Updating Uncertainties in Soil Shear Strength Parameters with Multivariate In-Situ and Laboratory Test Data

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1 The 1 th International Conferene of International Assoiation for Compter Methods and Advanes in Geomehanis (IACMAG) 1-6 Otober, 008 Goa, India Updating Unertainties in Soil Shear Strength Parameters with Mltivariate In-Sit and Laboratory Test Data Jianye Ching, Yi-Ch Chen Dept of Constrtion Engineering, National Taiwan University of Siene and Tehnology, Taiwan. Kok-Kwang Phoon Dept of Civil Engineering, National University of Singapore, Singapore. Keywords: nertainty, Bayesian analysis, empirial eqation, frition angle, ndrained shear strength ABSTRACT: In reent years, geotehnial engineers have paid mh attention on reliability-based design. The probability density fntions (PDFs) of the nertain soil parameters are reqired for the reliability-based design. One these PDFs are obtained, reliability analysis an be ondted, and reliability-based design an be ahieved. Among the soil parameters, soil strengths play important roles in geotehnial design, e.g. frition angle of sand and ndrained shear strength of lay. It is essential to obtain the PDFs of the soil strengths to failitate reliability-based design. How to obtain those PDFs based on all available information is an important researh sbjet. It is possible to establish the PDFs of the soil strengths based on reslts from sophistiated laboratory tests, e.g. drained test for sand to obtain its frition angle, bt those tests may be expensive and time onsming. On the other hand, some in-sit and laboratory tests are qik and onvenient, bt they sally do not provide diret information on the soil strengths. Nevertheless, they do sally provide indiret information of the soil strengths by orrelation, e.g. SPT-N is positively orrelated to frition angle of sand, and overonsolidation ratio (OCR) is also related to ndrained shear strength of lay. Therefore, it is possible to pdate the PDFs of the soil strengths by sing the in-sit and laboratory test data. 1 Introdtion Unertainties are abndant in geotehnial engineering. Possible sores of the nertainties may inlde inherent variabilities, measrement errors and modeling nertainties. In reent years, the probabilisti framework has been adopted to rigorosly qantify sh nertainties. The standard way of ahieving so is to transform the information of field or laboratory test data, denoted by test indies later, into the probability density fntions (PDF) of the qantities of interest. Qite freqently, transformation eqations between soil parameters are very sefl in onverting test index information into shear strength information throgh parametri orrelation. In the literatre, sh pairwise orrelations between varios test indies and shear strengths have been stdied. Klhawy and Mayne (1990) and Phoon (1995) both ontain fairly omprehensive reviews abot the pairwise orrelations between varios test indies and shear strengths. In reality, mltivariate information is sally available dring in-sit and laboratory tests. For instane, when boreholes are made, SPT-N vales are sally available; moreover, the information abot nit weight, plastiity index (PI), liqid limit (LL), water ontent, et. is also often available. Many of these test indies may be simltaneosly orrelated to the shear strengths of the soils. With the pairwise orrelations at hand, it is possible to pdate the PDF of the shear strength onditioning on a single test index (e.g.: SPT-N), bt it is not obvios how to pdate the PDF of the shear strength onditioning on mltivariate test indies (e.g.: SPT-N and OCR). Of orse, one an disard most test indies bt keep the most relevant test index to pdate the PDF. However, it is a pity to not inorporate all available information in the pdating proess bease the abandoned test indies may frther rede the nertainties. In this paper, the qestion of how to pdate the PDF of shear strengths with mltivariate test index information is addressed. Only the PDF pdating of soil shear strength parameters, i.e.: frition angle for sands and ndrained shear strength for lays, is of onern. Probabilisti graphial models are proposed for sands (and lays) to desribe the probabilisti orrelation between frition angle (and ndrained shear strength) and important test indies. With these models and the pairwise orrelations at hand, it is then possible to pdate the PDF of the frition angle (and ndrained shear strength) onditioning on mltivariate test index information throgh Bayesian analyses. The main prodt of this paper is a set of eqations whose inpts are the observed mltivariate test index vales and otpts are the pdated mean vales and oeffiients of variation (.o.v.) of the shear strength 1898

2 parameters. Graphial models for sandy and layey soils.1 Graphial model for sandy soils Figre 1(a) depits the probabilisti graphial model adopted for sandy soils. For the model of the frition angle of sandy soils, the following test indies are onsidered: (a) relative density (D r ), oeffiient of niformity (C ) and onfining pressre (σ n ); (b) modified SPT-N (standard penetration test) vale ((N 1 ) 60 ), defined as the eqivalent SPT-N vale when effetive vertial stress is eqal to 1P a (P a denotes the atmosphere pressre); () modified CPT (one penetration test) reading (Q ), defined as the CPT reading (in P a ) divided by the sqare root of the effetive vertial stress (in P a ). (a) sandy soils (b) layey soils Figre 1 Graphial models for sandy and layey soils. In the following, the probabilisti pairwise relationships between frition angle and the hosen test indies will be derived based on the pairwise data points in or database. These pairwise relationships are neessary for the mltivariate orrelation. In partilar, the following onditional PDFs are needed to qantify the nertainties in the pairwise relationships: (a) f(φ D r,c,σ n ); (b) f((n 1 ) 60 φ); () f(q φ)..1.1 Probabilisti model between φ and {D r,c,σ n } Relative density D r and oeffiient of niformity C are assmed to be fndamental indies affeting the vale of frition angle. This was sggested by Dnan (004) and is implemented herein. He proposed the following eqation to predit the vale of frition angle based on the knowledge of D r, C and effetive onfining pressre (σ n ): φ = A+ B Dr ( C+ D Dr) log 10 ( σn Pa) + εφ (1.) D r where A, B, C, D are oeffiients depending on C ; their vales an be fond in Table 1; ε φ Dr is the zero-mean predition error term. As shown in the table, the standard deviation of ε φ Dr, denoted by σ φ Dr, ranges from 3.1 o to 3. o depending on C. Table 1 Vales of the A, B, C, D oeffiients. Material type A B C D Standard deviation σ φ Dr Data nmber Gravel with C > o 69 Sand with C > o 6 Sand with C < o 30 By assming ε φ Dr to be Gassian, the probabilisti pairwise relationship an be represented as the PDF of φ onditioning on the knowledge of {D r,c,σ n } as follows: φ A B Dr + C + D Dr log10 σn Pa σ f ( φ Dr, C, = e ( π σ φ D ) (.) r where the A, B, C, D, σ φ Dr parameters depend on C and an be fond in Table 1; the D r vale rather than its perentage vale shold be diretly sed in this eqation. In the ase that the relative density and oeffiient of niformity information is not available, a prior PDF of φ is reqired for the analysis. In that ase, a broad Gassian PDF with mean vale eqal to 35 o and standard deviation eqal to 5 o is assmed based on ommon observation that frition angle of sandy soil mostly lies within the interval of [0 o, 50 o ]. 1899

3 .1. Probabilisti model between φ and modified SPT-N vale The orrelation between frition angle of sandy soils and the SPT-N vale is well known. The data from Hatanaka and Uhida (1996) are from high qality triaxial ompression tests on ndistrbed frozen samples. These data are taken to derive the probabilisti pairwise relationship between frition angle and modified SPT-N vale. A parametri regression model is assmed for the pairwise relationship, and the maximm likelihood method is employed to estimate the regression parameters as well as the standard deviation. For this speifi pairwise relationship, the following regression model is obtained: φ 7.5 log10 (( N1 ) ) = + ε (3.) 60 log 10 (( N1 ) 60 ) φ 9. where the standard deviation of ε log10((n1)60) φ is The PDF of (N 1 ) 60 onditioning on the knowledge of φ is therefore log 10( ( N1 ) 60 ) ( φ 7.5) f log ( N ) φ = e π (4.) ( 10 ( 1 60 ) ).1.3 Probabilisti model between φ and normalized CPT reading The orrelation between frition angle and the modified CPT reading has been stdied by Klhawy and Mayne (1990). The pairwise relationship proposed by Klhawy and Mayne (1990) is φ = log 10 ( Q ) + ε φ (5.) Q where Q =[Q /P a ]/[σ /P a ] 1/ is the modified CPT reading. The standard deviation of ε φ Q is estimated to be.8 o. The relationship provided by Klhawy and Mayne (1990) an be sed to obtain f(φ Q ); however, or goal is to derive f(q φ), i.e.: we need to onvert (5.) into ln ( Q ) = ( φ 17.6) ε (6.) ln( Q ) φ where ε ln(q ) φ is the predition error term. Note that the 10-basis logarithm in (5.) has been onverted into natral logarithm in (6.). In order to estimate the standard deviation of ε ln(q ) φ, the maximm likelihood method is taken, and the reslting estimate of the ε ln(q ) φ standard deviation is The PDF of Q onditioning on the knowledge of φ is therefore φ ( ) ( ln( Q ) φ) = π (7.) f Q e Q. Graphial model for layey soils Figre 1(b) depits the probabilisti graphial model adopted for layey soils. For the model of the ndrained shear strength (S ) of layey soils, the following test indies are onsidered: (a) overonsolidation ratio (OCR) and plastiity index (PI); (b) (nmodified) SPT-N vale (N); () adjsted CPT reading Q, defined as Q -σ. This model is similar to the model for sands exept that D r and C are now replaed by OCR and PI. In fat, the OCR and PI in lays play similar roles of the D r and C in sands. Similarly, the following onditional PDFs are needed to qantify the nertainties in the pairwise relationships: (a) f(s OCR,PI,σ ); (b) f(n S ); () f(q S )...1 Probabilisti model between S and {OCR, PI} Overonsolidation ratio OCR and plastiity index PI are assmed to be fndamental indies affeting S. The following eqation was mentioned in Phoon (1995) that orrelates {OCR, PI} to ndrained shear strength diretly obtained from a field vane shear test: VST VST S σ = [ PI] OCR or S σ p = PI (8.) where S VST denotes the ndrained shear strength determined from a field vane shear test; σ p denotes the preonsolidation stress; PI is in perentage (PI = 100 means PI is one hndred perent). The aray, i.e.: oeffiient of variation (.o.v.), of this eqation was reported to be arond 5% by Chandler (1988). By taking logarithm, the eqation an be onverted into the following eqation: VST ln S = ln PI + ln OCR + ln σ + ε (9.) VST ln( S ) OCR where the standard deviation of ε ln(s VST) OCR orresponding to the 5%.o.v. is A orretion fator μ is reqired to onvert S VST into the design ndrained shear strength (Klhawy and Mayne 1990; Phoon 1995): μ = PI (10.) and the standard deviation of ε μ is estimated to be Combining (9.) and (10.), we have ln S = ln PI + ln OCR + ln σ + ln PI + ε (11.) ln( S ) OCR where the standard deviation of ε ln(s) OCR is ( ) 0.5 =0.91 by assming independene between ε ln(s VST) OCR and ε μ. Let s frther assme ε ln(s) OCR is zero-mean Gassian, then the PDF of S onditioning on the knowledge of OCR, PI and σ is ln( S ) ln( ) ln ln( 0 ) ln + PI OCR σ v ( PI ) 0.91 f S OCR, PI, σ = e π 0.91 S (1.) 1900

4 In the ase that the OCR information is not available, a prior PDF of S is reqired for the analysis. In that ase, a lognormal PDF is assmed whose mean vale and.o.v. are estimated based on the database olleted by Phoon (1995). The overall average vale of S regardless the test type is roghly 173. kn/m and.o.v. eqal to Probabilisti model between S and SPT-N vale The orrelation between S of layey soils and the SPT-N vale is well known. the pairwise data omplied by Hara et al. (1985) (also see Phoon (1995) and Klhawy and Mayne (1990)), the following eqation is proposed for the relationship: UU ln N = ln S + ε (13.) UU ( ) ln where the standard deviation of ε ln(n) S UU is 0.407, and the nit of S UU is in kn/m. Aording to the database presented in Chen and Klhawy (1993), UU CIUC CIUC ln S S = ln S σ + ε (14.) N S ( ) ( ) where the standard deviation of ε is roghly Eqation (13.) implies that ln ( N) = ln ( S σ ) + ln ( S) + ε + ε UU = ln ( S ) 0.403ln ( 0) ln( N) S σv + εln( N) S (15.) where the standard deviation of ε ln(n) S is ( ) 0.5 =0.456, and both S and σ are in the nit of kn/m. By assming ε ln(n) S to be Gassian, the PDF of N onditioning on the knowledge of S is therefore ln ln 0.403ln( 0 ) N + S + σ v f ( N S, σ ) e π N = (16.)..3 Probabilisti model between S and CPT reading The orrelation between ndrained shear strength and the adjsted CPT reading Q =Q -σ has been stdied by several previos stdies. Phoon (1995) reported that: S Q (17.) for CIUC test reslts. Phoon (1995) frther reported that the nertainty of (17.) is roghly 35%. The probabilisti version of (17.) is, therefore, ln S = ln ln Q + ε =.54 + ln Q + ε (18.) ( ) ln ( ) ln S Q S Q where ε ln(s) Q is taken to be zero-mean Gassian with standard deviation eqal to 0.34, orresponding to the 35% nertainty of (17.). In other words, ln( Q ).54 ln( S) 0.34 f ( Q S ) = e π 0.34 Q (19.) 3 Bayesian inferene with mltivariate test data Bayesian analysis is a natral way of handling mltivariate information, or even onfliting information. The basi Bayes rle onsists of the following eqation: f ( x y) f ( y) (0.) f ( y x) = f ( x) where x and y an be both vetors; y is the nertain variable of interest, while x is the observed variable. f(y) is alled the prior PDF of y, whih qantifies the nertainties in y before observation on x is made, and f(x y) is alled the likelihood fntion of y given x. f(y x) is the pdated or posterior PDF of y onditioning on the information of x. In the following setion, an approximate soltion for the pdated mean vale and.o.v. based on the first-order seond moment method (Ang and Tang 1984) is proposed. 3.1 Bayesian analysis based on the first-order seond moment (FOSM) method Let x=q(y)+ε, where q(y) is in general nonlinear, and ε is zero-mean Gassian with variane-ovariane matrix eqal to Σ. The FOSM method simplifies the (x,y) fntional relationship by linearizing the q(y) fntion abot the mean vale of y, i.e.: x q E( y) + y q E( y) y E( y) + ε (1.) where y q[e(y)] is the gradient of the q(y) fntion evalated at the mean vale of y. If we frther assme x and y are jointly Gassian, it trns ot that T E( y x) = E( y) + Var( y) yq E( y) Var( x) x E( x) (.) T Var ( y x) = Var ( y) -Var ( y) yq E( y) Var ( x) yq E( y) Var ( y) where Var(y) is the variane-ovariane matrix of y; T E( x) q E( y) Var ( x) yq E( y) Var ( y) yq E( y) +Σ (3.) 1901

5 are the mean vale and variane-ovariane matrix of x, respetively. 4 Analysis reslts 4.1 Reslts for sandy soils For the ease of the presentation, let s denote A+B D r -(C+D D r )log 10 (σ n /P a ) by F A,B,C,D (D r,σ n ) Conditioning on Dr, C and σ n : Gravel C > 4 E φ D, C, σ = F D, σ Var φ D, C, σ = 3.1 (4.) 44,10,7, ( φ σ ) 39,10,3, ( σ ) ( φ σ ) ( φ σ ) ( σ ) ( φ σ ) r n r n r n Sand C > 6 E D, C, = F D, Var D, C, = 3. (5.) r n r n r n Sand C < 6 E D, C, = F D, Var D, C, = 3. (6.) 4.1. Conditioning on (N 1 ) 60 : r n 34,10,3, r n r n ( ( 1) ) 10 ( 1) E φ N = log N Var φ N = 3.90 (7.) Conditioning on Q : E φ Q ln Q Var φ Q.65 (8.) Conditioning on (N 1 ) 60, Dr, C and σ n : Gravel C > 4 E φ ( N ), D, C, σ = F D, σ log N Var φ N, D, C, σ =.58 ( 1 r n) 44,10,7,( r n) 10( ( 1) ) ( ( 1) r n) Sand C > 6 ( φ ( 1) r 39,10,3, ( r 10 (( 1) ) φ ( 1) r σn E N, D, C, = F D, log N Var N, D, C, =.58 Sand C < 6 ( φ ( 1) r 34,10,3, ( r 10 (( 1) ) φ ( 1) r σn E N, D, C, = F D, log N Var N, D, C, = Conditioning on Q, D r, C and σ n : Gravel C > 4 ( φ r 44,10,7, ( r ( ) ( φ r E Q, D, C, F D, ln Q Var Q, D, C,.0 Sand ' ' ' C > 6 ( φ r 39,10,3, ( r ( ) ( φ r E Q, D, C, F D, +.44 ln Q Var Q, D, C,.37 Sand ' ' ' C < 6 ( φ r 34,10,3, ( r ( ) ( φ r E Q, D, C, F D, +.44 ln Q Var Q, D, C,.37 ' ' ' (9.) (30.) (31.) (3.) (33.) (34.) Conditioning on (N 1 ) 60 and Q : E N, Q = log N ln Q Var N, Q =.67 (35.) ( φ ) ( φ ) Conditioning on (N 1 ) 60, Q, Dr, C and σ n : Gravel C > 4 E φ ( N ), Q, D, C, σ = 0.40 F D, σ log N ln Q ( 1 ) ( ) 60 r n 44,10,7, r n ( φ ( 1),,,, 60 r = ( φ 1 ) ( ) 60 r σn 39,10,3, r σ n ( φ ( 1),,,, 60 r = ( φ 1 ) ( ) 60 r σn 34,10,3, r σ n ( φ,, r,, = Var N Q D C Sand C > 6 E N, Q, D, C, = F D, log N ln Q Var N Q D C Sand C < 6 E N, Q, D, C, = F D, log N ln Q Var N Q D C 1 60 The above reslts only provide the estimates for the first two moments of the posterior PDFs of the frition angle. Note that these estimates are not exat bease the orrelation eqations between the frition angle and the test indies are nonlinear, hene the linearization proedre in FOSM may bring in ertain approximation errors. Also, the posterior PDFs are not of standard types, bt jdging from the small standard deviations, those posterior PDFs shold resemble Gassian PDFs. (36.) (37.) (38.) 190

6 4. Reslts for layey soils For the ease of presentation, let s denote ln( PI)+ln(OCR)+ln(σ )+ln( PI ) by H(OCR,PI,σ ) Conditioning on OCR, PI and σ : E ln S OCR, PI, σ = H OCR, PI, σ Var ln S OCR, PI, σ = Conditioning on N: = + + ( σ ) = E ln S N ln N 0.08 ln Var ln S N Conditioning on Q : E ln S Q = ln Q Var ln S Q = 0.30 (39.) (40.) (41.) 4..4 Conditioning on N, OCR, PI and σ : E( ln ( S) N, OCR, PI, σ ) = ln ( PI) ln ( OCR) ln ( PI ) ln ( σ ) ln ( N) ( Var ( ln ( S) N, OCR, PI, σ ) = ) 4..5 Conditioning on Q, OCR, PI and σ : E( ln ( S) Q, OCR, PI, σ ) = ln ( PI) ln ( OCR) ln ( σ ) ln ( PI ) 0.43 ln ( Q ).074 ( Var ( ln ( S) Q, OCR, PI, σ ) = ) 4..6 Conditioning on N and Q '': Eln S NQ, = ln N ln Q ln σ Varln S NQ, = 0.05 (44.) ( ) ( ) ( ) ( ) 4..7 Conditioning on N, Q, OCR, PI and σ : E ln ( S ) N, Q, OCR, PI, σ ( ) ( PI ) ( OCR) ( PI ) ( σ ) ( N ) ( Q ) ( ln,,,, σ ) = = ln ln + ln ln ln ln Var S N Q OCR PI (45.) 5 Case stdies This deep exavation site is extrated from O (006). SPT-N and CPT tests were ondted at this site at the hosen borehole loations. The soil profile inldes three thik layey layers and three thin sandy layers. The water table is m below the srfae. Cone penetration and vane shear test reslts were taken to estimate the ndrained shear strength of the lays. Moreover, several ndistrbed lay samples are extrated from the site, and laboratory tests, inlding UU and CK o U tests, were taken to determine the design ndrained shear strengths. Table smmarizes the design ndrained shear strengths as well as the other in-sit test indies of the lay at varios depths. Note that the ndrained shear strengths determined by the vane shear tests are mltiplied by the orretion fator to obtain the design S in the table. Bt, the ndrained shear strengths determined by the UU tests are diretly taken as the design S bease it is fond that the S vales determined by the UU tests are onsistent with the CK o U, VST and CPT test reslts. these data, the pdated mean vales and.o.v.s of S are listed in Table 3. It is evident that the pdated.o.v. of S dereases as more information is taken for the pdating. It is obvios that the pdated 95% onfidene interval based on {OCR, σ, PI} are sometimes onservative, i.e.: the design S vales are mostly smaller than the lower bond of the interval. Nonetheless, the pdated 95% onfidene interval based on either N or Q seems reasonable bease the design S vales are mostly inside the intervals. It is interesting to see when the information of {OCR, σ, PI} and N is ombined, the reslting onfidene interval performs satisfatorily althogh the onfidene intervals based on the former are sally not. Similarly, when the information of {OCR, σ, PI} and Q is ombined, the reslting onfidene interval also performs satisfatorily. Finally, when all information is ombined, the reslting onfidene intervals seem tighter, bt yet most of the design S vales fall into the intervals. 1903

7 Depth (m) Test S (kn/m ) Table In-sit data and eqation parameters. Design S (kn/m ) Test data Test type PI OCR (%) N σ (kn/m ) Q (kn/m ) 11.3 CK 0 U-AC UU VST UU CK 0 U-AC UU VST UU UU VST VST UU UU ** Converted throgh Q =Q -σ. Q ** (kn/m ) Table 3 mean vales,.o.v.s and onfidene intervals of S for varios test data ases. Soil profile Test depth for sample example depth (m) (m) Design S (kn/m ) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ mean vale of S (kn/m ) (lower bond; pper bond) of 95% onfidene interval Case 1 Case Case 3 Case 4 Case 5 Case 6 Case 7 {OCR} SPT-N Q {OCR}, N {OCR}, Q N, Q {OCR}, N, Q.o.v. = 0.91.o.v. = 0.56.o.v. = 0.30.o.v. = 0.01.o.v. = 0.1.o.v. = 0.05.o.v. = (15.6, 49.9) (59.1, 165.7) (35.9, 10.3) (35., 78.3) (3.6, 56.8) (5.5, 119.1) (37.0, 73.7) (15.0, 48.4) (5.5, 146.9) (35., 117.9) (3.1, 71.5) (.9, 55.1) (47.9, 108.9) (34.5, 68.7) (16.4, 5.5) (56.8, 159.) (30.9, 103.5) (35., 78.3) (.4, 54.1) (47.5, 107.8) (35., 70.8) (17.1, 55.1) (63.4, 175.9) (30.6, 101.5) (38.5, 85.6) (.9, 55.1) (50.4, 114.4) (37.3, 75.) (18., 58.0) (64.1, 179.5) (31.8, 106.7) (39.6, 88.) (4.0, 58.0) (51.9, 117.9) (38.9, 77.5) (18.5, 59.7) (64.7, 179.5) (33.4, 11.) (40.4, 90.0) (5.0, 60.9) (53.5, 11.5) (40.0, 79.8) (18.5, 59.7) (64.7, 181.3) (33.8, 11.) (40.4, 90.0) (5.3, 60.9) (54.1, 1.7) (40.4, 80.6) (19.7, 6.8) (66.0, 184.9) (37.7, 16.5) (4.1, 93.7) (7.7, 66.7) (57.4, 130.3) (4.9, 85.6) (19.7, 6.8) (66.0, 184.9) (35.9, 10.3) (4.1, 93.7) (6.8, 64.7) (56.3, 17.7) (4.1, 84.8) (19.9, 64.1) (66.7, 184.9) (38.1, 16.5) (4.5, 94.6) (7.9, 67.4) (58.0, 131.6) (43.4, 86.5) (0.9, 66.7) (74.4, 06.4) (40.9, 137.0) (46.5, 103.5) (30.0, 7.) (64.1, 145.5) (47.5, 94.6) (1.8, 69.4) (81.5, 5.9) (47.9, 159.) (49.9, 11.) (33.1, 80.6) (73.0, 165.7) (53.0, 105.6) (3.1, 73.7) (94.6, 6.4) (58.6, 194.4) (56.3, 16.5) (38.1, 9.8) (88., 00.3) (61.6, 14.0) 6 Disssion and onlsions A new framework is proposed to pdate the probability distribtions of shear strengths of sandy and layey soils based on in-sit and laboratory test data and indies. More speifially, a new method that is based on pairwise 1904

8 orrelations developed by previos researh is proposed to pdate the probability distribtions with mltivariate test data and indies. The shear strengths of interest inlde φ for sands and S for lays. For φ, relative density, oeffiient of niformity, SPT-N vale and CPT reading are the hosen test indies. For S, overonsolidation ratio, plastiity index, SPT-N vale and CPT reading are hosen. The main prodt of this paper is a set of eqations whose inpts are the observed mltivariate test index vales and otpts are the pdated mean vales and oeffiients of variation (.o.v.) of the shear strength parameters. One the PDFs of the nertain soil parameters are known, the reliability-based design an be ahieved. The reslts of this researh may be benefiial to reliability-based designs. A real ase stdy is employed to demonstrate the seflness of the proposed framework. The reslts show that the proposed framework offers satisfatory estimations of the ndrained shear strengths. 7 Referene Ang A.H.S., Tang W.H Probability Conepts in Engineering Planning and Design: Basi Priniples, John Wiley and Sons. Chandler R.J The in-sit measrement of the ndrained shear strength of lays sing the field vane. Vane Shear Strength Testing in Soils: Field and Laboratory Stdies (DTP1014), ASTM, Philadelphia, Chen Y.J., Klhawy F.H Undrained strength interrelationships among CIUC, UU, and UC tests. Jornal of Geotehnial Engineering, 119(11), Dnan J.M Frition angle for sand, gravel and rokfill. Notes of a Letre Presented at the Kenneth L. Lee Memorial Seminar, Long Beah, California. Hara A., Ohta T., Niwa M., Tanaka S., Banno, T Shear modls and shear strength of ohesive soils. Soils and Fondations, 14(3), 1-1. Hatanaka M., Uhida A. 1996, Empirial orrelation between penetration resistane and internal frition angle of sandy soils. Soils and Fondations, 36(4), 1-9. Klhawy F.H., Mayne P.W Manal on Estimating Soil Properties for Fondation Design, Report EL-6800, Eletri Power Researh Institte, Palo Alto. Ladd C.C., Foote R., Ishihara K., Shlosser F., Polos H.G Stress-deformation and strength harateristis. Proeedings of 9th International Conferene on Soil Mehanis and Fondation Engineering, Tokyo. O C.Y Deep Exavation Theory and Pratie, Taylor & Franis Grop, London (UK). Phoon K. K. 1995, Reliability-based Design of Fondations for Transmission Line Strtres. Ph.D. Dissertation, Cornell University, Ithaa, NY. 1905