LOADED PILES BY STATISTICAL ANALYSES YUSUKE HONJOi), …

SOILS AND FOUNDATIONS Vol. 45, No. 3, 51-70, June 2005

Japanese Geotechnical Society

ESTIMATION OF SUBGRADE REACTION COEFFICIENT FOR HORIZONTALLY

LOADED PILES BY STATISTICAL ANALYSES

YUSUKE HONJOi), YULVI ZAIKAii) and GYANESWOR POKHARELiii)

ABSTRACT

The purpose of the present study is to estimate horizontal subgrade reaction coefficient (kh) of pilesfrom SPT N-value that is required to design piles against horizontal loading. This information is much demandedrecently due to the introduction of the limit state design of pile foundations where quantified uncertainty in the prediction of pile behavior is required. For this purpose 52 horizontal pile loading test results were collected from literatures and reports, which were compiled in a database. The actual number of test results put to the statistical analyses was 38 cases with pile top displacement and 21 cases with bending moment measurements along a pile. The statistical analyses were carried out in the two folds; First, the inverse analysis procedure was applied to each loading test result to obtain the coefficients in different soil layers. Then the obtained coefficient values were related to observed SPT N-values of the layers by the regression analysis. The pronounced feature of the regression analysis employed in this study is that it takes into account the magnitude of estimation uncertainty as well as the correlation structure of estimated kh for every

pile automatically. The mean value of kh obtained from the recommended equation in this study gives very similar results to those of the major design equations used in Japan. The uncertainties associated with the recommended equation are presented, which are intended to be used in a reliability analysis of horizontally loadedpiles.

Key words: horizonal loading test of a pile, horizonal subgrade reaction coefficient, limit state design, pile design,reliability analysis, statistical analysis (IGC: E4/H1)

INTRODUCTION

In the process of the introduction of the reliability based design (RBD)1, it is recognized that quantification of the uncertainty in design is much more required. The purpose of the present study is to obtain horizontal subgrade reaction coefficient of piles based on previous loading test results with quantified uncertainties. The results of the study is intended to be used as a base for RBD of pile foundation (e.g. Zaika and Honjo, 2003).

Some statistical analyses based on pile loading test results to obtain the horizontal subgrade reaction coefficient has been done by Aoki and Sanada (1985) and Okahara and Takagi (1990). The most distinguished difference between these studies and the present study is the uncertainty (one can say reliability in this case.) in each loading test results are statistically quantified and thoroughly considered in the analyses.

Since the procedure taken in this study to accomplish this aim is somewhat complicated, it is considered appropriate to give the outline of the procedure at the outset:

Step 1: An extensive database on laterally loaded pile

tests is built to be put to statistical analyses.

The measurements on pile can be divided into

two types: the pile top displacement measure-ments and the bending moment measurements

along the pile axis. The analyses for the former

is termed Analysis A and for the latter Analysis

B throughout this study.

The statistical analyses consist of the following two

steps:

Step 2: Inverse analysis of horizontal subgrade reaction

coefficient of each lateral loading test.Of each loading test data, inverse analysis is

carried out. The analysis based on pile top

displacement measurements is called Inverse

Analysis A, whereas that based on the bending

moment measurements, Inverse Analysis B.

It should be noticed that, in Inverse Analysis

A, only one subgrade coefficient, kh, can be

estimated from each pile loading test because

only one layer is assumed for each pile. On the

other hand, in Inverse Analysis B, three kh's are

i) Professor, Gifu University, Japan ([email protected]).ii) Lecturer, Faculty of Engineering, Andalas University, Padang, Indonesia (formerly Graduate Student at Gifu University, Japan).iii) Formerly Research Associate, Gifu University, Japan.

The manuscript for this paper was received for review on January 7, 2004; approved on January 18, 2005.

Written discussions on this paper should be submitted before January 1, 2006 to the Japanese Geotechnical Society, 4-38-2, Sengoku,

Blinkvo-ku. Tokyo 112-0011. Japan. Upon request the closing date may be extended one month.1In this paper, the limit state design (LSD) or the load and resistance factor design (LRFD) are used almost the same sense as the reliability based

design.

51

52 HONJO ET AL.

estimated from each test because the ground influencing pile behavior in horizontal loading is divided into three layers.

The number of layers introduced in the analy-sis is determined from the stability of inverse analysis after some trial calculation. Since the bending moment measurements (information along a pile) contain more amount of informa-tion than the pile top displacement measure-ments (information at a single point of a pile), it was possible to divide the ground into multiple numbers of layers. As a result, Inverse Analysis B is believed to give more accurate estimation of kh compared to that of Inverse Analysis A.

It is, however, very important to recognize that the three kh's obtained for each pile in Inverse Analysis B are correlated. Cautious care should be taken when one relate the estimated kh's to, for example, SPT N-values of layers, which is explained below in Step 3.

It goes without saying that the estimated uncertainty, i.e. estimated variances of the estimated kh, are also quantified in the inverse analyses.

Step 3: Regression analyses to relate the estimated kh and SPT N-value of soil layers.

The estimated kh's are related to SPT N-value of ground by regression analyses. SPT N-value is chosen as an index parameter to estimate kh because (1) this is the most popular soil investi-

gation data in Japan, and (2) only this value was available for all the loading test results2.

One of the most original points introduced in this study is the employment of the weighted linear regression (WLR) method. Since all estimated kh values have different estimated variances, it is important to take into considera-tion this aspect in the regression analyses . Furthermore, in Analysis B, three estimated kh's from one loading test are correlated (as mentioned above), and this effect also should be taken into consideration in the WLR analyses.

As a result, kh value of larger variances are less weighted and the correlation structures among the estimated kh are properly handled in the WLR analyses introduced in this study.

In order to highlight the effects of the WLR analyses, the ordinary linear regression (OLR) analyses are also carried out. In the OLR analyses the error associated to each data value is assumed to be independently and identically distributed following a normal distribution with mean zero and a constant variance.

The obtained results in this paper are expected to improve our prediction capability of the behavior of a

laterally loaded pile. It should be emphasized that uncer-tainties in the estimated horizontal subgrade reaction coefficients are also quantified, which makes it possible to perform reliability based analysis and design of pile foundations.

DEVELOPMENT OF ESTIMATION METHODS OF HORIZONTAL SUBGRADE REACTION COEFFICIENT IN JAPAN

Historical development of horizontal subgrade reaction coefficients, kh, are summarized. One of the distinguished features of this section is that the develop-ment of kh in Japan is highlighted, which is not necessari-ly well known outside Japan. In this section, old units

(e.g. kgf/cm2) are preserved because many of the units are related to the empirical numbers introduced in the design equations.

Horizontal Subgrade Reaction Coefficient Specified in the Specifications of Highway Bridges kh Based on Elastic Theory

One of rational approach to obtain kh is to estimate it based on the linear elastic solution of rigid circular plate on half infinite elastic body (e.g. Terzaghi, 1943, p. 382).

(1)

where,

B: diameter of rigid circular plate

E: Young's modulus of soil from rigid circular plate

loading test of plate diameter B

Ip: shape coefficient (for circular shape, 4 = 0.79)v: Poisson's ratio of soil

kh: horizontal subgrade reaction coefficient for di-ameter B

One of application of this equation to the evaluation of subgrade reaction coefficient on horizontally loaded pile can be seen in Terzaghi (1955).Yoshinaka (1967)

Yoshinaka (1967) carried out some extensive researches on kh, which includes a large number of horizontal load tests with spherical and quadrangle plate in sand and loam soil. The relationship between horizontal subgrade reaction coefficient corresponding to diameter 30 cm (k30) and B cm diameter plate (kh), for sand and loam was derived as follows:

(2)

Obtaining k30 based on Eq. (1) and substituting it into Eq. (2), one obtains:

(3)

2 This is one of the weakness of a study based on statistical analysis that only popular soil investigation methods could be taken into account due to

the availability of the data.

ESTIMATION OF SUBGRADE REACTION COEFFICIENT 53

where,

B: plate diameter (where, B > 30 cm)k30: horizontal subgrade reaction for B = 30 cm plate

loading test (kgf /cm')En : Young's modulus of soil for plate diameter 30

cm (kgf/cm2)

Yoshinaka (1967) also obtained relationship between Young's modulus from bore hole tests (i.e. pressuremeter measurement; Ep) and from plate loading tests as E30 = 3E as well as relationship between SPT N-value and Ep as E,=7N (kgf /cm2).Yoshida and Adachi (1970)

Yoshida and Adachi (1970) and Adachi (1970) re-evaluated Yoshinaka (1967)'s test results and proposed equation for soil subgrade reaction coefficient for caisson foundation as follows:

(4)

Parameter a is 1 for normal condition and 2 for earth-

quake condition. Yoshida and Adachi (1970) also re-evaluated the relation between E30 and Ep as E30 = 4 .Ep.The Specifications for Highways Bridges, 1980 (Japan Road Association, JRA, 1980)

Based on the previous studies by Yoshinaka (1967) and Yoshida and Adachi (1970), the Specifications for Highway Bridges (1980) proposed to estimate the soil subgrade reaction coefficient kh, applied to pile founda-tion, as follows (Okahara and Takagi, 1990):

(5)

where,

kh: value of soil horizontal subgrade reaction coefficient along pile (kgf/cm3)

k0: value of kh for y =1 cm at the ground surface (kgf /cm3)

Eo: Young's modulus is obtained by various methods (see Table 1) (kgf/cm2)

D: diameter of pile (cm)αP: correction factor (see Table 1)

y: horizontal displacement at the ground surface (cm)

The original form of Eq. (5) came from the following equations:

in which

(6)

where,

k30: the value of k correspond to the value of plate

loading for D = 30 cm

Bh: converted load width of foundation, Bh = 'VA, (cm)

Ah: horizontal component of load area =D(1 //3) (cm2)

αk: correction factor (see Table 1) 13: characteristic value of a pile, fl =,41khDI4EI

EI: flexural stiffness of pile (kgf/cm2)

Although they have proposed non-linear subgrade reaction coefficient, kc, has been used in practical design in order to avoid non-linear analysis considering the fact that the horizontal displacement level of pile may not exceed far beyond 0.01 m.Specification for Highway Bridge IV, 1991 (JRA, 1991)

In 1991 revision of Specification for Highway Bridge IV (JRA, 1991), Eq. (6) is modified to accommodate kh for larger diameter piles as well as other type of foundations. At this revision, it was attempted to unify estimation procedure of kh to all types of foundations. The equation was modified to:

(7)

where,

β: characteristic value of a pile, f3=,, 41lchD 14E1EI: flexural stiffness of pile (kgf/cm2)

This formula has been used in JRA (1996) also.

Horizontal Subgrade Reaction Coefficient in Technical Standards for Port and Harbor Facilities (JPHA, 1999)

Technical Standards for Port and Harbor Facilities

(JPHA, 1999) are widely accepted technical standards to design port and habor structures in Japan. The Port and Harbor Research Institute (PHRI) Model has been employed in this standard.

Table 1. Relation between E0 and ce (JRA, 1996)

54 HONJO ET AL.

In PHRI model, the horizontal subgrade reaction coefficient is defined to be proportional to square root of displacement. This method is considered to represent the behavior of softer soil subgrade reaction coefficients better because of its capability to take into account of the softening force-displacement relationships of softer soil.The model was originally developed by extensive investi-

gation on piles used for port and harbor structures by Kubo (1964).

Based on PHRI model, the relation between the horizontal subgrade reaction stress p and displacement y is given as follows:

(8)

where,

z: depth from the soil surface (m)k: lateral resistance constant of soil (kN /m3.5 for S

Model, kN /m2.5 for C Model)m: index number 1 or 0 (m = 0: C Model; m= 1: S

Model)

y: horizontal displacement of a pile (m)

Different from Chang's model, which assume a linear relationship between p and y, PHRI model employed a non-linear relationship, thus no closed form solution is available. Compared to Chang's model, however, the PHRI model has more potential to simulate the actual behavior of a pile better. From Eq. (8) kh becomes:

(9)

(10)

C Model (Eq. (9)) is applied when SPT N-value is constant throughout the depth, such as highly overcon-solidated clay layer and dense sand. S type is applied when SPT N-value increases linearly with depth, e.g. lose uniform sand layer and normally consolidated clay layer.The relationship between ks and SPT N-value is obtained from analyses of field test data and JPHA provides charts which relate N-value and kc as well as ks.

Summary of the ReviewThe two groups of studies on kh introduced here are

two major streams of kh's used in Japanese pile design

practice. It is fair to say that both approaches are fundamentally based on a hyperbolic type force-displace-ment relationship such as Eq. (5) for pile top displace-ment and Eq. (9) for all points along the pile. In the following section, we have respected this fact, and tried to estimate parameters of the hyperbolic type model reviewed in this section. More specifically, we will try to estimate parameter kc in Eq. (9) as well as the quantified uncertainties associated to it in the following sections so as to provide basic information for the reliability analysis of the pile foundation.

DATABASE

Many lateral loading pile test results have been col-lected from various literatures (Table 2). The database included soil types, pile types and dimensions, loading conditions and geometric conditions such as loading and measurement points.

Soil ConditionSoil types vary from soft soil such as loam, ash, clay

and silt to hard soil such as sand and gravel (Table 2). In some cases (BCSS-600-6, BCSS-700-1, BCSS-800-13, SP-9-92, BCSC-500-111, BCSC-1000-107), top part of a

pile is embedded in an embankment of which the under-neath is soft clay ground.

Only in limited cases, ground water levels are given; they are not available in most of the cases. For this reason we could not obtain corrected SPT N-values which take into account the influence of the effective overburden

pressure.Variation of SPT N-value is between 0 and 30. Loam,

clay and ash usually have zero or very small SPT N-values. In some of these cases, undrained shear strength of soil is given.

Pile TypeFor the pile types, the following items can be observed:

(1) The database includes 52 piles which consist of 29steel piles, 11 PHC piles, 5 cast in place concrete

piles and 7 steel pipe reinforced cast in place con-

crete piles.

(2) The diameters of steel piles vary between 0.30 (m)and 2.00 (m), the PHC pile diameters are only be-tween 0.30 (m) and 0.50 (m) with one exception, whereas diameter of the cast in place concrete piles vary between 0.40 (m) and 1.20 (m).

(3) Length of piles varies between 15 (m) and 60 (m) for steel piles, 13 (m) and 28 (m) for PHC piles and 7.5 (m) and 30 (m) for cast in place concrete piles (Table 2).

(4) L/D (length/diameter) ratio shown in Fig. 1 ranges from 20 to 70 in steel piles, 20 to 57 in PHC piles, and 8 to 45 in cast in place concrete piles. It can be said that L/D ratios of steel piles tend to be larger than those of PHC piles and cast in place concrete

piles.(5) The cast in place concrete piles generally give smaller

(the maximum pile top horizontal displacement)/ (pile diameter), and dr./D, of less than 17%; on the other hand steel pipe reinforced cast in place con-crete piles have that of between 20% and 78%. PHC

piles exhibit dmax/D of less than 18% whereas steel piles, the maximum dmax/D is 52% except 69% of the experimental small diameter pile (SP-91-3-37A)

(Fig. 2). Note that dmax is the maximum pile top displacement and D is diameter of a pile.

(6) Relationship between averaged SPT N-values over the influential depth versus the influential depths is shown in Fig. 3. (Note that the influential depth is

56 HONJO ET AL.


11,6 where 13 is defined under Eq. (5).) The influential depth is deeper for the smaller SPT N-values, which can be naturally expected from the definition of the influential depth given as 1 /fl in Eq. (7).

Loading TestLateral load tests are usually used to examine the load

deflection behavior of piles. Since the load test result employed in this study are taken from different litera-tures, several different loading and measuring procedures are taken for different cases. There are 4 types of loading procedures: Type I: The load is applied in number of increments in

a stepwise fashion till the target value and then released in the same manner.

Type II: A sequence of loading and unloading cycles with some intervals in between the sequences. The load magnitudes are increased step by step at each sequence.

Type III: The same procedure as Type I, but loaded to the two opposite directions.

Type IV: Details of loading procedures are not clearly described.The types of the loading are indicated in Table 2 for

each test. For Type I, II and III of loading procedures, the measurements at the end of each loading steps are used for the analysis.

The measurements can be divided into two categories; (1) Horizontal displacement measurements at pile top

Fig. 1. Average SPT N-value in influence area vs ratio of length and

diameter

Fig. 2. Ratio of maximum pile top displacement and diameter vs ratio

of length and diameter

58 HONJO ET AL.

by dial gauges.

(2) Bending moments calculated from flexural stresses which are obtained by strain gauges.

There are 42 cases where the pile top displacements were measured: 21 steel piles, 10 PHC piles and 11 cast in

place concrete piles. The load vs. displacement relation-ships for all available data are presented in Figs. 4, 5 and 6. As for the bending moment measurement, there are 29

cases: 18 steel piles, 8 PHC piles and 3 cast in place concrete piles.

In cases of BCSS-600-111, only one loading step was applied, which are found difficult to estimate kc, thus are removed from further analysis. In some other cases, inconsistent observation results are found, i.e. the moment or displacement measured do not increase with the load increase. These cases are BCSS-700-1, BCSS-800-7, BCSS-800-12 and BCSB-800-101 for the moment measurements and BCSB-1000-7 and BCSB-1000-8 for the pile top displacement measurements. On the other hand, BF77-6-CP-C and BF77-6-CP-D piles have excessive loading that resulted the cracks in the piles. The one loading stage tests, i.e. BCSS-600-111, and the incon-sistent test results, i.e. BCSB-1000-7 and BCSB-1000-8 for the displacement measurement and BCSS-700-1, BCSS-800-7, BCSS-800-12, BCSB-800-101 for the moment measurements, have been discarded from further analysis.

As a result, 38 cases with pile top displacement measurements (21 steel piles, 8 PHC piles, 2 cast in place

concrete piles and 7 steel pipe reinforced cast in placeFig. 3. Average SPT N-value in influence depth vs influence depth

Fig. 4. Load vs pile top displacement (steel pile)

Fig. 5. Load vs pile top displacement (PHC pile)


concrete piles) and 22 cases with the moment measure-ments (14 steel piles and 8 PHC piles) are put to the

statistical analysis.

METHODS OF DATA ANALYSIS

IntroductionThe data presented in previous chapter is analyzed in

the two folds as stated in the introduction of this paper.The first step (Step 2 in the introduction) is to estimate

a parameter of horizontal subgrade reaction coefficient, lc, of Eq. (9), of each soil layer based on the observed data by an inverse analysis procedure.

Honjo et al. (1999) has studied the problems of fitting the hyperbolic type equation for the horizontal subgrade reaction coefficient of piles and found that a more important parameter to be estimated in this model is /cc than the power 0.5. In other words, the power was less insensitive to the result of calculation compared to per-turbation of /cc. This conclusion is inherited in this study. Inverse analysis A and B defined in the introduction are respectively carried out.

In the second step (Step 3 in the introduction), linear regression analyses are used to estimate the relationship between SPT N-value of each layer and the identified kc's by the inverse analyses.

The estimated kc's in the inverse analyses contain not only mean values but also variances which reflect the reliability of the estimated values. This information should be properly taken into account in the regression analyses.

To achieve this aim, a weighted linear regression

(WLR) analysis is introduced. The analysis is termed WLR in this paper. It should also be noticed that strong correlations among 3 different kg's estimated for each pile in Inverse Analysis B can also be handled in the WLR.The details of the method will be explained in the follow-ing section.

For the purpose of comparing the effects of considering

quantified uncertainties and correlations in WLR

analyses, the ordinary linear regression (OLR) method is also applied to the results of the Inverse Analyses.

Laterally Loaded Pile ModelA laterally loaded pile can be described by the

differential equation:

(11)

where:

z: depth from the soil surface

y: horizontal displacement at depth zp(y): soil reaction stress at depth z (kN/m2)

EI: flexural stiffness of pile (kN m2)

p is described as:

The Eq. (11) becomes:

(12)

where:

Note that /3 in Eq. (12) is a function of y.In this study, Eq. (12) is solved by finite element

method (Smith and Griffiths, 1988, pp. 107-112) where the variable stiffness method (i.e. tangent stiffness method) is employed to account for the non-linearity.

Inverse AnalysisFormulation of the Inverse Analysis

A type of inverse analysis employed in this study estimates model parameter values based on a physical model and observed data. The basic formulation taken here is very similar to the one employed by Honjo et al.

(1993). A model which relates to the observation data and the calculation results is given as follows:

Fig. 6. Load vs pile top displacement (Cast in place concrete pile)

60 HONJO ET AL.

(13)

where,

ア*i: field observation vector for i-th loading stage

(dimension k)f: laterally loaded pile model (dimension k) xi: applied load at i-th loading stageθ: model parameters vector,

i.e. k. of Eq. (9) of the layers (dimension m)εi : the errors which are assumed to follow

a multivariate normal distribution (dimension k)i: loading stage. i is 1 to (

k: total number of observation points

l : total number of loading stages

m: total number of estimating parameters

The bending moment and/or horizontal displacement are obtained by two means: the laterally loaded pile model and the observed data in the pile loading tests . The discrepancies between the two are considered as errors which may consist of two components: the measurement error and the modeling error. In practice , it is impossible to separate these two sources, thus it is as-sumed that the error simply follows a multivariate nor-mal distribution with mean vector p and covariance matrix V,:

(14)

As it is apparent from the covariance matrix , it is presumed that each observation is independent and has a constant error variance. The objective function that has to be minimized to obtain the maximum likelihood estimates of 0 derived from the log likelihood function of the assumed model is given as follows:

(15)

As the objective function is a non-linear function with respect to the model parameters (i .e. 0), a non-linear optimization technique is required. A type of the quasi Newton method so called BFGF (Broyden-Fletcher-Goldfaeb-Shanno Formula) is employed to minimize the function J (Ibaraki and Fukushima, 1991) . Evaluation of Parameter Uncertainties

In this study not only mean value of estimated

parameter but also uncertainties associated with the estimated parameters are quantified. The non-linear equation, Eq. (13), is linearized at 0, i.e. the estimate of 8, by Taylor expansion as follows:

(16)

where

The uncertainties associated with the estimated model

parameters can be obtained by the following covariance matrix (e.g. Chatterjee and Price, 1977):

(17)

where,

It can be stated that the model parameter vector 0 is a

normal random vector with mean 0 and covariance

matrix Ve.

The conditional number can quantify the severity of multicolinearity:

(18)

where )max and ,min are maximum and minimum eigen values of the covariance matrix Vo, respectively (Chatter-

jee and Price, 1977).

Weighted Linear Regression Analysis Formulation of the Regression Analysis

The results of the inverse analysis are related to SPT N-value of soil layers by a linear regression model . It is of essential importance here to take into account the uncertainties and the correlation structures that have been obtained in the inverse analyses into this regression analysis. The following regression model is employed to fulfill these requirements where the estimated kg's by the inverse analyses and SPT N-value are related as,

(19)

where,

lc e: the objective variable vector, i.e. the mean value vector of kg's from the inverse analyses (dimen-sion n)

H: the explanatory variable matrix (i .e. SPT N-values)


in which n is total number of k, and SPT N-value pairs.

iq: the regression coefficients

ζ: the residual error vector

The uncertainties and the correlation structures of the inverse analysis results are introduced to this regression model by assuming C to follow the multivariate normal distribution as follows:

(20)

where

σ1: the error associated with the regression on kc by

SPT N-value.

V-: the covariance matrix ( Vo includes as sub-matrice), in which the uncertainties and the correlation structures obtained in the inverse analyses.

The objective function that should be minimized in the regression analysis is given as follows:

(21)

which can be solved as,

(22)

where the covariance matrix is calculated as follow:

(23)

The unbiased estimator of Gq is obtained as follows:

(24)

As can be understood from the explanation given above, the estimated variances and correlation structures of kc's by the inverse analyses are taken into account in the regression analysis. This regression analysis formula-tion is termed the weighted linear regression (WLR) model through out this paper.

In order to compare the results for the case where no considerations are taken for these aspects, the ordinary linear regression (OLR) analyses are also carried out. In this case, the covariance matrix V( in Eq. (21) is replaced by an unit matrix.Prediction of kc

The final aim of this study is to estimate kc based on the

given SPT N-value. Based on the regression analysis formulated in the previous section, the estimated mean of kc for the given N is given as follows:

(25)

On the other hand, knowing that kc= + Co, the variance of kc is given as follows':

It is natural to assume that the error associated with the estimation of i, i.e. -and the error in the prediction, i.e. Co, are independent. Which implies E [hT(3 ri)] = 0.Thus,

(26)

A confidence interval can be obtained based on Eqs. (25) and (26). It is a well known fact that if all the observation errors in the regression model are independ-ently and identically distributed following a normal distribution, the confidence interval is obtained based on t-distribution. However, it is not the case in this study.

On the other hand, it is also a well known fact that t-distribution can be asymptotically approximated by a normal distribution if the degree of freedom (i.e. equivalent to the number of data minus the number of model parameters) increases. It is speculated that in the

present case, the confidence intervals can be approxi-mated using a normal distribution.

RESULTS AND DISCUSSION

Parameter Estimation of Inverse AnalysisInverse Analysis A

Based on the result from the inverse analyses calcula-tions as shown in Table 3, the following points can be observed: (1) When the displacement is small, sensitivity of kc is small in the inverse analysis, which resulted in a little change in kc from the initial value and larger standard deviation.(2) There are two considerably large values in the estimated kc, namely BCSB-1000-7 and BCSB-1000-8.This is considered due to very small clma„ID ratios (the maximum pile top displacement/diameter) compared to other pile test results.Inverse Analysis B

The result of inverse analysis is presented in Table 4.The following items can be observed:

(1) Almost in all cases, the conditional numbers are low enough to avoid serious multicollinearity problem.

3The derivation of prediction variance for the correlated linear regression is rarely seen in any standard statistic textbooks. Thus derivation here is

rather detailed.

62 HONJO ET AL.

(a) Steel pile

(b) PHC pile

However, there are considerably high negative correla-tions especially between kg's of the second and the third layers. The reason for such correlation is explained in detail in Honjo et al. (1993).

(2) The estimated kg in the first layer tends to be larger than that of the other two layers. It is also observed that the standard deviations are smaller in the first layer.

(3) When the sensitivity of the k, is small in the inverse

Table 3. Result of Inverse Analysis A (pile top displacement analysis)


(c) Cast in place concrete pile

(d) Steel pipe reinforced cast in place concrete pile

(a) Steel pile

Table 4. Result of Inverse Analysis B (moment analysis)

64 HONJO ET AL.

analysis, it tends to result in a little change in k. from the initial value and larger standard deviation.

(4) The estimated kc for zones with SPT N-value equal to zero, there are quite a number of variations in the estimated value (from 200 to 2493 kN/m2.5), also some of the estimated standard deviations are very large. The fact indicates that SPT N-value is not a appropriate parameter to determinate ice for very soft ground.(5) There are several steel piles where the standard deviation are considerably large, namely BCSS-600-8, SP-1294-2 and SP-360. The reason for these larger

standard deviation are mentioned in (3) and (4).

Regression AnalysisThe reason for choosing SPT N-values for an explana-

tory soil parameter to estimate /cc is that it is the only

parameter that is available for all the cases put to the inverse analyses in this study. It goes without saying that other soil parameters could have been used if they were available for many recorded cases.

In conducting the regression analysis, it is of essential importance to take into account the uncertainties and the

(continued)


(b) PHC pile

(c) Cast in place concrete pile

66 HONJO ET AL.

correlation structures that have been obtained in the in-

verse analyses into the regression analysis as explained in

the previous section. This can be done by properly setting

covariance matrix, of the regression equation given in

Eq. (20). The actual value of o in Eq. (20) are given in

Table 4: as standard deviation and V as estimated

correlation matrix. This type of regression analysis

presented here is called the weighted linear regression

analysis or WLR. To illustrate the magnitude of quanti-

fied estimation uncertainties for all cases, •} one standard

deviations from the means are presented in Figs. 7 and 8.

In order to compare the results of the regression anal-

yses that do not take into account the uncertainties and

the correlation structures of the inverse analyses, the

ordinary linear regression, OLR, are also performed. In

OLR, it is assumed that the matrix VV in Eq. (20) is simply

assumed to be an unit matrix of dimension n . In other

words, all estimated kc's are treated independently and

with the equal weight in OLR.

The Results of the Weighted Linear Regression (WLR)

Based on the results of the Inverse Analysis A for one

layer (Table 3) and Inverse Analysis B for three layers

(Table 4), the regression lines are estimated. The regres-

sion lines obtained by WLR are presented in Figs. 9 to 12.

The estimated regression lines are summarized as follows

together with •} one standard deviation for each data set

(Note that the standard deviations are N-value depend-

ent. The s.d.'s at SPT N-value is 15 are presented in the

following equations):

(1) Inverse Analysis A (The pile top displacements, for

all data.)

(27)

(2) Inverse Analysis B (The moment measurements, for each pile type and for all data.)

(a) Steel pile

(28)

(b) PHC pile

(29)

(c) All data

(30)

Before taking any closer look at these results , the results obtained from OLR are exhibited first so that it

Fig. 7. k, and •} one standard deviation vs SPT N-value in Inverse

Analysis B (steel pile)

Fig. 8. k, and •} one standard deviation vs SPT N-value in Inverse

Analysis B (PHC pile)

Fig. 9. k, vs SPT N-value with •} one standard deviation for WLR

based on Inverse Analysis A (all piles)


based on Inverse Analysis B (steel pile)


becomes easier to highlight the characteristics of WLR

results.

The Results of the Ordinary Linear Regression (OLR)

The results of OLR together with •} one standard

deviation interval at N= 15 are summarized below. Since

in OLR all estimated kc's are treated independently and

with equal weight, the results give the trend line in the

middle of scattered points of kc's.

(1) Based on the result of Inverse Analysis A:

(31)

(2) Based on the result of Inverse Analysis B: (a) Steel Pile

(32)

(b) PHC Pile

(33)

(c) All data

(34)

The regression lines and standard deviation are

presented in Figs. 13 and 14 respectively.

DiscussionComparison of WLR and OLR

In order to compare the difference between WLR and OLR, we first compare Figs. 12 and 14. These are the results based on all data employed in the regression analyses for the bending moment measurements (i.e. Inverse Analysis B).

Two distinguished features of the WLR results are apparent:

1. The WLR regression line does not hit the middle of the scattered data as is the case for the OLR regression line. The WLR line is more closer to the origin at SPT N-value is 0, and takes lager If, value for higher SPT N-values such as 20 and 25.

2. The standard deviation of the WLR analysis is considerably smaller than that of the OLR analysis.

The first point comes from the fact that the points with lager uncertainties are weighted less in WLR analysis.Those points with lager uncertainties are likely to be those obtained based on steel piles, and with higher Icc values in lower SPT N-values, such as N= 0, and lower kc values in higher SPT N-value range, such as N more than 20.


based on Inverse Analysis B (PHC pile)


based on Inverse Analysis B (all data)

Fig. 13. k, vs SPT N-value with •} one standard deviation for OLR

based on Inverse Analysis A

Fig. 14. k, vs SPT N-value with •} one standard deviation for OLR

based on Inverse Analysis B (all data)

68 HONJO ET AL.

The authors believe that the observed facts above is

relevant to our engineering intuition such as SPT N-value

is not very reliable indicator of very soft soil properties.

The second point above is deeply related to mathemati-

cal mechanism that are embedded in the weighted linear

regression analysis. Since the method gives more weight

to the more reliable data, the error involved in the final

results are more controlled by a limited number of more

reliable data, and the data with larger uncertainty are

automatically almost discarded from the analysis4.

It should be emphasized that no subjective judgments

are included in obtaining the results in WLR analysis.

Every procedure from the inverse analysis to the WLR

regression is automatic and based on a sound statistical

theory. For this reason, the authors believe that the

results by WLR should be used instead of the results by

OLR.

For comparison of WLR results for Inverse Analyses A

and B, one should compare Figs. 9 and 12. There are

quite large differences between the two results obtained.

Considering the fact that Inverse Analysis A is only

based on a single point measurement of a pile, i.e. the

pile top displacement, the authors consider the results

obtained in Inverse Analysis B are more reliable. There-

fore, this result is recommended to be used to estimate the

horizontal subgrade reaction coefficient in pile design.

An equation to be used in design

In the above discussion, it was recommended the

results obtained in Inverse Analysis B, be used in the

design, i.e. Eqs. (28), (29) and (30). It is further recom-

mended in this study that one should use Eq. (30) in

estimating horizontal subgrade reaction coefficient. The

reason for this is that there is no statistical difference

among the three equations, and Eq. (30) is representative

of the three because it is based on all the available data.`No statistical difference' implies if one carries o

ut a

statistical test for null hypothesis that Eq. (30) is statisti-

cally identical to Eq. (28), the hypothesis is accepted with

typical significance level of say 5% because Eq . (30) is

included in •} one standard deviation zone of Eq . (28)

(actually if the significance level is 5% for both sided test,

the zone is •} 1.96 standard deviation). The same is true

for Eqs. (29) and (30). Therefore the authors recommend

the use of Eq. (30) for the design.

Influence of pile diameter on kc

It is understood from the literature review on the

horizontal subgrade reaction coefficient that some assume

dependence of pile diameter on the subgrade reaction

coefficient such as Eqs. (1) and (2). If such dependence

exists, it is appropriate to include the diameter as an

explanatory variable to the regression analysis.

In order to check this dependence, the obtained kg's by

Inverse Analysis B divided by the estimated values by Eq. (30) so as to cancel the effect of SPT N-values are

plotted against pile diameters as shown in Fig. 15. It is observed from the figure that the effect of diameter on kc is hardly visible. Thus, it is not necessary to consider the dependence of kg on the diameter in the regression analysis.Obtained result vs JPHA (1999) and JRA (1996)

A comparison is made among the calculated kh by the

proposed equation, Eq. (30), JPHA (1999), JRA (1996) of diameter 0.5, 1.0, 1.5 and 2.0 respectively in Fig. 16: kh is calculated for 1 cm of pile displacement for Eq. (30)5 and JPHA (1999) for m = 0 ground (i.e. Eq. (9)); whereas for JRA (1996), since the Eq. (7) is a function of pile diameter, D and the flexural stiffness, EI, they are

parametrically altered: D is changed to 0.5, 1.0, 1.5 and 2.0 (m) and E is set to 2.0E+ 08, 2.5E+ 07 and 2.7 to 4.2E+ 07 (kN/m2) for steel pipe, cast in place concrete and PHC piles respectively. The skin thickness of steel

pipe piles are set between 0.016 and 0.025 (m) and that of PHC pile 0.13 to 0.08 (m). Since the calculated second moments, I, for steel pipes are one order smaller than those cast in concrete and PHC piles, resulting EI for all

piles having almost the same order for each diameter.It is observed from the first glance that the proposed

result, JPHA (1999) and JRA (1996) for the diameter 1.5 m are very similar. If one looks at the result more careful-ly, it is possible to say that the JPHA (1999) result is more closer to Eq. (30) for smaller SPT N-value range

(especially when SPT N-value is smaller than 10), whereas JRA (1996) with D=1.5 (m) is more closer to the

proposed equation in higher SPT N-value range. The justification for this may be that JPHA (1999) deals more with the softer soils along sea shore, whereas JRA (1996)

4All these mechanisms are embedded in Eq. (22). It is seen from this equation that estimated value with larger variances , i.e. diagonal elements ofthe estimated covariance matrix Vc, are less weighted because it is the inverse of this matrix that is finally taken into calculation . Furthermore, if there are strong correlations among the estimated parameters , they too result larger covariance in 170 which after all reduces the weights of those values in the linear regression.

5As we have seen in Chapter that horizontal subgrade reaction coefficient used in the specifications for highway bridges is originally developed for

that at 1 cm displacement.

Fig. 15. Ratio of ke from result of Inverse Analysis B to estimates by

Eq. (30) vs pile diameter (all data)


with stiffer soils ashore. Equation (30) seems to cover all the range of SPT N-value reasonably well.The parameter to be used in reliability analysis

As stated in the introduction, one of the purposes of this research is to obtain input kc for reliability analysis for horizontally loaded pile.

It is the recommendation of the authors that Eq. (30),

be used for the mean values of kc for a reliability analysis.The standard deviation for each given SPT N-value is

presented in Table 5 based on WLR analysis on all the data from Inverse Analysis B. The uncertainties included in this result is believed to include both modeling error and statistical uncertainty. It is assumed to follow a normal distribution.

It may be worth to notify that since this recommenda-tion is based on Inverse Analysis B, i.e. bending moment measurement, the result, in a strict sense, may only be used for a reliability analysis to deal with bending moment in a pile, and not to be used for pile displacement analysis6. If the result is to be used for a reliability analy-

sis of pile displacement, the evaluation may have a chance to be non-conservative and needs further considerations

(Zaika and Honjo, 2003).

CONCLUSION

The parabolic type horizontal subgrade reaction

coefficient, kh, is estimated based on the 22 pile loading test results. The two folds estimation procedure where the inverse analysis on observation data and the weighted linear regression analysis to relate estimated kh and SPT N-value is employed.

The recommended horizontal subgrade reaction coefficient model based on this study is given as follows:

(35)

where

(36)

The regression line with •} one standard deviation is

shown in Fig. 12 and Table 5.

In addition to the above recommendation, the follow-

ing points are found:

1. The recommended equation above almost coin-

cides with JPHA (1999) where SPT N-value is smaller than 10, and gives similar results to JRA

(1996) for diameter 1.5 (m) where SPT-N value is more than 20.

2. The uncertainties associated with the recom-

mended equation are presented, which are in-

tended to be used in a reliability analysis of

horizontally loaded piles.

3. No significant influence of pile diameter is ob-served on kc as far as the collected data are

concerned.

6It is generally known that the calculated bending moment of a pile is less sensitive to kh value set inpile design than the calculated pile displace-

ment. Actually, assuming ground to be a homogeneous linear elastic body, it is obtained that effect of kh on the pile displacement is proportional to the power of -3/4, whereas that on the bending moment is to the power of -1 /4 only.

Fig. 16. Comparison among estimated kh at 1 cm displacement by JPHA (1999) and by the proposed equation (Eq. (30)) and possible ranges of ki,

by JRA (1996) for D = 0.5, 1.0, 1.5 and 2.0 (m)

Table 5. Standard deviation of recommended kg's formula (k. = 217 + 191*N)

70 HONJO ET AL.

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