Tunnelling and Underground Space Technology 112 (2021) 103918
Contents lists available at ScienceDirect
Tunnelling and Underground Space Technology
incorporating Trenchless Technology Research
journal homepage: www.elsevier.com/locate/tust
Experimental investigation and numerical analysis of mechanical
behaviour of railway station reconstruction over twin tunnels
Feiyue Yan a, Wenge Qiu a, Yuchao Zheng a, *, Shuhua Jiang a, Hui Hu a, Gang-gang Gao a,
Yunjian Cheng b
a
b
Key Laboratory of Transportation Tunnel Engineering, Ministry of Education, School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, PR China
School of Civil Engineering and Geomatics, Southwest Petroleum University, Chengdu 610500, PR China
A R T I C L E I N F O
A B S T R A C T
Keyword:
Model test
Numerical modelling
Mechanical behavior
Railway station reconstruction
Twin tunnels
Railway station reconstruction adjacent to existing twin tunnels has been a challenging problem, as the former
can impose heavy loads on the latter. As a result, the existing twin tunnels may sustain damage. To address this
problem, in this study, a design plan for a railway station reconstruction over twin tunnels was improved by
optimising the cross-section form and installing a buffer layer (XPS). Model tests were employed to investigate
the mechanical behaviour of a railway station reconstruction over twin tunnels and the earth pressure in the
surrounding rock, and an assessment of the safety of the existing tunnel lining was also performed. The numerical
modelling was validated by an experiment based on the comparison of reinforcement strain, concrete stress, and
failure pattern, particularly, the tensile-shear failure. In addition, the parametric study considering the different
stiffness of the buffer layer was presented. These results comprehensively show the optimised design scheme not
only addressed the very small clearance ranging from 1.46 m to 3.06 m but also reduced the construction cost
including the decrease of concrete. As the stiffness of the buffer layer increases, the tunnel lining bears larger
forces. The relatively low 10 MPa elastic modulus of buffer layer was recommended in this project. The valuable
ideas, in this study, can potentially be generalized to similar projects for reducing the disturbance from the
adjacent construction to the existing tunnels.
1. Introduction
As many subway lines have been constructed in China’s urban areas,
subway tunnels have become critical structures that may be influenced
by other infrastructure construction projects. To date, a large number of
studies have focused on the mechanical behaviour of existing structures
during tunnelling. Ding et al. (2019); Moosazadeh et al. (2019); Mroueh
and Shahrour (2003); Zhang et al. (2013) analysed the interaction be­
tween existing buildings and tunnels, and discussed methods to mitigate
damage to the former during tunnelling. In addition, soil movement
caused by tunnel excavation can adversely affect or damage nearby
underground pipelines. Zhang et al. (2012) proposed a continuous
elastic analysis method in the finite difference form to simulate the
tunnel–soil–pipeline interaction. Xia et al. (2019) performed a safety
assessment of pipelines affected by blasting vibration during excavation,
based on fracture mechanics theory under different water pressure. Sun
et al. (2019) analysed the instability problem of slopes due to tunnel
excavation and blasting, and proposed monitoring methods and pro­
tection measures. Zhang et al. (2018) analysed the mechanical behav­
iour of the Chongwenmen subway station on Beijing Subway Line 5,
which was excavated beneath the existing Line 2 with a close clearance
of 1.98 m. The settlement of the latter was found to reach 31.3 mm
during construction, and compensation grouting was adopted accord­
ingly. Liu et al. (2020) investigated the influence of a large–scale base­
ment excavation on existing tunnels by numerical modelling and a
long–term field monitoring for four years. Moreover, through the anal­
ysis of numerical results and monitoring data, the treatment measure of
micro–disturbance grouting was performed.
However, investigatigations into the influence of new buildings on
existing tunnels are quite limited. F. C. Schroeder et al. (2004) studied,
under group pile loading, the maximum allowable tunnel deformation in
terms of both tunnel distortion and global movement, and provided
design guidelines. Lueprasert et al. (2017) presented a new assessment
method for the pile-soil-tunnel interaction mechanism when adjacent
* Corresponding author.
E-mail address: Yczh@swjtu.edu.cn (Y. Zheng).
https://doi.org/10.1016/j.tust.2021.103918
Received 2 November 2020; Received in revised form 1 March 2021; Accepted 4 March 2021
Available online 22 March 2021
0886-7798/© 2021 Elsevier Ltd. All rights reserved.
F. Yan et al.
Tunnelling and Underground Space Technology incorporating Trenchless Technology Research 112 (2021) 103918
Fig. 1. Geological condition profile.
traffic and thus it must be torn down and reconstructed. The complexity
of this project is the new railway station in very close proximity to the
existing twin tunnels in comparison with previous situation–the depth of
tunnels before reconstruction ranges from 12.8 m to 16.5 m. The
detailed space relation can be seen hereinafter. The geological condition
profile is shown in Fig. 1. It can be seen from this picture that the tunnels
are absolutely embedded into the sandy mudstone rocks. The integrity of
these rocks is very good and rock mass without geological tectonic ac­
tivities can be considered to be homogenous and isotropic. In this case,
the surface layers including plain fill, sandstone, and part of sandy
mudstone would be removed during the railway station reconstruction.
For simplifying these tests, the sandstone interlayer can be omitted. As a
result, in the experiment only sandy mudstone was presented. Based on
the design codes for railway tunnels in China, the surrounding rock mass
in situ was categorised as Grade IV. The in-situ mechanical properties of
the surrounding rock mass are summarised in Table. 4. The in-situ
stresses in the gravity stress field are approximately from 0.27 MPa to
0.35 MPa (12.8 m–16.5 m depth). When the previous station was torn
down and the new one would be reconstructed, the clearance between
the latter and the existing tunnels was reduced to less than 2 m, as shown
in Fig. 2(b). As a result, the in-situ stresses decrease more significantly
than before and thus the most of the loads acting on the existing tunnels
consist of dead and live ones derived from the new station. Moreover,
based on the geological survey, underground water is stored very little
and mainly consists of fissure water. Therefore, its influence on the
stability of this project can be neglected.
The plan view of the existing twin subway tunnels, Line 10 under­
crossing the reconstructed Chongqingbei railway station is shown in
Fig. 2(b). The special relation between the two structures will be clari­
fied clearly from the schematic three dimension diagram, Fig. 2(e). The
existing tunnels are oriented in south–north direction, as shown in Fig. 2
(a). The axial direction of Line 10 is orthogonal, with 11 station girders
of three different section types, (L1, L2, and L3), which have the same
height of 3.6 m and different widths of 2.4 m, 3.6 m, and 4.2 m,
respectively. L2 and L3 bear greater loads than L1. To guarantee the
security of existing tunnels and to bear the weight of the corridor and
station buildings as well as the live load of trains shown in Fig. 2(b), the
original girder–pile composite structure forms shown in Fig. 2(c) were
proposed. However, a drawback of this scheme is that the clearance
between station girders and the existing tunnels is extremely small,
approximately 1.46 m, as shown in Fig. 2(c). Another drawback is that
this scheme requires the consumption of a large quantity of building
pile loads are exerted on an existing tunnel. In particular, few of in­
vestigations on the interaction between railway station reconstruction
and existing twin tunnels in the vicinity have been reported. For a
railway station adjacent to existing twin subway tunnels, the deforma­
tion and stress of the tunnel lining will be disturbed inevitably, partly
due to load variation. Therefore, this issue is worth investigating.
Model tests have been extensively conducted to study engineering
problems (Chapman et al., 2007; Fang et al., 2016; Huang et al., 2013;
Idinger et al., 2011; Li et al., 2017; Liu et al., 2019; Weishen et al., 2011;
Xu et al., 2017; Zhang et al., 2016; Zhang et al., 2019; Zhu et al., 2010).
In this study, model tests were conducted to investigate the stress and
deformation of railway station girders and the existing tunnel lining, as
well as earth pressure in different test cases. The ultimate bearing ca­
pacity of the girders was analysed and a safety assessment of the tunnel
lining was performed. Based on analyzing the deformation and stress of
structures within different design plans, the optimal scheme was ob­
tained. Although the model test has been widely applied, it is circum­
scribed for parametric study. Undoubtedly, based on finite element
software ABAQUS, numerical modelling is a convenient method to
address this problem (Murthy et al., 2018; Wang et al., 2017). Compared
with other softwares, ABAQUS not only can more conveniently simulate
refined models such as reinforcements but also can more effectively
address contact problems. In this study, we developed a simple numer­
ical model validated by an experiment to perform a parametric study.
These tests aim to investigate the mechanical behaviour of railway
station reconstruction, to evaluate the safety of the existing twin tunnels
by considering structure–rock interaction, and to find out that which
stiffness of the buffer layer is much better. The main works in this study
are divided into three parts: (i) investigating the mechanical behaviour
of the railway station structures, (ii) calculating the safety factor of the
existing twin tunnels, (iii) discussing the functions of the buffer layer,
and performing parametric study based on its stiffness variation. The
study is organised as follows: Section 2 introduces the project back­
ground; Section 3 describes 3 experiment configuration, including
testing frame and loading method, material preparation, test cases, and
measurement; Section 4 presents and discusses the test results; Section 5
validates the numerical modelling and performs parameters study;
Section 6 concludes the analysis of experimental and numerical results.
2. Project overview
The previous railway station did not satisfy the increasing amount of
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Fig. 2. Layout of the project: (a) Plan view, (b) longitudinal section, (c) cross–section, (d) cross–section after optimization, (e) schematic three dimension diagram.
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Fig. 2. (continued).
materials. Hence, the design needs to be optimised. The improved
scheme, shown in Fig. 2(d), increases the clearance between the station
girders and the existing twin tunnels from 1.46 m to 3.06 m and fully
utilises mechanical merit of variable cross-section girder to reduce the
section area and thus the construction cost. The volume of concrete
decreased by approxiametely 2700 m3. In addition, a buffer layer con­
sisting of a rigid extruded polystyrene (XPS), foam board, is installed at
the bottom of the station girders to translate loads from the super con­
struction to the two sides of the tunnels. Thus, the buffer layer can
effectively reudce the force exerted on the tunnel crown.
Table 1
The scale of the experimental modes.
Parameters
Definition
Reduced scale
Length
Cl = Lp /Lm
20
Cσ = σ p / σ m
20
Displacement
Stress
Strain
Gravity
Elastic modulus
Poisson’s ratio
3. Experiment configuration
Inertial friction angle
Cohesion
3.1. Law of similarity
Time
The law of similarity provides a theorical connection between scale
models and prototype models. Rocha (1957) comprehensively described
a scale model for civil engineering problems in a 1 g gravity field.
Moncarz and Krawinkler (1981) noted that as long as the fundamental
features of a prototype were simulated, the difference between the
prototype and model would be acceptable. Fang et al. (2016), Hobbs
(1968), Hobbs (1969) determined that the basic physical parameters
include gravity (γ), elastic modulus (E), cohesion (c), the internal friction
Cδ = δp /δm
20
Cε = εp /εm
1
Cγ = γp /γm
1
Cμ = μp /μm
1
Cc = cp /cm
20
√̅̅̅̅̅̅
20
CE = Ep /Em
20
Cφ = φp /φm
1
CT = Tp /Tm
angle (φ), Poisson’s ratio (μ), geometric size (L), stress (σ), strain (ε),
displacement (δ), and time (T). When scale model tests are conducted, it
is important to determine the dimension scale, Cl . In this study, grav­
ity,γ, and the geometry, L, are assumed to be elemental physical quan­
tities, and the reduced scale of dimension Cl in the model tests was set as
20. Based on the uniformity and homogeneity of the dimensions, the
equation is used to determine these physical parameters, as follows:
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Fig. 3. Test frame (a) test box and acquisition system, (b) internal structure in the test box.
Fig. 4. Test process involving detailed experiment steps.
f (γ, E, c, φ, μ, L, σ, ε, δ, T) = 0
3.2. Testing frame and loading method
(1)
Based on the Buckingham pi theorem for dimensional analysis, Eq.
(1) can be rewritten as follows:
F(π1 , π2 , π3 , π4 , π5 , π6 , π7 , π8 ) = 0
(2)
π 1 = E/γL, π2 = c/γL, π3 = φ, π 4 = μ
π5 = σ /γL, π6 = ε, π7 = δ/L, π8 = L/T 2
(3)
Although the entire prototype construction can be scaled, the
resulting scale model will also be very large and expensive. Hence, the
local structure of the L2 girder in Fig. 2(b), which is located at the B axis
with maximal applied loads, was chosen. Due to the approximate sym­
metry of the cross–section in the dimension and applied loads, as shown
in Fig. 2(d), the right half was reserved because the east tunnel section is
slightly larger than the other section. Using pre–embedded steel plates
welded to the test box, we chose the fixed boundary condition at the
both ends of girder. Fig. 3 shows the test frame and the acquisition
system. The box has the dimensions of 1.6 m (length), ×1.5 m (height),
× 0.5 m (thickness).
Taking structure–rock interaction into account, these tests were
more concerned with investigating the mechanical behaviour of railway
where π1 , π 2 , π3 , π4 , π5 , π6 , π 7 andπ 8 are all dimensionless quantities. Ac­
cording to the similarity theory, the detailed scale of the experimental
models can be deduced, as shown in Table. 1. Moreover, the loading
condition and materials preparation concerning the scale model tests is
presented in Sections 3.2 and 3.3, respectively.
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Fig. 5. Micro–concrete uniaxial compression tests (a) four groups of test specimens (b) experiments under a servo Press.
Table 2
Compressive strength and elastic modulus of micro–concrete specimens.
No.
B
C
E
G
Mixing
proportion
(water:
cement:
sand:
hydrated
lime)
Compressive
strength
(MPa)
1.3:1:6:1.4
1.5:1:6:1.4
1.7:1:6:1.4
1.9:1:6:1.4
3.02
2.68
2.17
1.85
Error
relative to
the
expected
value (%)
Elastic
modulus
(GPa)
75.07
55.36
44.50
12.50
3.13
2.99
2.34
1.91
Table 3
Splitting tensile strength of micro–concrete.
Error
relative to
the
expected
value (%)
85.79
77.68
40.00
15.07
Reference
Equation
Splitting tensile
strength (MPa)
Average value
(MPa)
(ACI Committee and
318, 1999)
(Carino and Lew,
1982)
fsp = 0.56fc0.5
0.76
––
0.53
––
(Carneiro and
Barcellos, 1953)
fsp =
0.34fc0.735
fsp =
0.272fc0.71
0.41
––
0.45
––
0.68
––
0.47
0.55
(Oluokun and
Burdette, 1991)
(Gardner et al., 1988)
station reconstruction and evaluating the safety of the existing twin
tunnels. As a result, the existing twin tunnels excavation and foundation
pit excavation were not considered in the study, because their influence
on the research objective can be negligible. Meanwhile, according to a
geological survey of the prototype, tectonic stress was not apparent due
to the following reasons: (1) the depth of the tunnels is considerably
shallow, especially after the reconstruction, (2) there is no geological
tectonic activity and the rock mass is significantly homogenous and
isotropic. As a result, only the self–weight stress field was considered.
The entire test process was divided into five steps, as shown in Fig. 4.
In Step 1, the test materials and moulds were prepared. In Step 2, the
girder models (L2), tunnel models, and test soil were manufactured.
Simultaneously, sensors for measuring stress and strain were installed in
the corresponding design positions. The entire model installation was
completed in Step 3, in which the test soil was strictly compacted, and
earth pressure cells were installed at the bottom of the girders. In Step 4,
after the stress field stabilised for approximately one day, a step load of
0.25 MPa per step was applied until the bearing capacity of the model
structure reached its limit. In the step loading process, the hold time was
2 min, and the data were collected until girder damage occurred.
(Raphael, 1984)
fsp =
0.294fc0.69
fsp =
0.47fc0.59
fsp =
0.313fc0.667
Table 4
Mechanical properties of the surrounding rock mass.
Prototype (Grade IV)
Model
(
)
γ kN/m3
E(GPa)
μ
c(MPa)
φ(◦ )
20–23
20
1.30–6.00
0.18
0.3–0.35
0.35
0.2–0.7
0.019
27–39
33
mico–reinforced concrete and regarded it as a substitute for the model
experiments. To obtain the appropriate mix ratio of micro–concrete,
uniaxial compression tests on micro–concrete categorised into four
groups: B, C, E, and G, respectively, were performed. Each group con­
tained three test specimens, as shown in Fig. 5. Based on the design
scheme, C45 concrete girders and tunnel lining were used. It was diffi­
cult to meet the requirements for all the parameters based on the simi­
larity theorem. Thus, the elastic modulus and ultimate uniaxial
compression strength of the C45 concrete —34.5 GPa and 33 MPa,
respectively—were chosen as control parameters. Hence, the expected
values for the scale models were 1.725 GPa and 1.65 MPa, respectively.
Through uniaxial compression tests, the compressive strength was found
to be 1.85 MPa.
In the experimental results listed in Table. 2, the group G samples
approached the expected values, and the ratio of group G was selected as
the mix ratio for the micro–concrete. In addition to the compressive
strength and elastic modulus, the splitting tensile strength was also
important. Several researches (ACI Committee and 318, 1999; Carino
and Lew, 1982; Carneiro and Barcellos, 1953; Gardner et al., 1988;
Oluokun and Burdette, 1991; Raphael, 1984) have proposed empirical
equations for the relationship between the splitting tensile strength and
compressive strength of concrete. The splitting tensile strengths derived
from these empirical equations are listed in Table. 3. The ultimate
3.3. Material preparation
3.3.1. Concrete simulation
It is impossible to use conventional concrete to manufacture concrete
members with reduced dimensions. Meguid et al. (2008) used a variety
of modelling techniques to investigate the ground response to tunnelling
in his study, where artificial mixed materials played an important role in
simulating prototype materials such as conventional concrete. Typically,
a plaster–water mixture has been used as the simulated primary support
(Fang et al., 2016). However, a plaster–water mixture behaves in a more
brittle manner than conventional concrete, and it can not well indicate
the mechanical properties of concrete (Hobbs, 1966). Hence, micro­
–concrete has been widely used to replace the plaster–water mixture.
Busse and Empelmann (2018) investigated the bending behaviour of
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layer with a thickness of 10 cm (seeing front section) and an elastic
modulus of 10 MPa. For the scale model, the XPS was substituted by
expanded polystyrene (EPS) with a thickness of 0.5 cm and an elastic
modulus of 0.52 MPa.
Table 5
Mix proportions of the model stratum (mass ratio) (Fang et al., 2016).
Component
Barite
River
sand
Quartz
sand
Coal fly
ash
Viscous
oil
Rosin
Proportion
1
0.667
0.667
1.61
0.5
0.053
3.4. Test cases
splitting tensile strength had an average value of 0.55 MPa.
Four tests were performed on the scale model, as shown in Fig. 6. For
Case 1, the girder was optimised based on the original design shown in
Case 4. The mid–span height of the girder was decreased to 2 m, while
the clearance between the girder and the tunnel crown was increased to
3.06 m. The girder was close to the surrounding rock. The only differ­
ence between Case 1 and Case 2 was that a buffer layer was installed
beneath the girder. Moreover, Case 3 reserved the interspace at the
bottom of the girder. Based on the mechanical behaviour observed in the
tests, the optimal scheme was determined.
3.3.2. Surrounding rock and buffer layer simulation
Based on a geological survey, the lithology of the surrounding rocks
in situ includes sandstone and sandy mudstone. Based on the design
codes for railway tunnels in China, the surrounding rock mass in situ was
categorised as Grade IV. The mechanical properties of the surrounding
rock mass in prototype and model conditions are summarised in Table 4.
Different components such as barite powder, sand, plaster powder,
lime, cement, water, and liquid laundry detergent, were used to simulate
various soils and rocks. (Huang et al., 2013; Jeon et al., 2004). Fang et al.
(2016) applied a model test to investigate the behaviour of a highway
tunnel constructed under a mined–out thin coal seam. In that test, the
surrounding rock mass contained sandstone and siltstone and was
classified as Grade Ⅳ; these parameters are similar to the surrounding
rock mass parameters in the present study. Hence, in the experiments
conducted in this study, the same artificial mixing materials could be
selected to simulate the surrounding rock mass. The mix proportions are
listed in Table. 5. For the buffer layer, the prototype contained an XPS
3.5. Measurement
The monitored girder strain and rebar stress are shown in Fig. 7(a)
and (b), respectively. In Fig. 7(a), each girder contained seven sections
labelled as A to G, and each section included six monitoring points.
Thus, 42 concrete strain gauges glued onto the surfaces of the girder
were used to monitor the girder strain. To observe the stress of the main
reinforcement in the steel skeleton of the girder, 56 steel strain gauges
Fig. 6. Four test cases where the first three cases, except the original design Case 4, have the same optimized cross-section, but the contact relation between the
girder and the surrounding rock is rather different.
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Fig. 7. Layout of monitoring points: (a) girder strain measurement, (b) rebar stress measurement, (c) internal force of the tunnel lining.
were cemented onto the surface of the rebar, as illustrated in Fig. 7(b).
The earth pressure cells located at the bottom of the girder for each
target section were devoted to monitoring the pressure from the girder.
In the existing tunnel lining, the target plane was set in the middle of the
testing frame to mitigate the boundary effect. The internal force,
including the axial force and bending moment, measured through the
calculation of the strain, were to evaluate the security of the tunnel
lining. The arrangement of the monitoring points is shown in Fig. 7(c).
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Fig. 8. Load–strain curve of the main reinforcement: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
4. Test results and discussion
Case 1, when the load was up to 2.75 MPa, the strain of the top main
reinforcement continued to increase normally. However, at the given
point of the bottom main reinforcement, the strain increased steeply to
more than 50000με. It has been established that, from the perspective of
the mechanics of materials, this strain value is sufficiently large for the
reinforcements that it will be certainly damaged. Thus, in Case 1, the
limit failure load was 2.75 MPa. Similarly, in Fig. 8(b)–(d), the limit
failure loads were 2.5 MPa, 2.25 MPa, and 3.0 MPa, respectively. When
the buffer layer was used, the limit bearing capacity of the girder only
decreased by approximately 9%. Among the four cases, the original
design in Case 4, shown in Fig. 6, had the highest limit bearing capacity
as it had the greatest section height.
For the first three cases, the specimens had different limit failure
loads, and that of Case 1 was the largest, as shown in Fig. 8(a). The
reason for this is that when the girder was close to the surrounding rock,
the rock would bear the load together with the girder. As shown in Fig. 8
(c), if there was a void between the girder and the surrounding rock, the
limit failure load was the lowest due to the lack of support from the
surrounding rock. It can be observed from Fig. 8(b) that when the buffer
layer was present, the limit failure load was between cases 1 and 3. Thus,
the function of the buffer layer reduced the load exerted on the sur­
rounding rock, and the security of the tunnel lining was enhanced. As
shown in Fig. 8(d), the specimens in Case 4 had different failure posi­
tions due to the different section areas. Although Case 4 had the largest
limit failure load, the small clearance between the girder and sur­
rounding rock would be detrimental to the tunnel lining. To determine
the optimal design, the mechanical state of the tunnel lining should be
analysed as described below.
4.1. Internal force of the reinforcements
Under the ultimate bearing capacity condition, the designed com­
bination of dead and live loads originating from the upper column was
approximately 15 MPa, based on the code for the design of railway
passenger station buildings in China. For convenience, a designed ulti­
mate load of 0.75 MPa was applied in the tests according to similarity
theory. When the specimens were installed, the initial value was
recorded. The internal force of the reinforcement for the four cases is
shown in Fig. 8 (To make these curves brief and clear, only the important
points where the monitoring data vary distinctly, are reserved in the
chart.).
In these cases, a few strainmeters (e.g. S1–G2) were destroyed in the
installation process. It can be seen from these figures that the four cases
exhibited a fundamentally similar trend. The strains increased with step
loading. Meanwhile, when the load was less than 1.0 MPa, the load–­
strain curves of all points were almost linear, and the maximum strain
was 2204.39μεat point S3–G7. Thus, in the case of the design re­
quirements, none of the reinforcement yielded. However, due to the
materiality of station buildings, the complexities of the underground
environment, and the fatigue problem of the structure under train loads,
it is necessary to continue to load until the structure loses its capacity.
Throughout the test process, most of the top main reinforcement was
compressed, and the remaining was under tension; however, the bottom
main reinforcement was just conserved. Some points located at section D
and section E were under compression obviously. The reason for this
phenomenon was that the loading site located near the two sections. In
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Fig. 8. (continued).
4.2. Stress of the girders
capacity at that time. In Fig. 9(b) and (c), when the load was increased to
2.5 MPa and 2.25 MPa, respectively, some points reached limit failure.
There also was a regularity for the limit failure load as the reinforcement
strain analysis shows. This again confirmed the effect of the buffer layer.
For the first three figures, the failure is located at section D; however, in
Fig. 9(d), the failure is in section F. The reason for this may be that in
Case 4, the girder had a uniform section, and thus, the failure position
moved toward the pile.
Based on the analysis of the internal force of the reinforcement, it is
also meaningful to understand how the concrete stress varies. The
concrete stress monitors for all cases are shown in Fig. 9. None of the
compressive stresses of specimens exceeded the uniaxial compressive
strength of 1.85 MPa. The tensile stress was, however, larger than the
tensile strength at some points. Therefore, the failure pattern in the
specimens belonged to a tensile-shear failure. This may be because the
scale ratio was too small to sufficiently set hooping. For the first three
cases, the curves exhibited fundamentally similar trends. The original
values of these curves for the weight of the specimens approached null,
which is because the self–weight of the specimens was smaller than the
applied load in the tests. Note that in parts (a)–(c) in Fig. 9, all the stress
values are negative, which indicates that part of the girder was in a
compressive state. In contrast, in parts (f)–(g), the stress values are
positive. This is because these sections were near both the boundaries. In
addition, these values were fairly small in the two aforementioned sit­
uations as a result of the constraint effect of the two piles.
In particular, in section D and section E, the regularity of the stress
corresponding to the mechanical character of the compression–flexure
member indicates the top of the girder was compressive and that the
bottom was tensile. As the load increased, some locations on these
specimens began to produce crack, such as in Fig. 9(a), points S1–DR5,
and S1–DR6 split to some degree when the load reached 2.5 MPa.
Moreover, when the load was up to 2.75 MPa, severe cracks were
developed, and the stress values of these points were very large.
Certainly, these values were not meaningful in mathematically, and this
showed that the girder had been damaged and had lost its bearing
4.3. Earth pressure in surrounding rock
To investigate the interaction between the girder and surrounding
rock, earth pressure cells with a measuring range of 2.0 MPa were placed
in the surrounding rock with a certain interspace. The location of the
cells corresponded to signs A–G in Fig. 7(a). The pressures transmitted
from the girder to the surrounding rock were recorded. Under uniform
comparison conditions with a load of 2.25 MPa, the results are shown in
Fig. 10. It can be observed from this figure that the earth pressure in Case
3 was close to null, which was in agreement with the internal force of the
tunnel lining. At locations D and E, the earth pressures were larger than
those at the other locations. This confirms that the tunnel was com­
pressed eccentrically. In particular, the largest earth pressure was
observed in Case 4. Thus, this design scheme was not beneficial to the
tunnel.
4.4. Internal forces of lining
For new buildings over existing tunnels, the bearing capacity of the
new buildings, as well as the security of the existing tunnels should be
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Fig. 9. Concrete stresses of girder: (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4.
considered. In the model tests, the primary lining and bolts were
neglected. This is because the ultimate tunnel failure must be reflected
in the permanent lining, i.e., tunnel failure is determined by the per­
manent lining. Thus, to simplify the test, the only permanent lining was
installed. The internal forces of the permanent lining were monitored,
using many strain gauges. The axial force and moment have often been
used to evaluate concrete properties. The safety factor is a comprehen­
sive index for concrete structures. Based on the formulas in the code for
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Fig. 9. (continued).
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Fig. 9. (continued).
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Fig. 9. (continued).
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5.1. Modelling of concrete
The concrete girders were simulated with the C3D8R element, an 8node 3D solid element with reduced integration. The 4-node shell
element S4R with reduced integration was employed to simulate the
tunnel lining. To simulate the evolution of concrete crack, the concrete
damage constitutive model was used to describe concrete stress–strain
relation which is described as follows:
[
]
′
′
1 − dt
σc =
(5)
Dε
1 − dc e c
whereσc = the principal stress, andεc = the principal strain; De = the
elastic stiffness; dt anddc = the tension and compression parameters of
damage, respectively. Hibbit and Sorensen (2007); Huang et al. (2020)
depicted damage evolution paths of concrete for compression and ten­
sion, shown as follows:
⎧
(
)
⎨ σ = Ec : ε − εpl ∈ {σ|F(σ, κ)⩽0 }
pl
(6)
ε̇ = λ̇∇σ ϕ(σ )
⎩
κ̇ = λ̇H(σ, κ)
′
Fig. 10. Earth pressures in the surrounding rock.
whereσ = the effective stress tensor; εandεpl = the total strain tensor and
plastic strain tensor; Ec = the initial elastic modulus of the concrete; λ̇= a
given nonnegative function as the plastic consistency; κ= the plastic
damage index, which depicts damage condition of the concrete. The
total stress tensor σ can be given by
{
σ = [1 − d(κ, σ) ]σ
(7)
d(κ, σ ) = 1 − [1 − dc (κ) ][1 − s(σ )dt (κ) ]
the design of railway tunnels (State Railway Administration of the
People’s Republic of China, 2016), these safety factors were calculated
by following formula:
K = φαRa bh/N
′
(4)
where K is the safety factor, φ is the longitudinal bending coefficient of
member, α is eccentricity coefficient, Ra is the limit compressive
strength, b is the width of section and generally the unite one, h is the
height of section, and N is the axial force of section.
If the bearing capacity of concrete members with rectangular
cross–sections is controlled by the compressive strength, then the safety
factor will not be less than 2.4. These results are shown in Fig. 11 for a
load of 2.25 MPa. Their bearing capacity was determined by compres­
sive strength. It is clear from Fig. 11 that larger moments are generated
#2 or #3, i.e., the moments are not symmetric. The tunnel lining is in
eccentric compression. Fig. 11(c) shows that when the girder does not
contact the surrounding rock, the moments and axial forces are small,
and the safety factors are very large. If the buffer layer is present, the
internal force shown in Fig. 11(b) will be smaller than that in Fig. 11(a),
and the safety factor will also be greater. It can be observed from Fig. 11
(a) that the smallest safety factor was 4.5, and that an adequate margin
was not present. However, there was an adequate margin for the safety
factor, as shown in Fig. 11(b). The extremely large axial force and
moment in Fig. 11(d) are extremely unfavourable. In particular, the
safety factor for #2, #3, #4, #5, and #10 were 2.20, 2.67, 1.98, 2.27,
and 2.27, respectively, which did not satisfy the code prescript. The
reason for this was that in Case 4, the clearance between the girder and
surrounding rock was only 1.46 m. It can be concluded from the results
that the original design was unfavourable for the tunnel lining. When
the optimised design with the buffer layer was used, the smallest safety
factor was increased from 1.98 to 21.55.
whered = the stiffness degrading variable; ands = the unilateral effect
during the elastic unloading procedure from tensile condition to
compression condition, which can be expressed as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
s(σ ) = s0 + (1 − s0 )r( ̂
σ)
⎧
⎪
⎪
⎪
⎪
⎪
⎨
0σ=0
〈 〉
⎪
⎪
r( ̂
σ ) = ∑3 ̂
⎪
⎪
σi
⎪
⎪
⎪
i=1
⎪
⎪
⎪ ∑ ⃒ ⃒ otherwise
⎩
⎪
⎪
3 ⃒ ⃒
⎩
σ i⃒
⃒̂
i=1
(8)
where ̂
σ i = the principle value of effective stress; 〈∙〉 depicts the
McCauley bracket; s0 = the constant (0⩽s0 ⩽1)indicating the min value.
The stress–strain relation curves and the damage evolution paths are
shown in Fig. 12(a) and (b), respectively.
In the numerical simulations, we defined the compressive, tensile,
crushing strengths, initial elastic modulus of concrete asfc = 16.5MPa,
ft = 1.8MPa, fcu = 45MPa, Ec = 34.5GPa, respectively; the concrete
strains at fc , ft , fcu were defined as εc0 = 0.002, εtu = 0.001, εcu =
0.0033, respectively.
5.2. Modelling of reinforcement
5. Numerical study
For the refined simulation of reinforcement, the T3D2 element, a 2node 3D element, was used, which can sufficiently reflect the properties
of reinforcement. All of the reinforcements were embedded into the
concrete. The constitutive model proposed by Ziraba et al. (1994), an
isotropic strain hardening and uniaxial bilinear model, can be in accord
with the mechanical properties of reinforcement. This model can be
expressed as follows:
The finite element analysis software ABAQUS (ABAQUS, 2016) is
used as a modelling instrument in the chapter. We developed the same
refined model as that in Case 2 but in prototype condition. The nu­
merical model was validated by an experiment. Furthermore, the in­
fluence of the different elastic modulus of the buffer layer on structures
was investigated.
σ ∗ = bε∗ +
(1 − b)ε∗
1
(1 + ε∗R ) /R
(9)
where σ ∗ = the normalised stress, ε∗ = the normalised strain, b= the
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Fig. 11. Internal forces (axial force and moment) and safety factor of the tunnel lining: (a) Case 1 (b) Case 2 (c) Case 3 (d) Case 4.
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Fig. 12. Concrete damage constitutive model used in the numerical model: (a) stress–strain relation (b) damage evolution paths.
Fig. 13. The finite element model in the numerical analysis: the brown parts denotes the girders.
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Fig. 14. Comparison of the numerical and the experimental results: (a) comparison of strain-load curves of reinforcements derived from numerical and experimental
studies where the labels encompassing the capital “ N ” denote the numerical analysis, whereas no capital “ N ” represent the experimental analysis. Note that the
right axis depicts the step loads in the numerical model and that the left axis denotes the step loads in the experimental model, (b) comparison of stress-load curves of
concrete derived from numerical and experimental studies, (c) final failure pattern of the specimen.
density, elastic modulus, and Poisson’s ratio were set as 3000 kg/m3, 10
MPa, and 0.2, respectively. The Mohr-Coulomb constitutive model
widely employed to simulate geotechnical materials was used here. The
relative mechanical properties were obtained from Table 4 herein.
slope of the hardening extent, andR = the curvature of unloading.
In the numerical model, the yield strength of the main re­
inforcements HRB400 was fy = 400MPa. The elastic modulus and the
strain-hardening ratio were defined as E0 = 200GPa and 0.01,
respectively. The curvature of unloading, R, was set as 18.5.
5.4. Modelling of contact
5.3. Modelling of the buffer layer, surrounding rock
In this chapter, the surface-to-surface contacts were used to simulate
the contact behaviour among concrete, surrounding rock, and the buffer
layer. The hard contact and slip chosen represented the normal and
The 8-node 3D solid element was used to simulate the buffer layer,
XPS. And then the elastic constitutive model was employed here. The
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Fig. 15. Internal forces (axial force and moment) and safety factors of the tunnel lining under the different stiffness of buffer layer: (a) elastic modulus of 10 MPa (b)
elastic modulus of 15 MPa (c) elastic modulus of 20 MPa (d) elastic modulus of 30 MPa.
tangential contact behaviours, respectively. The tangential slip can be
expressed by Coulomb’s friction equation shown as follows:
τcritical = μ⋅p⩾τbond
the critical tangential force and the tangential force, respectively. If the
tangential force exceeds the critical tangential force, the slip at the
interface will occur. The friction factor was set as 0.1 between the buffer
layer and the surrounding rock; that between the buffer layer and con­
crete was defined as 0.2; that between surrounding rock and concrete
was set as 0.3.
(10)
where μ is the friction factor, which ranges from 0.1 to 0.3; p depicts the
normal force perpendicular to the contact surface; τcritical andτbond denote
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5.5. Validation of the numerical model by an experimental
numerical simulation was carried out, and the numerical simulation was
in good agreement with the experiment. Moreover, the parametric study
was performed. Based on these investigations, it is safe to draw the
following conclusions:
It is difficult to perform a parametric study based on experiments.
Thus, in the study, a numerical model corresponding to Case 2 was
presented, as shown in Fig. 13. In this simplified finite element model,
one of the main girders L2 illustrated in section 2 was considered. In
particular, the sub-girders connected with L2 was presented and the
main reinforcements and stirrups involved in the main girder were
developed. The other main girders and pillars imposing the constraints
on this sub-girder were simplified by fixed constraints on the bound­
aries. Based on St-venenat’s principle, this simplification was reliable for
the calculation of the main girder internal force. The step loads in the
numerical simulation were enlarged to 20 times according to similarity
theory. The comparison between reinforcement strains and concrete
stresses of Case 2 in section D and section E, derived from the numerical
and experimental results, is shown in Fig. 14(a) and (b), respectively.
However, when the step load was 2.5 MPa, the calculations did not
converge and thus the stress and strain did not obtain. For comparison
under the uniform condition, the experimental results of concrete stress
were transformed into the prototype model. It can be seen from these
two pictures that the numerical results agree mainly with the experi­
mental analysis, apart from a few points such as S2-D7, S2-E6 in Fig. 14
(a). At the early loading stage where the load is no more than 2.0 MP,
these curves approximately satisfies linear-elastic relation. At that stage,
numerical results are more in accord with experimental results, whose
maximum errors in terms of reinforcement strain and concrete stress are
490 με and 405 kPa, respectively. Following that stage, the specimen
experienced a short plastic course. Finally, it reached failure state. From
both numerical results and experimental results in Fig. 14(b) (on the
left), we see the failure position first occurred in section D. The final
failure pattern of this specimen in the numerical (on the left) and
experimental (on the right) studies is illustrated in Fig. 14(c). The cracks
were firstly generated in the tensile zone near the loading site and fastly
extended to the compressed zone. From the perspective of the reinforced
concrete principle, this failure pattern belonged to tensile-shear failure.
As this picture shows, the two failure patterns have a similar trend. Thus,
it is reliable that numerical modelling can be used to perform a para­
metric study.
(1) The original design scheme with a uniform cross–section had the
largest limit failure bearing capacity, but the existing tunnels had
the smallest safety factor, 3.52. The optimised design scheme not
only enlarged the clearance (from 1.46 m to 3.06 m) but also
reduced the construction cost including the decrease of concrete
approximately 2700 m3.
(2) The limit bearing capacity of the girders decreased slightly by
approximately 9% when the buffer layer (XPS) with elastic
modulus 10 MPa was used; however, the smallest safety factor of
the tunnel lining under eccentrically compressed state was
improved significantly up to approximately 11 times.
(3) The failure pattern of the girders belonged to tensile-shear fail­
ure. It had three stages: linear stage, crack developing stage, and
failure stage. In the crack developing stage, the cracks firstly
generated in the tensile zone near the loading site; in the failure
stage, the cracks fastly extended to the compressive zone and then
the failure state formed.
(4) As the stiffness of the buffer layer (XPS) increases, the tunnel
lining bears larger force. The relatively low 10 MPa elastic
modulus of buffer layer was recommended in this project. In
addition, the ideas, in this study, can potentially be generalized to
similar projects to reduce the disturbance from the adjacent
construction to existing tunnels.
CRediT authorship contribution statement
Feiyue Yan: Conceptualization, Methodology, Validation, Writing original draft, Writing - review & editing. Wenge Qiu: Writing - review
& editing, Supervision. Yuchao Zheng: Conceptualization, Methodol­
ogy. Shuhua Jiang: Investigation, Visualization. Hui Hu: Investigation,
Visualization. Gang-gang Gao: Investigation, Visualization. Yunjian
Cheng: Investigation, Visualization.
Declaration of Competing Interest
5.6. Parametric study
There is no conflict of interest.
It is expensive to consider the different stiffness of the buffer layer
through experiments while it is easy to do it through numerical
modelling. In this chapter, the elastic modulus of buffer layer set to 10
MPa, 15 MPa, 20 MPa, and 30 MPa, respectively, was performed to
discuss the safety of tunnel lining. Based on the method for calculating
the safety factors of tunnel lining hereinbefore, these results are shown
in the following Fig. 15. This picture depicts these curves have a similar
trend and that the safety factors decrease with the stiffness of the buffer
layer. This is because of the stiffer buffer layer and the larger load
transformation. In addition, it is proved again that the internal forces of
tunnel lining derived from numerical analysis in Fig. 15(a) are in accord
with experimental results in Fig. 11(b). To reduce the disturbance from
the adjacent construction to the existing twin tunnels, the relatively
softer buffer layer should be used. Through comprehensive analysis
containing both two sides, where the buffer layer can stay small defor­
mation when subjected to the gravity weight of the girders, and the
safety of tunnel lining had enough reserve. Thus, the elastic modulus 10
MPa of the buffer layer was recommended.
Acknowledgements
The authors gratefully acknowledge the financial support provided
by the National Key R&D Program of China (2017YFC0806000), the
National Natural Science Foundation of China (71942006), and the
National Natural Science Foundation of China (51991395).
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