Determination of partial factors for vertically loaded piles for a ...

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Determination of partial factors for vertically loaded piles
for a seismic loading condition based on reliability theory
Yusuke Honjo Shailendra Amatya Makoto Suzuki Masahiro Shirato
Gifu University,
Gifu, Japan
Gifu University,
Gifu, Japan
Shimizu Corporation,
Tokyo, Japan
1
Public Works
Research Institute,
Tsukuba, Japan
Abstract
The aim of this study is to establish a procedure in the limit state design format to
rationally determine the partial factors for a vertically loaded pile based on a sound
reliability theory. The frequency of the usage of pile types and dimensions are
investigated first. Several design examples are selected, and load ranges and
combinations on typical piles are studied. Based on these results, typical load
intensities, load combinations and soil profiles are set for the code calibration. Also
uncertainties involved in seismic loading are investigated based on the historical
seismic data using an extreme statistical analysis, so called POT (peaks over
threshold) analysis. Uncertainties concerning resistances of piles are taken from a
well-known study by Okahara et al (1991). FORM (first order reliability method)
analysis is carried out to find out the current level of reliability index. Finally, the
design value method is employed to determine the partial factors.
Keywords: pile design, vertical bearing capacity, partial factors, safety factor,
reliability analysis, FORM, extreme statistics, POT analysis, code calibration
1 Introduction
1.1 Background
It is recognized widely that one of the origins of the introduction of the limit design
concept started in Denmark in 1960s by the effort of Brinch-Hansen (1967). By
1970s, Danish geotechnical design code has been based on the limit state design
concept and partial factors were introduced. It is considered that Eurocode 7 (CEN,
1994) was much influenced by this code (Meyerhof 1993; Ovesen, 1993).
The introduction of the limit state design concept, which is sometimes called Load
and Resistance Factors Design (LRFD), is also coming to be popular in North
America. Such works includes Barker et al (1991), AASHTO (1994), Becker (1996)
and Phoon, Kulhawy and Grigoriu (1995 and 2000).
Gobel (1999) has summarized the works done in this area, especially in North
America, and concluded that most of the works done in 1990s still contain some
degree of vagueness in the determination of resistance factors. He further
emphasizes the use of available databases and rational probabilistic analysis to
determine the partial factors.

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Paikowsky and Stenersen (2000) who are working of revision of the AASHTO driven
pile design procedure are proposing a method to design piles based on
measurements during pile driving. They have carried out extensive statistical
analysis on a large database, and partial factors are proposed based on this result.
In Japan, conversions of the current design codes to ones based on the limit state
design concept are very much activated (see Honjo et al, 2000). All the major
foundation design codes seem to be aiming at the limit state design code and
performance based design concepts.
1.2 Objectives of this study
The objective of this study is to establish a procedure to rationally determine the
partial factors for a vertically loaded pile in the limit state design format based on a
sound reliability theory. We would employ pile design method specified in
“Specifications for Highway Bridges IV: Substructures” (JRA, 1996), denoted as SHB
(1996) hereafter.
This conventional design method can be rewritten in a simplified form as follows:
1
( Rt
+ Rs
) ≥ Sd
+ Se
(1)
Fs
where Rt and Rs are pile tip and pile side resistances, Rs being considered for two
cases: in sand and in cohesive soil, whereas, Sd and Se are vertical pile top forces
induced by dead load and seismic force.
In the partial factor format for which we are going to carry out calibration, the partial
factors are applied to dead and seismic loads as well as pile tip and side resistances
which can be formulated as the follows:
γ R + γ R ≥ γ S + γ S
(2)
Rt
t
Rs
s
Sd
d
Se
e
where γRt, γRs, γSd and γSe are partial factors for the resistances and the forces
respectively.
2 Frequent load ranges, combinations and soil profiles
2.1 Type and size of piles used in highway bridge foundations in
Japan
Fukui et al (1996) has carried out an investigation based on a questionnaire to study
the details of highway bridge foundations in Japan. The investigation period was
from November 1995 to January 1996. 4,995 bridge foundations were investigated
of which 2,441 were ordinary pile foundations.
Presented in Fig. 1(a) is distribution of pile types among these 2,441 cases where
about 70% consists of cast-in place piles, 24% steel piles and 5% PHC (pre-stressed
high strength concrete) piles. The frequency of the cast-in place pile is considerably
high due to the environmental restrictions during construction, such as noise and
vibration. Due to the extraordinary high usage frequency of cast in-situ pile, only
these types of piles are considered in this study.
2

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The distribution of pile diameters of the
1,734 cast in-situ piles is shown in
Fig. 1(b). Although some piles have
diameter of 200 cm, more than 95% of
them have diameter between 100 to 150
cm. Concerning the length of the cast-in
place piles, about 85% of them distribute
between 6 to 30 m as presented in
Fig. 1(c).
2.2 Characteristics of the load
ranges and combinations
and the chosen cases
Based on the investigation on usage frequency
of the piles for highway bridge
foundations, 8 pile foundation design
examples of cast-in place piles are selected
for more detailed study on load
ranges and combinations. The pile
diameter ranges between 100 to 150 cm
and the length 9.5 to 35.5 m. The
ranges conveniently cover the most
frequent dimensions of the commonly
used cast in-situ piles.
In 7 among the 8 examples chosen, the
critical loading condition is seismic loading
in L1 (Level 1) seismic situation 1 . In
only one case, the pile diameter is determined
by the L2 seismic force. Mainly
for this reason, the partial factors are
only calibrated for L1 case in this study.
Since L1 seismic loading situation is the
most critical for vertical loading of the
most of pile foundations, the vertical load
intensity and load combinations of this
situation for the most critical piles in
each foundation example are plotted in
Fig. 2 2 .
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1 In SHB(1996), two seismic situations are considered: L1 (Level 1)earthquakes whose occurrences
are expected to be several times during the service life of a structure. L2 (Level 2) earthquakes are
much larger earthquakes whose occurrences are not certain during the service life. SHB specifies that
seismic coefficient method be used for L1 earthquakes, whereas the ductility design method be
employed for L2 (JSCE, 2000).
2 The vertical dead and seismic loads for the most critical piles in chosen pile foundations are
calculated based on SHB specified calculation model. The model is basically rigid frame supported by
vertical and horizontal subgrade reaction springs, and the loads, namely vertical, horizontal and
moment, are applied at the center of footing as concentrated loads. In this calculation, the most
critical piles are ones in the front row of the pile foundation.
3
Ratio (%)
Frequency
Frequency
100
80
60
40
20
1000
800
600
400
200
500
400
300
200
100
0
0
0
69.2
630
1
23.5
773
(b) Pile diameter (cast in-situ pile)
3
4.7 2.6
Cast in- Steel piles PHC piles Other
situ piles
Pile types
(a) Pile types
types
266
1
58
100 110 120 124 150 180 200
Diameter (cm)
33
5 or less
416 387
6 to 10
11 to 15
317 364
16 to 20
21 to 30
144
31 to 40
Pile length (m)
(c) Pile length (Cast-in situ pile)
61
41 to 50
9
51 to 60
Fig. 1. Piles used in Japanese highway
bridge foundations (after Fukui et al, 1996)

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It is a common practice to check the pile
dimensions for two cases, namely the
case when buoyancy is considered
(assuming ground water level coincide
with the ground surface) and the case
when buoyancy is ignored. The load
ranges and load combinations for the
first case is shown in Fig. 2. It is observed
in the figure that the range of the
dead load lies from 500 to 2500 kN, so
that the seismic vertical load ranges
from 500 to 3,500 kN. The regression
line for the mean of the dead load vs. the
seismic load is Se = 1.2 Sd as indicated
in the figure with the one standard
deviation of Se = (1.2 ± 0.3) Sd.
Essentially the same results are
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obtained for the case where buoyancy is ignored: in this case, the regression line is
given as Se = (1.4±0.4) Sd.
Based on the results of the load range and the combinations found in Fig. 2, 21
loading cases are set in the following way:
1. The basic vertical loads are chosen. In regard to the load intensity range
observed in Fig. 2, vertical loads of 500, 1000, 1500, 2000 and 2500 kN are
chosen.
2. For each vertical load, seismic loads are determined based on the regression
lines presented in Fig. 2: 3 different kinds of seismic induced vertical forces are
set which are denoted by u, a and l (upper, average and lower). Thus, the cases
are denoted as, for example, 500.u, 2000.a etc.
3. For each case, a pile has to be designed for dead load and seismic induced
vertical load under conditions considering and ignoring the buoyancy: they are
indicated, for example, 500.u-bc (buoyancy considered), 2000.a-bn (buoyancy
neglected).
4. Two cases are set for 1500 and 2000 considering the higher frequency of the
usage of the piles in these, whereas only one case is set for the others, i.e. 500,
1000 and 2500.
5. The soil profiles are set for the loading cases based on the selected pile
foundation design examples described previously. They are believed to
represent the variety of profiles encountered in pile foundation design in Japan.
2.3 Designed pile lengths and diameters by the conventional
method
Piles are designed for each case set in the previous section based on the design
method specified in SHB (1996). The designed diameters of piles are not rounded
off as are commonly done in practice, but left in the order of centimetres (cm). In the
all cases, the diameters are determined for the cases pile buoyancy being neglected.
In other words, L2 seismic situations are not critical in determining the dimensions of
piles.
4
Se (kN)
5000
4000
3000
2000
1000
0
0 500 1000 1500 2000 2500 3000
Sd (kN)
Fig. 2. Dead loads vs. seismic loads

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3 Uncertainties in basic variables
3.1 Uncertainties in the loads
3.1.1 Dead load
An investigation was carried out by Japan Road Association on uncertainty involved
in the unit weight of reinforced concrete. It was concluded that the error in the unit
weight follows a normal distribution with mean 1.0 and s.d. 0.015 (JRA, 1989). In this
study, a normal distribution with mean 1.0 and s.d. 0.1 is assumed for the uncertainty
of dead load. It will be seen later that with this magnitude of uncertainty, the dead
load has very little influence on the results of the reliability analysis.
3.1.2 Seismic load
To estimate the uncertainties in the seismic load, based on Usami catalogue (Usami,
1997), the seismic data for the years 1600 to 1995 have been taken. Distances from
the location of Tokyo Metropolis Hall to the epicentres are calculated. The
accelerations for all the earthquakes from the data at distances deduced above are
then calculated using Fukushima-Tanaka (1991) attenuation model, and yearly
maximum accelerations extracted. Fig. 3 shows the yearly maximum acceleration
obtained for Tokyo.
The peaks over threshold (POT) analysis is employed in this study. In the POT
analysis, the data over a relatively high threshold value, say u, are fitted to an
exceedance distribution of u, F [u] (x). According to the theory by Balkema and de
Hann (1974) and Pickands (1975), this can be well approximated by a General
Pareto distribution, W(⋅), provided the threshold tends to the right endpoint of the
distribution of data (Reiss & Thomas, 1997).
F u [ ]
F1(
x)
− F(
u)
( x)
= P(
X ≤ x | X > u)
=
, x ≥ u
1−
F(
u)
−1
γ
[ ] ⎛ x − u ⎞ ⎛ x − u ⎞
F ( x)
= W ⎜ ⎟ = 1−
⎜1+
γ ⎟
⎝ σ ⎠ ⎝ σ ⎠
u
(3b)
From these, using Hazen plot identity, we can then derive the following equation as
the distribution function of the annual maximum:
5
(3a)
n k k ⎡
− 1
γ
+ 1−
⎛ x − u ⎞
⎤
F x
⎢
⎟ ⎥
1(
) = + 1−
⎜1+
γ ⋅
valid for x ≥ u
(4)
n + 1 n + 1⎢
⎣
⎝ σ ⎠ ⎥
⎦
where n = total number of observations, k = number of exceedances (or the number
of data with x ≥ u), γ = shape parameter and σ = scale parameter of General Pareto
distribution, u is the location parameter.
The threshold values, u, are decided using exploratory data analysis tools as mean
excess function (MEF) plot, γ-estimate plot and qq-plot.
Sample MEF plot, Fig. 4(b), is a precursor of the heavy-tailed distribution which
could be best fit to the data; the MEF of a Pareto distribution is a straight line with an
increasing trend. Hence, the threshold value in our case shall be decided from the

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portion of the plot within which the sample MEF “looks linear”, and leave out the
heavier-tailed portion deviating from it (Bassi, 1998).
The γ-estimate plots obtained for different estimates of General Pareto distribution as
maximum likelihood estimate (MLE), moment estimate (Moment) and Drees-
Pickands estimate (D-P) are actually used to choose the thresholds. As can be seen
in Fig. 4(a), these plots have “plateaus” or “peaks”. The regions of stable parameter
estimates are one of these (Reiss, 1997). XTREMES software (Reiss & Thomas,
1997) was used to estimate the distribution parameters.
The estimated parameters finally introduced in this study are as follows: n = 396, k =
99, γ = 0.070, u = 26.66 and σ = 80.45. Fig. 4(c) shows the cumulative distribution
functions of this estimated distribution and the sample data. The resulting 50, 100
and 500-year expected values are 320, 490 and 570 gals respectively (Honjo &
Amatya, 2001).
seismic acceleration, x (gals)
Fig. 3. Yearly maximum acceleration for Tokyo by Fukushima-Tanaka attenuation
model (1991) based on Usami catalogue.
-estimates
350
300
250
200
150
100
50
0
1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
1600
1620
1640
1660
1680
1700
Drees-Pickands estimates
Moment estimates
0 20 40 60 80 100 120 140 160 180 200
Number of exceedances, k
Fig. 4(a). γ-plots for yearly maximum acceleration for Tokyo
1720
1740
1760
1780
6
1800
1820
1840
1860
1880
1900
1920
MLE estimates
1940
1960
1980

Mean excess function, e (u )
100
80
60
40
20
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0 50 100 150 200
Excess acceleration, u (gals)
The obtained 100-year expected maximum acceleration is assumed to induce the
seismic force that is employed in SHB (1996) L1 situation. In the reliability analysis,
the reference period of a structure is assumed to be 100 years. Thus, the 100-year
maximum seismic force distribution is obtained from Eq.(4) as follows:
100
⎛ x ⎞ ⎛ S ⎡
⎤
e ⎞ ⎛ Se
⎞
100 ( S ) 100 ⎜
. 1 ⎟ = 100 ⎜ ⋅ 100 ⎟ = ⎢ 1 ⎜ ⋅ 100 ⎟
e = G ⋅S
e L F x F x ⎥ (5)
⎝ x100
⎠ ⎝ Se.
L1
⎠ ⎝ Se.
L1
⎠⎦
G
⎣
where G100(Se) is the 100-year maximum seismic force distribution, Se is the seismic
force, x is the maximum ground surface acceleration, x100 is the 100-year expected
maximum ground surface acceleration, Se.L1 is the seismic force used in SHB (1996)
for L1 seismic situation, F100 is the distribution function of 100-year maximum ground
surface acceleration, and F1 is the distribution function of the annual maximum
ground surface acceleration.
3.2 Uncertainties in resistances
Okahara et al (1991) tried to separately evaluate uncertainties in the SHB vertical pile
bearing capacity calculation formula on unit pile tip resistance, qd, and unit side
resistance, f, based on the available pile loading tests data with respect to SPT
N-values.
First they defined the failure load of a pile as the load when vertical displacement of
the governments and the public authorities. Due to the fact that not all the cases
provided all the necessary information, only 32 cases could be put into analysis to
evaluate the uncertainties for pile tip and side resistances for cast-in place pile.
Shown in Table 1 are the results for cast in-situ piles. It is one of the distinguished
features of their results that the biases of the calculated bearing capacity is small, i.e.
1.07 and 1.12 for pile tip and side resistances. The c.o.v.'s are very high: 0.46 and
0.63 respectively. They have carried out χ 2 fitness tests to the obtained distributions,
and found they can be approximated either by normal distribution or lognormal
distribution with the significance level of 5%.
7
F (x)
1.00
0.95
0.90
0.85
0.80
0.75
0.70
k = 99
data
MLE est.
0 100 200 300
x (gals)
400 500 600
Fig. 4(b). Sample MEF plot and Fig. 4(c). CDFs of estimated distribution and sample data

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Table 1. Uncertainties in the basic variables by Okahara et al (1991).
Notation Definition of variables
δf
δqd
δN
Side resistance estimated based on the
following formula:
f = 4 N (

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Table 2. Calculated partial factors by the design value method and the direct method
Case
Vertical
dead
load
Vertical
Diameter Length
seismic
of pile of pile
load
Sd (kN) Se (kN) D (m)
L(m
)
Reliability
Index
Partial factors by the design
value method
9
Partial factors directly
calculated from the design
values
β γSd γSe γRt γRs γ * Sd γ * Se γ * Rt γ * Rs
500.u 900 0.85 8.0 1.66 1.00 1.13 0.62 0.79 1.01 2.24 0.83 0.92
500.a 500 700 0.77 8.0 1.77 1.01 1.13 0.62 0.78 1.01 2.33 0.82 0.91
500.l
500 0.68 8.0 1.91 1.01 1.12 0.63 0.76 1.02 2.45 0.81 0.88
1000.u.1 1800 0.97 14.0 1.73 1.00 1.14 0.71 0.73 1.01 2.36 0.90 0.87
1000.a.1 1000 1400 0.86 14.0 1.81 1.01 1.13 0.72 0.71 1.02 2.43 0.90 0.85
1000.l.1
1000 0.75 14.0 1.92 1.01 1.12 0.73 0.70 1.02 2.57 0.89 0.82
1500.u.1 2700 0.91 27.5 1.64 1.00 1.13 0.79 0.69 1.01 2.22 0.96 0.85
1500.a.1 1500 2100 0.8 27.5 1.74 1.01 1.12 0.79 0.67 1.01 2.29 0.96 0.83
1500.l.1
1500 0.68 27.5 1.88 1.01 1.11 0.80 0.65 1.02 2.36 0.96 0.80
1500.u.2 2700 1.05 27.0 1.68 1.00 1.13 0.75 0.70 1.01 2.27 0.93 0.86
1500.a.2 1500 2100 0.92 27.0 1.76 1.01 1.13 0.76 0.69 1.01 2.32 0.93 0.84
1500.l.2
1500 0.79 27.0 1.90 1.01 1.12 0.76 0.67 1.02 2.42 0.93 0.81
2000.u.1 3600 1.45 22.5 1.70 1.00 1.13 0.69 0.73 1.01 2.29 0.88 0.88
2000.a.1 2000 2800 1.29 22.5 1.79 1.01 1.13 0.70 0.72 1.02 2.36 0.88 0.86
2000.l.1
2000 1.12 22.5 1.93 1.01 1.12 0.70 0.70 1.02 2.48 0.87 0.83
2000.u.2 3600 1.41 14.0 1.33 1.01 1.13 0.65 0.74 1.01 1.91 0.90 0.92
2000.a.2 2000 2800 1.26 14.0 1.41 1.01 1.12 0.66 0.73 1.01 1.96 0.89 0.90
2000.l.2
2000 1.09 14.0 1.50 1.01 1.10 0.66 0.70 1.02 1.99 0.88 0.88
2500.u 4500 1.41 23.5 1.67 1.00 1.13 0.73 0.71 1.01 2.27 0.92 0.86
2500.a 2500 3500 1.24 23.5 1.75 1.01 1.13 0.74 0.69 1.01 2.32 0.92 0.84
2500.l
2500 1.07 23.5 1.90 1.01 1.12 0.75 0.67 1.02 2.43 0.92 0.81
mean 1.73 1.01 1.13 0.71 0.71 1.02 2.30 0.90 0.86
s.d. 0.16 0.00 0.01 0.06 0.03 0.00 0.17 0.04 0.04
The sensitivity factors, α's, for the dead load, Sd, lies between -0.05 to -0.10, which
essentially can be considered insensitive in the current reliability analysis. In
contrast, those for the seismic load exhibit values between -0.82 to -0.87, which are
significant and fall in relatively narrow range. On the other hand, those of the pile tip
and side resistance, Rt and Rs, lie between 0.17 to 0.42 and 0.27 to 0.52 respectively.
In the present study, it is intended that the newly developed design formula should
have equivalent safety margin compared to the conventional design formula. For this
reason, the target reliability index, βT, is set to 1.73, which is the average of the
calculated reliability indices.
The design value method is used to calculate the partial factors for Rt and Rs
because the basic variables δqd and δf are assumed to follow the lognormal
distributions.
The calculated partial factors are presented in Table 2. In addition, the calculated
partial factors, namely γRt and γRs, are plotted against Rs/Rt and L/D in Fig. 5 to be
used in the further consideration where L and D being the length and the diameter of
each pile.
The following considerations can be made based on these results:

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1. The range of partial factors calculated
by the design value method are
1.00 - 1.01 for γSd, 1.10 - 1.14 for γSe,
0.62 - 0.80 for γRt and 0.65 - 0.79 for
γRs.
2. As supporting information, the range
of partial factors directly calculated
from the design values are given.
They are 1.01 - 1.02 for γSd’, 1.91 -
2.48 for γSe’, 0.81 - 0.96 for γRt’ and
0.81 - 0.91 for γRs’.
3. The range of γSd is very narrow and
can be essentially regarded as 1.0.
4. The ranges of γSe are very different
for the two methods; the fluctuation,
however, is relatively narrow
considering the thick tail of the
extreme value distribution. The
discrepancy between the results by
the two methods comes from the
inaccuracy in approximating this
distribution by a lognormal distribution.
It is therefore speculated that
the both methods fail to give
appropriate partial factors for the
seismic force. Some careful treatments
need to be taken in determining
the partial factor for the seismic
force.
5. γRt and γRs varies rather widely. In
order to understand reasons behind
these scatters, they are plotted
against Rs/Rt as shown in Fig. 5(a).
It is very clear from the figure that γRt
increases from 0.60 to 0.80 when
Rs/Rt increases from 1.0 to 5.0,
whereas γRs decreases from 0.80 to
0.65 in the same range of Rs/Rt.
Considering the convenience in the
design calculation, Rs/Rt is replaced
by L/D in Fig. 5(b). Essentially the
same characteristic mentioned
above is observed.
6. In order to find reasons for the
observed behaviour above, the
sensitivity factors of Rt and Rs are
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plotted against Rs/Rt in Fig. 6. αRt and αRs are exhibiting the same behaviour as
γRt and γRs in Fig. 5; therefore it is appropriate to conclude that the sensitivity
factors are responsible for the behaviour.
10
γRt and γRs
γRt and γRs
αRt and αRs
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
γRt
γRs
0.0 1.0 2.0 3.0 4.0 5.0
0.85
0.8
0.75
0.7
0.65
Rs/Rt
Fig. 5(a) R s/R t vs. γ Rt and γ Rs
0.6
0.55
0.5
γRt
γRs
0 10 20 30 40 50
L/D
0.60
0.50
0.40
0.30
0.20
Fig. 5(b) L/D vs. γ Rt and γ Rs
0.10 αRt
0.00
αRs
0.0 1.0 2.0 3.0
Rs/Rt 4.0 5.0 6.0
Fig. 6 R s/R t vs. α Rt and α Rs

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5 Conclusions
Based on the results obtained in the reliability analysis, a proposal is made to
determine partial factors for each basic variables:
Dead Load Sd: There is not much to discuss on determining the partial factor for Sd.
All results support that 1.0 should be adopted for γSd.
Seismic Load Se: Due to the extraordinary shape of the extreme value distribution, it
seems that neither the design value method nor the direct calculation gives an
appropriate partial factor value. For this reason, γSe is determined after the partial
factors for Rt and Rs are fixed. The value of 1.6 is finally recommended for γSe. The
reason for this choice is to keep the ratio of the total force to the total resistance to be
1:2, which is presently adopted in SHB (1996).
Pile tip resistance Rt and side resistance Rs: As discussed above, the partial factors
are much influenced by the relative magnitude of Rt and Rs, which is also almost
proportional to L/D ratio. Based on this fact, L/D ratio should be introduced in
determining the partial factors because L/D can be more conveniently used in design
calculation than Rs/Rt.
The proposed partial factors are given as follows:
γ
γ
Rt
Rs
=
=
0.
60
0.
75
+ 0
− 0
. 05
L
D
. 0025
L
D
L
( 10 ≤ ≤
D
40)
The proposed definitions are superposed on Fig. 5(b). If one feels some hesitation
in introducing such a new L/D dependent partial factors, an alternative is to use a
constant partial factor of 0.70 for both γRt and γRs. In this particular case, these two
components happen to compensate each other, and the uniform factor of 0.70 can
almost give uniform safety margin for considered combinations of Rt and Rs.
Note that the views stated in this paper are those of the authors’, and do not
necessarily reflect the resolutions of the working group in the JRA.
References
AASHTO (1994): AASHTO LRFD bridge design specifications, SI unit, First edition.
Barker, R.M., J.M. Duncan, K.B. Rohiani, P.S.K. Ooi, C.K. Tan and S.G. Kim (1991):
NCHRP report 343: manuals for the design of bridge foundations, Transportation
Research Board, National Research Council, Washington D.C., USA.
Bassi, F., Embrechts, P. & Kafetzaki M. (1998): Risk management and quantile
estimation. In R.J. Adler et al (eds), A practical guide to heavy tails, Basel:
Birkhäuser, pp111-130.
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