55 It should, however, be noted that the embed­ ment depth governs the mode of failure (punching shear failure) that is often observed in rock sockets in weak rocks. 7. Proposed probabilistic models Analysis of the load test data show that Ksi is related to Em, L, B, and qu. Because the deformation modulus of rock mass (Em) is related to the unconfined compressive strength (qu) (visual assessment of Fig. 3 and regression analysis in the previous sections), only Em is included in the predictive model. The proposed model is a function of hj(x) ¼ Em, L and B: ksi ¼

eθ 1

Emθ4 ¼1:00 θ2 ¼ θ3 ¼0:47

¼ 7:49

ðBLÞ

(13)

where Ksi is in units of MPa/mm. In Eq. (13), Em is in units of MPa, and B and L are in units of m. The standard deviation of model error (σ) is 0.32. The statistics for the model parameters θ1 through θ4 are summarized in Table 4. 10

P. Asem and P. Gardoni

International Journal of Rock Mechanics and Mining Sciences 124 (2019) 104129

Table 4 Statistics of model parameters for predictive model for initial shear stiffness (Ksi). Parameter

Mean 7.49 0.47 0.47 1.00 0.32

θ1 θ2 θ3 θ4

σ

Standard deviation

Correlation coefficient

0.12 0.03 0.03 0.02 0.02

1.00 0.14 0.41 0.96 0.03

θ1

θ2 1.00 0.46 0.03 0.00

θ3

1.00 0.26 0.03

θ4

1.00 0.03

Table 5 Statistics of model parameters for predictive model for initial normal stiffness (Kn).

¼ 4:43

Eθm2 ¼0:78

Km Kp

Standard deviation

θ1 θ2

4.43 0.78 0.79

0.3 0.06 0.05

Correlation coefficient θ1

θ2

σ

1.00 0.96 0.08

1.00 0.08

1.00

1.00

Table 6 Evaluation of the models for initial shear stiffness (Ksi) and initial normal stiff­ ness (Kn).

(14)

where Kn is in units of MPa/mm and Em is in units of MPa. Equation (14) may be used to predict the initial normal stiffness for rock sockets where B > ~0.5 m. Equation (14) is not a function of rock socket diameter because the visual assessment of the load test data indicates that Kn will not noticeably decrease after a rock socket diameter of B ¼ ~0.5 m is reached. The standard deviation of model error (σ ) is 0.79. The statistics for the model parameters θ1 through θ2 are summarized in Table 5. The additivity, normality and homoskedasticity assumptions for both equa­ tions are satisfied by using a natural logarithm as a variance stabilizing transformation. It should be noted that the posterior distributions of model parameters are asymptotically normal because the number of observations in our databases is quite significant.23,59,60 The correla­ tions between σ and other model parameters in Tables 4 and 5 are sta­ tistically insignificant and are essentially zero. One possible and trivial explanation for the non-zero posterior correlations of σ with other model parameters is the numerical inaccuracy due to the fitting algorithm. The following should be noted in relation to the proposed equations: (i) the functional forms of Eqs. (13) and (14) are determined using the Bayesian framework introduced in Sections 4 and 5, and (ii) both equations are functions of Em. This is because Em can account for rock mass properties (intact rock and secondary structure). Addition of the intact rock properties could potentially improve the model accuracy, however, it will add to the complexity of the proposed models. There­ fore, to keep the models as simple as possible, the intact rock properties (e.g., qu) are not included in the design equations. Different methods have been suggested for evaluation of the bias and accuracy of the predictive models.61,25,26,27 In this paper, we measure bias (λ) as: λ¼

Mean

σ

The proposed relationship for the initial normal stiffness (Kn) is shown in Eq. (14): Kn ¼ eθ1

Parameter

σ

Model

Bias (λ)

Mean Absolute Percentage Error (MAPE)

Initial shear stiffness (Ksi) Initial normal stiffness (Kn)

1.05 0.96

0.22 0.47

different variables on the measured values of Ksi and Kn were analyzed using the compiled load test databases. A Bayesian framework was used to develop probabilistic predictive models for Ksi and Kn. The following should be noted in relation to the probabilistic models proposed herein: (i) the proposed models are probabilistic in a sense that the uncertainties in the model parameters Θ ¼ (θ, σ ) are evaluated using a Bayesian framework and are reported in Tables 4 and 5, (ii) the uncertainty associated with anisotropy in the rock specimens, and variability in the test results from unconfined compression tests are not included because such information were not generally available, and (iii) measurement errors62,27,63 have not been accounted for in the analysis. Acknowledgments 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. Professor Emeritus James H. Long of University of Illinois at UrbanaChampaign has provided many useful comments and suggestions, which are acknowledged. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.ijrmms.2019.104129.

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where Km is the measured stiffness and Kp is the predicted stiffness. The model accuracy is measured using the Mean Absolute Percentage Error (MAPE)61 � n � 1 X jE½Kðx; ΘÞ� Ki j MAPE ¼ 100 (16) n i¼1 Ki where E½Kðx; ΘÞ� is the mean of the predicted stiffness and Ki is (i ¼ 1,…., n) are the corresponding measured or observed values. The analysis results are shown in Table 6. The results presented in Table 6 show that the models are accurate (i.e., represented by small calculated MAPEs) and generally unbiased (i.e., represented by bias λ being close to unity). 8. Conclusions An extensive literature review was conducted, and a comprehensive load test database was compiled. The load test data were used to esti­ mate the initial shear stiffness (Ksi) along the rock socket sidewalls and the initial normal stiffness (Kn) of the rock socket base. The effect of 11

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