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The characteristic strength of jet-grouted material
Article in Geotechnique · July 2017
DOI: 10.1680/jgeot.16.P.320
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The characteristic strength of jet-grouted material
Caterina Toraldo1; Giuseppe Modoni2; Maciej Ochmański3; Paolo Croce4
Abstract: In spite of being one of the most popular ground improvement techniques, jet grouting is often viewed
with suspicion, its properties underestimated and its role downgraded to provisional or subsidiary, mostly because of
the lack of reliable methods to manage uncertainty. The random composition and scattered properties of the material
observed at the small scale of laboratory tests lead designers to assume arbitrarily unjustified over-conservative
strength. A methodology is herein introduced to quantify uncertainty, characterising the different factors of variability
in the jet-grouted material and simulating their effects on representative structural elements. An experimental database
representative of a broad range of situations, consisting of laboratory tests on samples cored from jet-grouted columns
and sonic tomography scans performed on large blocks, is first examined to characterise the mechanical response of
the jet-grouted material and its variability. Probabilistic and autocorrelation functions are then introduced to simulate
the stochastic and spatially correlated variability and virtually reproduce realistic populations of samples. The virtual
simulation of uniaxial compression tests on these samples with a three-dimensional finite-element method code leads
to the statistical distribution of the strength being obtained and the mean and variance being computed. Repeating
systematically this process for the variable parameters of the original functions and for different shapes and
dimensions of the samples leads to general formulas expressing the statistical distribution of the uniaxial compressive
strength. The inferred relations are finally combined to calculate correcting factors giving the characteristic strength
of prismatic jet-grouted elements with variable dimensions and slenderness, starting from the distribution observed
at the laboratory scale.
DOI: 10.1680/jgeot.16.P.320. © 2017 ICE Publishing.
Author keywords: ground improvement; numerical modelling.
1. INTRODUCTION
Since its former application (Yahiro & Yoshida, 1973) and
through many decades jet grouting has gained popularity
among geotechnical engineers owing to its capability to
reinforce soils otherwise unable to withstand structural
functions (Croce et al., 2014). Foundation reinforcements
(Modoni & Bzówka, 2012), cut-off walls (Croce &
Modoni, 2006), tunnel canopies (Ochmański et al., 2015b)
or bottom plugs (Eramo et al., 2012; Liu et al.,
2015;Modoni et al., 2016) are just a few examples of the
many routinely performed applications. However, the lack
of standard codes often leads engineers to design
treatments with empirical rules of thumb, giving only
provisional or subsidiary roles to the jet-grouted structures
or largely underestimating mechanical properties. Indeed,
much of this caution stems from the variability of
mechanical properties seen in laboratory tests, which
leaves practitioners with the idea that treatments are
reliable but only to a limited extent. In general, for civil
construction, the common practice prescribes to quantify
strength and stiffness of a material with laboratory tests.
When representative statistical samples of data are
available, as in the case of artificial materials like concrete,
the characteristic strength is defined with statistical
criteria. Considering the statistical distribution of
properties, randomness is usually managed with semiprobabilistic approaches, that is defining characteristic
values as a prescribed percentile or using the variability in
probabilistic calculations (e.g. first-order reliability
method (FORM), first-order second-moment (FOSM) or
Monte Carlo method).
For geotechnical parameters, where variability is
higher and knowledge more limited, Eurocode 7 (BS EN
1997-2 (BSI, 2007)) states that the characteristic values
‘shall be selected as a cautious estimate of the value
affecting the occurrence of the limit state’ (BSI, 2007:
paragraph 2.4.5). This choice implies a certain degree of
subjectiveness that should be supported with more
experimental information and analysis. However, ‘if
statistical methods are used, the characteristic value
should be derived such that the calculated probability of a
worse value governing the occurrence of the limit state
under consideration is not greater than 5%’ (BSI, 2007:
paragraph 2.4.5.2). Schneider (2010) applies this
definition, computing the characteristic value, 𝑥 , of a
variable as follows
𝑥 = 𝑁 ∙ 𝑥 ∙ (1 ± 𝑘 ∙ 𝐶𝑉 )
(1)
1
Univ. of Cassino and Southern Lazio, via di Biasio 43, 03043 Cassino, Italy. E-mail: c.toraldo87@gmail.com
Associate Professor, Univ. of Cassino and Southern Lazio, via di Biasio 43, 03043 Cassino, Italy. E-mail: modoni@unicas.it
3
Silesian University of Technology, ul. Akademicka 5, 44-100 Gliwice, Poland (corresponding author).
E-mail: maciej.ochmanski@polsl.pl
4
Professor, Univ. of Cassino and Southern Lazio, via di Biasio 43, 03043 Cassino, Italy. E-mail: croce@unicas.it
2
Authors’ copy
1
where 𝑥 is the mean value estimated or measured from
laboratory or field tests; 𝑁 serves to correct the bias of the
measured mean; 𝑘 is a factor dependent on the accepted
fractile and probability distribution (for a normal
distribution and a tolerated fractile of 5%, k=1.645); and
𝐶𝑉 is the total coefficient of variation computed by
taking into account the overall variability and uncertainties
as well as the spatial extent of the zone governing failure.
This definition implies that the statistical distribution of
properties should be known on elements that are
sufficiently large to represent the whole structures as the
transition from small samples to larger elements may
result in unpredictable consequences for design (e.g.
Namikawa & Koseki, 2013; Liu et al., 2015). Indeed, this
issue is already addressed by existing standards, for
example Eurocode 7 (BS EN 1997-2 (BSI, 2007)), where
it is pointed out that the characteristic strength of a generic
material should be obtained from experiments and selected
as a cautious estimate considering the experimental
variability and the extent of the zone governing the
behaviour of the geotechnical structure with reference to a
considered limit state. With particular regard to this
governing zone, Eurocode 7 states that it ‘is usually much
larger than a test sample… Consequently the value of the
governing parameter is often the mean of a range of values
covering a large surface or volume of the ground’. In
conclusion, it states that ‘the characteristic value should be
a cautious estimate of this mean value’. Furthermore, it is
specified that ‘if statistical methods are employed in the
selection of characteristic values for ground properties,
such methods should differentiate between local and
regional sampling and should allow the use of a priori
knowledge of comparable ground properties’ (BSI, 2007:
paragraph 2.4.5.2).
Hence when applying the above statements to jet
grouting two main questions arise, one regards the
dimension of the governing zone for a geotechnical
structure, the second is connected with the measurement
of mechanical properties representative of such large
elements. Concerning the former issue, a decision cannot
be made for all possible situations, as it should involve the
type and geometry of the studied structure and the
minimum portion of material capable by its failure to
cause instability or impair its functionality. With regard to
the second issue, since the experimental evaluation on
large elements is nearly impossible due to evident
difficulties in handling large samples and equipment, a
methodology must be conceived to account for the change
of scale from small samples to larger portions. This
transition cannot be overlooked for jet grouting, as the
transformation from natural soil to cemented material can
be controlled only to a certain extent (Modoni et al., 2006;
Flora et al., 2013; Ochmański et al., 2015a) and a spatial
variation is unavoidable. Erosion, mixing and cementation
inherent with this technology normally take place in
limited soil portions around the injecting nozzle and thus
a memory of the heterogeneity of the original soil still
persists in the newly formed material. The new properties
change from point to point, but the variability observed at
Authors’ copy
the scale of the samples is too high to be directly assumed
in a calculation, as a design would then become so overconservative as to discourage users.
In fact, it is logical to assume that the stresses
acting on heterogeneous materials tend to distribute over
the strongest portions bypassing the weaker zones. It is
thus clear that small samples are not fully representative
and a change of scale accounting for the spatial
distribution of properties becomes necessary. In the
following a method based on the random field theory
(Fenton & Griffiths, 2008) is proposed to tackle this
transition in a rational yet practical way (Fig. 1). The
adopted strategy follows the typical sequence of the
random field theory, already adopted by Namikawa &
Koseki (2013) to evaluate the strength of cemented soil
volumes created with deep soil mixing. In particular, the
proposed method (Toraldo et al., 2016) aims to give
general functions to express the statistical distribution of
the uniaxial compressive strength (UCS) for elements of
any size and shape, representative of the different possible
jet-grouted structures.
Fig. 1. Flow chart of the implemented methodology
In the following a thorough discussion on the
mechanical response of the jet-grouted material is first
presented, starting from various experimental
observations to define a realistic constitutive model and a
number of fundamental parameters. Then, the role of
stochastic and spatial variability is analysed with
parametric numerical calculations and framed into charts
useful for design.
2
2. STRESS–STRAIN CHARACTERISATION OF
THE JET-GROUTED MATERIAL
As is typical for geotechnical engineering, the design of
composite structures including natural soil and jet-grouted
reinforcements should address the mechanical response
with regard to ultimate and serviceability limit states, the
former encompassing failure mechanisms, the latter more
focused on deformation under working conditions. In
principle, a tight interaction between the different
elements should be presumed for all analyses taking
simultaneously into account the stress–strain response of
all materials. However, the considerably different
stiffnesses and strengths of jet-grouted and natural soils
(see e.g. Fig. 2) suggests that further considerations need
to be developed. In fact, with such greatly different
responses, it is logical to assume larger stress variations in
the jet-grouted elements, while strains of the latter
contribute minimally to the deformation of the overall
system, which is more dominated by the response of the
surrounding soil. This analysis leads to greater importance
being placed on the strength and simple pre-failure
responses of the jet-grouted material (e.g. linear elastic)
being postulated. However, as non-linearity might
enhance the spatial variability of the stress distribution, a
non-linear response has been postulated in the following.
comparison between the compressive strength computed
for the case of nil confinement (uniaxial) and with a
confining stress of 200 kPa (here assumed as the upper
limit for many buried structures) reveals a rather limited
contribution from the frictional resistance.
As a practical consequence, the jet-grouted
material is often characterised neglecting the frictional
contribution and quantifying its resistance by the uniaxial
compressive strength, 𝑞 , this parameter being related to
the friction angle 𝜙 and cohesion 𝑐 as follows
(2a)
𝑞 =
where
𝛼=
(2b)
∙
The results reported in Table 1 show that, in
addition to being conservative, this assumption represents
a tolerable simplification.
It must be considered, however, that in some
applications (e.g. bottom plugs) the jet-grouted material
undergoes significant tensile stresses (Eramo et al., 2012),
and thus the limitation on the tensile portion of the stress
space must be explained. Based on a comprehensive
experimental observation on different jet-grouted soils,
Van der Stoel (2001) suggests the tension cut-off is
expressed by the following functions of the uniaxial
compressive strength for granular soils
𝑓 = −0.8 ∙ 𝑞 .
(3a)
𝑓 = −0.4 ∙ 𝑞 .
(3b)
for cohesive soils
With regard to the pre-failure behaviour, the
following hyperbolic function (Kondner, 1963) has been
here assumed
Fig. 2. Stress–strain relationship of a jet-grouted sand
subjected to unconfined compression and of natural
sand sheared in a triaxial apparatus at 100 kPa confining
stress
An overall assessment of the strength of the jetgrouted material can be made looking at Table 1. Here the
Mohr–Coulomb parameters found by various authors from
triaxial tests on different jet-grouted soils are reported. It
is immediately apparent that the obtained friction angles
and cohesions are unusual for the parent soils and it is thus
logical to assume that jet grouting produces totally
different materials that also have unusual friction angles
and cohesions.
A second non-negligible aspect is the predominant
role of the cohesion over the total resistance. In fact the
Authors’ copy
(
𝜀 =
)
(
)
(4)
where 𝐸 represents the initial tangent stiffness modulus
measured during uniaxial compression tests. The literature
on jet grouting includes numerous examples (see Table 2)
where authors suggest linear relationships between the
secant or tangent Young’s moduli computed at different
strain levels and the uniaxial compressive strength
obtained in the same test.
Here the same logical structure has been adopted
for the initial tangent stiffness
𝐸 =𝛽∙𝑞 =
∙𝑐
(5)
3
Table 1. Mohr–Coulomb parameters and compressive strengths for different jet-grouted materials
Reference
Soil type
𝜙: deg 𝑐: MPa 𝑞 : MPa at 𝜎 = 0 kPa
Bzówka (2009)
Croce & Flora (1998)
Mongiovì et al. (1991)
Mongiovì et al. (1991)
Mitchell et al. (1981)
Yahiro et al. (1982)
Miki (1982)
Yu (1994)
Fang et al. (1994.a)
Fang et al. (1994.b)
Fang and Chung (1997)
Fang et al. (2004)
Sandy
Silty sand
Gravel
Gravel
Clay
Sand and Clay
Various
Clay - Silty sand
Silty sand
Clay - Silty sand
Clay and silty sand
Silt and Sand
Clay & sand
Clay & sand
58.2
26.1
52
42
39.5
28.5
25
40.6
35
42
38.6
38.7
45
25
Nikbakhtan and Osanloo
(2009)
assuming constant values of 𝛽 and no dependency of the
stiffness on the confining stress, accordingly with the
assumption made on 𝑞 . In this way, the stiffness has been
considered as mainly dictated by the variable composition
of the material. The set of constitutive parameters is finally
completed assuming a constant Poisson ratio and zero
dilatancy at failure.
3. VARIABILITY
The mechanical properties of jet-grouted material result
from several concurrent factors including the
characteristics of the grout (type, quality and amount of
cement, water content, additives), original soil
composition (e.g. grain size distribution, organic content),
environmental factors (e.g. salt dissolved in the pore
water, temperature) and effectiveness of the mixing action.
For a comprehensive and detailed analysis of the influence
of these factors on the uniaxial compressive strength of
lime- or cement-treated soils reference can be made to
Terashi et al. (1983) and Terashi (1997). Considering that
erosion, mixing in place and cementation activated by jet
grouting generally take place within limited soil portions
around the nozzle, it must be expected that the properties
of the jet-grouted material vary from point to point.
Variability can be considered in statistical terms by
observing the distribution coming out from laboratory
tests on samples of small dimensions (typically cubic or
cylindrical specimens of a few centimetres size) randomly
cored within the jet-grouted element. However, these
discrete data are unable to capture the continuous variation
of the material composition from point to point, this aspect
being relevant to define a representative portion of
material. A comprehensive modelling of the variability is
required to account for this transition introducing a
dependency on the distance (spatial correlation). In the
Authors’ copy
2.3
3.2
2.1
0.3
0.58
0.7
0.8
1.1
4.2
4.2
0.8
0.7
0.6
0.77
𝑞 : MPa at 𝜎 = 200
kPa
16.1
10.3
12.2
1.3
2.5
2.4
2.5
4.8
16.1
18.9
3.3
2.9
2.9
2.4
18.4
10.6
13.7
2.2
3.2
2.7
2.8
5.5
16.7
19.7
4.0
3.6
3.9
2.7
following, the variation of 𝑞 observed on cored samples
has been thus used to quantify the spatially uncorrelated
variability of 𝑞 (the one taking place at larger distances),
while the measurement of sonic wave propagation
together with the tomographic reconstruction of the
velocities over the whole domain has been used to quantify
the spatial variability. The latter tests, being continuous
throughout the investigated zone, allow the
autocorrelation function to be estimated more precisely.
3.1. Uniaxial compressive strength of cored samples
The typical statistical distributions of 𝑞 on small samples
of jet-grouted material are summarised in Fig. 3 and in the
associated Table 3. It is soon evident that data are affected
by large dispersion, with variation coefficients reaching
values as high as 0.75. With such scattering, the classical
statistical approach adopted to define the characteristic
value as a percentile of the distribution (e.g. 5%) leads to
very low estimates, which are excessively onerous for use
in practice. Considering the skewness seen in all cases and
the natural requirement of the strength to assume positive
values, the log-normal probability function can be
assumed to model the asymmetric distribution of Fig. 3,
with parameters 𝜇
and 𝜎
defined by the following
relations
⎛
𝜇𝐿𝑜𝑔 = 𝑙𝑛 ⎜
⎝
σ
=
𝜇
𝜎
1+ 𝜇
ln 1 +
⎞
⎟
2
(6)
⎠
(7)
4
Table 2. Relation between Young’s modulus and 𝑞 from literature
Reference
Definition of 𝐸
Soil type
𝛽
Mongiovì et al. (1991)
tangent unspecified
gravel
280 – 1000
Lunardi (1992)
secant at 40% qu
gravel and sand
500 – 1200
Nanni et al. (2004)
tangent unspecified
gravel and sand
440 - 1000
Croce et al. (1994)
tangent unspecified
sandy gravel
210 – 670
Croce and Flora (1998)
secant at a =0.01%
silty sand
220 – 700
Nanni et al. (2004)
tangent unspecified
silty sand
330 – 830
Fang et al. (2004)
tangent at 50% qu
silty sand
300 – 750
Fang et al. (2004)
tangent at 50% qu
silty sand/silty clay
100 – 300
Lunardi (1992)
secant at 40% qu
silt and clay
200 – 500
Fig. 3. Statistical distribution of uniaxial compressive strength from small samples of jet-grouted soil: (a) cohesive
soil – Shibazaki et al. (1996); (b) clayey soil – Nikbakhtan & Osanloo (2009); (c) sandy soils – Shibazaki et al. (1996);
(d) silty sand – Croce & Flora (1998); (e) gravel – Croce et al. (1994); (f) gravel – Mongiovì et al. (1991))
where 𝜇 and 𝜎 are the mean and standard deviation of 𝑞
(Table 3). The Kolmogorov–Smirnov goodness-of-fit test
Authors’ copy
(Massey, 1951) confirms that this choice is acceptable for
almost all cases with a 0.05 significance level.
5
3.2. Spatial correlation
There are different methods to characterise and simulate
the spatial variation of a geotechnical property, the most
popular being regression (Baecher & Christian, 2003),
geostatistics (Krige, 1951; Matheron, 1965; Davis, 1986)
and random field theory (Honjo, 1982; Vanmarcke, 1983;
Matsuo, 2002). The first approach infers spatial relations
through data with the underlying assumption that values
have equal likelihood and are independent each other.
Random field and geostatistics, on the other hand, put
emphasis on the relative position among samples making
use of spatial statistics to quantify the influence of this
factor on the differential values of a selected variable. The
difference between the two latter methods basically stands
in the mathematical description of this dependency (e.g.
variograms for geostatistics, autocorrelation functions for
random field). In the present analysis, preference has been
given to the random field theory because of the wider
possibility offered to mimic variability and to the
existence of computer codes integrating this aspect into
mechanical analyses (Fenton & Griffiths, 2000).
Considering a field of data, the autocorrelation
function, 𝜌(𝑑) is introduced to quantify the similarity of
the data between intervals located at variable distances 𝑑
𝜌(𝑑) =
(
(
,
)∙
)
(
(8)
)
where 𝐶𝑜𝑣(𝑋 , 𝑋 ) = 𝐸[𝑋 − 𝜇 ] × 𝐸[𝑋
−𝜇 ]
is the covariance of the data taken at a mutual distance 𝑑
and 𝑉𝑎𝑟(𝑋 ) = 𝐸[(𝑋 − 𝜇 ) ] is the variance of the
series 𝑋. Value ρ ranges in the ±1 interval, being equal to
zero in case of nil correlation, to +1 or −1 for,
respectively, direct or inverse proportionality. The
function 𝜌(𝑑) starts from 1 (for 𝑑 = 0) and progressively
reduces to zero, meaning that values at large distances are
independent of each other. Such a trend can be captured
by a variety of theoretical functions, but the Markov
equation (Vanmarcke, 1983) adopted in the following is
particularly suitable because of its simple and iterative
form
(9)
𝜌(𝑑) = 𝑒𝑥𝑝 −
The only parameter characterising this function,
named the correlation length (𝜃), expresses the separation
distance beyond which the field is largely uncorrelated
(for 𝑑 = 𝜃 the autocorrelation function drops to 0.37).
Large 𝜃 values mean that the similarity of the studied
property persists over larger distances.
The above theory has been applied to characterise
the variation of the properties of jet-grouted materials. In
contrast with other authors (Asaoka & Grivas, 1982;
Soulié et al., 1990; Huber et al., 2009; Namikawa &
Koseki, 2013), who studied the spatial correlation of soil
properties on data sampled at variable spacing intervals,
the analysis has been carried out on the results of sonic
tomography tests. This technique makes use of inverse
analyses to give a complete distribution of the volume
waves’ (compressional or shear, depending on the adopted
technique) velocity over a block of jet-grouted material
starting from average values tracked along fixed
alignments.
It could be argued that wave velocities and
uniaxial compressive strength are different variables, and
that the spatial analysis of the wave velocity could lead to
a value of 𝜃 unrelated to qu. However, it is also logical to
assume that these two variables are correlated with each
other, as they both mainly depend on the composition of
the jet-grouted material. Therefore, as sonic tests are nondestructive and able to give continuous information on
large portions of material, it was preferred to base the
spatial analysis on these tests rather than on the equispaced
coring of samples, in this way getting rid of the uncertain
position of the samples and of the unavoidable damage
produced by coring. In particular, considering the initial
stiffness of the material to be proportional to the square of
the wave velocity and to the uniaxial compressive strength
(equation (5)), and considering that the Markovian
function of equation (9) applies to normally distributed
variables, the spatial analysis has been performed on the
logarithm of the squared wave velocity.
Table 3. Parameters of the statistical distribution of 𝑞
No.
Soil type
Number
of data
Mean:
MPa
a
Cohesive
Soil
Clay Soil
342
Sandy
soil
Silty sand
Gravel
Gravel
b
c
d
e
f
Variation
coefficient
Skewness
K–S goodness of
fit tests (*)
2.8
Standard
Deviation:
MPa
1.35
Reference
0.48
0.55
0.077 (0.074)
Shibazaki (1996)
55
2.14
1.33
0.75
0.49
0.090 (0.183)
298
12.75
5.55
0.44
0.84
0.084 (0.082)
Nikbakhtan & Osanloo
(2009)
Shibazaki (1996)
26
26
71
8.14
10.49
13.06
3.26
3.77
6.14
0.40
0.36
0.47
0.23
0.90
0.58
0.106 (0.267)
0.051 (0.267)
0.110 (0.161)
Croce & Flora (1998)
Croce et al. (1994)
Mongiovì et al. (1991)
(*) The values within brackets represent acceptance values with 0.05 significance level.
Authors’ copy
6
It could be argued that wave velocities and
uniaxial compressive strength are different variables, and
that the spatial analysis of the wave velocity could lead to
a value of 𝜃 unrelated to qu. However, it is also logical to
assume that these two variables are correlated with each
other, as they both mainly depend on the composition of
the jet-grouted material. Therefore, as sonic tests are nondestructive and able to give continuous information on
large portions of material, it was preferred to base the
spatial analysis on these tests rather than on the equispaced
coring of samples, in this way getting rid of the uncertain
position of the samples and of the unavoidable damage
produced by coring. In particular, considering the initial
stiffness of the material to be proportional to the square of
the wave velocity and to the uniaxial compressive strength
(equation (5)), and considering that the Markovian
function of equation (9) applies to normally distributed
variables, the spatial analysis has been performed on the
logarithm of the squared wave velocity.
Once the fields of this variable velocity have been
digitalised on an equispaced grid, data have been
processed to find the correlation length 𝜃 (equation (9))
that maximises the similarity between the theoretical and
experimental autocorrelation. To achieve this the
following maximum likelihood function 𝐿 (𝜃) defined by
Honjo & Kazumba (2002) has been used
L (θ) = − ln(2π) − ln V −
μ
V
log 𝑣
log 𝑣
−
(10)
−μ
where
log 𝑣
μ
log 𝑣 (r )
=
,
⋮
log 𝑣 (r )
μ
,
exp −
⋮
V =σ
=
(11)
μ
𝑛 is the number of values; 𝜇
and 𝜎
are,
respectively, the mean and standard deviation of the
logarithm of the squared compressional wave velocity 𝑣 ;
𝑉 is the matrix of the covariance; and 𝑟 is the space
vector at the point 𝑖. The above expression is valid for the
assumption that the following multivariate normal
distribution holds true
f
(2π)
μ
V
log 𝑣
exp −
log 𝑣
log 𝑣
−μ
V
Authors’ copy
θ =
−
In particular, the above analysis has been carried
out looking at four case studies a, b, c and d representative
of different subsoil conditions (Fig. 4). For each case, the
figure presents plots of compressional wave velocities
(plot 1), the likelihood function (plot 2) and the
experimental statistical correlation functions (plot 3),
together with the assumed theoretical autocorrelation
function (equation (9)).
Case a (Croce et al., 1994) reports the
compressional wave velocities measured in a 12 m deep,
circular shaft, buried in sandy gravels for the upper 7 m,
in silty soils for the lower part; here the analysis has been
performed for the upper stratum. Case b (Mongiovì et al.,
1991) refers to a 17 m deep, massive block forming the
foundation of a viaduct pier created in gravelly soils; here
the contour lines show a sub-horizontal direction
consistent with original subsoil layering that confirms the
memory of original soil composition still persists after
treatment. In fact, the anisotropy is transferred to the jetgrouted material as shown, for example in Fig. 4(a) plot 3,
by the variable autocorrelation functions computed in the
different directions. In particular, the autocorrelation
function computed along the vertical direction shows an
increasing trend for distances larger than 8 m, possibly
because of the periodicity of the velocity profile. Case c
(Arroyo et al., 2012) reports the sonic tomography
performed in a field trial where a group of columns is
created in an alternation of alluvial sandy and clayey soils
to see the efficiency of jet grouting to form the provisional
reinforcement of a tunnel to be excavated in the city of
Barcelona. Case d (Croce et al., 1990) refers to a shaft
excavated in sandy soils to found the 60 m tall pier of a
viaduct in northern Italy.
All the figures show a continuous transition of the
compressional wave velocities from the different zones, as
a result of the original heterogeneous subsoil composition.
For the cases where the detected portions had a significant
horizontal extension, the analysis has been performed
computing autocorrelation function separately along the
horizontal and vertical directions. The autocorrelation
functions calculated from experimental data (e.g. Fig. 4(d)
plot 3) reveal that a memory of the subsoil stratification
persists in the properties of the jet-grouted material. In all
cases, the assumed Markov equation (equation (8))
satisfactorily reproduces the spatial correlation with
correlation length 𝜃 within the order of a few metres
depending on the initial subsoil structure. It is worth
noting that the correlation length 𝜃 found in these
experiments is normally larger than that observed by
Namikawa & Koseki (2007) on deep soil mixing, ranging
between 0.2 and 2.2 m. A possible explanation consists in
the capability of jet grouting to extend its action and thus
homogenise the treated material over larger distances in
comparison with deep soil mixing.
(12)
7
(a)
(b)
Fig. 4. Spatial analysis from geophysical tests on four field trials. Plot 1 is the contour map of the compressional wave
velocity; plot 2 is the likelihood function; plot 3 is the statistical and theoretical autocorrelation function: (a) Croce et
al. (1994); (b) Mongiovì et al. (1991); (c) Arroyo et al. (2012); (d) Croce et al. (1990) (continued on next page)
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8
(c)
(d)
Fig. 4. Continued
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9
4. NUMERICAL CALCULATION
Once the above models (the lognormal distribution for the
stochastic variability and the exponential autocorrelation
function for the spatial variability) are defined, the
variable mechanical properties of the jet-grouted material
can be reproduced on samples of different shape and
dimensions. This step has been accomplished here using a
three-dimensional finite-element code (RFEM; Fenton &
Griffiths, 2000) that calculates boundary value problems
for geotechnical structures having properties that vary
from point to point. Published examples of such
calculations range from the analysis of material at the
sample scale (Namikawa & Mihira, 2007; Huang et al.,
2010) to the response of more complex geotechnical
structures (e.g. Fenton & Griffiths, 2005; Cho & Park,
2010; Chen et al., 2012). As this software is able to
generate multiple random scenarios (Smith & Griffiths,
1998), the output of the calculations forms a population of
results that can be interpreted with statistical criteria.
In the present study, this tool has been used just to
generate samples with spatially variable properties, but the
calculation has been performed importing the input data
from another software (Abaqus, 2013). This choice has
been made considering the greater flexibility of this code,
which enables the hyperbolic constitutive relation
(equation (4)) with a tension cut-off (equation (3)) to be
implemented instead of a simpler linear elastic stress–
strain relationship. In detail, each sample is divided into a
mesh of 20-node quadrilateral elements (Fenton &
Griffiths, 2000), each having a specific property: a random
field of the uniaxial compressive strengths 𝑞 is first
generated with the RFEM code (Fenton & Griffiths,
2000), then these values are used to compute the
constitutive parameters to be assigned to each cell for the
calculation with the adopted finite-element method (FEM)
software (Abaqus, 2013). In particular, the cohesion is
computed with equation (2) considering a constant value
of the friction angle, the tension cut-off is computed with
equation (3) and the initial stiffness 𝐸 with equation (5).
Fig. 5. Numerical calculations of uniaxial compression tests on sample of homogeneous and randomly variable
material: (a) distortional deformation at failure of (a1) homogeneous and (a2) inhomogeneous samples; (b) stress–
displacement responses
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10
The random generation of 𝑞 over the considered
samples is made with the local average subdivision
method (Fenton & Vanmarcke, 1990), a fast and accurate
algorithm able to produce realisations of a discrete ‘local
average’ random process statistically consistent with the
field resolution. As such, it is well suited for systems
represented by sets of clusters where the average
properties over each element are desired, like in the finiteelement calculation. The algorithm is based on a Markov
process having a simple exponential correlation function,
as well as by a fractional Gaussian noise process as defined
by Mandelbrot & van Ness (1968).
An example of the calculation for a square
prismatic sample with a 1 m base and 1.6 m high can be
seen in Fig. 5 where the two cases of homogeneous
samples with uniform properties and inhomogeneous
samples with randomly variable properties are compared.
The spatial distribution of distortional strains at failure
(Fig. 5(a)) is symmetric for the homogeneous sample,
being dictated by the boundary conditions (fixed end),
while it tends to produce concentrated strains in the
weaker zones of the variable material. These effects
become even more evident with regard to the stress–
displacement response of the material (Fig. 5(b)). In fact,
compared with the case of homogeneous material, the
samples with variable properties give higher or lower
ultimate strength depending on their random properties.
The curve for homogeneous material with a linear elastic–
perfectly plastic response without tension cut-off, reported
for reference, shows that non-linearity and tension cut-off
induce a small reduction of the ultimate strength. This
effect, rather limited in the present case because of the
mainly compressional stress state given to the sample, may
become meaningful when tensile stresses are applied to
the jet grout structures (e.g. jet-grouted bottom plugs).
4.1. Case study
The considered case study (Croce et al., 1994) includes a
relatively large number of uniaxial compression tests (see
Fig. 3(e) and Table 3) together with a sonic tomography
(see Fig. 4(a)) and thus represents an optimal example to
apply the above suggested calculation procedure. From the
combination of the available data, the parameters listed in
Table 4 have been found for each variable. In particular,
the cohesion 𝑐 has been assumed variable with a
lognormal probability function, similar to the uniaxial
compressive strength (Fig. 3(e)). The mean value 𝜇 and
standard deviation 𝜎 have been computed from the values
in Table 3 considering 𝑐 and 𝑞 to be linearly related to
each other (equation (2)) with the friction angle 𝜙 fixed as
constant (a value of 37° observed from triaxial tests has
been chosen giving 𝛼 = 0.25).
The constant of proportionality β expressing the
ratio between the mean values of 𝐸 and 𝑞 (equation (5))
has been computed from the experimental results as
follows. At first, 𝐸 has been related to the compressional
wave velocity 𝑣 with the following relation
𝐸 =𝜌𝑣
2.61
0.94
0.25
(13)
)
where ρ = 1800 kg/m3 and ν = 0.2 have been fixed. Then,
considering the mean compressional wave velocity
observed in the sonic tests (2739 m/s) and the mean value
of 𝑞 from laboratory tests (10.49 MPa), a value of 𝛽 equal
to 564 has been found. It is worth noting the similarity of
this value with the ranges reported in Table 2 and its
consistency with the laboratory investigations performed
in this case study (see Croce et al., 1994).
Finally, the correlation length, equal to 2.3 m,
found for the laboratory tests using equation (8) and the
logarithm of the squared compressional wave velocity 𝑣 ,
has also been assumed to describe the spatial variation of
strength and stiffness.
An example of the spatial distribution of the initial
Young’s modulus 𝐸 generated with the above parameters
over a square prismatic block of cemented material having
a base width of 12 m and height of 7.5 m (Fig. 6), shows
the typical clustering of material, with largely stronger or
weaker portions agglomerated into different spaces with a
continuous transition among these zones.
Table 4. Parameters adopted for the random field calculation
𝜙: deg
𝜇 : MPa
𝜎 : MPa
𝛼
𝛽
37
(

𝜚kg/m3
𝜈
𝜃m
1800
0.2
2.1
Fig. 6. Examples of clustering for the initial Young’s modulus 𝐸 computed with the adopted random field simulation
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11
The variability of the mechanical properties seen
in Fig. 5(b) has been extensively investigated, together
with the role of the portion of the material involved in the
failure mechanisms. To achieve this, several sets of
samples with variable dimensions have been artificially
created and their response to uniaxial compression tests
numerically simulated. In particular, seven classes of
square prismatic samples each have been considered, all
having the same fixed shape (height over base ratio
𝐻⁄𝐵 = 1.6) but different size (𝐵 equal to 0.1, 0.2, 0.5, 1,
2, 3 and 5 m). For each class, 100 samples have been
generated with the above random field procedure and the
statistical distribution of the strength 𝑞 computed with
the FEM analysis. The characteristics of the variability can
be summarily seen from the results plotted in Fig. 7. For
instance, it is immediately evident that the mean value of
strength does not change significantly from one case to
another, apart from some random effect given by the
relatively limited number of tested samples. On the
contrary, there is a progressive tendency to reduce
skewness, as the strength of the smallest samples is
scattered with an asymmetric statistical distribution,
consistently with the assigned log-normal distribution (see
Fig. 3), while the distributions for the larger samples
become progressively more symmetrical asymptotically
tending to the normal distribution.
However, the most evident factor is the reduced
dispersion (see also the standard deviation in the
accompanying table) as the size of the samples increases.
This effect is particularly relevant for the design of jetgrouted structures, being closely connected with the
definition of the characteristic strength. In fact, if the
percentile of the distribution is fixed (e.g. at 5%) the
corresponding 5% fractile of the strength increases with
the size of the considered element, as shown by the
calculated values in the case of spatial correlation plotted
in Fig. 8. For reference, the 𝑞 corresponding to the 5%
fractile is computed assuming a log-normal distribution
and the absence of spatial correlation and plotted as the
continuous line in Fig. 8. In this case, a sample of base B
has been assumed as formed by 𝑛 samples of dimensions
similar to that used in the laboratory tests (B =0.1 m), with
n equal to the ratio between the cross-sections of the
samples (𝑛 = (𝐵⁄𝑏) ). In this hypothesis, the standard
deviation of the mean over n samples is given by
𝜎 ∗ (𝐵) =
(
√
. )
=
(
( )
. )
(14)
The comparison in Fig. 8 between the dots and the
continuous line clearly highlights the role of the spatial
correlation. In fact, the clustering of material with similar
properties (θ = 2.1 m in the present case) tends to increase
the possibility of having a weaker response in comparison
with the case where the properties vary from point to point
in random and uncorrelated ways.
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5. PARAMETRIC ANALYSIS
To generalise the above observation and find a rule
applicable for design, a more extensive calculation has
been performed adopting the values of 𝜇 , 𝛼 and 𝛽 given
in Table 4 and varying systematically all the factors
capable of affecting the variability, that is the variation
coefficient of the cohesion 𝑐 (and consequently of 𝐸 , see
equations (2) and (5)), the correlation length 𝜃, the size
(𝐵) and shape (𝐻⁄𝐵) of the sample. The ranges assumed
for the mean value of 𝑐, the variation coefficients of 𝑐 and
𝐸 , and the correlation distance, 𝑞, listed in Table 5, are
chosen to cover the typical situations coming from
experiments (see Figs 3 and 4). For each combination of
parameters, 100 simulations have been carried out and the
obtained statistical distribution of 𝑞 is analysed.
It is noted that the calculation is implemented for
values of 𝛼 and 𝛽 derived from the examined case study
(𝛼 = 0.25 corresponding to 𝜙 = 37°; 𝛽 = 564) and
different values of these variables could in principle give
other results. However, it must be considered that the
assumed values are rather typical of jet-grouted structures
and that the calculation is more strongly dominated by the
variability terms than by these factors.
The simulations have been first carried out
considering samples of the same shape (𝐻⁄𝐵 = 1.6) and
different size (𝐵 from 0.1 to 5 m), then extending the
analysis to jet-grouted elements of different shapes (𝐻⁄𝐵
ranging from 1.6 to 8) with the same range of sizes.
With regard to the first issue, the mean and
standard deviation of 𝑞 computed for increasing 𝐵 are
shown in the plots of Fig. 9, each being representative of
a different correlation length 𝜃 and coefficient 𝐶𝑉
(respectively 0.1, 0.36 and 0.5). For each assigned set of
parameters, the distribution of 𝑞 showed the typical shape
of Fig. 7. The sequence of plots shows always asymmetric
distributions, even though skewness tends to reduce with
the sample dimension. Goodness-of-fit tests satisfactorily
confirmed that variation could be appropriately simulated
with log-normal probability functions.
With regard to the mean 𝑞 , a random scattering
is observed in Fig. 9, but without any meaningful
systematic (increasing or decreasing) trend. It is also worth
noting that the mean values of 𝑞 are not dissimilar from
the ratio 𝜇 ⁄𝛼 = 10.44 MPa, in accordance with equation
(2a) (and Table 4). The slightly larger values possibly stem
from the restriction given in the numerical model at the
base of the samples. On the contrary, the standard
deviations show a systematic decay with 𝐵, starting at the
lowest base dimensions from a value approximately equal
to the product between the assigned variation coefficient
(𝐶𝑉) and the mean uniaxial compressive strength 𝜇 .
In each plot, the curves form a fan governed by the
correlation lengths where the lower 𝜃 values produce more
rapid decay. From this observation it is deduced that a
sample of fixed base 𝐵 becomes more representative of the
material, with 𝑞 being less dispersed, for relatively lower
𝜃 values. On the contrary, larger 𝜃 values indicate that
12
Fig. 7. Statistical distributions of 𝑞 for samples of fixed shape (𝐻⁄𝐵 = 1.6) and increasing dimension 𝐵
Fig. 8. Values of 𝑞 corresponding to 5% percentile of the distribution for square prismatic samples (𝐻⁄𝐵 = 1.6) of
variable base dimension
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13
Table 5. Input parameters of the parametric analysis
Probabilistic
distribution for
𝜇 :
MPa
𝐶𝑉(𝑐) = 𝐶𝑉(𝐸 )
2.61
0.1-0.5
𝜃
( ) m
1
𝑓𝑜𝑟 𝐵 ≤ 𝜃
1−
𝑓𝑜𝑟 𝐵 > 𝜃
(15)
from Vanmarcke (1977), and
𝑐 and 𝐸
Log-normal
𝛤(𝐵, ) =
0.5-4
material with similar properties, whether weaker of
stronger, tend to agglomerate in larger portions and thus
samples of fixed dimensions become less representative.
In conclusion, the representativeness of the sample seems
to be determined by the ratio between 𝐵 and 𝜃.
Following the above consideration, all results in
Fig. 9 have been summarily plotted in Fig. 10, where the
variation coefficients computed with the FEM analysis for
each combination of 𝐵, 𝜃 and 𝐶𝑉(𝑐) = 𝐶𝑉(𝐸 ) values
have been scaled by 𝐶𝑉(𝑐) and reported as a function of
the ratio 𝐵⁄𝜃. It is quite clear that all the results tend to
follow the same alignment independently of the assigned
properties. For comparison, two theoretical functions
describing the variance reduction have also been presented
in Fig. 10
𝛤(𝐵, ) =
| |
+ 𝑒𝑥𝑝 −
| |
−1
(16)
from Fenton & Griffiths (2008).
The first function fails to capture the trend of the
calculated results as it imposes constant variance for
𝐵⁄𝜃 < 1; the mathematical structure of the second
function leads to a better approximation of the results,
although with a non-negligible underestimate of the
variance at larger 𝐵⁄𝜃. To correct the prediction, the
following simpler exponential function has been proposed
with coefficients chosen empirically to give a closer fit of
the numerical results
𝛤(𝐵, ) = exp −0.5
.
(17)
Fig. 9. Mean and standard deviation of 𝑞 for samples having slenderness ratio 𝐻⁄𝐵 = 1.6 (𝐶𝑉(𝑐) = 𝐶𝑉(𝐸 ) is
equal to (a) 0.1; (b) 0.36; (c) 0.5)
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14
Fig. 10. Variance reduction of 𝑞 as function of 𝐵/𝜃
The above calculation refers to jet-grouted
elements having different size but the same slenderness
ratio (𝐻⁄𝐵 = 1.6 has been taken for reference). It may be
argued that the above results are somehow dictated by the
choice made for 𝐻⁄𝐵 . For instance, jet-grouted structures
often consist of separated columns (e.g. foundation
reinforcement – Modoni & Bzówka (2012)) or the
transmission of forces occurs on longer portions of
material (e.g. bottom plugs – Modoni et al. (2016)) and
thus the constitutive units of jet-grouted structures, that is
those elements capable by their failure of impairing the
overall stability, are much longer than considered above.
Therefore, to understand the role of the shape of the jetgrouted elements and generalise the above results, the
calculation has been repeated, fixing the statistical
properties of the jet-grouted material (mean and variation
coefficient) and parametrically varying the base width,
height and correlation length of the element subjected to
uniaxial compression (Table 6).
Table 6. Range of parameters adopted for the numerical analysis of elements of variable size and slenderness ratio
𝐶𝑉(𝑐) = 𝐶𝑉(𝐸 )
Probabilistic distribution for 𝑐 𝜇 : MPa
𝐵: m
𝐻 ⁄𝐵 
𝜃 ( ) m
and 𝐸
Log-normal
2.61
0.36
0.5-2
0.5-2
1.6-8
Fig. 11. Variation of mean 𝑞 with the sample dimensions and (a) correlation length and (b) normalisation with the
slenderness ratio
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15
One of the most evident results obtained from this
calculation is the systematic reduction of the mean 𝑞
values with increase in the slenderness of the studied
elements. In fact, the results plotted in Fig. 11(a), reporting
the ratio between the mean 𝑞 for each slenderness and the
mean 𝑞 value obtained for the same set of parameters
with 𝐻⁄𝐵 = 1.6, reveal that longer elements tend on
average to have lower 𝑞 values and hence fail before
shorter ones, as the resistance is governed more by the
base width than by the correlation length 𝜃. This
observation can be explained by considering the longer
elements as formed by chains of smaller portions and
considering that failure is determined by the lowest 𝑞
value of one of these portions. As a consequence, the
probability of failure by the collapse of one of these is
generally higher for longer than shorter elements, being
governed by the lowest 𝑞 over a sequence of jet-grouted
segments. In order to find a general rule, the above-defined
mean 𝑞 ratio is represented as a function of the
slenderness ratio 𝐻⁄𝐵 in Fig. 11(b). The plot readily
shows the same decay for all data, with a trend well
captured by the following simple function
( , , )
( , ,
. )
= 𝑒𝑥𝑝 −0.25 ∙
− 1.6
.
(18)
A similar equation can be determined for the
variance. To achieve this, the variation coefficient
obtained from the above calculation, plotted in Fig. 12(a),
shows a decay ruled by the combination of base width,
height and correlation length. Scaling the variation
coefficient 𝐶𝑉(𝑞 ) computed for each combination of 𝐵,
𝐻 and 𝜃 by the value of the original distribution (𝐶𝑉(𝑐) =
𝐶𝑉(𝐸 ) = 0.36 given in Table 6) and grouping the
geometrical factors in a dimensionless variable reported in
the abscissa of Fig. 12(b), results in all the abscissa falling
along the same alignment. With the adopted dimensionless
variable, the data from this calculation tend to align with
those of Fig. 10 previously obtained on elements of the
same slenderness (𝐻⁄𝐵 = 1.6) and different size (Fig.
12(c)). The mathematical variance reduction function
assumed to fit this trend has the following expression
Γ(B, H, ) = exp −0.435
.
.
∙
(19)
It is not trivial to note that equation (19) becomes
equation (17) for 𝐻⁄𝐵 = 1.6 and thus equation (19)
represents a generalisation of equation (17) for the case of
jet-grouted elements of variable shape.
6. CHARACTERISTIC STRENGTH OF THE JETGROUTED MATERIAL
The assessment of the ultimate limit state in geotechnical
engineering is currently performed following two different
approaches: the load and resistance factor design (LRFD)
approach, also referred to as the American approach, or
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the partial factor (PF) approach, also known as the
European approach (e.g. Becker, 1997). Both approaches
compute design loads using multiplying factors dependent
on the type of action, but differences exist in the
calculation of design resistance. In the former case it is
obtained by factoring the ultimate resistance of the
structure calculated with unfactored strength parameters
of the material; in the latter case by adopting a double set
of scaling parameters, one applied on the resistance, the
other on the material’s parameters. Although Eurocodes
suggest to put these factors alternatively equal to 1, this is
not always the case in national standards (e.g. NTC, 2008),
where factors larger than 1 are simultaneously set to the
geotechnical properties and to the resistance.
Independently of the adopted approach, a characteristic
resistance or characteristic strength properties of the
material must be selected. In case of statistical inference
like the one expressed by equation (1) (Schneider, 2010),
the characteristic strength is defined as the value
corresponding to a certain percentile of the population,
chosen in order to keep the risk connected with failure
within tolerable levels. Equations (18) and (19) may thus
serve to meaningful for the jet-grouted structure than the
sample typically tested in the laboratory.
The statistical distributions observed on
laboratory samples (Fig. 3) or computed on larger
elements (Fig. 7) are typically found to be asymmetric,
although with a transition to more symmetric functions for
the larger elements. Considering that the correlation
distances experimentally found on jet-grouted structures
(see Fig. 4) are typically of the order of some metres, the
statistical distribution of 𝑞 becomes symmetric only for
blocks of very large size, unlikely to be considered as a
unit element for calculation. With the aim of finding a
general rule, the characteristic value of 𝑞 has then been
computed assuming it is distributed with a log-normal
function
𝑙𝑜𝑔𝑞 , = 𝜇
(𝑞 ) − 𝛿 ∙ 𝜎
(𝑞 )
(20)
where 𝛿 is the standard normal deviate computed from the
Gaussian distribution as a function of the tolerated
cumulative probability of failure 𝑃 (e.g. 𝛿 = 1.645 for
𝑃 = 5%); 𝜇 (𝑞 ) and 𝜎 (𝑞 ) can be computed
with equations (6) and (7) as a function of the mean value
and variation coefficients. By applying equations (18) and
(19) it is possible to compute the mean and variance (and
so the standard deviation) of the strength on elements
having variable shape (𝐻⁄𝐵 ), dimension (𝐵) and
correlation length (𝜃), starting from the statistical
distribution obtained in laboratory tests.
The above procedure has been implemented (Fig.
13) to compute the characteristic strength corresponding
to the 5% fractile 𝑞 _ (5) of jet-grouted elements of
variable correlation lengths 𝜃, base width 𝐵 and height 𝐻,
starting from the laboratory value 𝑞 _ (5) . More
practically, the graphs in Fig. 13 serve to determine from
16
the characteristic strengths obtained for laboratory
samples with a particular width and slenderness the
characteristic strength representative of zones of jetgrouted soil with larger widths and different slenderness
than the laboratory samples. The graphs immediately
show the positive role of the size representative of the jetgrouted structure, which is capable of attenuating the
reduction of characteristic strength given by the variability
at the sample scale (quantified by 𝐶𝑉). This scale effect is
more relevant for larger variation coefficients; that is, for
materials having a larger stochastic variation, where
correcting factors almost doubling resistance can be
inferred. However, this positive effect is counterbalanced
by the agglomeration of material into a larger portion of
space quantified by the correlation length 𝜃, and moreover
by the slenderness of the considered element 𝐻⁄𝐵 ,
because of the previously identified chain effect.
7. CONCLUSIONS
Some of the most relevant uncertainties affecting the
reliability of jet grouting, that is those related with the
inherent variability of strength, have been examined with
the aid of field observation and their role interpreted with
numerical analyses to infer a statistical meaning to the
characteristic strength of the cemented material. At the
present time, Eurocode 7 states that ‘the characteristic
value of a geotechnical parameter shall be selected as a
cautious estimate of the value affecting the occurrence of
the limit state’ (BSI, 2007: paragraph 2.4.5.2). The
proposed procedure makes it possible to quantify on a
statistical basis the caution claimed in the above definition
of the characteristic strength (Phoon & Kulhawy, 1999).
Fig. 12. (a) Variation coefficients of 𝑞 for samples of variable base width, height and correlation length; (b)
cumulative dimensionless plot of the variance reduction and fitting with equation (19)
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17
Fig. 13. Scale correction factor for the 5% percentile characteristic strength of the jet-grouted material
With this aim, experimental results and numerical
models have been coupled to rationalise the variability of
mechanical properties within bodies of jet-grouted
material of different shape and dimension and to define
characteristic values useful for design. The constitutive
model for the jet-grouted material has been simplified
looking at the evident difference between treated and
natural soils, quantifying resistance with the uniaxial
compressive strength (i.e. independent of the confining
stress) and relating this quantity to the stiffness and tensile
strength. The experimental evidence has in fact suggested
that jet-grouted material may be considered as a purely
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cohesive material, given the increase of strength inferred
by the injection procedure (namely single, double and
triple fluid systems) and parameters (the water/cement
ratio) and the limited resemblance of the jet-grouted soil
to the original soil.
A procedure is thus suggested to define properties
based on the results of laboratory tests and sonic
tomography, giving particular significance to the latter
test, which is very useful to depict the spatial variation in
a continuous and precise way, and which is normally used
to give just a broad feeling of the treatment effectiveness.
18
The large variability observed at the sample scale
cannot be directly used to determine characteristic values,
but must be somehow averaged considering that larger
portions of materials are involved in the failure of a jetgrouted structure and interaction between weaker and
stronger portions might play some role. The implemented
random field calculation has allowed this scale transition
to be managed comprehensively, taking account of the
spatial structure of the mechanical properties.
With the proposed approach, general formulas
(equations (18) and (19)) have been provided to compute
the probabilistic distribution of the uniaxial compressive
strength qu for jet-grouted elements of different size and
shape, starting from the distribution seen on small
laboratory samples. These formulas can be used to
determine a characteristic strength in semi-probabilistic
calculation (e.g. Modoni et al., 2016), that is, computing
partial factors, or to include strength in probabilistic
calculation, that is, those performed with multiple
generation of random scenarios (e.g. Modoni & Bzówka,
2012; Liu et al., 2015). Considering the first approach, the
characteristic strength computed from laboratory tests
assigning a percentile of the distribution (e.g. 5%) must
then be multiplied by a factor dependent on the ratio
between the size of the elements’ characteristics for the
jet-grouted structure and the correlation length. Values of
the correcting factor as high as 2 are seen for short
elements, while the increase of slenderness tends to give
strong reductions because of an unavoidable chain effect.
NOTATION
𝐵
base width of the sample
𝑏
cross-section size
𝑐
cohesion
𝑑
distance between two points
𝐸
Young’s modulus
𝐸
initial tangent stiffness modulus
𝐸
secant modulus calculated at 50% of the failure
𝑓
tensile strength
𝐻
height of the sample
𝑘
factor dependent on the accepted fractile and
probability distribution
(𝜃)
𝐿
likelihood function
𝑁
correcting factor for the bias of the mean estimate
𝑛
number of elements
𝑞
deviator stress
𝑞
compressive strength
𝑞
uniaxial compressive strength
𝑞 ,
characteristic values of uniaxial compressive
strength
𝑟
space vector at the generic point 𝑖
𝑉
terms of the covariance matrix
𝑉
propagation velocity of the compression waves
𝑥
characteristic value of a generic variable
𝑥
mean value of a generic variable
𝛼
proportionality coefficient between cohesion and
unconfined compressive strength
𝛽
correlations coefficient between 𝐸 and 𝑞
Authors’ copy
𝛤
𝛿
𝜀
𝜃
𝜇
𝜇
𝜈
𝜌
𝜌(𝑑)
𝜎
𝜎′
𝜎
𝜎
𝜙
variance reduction function
standard normal deviate
axial strain
correlation length
mean value
mean of lognormal probability function
Poisson coefficient
density
autocorrelation function
standard deviation
effective confining stress
standard deviation of lognormal probability
function
radial stress
friction angle
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