STUDY OF STRESS PATHS IN ARCHING
EFFECT USING FRICTIONAL STRAIN
HARDENING AND SOFTENING IN FINE
SAND
Alireza Abbasnejad*
Assistant Professor, University of Tabriz, Tabriz, Iran
abbasnejad@tabrizu.ac.ir
Mahyar Soltani
MSc Student, University of Tabriz, Tabriz, Iran
maahyarsoltani@gmail.com
Reception: 18/02/2023 Acceptance: 15/04/2023 Publication: 02/05/2023
Suggested citation:
Abbasnejad, A. and Soltani, M. (2023). Study of Stress Paths in Arching
Effect Using Frictional Strain Hardening and Softening in Fine Sand. 3C
TIC. Cuadernos de desarrollo aplicados a las TIC, 12(2), 15-58. https://doi.org/
10.17993/3ctic.2023.122.15-58
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ABSTRACT
Arching is one of the most common phenomena that occur in most geotechnical
structures. To determine the properties and quality of this phenomenon, a physical
model has been designed and constructed. The apparatus comprises rectangular
trapdoors with different widths that can yield downward while stresses and
deformations are recorded simultaneously. As the trapdoor starts to fail, the whole soil
mass deforms elastically. However, after an immediately specified displacement,
depending on the width of the trapdoor, the soil mass behaves plastically. This
behavior of sand occurs due to the flow phenomenon and continues until the stress on
the trapdoor is minimized. Then the failure process develops in the sand, and the
measured stress on the trapdoor shows an ascending trend. This indicates a gradual
separation of the yielding mass from the whole soil body. Finally, the flow process
leads to the establishment of a stable vault of sand called the arching mechanism or
progressive collapse of the soil body. To simulate this phenomenon with continuum
mechanics, the experimental procedure is modeled in ABAQUS software using stress-
dependent hardening in an elastic state and plastic strain-dependent frictional
hardening-softening with Mohr Coulomb failure criterion applying user sub-routine.
The results show that the experimental data have an acceptable corresponding to the
numerical analysis data. So the selected soil behavior could indicate the main aspects
of the arching effect, such as the flow that occurs in specific periods of strains. In the
following, the stress path in p, q, and p, ν
space was extracted from numerical
analysis, and the results have been discussed.
KEYWORDS
ABAQUS, Arching Effect, Stress Path, PIV, Frictional Strain Hardening and Softening,
Fine Sand.
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INDEX
ABSTRACT
KEYWORDS
1. INTRODUCTION
2. PHYSICAL MODEL
2.1. Soil under test
2.2. Specifications of the built physical model
3. NUMERICAL MODEL
3.1. Behavioral model governing the arcing phenomenon
3.2. Rupture cover
3.3. Friction hardening
3.4. Frictional softening
3.5. The dependence of the friction angle and expansion on the stress level
3.6. The parameters of the behavioral model obtained from the experiments
3.7. Calibration of the behavioral model
3.8. Numerical model of arc phenomenon and modeling assumptions
3.9. Eliminating the effect of the arc in the place of stress gauges
4. EXPERIMENTS CONDUCTED ON THE PHYSICAL MODEL
4.1. The results of the data of the stress gauge installed on the valve and in the
middle of it (S1).
4.2. Comparison of the laboratory results with the numerical model, taking into
account the data of the stress gauge installed on the valve and in the middle of
it.
4.3. The results obtained from the PIV method concerning the measurement of
strains during the occurrence of the arcing phenomenon
4.4. The results of strain analysis
5. THE NATURE OF THE FLOW PHENOMENON IN THE ARCING PHENOMENON
6. TENSION SPACE IN THE SELECTED BEHAVIORAL MODEL
6.1. The stress space of area 1 (elastic area)
6.2. The stress space of zone 2 (softening zone)
6.3. The stress space of the area near the valve (hardening and softening
expansion area)
6.4. The stress space of zone 3 (hardening zone) - farther from the valve
7. CONCLUSION
REFERENCES
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1. INTRODUCTION
The arch phenomenon is one of the most important phenomena that we deal with
in geotechnical engineering. The impact of this phenomenon can be clearly seen in
underground and buried structures, so that the amount of force applied to the
structure depends on the redistribution of the stress caused by overburden or
overburden. Many scientists and researchers have worked in this field in a laboratory
and theoretical manner and have written numerous articles. This phenomenon was
first observed in the gunpowder storage silos belonging to the French army, and in
1895 John Sen presented the theory of silos. This phenomenon was shown for the
first time in a scientific way by an experiment on sand with an open valve conducted
by Terzaghi. It was proposed by him in geotechnical engineering. Usually, this
phenomenon occurs in places where there are sudden differences in the type of
materials in the soil mass. In other words, in the mass, two types of materials with a
different modulus of elasticity come into contact and exchange stress with each other.
In general, it can be said that arching occurs wherever there is a change of location in
the soil mass enclosed between stable supports, whether horizontal or vertical. Also,
this phenomenon can exist in all extents of deformation in the soil, so it starts with the
occurrence of elastic shear deformations and continues until the irreversible (plastic)
deformations and the breaking of parts of the soil mass.
To investigate the arching phenomenon in underground structures, Terzaghi
conducted an experiment in which a horizontal valve was lowered. When it was
moved, the amount of stress applied to the center of the valve was simultaneously
read. Using the results of these experiments and assuming plastic behavior for soil,
Terzaghi presented the theory of shear plates. In fact, by considering the balance of
forces in the plastic state of the soil, Tarzaghi was able to make the arch phenomenon
mathematically legal. After V. Finn, he modified the hypothesis related to Terzaghi's
theory and considered the elastic state of the soil. In the following years, many
experimental and numerical studies have been conducted to investigate the failure
mechanisms of the soil mass above the tunnel (Atkinson and Potts, 1977; Jiang and
Yin, 2012; Guo and Zhou, 2013; Han et al., 2017; Franza et al., 2018; Chen et al.,
2018; Jin et al., 2021; Zheng et al., 2021). These studies play an important role in
improving the understanding of the interaction between tunnels and soil and create a
solid foundation for developing theoretical models. After that, the parameters of silo
width and lateral stress ratio in Terzaghi theory were modified by many researchers
based on model tests and numerical analyzes (Stein et al., 1989; Hendi, 1985; Chen
et al., 2015; Zhang et al., 2016). Although many significant modifications have been
made based on Terzaghi's loose earth pressure theory, most of the research has
focused on shallow tunnels where the failure zone at the top of the tunnel extends to
the ground surface when the soil mass is in a limited state. However, for deep tunnels,
local failure occurs at the top of the tunnel according to many laboratory tests (Jacobs,
2016; Song et al., 2018) to evaluate the earth pressure on deep tunnels. Based on the
existing theoretical models for the shallow tunnel, a limited height of the silo was
further considered. Chen and Peng (2018) assumed that the height is 1.5 times the
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radius of the tunnel based on the numerical results in the soft ground of Shanghai.
The local failure height for the deep tunnel is related to the ground subsidence, the
phenomenon of soil expansion (change), and the depth of the tunnel cover in the light
of the previous study. Zhang et al. (2016) obtained a formula for calculating the local
failure height caused by the construction of the deep jacked pipe, in which the
relationship between the local failure height, the volume of loosened soils, and soil
bulking factors are simultaneously considered, was taken. Zhang's method assumed
that the shear bands start from the spring lines of the tunnel section. However,
according to the numerical results of Lin et al. (2019), the shear bands developed
diagonally from the bottom of the tunnel in the sandy ground. Therefore, the formula
presented by Zhang et al. (2016), used to calculate the local failure height, is limited to
sand.
However, the previous models for calculating the earth pressure in deep tunnels
assumed that the soil mass above the failure zone was not disturbed by the
construction of the tunnel and that the earth pressure applied above the failure zone is
the stress caused by the weight of the soil above, in fact, soil arching in occurs above
the failure zone, which leads to the transfer of earth pressure to both sides of the
failure zone. When the arch-bearing capacity of the soil is greater than or equal to the
weight of the soil above the failure zone, a cavity can be created above the failure
zone. The failure zone is zero. Such an arching effect of the soil above the failure
zone has been neglected by previous analytical approaches, which may consequently
overestimate the earth pressure exerted on the silo. A new 3D model considering the
arch effect above the fracture zone was developed by Chen et al. in 2019. To predict
the confining pressure exerted on the surface of the deep shield tunnel. The
calculated results by obtaining the new model agree with the experimental results.
However, it is assumed that the earth pressure distribution in the loosened zone is
uniform. The distribution of earth pressure on the tunnel can be different due to the
difference in the distribution of earth losses caused by the construction of the tunnel.
For the circular tunnel, Chen and Teng in 2018 pointed out that the vertical earth
pressure distribution shows a concave curve; that is, it is smaller in the center line and
increases with the increase of the horizontal distance from the center of the tunnel.
However, the current tunnel design for tunnels generally assumes vertical earth
pressure of uniform width. To reflect the uneven distribution of the vertical earth
pressure in the tunnel, Chen and Teng in 2018 assumed that the vertical earth
pressure on the tunnel conforms to the distribution of the Gaussian function, and then
presented a formula for calculating the earth pressure. However, there is a relatively
large error between the model results and the numerical analysis. Based on particle
flow theory, Wu et al. in 2019 obtained a modified formula by assuming that the
vertical earth pressure on the tunnel corresponds to a trapezoidal distribution. But, in
fact, the vertical pressure distribution of the earth is a smooth concave curve.
In this article, in order to determine the characteristics and how this phenomenon
occurs, a physical model has been designed and built that can model the arcing
phenomenon in a laboratory. In this physical model, valves with different widths have
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been installed to investigate the effect of valve width on the occurrence of this
phenomenon. To determine the pattern of stress changes during the event of the
arcing phenomenon, miniature stress gauges with a diameter of 14 mm have been
used, and to determine the pattern of strain changes, the PIV method has been used.
Also, numerical modeling of this phenomenon has been done in Abaqus software and
the finite element method. To model the arch phenomenon, the hardening behavior
dependent on the stress level in the elastic range and the hardening and softening
depending on the plastic strain in the plastic range were used; for this purpose, a
program was written in the form of a subroutine in the Fortran environment and then
with the help of a compiler Visual Studio has been introduced to Abaqus software.
2. PHYSICAL MODEL
In this research, to model the arcing phenomenon, a physical model was designed
and built to provide the ability to model the arcing phenomenon on a laboratory scale.
2.1. SOIL UNDER TEST
A type of non-sticky silty sand passed through a grade 10 sieve, with a constant
humidity of 2% and a specific gravity of 2.62, was used for the experiments. The
granulation diagram and characteristics of the soil used are presented in Fig. 1 and
Table 1, respectively.
Figure 1. Sand grading curve
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Table 1. Characteristics of the soil used in the research
2.2. SPECIFICATIONS OF THE BUILT PHYSICAL MODEL
To physically model the arcing phenomenon, a device was designed and built. Fig.
2 shows the fabricated device and test details. The physical model consists of a
metal skeleton, in the upper part of which is a rectangular cube tank with a steel frame
with internal dimensions of 400 x 1830 and a height of 1250 mm. Its main skeleton is
made of steel plates with a thickness of 10 mm and a stud profile. 200 were made. In
the outer part of the side plates, five rows of 80-grade corner type hardeners have
been used. To increase the strength, the two parts of the skeleton are welded
together using 100-grade stud material to eliminate lateral deformation in the skeleton
due to the lateral pressure of the soil and the overhead pressure caused by the
loading jack. The lower part of the structure consists of three rows of grade 200 stud
profiles, along with a 100 mm thick plate welded, and the upper skeleton is made of
100- grade studs with the help of four legs to this 3000 x 600 mm plate is connected.
In total, the height of the device is 2200 mm. Two 10 mm thick steel plates are
installed on the bottom of the machine's tank, which can be moved to the sides in a
sliding manner, and as a result, the distance between the two plates, which is equal to
the width of the valve, can be adjusted. Four separate rectangular pieces with widths
of 10, 20, 30, and 35 cm were used for movable valves. When using each valve, the
two side plates are opened to the sides as wide as the valve, and the desired valve is
placed between the two jaws embedded in the side plates, which serve both as a
stiffener and as a guide for the valve movement would take. The valve is secured in
place using eight provided screws. These valves are made of a 10 mm thick sheet
and could move down up to 40 mm. On both sides of the tank, two transparent
Plexiglas plates with a thickness of 30 mm were installed to observe the changes in
soil locations. To increase the rigidity, a steel grid with stiffeners of 50 mm height and
20 mm thickness was installed, which could be opened, and each of them was
connected to the skeleton by 20 screws.
According to Fig. 2, in order to measure the displacements applied to the valves, a
strain gauge was installed under the valves. Also, to measure the stresses involved
on the valves by the sand, a stress gauge was also established under the device, so
that the applied stress from the valves is directly transferred to the stress gauge.
γ max 17.35 (kN/m3) Fc9.5%
γd max 17.01(kN/m3) Gs2.62
γmin 12.88 (kN/m3) M(moisture) 2 %
γd min 12.63 (kN/m3) Classification SP-SM
Dmax 2 mm
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To apply displacement to the valve, a 10-ton hydraulic jack with a stroke of 70 mm
with the ability to adjust the speed was installed under the stress gauge, which
enables displacement be applied to the valves. Also, a 20-ton hydraulic jack with a 50
mm stroke with a rigid pressure plate was designed and built to spread the overhead
pressure. This overhead pressure system was able to be installed on the device
using a roof crane made for this purpose. This capability provided the possibility of
filling the tank by installing the sand precipitation system and then re-installing the
loading system.
The dimensions of the tank have been chosen according to the maximum width of
the valve and the type of soil used, and other physical models have been designed in
such a way that while removing the effect of the side borders (walls of the tank) on the
results, it is not too large as required so that It is possible to fill and empty the tank.
Researchers have chosen physical models with different dimensions to study the
arcing phenomenon. In this research, the criteria considered for the study of
underground tunnels have been used to select the size of the tank size. In the model
made by Branko and his colleagues, the distance from the center of the tunnel with a
diameter of 55 cm to the lateral borders of the model is 1.2 times the diameter of the
tunnel. in the model made by Kim and his colleagues, for a tunnel with a diameter of 7
cm, this distance is 7 times The diameter of the tunnel selected. On average, in most
of the designed modes, the distance from the side walls to the center of the tunnel is
in the range of 4 to 6 times the diameter of the tunnel. In this research, taking into
account the recommendations of previous researchers and the difficulties caused by
building a physical model with large dimensions as well as filling and emptying the
tank for multiple tests, the distance from the side walls to the center of the valve with a
maximum width (35 cm) is more than 5 times the width of the valve (exactly 5.22
times) was chosen. The width of the built model is equal to 183 cm and equal to the
length of the full Plexiglas sheet.
After designing the initial dimensions of the model, the effect of the distance of the
side walls selected for the tank was investigated using a numerical model. In this
way, in the built numerical model corresponding to the dimensions of the physical
model, the distance of the lateral borders of the model was increased to 1.5 and 2
times the initial value, and the results were compared with the values obtained from
the selected state. Based on the results obtained from the numerical model, the
stresses and strains created as a result of the occurrence of the arch phenomenon did
not change with the increase in the distance of the lateral borders. Therefore the
distance of 5 times the width of the valve is acceptable for the lateral boundaries of
the model.
In the construction of the physical model, the issue of model rigidity and removal of
model deformations should be considered. The physical model should have been
designed and built so that it would not show any deformation due to the overhead jack
force. This issue is significant, especially in the strain measurement results using the
image speed measurement method. For this purpose, the box’s design was done
with great care, and steel sheets with the required hardeners were used to make the
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box. To check the rigidity, the designed physical model was also modeled in the
Abaqus software environment. The displacements caused by the lateral pressure
caused by the soil and the force applied by the overhead jack were calculated. It was
found that the amount of horizontal displacement caused by using a uniform pressure
of 2 kg/cm2 on the vertical sides of the steel tank and the Plexiglas plate, which is
more than the maximum pressure applied to the walls during the tests, is at most 1
mm. In general, the amount of deformations is minimal and can be ignored, and
therefore the built physical model has sufficient rigidity against the applied loads.
Modeling of the machine body in Abaqus software is shown in Fig. 3.
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Figure 2. Built physical model and used strain gauges
Figure 3. Modeling of the machine body in Abaqus software
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3. NUMERICAL MODEL
Abaqus software has been used for the numerical modeling of the arcing
phenomenon. To choose the appropriate element classification, both element type
and mesh density have been investigated. To select the proper element type in the
base model, the types of introduced elements were checked. The selection of the
proper kind of element is made based on the criteria of producing stress conditions.
This means that the elements that can establish the stress conditions in place based
on the geotechnical characteristics of the site with the highest accuracy and the
lowest computational cost have been selected as the appropriate elements to
continue the research. Based on the results obtained from these models and
comparing the trends of crown changes with each other, the CPE4 element was
selected as the most suitable type of element.
In addition to studying the element type, the effect of meshing density on the results
has also been investigated. In this way, for different kinds of elements, different
meshing densities were also checked and matched with the values of the in-place
stresses. To choose the best density of meshing, triangular and quadrilateral
elements with different thicknesses were tested, and the changes in the accuracy of
the results and their quality were investigated in different modes. Based on these
studies, models with coarser meshing do not have smooth behavior in presenting
results, and finer meshing gives smoother results. For example, according to the
results of this review and preliminary explanations, CPE4 elements with the meshing
shown in Figure 4 were selected as the most suitable mode in the model.
Figure 4. Selected elements in the numerical model
3.1. BEHAVIORAL MODEL GOVERNING THE ARCING
PHENOMENON
The behavior before yielding is defined as linear elastic with the secant shear
modulus as follows:
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(1)
Where and are the strain and shear stress at the yield point, respectively, which
can be obtained directly from the test data (the shear stress corresponding to
calculating, the thickness of the shear zone must be assumed. Before the formation
of the shear band, the shear strain can be seen to have an almost uniform distribution
throughout the height of the specimen ( ). As a result, it can be defined as follows:
(2)
The same can be considered for the peak shear strain as follows (assuming that
the shear band has not yet formed):
(3)
As a result, the plastic shear strain at the peak point will be as follows:
(4)
Young's modulus ( ) is obtained from the following equation:
(5)
Where is the bulk modulus. Both the bulk modulus ( ) and the secant shear
modulus ( ) are dependent on the stress level, and to consider this dependence, the
following equations have been used:
(6)
(7)
Where is the reference pressure for which and Pressure power ( ) is a
component that expresses the change of elastic module with isotropic pressure. The
value of b ranges from 0.435 in small strains to 0.765 in large strains according to the
research of Root et al. The appropriate value for the b parameter to show the increase
in shear hardness depending on the stress level is 0.5.
Poisson's ratio can be defined using the following equation :
G
s=
τy
γ
y
D
γ
y=
δχ
y
D
γ
p=
δχ
p
D
γ
p
p=
δχ
p
−δχ
y
D
E
E
=
9KG
s
3K+Gs
K
K
Gs
K
=K0
(
P
Pref
)b
G
s=G0
(
P
Pref
)b
Pref
b
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(8)
In Figs. 5 and 6, respectively, the diagrams of changes in bulk modulus and
modulus of elasticity against the shift in average effective stress are presented.
Figure 5. Diagram of bulk modulus against average effective stress
Figure 6. Diagram of modulus of elasticity - average effective stress
v
=
3K−2G
s
2(3
K
+
G
s)
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3.2. RUPTURE COVER
In this research, the behavior model used for sand is the elastoplastic behavior
model with the Mohr-Coulomb rupture criterion. The hardening and softening behavior
dependent on isotropic strain is considered, and to apply this behavior model to the
quiet finite element environment ABAQUS 2012 software is coded in FORTRAN
language and introduced in ABAQUS software using Visual Studio compiler.
3.3. FRICTION HARDENING
Vermeer and de Borst proposed the equation (9) for the friction-hardening behavior
of geotechnical materials, in which the mobilized friction angle depends on the
plastic strain and gradually increases until it reaches the peak friction angle:
(9)
Where the plastic shear strain at the peak friction angle is .
The equation (10) for the variable expansion angle is presented by Rowe, which is
called the stress expansion equation and is as follows:
(10)
(11)
Where and are the mobilized expansion angle and the mobilized friction
angle, respectively, is the critical friction angle or constant volume friction angle.
The mobilized expansion angle is initially hostile and increases with the increase of
plastic strain. To avoid this considerable negative value of the expansion angle in
minor strains, the following equation was presented by Surid et al.
(12)
Power controls the shape of the mobilized expansion angle and is considered 1
in this study. The change in the mobilized expansion angle with plastic strain is shown
in Figure 7 for different values of power.
(φm)
(γp)
Sin
φm= 2
γp×γp
p
γp+γp
p
Sin φ
p
γp
p
φp
Sin
Ψm=
sin φ
m
−sin φ
cr
1−sin φmsin φcr
Sin
φcr =
sin φ
p
−sin Ψ
p
1−sin
φ
psin Ψp
Ψm
φm
φcr
Sin
Ψ′
m= sin Ψm
(
sin φm
sin φcr
)p
P
P
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Figure 7. Change of the mobilized expansion angle with P power
3.4. FRICTIONAL SOFTENING
In the two-block shear model of Shibuya et al., it is assumed that after the formation
of the shear band (just after the peak point), all plastic shear deformation is formed
within the shear band, while the rest of the soil mass remains elastic. Stays assuming
that the shear band width ( ) is , where is the average sand particle size,
the plastic shear strain at which softening is complete ( ) will be:
(13)
The strain-dependent softening with the decrease of the mobilized friction angle
and the mobilized expansion angle with the increase of the plastic shear strain is
as follows:
(14)
(15)
(16)
dB
16d50
d50
γp
f
γ
p
f=γp
p+
δχp
p
−δχ
y
16d50
=
δχp
p
−δχ
y
D
+
δχp
p
−δχ
y
16d50
φm
Ψm
φ
m=φp−
φ
p
−φ
cr
γp
s
γp
oct for γp
p≤γp
oct <γp
f
φcr for γp
oct >γp
f
Ψ
m=Ψp
(
1−γp
oct
γp
s
)
for γp
p≤γp
oct <γp
f
Ψcr for γp
oct >γp
f
γ
p
oct =
2
3[(
εp
1−εp
2
)
2+
(
εp
2−εp
3
)
2+
(
εp
3−εp
1
)
2
]1/2
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Where and are the peak friction angle and critical friction angle, respectively,
is the peak expansion angle and the plastic shear strain at the end of softening.
Also, plastic strains are the main ones.
3.5. THE DEPENDENCE OF THE FRICTION ANGLE AND
EXPANSION ON THE STRESS LEVEL
Since the fact that the friction and expansion angles are dependent on the stress
level, and this fact has also been observed in laboratory tests, to determine the shear
strength components of sand corresponding to the stress level, direct shear tests
under different stress levels were performed. In a particular soil, the value of the
internal friction angle depends on the amount of stress applied to it. The lower the
normal stress, the greater the internal friction angle . According to the theory of
stress-expansion ratio, the amount of porosity, moisture and expansion are as
necessary as the effective normal and shear stresses in analyzing the results and soil
behavior. The stress-expansion ratio equation is presented as follows:
(17)
In the above equation, the expansion angle depends on the initial conditions of
the soil. In this research, the results obtained from the experiments have been
modified based on the theory of the stress-expansion ratio.
The relationship between the peak friction angle and the critical friction angle can
be approximated by the following equation:
(18)
Where is a constant value, Shibuya and his colleagues have shown that simple
shear behavior in the soil is only possible within the range of the shear band. In the
box of a simple straight cutting machine (no rotation of the loading plate, smooth
walls, adjustable distance between the upper and lower metal plates equal to the
thickness of the cutting strip), α can be considered equal to 1. The peak friction angle
in-plane strain can be calculated as follows:
(19)
Figure 8 Changes in stress ratio and volume change (obtained through the vertical
displacement of the upper metal plate) with the horizontal displacement in the
direct shear test on "Toyura sand" in a dense state (based on the data of Shibuya et
al.) is showing. According to Figure 8, soil behavior can be divided into 4 distinct parts:
φp
φcr
Ψp
φ
φ
τ
σ
= tan
(
φcr +Ψ
)
Ψ
tan φp= tan φcr +αtan ψp
α
Sin
φp=
tan φ
p
sin Ψp+ sin Ψptan
φ
p
δy
δx
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Quasi-elastic behavior (OA): Up to point A, the soil deforms quasi-elastically. The
nonlinear behavior of the soil without any expansion is observed. is the horizontal
displacement at .
Hardening behavior (AB): From points A to B, the soil "yields" (enters the plastic
zone) and expands. Point B corresponds to the peak shear stress and shows the
hardening behavior of the soil. is the horizontal displacement at .
Softening behavior (BC): From points B to C, the soil experiences softening
behavior. Just after the peak point, a horizontal shear band extends through the
middle of the sample. Softening is completed at point C.
is the horizontal
displacement at .
Residual behavior (CD): shear accumulates along the entire length of the shear
band.
Figure 8. Changes in the ratio of stress and volume with horizontal displacement in the
cutting test on Toyoura sand
δxy
δy/δx= 0
δxp
δy/δx= 0
δxf
δy/δx= 0
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3.6. THE PARAMETERS OF THE BEHAVIORAL MODEL
OBTAINED FROM THE EXPERIMENTS
The characteristics of the sand used in the research, such as modulus of elasticity,
friction angle, and expansion angle, were extracted from the results of direct and
triaxial cutting tests. In Tables 2-4, the properties of sand for three different relative
densities of loose, medium, and dense are shown. In these tables, , ϭn, τp, and τcr
are dry density, applied normal stress, maximum shear stress, and critical shear
stress, respectively. Also, , , and
are the peak friction angle, critical friction
angle, and expansion angle, respectively, which were obtained directly from the
results of the experiments. is the corrected friction angle obtained from Equation
(18) using and .
is the plane strain friction angle used in the simulation and
is obtained from Equation (19). , , and
are the horizontal displacements of
the shear box at the yield, peak, and critical points, and the strains corresponding to
yield points are peak and critical. For example, the diagrams related to the behavioral
model and the changes in the internal friction angle and the expansion angle with
plastic strain for a relative density of 25% are presented in Figures 9 to 11.
Table 2. Material properties from experimental tests for dense sand (Dr=92%).
yd
φp
φcr
Ψ
φcor
Ψ
φcr
φpl
δxy
δxp
δxf
No Dr Ψ
φcor
(corrected)
φpl
(plane strain)
1 92 6.27 47.80 57.36 0.00 0.10 0.27
2 92 5.03 44.70 53.65 0.00 0.11 0.21
3 92 4.81 36.89 44.26 0.00 0.11 0.26
4 92 4.80 41.15 49.38 0.00 0.11 0.27
5 92 4.74 46.28 55.53 0.01 0.11 0.27
6 92 4.54 41.48 49.78 0.01 0.12 0.24
7 92 4.56 36.52 43.83 0.01 0.10 0.26
8 92 4.45 36.25 43.50 0.01 0.16 0.26
9 92 4.36 35.53 42.63 0.01 0.13 0.26
10 92 3.59 34.56 41.41 0.01 0.16 0.24
γy
γp
γf
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Table 3. Material properties from experimental tests for sand with medium density (Dr=68%).
Table 4. Material properties from experimental tests for loose sand (Dr=25%).
No Dr Ψ
φcor
(corrected)
φpl
(plane strain)
1 68 6.92 45.45 54.55 0.00 0.26 0.35
2 68 6.63 45.20 54.24 0.00 0.22 0.28
3 68 5.47 42.00 50.40 0.00 0.16 0.24
4 68 4.04 43.83 52.60 0.00 0.14 0.22
5 68 3.85 43.86 52.63 0.01 0.16 0.29
6 68 3.01 36.87 44.24 0.01 0.11 0.27
7 68 2.53 41.04 49.25 0.01 0.15 0.24
8 68 2.04 39.31 47.18 0.01 0.14 0.32
9 68 1.74 39.81 47.77 0.01 0.14 0.28
10 68 0.70 38.51 46.22 0.01 0.14 0.22
γy
γp
γf
No Dr Ψ
φcor
(corrected)
φpl
(plane strain)
1 25 8.87 40.07 48.09 0.00 0.22 0.30
2 25 7.41 35.13 40.54 0.00 0.24 0.29
3 25 7.33 34.08 38.87 0.00 0.21 0.31
4 25 4.22 30.60 34.63 0.00 0.17 0.27
5 25 2.47 38.97 51.48 0.01 0.22 0.29
6 25 1.81 35.71 44.69 0.01 0.17 0.29
7 25 1.05 34.61 42.97 0.01 0.19 0.32
8 25 0.51 33.93 41.97 0.01 0.17 0.31
9 25 0.82 33.05 40.14 0.01 0.27 0.32
10 25 0.10 32.17 38.93 0.01 0.32 0.42
γp
γy
γf
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Figure 9. Changes of internal friction angle and expansion angle for relatively loose sand with
low head load. (Dr=25%, ϭn=18.13KN/m3)
Figure 10. Changes of internal friction angle and expansion angle for relatively loose sand
with medium overhead. (Dr=25%, ϭn=146.82KN/m3)
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Figure 11. Changes of internal friction angle and expansion angle for relatively loose sand
with high overhead. (Dr=25%, ϭn=232.61KN/m3)
3.7. CALIBRATION OF THE BEHAVIORAL MODEL
To calibrate the presented behavioral model, the direct cutting test was simulated in
Abaqus software, and the obtained results were compared with the data obtained
from the tests. In the simulation of the direct shear test, due to the lack of change in
the stress level during a trial, the elastic modulus is defined as a constant value during
the analysis, Still, in the simulation of different tests, the elastic modulus is changed
based on the normal stress level. The finite element model and plastic strain gauge
are shown in Fig. 12. As an example, Fig. 13 shows the graphs of shear stress and
horizontal displacement as well as the vertical displacement and horizontal
displacement in experimental tests and numerical modeling under different normal
stresses for a relative density of 95%. The comparison between simulated and
experimental curves shows their agreement. Therefore, the strain-dependent
hardening-softening model should be used in simulating the behavior of granular soils
in conditions where deformations are similar to direct shear tests.
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Figure 12. Meshing and plastic strain counters in the numerical model of direct cutting
Figure 13. Comparison between shear stress-horizontal displacement and vertical
displacement-horizontal displacement in the experiment and numerical results for dense sand
(Dr=95%)
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3.8. NUMERICAL MODEL OF ARC PHENOMENON AND
MODELING ASSUMPTIONS
Numerical modeling of the arcing phenomenon has been done in Abaqus software,
2012 edition. The numerical model was created in the graphical environment of the
software in such a way that it accurately evokes the laboratory conditions. This
numerical model includes three separate parts of the soil mass, tank, and valve, which
are mounted together in the Assembly section of the software. The lateral boundaries
include supports with degrees of freedom in the vertical direction, and the bottom part
of the soil rests on the defined tank. The interaction between different elements is
modeled according to actual conditions. In this way, the interaction between the tank
and the soil is frictional for sliding movement and hard contact for motion
perpendicular to the plane. The coefficient of sliding friction between the tank and the
soil is considered equal to 0.3. This number is obtained from the tensile test of the
painted metal plate buried in the test sand. A measurement of the rigid contact type
leads to the soil elements not passing through the tank elements. Such an interaction
is also considered for the soil and valve interaction. To remove the stress
concentration in the sharp corner after lowering the valve, the sharpness of this part of
the tank has been removed with the help of bending with a minimal radius.
Among the basic cases in modeling the phenomenon of arcing of elements,
especially in the areas around the valve. The large size of the components leads to
the occurrence of errors in the analysis and the failure of the flow phenomenon during
the event of the arcing phenomenon, at the same time, the smallness of the elements
also leads to the bulkiness of the analysis and the excessive increase of the analysis
time. Therefore, choosing the right size for the details requires trial and error and
comparing the results. Of course, the meshing was done according to the capabilities
of the software in such a way that the size of the elements around the valve and
places with higher strain is smaller and gradually towards the boundaries of the
model, with a lower strain rate, the size of the elements increases.
The analysis has been done in two stages, which can be defined in the Step
section of the software:
1. The stage of establishing the initial stress is defined as Geostatic in the model.
2. The opening stage of the valve and the beginning of the arcing phenomenon.
In the Abaqus software, it is impossible to define the modulus of elasticity and
variable friction angle directly in the graphic part. Of course, the elasticity modulus
depending on the stress level, can be determined through the Edit Keywords module.
To apply frictional hardening and softening depending on the plastic strain, a program
was written in FORTRAN space. This program is compiled through the Visual Studio
interface and used in ABAQUS software. In this way, the variable internal friction
angle and dependent on plastic strain can be defined in the software. Figs. 14 and
15, respectively show the numerical model along with the boundary conditions and the
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valve along with the reservoir defined with the curvature applied to reduce the stress
concentration.
Figure 14. Numerical model and boundary conditions
Figure 15. Numerical model and valve modeling details
3.9. ELIMINATING THE EFFECT OF THE ARC IN THE PLACE
OF STRESS GAUGES
The research conducted on strain gauges and sensors installed in dams and roads
shows an arc effect in their readings, which should be considered in the data used.
Islam and his colleagues, in 2014, with finite element analysis and modeling of the
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strain gauge installed on the road, estimated the amount of arc effect in the strain
gauge reading from 2.21 to 2.72 percent. In this research, by using two strain gauges
installed in the middle part of the tank and comparing the initial stresses shown by the
strain gauges with the calculated stress, the value of the arcing effect was obtained in
about 2.5 to 2.8 percent, and the reading data has been modified accordingly.
4. EXPERIMENTS CONDUCTED ON THE PHYSICAL
MODEL
Table 5 summarizes the conditions governing the experiments and the results
obtained. In the first column of the table (Test No.), the number of the test is inserted
in the order it was performed. In the second column (Trapdoor width), the width of the
valve used in the experiment is shown. In the third and fourth columns (γ, γ
d), the
amount of wet and dry density of sand in terms of kilonewtons per cubic meter is
included in each test. In the fifth column, the relative density (Dr), and the sixth
column (σ
0), the value of the initial stress applied to the valve without any
displacement to the valve or strain in the sand, i.e., γh where γ
is the density of the
sand and h is the height of the sand to its surface. In the eighth column (σ
min), the
value of the minimum stress applied to the valve, which was read on the strain gauge
during the test, is entered. In the ninth column (Surcharge Pressure), the amount of
overhead applied in different tests is presented.
In the following, the results obtained from the experiments are presented as
graphs. In these diagrams, the stress applied to the valve, is measured in
kilonewtons, is read from the strain gauge installed under the valves and plotted
against the displacement of the valve. Because the graph in the direction of Δ
H was
very long and it was not possible to reach a conceptual and understandable diagram,
therefore a logarithmic scale was used in the direction of ΔH. Also, in the direction of
Δ
H, the numbers read from the chart should be divided by 100 to get the actual
displacement of the valve in millimeters.
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Table 5. Specifications governing each experiment and the results obtained
4.1. THE RESULTS OF THE DATA OF THE STRESS GAUGE
INSTALLED ON THE VALVE AND IN THE MIDDLE OF IT
(S1).
In Fig. 16, the results of the tests performed in three relative densities of 25%, 68%,
and 92% for the valve with a width of 10 cm are presented. As can be seen, after the
start of valve movement, the stress immediately drops to about 4% of the initial
stress. This minimum stress occurs when the valve moves about 0.06 mm. Then, as
the valve continues to move, the tension is constant, and when the valve moves about
1 mm, the tension increases up to 6% of the initial tension and remains almost
constant. With the change of the relative density, this stress ratio has changed, so the
stress ratio is the highest in the relative density of 25% and the lowest in the relative
density of 92%. In this case, it can be said that a stable arc is formed in all tests.
Test
NO.
Trapdoor
Diameter
γ
(kN/m3)
γd
(kN/m3)
Dr
(%)
σ0
(kpa)
Surcharge
Pressure
(kN/m
2
)
1 10 13.80 13.16 25 8.23 ---
2 10 15.60 12.75 68 7.95 ---
3 10 16.90 14.21 92 8.88 ---
4 20 13.80 14.88 25 8.01 ---
5 20 15.60 14.88 68 9.3 ---
6 20 16.90 12.75 92 8.01 ---
7 30 13.80 12.83 25 8.01 --
8 30 15.60 14.88 68 9.35 ---
9 30 16.90 14.97 92 9.35 ---
10 35 13.80 12.7 25 9.35 ---
11 35 15.60 13.68 68 8.54 ---
12 35 16.90 13.68 92 8.54 ---
13 35 15.60 14.02 68 8.76 128
14 35 15.60 14.09 68 8.8 212
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Figure16. Stress diagram in the middle of the 10 cm valve against the change in valve
location
In all the graphs, the stress value has decreased sharply at low strains, and after
reaching a minimum value, it has taken an upward trend. The minimum amount of
stress applied to the valve occurred in the displacement of 0.02 to 0.3 mm of the
valve.
As it is clear from these diagrams, four distinct phases can be distinguished in the
arc phenomenon:
The first phase: This phase starts immediately after the start of the valve drop, so
that the tension applied to the valve is immediately and strongly reduced and reaches
the minimum value. In this phase, the soil behaves elastically, and the deformations
are small, while the stress changes are huge. At the beginning and before moving the
valve, the shear stress values in the sand mass and around the valve are zero, and
the principal stresses are vertical and horizontal. As the valve begins to move around
the arch area, the shear stress starts to increase, and as a result, the principal
stresses deviate from the vertical and horizontal state and rotate. In fact, in this
phase, in arc development, the principal stress plane turns by creating shear
stresses. So that the value of the principal stresses is reduced to the minimum and
the principal stresses are increased to the maximum. In other words, the diameter of
Mohr's circle increases in the arched area. This process of changes in the plane of
principal stresses continues until the formation of a stable arch.
The Second phase: This phase starts after the minimum stress point. This phase
occurs in a wide range of valve displacement and the range of development of plastic
strains. In the second phase, plastic strains and, finally, warping start from the side of
the valve and proceed in the direction perpendicular to the two ends of the valve
towards the surface. Convergence or divergence of this development of plastic
strains and rupture depends on the density of the soil and the width of the valve,
which determines the stable or unstable arch. In this phase, the phenomenon of flow
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occurs in the soil mass, so that despite high strains, there is no significant change in
the applied stress on the valve.
The third phase: This phase begins with an increase in the applied tension on the
valve. At this stage, the separation and formation of a stable arc take place. If there
is no stable arc, the rising trend of tension continues.
The fourth phase: In this phase, the total weight of the stable arc is applied to the
tension gauge, and the applied tension on the valve remains almost constant. In case
of an unstable arc in this phase, the tension will be steady and increasing.
These separate phases are presented in Fig. 17 .
Figure 17. Four distinct phases of stress changes
4.2. COMPARISON OF THE LABORATORY RESULTS WITH
THE NUMERICAL MODEL, TAKING INTO ACCOUNT
THE DATA OF THE STRESS GAUGE INSTALLED ON
THE VALVE AND IN THE MIDDLE OF IT.
The S16 strain gauge was installed in the middle of the valve and recorded the
changes in tension during the movement of the valve. To compare the results
obtained from the physical model and numerical modeling, the results have been
drawn in a single diagram for the stress applied in the middle of the valve. In the
numerical model, the element, such as the location of the stress gauge S16, is
selected, and its results are presented. Fig. 18 shows an example of graphs obtained
from laboratory data and numerical analysis results for a valve with a width of 10 cm
and a relative density of 25%. As can be seen, the trend of changes in stress ratio
with valve drop in both graphs obtained from experiments and numerical analysis is
almost the same. Of course, there is a small difference of about 5% to 10% between
the values of the stress ratios, which is caused by the error of the experiment and
numerical analysis. It should be noted that with the increase in relative density, the
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difference between the results of experiments and numerical analysis has decreased.
This is due to the reduction of the error caused by the uniformity of the sand, as well
as the more excellent compatibility of the parameters obtained from the direct cutting
tests and the arc mechanism. As can be seen, in the diagram related to the numerical
model, as well as the physical model, the four regions defined in the previous section
can be distinguished. Of course, to reduce the error in low relative densities, the size
of the elements was increased so that the dimension of the component is as close as
possible to the width of the cutting area. However, this reduced the slope of the
elastic region and the rate of displacements and strains, especially in the part of the
flow phenomenon. In other valves, the agreement between the results of the
experiments in the physical model and the numerical model is evident.
Figure 18. Comparison of the results of the physical model and the numerical model for a
valve with a width of 10 cm and a relative density of 25%.
According to these results, it can be seen that by moving the valve downwards, the
stress applied on the strain gauge drops drastically, and in the elastic range, this
stress drop is very severe. As the valve moves, this drop continues with a gentler
slope until it reaches a minimum value and an insignificant number near zero. Of
course, this state occurs in a stable arc, and when an unstable arc occurs, the amount
of increase in the stress applied to the strain gauge can be seen within the range of
10 mm displacements of the valve. Also, the agreement of the numerical modeling
results with the results obtained from the physical model increases with increasing
density.
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4.3. THE RESULTS OBTAINED FROM THE PIV METHOD
CONCERNING THE MEASUREMENT OF STRAINS
DURING THE OCCURRENCE OF THE ARCING
PHENOMENON
The PIV method has been used to measure the strains created during the arcing
phenomenon. Fig. 19 shows an example of the results of the analysis of strains
during the event of the arch phenomenon for a displacement of 10 mm of the valve.
In this image, the upper and left figures are the color counters of the strains, and the
upper right figures are the alignment lines of the strains. Also, in the diagram shown,
the diagrams related to the strain changes corresponding to the location of stress
gauges S18 to S23 (valve sides) are presented. As can be seen, the shear strain
meters show a stable arc. Also, in these figures, the graphs related to the strain
changes at 5 points where the strain gauge is installed, against the displacement of
the valve, show that the relationship between the strain changes and the valve
displacement is almost linear. Also, most strains are related to the sides of the valve.
In the case of not creating a stable arch, the shape, and development of shear
counters are different. Fig. 20 shows the shear strain counters of the unstable arch
for a valve with a width of 30 cm and a relative density of 25%. The development of
shear strains in this form is initially in the converging direction and towards the
formation of a stable arch, still the continuation of the process of moving the valve, the
shear strains developed towards the sand surface and led to the instability of the arc.
While with the densification of the sand, the rate of formation of a stable arc is higher,
and it does not allow the total rupture of the sand mass and a stable arc is formed.
The development of shear strains first converged towards the formation of a stable
arc, but at the same time, the vertical component of the shear strains developed
towards the sand surface and led to the non-formation of a stable arc. Of course, with
the densification of the sand, the strains are reduced, and the tendency to converge
and form a stable arc is evident. Also, with the densification of sand, the growth rate
of shear strains decreases.
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Figure 19. The results of PIV method for a valve with a width of 10 cm and a relative density
of 25% .
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Figure 20. The results of the PIV method for a valve with a width of 30 cm and a relative
density of 25%.
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4.4. THE RESULTS OF STRAIN ANALYSIS
Based on the results of experiments and numerical modeling, after the occurrence
of the arch phenomenon, four separate areas can be distinguished in terms of the
behavior of the sand mass.
The first area: In this area, the soil mass behaves elastically, and the sand mass
moves in general, and the particles do not have any relative movement to each other.
The second area: In the second area, a shear band is formed, and during the
expansion of the arch phenomenon, it develops towards the crown of the arch, and
the sand shows softening behavior.
The third area: the stresses in this area increased with the development of the arch
phenomenon, and the sand mass shows hardening behavior. This area acts as a
column to transfer stress to the foundation. Of course, this area is moved to the sides
with the movement of the valve, and its position depends on the rate of motion of the
valve.
The fourth area: This part has not been affected by the arching phenomenon, and
the stresses and strains in this area remain almost constant during the arching
phenomenon.
These four areas are shown in Fig. 21 .
Figure 21. Four separate areas in terms of the behavior of the sand mass
5. THE NATURE OF THE FLOW PHENOMENON IN THE
ARCING PHENOMENON
As mentioned, when the valve starts to move downward, a sharp and rapid drop in
the amount of stress on the valve is immediately observed, which is due to the
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transfer of stresses to the fixed sides in the range of elastic strain. Then, by
minimizing the stress on the valve by moving it more and applying more strains to the
soil, the amount of stress on the valve does not change, and the stress remains
constant in a wide range of plastic strains, especially at the edges of the valve. This
phenomenon is called flow, which isn't associated according to the definition of the
expansion angle dependent on plastic strain in numerical modeling and the
acceptable agreement of the results of the numerical model with the results of the
experiments. Also, the phenomenon of flow and application of large strains, which is
the nature of the arch phenomenon in the sand, is well modeled in the finite element
method, in Fig. 22, these large deformations in the elements can be seen well.
Figure 22. Large deformations applied to the elements in the finite element method
6. TENSION SPACE IN THE SELECTED BEHAVIORAL
MODEL
The selected behavioral model for numerical analysis can be divided into two
separate parts, elastic and plastic. In the flexible section, with the increase in the
stress level, the modulus of elasticity increases until the sand reaches the yield point.
When the sand comes to the yield point, the plastic stage begins, and the behavior
leaves the elastic region. In the plastic area, depending on the plastic strain rate,
there are two behaviors hardening and softening. According to the mentioned zoning,
depending on the position considered for it, in each of the areas, the sand element
can have a hardening or softening behavior or both behaviors. In the numerical
model, according to the definition of the plane strain state, the stress path at different
points is obtained using the following relations.
Plain Strain;
ε2= 0
p′ =
σ′
1
+σ′
2
+σ′
3
3
and p=
σ
1
+σ
2
+σ
3
3
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As stated in the previous sections, four separate areas can be distinguished in the
arcing phenomenon. These areas are shown in Fig. 17. To investigate the approach
to stress in each area, a point has been selected, and the approach to stress in these
points has been discussed. For example, numerical modeling results have been
chosen for a relative density of 25% and a valve width of 10 cm.
6.1. THE STRESS SPACE OF AREA 1 (ELASTIC AREA)
In Fig. 23, the diagram related to the path of stress in the space of P and q is
presented at a point located in zone 1 in a valve with a width of 10 cm and a relative
density of 25%. Also, in Fig. 23, specific volume changes in P and ν space are
shown.
According to Fig. 22, P and q decrease rapidly at the beginning of the experiment.
After reaching a minimum value, the reduction rate of q decreases, and the sand
enters the flow phenomenon. In the range of occurrence of the flow phenomenon, the
value of q has decreased much less, while the rate of decrease of p is very high. As
the valve continues to go down, the average and shear stresses start to decrease
again, this process continues until the end of the test, until it finally reaches a constant
value in the shear stress. This is although the specific volume has not changed much
during the trial and is almost constant. In the area of a mass of sand, it is entirely and
without strain changed by the displacement of the valve, and the average and shear
stresses are reduced at different rates depending on the amount of displacement of
the valve. It can be said that Region 1 remains in the elastic range due to the meager
amount of strains generated during the arch phenomenon, and only changes in the
modulus of elasticity depending on the stress level are effective in this range.
In the graphs related to Figs. 23 and 24, the direction of the arrow represents the
direction of changes in stresses and specific volume during the test.
p′ =p−u
q
=q′ =
1
2
[(σ′
1−σ′
3)+(σ′
1−σ′
3)+(σ′
1−σ′
3)]1/
2
εp=ε1+ε3
ε
q=
2
3(
ε2
1+ε2
3−ε1ε3
)1/2
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Figure 23. Diagram of stress path in p and q space for the point located in area 1
Figure 24. Diagram of stress path in p and ν space for the point located in area 1
6.2. THE STRESS SPACE OF ZONE 2 (SOFTENING ZONE)
In Fig. 25, the diagram related to the path of stress in the space of P and q is
presented at a point located in the expansion of the plastic points and the place of
rupture (region 2) in the valve with a width of 10 cm and a relative density of 25%.
Also, in Fig. 25, specific volume changes in P and ν space are presented.
According to Fig. 25, P and q decrease rapidly at the beginning of the experiment.
After reaching a specific value, the reduction rate of q is reduced due to the
phenomenon of flow, the rate of extreme changes is related to the average stress. In
the area where the flow phenomenon occurs, the value of q has decreased much
less, while the rate of decrease of p is very high. As the valve goes down, the
stresses continue in a straight line and on the critical line (CSL) until the end of the
test. In this area, according to the diagram presented in the plastic strain range, the
sand shows a softening behavior on the critical state line with an almost constant
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slope that can be calculated from the internal friction angle of the sand in the residual
state, the path continues itself. In Fig. 26, with the beginning of the test and with the
reduction of the average stress, the specific volume increases upon reaching the
critical line, and the rate of growth in the particular volume increases until it reaches
the maximum value. In fact, in this diagram, in the area where the flow phenomenon
occurs, the specific volume change rate is much lower than in other areas.
In the graphs related to Figs. 25 and 26, the direction of the arrow represents the
direction of changes in stresses and specific volume during the test.
Figure 25. Diagram of stress path in p and q space for the point located in area 2
Figure 26. Diagram of stress path in p and ν space for the point located in area 2
6.3. THE STRESS SPACE OF THE AREA NEAR THE VALVE
(HARDENING AND SOFTENING EXPANSION AREA)
In Fig. 27, the diagram related to the path of stress in the space of P and q is
presented at a point located in the area of expansion of hardening and softening
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(region 3 - areas near the valve) for a valve with a width of 10 cm and a relative
density of 25%. Also, in Figure 28, specific volume changes in P and ν
space are
shown.
In Fig. 27, with the start of the test and moving the valve, and transferring the
stresses to the fixed parts of the model, the stresses P and q increase, this stress
transfer leads to the occurrence of hardening behavior in the sand, and this process
continues until reaching the ultimate point, after reaching the ultimate point, the
average and shear stresses decrease, and the sand shows a softening behavior. It
can be seen in Fig. 28 that the specific volume is almost constant at the beginning of
the test, and near the maximum point of the average stress, the specific volume
increases. After the ultimate point of moderate stress, the increase rate of specific
volume increases as it decreases. In the graphs related to Figs. 27 and 28, the
direction of the arrow represents the direction of changes in stresses and specific
volume during the test.
Figure 27. Diagram of stress path in p and q space for the point located in area 3 - areas near
the valve
Figure 28. Diagram of the path of stress in p and ν space for the point located in area 3 -
areas near the valve
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6.4. THE STRESS SPACE OF ZONE 3 (HARDENING ZONE) -
FARTHER FROM THE VALVE
In Fig. 29, the graph related to changes in the stress path in the environment p and
q for the point located in the hardening area (area 3 - areas far from the valve) is
presented. As you can see, as the valve goes down, the amount of average stress
and shear stress increases. With the increase of moderate and shear stresses, the
value of specific volume decreases (Fig. 30). Of course, this reduction in particular
volume is minimal. In this area, sand exhibits elastic and plastic hardening behavior.
So that in the flexible region, it is related to the hardening caused by the increase in
the modulus of elasticity, which is dependent on the stress level, in the plastic region,
it is related to the hardening caused by the increase in the internal friction angle,
which is dependent on the plastic strain. In the graphs related to Figs. 29 and 30, the
direction of the arrow represents the direction of changes in stresses and specific
volume during the test.
Figure 29. Diagram of stress path in p and q space for the point located in area 3 - areas far
from the valve
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Figure 30. Diagram of stress path in p and ν space for the point located in region 3 - regions
far from the valve
7. CONCLUSION
Based on the studies conducted in this research, the following can be stated as the
main results:
1. Four distinct phases can be distinguished in the arcing phenomenon:
•
The first phase: This phase starts immediately after the start of the
valve drop, so that the tension applied to the valve is immediately and
strongly reduced and reaches the minimum value. In this phase, the soil
behaves elastically, and the deformations are small, while the stress
changes are huge. At the beginning and before moving the valve, the
shear stress values in the sand mass and around the valve are zero, and
the principal stresses are vertical and horizontal. As the valve begins to
move around the arch area, the shear stress starts to increase, and as a
result, the principal stresses deviate from the vertical and horizontal state
and rotate. In fact, in this phase, in arc development, the principal stress
plane turns by creating shear stresses. So that the value of the principal
stresses is reduced to the minimum and the principal stresses are
increased to the maximum. In other words, the diameter of Mohr's circle
increases in the arched area. This process of changes in the plane of
principal stresses continues until the formation of a stable arch.
•
The second phase: This phase begins after the minimum stress
point. This phase occurs in a wide range of valve displacement and the
range of development of plastic strains. In the second phase, the plastic
strains, and finally, the rupture starts from the side of the valve and
proceeds in the direction perpendicular to the two ends of the valve
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towards the surface. Convergence or divergence of this development of
plastic strains and rupture depends on the density of the soil and the
width of the valve, which determines the stable or unstable arch. In this
phase, the phenomenon of flow occurs in the soil mass, so that despite
high strains, there is no significant change in the applied stress on the
valve.
•
The third phase: This phase begins with an increase in the applied
tension on the valve. At this stage, the separation and formation of a
stable arc take place. If there is no stable arc, the rising trend of tension
continues.
•
Fourth phase: In this phase, the total weight of the stable arc is
applied on the tension gauge, and the applied tension on the valve
remains almost constant. In case of an unstable arc in this phase, the
tension will be steady and increasing.
2.
Based on the results of experiments and numerical modeling, after the
occurrence of the arch phenomenon, four separate areas can be distinguished
in terms of the behavior of the sand mass :
•
The first area: In this area, the soil mass behaves elastically, so that
the mass moves as a whole and the particles do not have any relative
movement with each other, that is, the amount of strains in this area is
minimal.
•
The second area: In the second area, a shear band is formed, and
during the expansion of the arch phenomenon, it develops towards the
crown of the arch, and the sand shows softening behavior.
•
The third area: the stresses in this area increased with the
development of the arch phenomenon, and the sand mass shows
hardening behavior. This area acts as a column to transfer stress to the
foundation. Of course, this area is moved to the sides with the
displacement of the valve, and its position depends on the displacement
of the valve.
•
The fourth area: this part has not been affected by the arch
phenomenon, and the stresses and strains remain almost constant
during the occurrence of the arch phenomenon.
3.
Due to the wide application of the simple straight-cut test and the ease of
performing this test, the resistance parameters obtained from the simple
straight-cut test can be used with acceptable accuracy for the numerical
modeling of the arcing phenomenon.
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4.
The best behavioral model governing the phenomenon can be defined as
follows:
•
Elastic part: hardening depending on the stress level by defining the
change of the modulus of elasticity against the stress change.
•Plastic part: Hardening depending on the plastic strain up to the peak
point (peak) by defining the increase of the internal friction angle up to
the peak internal friction angle and softening depending on the plastic
strain up to the critical limit point by defining the decrease of the internal
friction angle up to the critical limit internal friction angle.
5.
In the arched area, the lateral pressure coefficient of the sand is not constant
and increases with the displacement of the valve. This rate of increase
depends on the relative density of sand and shows an increase up to 2 times
the initial value.
6.
According to the definition of the expansion angle dependent on the plastic
strain in numerical modeling and the acceptable agreement of the results of the
numerical model with the results of the experiments, the flow in the arc
phenomenon has no associated nature.
7. In examining the path of stress during the occurrence of the arch phenomenon,
three separate areas can be identified in terms of changes in average and
shear stress and specific volume :
•
Area 1: rapid decrease of p and q, approximate constancy of specific
volume (elastic).
•
Area 2: reduction of p and q, flow phenomenon area, critical line
(CSL), an increase of specific volume (softening area).
•Area 3 and in the areas near the valve: increase and then decrease p
and q, hard behavior and then softening, stability and then increase of
specific volume (expansion area of hard and softening).
•
Area 3 and in areas further from the valve: increasing p and q,
reducing the specific volume to a minimal amount (area of hardening) .
REFERENCES
(1) Terzaghi, K. (1943). Theoretical Soil Mechanics. John Wiley and Sons. Inc. New
York, U.S.A.
(2)
Janssen, H. A. (1895). "Versuche fiber Getreidedruck in Silozellen". Zeitschrift
Verein Deutscher Ingenieure, Bd XXXIX, 1045-1049.
(3) Finn, W. D. L. (1963). "Boundary Value Problems of Soil Mechanics". Journal of
the Soil Mechanics and Foundation Division, ASCE, Vol. 89, No. SM5, 39-72.
https://doi.org/10.17993/3ctic.2023.122.15-58
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed.43 | Iss.12 | N.2 April - June 2023
56
(4)
Amimoto, K., Okamoto, T., & Yamada, S. (1959). "Measurement of Earth
Pressure Acting on Earth Retaining Walls for-Subway-Construction". Tsuchito-
Kiso, JSSMFE, Vol.7, No.4, 21.
(5)
Atkinson, J. H., & Potts, D. M. (1977). Stability of a shallow circular tunnel in
cohesionless soil. Geotechnique, 27(2), 203–215.
(6)
Jiang, M. J., & Yin, Z. Y. (2012). Analysis of stress redistribution in soil and earth
pressure on tunnel lining using the discrete element method. Tunnelling and
Underground Space Technology, 32(6), 251–259.
(7)
Guo, P., & Zhou, S. (2013). Arch in granular materials as a free surface problem.
International Journal of Numerical Analysis and Methods in Geomechanics,
37(9), 1048–1065.
(8)
Han, L., Ye, G. L., Chen, J. J., Xia, X. H., & Wang, J. H. (2017). Pressures on the
lining of a large shield tunnel with a small overburden: a case study. Tunnelling
and Underground Space Technology, 64, 1–9.
(9)
Franza, A., Marshall, A. M., & Zhou, B. (2018). Greenfield tunnelling in sands:
the effects of soil density and relative depth. Geotechnique, 1–11.
(10)
Chen, K. H., & Peng, F. L. (2018). An improved method to calculate the vertical
earth pressure for deep shield tunnel in Shanghai soil layers. Tunnelling and
Underground Space Technology, 75, 43–66.
(11)
Jin, D. L., Zhang, Z. Y., & Yuan, D. Y. (2021). Effect of dynamic cutterhead on
face stability in EPB shield tunneling. Tunnelling and Underground Space
Technology, 110(1), 103827.
(12)
Zheng, H. B., Li, P. F., & Ma, G. W. (2021). Stability analysis of the middle soil
pillar for asymmetric parallel tunnels by using model testing and numerical
simulations. Tunnelling and Underground Space Technology, 108, 103686.
(13)
Stein, D., Mollers, K., & Bielecki, R. (1989). Microtunnelling. Berlin, Germany:
Ernest & Sohn, pp. 237–243.
(14)
Handy, R. L. (1985). The arch in soil arching. Journal of Geotechnical
Engineering, 111(3), 302–318.
(15)
Chen, R. P., Tang, L. J., Yin, X. S., Chen, Y. M., & Bian, X. C. (2015). An
improved 3D wedge-prism model for the face stability analysis of the shield
tunnel in cohesionless soils. Acta Geotechnica, 10(5), 683–692.
(16)
Zhang, H. F., Zhang, P., Zhou, W., Dong, S., & Ma, B. S. (2016). A new model to
predict soil pressure acting on deep burial jacked pipes. Tunnelling and
Underground Space Technology, 60, 183–196.
(17)
Jacobsz, S. W. (2016). Trapdoor experiments studying cavity propagation. In:
Proceedings of the first Southern African Geotechnical Conference, pp. 159–
165.
(18)
Song, J. H., Chen, K. F., Li, P., Zhang, Y. H., & Sun, C. M. (2018). Soil arching in
unsaturated soil with different water table. Granular Matter, 20(4), 78.
(19)
Chen, R. P., Yin, X. S., Tang, L. J., & Chen, Y. M. (2018). Centrifugal model tests
on face failure of earth pressure balance shield induced by steady-state seepage
in saturated sandy silt ground. Tunnelling and Underground Space Technology,
81, 315–325.
https://doi.org/10.17993/3ctic.2023.122.15-58
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed.43 | Iss.12 | N.2 April - June 2023
57
(20)
Lin, X. T., Chen, R. P., Wu, H. N., & Cheng, H. Z. (2019). Three-dimensional
stress-transfer mechanism and soil arching evolution induced by shield tunneling
in sandy ground. Tunnelling and Underground Space Technology, 93.
(21)
Chen, R. P., Lin, X. T., & Wu, H. N. (2019). An analytical model to predict the limit
support pressure on a deep shield tunnel face. Computers and Geotechnics,
115, 103174.
(22)
Costa, Y. D., Zornberg, J. G., Bueno, B. S., & Costa, C. L. (2009). Failure
mechanisms in sand over a deep active trapdoor. Journal of Geotechnical and
Geoenvironmental Engineering, 135(11), 1741–1753.
(23)
Wu, J., Liao, S. M., & Liu, M. B. (2019). An analytical solution for the arching
effect induced by ground loss of tunneling in sand. Tunnelling and Underground
Space Technology, 83, 175–186.
(24)
Adrian, R. J. (1991). Particle imaging techniques for experimental fluid
mechanics. Annual Review of Fluid Mechanics, 23, 261–304.
(25)
White, D. J., Take, W. A., & Bolton, M. D. (2003). Soil deformation measurement
using particle image velocimetry (PIV) and photogrammetry. Geotechnique,
53(7), 619–631.
(26) ABAQUS 2012 User’s Documentation, Hibbit, Karlsson and Sorensen, Inc.
(27)
Shibuya, S., Mitachi, T., and Tamate, S. (1997). Interpretation of direct shear box
testing of sands as quasi-simple shear. Geotechnique, 47(4), 769–790.
(28)
Davis R. O., & Selvadurai, A. P. S. (2002). Plasticity and Geomechanics.
Cambridge University Press
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3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
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