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Reliability-based design for transmission line structure foundations
Computers and Geotechnics 26 (2000) 169±185 www.elsevier.com/locate/compgeo
Reliability-based design for transmission line structure foundations Kok-Kwang Phoon a,*, Fred H. Kulhawy b, Mircea D. Grigoriu b a
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b School of Civil and Environmental Engineering, Hollister Hall, Cornell University, Ithaca, NY 14853-3501, USA Received 4 January 1999; received in revised form 18 May 1999; accepted 24 May 1999
Abstract This paper presents one of the geotechnical initiatives in reliability-based code development that has been sponsored by the Electric Power Research Institute for transmission line structure foundations. The framework for the development of a practical reliability-based design approach is illustrated using the design of drilled shafts (bored piles) for uplift under undrained loading. A target reliability index of 3.2 is selected based on the reliability indices implied by existing working stress designs. Two simple design formats (load and resistance factor design and multiple resistance factor design) are rigorously calibrated using the ®rstorder reliability method to produce designs that achieve a known level of reliability consistently. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction During the past 40 years, signi®cant progress has been made in the ®eld of structural safety. The primary theme in structural safety is reliability analysis, which can be de®ned as the consistent evaluation of design risk using probability theory. Any design methodology that incorporates the principles of reliability analysis, either explicitly or otherwise, may be classi®ed as reliability-based design (RBD). Much of the impetus for this innovation arose from the widespread rethinking of the whole * Corresponding author. 0266-352X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(99)00037-3
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design process that was brought about by the proliferation of new technology and new materials accompanying the boom in construction following World War II. In the ensuing years, several structural RBD codes were developed and implemented in short succession around the world, for example, in the UK in 1972 [1], in Canada in 1974 [2], in Denmark in 1978 [3], and in the US in 1983 for concrete [4] and in 1986 for steel [5]. More recent developments include the introduction of API RP 2ALRFD [6], Eurocode 1 [7], and ISO 2394 [8]. In the ®eld of transmission line design, the Electric Power Research Institute (EPRI) has sponsored research studies directed toward the implementation of these new safety concepts for the design of transmission line structures [9]. Parallel research and development eorts in this ®eld also have been undertaken by the ASCE Task Committee on Structural Loadings [10] and the IEC Technical Committee 11 [11]. Most of the impetus in the development of this new design methodology (RBD) arises from the structural engineering community. In recent years, however, there is a growing awareness and interest among geotechnical engineers in these new safety concepts and design methodologies [12±14]. It is anticipated that the geotechnical design community will become more involved in this code evolution process as the need arises to maintain compatibility between structural and geotechnical design codes and the trends towards code standardization between dierent countries become more prevalent [15,16]. To date, only a few major geotechnical initiatives in reliability-based code development have been undertaken [17, 18], but the verdict on their usefulness is still out in view of their relative immaturity. This paper presents one of these major initiatives [18], sponsored by the Electric Power Research Institute for transmission line structure foundations, with the aim of encouraging more discussion in this arena. The framework for the development of a practical reliability-based design approach for transmission line structure foundations will be illustrated using the design of drilled shafts (bored piles) for uplift under undrained loading. 2. Geotechnical design process 2.1. Working stress design The presence of uncertainties and their signi®cance in relation to design has long been appreciated [19]. The engineer recognizes, explicitly or otherwise, that there is always a chance of not achieving the design objective, which is to ensure that the system performs satisfactorily within a speci®ed period of time. Traditionally, the engineer relies primarily on factors of safety to reduce the risk of adverse performance (collapse, excessive deformations, etc.) at the design stage. Factors of safety between 2 and 3 generally are considered to be adequate in foundation design [20]. Important considerations that aect the factor of safety include variations in the loads and material strengths, inaccuracies in design equations, errors arising from construction, and the consequences of failure. Traditionally, the engineer does not actually go through the process of considering each of these factors separately and
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in explicit detail. The selection of an appropriate factor of safety is essentially subjective, requiring only global appreciation of the above factors against the backdrop of previous experience. For foundation design, the global factor of safety is generally applied to the geotechnical capacity as shown below: Fn <
Qn FS
1
in which Fn nominal load, Qn nominal capacity, and FS factor of safety. This procedure is sensible because geotechnical capacity is much more uncertain than the loading. The working stress approach [Eq. (1)] is simple to use and has worked well for many years. However, the traditional approach has a number of well-recognized and signi®cant limitations. In particular, the factor of safety is generally not accompanied by a carefully prescribed standard procedure for de®ning capacity (e.g. net or gross capacity), for carrying out the analysis (e.g. empirical or rational method), and for deriving the pertinent design soil properties (e.g. correlation or direct measurement). As a result, the same numerical factor of safety can imply different safety margins in the actual design. Another signi®cant source of ambiguity lies in the relationship between the factor of safety and the underlying level of risk. A larger factor of safety does not necessarily imply a smaller level of risk, because its eect can be negated by the presence of larger uncertainties in the design environment. There is clearly a need to reformulate the traditional design approach in clearer terms and to rationalize those aspects dealing with uncertainties. In the authors' opinion, this rationalization process is inevitable in view of the increasing need to: (a) convey safety issues clearly to the public and regulatory authorities [21], (b) handle complex civil facilities (system reliability), (c) keep up with the rapid evolution in materials and construction methodology, and (d) share design experience across national boundaries (code standardization). 2.2. Limit state design An extensive research study (EPRI Report TR-105000 [18]) was undertaken as part of this rationalization process to develop practical reliability-based design (RBD) equations for transmission line structure foundations. Ultimate and serviceability limit state RBD equations for drilled shafts and spread foundations subjected to a variety of loading modes (uplift, compression, lateral-moment loading) were developed as part of this study. The framework for the development of this RBD approach will be illustrated below using the design of drilled shafts for uplift under undrained loading. A major underpinning for the RBD approach is limit state design. In this paper, limit state design refers to a design philosophy that entails the following three basic requirements: (a) identify all potential failure modes or limit states, (b) apply separate checks on each limit state, and (c) show that the occurrence of each limit state is suciently improbable. Conceptually, limit state design is not new. It is merely a
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clearer formulation of the traditional design approach that would help facilitate the explicit recognition and treatment of engineering risks. In recent years, the rapid development of RBD in the structural engineering community has overshadowed considerably the fundamental role of limit state design. Much attention has been focused on the consistent evaluation of safety margins using advanced probabilistic techniques [22]. Although the achievement of consistent safety margins is a highly desirable goal, it should not be overemphasized to the extent that the importance of the principles underlying limit state design become diminished. A detailed exposition of this important issue and its impact on foundation design is given elsewhere [23]. Foundations must be designed to: (a) ensure sucient safety against ultimate failure, and (b) limit foundation deformations to allowable limits for the superstructure in question. The ®rst condition relates to the ultimate limit state, while the second relates to the serviceability limit state. Following the principles of limit state design, the occurrence of these two limit states must be checked and shown separately to be suciently improbable. At the ultimate limit state, the uplift capacity of drilled shafts under undrained loading conditions is governed by the following general equation [24]: Qu Qsu Qtu W
2
in which Qu =uplift capacity, Qsu =side resistance, Qtu =tip resistance, and W= weight of foundation. The side resistance (Qsu ) can be calculated using the total stress method as follows: Qsu B
D 0
su zdz
3
in which B=shaft diameter, D=shaft depth, =adhesion factor, su =undrained shear strength, and z=depth. The tip resistance under undrained loading conditions develops from suction forces and can be estimated by [25]: Qtu ÿu ÿ ui Atip
4
in which ui =initial pore water stress at the foundation tip or base, u=change in pore water stress caused by undrained loading, and Atip =tip or base area. Note that no suction force develops unless ÿu exceeds ui . Extensive comparisons with fullscale load tests have shown that this overall model provides realistic predictions of the uplift capacity for drilled shafts under undrained loading conditions [25,26]. The criterion of realism is crucial for reliability-based design. If the model is conservative, it is obvious that the probabilities of failure calculated subsequently will be biased, because those design situations that belong to the safe domain will be assigned incorrectly to the failure domain, as a result of the built-in conservatism. Detailed discussions of this predictive model are given elsewhere [24±26]. The ultimate limit state is de®ned as that in which the undrained uplift capacity is equal to the ultimate applied load. Clearly, the drilled shaft will fail if the undrained
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uplift capacity is less than this applied load. Conversely, the drilled shaft should perform satisfactorily if the applied load is less than the undrained uplift capacity. These situations can be described concisely by a single performance function P, as follows: P Qu ÿ F
5
in which Qu =uplift capacity determined from Eqs. (2)±(4) and F=applied foundation load. Mathematically, the above situations simply correspond to the three possible conditions of P 0, P < 0, and P > 0. Finally, the occurrence of each limit state must be shown to be suciently improbable. The philosophy of limit state design does not entail a preferred method of ensuring safety. Since all engineering quantities (e.g. loads, strengths) are inherently uncertain to some extent, a logical approach is to model the above performance function using probabilistic means. The mathematical formalization of this aspect of limit state design constitutes the main thrust of RBD. Aside from probabilistic methods, less formal methods of ensuring safety, such as the partial factors of safety method [27,28], have also been used within the framework of limit state design. 3. Reliability-based design The basic objective of RBD is to ensure that the probability of failure of a component does not exceed an acceptable threshold level. While the above objective clearly is satis®ed if the probability of failure of a component lies far below the threshold, it is equally clear that the design is not economical. Therefore, a realistic interpretation of the design objective would include the implicit requirement that the probability of failure does not depart signi®cantly from the threshold. For the design of drilled shafts under undrained uplift loading, the RBD objective can be formally stated as follows: pf Prob Qu < F4pT or pf Prob P < 04pT
6
in which Prob(.)=probability of an event, pf =probability of failure and pT = acceptable target probability of failure. The left hand side of the inequality is evaluated using reliability analysis while the target probability of failure on the right hand side is primarily selected based on the probability of failure implied by existing working stress designs. 3.1. Reliability analysis Reliability analysis attempts to resolve the problem associated with the traditional method of ensuring safety by rendering broad, general concepts, such as uncertainties
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and risks, into precise mathematical terms that can be operated upon consistently. Uncertain engineering quantities (e.g. load and soil strength) are modeled by random variables, while design risk is quanti®ed by the probability of failure. Note that the term ``failure'' used herein is synonymous with ``adverse performance'' and does not necessarily refer to a collapse event. From the de®nition of the limit state performance function given in Eq. (5), it can be seen that the main uncertain parameters are: (a) adhesion factor ( ), (b) undrained shear strength (su ), (c) tip suction (Qtu ), and (d) foundation load (F). In this study, the adhesion factor was determined by the following regression equation [26]: pa 7 0:31 0:17 " su in which pa =atmospheric pressure 100 kN/m2, su =undrained shear strength determined from consolidated isotropic undrained triaxial compression tests, and "=uncertainty about the regression equation. It is important to note that the de®nition of su in Eqs. (3) and (7) should be consistent, because su is not a fundamental material property and depends on many factors, such as the mode of testing. A detailed discussion on the importance of evaluating soil properties speci®cally within a particular design context has been given elsewhere [29]. 3.1.1. Adhesion factor and undrained shear strength The uncertainty in the adhesion factor is represented by ", which can be modeled as a normal random variable with zero mean. The standard deviation of " for the regression model given in Eq. (7) is 0.1 [26]. The undrained shear strength (su ) can be modeled as a log-normal random variable [18]. The typical range for the mean undrained shear strength (msu ) can be taken to lie between 25 and 200 kN/m2. Based on an extensive statistical analyses of soil data [18], a realistic range for the coecient of variation of the undrained shear strength (COVsu ) is estimated to be between 10 and 70%. It is very important to note that unlike the variability of manufactured materials used in structures, geotechnical variability is a complex attribute that results from many disparate sources of uncertainties. The three primary sources are: (a) inherent soil variability, (b) measurement error, and (c) transformation uncertainty. The relative contribution of these sources to the overall uncertainty in the design soil property clearly depends on the site condition, degree of equipment and procedural control, and precision of the correlation model. A systematic approach for evaluating the variability of design soil properties that is suitable for general use is given elsewhere [18]. 3.1.2. Tip suction Tip suction is a complex phenomenon that is not understood well. One approximate solution for the change in pore water stress caused by undrained loading (u) is as follows [25]: u ÿ
W Atip
8
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175
in which W=total foundation weight. By substituting Eq. (8) into Eq. (4), the tip or base suction force can be estimated. It is important to note that the suction stress acting over the tip area can not exceed one atmosphere, which is about 100 kN/m2 [25]. This physical constraint must be imposed on the above tip suction calculation. Some measurements of tip suction stress have been performed on ten model drilled shafts subjected to undrained uplift loading [30]. All ten of the shafts had a diameter of 89 mm and depth to diameter ratios equal to 6.7. The measured tip suction forces for these nominally identical drilled shafts were found to vary erratically from 0 to 424 N. For comparison, the maximum tip force based on one atmosphere of suction stress is 622 N, while the solution obtained from Eqs. (4) and (8) is 53 N. Tip suction apparently is dicult to predict accurately because it is sensitive to the drainage conditions in the vicinity of the tip. The erratic tip suction measurements noted above might be attributed to variable concrete porosity and the presence of small air voids at the shaft tip and possibly within the soil [30]. It also should be noted that tip suction could be eectively eliminated if there is a signi®cant thickness of cohesionless material beneath the foundation tip [25]. In the absence of a rigorous model that can account for these important construction eects and the lack of statistical data, it is reasonable to assume that the tip suction stress is uniformly distributed between zero and one atmosphere. A less diuse probability distribution can be assumed when additional information is available. 3.1.3. Foundation load In the ASCE reliability-based design approach [10], the load eect (F) is modeled as follows: F kV2
9
in which k=deterministic constant and V=Gumbel random variable with an implied coecient of variation of 30%. This simple probabilistic model is applicable to weather-related loads. The use of more sophisticated probabilistic load models for the weather-related loads probably was not considered because the statistical data on ice thickness and combined wind and ice events are very limited [18]. The probabilistic load model given by Eq. (9) is adopted in this study to maintain statistical compatibility between the structural and foundation RBD procedures for transmission line structures. 3.1.4. Evaluation of design risk Once the performance function (P) and the underlying random variables have been de®ned, the design risk or probability of failure (pf ) can be evaluated using a variety of techniques. The calculation of pf for the general case in which Qu and F are modeled as nonlinear functions of basic random variables that are not necessarily normally distributed is quite involved. No closed-form solutions are available for these general cases, which are typical for foundation design. A practical approach is to evaluate pf approximately using numerical techniques. The most commonly used technique is the ®rst-order reliability method (FORM), which provides
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good approximate solutions for most engineering problems [31]. A description of FORM is given elsewhere [18] and is beyond the scope of this paper. The probability of failure is cumbersome to use when its value becomes very small, and it carries the negative connotation of ``failure''. A more convenient measure of design risk is the reliability index ( ), which is de®ned as: ÿÿ1 pf
10
in which ÿ1 =inverse standard normal cumulative function. Numerical values for are tabulated in many standard texts on reliability theory [31]. Note that is not a new measure of design risk. It merely represents an alternative means of presenting pf on a more convenient scale. The reliability indices for most structural and geotechnical components lie between 1.5 and 4, corresponding to probabilities of failure ranging from about 6.7 to 0.003%, as illustrated in Table 1. Note that pf decreases as increases, but the change is not linear. A proper understanding of these two terms and their interrelationship is essential because they play a fundamental role in RBD. 3.2. Target reliability In principle, the most economical target probability of failure (pT ) can be determined by conducting a cost±bene®t analysis, as shown in Fig. 1. By studying the variation of the initial cost, maintenance costs, and the expected failure costs with pf , it is possible theoretically to arrive at the most economical target probability of failure for design [32]. At present, such an approach is not yet practical because of the diculties in evaluating failure costs (e.g. cost of human lives) and the eect of component failure on the system. Another approach is to set the value of pT at a level that is comparable with the failure rates estimated from actual case histories (Fig. 2). However, comparing the theoretical probability of failure derived from reliability computations with a value established by actual case histories is not straightforward. It has been noted that the theoretical probability of failure usually is signi®cantly smaller than the actual failure rate [33]. This result is not surprising, because the safety of a design is not aected by uncertainties underlying design Table 1 Relationship between the reliability index ( ) and the probability of failure (pf ) Reliability index
Probability of failure [ pf ÿ ]a
1.5 2.0 2.5 3.0 3.5 4.0
0.0668 0.0228 0.00621 0.00135 0.000233 0.0000316
a
( . )=Standard normal probability distribution.
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177
Fig. 1. Illustrative cost±bene®t analysis.
Fig. 2. Empirical rates of failure for civil engineering facilities. Source: Baecher, 1987 [35].
calculations alone. It also can be severely compromised by factors such as poor construction and human errors. At present, the most widely used approach of selecting a target probability of failure for design is to calculate the theoretical probabilities of failure implicit in existing working stress designs and to use those values as a basis for selecting an appropriate value of pT [34]. While this approach is empirical, it does possess a
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major advantage of keeping the new design methodology compatible with the existing experience base. This approach is consistent with the evolutionary nature of codes and standards that require changes to be made cautiously and deliberately. The general approach for the calibration of pT involves the following steps [18]: a. Select a set of representative design problems. b. Determine an acceptable solution to each problem based on existing methodology, such as the working stress method. c. Evaluate the probability of failure for each design solution generated by Step (b) using a common reliability calculation scheme (e.g. FORM) and a common set of probabilistic models. d. Based on the range of pf determined by Step (c), select an appropriate value for pT . Currently, drilled shafts subjected to undrained uplift loading typically are designed using the working stress design approach as shown below: F50
Qun FS
11
in which F50 =50-year return period design load (typical for transmission line structures), Qun =nominal uplift capacity, and FS=global factor of safety. The nominal uplift capacity is given by: Qun Qsun Qtun W
12
in which Qsun =nominal uplift side resistance and Qtun =nominal uplift tip resistance. The nominal uplift side resistance was de®ned as follows: Qsun BD n msu
13
in which n =nominal adhesion factor determined from the mean undrained shear strength (msu ) using Eq. (7). The nominal tip resistance (Qtun ) was evaluated using Eqs. (4) and (8) as follows: W ÿ ui Atip 14 Qtun Atip The global factor of safety (FS) used in the working stress design approach [Eq. (11)] generally lies between 2 and 3. The reliability levels implicit in these designs are highly variable, as illustrated in Fig. 3. It can be seen that the reliability indices lie in an approximate range of 2.8± 3.6, and the average reliability index is about 3.2. For reliability-based design, a target reliability index of 3.2 was chosen based on the following important considerations: a. It is representative of the range of reliability indices implicit in existing ultimate limit state design of drilled shafts under undrained uplift loading.
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179
Fig. 3. Reliability levels implicit in existing ultimate limit state design of drilled shafts in undrained uplift.
b. It is applicable to the ultimate limit state of all the loading modes considered throughout this study [18]. c. The target reliability index for foundations should exceed that of the structures in transmission line structure design. This approach is consistent with the design philosophy that foundations should be safer than the structures, because foundation repairs are more dicult and costly [9,11]. d. It is consistent with the empirical rates of failure shown in Fig. 2 after an appropriate adjustment has been made to account for the one order of magnitude dierence between the actual and theoretical rates of failure.
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The target reliability index of 3.2 has been chosen based on extensive reliability studies of existing ultimate limit state foundation designs. The reliability index of 3.2 also has been chosen to be consistent with the empirical evidence on actual foundation failure rate and to conform with the widely accepted transmission line design philosophy of making foundations safer than structures. Therefore, there are strong theoretical, empirical, and philosophical reasons for using a target reliability index of 3.2. This target can be considered to be optimal within the scope of this study, and it is recommended for future transmission line structure foundation design. This target reliability index of 3.2 is incorporated into all the ultimate limit state reliability-based design equations presented in the EPRI study [18]. 4. Simpli®ed RBD formats Reliability-based design in the form of Eq. (6) involves the repeated use of fairly complicated reliability assessment routines, such as FORM, to evaluate the probabilities of failure of trial designs until the computed probability of failure is reasonably close to the chosen threshold level. While the approach is rigorous, it may not be suitable for designs that are carried out on a routine basis. Routine designs would include conventional types of structures and foundations with no abnormal risks or unusual or exceptionally dicult ground or loading conditions. In the EPRI study [18], a simpli®ed RBD approach was developed that involves the use of conventional lumped or multiple-factor formats for checking foundation designs, as given below for uplift loading of a drilled shaft: F50 u Qun
15a
F50 su Qsun tu Qtun w W
15b
in which u , su , tu , and w =resistance factors. The resistance factors in Eqs. (15a) and (15b) were calibrated rigorously using FORM to produce designs that achieve a known level of reliability consistently. Comparable forms of the design equations are available for other loading modes as well. Details of the calibration process are given in the EPRI study [18]. It should be noted that consistently better designs were obtained with the multiple-factor formats. The simpli®ed RBD approach has a number of important advantages. First, the multiple-factor formats are familiar to most engineers. It is easy for practicing engineers to develop a physical feel for these new design formats, because they can see directly how the traditional global factor of safety is separated into the resistance and load factors. Second, the simpli®ed RBD approach is easy to use because the engineer is not directly involved in the elaborate probability computations that are used to produce the factors in the format. Finally, the simpli®ed RBD approach satis®es the RBD objective shown in Eq. (6). The resistance and load factors in the format are calibrated to produce designs that achieve a known level of reliability consistently. The main disadvantage is the loss of ¯exibility, because the engineer
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can not freely change the predictive model, the underlying probability distributions for loads and strengths, and the target probability of failure. However, such a disadvantage is more apparent than real. As noted earlier, the ¯exibility associated with the traditional factor of safety approach produces inconsistent safety margins. In principle, any format can be used for reliability calibration. The selection of an appropriate format is unrelated to reliability analysis. However, practical issues such as simplicity, familiarity, and compatibility with the existing design approach, are important considerations that will determine if the simpli®ed RBD design approach can gain ready acceptance among practicing foundation engineers. The formats shown in Eqs. (15a) and (15b) clearly satisfy these considerations. For example, the resistance factor u in Eq. (15a) corresponds to the reciprocal of the factor of safety (FS) in the traditional working stress design approach [compare with Eq. (1)]. Eq. (15a) also is known as the load and resistance factor design or LRFD format. It already has been adopted widely in the structural community for reliability-based design [4,5,34], and it appears in a number of the recently proposed RBD codes for foundations [14,17]. Eq. (15b) is a broad generalization of Eq. (15a) (called multiple resistance factor design or MRFD) that involves the application of one resistance factor to each component of the capacity rather than the overall capacity, and it is the preferred and more physically meaningful format for foundation engineering. The results of an extensive reliability calibration study for ultimate limit state design of drilled shafts under undrained uplift loading are presented in Tables 2 and 3 and are to be used with Eqs. (15a) and (15b), respectively. All other limit states, foundation types, loading modes, and drainage conditions addressed in the EPRI study [18] have similar types of results, with simple RBD equations and corresponding tables of resistance factors. Note that the resistance factors for undrained
Table 2 Undrained ultimate uplift resistance factors for drilled shafts designed using F50 u Qun a,e Clay
COV of su (%) b
u
Medium
10±30 30±50 50±70
0.44 0.43 0.42
Stic
10±30 30±50 50±70
0.43 0.41 0.39
Very stid
10±30 30±50 50±70
0.40 0.37 0.34
a b c d e
Target reliability index=3.2. Mean su =25±50 kN/m2. Mean su =50±100 kN/m2. Mean su =100±200 kN/m2. Source: Phoon et al. (1995) [18].
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Table 3 Undrained uplift resistance factors for drilled shafts designed using F50 su Qsu tu Qtun w Wa,e Clay
COV of Su , (%)
su
tu
w
Medium
10±30 30±50 50±70
0.44 0.41 0.38
0.28 0.31 0.33
0.50 0.52 0.53
Stic
10±30 30±50 50±70
0.40 0.36 0.32
0.35 0.37 0.40
0.56 0.59 0.62
Very stid
10±30 30±50 50±70
0.35 0.31 0.26
0.42 0.48 0.51
0.66 0.68 0.72
b
a b c d e
Target reliability index=3.2. Mean su =25±50 kN/m2. Mean su =50±100 kN/m2. Mean su =100±200 kN/m2. Source: Phoon et al. (1995) [18].
uplift of drilled shafts depend on the clay consistency and the coecient of variation (COV) of the undrained shear strength (su ). The clay consistency is classi®ed broadly as medium, sti, and very sti, with corresponding mean su values of 25±50, 50±100 and 100±200 kN/m2, respectively. Some of the trends in the resistance factors presented in Tables 2 and 3 can be explained readily. For example, both u and su decrease with increasing COVsu because an increase in the uncertainty of the undrained shear strength produces a less reliable side resistance estimate, which makes the overall capacity estimate less reliable as well. The eect of msu on u and su is caused by the decrease in the mean adhesion factor as the mean undrained shear strength increases. This resulted in an increase in the uncertainty of the adhesion factor relative to its mean value because the standard deviation of the adhesion factor does not change. The increase in tu and w as msu and COVsu increase can be explained partially by noting that the relative contribution of the uncertainty in the side resistance to the overall uncertainty in the capacity increases as the above undrained shear strength statistics increase. As a result, it becomes less eective to use tip resistance and weight factors to control the reliability of the design. Foundations are designed using these new RBD formats in the same way as in the traditional approach, with the exception that the rigorously-determined resistance factors shown in Tables 2 and 3 are used in place of an empirically-determined factor of safety. The performance of the two simpli®ed RBD formats is examined by redesigning the examples shown in Fig. 3 using the resistance factors given in Tables 2 and 3. The results of this performance study are shown in Fig. 4. A comparison between Figs. 3 and 4 illustrates clearly the improvement in the uniformity of the reliability levels resulting from use of the new RBD formats.
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Fig. 4. Performance of ultimate limit state RBD formats for drilled shafts in undrained uplift.
5. Summary There is an increasing pressure on the geotechnical engineering community to evolve from the traditional working stress design approach to a more rational limit state design and reliability-based design approach. This paper presents one of the few geotechnical initiatives in reliability-based code development, and it is sponsored by the Electric Power Research Institute for transmission line structure foundations. The framework for the development of a practical reliability-based design approach is illustrated using the design of drilled shafts for uplift under undrained
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loading. A target reliability index of 3.2 is selected based on the reliability indices implied by existing working stress designs. Two simple design formats (load and resistance factor design and multiple resistance factor design) are rigorously calibrated using the ®rst-order reliability method to produce designs that achieve a known level of reliability consistently. The resulting resistance factors are summarized in a simple tabular form that can be easily applied to practice. Acknowledgements This paper is based, in part, on research sponsored by the Electric Power Research Institute. The EPRI project manager is A. Hirany. References [1] Code of practice for structural use of concrete. London: British Standards Institution, 1972 [CP110 (Pt 1)]. [2] Cold-formed steel structural members. Committee on Design of Light Gauge Steel Structural Members (N. C. Lind, Chair), Standard S136, Canadian Standards Association, Rexdale, 1974. [3] Recommendations for loading & safety regulations for structural design. Report 36, Nordic Committee on Building Regulations (NKB), Copenhagen, 1978. [4] ACI Committee 318, Building code requirements for reinforced concrete. ACI 318±83, American Concrete Institute, Detroit, 1983. [5] Load & resistance factor design manual steel construction. 1st ed. Chicago, American Institute of Steel Construction, 1986. [6] Recommended practice for planning, designing and constructing ®xed oshore platforms Ð load and resistance factor design. API RP 2A-LRFD, American Petroleum Institute, 1993. [7] CEN/TC250. Basis of design and actions on structures Ð Part 1, basis of design. Eurocode 1, ENV1991-1, European Committee for Standardisation (CEN), 1993. [8] General principles on reliability for structures. ISO 2394, International Standard Organisation, 1998. [9] Criswell, M.E., Vanderbilt, M. Reliability-based design of transmission line structures: methods. Report EL-4793 (1), Electric Power Research Institute, Palo Alto, 1987. [10] Task Committee on Structural Loadings. Guidelines for electrical transmission line structural loading. Manual & report on engineering practice 74, New York: ASCE, 1991. [11] IEC Technical Committee 11. Loading & strength of overhead transmission lines. Technical Report 826, International Electrotechnical Commission, Geneva, 1991. [12] Meyerhof GG. Development of geotechnical limit state design. Canadian Geotechnical Journal 1995;32(1):128±36. [13] Becker DE. Limit states design for foundations. Part I. An overview of the foundation design process. Canadian Geotechnical Journal 1996;33(6):956±83. [14] Becker DE. Limit states design for foundations. Part II. Development for the national building code of Canada. Canadian Geotechnical Journal 1996;33(6):984±1007. [15] CEN/TC250. Geotechnical design Ð Part 1. General rules. Eurocode 7, ENV-1997-1, European Committee for Standardisation (CEN), 1994. [16] Research Committee on Present & Future of Japanese Foundation Design and Soil Investigation in View of International Equivalency (O. Kusakabe, Chair). Foundation design standards in the world Ð toward performance-based design. Japanese Geotechnical Society (1998). [17] Barker RM, Duncan JM, Rojiani KB, Ooi PSK, Tan CK, Kim S.G. Manuals for design of bridge foundations. NCHRP Report 343, Transportation Research Board, Washington, 1991.
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[18] Phoon K-K, Kulhawy FH, Grigoriu MD. Reliability-based design of foundations for transmission line structures. Report TR-105000, Electric Power Research Institute, Palo Alto, 1995. [19] Casagrande A. Role of calculated risk in earthwork and foundation engineering. Journal of Soil Mechanics & Foundations Division (ASCE) 1965;91(SM4):1±40. [20] Focht Jr JA, O'Neill MW. Piles & other deep foundations. Proceedings, Eleventh International Conference on Soil Mechanics & Foundation Engineering, San Francisco, 1985;1:187±209. [21] Vick SG. Risk in geotechnical practice. Geotechnical News 1992;10(1):55±7. [22] MacGregor JG. Problem from viewpoint of structural engineer. Symposium on Limit States Design in Foundation Engineering, Canadian Geotechnical Society, Toronto, 1989, 1±16. [23] Kulhawy FH, Phoon KK. Engineering judgment in the evolution from deterministic to reliabilitybased foundation design. Uncertainty in the geologic environment Ð from theory to practice (GSP 58), ASCE, New York, 1996. p. 29±48. [24] Kulhawy FH, Trautmann CH, Beech JF, O'Rourke TD, McGuire W, Wood WA, et al. Transmission line structure foundations for uplift-compression loading. Report EL-2870, Electric Power Research Institute, Palo Alto, 1983. [25] Stas CV, Kulhawy FH. Critical evaluation of design methods for foundation under axial uplift & compression loading. Report EL-3771, Electric Power Research Institute, Palo Alto, 1984. [26] Chen Y-J, Kulhawy FH. Case history evaluation of behavior of drilled shafts under axial & lateral loading, Report TR-104601, Electric Power Research Institute, Palo Alto,1994. [27] Code of Practice for Foundation Engineering. Bulletin 36. Copenhagen: Danish Geotechnical Institute, 1985. [28] Technical Committee on Foundations. Canadian foundation engineering manual. Richmond: Canadian Geotechnical Society, 1992. [29] Kulhawy FH. On evaluation of static soil properties. Stability & performance of slopes and embankments II (GSP 31), ASCE, New York, 1992. p. 95±115. [30] McManus KJ, Kulhawy FH. Cyclic axial loading of drilled shaft foundations in cohesive soil for transmission line structures. Report EL-7161, Electric Power Research Institute, Palo Alto, 1991. [31] Melchers RE. Structural reliability, analysis, & prediction. New York: Wiley, 1987. [32] Baecher GB, Pate ME, De Neufville R. Risk of dam failure in bene®t-cost analysis. Water Resources Research 1980;16(3):449 Ð 456. [33] Construction Industry Research & Information Association. Rationalization of safety & serviceability factors in structural codes. Report 63, CIRIA, London, 1977. [34] Ellingwood B, Galambos TV, MacGregor JG, Cornell CA. Development of probability-based load criterion for American national standard A58. Special Publication 577, National Bureau of Standards, Washington, 1980. [35] Baecher GB. Geotechnical risk analysis user's guide. FHWA/RD-87-011, Federal Highway Administration, McLean, 1987.
Reliability-based design for transmission line structure foundations Kok-Kwang Phoon a,*, Fred H. Kulhawy b, Mircea D. Grigoriu b a
Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b School of Civil and Environmental Engineering, Hollister Hall, Cornell University, Ithaca, NY 14853-3501, USA Received 4 January 1999; received in revised form 18 May 1999; accepted 24 May 1999
Abstract This paper presents one of the geotechnical initiatives in reliability-based code development that has been sponsored by the Electric Power Research Institute for transmission line structure foundations. The framework for the development of a practical reliability-based design approach is illustrated using the design of drilled shafts (bored piles) for uplift under undrained loading. A target reliability index of 3.2 is selected based on the reliability indices implied by existing working stress designs. Two simple design formats (load and resistance factor design and multiple resistance factor design) are rigorously calibrated using the ®rstorder reliability method to produce designs that achieve a known level of reliability consistently. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction During the past 40 years, signi®cant progress has been made in the ®eld of structural safety. The primary theme in structural safety is reliability analysis, which can be de®ned as the consistent evaluation of design risk using probability theory. Any design methodology that incorporates the principles of reliability analysis, either explicitly or otherwise, may be classi®ed as reliability-based design (RBD). Much of the impetus for this innovation arose from the widespread rethinking of the whole * Corresponding author. 0266-352X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(99)00037-3
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design process that was brought about by the proliferation of new technology and new materials accompanying the boom in construction following World War II. In the ensuing years, several structural RBD codes were developed and implemented in short succession around the world, for example, in the UK in 1972 [1], in Canada in 1974 [2], in Denmark in 1978 [3], and in the US in 1983 for concrete [4] and in 1986 for steel [5]. More recent developments include the introduction of API RP 2ALRFD [6], Eurocode 1 [7], and ISO 2394 [8]. In the ®eld of transmission line design, the Electric Power Research Institute (EPRI) has sponsored research studies directed toward the implementation of these new safety concepts for the design of transmission line structures [9]. Parallel research and development eorts in this ®eld also have been undertaken by the ASCE Task Committee on Structural Loadings [10] and the IEC Technical Committee 11 [11]. Most of the impetus in the development of this new design methodology (RBD) arises from the structural engineering community. In recent years, however, there is a growing awareness and interest among geotechnical engineers in these new safety concepts and design methodologies [12±14]. It is anticipated that the geotechnical design community will become more involved in this code evolution process as the need arises to maintain compatibility between structural and geotechnical design codes and the trends towards code standardization between dierent countries become more prevalent [15,16]. To date, only a few major geotechnical initiatives in reliability-based code development have been undertaken [17, 18], but the verdict on their usefulness is still out in view of their relative immaturity. This paper presents one of these major initiatives [18], sponsored by the Electric Power Research Institute for transmission line structure foundations, with the aim of encouraging more discussion in this arena. The framework for the development of a practical reliability-based design approach for transmission line structure foundations will be illustrated using the design of drilled shafts (bored piles) for uplift under undrained loading. 2. Geotechnical design process 2.1. Working stress design The presence of uncertainties and their signi®cance in relation to design has long been appreciated [19]. The engineer recognizes, explicitly or otherwise, that there is always a chance of not achieving the design objective, which is to ensure that the system performs satisfactorily within a speci®ed period of time. Traditionally, the engineer relies primarily on factors of safety to reduce the risk of adverse performance (collapse, excessive deformations, etc.) at the design stage. Factors of safety between 2 and 3 generally are considered to be adequate in foundation design [20]. Important considerations that aect the factor of safety include variations in the loads and material strengths, inaccuracies in design equations, errors arising from construction, and the consequences of failure. Traditionally, the engineer does not actually go through the process of considering each of these factors separately and
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in explicit detail. The selection of an appropriate factor of safety is essentially subjective, requiring only global appreciation of the above factors against the backdrop of previous experience. For foundation design, the global factor of safety is generally applied to the geotechnical capacity as shown below: Fn <
Qn FS
1
in which Fn nominal load, Qn nominal capacity, and FS factor of safety. This procedure is sensible because geotechnical capacity is much more uncertain than the loading. The working stress approach [Eq. (1)] is simple to use and has worked well for many years. However, the traditional approach has a number of well-recognized and signi®cant limitations. In particular, the factor of safety is generally not accompanied by a carefully prescribed standard procedure for de®ning capacity (e.g. net or gross capacity), for carrying out the analysis (e.g. empirical or rational method), and for deriving the pertinent design soil properties (e.g. correlation or direct measurement). As a result, the same numerical factor of safety can imply different safety margins in the actual design. Another signi®cant source of ambiguity lies in the relationship between the factor of safety and the underlying level of risk. A larger factor of safety does not necessarily imply a smaller level of risk, because its eect can be negated by the presence of larger uncertainties in the design environment. There is clearly a need to reformulate the traditional design approach in clearer terms and to rationalize those aspects dealing with uncertainties. In the authors' opinion, this rationalization process is inevitable in view of the increasing need to: (a) convey safety issues clearly to the public and regulatory authorities [21], (b) handle complex civil facilities (system reliability), (c) keep up with the rapid evolution in materials and construction methodology, and (d) share design experience across national boundaries (code standardization). 2.2. Limit state design An extensive research study (EPRI Report TR-105000 [18]) was undertaken as part of this rationalization process to develop practical reliability-based design (RBD) equations for transmission line structure foundations. Ultimate and serviceability limit state RBD equations for drilled shafts and spread foundations subjected to a variety of loading modes (uplift, compression, lateral-moment loading) were developed as part of this study. The framework for the development of this RBD approach will be illustrated below using the design of drilled shafts for uplift under undrained loading. A major underpinning for the RBD approach is limit state design. In this paper, limit state design refers to a design philosophy that entails the following three basic requirements: (a) identify all potential failure modes or limit states, (b) apply separate checks on each limit state, and (c) show that the occurrence of each limit state is suciently improbable. Conceptually, limit state design is not new. It is merely a
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clearer formulation of the traditional design approach that would help facilitate the explicit recognition and treatment of engineering risks. In recent years, the rapid development of RBD in the structural engineering community has overshadowed considerably the fundamental role of limit state design. Much attention has been focused on the consistent evaluation of safety margins using advanced probabilistic techniques [22]. Although the achievement of consistent safety margins is a highly desirable goal, it should not be overemphasized to the extent that the importance of the principles underlying limit state design become diminished. A detailed exposition of this important issue and its impact on foundation design is given elsewhere [23]. Foundations must be designed to: (a) ensure sucient safety against ultimate failure, and (b) limit foundation deformations to allowable limits for the superstructure in question. The ®rst condition relates to the ultimate limit state, while the second relates to the serviceability limit state. Following the principles of limit state design, the occurrence of these two limit states must be checked and shown separately to be suciently improbable. At the ultimate limit state, the uplift capacity of drilled shafts under undrained loading conditions is governed by the following general equation [24]: Qu Qsu Qtu W
2
in which Qu =uplift capacity, Qsu =side resistance, Qtu =tip resistance, and W= weight of foundation. The side resistance (Qsu ) can be calculated using the total stress method as follows: Qsu B
D 0
su zdz
3
in which B=shaft diameter, D=shaft depth, =adhesion factor, su =undrained shear strength, and z=depth. The tip resistance under undrained loading conditions develops from suction forces and can be estimated by [25]: Qtu ÿu ÿ ui Atip
4
in which ui =initial pore water stress at the foundation tip or base, u=change in pore water stress caused by undrained loading, and Atip =tip or base area. Note that no suction force develops unless ÿu exceeds ui . Extensive comparisons with fullscale load tests have shown that this overall model provides realistic predictions of the uplift capacity for drilled shafts under undrained loading conditions [25,26]. The criterion of realism is crucial for reliability-based design. If the model is conservative, it is obvious that the probabilities of failure calculated subsequently will be biased, because those design situations that belong to the safe domain will be assigned incorrectly to the failure domain, as a result of the built-in conservatism. Detailed discussions of this predictive model are given elsewhere [24±26]. The ultimate limit state is de®ned as that in which the undrained uplift capacity is equal to the ultimate applied load. Clearly, the drilled shaft will fail if the undrained
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uplift capacity is less than this applied load. Conversely, the drilled shaft should perform satisfactorily if the applied load is less than the undrained uplift capacity. These situations can be described concisely by a single performance function P, as follows: P Qu ÿ F
5
in which Qu =uplift capacity determined from Eqs. (2)±(4) and F=applied foundation load. Mathematically, the above situations simply correspond to the three possible conditions of P 0, P < 0, and P > 0. Finally, the occurrence of each limit state must be shown to be suciently improbable. The philosophy of limit state design does not entail a preferred method of ensuring safety. Since all engineering quantities (e.g. loads, strengths) are inherently uncertain to some extent, a logical approach is to model the above performance function using probabilistic means. The mathematical formalization of this aspect of limit state design constitutes the main thrust of RBD. Aside from probabilistic methods, less formal methods of ensuring safety, such as the partial factors of safety method [27,28], have also been used within the framework of limit state design. 3. Reliability-based design The basic objective of RBD is to ensure that the probability of failure of a component does not exceed an acceptable threshold level. While the above objective clearly is satis®ed if the probability of failure of a component lies far below the threshold, it is equally clear that the design is not economical. Therefore, a realistic interpretation of the design objective would include the implicit requirement that the probability of failure does not depart signi®cantly from the threshold. For the design of drilled shafts under undrained uplift loading, the RBD objective can be formally stated as follows: pf Prob Qu < F4pT or pf Prob P < 04pT
6
in which Prob(.)=probability of an event, pf =probability of failure and pT = acceptable target probability of failure. The left hand side of the inequality is evaluated using reliability analysis while the target probability of failure on the right hand side is primarily selected based on the probability of failure implied by existing working stress designs. 3.1. Reliability analysis Reliability analysis attempts to resolve the problem associated with the traditional method of ensuring safety by rendering broad, general concepts, such as uncertainties
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and risks, into precise mathematical terms that can be operated upon consistently. Uncertain engineering quantities (e.g. load and soil strength) are modeled by random variables, while design risk is quanti®ed by the probability of failure. Note that the term ``failure'' used herein is synonymous with ``adverse performance'' and does not necessarily refer to a collapse event. From the de®nition of the limit state performance function given in Eq. (5), it can be seen that the main uncertain parameters are: (a) adhesion factor ( ), (b) undrained shear strength (su ), (c) tip suction (Qtu ), and (d) foundation load (F). In this study, the adhesion factor was determined by the following regression equation [26]: pa 7 0:31 0:17 " su in which pa =atmospheric pressure 100 kN/m2, su =undrained shear strength determined from consolidated isotropic undrained triaxial compression tests, and "=uncertainty about the regression equation. It is important to note that the de®nition of su in Eqs. (3) and (7) should be consistent, because su is not a fundamental material property and depends on many factors, such as the mode of testing. A detailed discussion on the importance of evaluating soil properties speci®cally within a particular design context has been given elsewhere [29]. 3.1.1. Adhesion factor and undrained shear strength The uncertainty in the adhesion factor is represented by ", which can be modeled as a normal random variable with zero mean. The standard deviation of " for the regression model given in Eq. (7) is 0.1 [26]. The undrained shear strength (su ) can be modeled as a log-normal random variable [18]. The typical range for the mean undrained shear strength (msu ) can be taken to lie between 25 and 200 kN/m2. Based on an extensive statistical analyses of soil data [18], a realistic range for the coecient of variation of the undrained shear strength (COVsu ) is estimated to be between 10 and 70%. It is very important to note that unlike the variability of manufactured materials used in structures, geotechnical variability is a complex attribute that results from many disparate sources of uncertainties. The three primary sources are: (a) inherent soil variability, (b) measurement error, and (c) transformation uncertainty. The relative contribution of these sources to the overall uncertainty in the design soil property clearly depends on the site condition, degree of equipment and procedural control, and precision of the correlation model. A systematic approach for evaluating the variability of design soil properties that is suitable for general use is given elsewhere [18]. 3.1.2. Tip suction Tip suction is a complex phenomenon that is not understood well. One approximate solution for the change in pore water stress caused by undrained loading (u) is as follows [25]: u ÿ
W Atip
8
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175
in which W=total foundation weight. By substituting Eq. (8) into Eq. (4), the tip or base suction force can be estimated. It is important to note that the suction stress acting over the tip area can not exceed one atmosphere, which is about 100 kN/m2 [25]. This physical constraint must be imposed on the above tip suction calculation. Some measurements of tip suction stress have been performed on ten model drilled shafts subjected to undrained uplift loading [30]. All ten of the shafts had a diameter of 89 mm and depth to diameter ratios equal to 6.7. The measured tip suction forces for these nominally identical drilled shafts were found to vary erratically from 0 to 424 N. For comparison, the maximum tip force based on one atmosphere of suction stress is 622 N, while the solution obtained from Eqs. (4) and (8) is 53 N. Tip suction apparently is dicult to predict accurately because it is sensitive to the drainage conditions in the vicinity of the tip. The erratic tip suction measurements noted above might be attributed to variable concrete porosity and the presence of small air voids at the shaft tip and possibly within the soil [30]. It also should be noted that tip suction could be eectively eliminated if there is a signi®cant thickness of cohesionless material beneath the foundation tip [25]. In the absence of a rigorous model that can account for these important construction eects and the lack of statistical data, it is reasonable to assume that the tip suction stress is uniformly distributed between zero and one atmosphere. A less diuse probability distribution can be assumed when additional information is available. 3.1.3. Foundation load In the ASCE reliability-based design approach [10], the load eect (F) is modeled as follows: F kV2
9
in which k=deterministic constant and V=Gumbel random variable with an implied coecient of variation of 30%. This simple probabilistic model is applicable to weather-related loads. The use of more sophisticated probabilistic load models for the weather-related loads probably was not considered because the statistical data on ice thickness and combined wind and ice events are very limited [18]. The probabilistic load model given by Eq. (9) is adopted in this study to maintain statistical compatibility between the structural and foundation RBD procedures for transmission line structures. 3.1.4. Evaluation of design risk Once the performance function (P) and the underlying random variables have been de®ned, the design risk or probability of failure (pf ) can be evaluated using a variety of techniques. The calculation of pf for the general case in which Qu and F are modeled as nonlinear functions of basic random variables that are not necessarily normally distributed is quite involved. No closed-form solutions are available for these general cases, which are typical for foundation design. A practical approach is to evaluate pf approximately using numerical techniques. The most commonly used technique is the ®rst-order reliability method (FORM), which provides
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good approximate solutions for most engineering problems [31]. A description of FORM is given elsewhere [18] and is beyond the scope of this paper. The probability of failure is cumbersome to use when its value becomes very small, and it carries the negative connotation of ``failure''. A more convenient measure of design risk is the reliability index ( ), which is de®ned as: ÿÿ1 pf
10
in which ÿ1 =inverse standard normal cumulative function. Numerical values for are tabulated in many standard texts on reliability theory [31]. Note that is not a new measure of design risk. It merely represents an alternative means of presenting pf on a more convenient scale. The reliability indices for most structural and geotechnical components lie between 1.5 and 4, corresponding to probabilities of failure ranging from about 6.7 to 0.003%, as illustrated in Table 1. Note that pf decreases as increases, but the change is not linear. A proper understanding of these two terms and their interrelationship is essential because they play a fundamental role in RBD. 3.2. Target reliability In principle, the most economical target probability of failure (pT ) can be determined by conducting a cost±bene®t analysis, as shown in Fig. 1. By studying the variation of the initial cost, maintenance costs, and the expected failure costs with pf , it is possible theoretically to arrive at the most economical target probability of failure for design [32]. At present, such an approach is not yet practical because of the diculties in evaluating failure costs (e.g. cost of human lives) and the eect of component failure on the system. Another approach is to set the value of pT at a level that is comparable with the failure rates estimated from actual case histories (Fig. 2). However, comparing the theoretical probability of failure derived from reliability computations with a value established by actual case histories is not straightforward. It has been noted that the theoretical probability of failure usually is signi®cantly smaller than the actual failure rate [33]. This result is not surprising, because the safety of a design is not aected by uncertainties underlying design Table 1 Relationship between the reliability index ( ) and the probability of failure (pf ) Reliability index
Probability of failure [ pf ÿ ]a
1.5 2.0 2.5 3.0 3.5 4.0
0.0668 0.0228 0.00621 0.00135 0.000233 0.0000316
a
( . )=Standard normal probability distribution.
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177
Fig. 1. Illustrative cost±bene®t analysis.
Fig. 2. Empirical rates of failure for civil engineering facilities. Source: Baecher, 1987 [35].
calculations alone. It also can be severely compromised by factors such as poor construction and human errors. At present, the most widely used approach of selecting a target probability of failure for design is to calculate the theoretical probabilities of failure implicit in existing working stress designs and to use those values as a basis for selecting an appropriate value of pT [34]. While this approach is empirical, it does possess a
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major advantage of keeping the new design methodology compatible with the existing experience base. This approach is consistent with the evolutionary nature of codes and standards that require changes to be made cautiously and deliberately. The general approach for the calibration of pT involves the following steps [18]: a. Select a set of representative design problems. b. Determine an acceptable solution to each problem based on existing methodology, such as the working stress method. c. Evaluate the probability of failure for each design solution generated by Step (b) using a common reliability calculation scheme (e.g. FORM) and a common set of probabilistic models. d. Based on the range of pf determined by Step (c), select an appropriate value for pT . Currently, drilled shafts subjected to undrained uplift loading typically are designed using the working stress design approach as shown below: F50
Qun FS
11
in which F50 =50-year return period design load (typical for transmission line structures), Qun =nominal uplift capacity, and FS=global factor of safety. The nominal uplift capacity is given by: Qun Qsun Qtun W
12
in which Qsun =nominal uplift side resistance and Qtun =nominal uplift tip resistance. The nominal uplift side resistance was de®ned as follows: Qsun BD n msu
13
in which n =nominal adhesion factor determined from the mean undrained shear strength (msu ) using Eq. (7). The nominal tip resistance (Qtun ) was evaluated using Eqs. (4) and (8) as follows: W ÿ ui Atip 14 Qtun Atip The global factor of safety (FS) used in the working stress design approach [Eq. (11)] generally lies between 2 and 3. The reliability levels implicit in these designs are highly variable, as illustrated in Fig. 3. It can be seen that the reliability indices lie in an approximate range of 2.8± 3.6, and the average reliability index is about 3.2. For reliability-based design, a target reliability index of 3.2 was chosen based on the following important considerations: a. It is representative of the range of reliability indices implicit in existing ultimate limit state design of drilled shafts under undrained uplift loading.
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179
Fig. 3. Reliability levels implicit in existing ultimate limit state design of drilled shafts in undrained uplift.
b. It is applicable to the ultimate limit state of all the loading modes considered throughout this study [18]. c. The target reliability index for foundations should exceed that of the structures in transmission line structure design. This approach is consistent with the design philosophy that foundations should be safer than the structures, because foundation repairs are more dicult and costly [9,11]. d. It is consistent with the empirical rates of failure shown in Fig. 2 after an appropriate adjustment has been made to account for the one order of magnitude dierence between the actual and theoretical rates of failure.
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The target reliability index of 3.2 has been chosen based on extensive reliability studies of existing ultimate limit state foundation designs. The reliability index of 3.2 also has been chosen to be consistent with the empirical evidence on actual foundation failure rate and to conform with the widely accepted transmission line design philosophy of making foundations safer than structures. Therefore, there are strong theoretical, empirical, and philosophical reasons for using a target reliability index of 3.2. This target can be considered to be optimal within the scope of this study, and it is recommended for future transmission line structure foundation design. This target reliability index of 3.2 is incorporated into all the ultimate limit state reliability-based design equations presented in the EPRI study [18]. 4. Simpli®ed RBD formats Reliability-based design in the form of Eq. (6) involves the repeated use of fairly complicated reliability assessment routines, such as FORM, to evaluate the probabilities of failure of trial designs until the computed probability of failure is reasonably close to the chosen threshold level. While the approach is rigorous, it may not be suitable for designs that are carried out on a routine basis. Routine designs would include conventional types of structures and foundations with no abnormal risks or unusual or exceptionally dicult ground or loading conditions. In the EPRI study [18], a simpli®ed RBD approach was developed that involves the use of conventional lumped or multiple-factor formats for checking foundation designs, as given below for uplift loading of a drilled shaft: F50 u Qun
15a
F50 su Qsun tu Qtun w W
15b
in which u , su , tu , and w =resistance factors. The resistance factors in Eqs. (15a) and (15b) were calibrated rigorously using FORM to produce designs that achieve a known level of reliability consistently. Comparable forms of the design equations are available for other loading modes as well. Details of the calibration process are given in the EPRI study [18]. It should be noted that consistently better designs were obtained with the multiple-factor formats. The simpli®ed RBD approach has a number of important advantages. First, the multiple-factor formats are familiar to most engineers. It is easy for practicing engineers to develop a physical feel for these new design formats, because they can see directly how the traditional global factor of safety is separated into the resistance and load factors. Second, the simpli®ed RBD approach is easy to use because the engineer is not directly involved in the elaborate probability computations that are used to produce the factors in the format. Finally, the simpli®ed RBD approach satis®es the RBD objective shown in Eq. (6). The resistance and load factors in the format are calibrated to produce designs that achieve a known level of reliability consistently. The main disadvantage is the loss of ¯exibility, because the engineer
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can not freely change the predictive model, the underlying probability distributions for loads and strengths, and the target probability of failure. However, such a disadvantage is more apparent than real. As noted earlier, the ¯exibility associated with the traditional factor of safety approach produces inconsistent safety margins. In principle, any format can be used for reliability calibration. The selection of an appropriate format is unrelated to reliability analysis. However, practical issues such as simplicity, familiarity, and compatibility with the existing design approach, are important considerations that will determine if the simpli®ed RBD design approach can gain ready acceptance among practicing foundation engineers. The formats shown in Eqs. (15a) and (15b) clearly satisfy these considerations. For example, the resistance factor u in Eq. (15a) corresponds to the reciprocal of the factor of safety (FS) in the traditional working stress design approach [compare with Eq. (1)]. Eq. (15a) also is known as the load and resistance factor design or LRFD format. It already has been adopted widely in the structural community for reliability-based design [4,5,34], and it appears in a number of the recently proposed RBD codes for foundations [14,17]. Eq. (15b) is a broad generalization of Eq. (15a) (called multiple resistance factor design or MRFD) that involves the application of one resistance factor to each component of the capacity rather than the overall capacity, and it is the preferred and more physically meaningful format for foundation engineering. The results of an extensive reliability calibration study for ultimate limit state design of drilled shafts under undrained uplift loading are presented in Tables 2 and 3 and are to be used with Eqs. (15a) and (15b), respectively. All other limit states, foundation types, loading modes, and drainage conditions addressed in the EPRI study [18] have similar types of results, with simple RBD equations and corresponding tables of resistance factors. Note that the resistance factors for undrained
Table 2 Undrained ultimate uplift resistance factors for drilled shafts designed using F50 u Qun a,e Clay
COV of su (%) b
u
Medium
10±30 30±50 50±70
0.44 0.43 0.42
Stic
10±30 30±50 50±70
0.43 0.41 0.39
Very stid
10±30 30±50 50±70
0.40 0.37 0.34
a b c d e
Target reliability index=3.2. Mean su =25±50 kN/m2. Mean su =50±100 kN/m2. Mean su =100±200 kN/m2. Source: Phoon et al. (1995) [18].
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Table 3 Undrained uplift resistance factors for drilled shafts designed using F50 su Qsu tu Qtun w Wa,e Clay
COV of Su , (%)
su
tu
w
Medium
10±30 30±50 50±70
0.44 0.41 0.38
0.28 0.31 0.33
0.50 0.52 0.53
Stic
10±30 30±50 50±70
0.40 0.36 0.32
0.35 0.37 0.40
0.56 0.59 0.62
Very stid
10±30 30±50 50±70
0.35 0.31 0.26
0.42 0.48 0.51
0.66 0.68 0.72
b
a b c d e
Target reliability index=3.2. Mean su =25±50 kN/m2. Mean su =50±100 kN/m2. Mean su =100±200 kN/m2. Source: Phoon et al. (1995) [18].
uplift of drilled shafts depend on the clay consistency and the coecient of variation (COV) of the undrained shear strength (su ). The clay consistency is classi®ed broadly as medium, sti, and very sti, with corresponding mean su values of 25±50, 50±100 and 100±200 kN/m2, respectively. Some of the trends in the resistance factors presented in Tables 2 and 3 can be explained readily. For example, both u and su decrease with increasing COVsu because an increase in the uncertainty of the undrained shear strength produces a less reliable side resistance estimate, which makes the overall capacity estimate less reliable as well. The eect of msu on u and su is caused by the decrease in the mean adhesion factor as the mean undrained shear strength increases. This resulted in an increase in the uncertainty of the adhesion factor relative to its mean value because the standard deviation of the adhesion factor does not change. The increase in tu and w as msu and COVsu increase can be explained partially by noting that the relative contribution of the uncertainty in the side resistance to the overall uncertainty in the capacity increases as the above undrained shear strength statistics increase. As a result, it becomes less eective to use tip resistance and weight factors to control the reliability of the design. Foundations are designed using these new RBD formats in the same way as in the traditional approach, with the exception that the rigorously-determined resistance factors shown in Tables 2 and 3 are used in place of an empirically-determined factor of safety. The performance of the two simpli®ed RBD formats is examined by redesigning the examples shown in Fig. 3 using the resistance factors given in Tables 2 and 3. The results of this performance study are shown in Fig. 4. A comparison between Figs. 3 and 4 illustrates clearly the improvement in the uniformity of the reliability levels resulting from use of the new RBD formats.
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Fig. 4. Performance of ultimate limit state RBD formats for drilled shafts in undrained uplift.
5. Summary There is an increasing pressure on the geotechnical engineering community to evolve from the traditional working stress design approach to a more rational limit state design and reliability-based design approach. This paper presents one of the few geotechnical initiatives in reliability-based code development, and it is sponsored by the Electric Power Research Institute for transmission line structure foundations. The framework for the development of a practical reliability-based design approach is illustrated using the design of drilled shafts for uplift under undrained
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loading. A target reliability index of 3.2 is selected based on the reliability indices implied by existing working stress designs. Two simple design formats (load and resistance factor design and multiple resistance factor design) are rigorously calibrated using the ®rst-order reliability method to produce designs that achieve a known level of reliability consistently. The resulting resistance factors are summarized in a simple tabular form that can be easily applied to practice. Acknowledgements This paper is based, in part, on research sponsored by the Electric Power Research Institute. The EPRI project manager is A. Hirany. References [1] Code of practice for structural use of concrete. London: British Standards Institution, 1972 [CP110 (Pt 1)]. [2] Cold-formed steel structural members. Committee on Design of Light Gauge Steel Structural Members (N. C. Lind, Chair), Standard S136, Canadian Standards Association, Rexdale, 1974. [3] Recommendations for loading & safety regulations for structural design. Report 36, Nordic Committee on Building Regulations (NKB), Copenhagen, 1978. [4] ACI Committee 318, Building code requirements for reinforced concrete. ACI 318±83, American Concrete Institute, Detroit, 1983. [5] Load & resistance factor design manual steel construction. 1st ed. Chicago, American Institute of Steel Construction, 1986. [6] Recommended practice for planning, designing and constructing ®xed oshore platforms Ð load and resistance factor design. API RP 2A-LRFD, American Petroleum Institute, 1993. [7] CEN/TC250. Basis of design and actions on structures Ð Part 1, basis of design. Eurocode 1, ENV1991-1, European Committee for Standardisation (CEN), 1993. [8] General principles on reliability for structures. ISO 2394, International Standard Organisation, 1998. [9] Criswell, M.E., Vanderbilt, M. Reliability-based design of transmission line structures: methods. Report EL-4793 (1), Electric Power Research Institute, Palo Alto, 1987. [10] Task Committee on Structural Loadings. Guidelines for electrical transmission line structural loading. Manual & report on engineering practice 74, New York: ASCE, 1991. [11] IEC Technical Committee 11. Loading & strength of overhead transmission lines. Technical Report 826, International Electrotechnical Commission, Geneva, 1991. [12] Meyerhof GG. Development of geotechnical limit state design. Canadian Geotechnical Journal 1995;32(1):128±36. [13] Becker DE. Limit states design for foundations. Part I. An overview of the foundation design process. Canadian Geotechnical Journal 1996;33(6):956±83. [14] Becker DE. Limit states design for foundations. Part II. Development for the national building code of Canada. Canadian Geotechnical Journal 1996;33(6):984±1007. [15] CEN/TC250. Geotechnical design Ð Part 1. General rules. Eurocode 7, ENV-1997-1, European Committee for Standardisation (CEN), 1994. [16] Research Committee on Present & Future of Japanese Foundation Design and Soil Investigation in View of International Equivalency (O. Kusakabe, Chair). Foundation design standards in the world Ð toward performance-based design. Japanese Geotechnical Society (1998). [17] Barker RM, Duncan JM, Rojiani KB, Ooi PSK, Tan CK, Kim S.G. Manuals for design of bridge foundations. NCHRP Report 343, Transportation Research Board, Washington, 1991.
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