Extent of geotechnical testing for pile excavation in port of Dubrovnik
M. Bandić & B. Galjan
Investinženjering d.o.o., Zagreb, Croatia
I. Barbalić & N. Štambuk Cvitanović
Institut IGH d.d., Split, Croatia
I. Vrkljan
Institut IGH, Sector IGH Institut; Department IGH Laboratory, Croatia; University of Rijeka, Faculty of
Civil Engineering, Rijeka, Croatia
ABSTRACT: After noticing poor rock penetration performance during pile excavation works, contractor induced additional site investigation to support his claim on higher rock strength than previously inspected. Results of 115 point load tests and 97 uniaxial tests were fitted to Weibull and Gamma probability density functions. One-sided confidence interval with confidence level of 95% (CI95) for strength value which is higher
than 95% of population (i.e. 95% percentile) was obtained with "Likelihood Ratio Confidence Bounds" method. Suggestion on number of specimens is given for percentile interval estimation in future geotechnical investigation works.
0.015
1 INTRODUCTION
0.01
0.005
Number of specimens
30
Probability density, f(x;)
Poor rock penetration performance plus high wearing of equipment were noted during pile excavation
works for a reconstruction of passenger port Dubrovnik, Croatia. Piles of 1800 mm diameter were
excavated trough surface deposits, mud, hard to stiff
clay and penetrated for 4.50 m in carbonate limestone/dolomite rock. Contractor expressed doubts on
range of previously obtained rock strengths from geotechnical report, which gave unconfined compression strength (UCS) of 7 specimens, in range from
31 to 137 MPa. Reported rock strengths, together
with rock quality designation (RQD) were basis for
excavation equipment selection. To explore his
doubts, 11 additional investigation boreholes were
performed.
Ten investigation boreholes were positioned on a
zone previously described as compact limestone
rock mass were poor excavation performance encountered and one in zone previously described as
highly
fragmented
and
weathered
limestone/dolomite rock mass (Vrkljan et al. 2007).
Overall 97 cylindrical samples were uniaxially
tested in compression with results in range from 21
to 262 MPa. In addition to UCS, 115 point load tests
(PLT) were reported (from 15 to 185 MPa UCS
equivalent), but tests specimens for both test methods were not all obtained from same locations. All
UCS strengths were corrected to L / D = 2 standard
specimen ratio (Thuro at al. 2001), and PLT specimens were transformed to UCS strengths with conversion factor 24 (Rusnak et al. 1999).
25
20
15
10
5
0
0
0
40
80
120 160 200 240
PLT correlated to UCS [MPa]
280
Figure 1. PLT correlated to UCS histogram and applied fits
(dash line Weibull and continuous line Gamma).
Excavation performance is mainly influenced by
several factors as; excavation method, machinery
features, working place (on shore/off shore), periods
of breakdown and maintenance, weather conditions,
presence of boulders and fissures, slope of bedding
planes (rock surface), supervision and motivation,
intact rock characteristics, rock mass characteristics
and experience (competence) of operator and his
crew (Bunning et al. 2007). This article deals only
with intact rock characteristics and offers statistical
method for estimation of strength value, which is
with certain confidence level (e.g. 95%) higher than
selected percent of intact rock population (e.g. 95%).
Described procedure could be used in future projects
where intact rock strength plays important role in
excavation equipment selection.
0.9
0.8
0.7
0.6
3 ESTIMATION OF PARAMETERS FOR
WEIBULL AND GAMMA DISTRIBUTION
0.5
0.4
0.3
0.2
UCS95
0
0
40
80
120
160
200
240
280
PLT correlated to UCS [MPa]
Figure 2. PLT correlated to UCS cumulative distribution (dots)
and applied fits (dash line Weibull and continuous line Gamma).
2 WEIBULL AND GAMMA DISTRIBUTION
FUNCTIONS
Results of PLT and UCS tests were fitted to Gamma
and Weibull probability density functions. The probability density functions (PDF) f(x;θ), for both PLT
and UCS data are assumed to be continuous, non
negative and two parameter. Weibull PDF (Figs 1,
3) is defined with relation:
 kW

f ( x;  , kW )   
0

x
 

kW 1
e ( x /  )
kW
x0
(1)
x0
x

f ( s; )ds
A
kW = 2.474
B
kW =2.328
 = 89.657
= 111.965
L = 3.5310-247
L = 1.5010-218
Gamma fit
C
kG = 4.595
D
kG = 5.095
G= 17.289
G = 19.436
-248
L
=
3.4110
L = 6.7110-217
___________________________________________________
0.015
(2)
x0
where  = Gamma function; kG > 0 is shape parameter and θG > 0 scale parameter.
The cumulative distribution function (CDF)
F(x;θ), is defined as integration of PDF f(s;θ) (Figs
2, 4), from -∞ up to a variable x:
F ( x; ) 
Weibull fit
x0
where x = random variable; e = the base of the natural logarithm; kW > 0 is shape parameter and  > 0
scale parameter. Gamma PDF is defined with relation:
 kG 1 e  ( x / G )
x
k
f ( x; kG ,  G )  
G G (kG )
0

Table 1. Maximum likelihood parameters for Weibull and
Gamma
PDF fitted to PLT and UCS data.
___________________________________________________
Data sets
PLT correUCS
lated
to UCS
____________
_______________
No.
specimens
n
=
115
n = 97
___________________________________________________
(3)
25
0.01
0.005
Number of specimens
0.1
Parameters of Gamma and Weibull distribution fits
(See Figs 1, 3) were estimated by the "Maximum
Likelihood" (ML) method, as better estimator of
parameters for large samples than "Matching
Moments Method" (Lloyd & Lipow 1962). Weibull
distribution has better fit than Gamma for PLT correlated to UCS data, while Gamma distribution has
better fit than Weibull for UCS data, all in terms of
greater likelihood function value. ML parameters for
Weibull and Gamma PDF fitted to PLT and UCS data are shown in Table 1, where L is value of likelihood function.
Probability density, f(x;)
Cumulative distribution of PLT, F(x;)
1
where symbol  presents Weibull parameter list values {kW, } or Gamma parameter list values {kG,
θG}.
The Weibull and Gamma distributions are used in
engineering for reliability analysis which deals with
failure times in mechanical systems (i.e. life data
analysis). In this article, PLT correlated to UCS and
UCS data corresponds to random variable x.
20
15
10
5
0
0
0
40
80
120 160
UCS [MPa]
200
240
280
Figure 3. UCS histogram and applied fits (dash line Weibull
and continuous line Gamma).

Likelihood L of observed n independent specimen
values x1, x2,…, xn, from the same rock type, in vol-
ume area of interest, with continuous PDF f(x;),
with parameters , is defined with relation:
n
L( x1 , x2 , , , xn |  )   f ( xi |  )
(4)
i 1
Principle of maximum likelihood states that best
parameters  are those which maximise probability
of observing already observed UCS specimen values. ML parameters are those which satisfy relation:
L( x1 , x2 , , , xn |  *)  L( x1 , x2 , , , xn |  )
(5)
where θ* are parameter values which maximize the
L function. If derivatives of the likelihood function
exist then it can be done using relation:
dL( x1 , x2 , , , xn |  )
0
d
(6)
Solution of this equation will give ML values of .
Cumulative distribution of UCS, F(x;
1
0.9
selection. Or, we are interested in UCS value estimation which is lower than 95% (UCS5) of all expected
UCS values, with confidence level of 95%, for roof
support or pillar design calculations.
Estimated interval of possible percentile xF(x;θ)
values is called Confidence Interval (CI) (Hudson
1997). As width of CI depends of confidence level δ,
it is usual to denote it as CIδ. "Confidence bounds"
are the lower and upper boundaries/values of a CI.
Two-sided confidence bounds, define interval in
which wanted percentile xF(x;θ) is likely to lie, while
one-sided bounds defines the values from where a
certain percentage of the population is either higher
or lower. For two examples of interest from previous
paragraph, determination of one-sided upper confidence bound estimation is required in former case
and one-sided lower confidence bound estimation is
required in latter case.
Likelihood ratio (LR) test statistic with asymptotic Chi-square distribution 2 provides an approximate method for obtaining CI both for parameters as
well as for variable estimation around certain percentile:
 L( ) 
  2;k
 2  ln 
 L( *) 
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
UCS95
0
0
40
80
120
160
200
240
280
UCS [MPa]
Figure 4. Cumulative distribution of UCS (dots) and applied
fits (dash line Weibull and continuous line Gamma).
4 LIKELIHOOD RATIO CONFIDENCE
BOUNDS
When some extent of UCS data are collected, then
there is natural interest of interval estimation which
would, with certain confidence level, "cover" true
unknown UCS value, which is higher or lower than
certain percent of all expected UCS specimens. UCS
variable value which is higher than certain percent
of all expected UCS samples is called percentile
xF(x;θ) and is commonly represented with cumulative
distribution function F(x;θ) (Figs 2, 4).
For example we are interested in UCS percentile
estimation which is higher than 95% (F(x;θ) = 0.95,
denoted as UCS95) of all expected UCS values, with
confidence level of 95%, for excavation equipment
(7)
where L(θ) = the likelihood function, consisting the
true unknown parameter list values θ, from the same
rock type, in space volume of interest, L(θ*) = the
likelihood function value calculated with parameters
estimated by means of ML method, k = the chisquared statistic with probability  and k degrees of
freedom.
For large number of specimens with known parameter values θ, function -2[ln(L(θ)) – ln(L(θ*))] is
approximately distributed as 2 random variable
with one degree of freedom (k = 1). Chi-square statistics is calculated accordingly to purpose of estimation. When dealing with one set of specimen data
and estimating CI, than k = 1. If comparing two data
sets to determine whether the two sets are significantly different, than k = 2. Probability  equals
confidence level δ for two-sided, and α = (2δ – 1) for
one-sided confidence bounds estimation. At the prescribed δ confidence level for parameters estimation,
CI consist of all parameter values that would not be
rejected at the (δ – 1) significance level.
The values of the parameters that satisfy Equation
7 will change based on the desired confidence level
δ, and degrees of freedom k, but at a given value of δ
and k, there is only a certain region of values for θ
for which Equation 7 holds true.
4.1 Likelihood Ratio Confidence Bounds on
Parameters
Parameters of estimated distribution fits and their
confidence intervals are not of great interest for rock
4.2 Likelihood Ratio Confidence Bounds on
variables (UCS)
For estimation of UCS confidence bounds trough
contour plot with certain F(x;θ) value, unknown
term L(θ) in Equation 7 should be rewritten in manner that one parameter (e.g. kW) stays in likelihood
function and other parameter () should be related
trough desired F(x;θ) and UCS value. Finding points
of contour is an iterative process that requires setting
the value of first parameter (kW) and calculating the
appropriate values of UCS for desired F(x;θ), or vice
versa. Example of similar procedure for Weibull
PDF is shown at "Likelihood Ratio Confidence
Bounds" chapter (ReliaSoft Corporation, 2006). In
this calculation procedure, set of values in list {kW,
UCS} or {kG, θG, UCS} are calculated together for
one contour point of wanted CI. For one UCS value
on higher or lower boundary of CI, only one set of
parameters θ is obtainable, while for UCS values inside of boundaries of CI two sets of variable values
are obtainable.
Figure 5 shows one-sided confidence bounds for
PLT and UCS data sets fitted to Weibull (upper) and
Gamma (lower graph) distribution. One-sided CI95
are obtained for UCS95 ( = 0.90 and k = 1). Upper
graph shows that PLTL = 130.86 MPa is Weibull estimation of one-sided lower bound of which 5% of
all PLT correlated to UCS samples are higher, with
confidence level of 95%. PLTU = 150.74 MPa presents Weibull estimation of one-sided upper bound
of which 95% of all PLT correlated to UCS samples
are lower, with confidence level of 95%. Similar
statements are valid for UCS sample values with
UCSL = 166.66 MPa and UCSU = 195.38 MPa.
Lower graph in Figure 5 shows one-sided confidence bounds for PLT and UCS data sets fitted to
Gamma distribution. Values of one-sided confidence
bounds for both distributions are given in Table 2.
Weibull parameter, kW
2.8
PLTL
UCSL
2.6
2.4
2.2
PLTU
UCSU
2.0
120
130
140
7.0
Gamma parameter, kG
engineering purposes, but understanding procedure
of calculation could be helpful when dealing with
UCS CI. Since the values of the parameter estimates
θ* have been calculated using ML methods, the only
unknown term in Equation 7 is the L(θ) term in the
numerator of the ratio. For Weibull and Gamma distributions with two parameters, the values of these
two parameters can be varied in order to satisfy
Equation 7. Boundary of region which satisfies
Equation 7 could be graphically represented by contour plot. Finding points of contour is an iterative
process that requires setting the value of first parameter (e.g. kW) and calculating the appropriate value of
other parameter (), and vice versa. Note that 2 is
calculated with degrees of freedom k = 1, as dealing
with only one set of data.
6.0
150 160 170
UCS [MPa]
180
190
200
UCSL
PLTL
5.0
4.0
PLTU
3.0
120
130
UCSU
140
150 160 170 180 190 200
UCS [MPa]
Figure 5. UCS95 one-sided CI95 for two data sets. PLT samples
correlated to UCS are presented with dash line and UCS samples with continuous line.
Table 2. Weibull and Gamma estimation of UCS95 one-sided
CI
95 confidence bounds for two data sets.
________________________________________________
Data sets
PLT correUCS
lated
to
UCS
__________
_______
MPa
MPa
________________________________________________
U
150.74
B
U

L
130.86
L
166.66
Gamma fits C
U
163.35
D
U
199.96
L
136.57
L
164.98
________________________________________________
U = one-sided upper bound, L = one-sided lower bound.
Weibull fits
A
4.3 Inspecting difference between two data sets
From presented one-sided CI, differences between
two estimated data sets are evident. In order to inspect significance of difference, one parameter (e.g.
kW) versus other parameter () CIδ contour plot, for
k  2 statistics, should be shown together in the
same graph as in Figure 6. Two degrees of freedom
is used for allowance of including the two data sets
joint confidence region.
If there are no intersection between two plots,
than two data sets with the assumed distribution are
significantly different at confidence level δ. Increasing confidence level is widening contour plot around
ML parameter estimation (centre), thus probability
for no-significant difference gets higher i.e. probability of intersection gets higher.
Reason for significant difference, with 95% confidence level, lies in fact that specimens for different
test methods were not obtained from same locations,
and were not evenly distributed on site of interest.
PLT correlated to UCS confidence bounds
UCS confidence bounds
2.8
2.4
2.0
Gamma parameter, kG
7.0
80
90
100
110
Weibull parameter, 
120
130
6.0
5.0
4.0
3.0
10
20
Gamma parameter, G
Figure 6. Comparing difference between two data sets.
30
5 NUMBER OF SPECIMENS DISCUSSION
What number of specimens n should be taken to
obtain enough narrow and accourate CI? Intuitivly,
as greater number of specimens, as narrower and
accurate CI is. Simple relation between number of
specimens and approximate mean value CI for
Normaly distributed population is expressed with:
CI  2  z

n
In tribute to number of specimens discussion for
UCS95 one-sided CI95 estimation, random sample
strengths (n = 50, 100, 200 and 800 samples) which
follow the Weibull and Gamma distributions were
generated. Parameters of two Weibull and two
Gamma distributions for random UCS generation
are given in Table 1. Confidence bounds of Weibull
generated random specimens are only fitted to
Weibull distribution and Gamma generated specimens are only fitted to Gamma PDF. For each independent group of specimens (16 groups) ML estimation of group parameters as well as UCS95 one-sided
CI95 was obtained. Figure 7 shows four one-sided
CI95 for different number of specimens generated
with parameters A from Table 1. As greater number
of specimens, as narrower CI95, and closer to true
value, but true UCS95 is not always precisely targeted, owing that to allowance of prescribed confidence
level and nature of generating random events. Values of obtained CI95 widths in relation to number of
specimens are shown as hollow circles in Figure 8.
Same procedure was done with parameters B, C and
D.
Widths of UCS95 one-sided CI95 for each independent group of specimens is graphically related to
number of specimens in Figure 8. Two hatched areas
are bounding spread of Weibull and Gamma confidence intervals. Note that, in general, for same number of specimens Weibull distribution gives narrower confidence interval, as happened for two real data
sets from Figure 5, where UCS95 confidence interval
CI(A) < CI(C) and CI(B) < CI(D).
(8)
where z = represent the point on the standard Normal PDF such that the probability of observing a
value greater than z is equal to (1 - ) / 2 (for example, if  = 95%, than z = 1.96 ), and  = standard
deviation.
Often in practice, estimation of UCS95 or any other UCS percentile estimation would be of more interest than mean value CIδ estimation. Rock strength
population of interest, in reality has empirical distribution (Pauše 1993), which is neither Weibull, neither Gamma, but is approximately Weibull like, or
Gamma like, or from group of any similar nonnegative distribution like. Principally, rock UCS
population distribution could not possibly be Normal
nor of any left-tailed distribution type. Thus, UCS
mean value, and standard deviation estimation
should be understood only as collected specimens
average and measure of spread.
Gathering several specimens and averaging into
samples is not appropriate if percentile estimation is
of concern. Averaging would reduce spread of extreme specimen values, and distribution of samples
would tend to look like Normal, which is expected
outcome due to "Central limit theorem".
3.4
n = 50
n = 100
n = 200
n = 800
3.3
3.2
Weibull distribution parameter, kW
Weibull parameter, kW
3.2
3.1
3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
110
120
130
140
150
UCS [MPa]
Figure 7. Contour plots for UCS95 one-sided CI95. Random
UCS values are generated from Weibull PDF with parameters
A (Table 1). Coordinate position of symbol + is placed on true
(UCS95, kW ) value.
Presented graph in Figure 8 could help when
planning geotechnical exploration works, as proposi-
900
800
700
Number of specimens, n
tion what number of specimens should be prescribed
to satisfy stipulated width of UCS95 one-sided CI95.
Specimen number of 50-100 is minimal when
making evaluation of any confidence interval, which
could be too wide for reasonable engineering decision making. Range of 100-200 specimens ought to
give enough accurate confidence interval for rock
excavation equipment selection, and probably for
other engineering purposes. More than 200 specimens would give narrower CI, but from there on
narrowing of CI would be slow with specimen number growth. Thus, with specimen number growth,
economical benefits of cheaper tests like PLT or
Schmidt hammer rise.
Weibull random specimens A
Weibull random specimens B
Gamma random specimens C
Gamma random specimens D
600
500
Gamma UCS95 one-sided CI95 area
Weibull UCS95 one-sided CI95 area
400
300
200
100
6 CONCLUSION
For any percentile estimation in geotechnical investigation works, each possible specimen should have
equal chance of being included in analysis. This
could be done by planning investigation works in
manner that extracted test specimens are evenly distributed in the one rock type volume formation of interest, each representing one volume part. If many
specimens are extracted from one volume part than
one random should be chosen for further percentile
analysis.
Range of 100-200 specimens would be enough
for percentile interval estimation. Specimens should
be together presented with histogram, cumulative
plot and best fitted to non-negative distribution type
by means of ML method. Use of Weibull and Gamma two parameter distributions are appropriate.
"Likelihood ratio confidence bounds" is recommended method when estimating one-sided or twosided percentile confidence interval.
Authors are encouraging engineers in effort to incorporate same method in estimation of other rock
and geotechnical natural random properties (e.g.
rock block volume distribution).
0
10
20
30
40
50
60
70
Width of estimated one-sided CI95 for UCS95 [MPa]
Figure 8. Widths of UCS95 one-sided CI95 for 16 groups of random specimens in relation to number of specimens. Specimens
generated from parameters A, B, C and D (Table 1) are denoted as ○, ×, ◊ and □ respectively. Confidence intervals of two
real data sets from Figure 5 are denoted as A, B, C and D.
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