Calibrated Partial Factors for Support of Wedges Exposed in Tunnels

Available online at www.sciencedirect.com

ScienceDirect Procedia Engineering 191 (2017) 802 – 810

Symposium of the International Society for Rock Mechanics

Calibrated Partial Factors for Support of Wedges Exposed in Tunnels A. El Matarawi, J. P. Harrison* University of Toronto, 35 St George St, Toronto, ON M5S 1A4, Canada

Abstract Geotechnical design is evolving to adopt the limit state design (LSD) philosophy, also known as reliability-based design (RBD). This is evident by its inclusion in geotechnical design codes (e.g. Eurocode 7). Partial factors are often used in design codes to overcome the difficulty in performing probabilistic analysis suggested by the RBD. The increasing use of RBD suggests a need to investigate the applicability of design with partial factors for various rock engineering structures; this paper will investigate their application in the design of support for a rock wedge in an underground opening. The paper provides a critical overview of the design philosophy of RBD, the components necessary for its application, and the methods by which the probability of failure may be computed. In addition, it discusses how partial factors are calibrated from RBD and how code development can be subsequently performed. This is put into context with a design example for the support of a rock wedge. © 2017 2017The A. Authors. El Matarawi, J. P. by Harrison. Ltd.article under the CC BY-NC-ND license © Published ElsevierPublished Ltd. This isby anElsevier open access Peer-review under responsibility of the organizing committee of EUROCK 2017. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017 Keywords: partial factor design and calibration; tunnel design; limit state design

1. Introduction Rock engineering design customarily uses deterministic methods with factors of safety. However, this is inappropriate as these methods do not appropriately account for the variable conditions prevalent in rock engineering. In addition, it is known that factors of safety do not always lead to safe designs [1].In the presence of variable conditions, probabilistic design – which asks “does the design satisfy a specified probability of failure given the observed variability?” – is known to be more appropriate. This has long been recognised by the structural

* Corresponding author. Tel.: +1-416-978-1634. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of EUROCK 2017

doi:10.1016/j.proeng.2017.05.247

A. El Matarawi and J.P. Harrison / Procedia Engineering 191 (2017) 802 – 810

engineering community, and fuelled the development of structural reliability – also known as reliability based design (RBD) or limit state design (LSD) – as a formal design philosophy that quantifies the probability of a structure behaving as intended. This includes ensuring stability and satisfying deformation limits, as well as any other requirements of the structure. Presently, this design approach is used globally in structural engineering, and is currently also being adopted by the geotechnical community [2, 3]. For various reasons, but mostly those associated with ease-of-use by practicing engineers, RBD was introduced into structural engineering in a deterministic format that used partial factors to account for variability. These factors were initially selected so as to produce designs that were similar to those generated by earlier approaches, but have been re-calibrated over time to better account for observed variability and hence produce more reliable and economic designs. The partial factor approach is now prevalent in RBD design codes worldwide. The adoption of RBD by the geotechnical community, together with the prevalence of the partial factor approach, suggests that there is a clear need to investigate the use of partial factors in rock engineering. This is tackled here, firstly by reviewing fundamental concepts in structural reliability – including the components needed to perform RBD and how to compute the probability of failure – and then investigating the process of calibrating partial factors and code development. The procedures are demonstrated with the aid of an example involving the design of support against wedge failure in an underground opening. 2. Structural reliability 2.1. Components of RBD RBD formalizes the approach to design by requiring x the setting of a target probability of failure Pf ,T to design for; x identification of the random variables (RVs), defined by the random vector X > X 1 , X 2 , X n @ where X i is a random variable, that govern the behaviour of the structure; and x mathematical definition of the failure state of the structure by a limit state function (LSF) g  X 0 . These are the fundamental steps of RBD. The simplest LSF is g R  S 0 , where R is the resistance and S is the load. It is convention to define the LSF such that g ! 0 defines a satisfactory state, g  0 defines an unsatisfactory state and g 0 defines the limiting state; this is shown in Fig. 1(a) for the LSF R  S 0 . The random variables within the LSF are defined by their marginal distributions and a covariance matrix. In R  S 0 , the probability density functions (pdfs) of the marginal distributions R and S may be combined to produce a joint pdf as shown in Fig. 1(b). If the LSF and the joint pdf of the RVs are superimposed as shown in Fig. 1(c), the shaded area represents the probability of failure Pf . The design objective is to ensure that Pf satisfies the target probability of failure Pf ,T , i.e. Pf d Pf ,T .

Fig. 1. Components of reliability-based analysis of LSF R − S = 0.

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2.2. Methods for computing the probability of failure Currently the probability of failure Pf may be calculated by analysis, simulation or approximation (Fig. 2). The first of these requires the joint pdf to be integrated analytically, and as this is generally not possible for the non-normal RVs encountered in practice, the method is seldom used. On the other hand, simulation methods, such as the widely-used Monte Carlo simulation (MCS), are only realistic for computationally economical problems as they may require more than 1×104 simulations to evaluate the probabilities across the domain of interest. These shortcomings mean that approximation methods are predominant. In particular, reliability methods are widely used due to their robustness. Other approximation methods include moment methods [4] and quadrature methods [5, 6]. Methods for computing Pf Analytical

Approximation

Simulation Uncertainty propagation methods

Moment methods (e.g. first-order second moment, FOSM)

Reliability-based methods (e.g. first-order reliability method, FORM)

Quadrature methods (e.g. point estimate method, PEM)

Fig. 2. Methods for computing the probability of failure in structural reliability.

2.2.1. Reliability-based methods Reliability-based methods use the reliability index E as a measure of Pf , as the two are linked via the cumulative normal distribution through the relation Pf )  E . Two methods exist within this class: first-order reliability method (FORM) and, its extension, second-order reliability method (SORM); FORM has been found to be sufficiently accurate for most applications [7], and thus is discussed here exclusively. The reliability index E is defined as the distance in standardized and uncorrelated normal space between the origin and the most probable point of failure on the limit state surface h  U 0 [8]. The function h is the mapping of the function g  X from the space of the physical RVs X to the standardized and uncorrelated normal space of the RVs U , where U >U1 , ,U n @ . The most probable point of failure, i.e. the failure state that is closest to the mean, is known as the design point. This is given by vector x d in physical space and vector u d in the standardized and uncorrelated normal space. An optimisation scheme is used to find u d , and can be summarised as

find

ud

to minimize E= UU T subject to

h ud

(1)

0

Although a closed-form solution for E exists if g  X 0 is linear and X is normal, optimization schemes are required to solve g  X 0 in nonlinear cases. Amongst the methods for this are HLRF [9] and the Excel Solver add-in, the latter of which has found popularity within the geotechnical community [10-12]. 2.3. Derivation of partial factors from RBD and code calibration The use and role of factors of safety was briefly introduced in Section 1. These factors may be applied to R or S (or both), or applied to specific values of the variables in X ; the first case represents a global factor of safety, whereas the latter represents partial factors. Historically, global factors have been selected somewhat empirically,

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in order to generate designs that are considered acceptable in terms of some, possibly subjective, assessment. However, partial factors are calibrated from RBD to ensure the design satisfies a prescribed probability of failure. This allows the design of safe structures in lieu of performing a full reliability analysis using the methods discussed above. The factors are applied to each X i , with their magnitude reflecting the variability of the particular random value and its sensitivity on the probability of failure. The design analysis proceeds deterministically with these factored values. Now, the design point x d , in addition to being the most probable point of failure, exists on the limit state surface g  X 0 . Thus, the vector X that corresponds to x d contains values of the random values that, in combination, satisfy the target probability of failure. Because the target probability is very small (often of the order of 1×10 -7), each xd ,i will be an extreme value of the distribution of xi . If the distribution of xi is characterised by one particular value (say, the mean), then the partial factor can be calibrated such that if applied to the characteristic value it will return the design value. By this definition, the partial factor J X i associated with RV X i is given by

J Xi

xd ,i xk ,i

(2)

where xd ,i is the design value for X i and xk ,i is a representative value of X i known as the characteristic value. The designer applies partial factors to the characteristic values of the RVs, and proceeds with a deterministic analysis to obtain a design for which g 0 . If the partial factors have been correctly calibrated, this design will be the same as that which would be obtained from the reliability analysis used in the calibration. The convention in structural reliability is to define partial factors for resistance-type variables as J X i xk ,i xd ,i and J X i xd ,i xk ,i for load-type variables, such that J X i ! 1 . However, in rock engineering, and particularly in tunnel design, the distinction between load and resistance-type variables is not clear [13]. As a result, and to avoid confusion, partial factors should be defined as in Eq. (2), so that the factors for favourable and unfavourable variables are d 1 and t 1 respectively. The process by which partial factors are obtained is known as partial factor code calibration, and is used by code authorities to produce sets of rules for similar structures that optimise designs to a particular objective, often reliability [14, 15]. These sets of rules, known as a ‘code format’, are applied to a class of structures, known as a ‘structure class’. These code formats can be as simple as the code authority choosing a set of RVs > X1 , X 2 , , X n @ , corresponding characteristic values ¬ª xk ,1 , xk ,2 , , xk ,2 ¼º and partial factors ª¬ J X1 , J X 2 , , J X n º¼ . These rules are written so that their application to a given structure class produces designs with similar levels of reliability. The code calibration process starts by defining structure classes that are mechanically and statistically similar. Representative cases of the class are chosen, and reliability analysis is performed on each case in order to obtain the relevant partial factors. These factor values are used as the basis for finding the ‘optimal’ set of partial factors which represents the entire structure class. By this means one set of partial factors is developed that minimises the difference, across the class, between the reliability of designs obtained using partial factors and the target reliability. 3. RBD application and code calibration to design of rock wedge support The design of support for a rock wedge in an underground circular opening is used here to demonstrate the application of RBD, the calibration and application of partial factors, and the steps involved in the development of a design code. A 2D case is considered, where the wedge is formed by two discontinuities defined by their dip angles D1 and D2 (Fig. 3). In this example, a tunnel radius R0 = 6 m is used, and the largest wedge is formed by the intersection of two persistent discontinuities tangent to the tunnel perimeter. The support consists of rock bolts to support the self-weight of the rock wedge, where loading on the rock bolts is assumed only to be in tension.

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3.1. Application of RBD The LSF for this case is

g

R  J rVw

0

(3)

where R is the tensile resistance provided by the rock bolts (kN), J r is the unit weight of the rock (kN/m3), and Vw is the volume of the rock wedge per unit length of excavation (m3/m). The mean R is the design parameter, and a value is required that gives a target probability of failure of Pf ,T 0.35% , i.e. a reliability index ET 2.7 . Variability is assumed in the resistance R (i.e. the rock bolt’s tensile capacity), the unit weight of the rock J r , and in the volume of the rock wedge Vw . Variability in the rock wedge volume is a function of the variability of the dip angles D1 and D2 of the discontinuities. For this demonstration R and J r are assumed to be normally distributed, and the discontinuity dip angles D1 and D2 are assumed to follow a bounded four-parameter beta distribution to ensure that 0 d D d 90q . The parameters of the RVs are shown in Table 1. Table 1. Random variable characteristics for wedge example. coefficient of variation, CoV 9 P

[lower bound, upper bound]

To be determined

0.1

N/A

27 kN/m3

0.02

N/A

4-parameter beta

30°

0.2

[0°, 90°]

4-parameter beta

45°

0.1

[0°, 90°]

Distribution type

Mean μ

Resistance R

Normal

Rock unit weight γr

Normal

Discontinuity 1 dip angle α1 Discontinuity 2 dip angle α2

Random variable

A

α1

α2

discontinuity 2

discontinuity 1

ROCK WEDGE B

R0

D C

90 − α1

α1 + α2

90 − α 2

Wedge volume D D 1 Vw  AC  BD  SR02 ¨§ 1 2 ¸· , 2 © 360 ¹ where sin  D1  D 2 BD R0 , cos  12  D1  D 2

AB

sin  12  D1  D2 sin  D1  D2

AC Fig. 3. Schematic of rock wedge example.

AB 2  R0 2

˜ BD

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A. El Matarawi and J.P. Harrison / Procedia Engineering 191 (2017) 802 – 810

Fig. 4. Distributions of the discontinuity dips and the associated wedge volumes.

The distributions of the discontinuity dip angles and simulated wedge volumes (found using MCS), together with the normal Q-Q plot of log Vw , are shown in Fig. 4. The Q-Q plot show pairs of theoretical and corresponding simulated quantiles, with the quantiles being equal for points that lie on the dashed line. As the Figure shows, log Vw follows a normal distribution, and hence the simulated wedge volumes follow a lognormal distribution. The computed mean and standard deviation for log Vw are PlogVw 1.39 m3/m and 9logVw 0.34 m3/m, and these together with the mean values in Table 1 are used as inputs to a FORM analysis to find the mean R that satisfies the design requirement of ET = 2.7. FORM is undertaken by applying the Excel Solver add-in to Eq. (1). As FORM is implemented in standardised and uncorrelated normal space, the LSF g  X 0 of Eq. (3) has to be transformed to Jw

R

h U

PR  U R 9R  PJ

r

Vw

 U Jr 9 Jr







ªexp P  U log Vw 9 log Vw º log Vw ¬ ¼

0 (4)

The FORM results are tabulated in Table 2. An assessment of different mean resistances found that PR = 282 kN gave ET = 2.7. As the design resistance xd , R is less than PR , this is a favourable variable for the LSF and thus needs to be factored down. Conversely, the design values for the unit weight of the rock xd , J r and the wedge volume xd ,Vw are greater than their means, showing that these are unfavourable and thus must be factored up. Table 2. Results of FORM for wedge support example. Parameter Value

Standardized design point ud ud

¬ªud ,R , ud ,Jr , ud ,Vw ¼º

> 0.84, 0.38, 2.55@

Physical design point xd

xd

¬ª xd ,R , xd ,Jr , xd ,Vw ¼º

ª¬258.4 kN, 27.5 kN/m , 9.4 m /mº¼ 3

3

Reliability index β (Pf) 2.69 (0.36 %)

For computing the partial factors using Eq. (2), we use mean values as characteristic values for the normal variables R and J r , and the median OVw for the lognormal variable Vw . This is because computing Vw using the mean values of D1 and D2 returns the 50th percentile. The partial factors, computed using the design values in Table 2 are presented in Table 3. These values confirm that resistance is factored down, and the unit weight

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of rock and wedge volume are factored up in the design. As noted earlier, applying these partial factors to the characteristic values of the RVs produces a design that satisfies g 0 . Hence, the design satisfies the relation

 J RPR   J J PJ r

r

J

Vw

OVw



0

(5)

which can be rearranged to give the mean resistance as

PR

xk , R

P

Jr

OVw

 J

Jr

JVw J R



(6)

Table 3. Computed partial factors for rock wedge example.

Parameter

Partial factors

Value

γ

ª¬ J R , J Jr , JWw º¼

[0.92, 1.02, 2.31]

For demonstration, OVw computed using the equations shown in Fig. 3 using D1 30q and D2 45q is 4.06 m3/m. Using P J w = 27 kN/m3 as shown in Table 1 and the partial factors of Table 3, Eq. (6) returns a mean resistance value of 281 kN/m. This is the same value as that used in the partial factor calibration, and which is known to satisfy the target reliability of ET = 2.7. This demonstrates how a partial factor calculation can replicate a full reliability analysis of this particular design example. However, this is only one specific design scenario, and so it is now necessary to move on to develop a code format that could be applied to many scenarios. 3.2. Code calibration for design of wedge support In this demonstration, only two structure classes will be calibrated. The classes will be characterized on the basis of discontinuity dip angle, and will assume rock unit weight characterised by a mean and standard deviation of 27 kN/m3 and 0.54 kN/m3 respectively, and a resistance with a coefficient of variation (CoV) of 0.1. The structure class definitions are presented in Table 4, and show that SC2 differs from SC1 only in the variability of D2 . These structure classes have been arbitrarily chosen, simply to allow the process to be demonstrated. In practice, the setting of structure classes is part of the code development process: if the domains are too small, then many structure classes will be required and this will result in a cumbersome design code; on the contrary, domains that are too large will lead to sets of partial factors that may produce uneconomic designs in certain parts of the parameter space. Table 4. Example structure class definitions. Discontinuity dip 1

Discontinuity dip 2

Structure class

mean

CoV

mean

CoV

SC1

30° – 45°

0.2 – 0.3

45° – 60°

0.1 – 0.2

SC2

30° – 45°

0.2 – 0.3

45° – 60°

0.2 – 0.3

To calibrate a set of partial factors for a structure class, representative cases within the domain are analysed. Here, as the number of random variables is small, extreme cases comprising combinations of the upper and lower bounds of each of the parameters are used as design scenarios (Table 5 and Table 6). Partial factors were found using the procedure given above, and those values that satisfy ET = 2.7 are also shown in these Tables.

809

CoV

mean

CoV

1

30

0.2

45

0.1

0.92

1.02

2.31

2

30

0.2

45

0.2

0.94

1.01

3.38

3

30

0.2

60

0.1

0.92

1.02

4

30

0.2

60

0.2

0.94

5

30

0.3

45

0.1

6

30

0.3

45

0.2

7

30

0.3

60

8

30

0.3

9

45

10 11

Rock unit weight γw

mean

Resistance R

Design Discontinuity 1 Discontinuity 2 Partial factors scenario dip α1 dip α2 Wedge volume Vw

Design Discontinuity 1 Discontinuity 2 Partial factors scenario dip α1 dip α2 Rock unit weight γw

Table 6. SC2 design scenarios and computed partial factors.

Resistance, R

Table 5. SC1 design scenarios and computed partial factors.

Wedge volume Vw

A. El Matarawi and J.P. Harrison / Procedia Engineering 191 (2017) 802 – 810

mean

CoV

mean

CoV

1

30

0.2

45

0.2

0.94

1.01

3.38

2

30

0.2

45

0.3

0.95

1.01

5.04

2.32

3

30

0.2

60

0.2

0.94

1.01

3.52

1.01

3.52

4

30

0.2

60

0.3

0.95

1.01

4.50

0.94

1.01

3.28

5

30

0.3

45

0.2

0.95

1.01

4.32

0.95

1.01

4.32

6

30

0.3

45

0.3

0.96

1.01

6.01

0.1

0.94

1.01

3.16

7

30

0.3

60

0.2

0.95

1.01

4.37

60

0.2

0.95

1.01

4.37

8

30

0.3

60

0.3

0.96

1.01

5.45

0.2

45

0.1

0.93

1.02

2.80

9

45

0.2

45

0.2

0.94

1.01

3.71

45

0.2

45

0.2

0.94

1.01

3.71

10

45

0.2

45

0.3

0.96

1.01

5.28

45

0.2

60

0.1

0.93

1.02

2.89

11

45

0.2

60

0.2

0.95

1.01

4.16

12

45

0.2

60

0.2

0.95

1.01

4.16

12

45

0.2

60

0.3

0.96

1.01

5.27

13

45

0.3

45

0.1

0.95

1.01

4.25

13

45

0.3

45

0.2

0.96

1.01

5.29

14

45

0.3

45

0.2

0.96

1.01

5.29

14

45

0.3

45

0.3

0.96

1.01

7.10

15

45

0.3

60

0.1

0.95

1.01

4.39

15

45

0.3

60

0.2

0.96

1.01

6.03

16

45

0.3

60

0.2

0.96

1.01

6.03

16

45

0.3

60

0.3

0.96

1.01

7.66

It is now necessary to determine one set of partial factors that is representative of the entire structure class. There are a number of approaches to this [15]; here, we have elected to use the simplest approach of using the mean partial factor for each of the random variables. The resultant code format for SC1 and SC2, which are both calibrated in the manner given above, is presented in Table 7. It is seen that due to the greater variability in the dip of discontinuity 2, which translates to greater variability in the wedge volume, the partial factors for SC2 are greater in magnitude than those for SC1. In addition, it is seen that cases within either structure class designed with the corresponding partial factors display a large range of probabilities of failure, of up to three orders of magnitude. This means that the design may be either significantly conservative from the intended probability of failure of 0.35 %, or under-designed. This implies that smaller structure classes are needed to create structure classes of more uniform levels of reliability. These results demonstrate how code calibration may be performed for a simple and idealised 2D wedge problem. However, it is seen that a single set of partial factors is unable to capture the reasonably small structure classes defined. Many more classes will be needed if their domains are to be reduced, or if the additional variables required for 3D wedges are introduced. These will significantly increase the complexity of the code, and so it seems that the development of a comprehensive and simple design code using a partial factor format is impractical.

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A. El Matarawi and J.P. Harrison / Procedia Engineering 191 (2017) 802 – 810 Table 7. Optimized partial factors for design code.

Structure class

Rock bolt Rock unit weight tensile strength

Discontinuity dip 1

Discontinuity dip 2

Partial factors

Pf achieved (%)

Mean

mean

JR

min

mean CoV

(kN/m3)

CoV

SC1 0.1 SC2

27

CoV

CoV

J Jr

JVw

max

30° – 45° 0.2 – 0.3 45° – 60° 0.1 – 0.2 0.94 1.01 3.76 0.0033

2.0

30° – 45° 0.2 – 0.3 45° – 60° 0.2 – 0.3 0.95 1.01 5.07 0.022

1.4

0.02

4. Conclusions Reliability-based design is necessary for proper quantification of structural safety when uncertainty is present, as is the case with rock engineering. For routine application by practicing engineers it has been found useful to cast RBD into a deterministic form, but this requires calibration of partial factors such that the resulting designs have prescribed probabilities of failure. In this paper, the calibration of partial factors from RBD, their application in design and the method by which code calibration is performed were demonstrated for a simple design problem of wedge support in an underground opening. In addition, the efficacy of the calibrated partial factors was demonstrated with a numerical example. However, it is apparent from even this simplest of designs that the traditional partial factor format will not be reasonable for code development in rock engineering. Either rock engineering design will require full reliability assessment on a case-by-case basis, or an alternative bridge between RBD and deterministic analysis needs to be developed. Further investigations by the rock engineering community are needed in order to allow reliability assessment of safety to be adopted.

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