Some Observations on Reliability-Based Design of Rock Footings

Widjojo A. Prakoso, A.M.ASCE 1 and Fred H. Kulhawy, Dist.M.ASCE 2 1Lecturer, Civil Engineering Department, University of Indonesia, Depok 16424, Indonesia; email: wprakoso@eng.ui.ac.id 2Professor Emeritus, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY, USA; email: fhk1@cornell.edu ABSTRACT

This paper continues our development of a simplified reliability-based design method for rock footings in an LRFD format. Two issues are addressed herein: the robustness of the empirical model and the proper approach to evaluate the rock strength variability. Results of simulations using three calculation models indicate that the empirical model is reasonably conservative and that the strength variability to be used in the calculations is that of the rock mass, not the intact rock. INTRODUCTION

The development of a reliability-based design (RBD) method for the ultimate

limit state (ULS) of rock footings in compression, calibrated for transmission line or similar structures subjected to ASCE wind load provisions (Task Committee 1991), was described briefly by Prakoso and Kulhawy (2003). The proposed method was in a Load and Resistance Factor Design (LRFD) format and consisted of two compo-nents: an empirical calculation model and the intact rock strength and its variability.

This paper addresses two issues that were not discussed in detail in the 2003 paper: the robustness of the empirical model and the proper approach to evaluate the rock strength variability. After presenting the proposed LRFD model, the variability of the rock strength is discussed, and the calculation models are assessed. Recom-mendations then are given, particularly for evaluating the rock strength variability.

CAPACITY OF ROCK FOOTINGS

The compression capacity of rock footings (Qc) is a function of the tip resis-

tance (Qtc), side resistance (Qsc), and footing weight (W). Because of their relative magnitude, Qsc and W commonly are neglected in practice, so Qc is given by: Qc = Qtc + Qsc - W ≈ Qtc (1)

The compression capacity involves a complex interaction among the intact structure, discontinuities, and imposed stress effects in the rock mass within the influence zone of the footing. This interaction typically is difficult to measure and estimate, and consequently the use of theoretical solutions require considerable

judgment. As a practical alternative, it is assumed commonly that Qc can be related directly to the intact rock uniaxial compressive strength (qu), as follows:

Qc = 0.25 π B2 Nc

* qu (2) in which B = footing diameter and Nc

* = empirical bearing capacity or resistance factor. The inherent assumptions within this model are that Nc

* represents the “failure mode” and that qu represents the material conditions. It is recognized that the uncertainties associated with the calculation model and the intact rock strength should be evaluated at every specific site. However, in the absence of site-specific data, guidelines on the probable variation resulting from the calculation models are useful as first-order approximations.

The development of Nc* was described by Prakoso and Kulhawy (2002),

using twelve field axial load tests on circular footings in four fine-grained sedimen-tary and igneous pyroclastic rock sites. The back-calculated Nc

* ranged from 2.40 to 14.8. Three tests yielded very high Nc

* values (Nc* = 8.35 – 14.8) and, to retain some

degree of conservatism, these values were not considered in further statistical and regression analyses. The remaining nine test data ranged from 2.40 to 5.21, and the mean of the footing resistance factor (mNc*) and its coefficient of variation (COV) or COVNc* are 3.46 and 26.8%, respectively, as shown in Figure 1.

LRFD FOR ROCK FOOTINGS

A reliability-based design equation in the following LRFD format was used in the calibration of the ULS for rock footings:

F50 = Qcn ΨcULS (3)

in which F50 = 50-year return period wind load (Task Committee 1991) and Ψc

ULS = resistance factor for ULS. The nominal compression capacity then is defined as:

Qcn = 0.25 π B2 mNc* mqu (4) in which mNc* = nominal empirical footing resistance factor and mqu = mean intact rock uniaxial compressive strength.

The calibration of ΨcULS was performed for three domains of COV of qu

(COVqu) that represent the overall rock strength variability, and each was calibrated to achieve a target reliability index (βT) of 3.2. The results are given in Table 1 and show that Ψc

ULS decreases with increasing COVqu. A higher value of COVqu gives a less reliable capacity estimate, and therefore a lower Ψc

ULS is required to achieve the design βT. The calibration details are discussed by Prakoso and Kulhawy (2003).

In traditional allowable stress practice, rock footings are designed as follows:

F50 = Qcn / FS (5)

0 1 2 3 4 5 6

Bearing Capacity Factor, Nc*

0

2

4

6

No.

Obs

erva

tions

Mean =COV =

m =

3.4626.8%9

0

Log-NormalDistribution

0 40 80

COV of qu (%)

1

2

3

4

Rel

iabi

lity

Inde

x, β

FS = 2FS = 3

Figure 1. Footing empirical Nc

* Figure 2. Effect of COVqu on β (Prakoso & Kulhawy 2002)

Table 1. Compression resistance factors (Prakoso & Kulhawy 2003) COVqu (%) Ψc

ULS 10 – 30 0.36 30 – 50 0.29 50 – 70 0.23

in which FS = global factor of safety (typically between 2 and 3). However, the implicit reliability index (β) in practice is shown in Figure 2. As can be seen for a wide range of COVqu, the range of β is between 2.06 (probability of failure, pf = 1.99%) to 3.44 (pf = 0.03%), and β is higher for a higher safety factor. Furthermore, Figure 2 shows that β is influenced significantly by COVqu.

The empirical Nc* might change with a change in the data base, and therefore

the effect of Nc* statistics on β also was evaluated. For changes of mNc* from 2.50 to

4.50, β changes from 2.752 to 2.726 (empirical mNc* = 3.46, β = 2.723). For changes of COVNc* from 20% to 35%, β changes from 2.816 to 2.637 (empirical COVNc* = 26.8%, β = 2.796). These results show that the statistics of Nc

* do not affect β significantly, specifically compared to the effect of COVqu. Accordingly, the proposed RBD equations should not change significantly as more data on footing capacity become available and the statistics of Nc

* change.

VARIABILITY OF ROCK MECHANICAL PROPERTIES

The variability of rock mechanical properties can be evaluated in terms of the intact rock or the rock mass. For the intact rock, the variability has been assessed by Prakoso and Kulhawy (2011). The COV values for ranges of mean property values (strength properties qu, qt-Brazilian, and Is, and deformation property Et-50) are shown as Figure 3. Overall, the mean COV values tend to decrease with increasing mean property values, from 10 - 45% for weaker rocks to 5 - 15% for stronger rocks.

Mean Mech. Property Values (MPa)

0

10

20

30

40

50M

ean

CO

V of

Inhe

rent

Var

. (%

)

Weaker Stronger

0 20 40 60 80

COV of Inherent Variability of Em (%)

0

10

Num

ber o

f Obs

erva

tions

Mean =S.D. =

m =

47.9 %21.8 %14

Figure 3. Mean and COV mean Figure 4. Variability of rock mass of intact rock mechanical properties modulus (Prakoso & Kulhawy 2004b) (Prakoso & Kulhawy 2011)

0 4 8 12 16Confining Stress, σ3 (MPa)

0

20

40

60

80

CO

V of

(σ1

- σ 3

) f (%

)

0 30 60 90Discontinuity Angle, θ

0.0

0.2

0.4

0.6

0.8

1.0

SRbl

ock

*

o o o

0 2 4 6 8 10Number of Discontinuity

0.0

0.5

1.0

1.5SR

bloc

k

Figure 5. Artificial rock block strength variability (Prakoso & Kulhawy 2004b)

For the rock mass, the property variability has been examined by Prakoso and Kulhawy (2004b). Two data groups were evaluated: field load tests and laboratory, artificial rock block tests. For the field tests, fourteen sets of rock mass Young’s modulus (Em) were back-calculated. The COV of Em (COVEm) ranged from 10 to 80%, and the mean and S.D. of COVEm are 47.9% and 21.8%, respectively, as shown in Figure 4. For the artificial rock block tests, the effect of rock mass structure on its strength could be estimated, as shown in Figure 5. These data show: (a) variability is most significant for rock masses with low confining stress, in which COV varies from 10 to 75%, (b) variability is most significant for discontinuity angles between 40° and 60°, and (c) variability tends to increase with increasing number of discontinuities.

Overall, the mechanical property variability for intact rock tends to be lower than that for the rock mass. The bearing capacity is actually controlled by the rock mass, and therefore the variability of the rock mass should be assessed, in addition to that of the intact rock, in the determination of the COVqu for Ψc

ULS.

EVALUATION OF BEARING CAPACITY FACTOR

Forty sets of Monte Carlo simulations were performed on three theoretical bearing capacity models to evaluate the possible range of Nc

* and to evaluate the significance of variability of strength parameters on Nc

*. The models are shown in Figure 6 and include: (a) the general wedge (Model 1) (Prakoso & Kulhawy 2006), (b) the rock mass with two ubiquitous, closed discontinuity sets (Model 2) (Prakoso & Kulhawy 2004a), and (c) the rock mass with open vertical discontinuities that consists of three transitional failure modes from uniaxial to splitting to wedge (Model 3) (Prakoso & Kulhawy 2006). Two thousand simulations were performed for each set, using the parameters given in Table 2. For all three models, the rock

B

Sj

Sj >> B General Wedge

mθ2 = 90°

Discontinuity Set 2

B

mθ1 = 0° Discontinuity Set 1

a) Model 1 b) Model 2

B

Sj

Sj < B Uniaxial Compression

B

Sj

Splitting Sj > B

B

Sj

Sj >> B General Wedge

c) Model 3

Figure 6. Theoretical models for bearing capacity of rock footings

Table 2. Discontinuity parameters for Monte Carlo simulations Model Discontinuities

1. General Wedge - 2. Two Ubiquitous,

Closed Discontinuity Sets

Orientation of discontinuities θ1 = -10° to 10°, θ2 = 80° to 100° Strength of discontinuities φj/φr = 0.5-0.7 to 0.8-1.0 cj/cr = 0.05-0.15 to 0.2-0.3

3. Open Vertical Discontinuities

Vertical discontinuity spacing Mean Sj/B = 2 to10 COV Sj/B = 20% to 80%

friction angle is characterized by a beta distribution with mean φr = 35°-50°, COV φr = 5-15%, lower bound = 27°, and upper bound = 70°. The discontinuity properties for Model 2 follow a uniform distribution, while those for Model 3 follow a log-normal distribution; detailed descriptions of the properties are given in each respective reference. For each simulation, Nc

*sim was calculated using the following:

Nc*

sim = qult-sim / qu-sim (6)

in which qult-sim = simulation bearing capacity and qu-sim = simulation intact rock uniaxial compressive strength. Subsequently, the mean and COV of Nc

* sim (mNc*sim

and COVNc*sim) of each set were evaluated. For Models 2 and 3, each mNc*sim value represents a different “failure mode” or, in terms of the proposed RBD method, each mNc*sim represents a possible mNc*. On the other hand, each COVNc*sim value repre-sents the effect of rock strength variability or, in terms of the proposed RBD method, each COVNc*sim would represent a possible value of COVqu.

Values of mNc*sim for the models are given in Figure 7, which shows that as mφ increases, mNc*sim increases. However, the increase for Model 1 is much more significant than for the other models, suggesting that, once discontinuities are considered explicitly, the intact rock properties do not control the behavior too significantly. The range of mNc*sim shown in Figure 7 is similar to the empirical Nc

* values discussed previously. The results of Models 2 and 3 suggest that mNc* of 3.46 is reasonably conservative for a wide range of rock types and conditions.

The effect of discontinuity properties on mNc*sim was evaluated as well, as shown in Figure 8 for Model 2 and in Figure 9 for Model 3. These figures show that the discontinuity properties have a relatively small effect on mNc*sim, which suggests that mNc* of 3.46 is reasonably conservative for a wide range of rock structures.

The effect of rock friction angle variability (COVφ) on COVNc*sim is given in

35 40 45 50

Mean Rock Friction Angle, mφ (o)

0

5

10

15

20

25

Mea

n of

Nc* si

m

Model 1Model 2Model 3

Discontinuity Strength Parameters

0

5

10

15

20

25

Mea

n of

Nc* si

m

φj / φrcj / cr

φj / φrcj / cr

0.5-0.7 0.05-0.15

0.8-1.0 0.2-0.3

= =

0.6-0.8 0.1-0.2

0.7-0.9 0.15-0.25

Figure 7. Effect of models on mNc*sim Figure 8. Effect of Model 2 discon- tinuity strength parameters on mNc*sim

2 4 6 8 10

Mean Spacing (Sj / B)

0

5

10

15

20

25M

ean

of N

c* sim

0 5 10 15 20

COV of Rock Friction Angle, COVφ (%)

-20

-10

0

10

20

Cha

nge

in C

OV N

c*si

m (%

)

Model 1Model 2Model 3

Figure 9. Effect of Model 3 discon- Figure 10. Effect of Model 2 discon- tinuity spacing on mNc*sim tinuity strength parameters on mNc*sim

Discontinuity Strength Parameters

0

20

40

60

80

100

CO

V of

Nc* si

m (%

)

φj / φrcj / cr

φj / φrcj / cr

0.5-0.7 0.05-0.15

0.8-1.00.2-0.3

= =

0.6-0.8 0.1-0.2

0.7-0.9 0.15-0.25

Disc. Orientation & Spacing Param.

0

20

40

60

80

100

CO

V of

Nc* si

m (%

)

Model2: θ1 Model3: COVSj

θ1 COVSj

-5o - 5o

20%-15o - 15o

80%

= =

-10o - 10o

40%

Figure 11. Effect of discontinuity Figure 12. Effect of discontinuity strength parameters orientation and spacing parameters Figure 10 and shows a significant effect for Model 1, but not for Models 2 and 3. Furthermore, as shown as Figures 11 and 12, the discontinuity properties have significant effects on COVNc*sim. This observation suggests that, for most cases, the variability of intact rock strength would not affect significantly the variability of the bearing capacity or, in terms of the proposed RBD method, the COVqu range to be used in determining Ψc

ULS should not be determined solely based on the intact rock strength variability. On the other hand, the variability of discontinuity properties would affect the variability of bearing capacity significantly and would have to be considered in the determination of the COVqu for Ψc

ULS.

SUMMARY AND CONCLUSIONS

This paper continues our development of a simplified reliability-based design method for rock footings in an LRFD format. Two issues were addressed herein: the robustness of the empirical model and the proper approach to evaluate the rock strength variability. The results of simulations indicate that the empirical calcula-tion model is reasonably conservative. In any case, the footing reliability would not be influenced significantly by the model statistics. The results of an extensive evaluation of rock properties, and the simulation results, suggest that the variabil-ity of rock mass strength should be used in evaluating the rock variability in the proposed LRFD format. REFERENCES Prakoso, W.A. and Kulhawy, F.H. (2002). Uncertainty in capacity models for foun-

dations in rock. Proc. 5th North American Rock Mech. Symp., Toronto, 1241-1248.

Prakoso, W.A. and Kulhawy, F.H. (2003). Reliability-based design for rock footings. Proc. Int. Workshop on Limit State Design in Geotech. Eng. Practice, Cam-bridge, 23-24 (extended abs) + 5 p (on CD).

Prakoso, W.A. and Kulhawy, F.H. (2004a). Bearing capacity of strip footings on jointed rock masses. J. Geotech. Eng., ASCE, 130(12), 1347-1349.

Prakoso, W.A. and Kulhawy, F.H. (2004b). Variability of rock mass engineering properties. Proc. 15th Southeast Asian Geotech Conf., Bangkok, 97-100.

Prakoso, W.A. and Kulhawy, F.H. (2006). Capacity of foundations on discontinuous rock. Proc. 41st U.S. Symp. Rock Mech., Golden, Paper 06-972, 7 p. (on CD).

Prakoso, W.A. and Kulhawy, F.H. (2011). Variability of intact rock mechanical properties. (to be submitted)

Task Committee on Structural Loadings. (1991). Guidelines for electrical transmis-sion line structural loading. Manual & Report on Eng. Practice 74. ASCE, Reston, 139 p.