Structural satety - 九州大学（KYUSHU UNIVERSITY）

ELSEVIER Structural Safety 14 (1994) 103-130

Structural satety

Performance factor for bearing resistance of bored friction piles **

Hidetosh i Ochiai ,,a, Jun Otani b, Kenji Matsui c

a Department of Civil Engineering, Kyushu University, Fukuoka, Japan b Department of Civil and Environmental Engineering, Kumamoto University, Kumamoto, Japan

c CTI Engineering Co., Ltd., Fukuoka, Japan

Abstract

This paper describes a method for evaluating the reliability of the vertical bearing resistance of bored friction piles, in which a semi-empirical expression using the SPT-N value is focused on. Two uncertainties, which are the spatial distribution of SPT-N value and the resistance coefficients in the bearing resistance expression, are quantitatively evaluated by statistical analyses of various soil investigations and loading test results. Based on these evaluations, a performance factor for the Limit State Design is proposed for the design of bored friction piles. The performance factor proposed here is also applied to a bridge foundation design.

Key words: Bearing resistance; Bored friction piles; In-situ loading test; Spatial distribution; Statistical analysis; SPT-N value; Uncertainty.

I. Introduction

Pile foundations are classified into two major categories, i.e. end bearing and friction piles [1], depending on the mechanisms of load transfer to the ground. End bearing type of pile foundations has been conventionally used in Japan. Friction piles, however, may be particularly efficient depending on the scale of the structures and soil conditions. For these friction piles, the friction around the pile shaft is the dominant resistance for the estimation of the bearing capacity. The uncertainties in evaluating this shaft resistance largely depend on the accuracy of the bearing resistance expression as well as that of the soil property of the ground. In Japan, a semi-empirical bearing resistance expression based on the N value from a Standard Penetra- tion Test (SPT) has been used for these friction piles. The basic idea of this semi-empirical

* Corresponding author. ** Discussion is open until November 1994 (please submit your discussion paper to the Editor, Ross B. Corotis).

0167-4730/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0167-4730(93)E0045-D

104 H. Ochiai et al. /Structural Safety 14 (1994) 103-130

form is that the resistance is linearly dependent on the SPT-N value for both the tip and shaft of piles. Design methods for pile foundations are shifting from Working Stress Design to Limit State Design concepts [2-4]. In order to introduce the Limit State Design, a reliability design is indispensable for the determination of a performance factor.

In this paper, the authors develop a method for evaluating the reliability of the bearing resistance of bored friction piles, which is based on the results of in-situ loading tests and consideration of the spatial distribution of soil properties by a statistical analysis. A performance factor for the Limit State Design is then proposed using an evaluation method developed here. Finally the proposed performance factor is applied to a bridge foundation design.

2. Uncertainties in the bearing resistance expression

2.1. Semi-empirical bearing resistance expression

An ultimate bearing resistance of vertically loaded piles, Ru, is generally expressed as a sum of pile tip and shaft resistances:

M

R u = R p + R F = q d A p + U Y'~ ( l i f i ) (1) i=1

in which R p = resistance at pile tip, R e = resistance around pile shaft, qd = unit tip resistance, fi = unit shaft resistance in the ith layer of the ground penetrated by the pile, Ap = pile tip area, U = pile perimeter, l i = thickness of the ith layer, and M is a total number of layers. When the values, qa and ft, in Eq. (1) are properly determined, the ultimate bearing resistance, Ru, is calculated properly by Eq. (1).

A semi-empirical expression for the resistance based on SPT-N value has been widely used for the design of pile foundation in Japan. Equation (1) is then rewritten as [1]

M

Ru =apUpAp + U Y'~ (olfiliNi) (2) i= l

w h e r e a p = tip resistance coefficient, afg = shaft resistance coefficient in the ith layer, Np = SPT-N value near the pile tip, and N, = SPT-N value in the ith layer. In order to predict the bearing resistance properly by Eq. (2), the quantitative evaluations of the SPT-N value and the resistance coefficients, ap and aft, should be taken into account [5].

2.2. SPT-N value

The soil properties, such as SPT-N values, exhibit considerable variation. This is usually caused by the following three types of uncertainty [6]:

I: natural heterogeneity or the in-situ variability of the soil; II: limited availability of information about subsurface conditions;

III: measurement errors due to human factors.

H. Ochiai et al. / Structural Safety 14 (1994) 103-130

Table 1 The first- and second-order statistics of the resistance coefficients and the resistance ratios

105

Sample Mean value Standard Coefficient of size deviation variation

Resistance ctf(sandy) 40 0.412 0.252 0.611 coefficient a f(clayey) 35 1.58 1.06 0.672

a o 16 10.6 5.74 0.541

Resistance PF 16 1.01 0.365 0.361 ratio Pu 16 0.991 0.294 0.296

Fujita et al. [7] have investigated the difference of SPT-N values related to the distance between two boring points by using several boring data conducted in Japan. According to the results of this study, the measurement errors due to human factors was negligible in the estimation of the SPT-N value, so that it was concluded that the uncertainty of the SPT-N value is mainly caused by its spatial distribution due to the natural heterogeneity of the soil.

2.3. Resistance coefficient

According to the Japanese Specifications for Substructures [1], the ultimate bearing resistance of the pile is estimated by the unit tip resistance, qd, and unit shaft resistance, f,., with SPT-N values based on in-situ loading test data [8] by Eq. (2). The total number of loading test results of bored piles is 32 which includes 16 cases of end bearing piles and 16 cases of friction piles. The ultimate bearing resistance, Rul 0, is defined as the pile head load at a displacement level of 10% of the pile diameter [8-10]. Based on these results, Table 1 shows the first- and second-order statistics for the resistance coefficients, ap and aft, and the resistance ratios, PF and Pu, which are defined as follows:

Otfi = f i m l N i , O~p = q d m / N p (3a,b)

and

P F = R F m / R F c Pu = Rum//Ruc (4a,b)

where subscripts m and c stand for measured and calculated values, respectively. The variation of the resistance coefficients ranges from 54% to 67% in terms of the coefficient of variation (COV). According to the X2-statistical test with a significance level at 5%, the distributions of the coefficient resemble either a normal distribution or a log-normal distribution. The mean values of both resistance ratios, PF and Pu, are close to one, so that it may be considered that the bearing resistance expression by means of the resistance coefficients, ap and af, could estimate the measured values without bias. However, the COV of PF at 36%, and the COV of Pu at 30% show large variations. In order to improve the reliability of bearing resistance, it is most effective to use the results of in-situ loading tests at each individual site.

106 H. Ochiai et al. / Structural Safety 14 (1994) 103-130

3. Estimation of spatial distribution of SPT-N value

3.1. Auto-correlation function of SPT-N value

Uncertainties in soil properties have conventionally been expressed in terms of mean value and variance of random variables [11-14]. However, this is not sufficient to express the spatial distribution of soil properties. The spatial distribution of the soil property is expressed by introducing an auto-correlation coefficient in addition to the mean value and the variance [6]. The concept of a random field is based on the presumption of a population. When a site is specified, there exists a sample field. It is, however, impossible to determine this sample field completely, and the soil properties at points excluding the sample points must be estimated. Such estimation introduces errors resulting from the spatial distribution of the soil properties. Here in this paper, the Kriging technique is used for the estimation of the sample field [15,16].

Kriging technique has been used by Christakos [17] and Honjo et al. [18] for estimating the plane distribution of settlement, and by Honjo and Matsunaga [19] for compaction control of soils in the geotechnical engineering. Furthermore, Suzuki and Ishii [20] have tried to apply this method in the stochastic finite element method and have confirmed its effectiveness. In the Kriging technique the estimator is generally obtained by a weighted linear sum of the sample values considering the unbiasness of the estimator and to minimize the estimation error. Supposing that an estimator, Z(x), is expressed by a linear sum of sample value, z(xi), obtained at the sample points xj ( j = 1, 2 , . . . , m) of SPT-N value, the estimator is obtained by

m

2(x)= E (5) j = l

where A t =weighted value applied to the j th data. The hj's are determined by using the Lagrange multiplier method considering the following two conditions:

E[ Z(x) - Z(x)] -- 0 (unbiasness of estimator) (6)

~r2=minE[{Z(x)-Z(x)} 2 ] (minimum variance of estimation error). (7)

Then, the auto-correlation function, p(Ax), is expressed by

p(Ax) = C(Ax)/C(O) (8a)

where

C(Ax)=E[Z(x + Ax)Z(x)] -E[Z(x)]: (covariance), (8b)

C(0) = E[ Z(x)2] _ E[ Z(x)]2 (variance) (8c)

with x = position coordinate, Ax = distance between two sample points, Z ( . ) = random variable of soil properties, and E[-] = expected value. The mean value and the variance can be easily obtained, but it is often difficult to estimate the auto-correlation function because of the limitation in the number of samples. The following one-dimensional auto-correlation model is used in this research [14,21]:

p(Ax) = exp( -- Ax/A) (9)

v

=- O

' 6

e~

c -

O

4

400

300

200

100


14'

~" 12 (ao) Sandy / I •

• Clayey _ O :

Sample points spacing, L (m)

6

4 . , - i

0

(b)

- - o - - - - Sandy - - e - - Clayey

107

2 0 . . 0 °

I I I I I I I I

0 100 200 300 400 500 0 100 200 300 400 500 Sample points spacing, L (m)

Fig. 1. (a) Relationships between correlation parameter of SPT-N value and sample points spacing in horizontal direction; (b) Relationships between correlation parameter of SPT-N value and sample points spacing in vertical direction.

in which A = correlation parameter indicating the degree of reduction of correlation characteristics.

The characteristics of the parameters of auto-correlation functions for both horizontal (x-direction) and vertical (z-direction) directions are investigated separately on the basis of the sampling data of SPT-N value, where the parameter for z-direction is indicated as B, whereas for x-direction as A. Figure l(a) shows the relationships between the average value of the spacing, L, in each adjacent two sample points and the correlation parameter, A, which is obtained using the sampling data of SPT-N value in each soil layer at the same depth. As seen from Fig. l(a), the value of A decreases exponentially when the spacing, L, becomes small. This tendency is independent of the soil type and is expressed by

A = exp(2.79 + 0.00736L). (10) n

The parameter A should be independent of the spacing, L. In order to obtain the unique value of the parameter, A, the sample points spacing should be small where the correlation is not affected. Now, supposing Z = 0, a value for A of approximately 15 m is obtained. It is appropriate for this value to take the real value of the parameter of SPT-N value in horizontal direction.

The relationships between the sample point spacing, L, and parameter B in the z-direction are shown in Fig. l(b). Parameter B is not related to the sample point spacing and is a unique value for that ground. The variation of the B values due to the soil type is also not clear and ranges from 1 m to 5 m. The value of B is less than one-third of the value of A ( = 15 m) for the SPT-N value and this is the same tendency as for the other soil properties.


Bridge length 1,463m El. +20rn

+ l Q m

+ 0....~m

STEP ] O 2 o

~-value N-value 2 0 4 0 ~ ' ~ . ~ . . ~ , , ~ _ . . ~ ~ , , ~ N-value

0 2 0 4 0 6 0 Dc

O O O O "0 O 0 O C O O O C JO O ~ O O

4

Dc : Di luvial clay Ds : Di luv ia l sand Dt : Tu f f sandy soil

I I 1 I I 1 I . I I 400 030 800 1000 1200 1400 16(30 18130

Station(m)

Fig. 2. Soil profile.

3.2. Spatial distribution of SPT-N value due to Kriging technique

The spatial distribution of SPT-N value estimated using the Kriging technique is compared with the measured value in order to verify the effectiveness of the Kriging technique. The soil profile at the investigation site is shown in Fig. 2, which is ground composed of diluvial clay (Dc layer), sand (Ds layer), and tuff sandy soil (Dt layer). The soil investigation has been carried out in four steps (designated STEP-1 to STEP-4) at the site. Here, the SPT-N value at the Dt layer in STEP-3 and STEP-4 is estimated from the investigated values in STEP-2 and STEP-3, respectively.

The estimation method of parameter A in horizontal direction of the Dt layer is indicated in Fig. 3. As shown in Fig. l(a), parameter A is a function of the spacing L and decreases as becomes small. The value of A at Z = 0 m is obtained using a least square method with known values of the parameters. As shown in Fig. 3, the results from STEP-1 and STEP-2 are used for obtaining the value of A for STEP-2 while those from STEP-I, STEP-2 and STEP-3 are used for that for STEP-3. Thus, the values of A = 32 m and 18 m are obtained for STEP-2 and STEP-3, respectively.

The auto-correlation function is assumed to be the sum of both the effects in horizontal and vertical directions [6]:

in which Ax and A z = the distances between two sample points in the horizontal and vertical

H. Ochiai et al. / Structural Safety 14 (1994) 103-130 1 0 9

E loo v

e, _ 2(

O r..) .~.

" S T E P - I 1 1

A = 3 2 m ( S T E P - 2 ) S T E P - 2

I ~ ~ ( ~ ) S T E P - 3

A = I 8 m ( S T E P - 3 )

I I I I 5 0 JCO 150 2 0 0

Sample points spacing, L(m)

Fig. 3. Decision of correlation parameter A.

I 2 5 0

directions, respectively, and A and B = correlation parameters. The analysis conditions of SPT-N value in the Dt layer are listed in Table 2. The values of N and V N are obtained using investigated SPT-N values in STEP-2 and STEP-3. The parameter A is determined from Fig. 3 while B is the parameter in the vertical direction obtained by Eq. (9) in STEP-2 and STEP-3, respectively.

Both results of the estimation by the Kriging technique and the measured values are shown in Fig. 4, in which the estimated SPT-N value at each point in the Dt layer is plotted at the stations. The estimated value (mean value), and this with + estimation error (standard deviation), are plotted in the figure. The estimated values approximately agree with the measured values, but the estimation errors are not reduced to zero. It is considered that this is caused by the variation associated with the distribution of the N-value in vertical direction. Needless to say, these estimation errors are largest at a point half way between two sample points. Thus, it is concluded that the spatial distribution of SPT-N value may be estimated in the range of mean value + standard deviation (lo') using the Kriging technique.

3.3. Simplified expression

In order to propose a performance factor with consideration of the spatial distributions of SPT-N value, a simplified expression for the estimation method is formulated [5]. The correlation in the horizontal direction is relatively strong, so that the estimated value is approximated by connecting the measured values at two sample points with a straight line. It

Table 2 Analysis conditions of N-values for Dt layer

STEP Sample Mean-value COV Parameter (m)

size (N) (V N) A B

2 136 19.8 0.462 32 3.7 3 261 22.0 0.415 18 4.5


>

z

E .=

40

30

20

10

0 ~00

(a) STEP-2

&

t i

Mean A ~ , " \ t"

Mean -Estimation error

I

1000 I

1200

Station (m)

A Mean j +Estimation error

~" . . x . . . ~ ! ( / /

0 Sample values r s Measured values

I

1400 1600

40

30

>

z

o~ 20

E r¢l

10

(b) STEP-3

... ~ i \

• I t

Mean\ /~. i V

Mean

M e a n

,~ +Estimation erro

A / ~ d!i',.. "<,,_2, it \7 i(,i<'

-Estimation error \

0 Sample values Measured values

0 I I I

~00 1000 1200 1400 1600

Station (m)

Fig. 4. Estimation of spatial distribution of N-value (Dt layer).

may be considered that the COV of the est imated SPT-N value reaches to that of the N value in that layer when the distance of each sample point spacing is long enough. Figure 5 shows the relationships between VNik/VNs and A/A based on the results from Kriging technique, in

H. Ochiai et al. / Structural Safety 14 (1994) 103-130 111

1.50

o¢

"d 1.25

1.00

~i 0.75

0.50 > © L) ~6 0.25 o

V~i k / VNi =0.331 +0.264(~,/A)

- - . . " \ /

0 ° 8 : o 'o'OOoo oo o

-io-io o 0 0 ~ ~ 0 o o

0.00 i i i i o.0 0.5 1.0 1.5 2.0 2.5

Normalized distance, L / A

Fig. 5. Relationships between COV ratio and normalized distance.

which VNi k is the COV of the SPT-N value at estimation point k and VNi is the COV of the SPT-N value in ith layer while A is the distance between a sample point and an estimation point. There is a linear relation between VNik/VNi and A/A:

VNiJVNi----- 0.331 + 0.264 A/A. (12)

It is noted that this is a unique relation for different soil types. When the value of A/A is equal to 2.5, VNik /VNi is approximately equal to 1.0, so that VNi k may be the same value as the COV of the SPT-N value at the layer for the condition of A = 15 m with A = 40 m. Therefore , the est imated value and the estimation error of the spatial distribution of SPT-N value are easily obtained using this proposed simplified expression with the idea of est imated value described above.

For the case of the spatial distribution of SPT-N value in STEP-2 at the Dt layer as shown in Fig. 4(a), the est imated values by the Kriging technique are compared with the results using the proposed simplified expression in Fig. 6. Both estimated value and estimation error show a good agreement between these two results. It means that the proposed simplified expression for the estimation method may have satisfactory accuracy.

4. E v a l u a t i o n o f r e s i s t a n c e c o e f f i c i e n t b a s e d o n l o a d i n g t e s t s

4.1. Probability model of bearing resistance

The dominant uncertainties in Eq. (2) are possibly attr ibuted to the SPT-N value and the resistance coefficients, ap and afi. The values Av, U and l i in Eq. (2) are considered to be


>

z

z E "4

>

z

(a) Mean (Estimated value) 5O

4O

30

2O

10

0 ~0q

• - . t-by Kriging -O-proposed

I l I

1000 1200 1400 1600

Station (m)

(b) Mean+_Estimation error 50

- t - by Kriging -O- proposed

Mean ation error

30

10 Mean -Estimation error

0 I i I

~00 1000 1200 1400

Station (m)

Fig. 6. Comparison of estimated N-values.

1600

11. Ochiai et al. / Structural Safety 14 (1994) 103-130 113

1.o 8 >

= 0.8

~2 ~ 0.6

g .~ 0.4

> 0.2 ©

0.0

/ / o Sandy /

/ • Clayey /

/ / o ~ ' X ~ ~x

/ / o J • / j •

1 2 3 4 5

Mean-value of resistance coeff icient, ~fi

Fig. 7. V,~ i - fffi re la t ionsh ips .

constant values, while the resistance coefficients, ap and afi , and SPT-N values, Np and N/, are random variables.

Supposing that there is no correlation among each layer, the estimated value of the resistance, Ru, and its variance of estimation error, 002, can be derived by Eq. (2):

M

/~u = ~ ,p/VpAp + U E ( f f f i l i N i ) , (13) i=1

M 002 2 2 U 2 O'pAp -{- E 2 2 = li 00ii (14)

i=1

in which

002 = --2 2 00gp~2 , 00fi = Ogfi001(li "~ 0' p 00~p -'}- 2 --2 2 00gf i ~ . 2

where ffp and 0-2p = mean value and the variance for the resistance coefficient, ap, at the pile tip, Np and 002p = estimated value and variance of estimation error for the SPT-N value at the pile tip, fffi and 0-2ei = mean value and variance for the resistance coefficient, afi, at the pile shaft in ith layer, and N,,. and 002 i = estimated value and variance of estimation error for the SPT-N value at the pile shaft in the ith layer. Furthermore, the C OV of the ultimate bearing resistance, K n, is obtained by

VR= /Ru. (15)

The first- and second-order statistics of the resistance coefficients, ap and o~fi , in the Japanese Specifications for Substructures have been given in Table 1. The values of C O V shown in Table 1 indicate a wide range of variation from 0.5 to 0.7 because of the wide variety of soils where the loading tests were conducted. Therefore, when a pile loading test is conducted at a site and also the results from the test are reflected in the evaluation of bearing resistance of the pile, it is expected that the uncertainty in the resistance coefficient is reduced.

114 I-1. Ochiai et al. / Structural Safety 14 (1994) 103-130

4.2. Variation of resistance coefficients

The relation between the mean value of the resistance coefficients, ~fi, and that for the COV, V,i, is plotted in Fig. 7 based on the loading test data at the same site. There is a linear relation (V,~ i = 0.158 ~'fi ) between these two values, and so the value of V,g increases as the value of fff~ becomes large. The difference in soil type does not appear in this relation.

In-situ loading tests are usually conducted on one pile (single loading test) at one construction site even though more than one pile is used for the foundation at that site. There are few cases where a number of pile loading tests (plural loading test) are carried out at one site. For the case of single loading test, although the mean value of the coefficient, aft, is obtained, its variance is not always clarified. In order to estimate the COV, V,,~, the relationship obtained in Fig. 7 can be used. However, it is noted that there are some cases that the application of this relation to each construction site may tend to be unsafe. As realized from Fig. 7, the regression with nonconstant variance increases as the mean value becomes large. When the probability of exceedance is assumed to be 1% on this relation, the relation between ~fi and V,~ i is finally [22]

Vai = 0 . 3 ~ ' f i . (16)

4.3. Accuracy of the proposed model

First, the case of a plural loading test is discussed. The reliability of the mean value and the COV obtained by a plural loading test is compared with that from a single loading test. When plural loading tests are conducted at the site, the mean value and the COV of the resistance coefficients are usually obtained from the statistical analysis based on those test results (CASE-l) . The most reliable estimation, on the other hand, is expected in the case that the bearing resistance of the test pile is est imated using the resistance coefficient obtained by its own pile loading test data. This case (CASE-2) is called "own pile estimation" in this paper while the case in which the test pile and the investigated pile are different is called "o ther pile estimation". The COV of the shaft resistance coefficient due to "own pile estimation" is obtained by Eq. (16), while the values shown in Table 1 are expediently applied to the COV of

I0

8

e -

~ 6

e ~

2

O

CASE-2

/

o.z 0.4 o.6 o.o

E~orprobabi l i typ

Fig. 8. P D F o f t h e e r r o r p r o b a b i l i ~ p.

CASE-3

r L2


Table 3 Estimated values (-~u) and the estimation errors (~r R) of the bearing resistance

115

A-region B-region

P12 P31 P56 P6 A2

Measured 6.468 6.860 6.860 6.468 6.468

CASE-1 Ru 6.655 6.986 7.134 6.451 6.495 o" R 1.353 1.346 0.929 1.256 1.121

CASE-2 Ru 6.476 6.629 6.935 6.194 6.278 or R 1.778 1.483 1.811 1.651 1.160

CASE-3 K' u 8.540 4.031 6.169 7.037 5.062 o- R 2.624 1.500 2.321 2.377 1.581

unit: MN

tip resistance coefficients. For the case of plural loading test, the mean value, Ru, and the COV, VR, for each bearing resistance are calculated by Eqs. (13) and (15) and are shown in Table 3, in which CASE-3 is the case based on the current Specifications [1]. The number of pile loading test data is five from two regions (A-region = P12, P31 and P56; B-region = P6 and A2). It is difficult to compare the calculated values with the measured ones in terms of estimation accuracy, so the authors introduce the parameter p (called "error probability" herein) in each pile test result as follows:

u101 /lix ut 2) =f_" - - - dx . (17) P oo 272q~ O'R exp 2" O" R

The value of p shows the probability for the calculated bearing resistance to be less than the measured resistance, Rul0; p = 0.5 for Rul 0 = R u. The first- and second-order statistics of the error probabilities are shown in Table 4 based on the results of five test piles in each case, and then the probability density function for these test data is shown in Fig. 8. It is easily realized that CASE-1 has almost the same estimation accuracy as CASE-2. In other words, when a number of loading tests are carried out at the site, the reliability of the estimation of the bearing resistance in the surrounding ground (CASE-l) is as high as that of CASE-2 by using the mean value and the COV of the resistance coefficients based on these loading test data. It is also clearly shown that CASE-3 is much less accurate compared to CASE-1 and CASE-2. Thus, it is concluded that the resistance expression in the Japanese Specifications for Substruc-

Table 4 The first- and second-order statistics of error probabilites (CASE-I-3)

Sample Mean-value Standard size (~) deviation (~v)

CASE-1 5 0.456 0.047 CASE-2 5 0.536 0.040 CASE-3 5 0.604 0.304

116 H. Ochiai et aL / Structural SaJety 14 (1994) 103-130

Table 5 Est imated values (/~u) and the est imation errors (~r R)

P12 P31 P56 P6 A2

Measured 6.468 6.860 6.860 Measured 6.468 6.468

from P12 test Ru 6.476 3.111 4.594 from P6 test Ru 6.194 * 5.970 * gR 1. 778 0.731 1.465 cr R 1.651 1.885

from P31 test R u 9.550 * 6.629 * 7.850 * from A2 test Ru 7.393 * 6.278 *

~r R 2.051 1.483 2.113 cr R 1.648 1.160

from P56 test Ru 8.218 * 7.432 * 6.935 *

o- R 1.507 3.259 1.811

unit: MN italic: own pile estimation. * selected data for CASE-4.

tures based on the loading test data collected from all over Japan is not always the optimum estimation for each individual site.

Second, the evaluation of the resistance coefficient based on the single loading test result (CASE-4) is discussed through comparison with that in CASE-1 of the plural loading test. Both the results from own pile estimation and other pile estimation are included in CASE-4 and are listed in Table 5. In CASE-4, since the number of piles is always one, Eq. (16) is used for the calculation of COV of the shaft resistance coefficients. Besides, the mean value and COV of the tip resistance coefficients are unknown, so that the estimation values by the Japanese Specifications for Substructures [1] are applied for these values. When the loading test results are used on the estimation of the other pile resistance, it is necessary to check whether the loading test results are applicable to surrounding ground conditions. According to the results shown in Table 5, the estimation related to the result of P12 is not as good as those for other pile results. Therefore, the cases related to the pile P12 are excluded from CASE-4; so the total number of data for CASE-4 is eight, indicated by an asterisk in Table 5. Based on these results, the first- and second-order statistics of the error probability for CASE-4 are p = 0.477 and ore = 0.121, respectively and are shown in Fig. 9 with the results of CASE-1 and CASE-3 in order to compare between the single loading test and plural loading test. Finally, it is

Io

CASE-4

0 0.2 0 .4 0.6 0.8 I

Error probability p

Fig. 9. PDF of the error probability p.


concluded that the accuracy of the resistance estimation for CASE-4 is considered to be the result for the single loading test case. Note that the COV of error probabilities for the single loading test case is 2.5 times as large as those for the plural loading test case.

5. Performance factor for the Limit State Design

5.1. Performance factor

When a performance factor for the Limit State Design (reliability-based design level I) is determined, it is necessary to consider the design level. In this paper, a method based on the calibration to the current design method is employed [23]. This means that the performance factor for the Limit State Design is determined to be equivalent to the total safety factor in the current design method (Working Stress Design). The design criterion for the bearing resistance of piles in the working stress condition is expressed by

Ruin >1 S d (18)

in which R u = nominal value (mean value) of ultimate bearing resistance, Sd = nominal value (mean value) of load at a pile top, and n = total safety factor. On the other hand, the design criterion in the Limit State Design can be expressed by

R u / F R >~ FsS d (19)

in which F R = resistance factor (>/1.0), and F s = load factor (>/1.0). Denoting Z = In R u - In S d as the performance function, an allowable safety index,/3 a, is defined as follows [24,25]:

m

[~a = Z / ° z " (20)

Then,

f l a = l n ( R u l S d ) l ~ V 2 + V 2 (21)

where Ru and V a - -mean value and COV of ultimate bearing resistances, and So and V s = mean value and COV of loads at pile top. Comparing Eq. (19) with Eq. (21) with introducing the relation ~/V 2 + V 2 = a ' ( V R + V s) where a' is a separation coefficient, both coefficients, F R and F s, are obtained by

F R = exp(ce ' f laVR), F s = exp(ot ' f laVs) . (22)

If the load factor is independent of the uncertainties of the spatial distribution of soil properties and of the resistance coefficients, these two factors could be conveniently expressed by a unique total safety factor. In other words, F R and F s can be collectively expressed by one performance factor F~t which is the same expression as the total safety factor in the current Japanese Specifications for Substructures:

F~ = F R F s = exp( a ' •aV R ) exp(a'flaV s) = exp( j~a~/V 2 "J- V 2 ). (23)

Hence, the factored resistance, R f , in the Limit State Design and the allowable bearing resistance, Ra, in the Working Stress Design can be expressed by the following equation using


R u and F~ or n, with the performance factor, F~, being completely equivalent to the total safety factor:

Rf = R u / F ~ , R a = R u / n . (24a,b)

There are two ways for the evaluation of V R in Eq. (23): (1) use of the values in the Japanese Specifications for Substructures, and (2) use of loading test results. The value of V R in the former case complies with the results in Table 1 while that in the

latter case is expressed by the following equation in which the spatial distribution of SPT-N value and the uncertainty of the resistance coefficient are taken into account for the evaluation of the unit shaft resistance, fi:

VR = ~/VRZl + V~2 (25)

in which VR1 = COV of SPT-N values at the estimation point for the estimation of the spatial distribution of SPT-N value, VR2 = c o g of the resistance coefficients.

The remaining variables at the right-hand side of Eq. (23) are /3 a and V s. The standards in most countries specify the range of/3 a for general building and highway bridge structures to be 2.0-3.5 [23]. For example the A-58 criterion specifies the value of/3a normally to be 3.0 (2.5 for the case of wind loads and 1.75 for earthquakes) [26]. Also, Yamada et al. [27] have proposed that the safety index/3 due to the current design method for the foundation structures in Japan is roughly 3 for an ordinary state (and 1.5 for earthquake loads). These latter values are used here in this study. The COV of the load, V s, is largely affected by the dead load for an ordinary state, so that a value of 0.1 is used for V s because of the small variance of the load. For the case of earthquake loads, there are still many uncertainties in the value of COV. Here, it is determined by conducting a calibration with the current design method. Based on these concepts, the authors formulate the performance factor taking due account of the variations of the spatial distribution of SPT-N value and the resistance coefficients.

5.2. Discussions

The evaluation method of the performance factor in the Japanese Specifications for Substructures is discussed first. From the COV of the resistance coefficients related to the shaft resistance of the friction pile shown in Table 1, the value of V R is equal to 0.361. Since the value of VR1 in Eq. (25) is equivalent to Vui in Eq. (12), the value of VR1 can be expressed as the function of the distance a between a sample point and an estimation point, horizontal parameter A, and the COV, V N, of the SPT-N values in the layer:

VR1 = (0.331 + 0 . 2 6 4 A / A ) V u (26)

V u for the ground may be roughly 0.4 in general. And, in the case of loading test, the position of the test pile nearly corresponds to the position of sample points, so that the effect of A / A is neglected. Hence, the value of VR2 is obtained as follows based on Eq. (25):

VR2 = ~ V 2 - V21 = ~/V 2 - 0.3312VN 2 = V/0.3612 -- 0.3312. 0.42 = 0.336. (27)


o B

6~

5

4

3i

2

/ v ' (a) ordinary state ,-/' s=0.6

~ ~ 0.2

I 2

Normalized distance, k/A

6

,." 5

o 4

i i (b) Earthquakes

Vn=0.6 3

~ ~ 0.g

I 0 I 2

Nonnalized distance, X/A

Fig. 10. Relation between performance factor (F~) and normalized distance.

The performance factor based on the current Specifications, F~, can be expressed by Eq. (28) by substituting Eqs. (26) and (27) into Eq. (23):

F ~ = exp[/ia¢(0.331 + 0.261A/A)2VN 2 + 0.3362+ V 2 ]. (28)

The relationships between F~ and A/A obtained from Eq. (28) under the conditions that /3 a = 3 and V s = 0.1 (for an ordinary state), and /3 a = 1.5 and V s -- 0.3 (for earthquakes), are shown in Fig. 10. Obviously, the performance factors F~ of both ordinary state and earthquakes for A/A = 0 and Vs = 0.4 is 3 and 2, respectively which are the same values as the total safety factor. The performance factor for an ordinary state becomes large as the estimation point gets farther away from the sample point, and this trend is more pronounced as the COV of the SPT-N values of the layer becomes large. As is conditioned in Eq. (12), the COV of e s t i m a t i o n , V N i k , at the estimation point corresponds to the COV, VNi , of that layer with A/A = 2.5, because the performance factors become constant for the range A/A >1 2.5. Com- pared with the performance factor for an ordinary state, the performance factor for earthquakes is hardly affected by the normalized distance A/A and the Vui of the SPT-N values.

The performance factor taking account of the resistance coefficient based on in-situ loading tests is discussed next. The value of VRi at the right-hand side of Eq. (25) is equivalent to Eq. (26) while the second term VR2 for the case of the loading test corresponds to the COV of the shaft resistance coefficient, V,~, for the case of friction piles. Then, Eq. (16) is used for the value


6

r, ~ 5

4

8 g 3

g. I o

(a) Ordinary state

- - ( ~fi = 0.5) - I

/ /

S / / . /

v:o6

f j

I 2

Normalized distance, X/A

0.4

0.2

6

5 2 t j

4 8

3

I I (b) Earthquakes

( etf~ = 0.5) I

VN=0.6 - 0.4

• . ~ 0.2 I o [ 2

Normalized distance, ~./A Fig. 11. Relation between performance factor (F~]) and normalized distance (in case of fffi = 0.5).

of VR2. Therefore, the value of V R for the ult imate bearing resistance can be expressed by the following equation:

VR= ~/V21 --t- V22 = ~/(0.331 -t-0.261A/A)2V 2 + (0.3fff/) 2 . (29)

Substituting this equat ion into Eq. (23), the per formance factor F~I taking account of the spatial distribution of the SPT-N value and loading test data is expressed as

F~I = exp[/3a~/(0.331 + 0.261A/A)2V2 q'- ( 0 . 3 f f f i ) 2 q - V 2 ]. (30)

The per formance factor F~I varies with the mean value, fffi, of the loading test data. Here, the per formance factor F~I for the case of fffi = 0.5 is shown in Fig. 11. The per formance factor F~I is approximately 2 at A / A = 0 and is about 50% smaller than the value of F~. (-- 3) shown in Fig. 10. This may be at tr ibuted to carrying out the loading test to increase the reliability of the bearing resistance. Similarly, the per formance factor for ear thquakes can be reduced to 80% of the value of F t .

6 . A p p l i c a t i o n t o a b r i d g e f o u n d a t i o n [ 2 8 ]

6.1. In-situ pile loading tests

The bridge under considerat ion here is a 190 m long hollow slab bridge composed of three prestressed concrete spans and six reinforced concrete spans. The foundat ions are suppor ted

tt. Ochiai et al. / Structural Safety 14 (1994) 103-130 121

2~o I00 150 I 20O

A I

210-

aary tuffaceous sand

~1 sand

al gravel

Dc : Diluvial clay

De' : Diluvial thin clayey layers

Station (in)

Fig. 12. Soil profile and embedded depth of piles.

by bored cast-in-place concrete piles each of diameter 1.2 m. Figure 12 shows the soil profile and the embedded depth of the foundation piles below the surface of ground at the bridge site.

In order to ascertain how the bearing capacities of friction piles are influenced by such differences in pile length and subsurface soil composition, two piles, P2 and P5, having different lengths were subjected to the in-situ vertical loading tests with the maximum loads, Pmax, of 12 MN for P2 and 11 MN for P5. Figure 13 shows the boring logs and the distributions of the axial force in the vertical direction. These results apparently indicate that Layer Dc' gives large pile shaft resistance and that the friction piles have very small tip resistance.

Figure 14 shows the relationships between the load at the pile top, Po, and the normalized settlement, So/D ( D ' p i l e diameter). The bearing resistances of the piles turned out to be much larger than had been expected before the tests and thus the pile settlement under maximum loading, Pma~, did not exceed about 2% of the pile diameter for these two piles. As clarified by Okahara et al. [8], the ultimate bearing resistance of a bored pile is generally reached when the pile settlement is equal to about 10% of the pile diameter. If the ultimate bearing resistance is estimated by the Weibull distribution curve equation as proposed by Uto et al. [29], it is presumed that the ultimate bearing resistance, Ru, would be approximately 17 MN to 19 MN for both the piles.

Figure 15 shows the relationships between the normalized shear resistance, r /N/ , and relative settlement, S, along the pile shaft for each layer. In this relation, ~-/N/ is a shear

SPT-N value I0 2 0 3O 40 5 0

< ,~S I

m m

1 200

'11

I

,

Axial force, P(MN)

0 2 4 6 8

• Strain guage o Settlement gauge

10 12

To

Si

D$

+

0¢

SPT-N value I 0 2 0 3 0 4O SO

I

f

\

4 A

S'

mm Axial force, P(MN) 1 200 0 2 4 6 8 10

r l 1 F r-1 r

i

tol

I !

~ ~, • Strain guage o Settlement gauge

Fig. 13. Boring logs and axial distributions of test piles.

H. Ochiai et aL / Structural Safety 14 (1994) 103-130

o 1

"~" 2

o 3

Load, Po(MN)

4 8 12 16

r.,,O

Fig. 14. Load-normalized settlement curves of test piles.

123

m

resistance, ~-, divided by average SPT-N value, N~, for that layer, and S is a relative settlement of the pile and the ground in that soil layer. The value of S is calculated using measured strain of pile under the assumption of elastic material. Si and Ds + Dg show the similar 'r/~-S curves for both piles P2 and P5. The layers Dc and Dc' were encountered by pile P2 only. The resistance coefficient for Dc, which is a layer that spreads continuously and extensively at a level below El. + 220 m is remarkably different from that for Dc'.

Table 6 shows the resistance coefficients for various layers. Since it is realized from Fig. 15 that since ~- seems to gradually approach the peak values, ~- at the maximum loading may be taken as an unit shaft resistance, fi. Although two different values of resistance coefficients are obtained for Si and Ds + Dg, since they are characterized by a similar ~-/Ni-S relation, the value for pile P5 which gave a larger S value was used as a design values.

I~- / ~ P 2 , / : . . . . . . P5 j . / a . (= r °.~ ~q,)

2[ .,-c" / , \ =2.63

> - / / De'

~-o ~. / Ds+Dg < / Dc 0.573./]

. . . . . .

/ .a- ~ ~ ~tr "~

I ~ ~ . I O 0 6 12 18 24

Relative settlement, S(cm)

Fig. 15. Normalized shear resistance-relative settlement curves for each layer.


Table 6 Resistance coefficients for each layer

Layer Pile Mean-value Unit shaft resistance Resistance coefficient ( Ni) fi (kPa) O~fi

Si P2 13.0 38.9 - P5 17.9 72.4 0.413

Ds + Dg P2 41.0 160 - P2 27.8 127 - P5 45.5 227 0.511

Dc P2 11.0 284 2.63

Dc' P2 13.8 77.5 0.573

6.2. Estimation of spatial distribution of SPT-N value

Table 7 shows the first- and second-order statistics of the SPT-N values in each layer of the ground. The C O V of each layer is approximately from 36 to 45%, and this is within a normal range for a SPT-N value. Figure 16 shows the estimated spatial distribution of SPT-N value for Si and Dc layers as obtained by the statistical analysis where the correlation parameter A = 10 m. In this connection, the estimation errors were obtained by multiplying the C OV of estimation, Vui ~, by the mean value of the estimation, N/. This figure clearly shows that there are no soil investigations between boring points c and e in the case of the Dc layer. As a result, the estimation errors for any points in between inevitably become large.

6.3. Evaluation of bearing resistances of bored friction piles

The ultimate bearing resistance, R u, the factored bearing resistance, Rf, and the allowable bearing resistance, Ra, for each foundation obtained by Eqs. (24a) and (24b) are shown in Fig. 17. The total safety factor used in calculating allowable bearing resistances is taken as 3 for an ordinary state and 2 for ear thquakes as recommended in the Japanese Specifications for Substructures [1]. Since the allowable bearing resistance for an ordinary state is equal to the ultimate bearing resistance divided by 3 (i.e. Ru/3) , the bearing resistance of each foundation is determined almost solely by the magnitude of its ultimate bearing resistance and is irrelevant to the uncertainty of the soil properties.

Table 7 The first- and second-order statistics of N-values for each layer Layer Sample Mean-value COV

size (~ ) ( V N) Si 85 18.0 0.364 Ds + Dg 84 41.8 0.452 Dc 39 11.2 0.413

H. Ochiai et aL / Structural Safety 14 (1994) 103-130 125

>

Z

E ~J

30

20

10

• M e ' a n " '

+Estimation error ~ r / o Mean

( a ) ~

V ~ MEei~imation error

Bor.a b c d e f g h i

0 , I = ! i

50 150 250 350

Station (m)

30 | ,

(b) Dc layer Mean +Estimation error

20 /

"~ 10 Mean

t~ -Estimation error

0" Bor .a . -b ..... c . . . . . . . . . . . e f ..... g ........ h-- ' i ...............

, I , I , i

50 150 250 350 Station (m)

Fig. 16. Est imated spatial distribution of N-value in cases of Si and Dc layers.

On the other hand, the evaluation of Rf is dependent on the uncertainty of the SPT-N value of each foundation. For example, in the case of pile P3, Rf is given lower evaluation than R a.

This is because the SPT-N value of Dc and Ds + Dc below it are estimated on the basis of the sample values at the positions of piles P2 and P5 and consequently these SPT-N values are liable to large estimation errors which cause the performance factors of these layers to be high values. In the same way, pile P8 is assessed to have a factored bearing resistance that is smaller than the allowable bearing resistance because no soil investigation was conducted at the bridge foundation location. The same comment can also be made about the bearing resistance for earthquake.

Figure 18 shows the ratio of the load, S d, to the bearing resistance, Rf or R~, by the aforementioned two kinds of safety factor. In this figure, Sd/R f by the performance factor or


14

Z

¢~ 10

~ 8

o 6

~ 4 .r~

m 2

sOss 0 - ~ ~ - O

" \ Ra) Ordinary state

0 I I ' ' i , i , , I

A1 Pl P2 P3 P4 P5 P6 P7 P8 A2 Foundations

Fig. 17. Ultimate, factored and allowable values of bearing resistance for each foundation.

S d / R a by the safety factor is given on the ordinate where this performance factor takes into account the uncertainties of the soil properties. Sd/R f is smaller than that for all the foundations, either in an ordinary state or when considering earthquake loading. Thus, it is believed that the use of the proposed performance factors enables the safety of bearing resistance to be assessed in a more rational manner and makes it possible to achieve more economical designs.

1.4

. ~ 1.2 ¢D

¢_) ~ l . u

-~ r~0. 6 o O o 0.4

0.2

0.0

q ~ Earthquakes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

',,, "0 . . . , . .

• S d / R a

O S d / R f

I I I I I I I I I

A1 P1 P2 P3 P4 P5 P6 P7 P8 A2 Foundations

Fig. 18. Ratio of load to bearing resistance for each foundation.


7. Concluding remarks

The semi-empirical bearing resistance expression for bored friction piles was focused on, and a performance factor for the Limit State Design of these piles was proposed taken into account the uncertainties of SPT-N value and the resistance coefficient.

The main conclusions drawn from this study are summarized as follows: (1) The spatial distribution of SPT-N value was well estimated using the Kriging technique

and the simplified expression for this estimation method was proposed in order to formulate the performance factor, explicitly.

(2) The difference of the quantitative evaluation of the resistance coefficient with and without in-situ loading tests was clarified. The COV of the resistance coefficient, 1/~, was estimated using its mean value, d e, when the in-situ loading test was conducted.

(3) The performance factor taking into account both the uncertainties on SPT-N value and the resistance coefficient was expressed by proposed normalized parameter A/A and the COV of SPT-N value, V u.

(4) It was demonstrated that the proposed performance factor provides a more rational and efficient basis for bridge foundation design.

Finally, the proposed performance factor could evaluate the reliability of the bearing capacity of bored friction piles quantitatively so that this factor could suitably be applied in Limit State Design procedures.

8. Acknowledgment

The authors would like to express their sincere gratitude to Dr. K. Ishii and Dr. M. Suzuki of Shimizu Corporation for their valuable suggestions in this study.

9. Appendix: Notation

The following symbols are used in this paper: A p = pile tip area, A = correlation parameter in horizontal direction, B = correlation parameter in vertical direction, COV = coefficient of variation, C(A x) = covariance, C(0) = variance, D = pile diameter, E[ ] = expected value, exp( ) = exponential, fi = unit shaft resistance in the ith layer of the ground, fm = measured value of unit shaft resistance, Fa = resistance factor (>1 1.0),

! F~ = performance factor,


F~I Fs L

li m M n

Ui Up P PF eu eo qd qdm Ra Rf

RF RFc RFm Rp Ru

Rue Rum Rulo

S

Sd So U Vui Vuik VR

VR2 X j

z( ) z( ) Ol r

Olp

O~fi ~a Ax

= performance factor taking account of pile loading test results, = load factor (~< 1.0), = sample points spacing, = thickness of the ith layer of the ground, = total number of sample points, = total number of layers, = total safety factor, = SPT-N value in the ith layer of the ground, = SPT-N value near the pile tip, = error probability defined by Eq. (17), = resistance ratio defined by Eq. (4a), = resistance ratio defined by Eq. (4b), = load at the pile top, = unit tip resistance, = measured value of unit tip resistance, = allowable bearing resistance in the Working Stress Design, = factored resistance in the Limit States Design, = resistance around pile shaft, = calculated value of R v, = measured value of R v, = resistance at pile tip, = ultimate bearing resistance of pile, = calculated value of R u, = measured value of R u, = ultimate bearing resistance (pile head load at a displacement level of 10% of the

pile diameter), = relative settlement, = load at the pile top, = set t lement at the pile top = pile perimeter, = COV of SPT-N value in ith layer, = COV of SPT-N value in ith layer at estimation point k, = C O V of ultimate bearing resistance, = COV of SPT-N value at estimation point for the estimation of the spatial

distribution of SPT-N value, = COV of resistance coefficient, = j th sample point, = random variable, = sample value, = separation coefficient, = resistance coefficient for qa, = resistance coefficient for f,. in the ith layer, = allowable safety index defined by Eq. (21), = distance between two sample points,


A

A t T p( )

2 Orfi

= dis tance b e t w e e n sample po int and es t imat ion point , = w e i g h t e d value appl ied to the i th data, = shear resistance, = auto-corre lat ion funct ion, = m i n i m u m variance o f e s t imat ion error, = variance of es t imated error for shaft resistance, = variance o f es t imated error for tip resistance, = variance o f es t imated error for ul t imate bearing resistance, = m e a n value.

References

[1] Japan Road Association, Specifications for Substructures, Specifications for Highway Bridges (Part. 4), 1990. [2] Ministry of Transportation and Communications, Ontario, Ontario Highway Bridge Design and Commentary

(2nd ed.), 1983. [3] Canadian Geotechnical Society, Canadian Foundation Engineering Manual (2nd ed.), 1985. [4] Japan Society of Civil Engineers, Standard Specification for Design and Construction of Concrete Structures,

Part 1 (Design), 1991. [5] K. Matsui and H. Ochiai, Bearing capacity of friction piles with consideration of uncertainty of soil properties

(in Japanese, with English abstract), Proc. Jpn. Soc. Civil Eng., 445/III-18 (1992) 83-92. [6] E.H. Vanmarcke, Probabilistic modeling of soil profile, ASCEJ. Geotech. Engrg. Div., 103(11) (1977) 1227-1246. [7] K. Fujita, An explanation and application of SPT N-values (in Japanese), Found. Engrg. Equipment, 18(3) (1990)

19-29. [8] M. Okahara, S. Nakatani, K. Taguchi and K. Matsui, A study on vertical bearing characteristics of piles (in

Japanese, with English abstract), Proc. Jpn. Soc. Civil Eng., 418/1II-13 (1990) 257-266. [9] Japanese Society of Soil Mechanics and Foundation Engineering, The Standard Method for Vertical Pile I~ad

Tests, (in Japanese) 1993. [10] Deutsches Institut fur Normung, DIN 4014, 1990. [11] G.G. Meyerhof, Limit states design in geotechnical engineering, Struct. Safety, 1 (1982) 67-71. [12] G.G. Meyerhof, Safety factors and limit states analysis in geotechnical engineering, Can. Geotech. J., 21 (1984)

1-7. [13] M. Matsuo and K. Kuroda, Probabilistic approach to the design of embankments, Soils Found., 14 (2) (1974)

1-6. [14] M. Matsuo and A. Asaoka, Probability models of undrained strength of marine clay layer, Soils Found., 17 (3)

(1977) 53-68. [15] A.G. Journal and Ch. J. Huijbregts, Mining Geostatistics, Academic Press, New York, 1978. [16] B.D. Riplay, Spatial Statistics, Wiley, New York, 1981. [17] G. Christakos, Modern statistical analysis and optimal estimation of geotechnical data, J. Engrg. Geol., 22 (2)

(1985) 175-200. [18] Y. Honjo, S. Sakaguchi and A. Morishima, Prediction of the spatial distribution of residual settlement in a

reclaimed area (verification of the proposed model by actual measurement) (in Japanese), Proc. Syrup. on the Risk Assessment Technique in Geotechnical Engineering, Japanese Society of Soil Mechanics and Foundation Engineering (1987) 21-28.

[19] Y. Honjo and M. Matsunaga, A reflection on compaction control (in Japanese), Proc. Syrup. on the Risk Assessment Technique in Geotechnical Engineering, Japanese Society of Soil Mechanics and Foundation Engi- neering (1987) 109-116.

[20] M. Suzuki and K. Ishii, Stochastic finite element method using estimations of spatial distributions of soil properties (in Japanese, with English abstract), Proc. Jpn. Soc. Civil Eng., 394/III-9 (1988) 97-104.


[21] E.E. Alonso and R.J. Krizek, Stochastic formulation of soil properties, Proc. 2nd Int. Conf. on Application Statistics and Probability in Soil and Structural Engineering, Aachen (1975) 9-33.

[22] A.H.-S. Ang and W.H. Tang, Probability Concepts in Engineering Planning and Design, Volume I (Basic Principles), Wiley, New York, 1977.

[23] M. Hoshiya and K. Ishii, Reliability-Based Design Method of Structures (in Japanese), Kajima Press, Tokyo, 1986. [24] C.A. Cornell, A normative second-moment reliability theory for structural design, Solid Mechanics Division,

Univ. of Waterloo, 1969. [25] N.C. Lind, Consistent partial safety factors, ASCE J. Struct. Engrg. Die., 97 (6) (1971) 1651-1669. [26] U.S. Department of Commerce, Development of a Probability Based Load Criterion for American National

Standard A58, 1980. [27] Z. Yamada, T. Matsumoto, S. Emi and K. Oshima, Safety evaluation for the design of the foundation structures

(in Japanese), Bridge Found. Engrg., 17(5) (1983) 10-16. [28] H. Ochiai and K. Matsui, Evaluation of bearing capacity of friction pile based on uncertainty of soil properties,

Proc. 3rd Int. Conf. on Case Histories in Geotechnical Engineering Vol. I (1993) 119-126. [29] K. Uto and M. Fuyuki and M. Sakurai, An exponential mathematical model to geotechnical curves, Proc. Int.

Symp. on Penetrability and Drivability of Piles, San Francisco (1985) 1-6.

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