RELIABILITY-BASED DESIGN INCORPORATING MODEL UNCERTAINTIES. K. K. Phoon Department of Civil Engineering, National University of Singapore, Singapore

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1 3-4 Agst 2, Samarang, Indonesia RELIABILITY-BASED DESIGN INCORPORATING MODEL UNCERTAINTIES K. K. Phoon Department of Civil Engineering, National University of Singapore, Singapore ABSTRACT: This paper presents a critical evalation of model factors for laterally loaded rigid drilled shafts. A comparison between controlled laboratory tests and fll-scale field tests reveals that the model factors have wider applicability beyond conditions implied by database. The log-normal probability density fnction appears to fit the data rather well. There are hidden statistical dependencies between the model factor and the compted capacity that can be removed sing a proposed generalization based on linear regression theory. The impact of the model factor and its potential correlation with the compted capacity on reliability-based design is discssed. INTRODUCTION A recent srvey for the National Cooperative Highway Research Program (NCHRP) on LRFD implementation in North America indicated a general relctance among practicing engineers to change the existing design procedre (Goble 1999). It also was noted that the resistance factors in most of the crrent LRFD (Load and Resistance Factor Design) implementations were determined by direct modification of the global factor of safety or by engineering jdgment. Only a few of the cases were based on probabilistic analysis that is fonded on robst statistics. A comprehensive srvey by the Federal Highway Administration (FHWA) also noted an absence of strong analytical calibration and verification of either the Erocode or the Ontario Highway Bridge Design Code (DiMaggio et al. 1999). The research effort reqired to collate robst geotechnical statistics is considerable. However, it is absoltely necessary if practicing engineers are to be convinced of the merits and sondness of the new reliability-based design (RBD) methodology. There is little dispte that the crrent geotechnical design process cold be improved significantly by integrating the varios design components (loads, soil parameters, calclation models, and factors of safety) in a more consistent way within a framework that recognizes ncertainties explicitly. Reliability-based design is the only methodology available to date that can ensre self-consistency from both physical and probabilistic reqirements and is compatible with the theoretical basis nderlying other disciplines sch as strctral design. In addition, geotechnical design will be sbjected to increasing codification as a reslt of code harmonization across material types and national bondaries. It also is clear that reglatory pressre eventally will bring geotechnical design within an mbrella framework predominantly established by strctral engineers. For example, EN199:22 (British Standards Institte 22) describes the principles and reqirements for safety, serviceability, and drability of strctres, and the basis for their design and verification; it also gives gidelines for related aspects of strctral reliability in the annexes. In the United States, this process also is nderway for highway bridge design (e.g., Withiam 23). Therefore, there are strong practical reasons to consider geotechnical LRFD as a simplified reliabilitybased design procedre, rather than an exercise in rearranging the original global factor of safety. Philosophically, this approach calls for a willingness to accept reliability analysis as a necessary basis for geotechnical LRFD calibrations (Phoon et al. 23). 191

2 3-4 Agst 2, Samarang, Indonesia Two main sorces of geotechnical ncertainties can be distingished. The first arises from the evalation of design soil properties, sch as ndrained shear strength and effective stress friction angle. This sorce of geotechnical ncertainty is complex and depends on inherent soil variability, degree of eqipment and procedral control maintained dring site investigation, and precision of the correlation model sed to relate field measrement with design soil property. A series of five stdies on geotechnical variabilities (Spry et al. 1988, Orchant et al. 1988, Filippas et al. 1988, Klhawy et al. 1992, Phoon et al. 199) have been condcted to establish realistic statistical estimates of the variability of design soil properties and to provide gidelines for the calibration of geotechnical RBD eqations. Details are smmarised in Phoon & Klhawy (1999a, 1999b). The second sorce arises from geotechnical calclation models. Althogh many geotechnical calclation models are simple, reasonable predictions of fairly complex soil-strctre interaction behavior still can be achieved throgh empirical calibrations. Becase of or geotechnical heritage that is steeped in sch empiricisms, model ncertainties can be significant. Even a simple estimate of the average model bias is crcial for reliability-based design. If the model is conservative, it is obvios that the probabilities of failre calclated sbseqently will be biased, becase those design sitations that belong to the safe domain will be assigned incorrectly to the failre domain, as a reslt of the bilt-in conservatism. Robst model statistics can only be evalated sing: (1) realistically large scale prototype tests, (2) a sfficiently large and representative database, and (3) reasonably high qality testing where extraneos ncertainties are well controlled. With the possible exception of fondations, insfficient test data are available to perform robst statistical assessment of the model error in many geotechnical calclation models. The development of a flly rigoros RBD code that can handle the entire range of geotechnical design problems is crrently impeded by the scarcity in these important statistics. Sidi (1986) was among the first to report model statistics that were firmly established sing a large load test database assembled by Olson & Dennis (1982). The focs of the stdy was on friction pile fondations in clay deposits sbjected to axial loadings. Briad and Tcker (1988) condcted a similar stdy sing a 98-pile load test database obtained from the Mississippi State Highway Department. A more recent stdy was condcted to spport the calibration of deep fondation resistance factors for AASHTO (American Association of State Highway and Transportation Officials) (Paikowsky 22). A sbstantial part of the stdy pertains to the evalation of driven pile axial capacity sing dynamic methods. Model statistics for serviceability calclations are absent from these stdies. This paper discsses important statistical isses nderlying the definition of model factor and the significance of incorporating model factors in reliability-based design. Model-scale and fll-scale rigid drilled shafts sbjected to lateral-moment loading are analyzed sing different capacity calclation models to illstrate the above concerns. Serviceability calclations are potentially more critical, bt otside the scope of this paper. Details are given elsewhere (Phoon et al. 26). MODEL FACTOR The force system of a laterally loaded drilled shaft is complex and three-dimensional. Althogh passive lateral soil resistance dominates, there are shearing and axial force components at the tip and along the front, back, and side faces of the shaft. The cstomary approach is to simplify this actal force system to a two-dimensional distribtion of lateral soil resistance (Fig. 1). The ltimate lateral capacity (H ) can be calclated analytically sing limit eqilibrim analysis once this simplification is made: 192

3 3-4 Agst 2, Samarang, Indonesia H H e e p 3γ BDK p z r D (a) D (b) p N p s B Figre 1. Calclation of ltimate lateral capacity in (a) clay and (b) sand. B N p p /s 2 1 N p p /s N p p /s 1 1.B 3B 3B 3B 6B 6B 6B z 12 z (a) (b) (c) Figre 2. Ultimate lateral soil stress distribtion proposed by: (a) Reese (198), (b) Randolph & Holsby (1984), and (c) Broms (1964a) z 9 H M zr p B dz H e zr p B dz p Bz dz + D zr D zr p Bz dz (1) in which B shaft diameter, D shaft depth, p ltimate lateral soil stress, and z r depth of rotation. It is common to correct for simplifications in the calclation model sing the following mltiplicative form: H m M H (2) in which H m measred lateral capacity (more precisely, capacity interpreted from load test), H ltimate lateral capacity compted sing limit eqilibrim analysis, and M model factor, typically assmed to be an independent log-normal random variable. It is well known that many different models exist for the comptation of H. Fig. 2 shows three 193

4 3-4 Agst 2, Samarang, Indonesia common distribtions of p for clay. Fig. 1b shows the distribtion of p sed in the simplified Broms (1964b) approach for sand (z r assmed at pile tip). There are also varios methods of interpreting the lateral capacity from load tests, sch as the displacement limit, rotation limit, lateral or moment limit, and hyperbolic capacity (Fig. 3). The displacement and rotation limits are rather arbitrary and do not relate directly to soil-shaft behavior. The lateral or moment limit (H L ) is based on the mode of soil-shaft failre and essentially represents a first yield or lower bond (Hirany and Klhawy 1988; 1989). The hyperbolic capacity (H h ) represents an ltimate limit or pper bond becase it is the asymptotic limit of the load-displacement crve. However, it reqires extrapolation from measred data and the asymptote is compted mathematically sing a hyperbolic eqation with no reference to actal shaft behavior. This stdy focses on free-head rigid drilled shafts becase fll mobilization of soil strength, as illstrated in Fig. 1, only is applicable if plastic hinges do not form anywhere along the shaft. The database consists of 7 load tests in clayey soils and 77 load tests in sandy soils. Details of this database are given by Chen & Klhawy (1994). The distribtions of the model factors for H m determined sing 2 different criteria (H L or H h ) and H compted from 4 different lateral soil stress models are shown in Fig. 4. Note that M < 1 implies that the calclated capacity is larger than the measred capacity, which is nconservative. If H m is defined as the hyperbolic capacity (H h ), M < 1 is most likely nsafe as well since there is no reserved capacity beyond H h and it is mobilized at very large displacements. It is immediately clear that the traditional factor of safety is not a meaningfl index when drilled shafts designed sing different models are compared. For illstration, consider a simple design example: B 1 m, D/B, e. m, s kpa, and applied load (F) 2 kn. The factors of safety (H /F) compted from Reese, Broms, and Randolph & Holsby models are 3.1, 1.7, and 3.4, respectively. If one assmes that the compted capacity adjsted by the mean model factor is close to the actal capacity, then the revised factors of safety (H m /F) are 2.8, 2.6, and 2.9, respectively for H m H L and 4.3, 4., and 4., respectively for H m H h. A drilled shaft designed sing Broms method has to be significantly larger to achieve the same nominal factor of safety, becase the degree of conservatism intrinsic in the method is not inclded. Hence, it is important to consider the model factor, even within or existing working stress design framework. Figre 3. Criteria to interpret lateral load capacity (Hirany & Klhawy 1988). 194

5 Capacity model (soil type) Reese (198) Broms (1964a) Randolph & Holsby (1984) Broms (1964b) (sand) 3 rd International Conference on Geotechnical Engineering combined with 3-4 Agst 2, Samarang, Indonesia Lateral or moment limit (H L ) Hyperbolic capacity (H h ) Mea H L /H (Reese) Mea H L /H (Broms) H L /H (Randolph & Holsby) Figre 4. Distribtion of model factors. Mea Mea H L /H (simplified Broms) Mea H h /H (Reese) Mea H h /H (Broms) Mea H h /H (Randolph & Holsby) Mea H h /H (simplified Broms) Another aspect clearly exhibited by Fig. 4 is that model factors are scattered [coefficient of variation (COV) between 3% and 4%] and this ncertainty complicates the selection of a single representative nmber for design. Althogh the mean model factor is a somewhat natral choice, one is rightflly concerned abot smaller vales, particlarly those less than one. For example, abot 2% of the Randolph & Holsby model factors are less than one. It is tempting to choose the smallest vale, rather than the mean, bt this is clearly very neconomical. For example, the smallest Randolph & Holsby model factor prodced by the database is.67, which is only half of the mean vale. A more sensible and realistic 19

6 3-4 Agst 2, Samarang, Indonesia approach is to consider the model factor as a random variable and to apply the reliability approach for design. The following section examines the robstness of the model statistics and the validity of applying the model factor as an independent random variable in reliability analysis. Reliability-based design is illstrated in the last section. MODEL STATISTICS Are model factors site-specific? The model-scale load tests were condcted in niform kaolinite clay and filter sand deposits prepared nder controlled laboratory conditions. Hence, ncertainties arising from evalation of soil parameters are minimal. In addition, constrction variabilities and measrement errors associated with load tests also are minimal. Therefore, model ncertainties compted from laboratory tests shold be an accrate indicator of errors arising from simplifications in the calclation models. The main concern is whether the model factors are applicable beyond the niform profile and specific soil type sed in the laboratory. Model factors from field tests are expected to be more general becase they are compted from load tests condcted in more diverse site environments. However, it is reasonable to qery if the statistics of sch model factors are lmped statistics, in the sense that extraneos sorces of ncertainties (e.g., constrction variabilities, measrement errors incrred dring load test, ncertainties in soil parameter evalation) are inextricably inclded in the comptation. A comparison between laboratory and field data sch as those shown in Fig. is illminating. Laboratory and field reslts are plotted as white and shaded histograms in the left panel of each figre, respectively. Visal inspection and simple statistics [mean, standard deviation (S.D.), coefficient of variation (COV)] show that the histograms are similar. The p- vales from the Mann-Whitney test (p MW ) formally show that the nll hypothesis of eqal medians can not be rejected at the cstomary % level of significance. Therefore, it is reasonable to combine the data and arge that the reslts presented in Fig. 4 have wider applicability beyond the conditions implied by the nderlying databases, and the model ncertainties are cased mainly by idealizations intrinsic to the respective analytical models. This observation is qite interesting becase the geometric and geotechnical parameters in the laboratory and field databases are noticeably different. For example, the average diameter is abot one order of magnitde smaller in the laboratory tests. Moreover, the ndrained shear strengths and in-sit horizontal soil stress coefficients in the respective ndrained and drained laboratory tests are in the lower ranges of those fond in the field tests, while the effective stress friction angles in the drained laboratory tests are in the pper range of those fond in the field tests. Are model factors log-normally distribted? The log-normal probability density fnction is plotted over the combined laboratory and field data (as a dashed line) for visal comparison in Fig. 4. It appears to be a reasonable probability model for M. However, for sample sizes commonly encontered in load test databases (say n 7 or so), histograms are known to be notoriosly misleading in identifying the nderlying probability model. In fact, it can be proven that histograms derived from small sample sizes are not expected to look like the poplation probability model. Histograms that provide a good fit are sspicios. At present, it is safe to say that the log-normal probability model is more of a reasonable hypothesis, rather than a fact firmly established by empirical data. Using the Anderson-Darling goodness-of-fit test, one cold formally state that there is no evidence to reject the nll hypothesis of lognormality at the cstomary % level of significance, becase the p-vales ( ) are larger than.. 196

7 Capacity model (soil type) Reese (198) 3 rd International Conference on Geotechnical Engineering combined with 3-4 Agst 2, Samarang, Indonesia Lateral or moment limit (H L ) Hyperbolic capacity (H h ) 2 1 Mea p MW Lab Field Mea p MW Lab Field Broms (1964a) Randolph & Holsby (1984) Broms (1964b) (sand) H L /H (Reese) Mea p MW Lab Field H L /H (Broms) 2 Lab Field Mea p MW H L /H (Randolph & Holsby) 2 1 Mea p MW Lab Field H h /H (Reese) 2 Lab Field Mea p MW H h /H (Broms) 2 Field 1 Mea p MW Lab H h /H (Randolph & Holsby) 2 1 Mea p MW Lab Field H L /H (simplified Broms) H h /H (simplified Broms) Figre. Comparison of model factors from laboratory and field tests. GENERALISED MODEL FACTOR By virte of the definition shown in Eq. 1, it is natral to qestion if M is negatively correlated to H. An inspection of the scatter plots wold reveal that sch correlations exist for the Broms method in clay and the simplified Broms method in sand (Fig. 6). The rank correlation provides a simple method to verify statistical independence. For p-vales <. shown in Fig. 6, the nll hypothesis of zero rank correlation is rejected at % significance level. Hence, part of the variation in M is explainable by variations in H. 197

8 Capacity model (soil type) Reese (198) 3 rd International Conference on Geotechnical Engineering combined with 3-4 Agst 2, Samarang, Indonesia Lateral or moment limit (H L ) Hyperbolic capacity (H h ) 2. rank correlatio -.48 p-vale rank correlatio -. p-vale.374 M HL/H M Hh/H Broms (1964a) ln[h (Reese) (kn)] 3. rank correlatio p-vale ln[h (Reese) (kn)] 4. rank correlatio p-vale. M HL/H M Hh/H Randolph & Holsby (1984) M HL/H ln[h (Broms) (kn)] M Hh/H ln[h (Broms) (kn)] rank correlatio -.9 p-vale.617 rank correlatio -.12 p-vale Broms (1964b) (sand) ln[h (Randolph & Holsby) (kn)] 1. rank correlatio p-vale.6 ln[h (Randolph & Holsby) (kn)] 2. rank correlatio p-vale.13 M HL/H 1.. M Hh/H ln[h (simplified Broms) (kn)] ln[h (simplified Broms) (kn)] Figre 6. Correlation between the model factor and the compted capacity 198

9 3-4 Agst 2, Samarang, Indonesia To remove the partial dependence between M and H, a generalized model factor can be defined as follows: H h M H α (3) It is important to emphasize that the generalized model factor in Eq. 3 is not dimensionless in contrast to the model factor in Eq. 2. The force nit adopted throghot this paper is kn and this nit mst be sed in conjnction with Eq. 3. Eq. 3 has a sond theoretical basis in standard linear regression theory becase: ln(h h ) λ + α ln(h ) + ε (4) in which λ and α regression constants and ε normal random variable with zero mean and variance ξ 2. Comparing Eqs. 3 and 4, it can be shown that: M exp( λ + ε) μ σ M 2 M exp( λ +. ξ μ 2 M [exp( ξ 2 2 ) ) 1] () in which μ M and σ M mean and standard deviation of M, respectively. Reslts from linear regression are shown in Table 1. The generalized model statistics are comparable with those shown in Fig. 4 becase the exponent α is close to 1. However, rank correlation between the regression error and the compted capacity (ρ ε,h ) are removed as shown in Table 1. In particlar, the correlations between M and H for the Broms method in clay and the simplified Broms method in sand (Fig. 6) are no longer significant when ε and H are considered (Fig. 7). Table 1. Regression parameters and statistics of generalised M. Capacity model Lateral or moment limit (H L ) Hyperbolic capacity (H h ) (soil type) Regression Statistics of M Regression Statistics of M parameters parameters Reese (198) λ α.993 ξ.29 R μ M.92 σ M ρ ε,h.1 λ.31 α.993 ξ.284 R μ M 1.42 σ M ρ ε,h -.3 Broms (1964a) Randolph & Holsby (1984) Broms (1964b) (sand) λ.32 α.96 ξ.36 R λ -.2 α.997 ξ.288 R λ α.949 ξ.383 R μ M 1.48 σ M.6.38 ρ ε,h -.2 μ M.8 σ M ρ ε,h.2 μ M.86 σ M.3.4 ρ ε,h -.7 λ.73 α.968 ξ.343 R λ.239 α.997 ξ.28 R λ.189 α.967 ξ.3 R μ M 2.26 σ M.8.38 ρ ε,h -. μ M 1.32 σ M ρ ε,h -.3 μ M 1.29 σ M ρ ε,h

10 Capacity model (soil type) Broms (1964a) ε ln(hl) ln(h) 3 rd International Conference on Geotechnical Engineering combined with 3-4 Agst 2, Samarang, Indonesia Lateral or moment limit (H L ) Hyperbolic capacity (H h ) rank correlatio -.23 p-vale.846 e ln(hh) ln(h) rank correlatio -. p-vale Broms (1964b) (sand) ε ln(hl) ln(h) ln[h (Broms) (kn)] rank correlatio -.7 p-vale.3 ε ln(hh) ln(h) ln[h (Broms) (kn)] rank correlatio p-vale ln[h (simplified Broms) (kn)] -1. ln[h (simplified Broms) (kn)] Figre 7. Correlation between the generalized model factor and the compted capacity RELIABILITY-BASED DESIGN The ltimate limit state is defined as that in which the hyperbolic capacity of a laterallyloaded drilled shaft is eqal to the ltimate applied load. Clearly, the drilled shaft will fail if the hyperbolic capacity is less than this applied load. Conversely, the drilled shaft shold perform satisfactorily if the applied load is less than the hyperbolic capacity. These three sitations can be described concisely by the following performance fnction G, as follows: G H h - F (6) in which F fondation load. The ltimate failre of drilled shafts nder lateral load is rare, bt a proper assessment of this condition still is needed as part of a rational limit state design approach. The basic objective of reliability-based design (RBD) is to ensre that the probability of failre of a component does not exceed an acceptable threshold level. This objective can be stated concisely sing the performance fnction as follows: p f Prob(G < ) p T (7) in which Prob( ) probability of an event, p f probability of failre, and p T acceptable target probability of failre. A more convenient alternative to the probability of failre is the reliability index (β), which is defined as: β - Φ -1 (p f ) (8) 2

11 3-4 Agst 2, Samarang, Indonesia in which Φ -1 ( ) inverse standard normal cmlative fnction. The probability of failre and the reliability index can be compted easily sing EXCEL (Phoon 24). For illstration, consider a simple design example: B 1 m, e. m, s is lognormally distribted with mean kpa and 3 to %, and F is lognormally distribted with mea 2 kn and %. The depth to diameter ratios (D/B) and nominal factors of safety (FS H /F) reqired to achieve a target reliability index of 3 for different calclation models are shown in Table 2. For the Broms method, the drilled shaft can be over-designed if the model factor is not inclded. It has been shown in the previos section that the standard model factor for Broms method is correlated to the compted capacity. If this correlation is ignored in the reliability analysis, the drilled shaft wold be nder-designed (as compared to D/B obtained sing the generalized model factor). Note that the D/B ratios prodced by different calclation models are more consistent when the model factor is inclded. As to be expected, a longer drilled shaft is needed when the ncertainty in the ndrained shear strength (s ) increases from a COV of 3% to %. Finally, the nominal factor of safety is back-calclated from the designs obtained from reliability analyses. It varies over a wide range from abot 2 to, depending on the calclation method and the ncertainty in the design soil parameter. Hence, the application of a single nominal factor of safety, say FS 3, wold prodce non-niform reliability over the range of calclation methods and ncertainties in the design soil parameter. CONCLUSIONS This paper discsses important statistical isses nderlying the definition of model factor and the significance of incorporating model factors in reliability-based design. Model-scale and fll-scale rigid drilled shafts sbjected to lateral-moment loading are analyzed sing different capacity calclation models to illstrate the above concerns. Three important statistical isses were examined rigorosly: (1) generality of model factors beyond conditions implied by database, (2) log-normality of data, and (3) statistical dependencies between the model factor and the compted capacity. The first aspect reqires a comparison between controlled laboratory tests and fll-scale field tests. The log-normal probability density fnction appears to fit the data rather well. There are hidden dependencies arising from the standard model factor definition that can be removed sing a proposed generalization based on linear regression theory. The impact of the model factor and its potential correlation with the compted capacity in reliability-based design is discssed. Table 2. Depth to diameter ratios (D/B) and nominal factors of safety (FS H /F) reqired to achieve a target reliability index of 3 for different calclation models. Reese Broms Randolph & Holsby D/B FS D/B FS D/B FS COV of s 3% w/o model factor Model factor Generalized model factor COV of s % w/o model factor Model factor Generalized model factor

12 REFERENCES 3 rd International Conference on Geotechnical Engineering combined with 3-4 Agst 2, Samarang, Indonesia Briad, J. L., and Tcker, L. M Measred and predicted axial response of 98 piles. Jornal of Geotechnical Engineering, ASCE 114(9): British Standards Institte 22. Erocode: Basis of strctral design. EN 199:22, London. Broms, B. B. 1964a. Lateral resistance of piles in cohesive soils. Jornal of Soil Mechanics and Fondations Division, ASCE 9(SM2): Broms, B. B. 1964b. Lateral resistance of piles in cohesionless soils. Jornal of Soil Mechanics and Fondations Division, ASCE 9(SM3): Chen, Y-J. & Klhawy, F. H Case history evalation of behavior of drilled shafts nder axial and lateral loading. Report TR-1461, Electric Power Research Institte, Palo Alto. DiMaggio, J., Saad, T., Allen T., Christopher, B. R., Dimillio, A., Goble, G., Passe P., Shike, T. and Person, G Geotechnical engineering practices in Canada and Erope. Report FHWA-PL-99-O, Federal Highway Administration, Washington. Filippas, O. B., Klhawy, F. H. and Grigori, M. D Reliability-based fondation design for transmission line strctres: Uncertainties in soil property measrement. Report EL-7(3), Electric Power Research Institte, Palo Alto. Goble, G Geotechnical related development and implementation of Load and Resistance Factor Design (LRFD) methods. NCHRP Synthesis 276, Transportation Research Board, Washington. Hirany, A. & F.H. Klhawy Condct and interpretation of load tests on drilled shaft fondation: detailed gidelines. Report EL-9 (1). Electric Power Research Institte, Palo Alto. Hirany, A. and Klhawy, F. H Interpretation of load tests on drilled shafts - Part 3: lateral & moment. Fondation. Engineering: Crrent Principles & Practices (GSP 22), Ed. F. H. Klhawy, ASCE, New York, Klhawy, F. H., Birgisson, B., and Grigori, M. D Reliability-based fondation design for transmission line strctres: Transformation models for in-sit tests. Report EL- 7(4). Electric Power Research Institte, Palo Alto. Olson, R. E. and Dennis, N. D Review and compilation of pile test reslts, axial pile capacity. Geotechnical Engineering Report CR83-4, Department of Civil Engineering, University of Texas, Astin. Orchant, C. J., Klhawy, F. H. and Tratmann, C. H Reliability-based fondation design for transmission line strctres: Critical evalation of in-sit test methods. Report EL-7(2). Electric Power Research Institte, Palo Alto. Paikowsky, S. G. 22. Load and resistance factor design (LRFD) for deep fondations. In Y. Honjo, O. Ksakabe, K. Matsi, M. Kosa & G. Pokharel (eds.); Proc. International Workshop on Fondation Design Codes and Soil Investigation in view of International Harmonization and Performance Based Design, Hayama, Japan: Phoon, K. K., Klhawy, F. H., and Grigori, M. D Reliability-based design of fondations for transmission line strctres. Report TR-. Electric Power Research Institte, Palo Alto. Phoon, K. K. & Klhawy, F. H. 1999a. Characterization of geotechnical variability. Canadian Geotechnical Jornal 36(4): Phoon, K. K. & Klhawy, F. H. 1999b. Evalation of geotechnical property variability. Canadian Geotechnical Jornal 36(4): Phoon, K. K., Becker, D. E., Klhawy, F. H., Honjo, Y., Ovesen, N. K. & Lo, S. R. 23. Why consider reliability analysis in geotechnical limit state design? Proc. International 22

13 3-4 Agst 2, Samarang, Indonesia Workshop on Limit State design in Geotechnical Engineering Practice (LSD23), Cambridge, MA (CDROM). Phoon, K. K. 24. General non-gassian probability models for First-Order Reliability Method (FORM): A state-of-the-art report. ICG Report (NGI Report ), International Centre for Geohazards, Oslo. Phoon, K. K., Chen, J. R. & Klhawy, F. H. 26. Characterization of model ncertainties for agered cast-in-place (ACIP) piles nder axial compression. GeoShanghai, Shanghai, China. Randolph, M. F. & Holsby, G. T Limiting pressre on a circlar pile loaded laterally in cohesive soil. Geotechniqe 34(4): Reese, L. C Discssion of Soil modls for laterally loaded piles. Transactions, ASCE 123: Sidi, I. D Probabilistic prediction of friction pile capacities. PhD Thesis, University of Illinois, Urbana-Champaign. Spry, M. J., Klhawy, F. H. and Grigori, M. D Reliability-based fondation design for transmission line strctres: Geotechnical site characterization strategy. Report EL- 7(1), Electric Power Research Institte, Palo Alto. Withiam, J. L. 23. Implementation of the AASHTO LRFD bridge design specifications for sbstrctre design. Proc. International Workshop on Limit State design in Geotechnical Engineering Practice (LSD23), Cambridge, MA (CDROM). 23