Transportation Geotechnics 44 (2024) 101171
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Transportation Geotechnics
journal homepage: www.elsevier.com/locate/trgeo
A mechanical analysis model for tunnels under strike-slip faulting
considering the fault zone width and nonlinear tunnel-stratum interaction
Xiao Zhang a, b, c, Li Yu a, b, Mingnian Wang a, b, *, Henghong Yang a, b
a
School of Civil Engineering, Southwest Jiaotong University, Chengdu 610036, China
Key Laboratory of Traffic Tunnel Engineering, Ministry of Education, Southwest Jiaotong University, Chengdu 610036, China
c
School of Transportation and Logistics, Southwest Jiaotong University, Chengdu 610036, China
b
A R T I C L E I N F O
A B S T R A C T
Keywords:
Tunnel engineering
Strike-slip fault
Fault zone width
Nonlinear tunnel-stratum interaction
Mechanical response
Tunnels crossing active fault zones are seriously damaged under the action of faulting. However, the influence of
the fault zone width or nonlinear tunnel-stratum interaction is neglected in existing analytical models, resulting
in incorrect estimation of the tunnel mechanical response. To aid in the seismic design of tunnels, a mechanical
analysis model for tunnels under strike-slip faulting considering the effect of fault zone width and nonlinear
tunnel-stratum interaction is first proposed in this work. The calculation results of the proposed model agree well
with those obtained from the model test and numerical simulation, and the accuracy is obviously higher than
that of the existing analytical methods. Compared with the FEM numerical model, the average errors of the peak
bending moment, shear force and axial force calculated by the proposed model are 6.37 %, 11.00 % and 9.91 %,
respectively. Neglecting the effect of fault zone width or nonlinear tunnel-stratum interaction, the tunnel
response will be misestimated, e.g., when the fault displacement is 3 m and the crossing angle β is 90◦ , the
overestimated value is approximately 40–110 %. Finally, the results of parameter analysis show that under the
condition of 0◦ < β < 90◦ , the tunnel failure scope caused by the left-lateral strike-slip faulting is obviously larger
than that caused by the right-lateral strike-slip faulting, while the conclusion is opposite under 90◦ < β < 180◦ ;
the maximum strain within the tunnel correlates directly with fault displacement and the strength of the fault
zone, exhibiting an inverse correlation with the width of the fault zone; meanwhile, the scope for tunnel failure is
directly influenced by fault displacement, fault zone width, and the crossing angle (where 90◦ < β < 180◦ ), while
inversely affected by the crossing angle (where 0◦ < β < 90◦ ) and the strength of the fault zone.
t
D
C
L
n
LF
LH
β
V
ux
uy
A
E
I
Thickness
Equivalent outer diameter
Buried depth
Length
Number of elements
Length in the footwall
Length in the hanging wall
Crossing angle
Shear force
Axial displacement
Transverse displacement
Cross-section area
Young’s modulus
Inertia moment
l
N
M
W
WF
WH
Δf
Δfx
Δfy
δx
δy
K0
Δx
Δy
χ
Pxu
Element length
Axial force
Bending moment
Fault zone width
Fault zone width in the footwall
Fault zone width in the hanging wall
Strike-slip displacement
Axial displacement
Transverse displacement
Axial relative tunnel-stratum displacement
Transverse relative tunnel-stratum displacement
Lateral pressure coefficient
Axial yield displacement
Transverse yield displacement
Axial bearing capacity
* Corresponding author at: School of Civil Engineering, Southwest Jiaotong University, Chengdu 610036, China.
E-mail addresses: xiao.zhang@swjtu.edu.cn (X. Zhang), yuli_1026@swjtu.edu.cn (L. Yu), 19910622@163.com (M. Wang).
https://doi.org/10.1016/j.trgeo.2023.101171
Received 18 July 2023; Received in revised form 9 November 2023; Accepted 4 December 2023
Available online 10 December 2023
2214-3912/© 2023 Elsevier Ltd. All rights reserved.
X. Zhang et al.
Pχyu
kχy
Eχ
Px
Py
f1
γχ
cχ
φχ
kxx
νχ
Nch , Nqh
χ
χ
Transportation Geotechnics 44 (2024) 101171
and cracking may occur. Additionally, Zhong et al. [21] pointed out that
the increase in stratum stiffness in the fault zone reduces the extension
but increases the severity of structural damage to the tunnel lining,
which indicates that the fault zone width should be considered in tunnel
response analysis. This point has also been widely recognized by many
authors [17,22].
However, the numerical simulation of a tunnel crossing an active
fault zone takes several hours even in a high-performance computer,
which is not applicable to engineering design. Therefore, a series of
analytical models have been proposed for the mechanical response
analysis of a tunnel or pipeline under faulting. Newmark and Hall [23]
first proposed a cable-like model for a pipeline under a strike-slip fault
without considering the contribution of lateral stratum resistance.
Kennedy [24] extended this model by considering stratum resistance,
but the pipeline’s flexural stiffness is still not considered. Later, Wang
and Yeh [25] partitioned the pipeline into four segments to compensate
for the limitation of the cable-like models in neglecting the contribution
of the pipeline’s flexural stiffness. In this model, the two high-curvature
segments proximal to the fault are treated as circular arcs, while the two
segments at the far ends are treated as elastic beams on an elastic
foundation. After that, this model was improved and developed by many
authors [26–31], and some shortcomings have been overcome. How­
ever, in these analytical models, the fault zone is simplified to a thin line,
which is applicable to shallow buried pipelines (buried a few meters in
the soft stratum), as shown in Fig. 3(a). The highway, railway and hy­
draulic tunnels crossing active fault zones are usually buried tens to
hundreds of meters, and the fault zone width has a significant effect on
the tunnel response [21], as shown in Fig. 3(b). Therefore, these models
are not applicable to tunnels crossing active fault zones due to the
neglect of the influence of fault zone width.
Additionally, to assess the tunnel response under the action of fault
zone displacement, a series of analytical models have been developed
[32–37], and the effect of fault zone width is considered based on the
elastic foundation beam theory. In these models, a tunnel crossing an
active fault zone is simplified as an elastic beam acting on an elastic
foundation, and the tunnel-stratum interaction is assumed to be linear.
However, the relevant model tests [38] and numerical simulations [39]
showed that the tunnel-stratum interaction stress increases nonlinearly
with increasing fault displacement, indicating that the tunnel-stratum
interaction has obvious nonlinearity. Therefore, this assumption is un­
realistic and can lead to incorrect estimation of the internal force of the
tunnel under faulting.
To this end, a mechanical analysis model for tunnels under strike-slip
faulting considering the fault zone width and nonlinear tunnel-stratum
interaction is proposed in this work, and the above weaknesses have
been fixed.
Transverse bearing capacity
Transverse stiffness
Young’s modulus
Axial stratum stress
Transverse stratum stress
Coating dependent factor
Unit weight
Cohesion
Friction angle
Axial stiffness
Poisson’s ratio
Bearing capacity factor
Note: Considering the stratum parameters of the fault zone and host
rock, the parameter variable χ is introduced. The parameters of the fault
zone for χ = f and host rock for χ = h.
Introduction
Active fault zones are widely distributed around the world, espe­
cially in China, the USA, Turkey and Japan. Generally, a fault zone
consists of a fault core and a damage zone, and its width varies from tens
to hundreds of meters [1,2], as shown in Fig. 1.
With the development of infrastructure construction, it is inevitable
that some tunnels will cross active fault zones, especially in seismically
active regions. However, the displacement of the active fault zone would
result in severe damage to tunnels, i.e., the 1906 San Francisco earth­
quake caused severe damage to two tunnels crossing the San Andreas
fault [3]; the 1999 Chi-Chi earthquake in Taiwan caused significant
damage to many nearby tunnels due to the displacement of the Che­
longpu fault [4]; the 2008 Wenchuan earthquake in China resulted in
severe damage to several tunnels along the Duwen expressway [5]; and
the 2016 Kumamoto earthquake in Japan severely damaged the Toyama
tunnel near the fault [6]. Most notably, during the 2022 Menyuan
earthquake in China, the Daliang tunnel suffered severe damage due to a
3 m strike-slip fault displacement [7,8], as plotted in Fig. 2. Therefore,
the mechanical response of tunnels under faulting has been widely
investigated.
A series of model tests [9–14] and numerical simulations [15–20]
have been conducted to investigate the tunnel response under faulting.
The results show that the tunnel is subjected to a combination of tensile
(compressive), bending and shearing and experiences an elongated ‘S’shaped deformation along the tunnel axis; the tunnel internal forces
increase with increasing fault displacement, stratum stiffness and tunnel
stiffness; under the action of fault zone displacement, the tunnel will be
severely damaged, especially near the fault zone, where tunnel collapse
Establishment of the proposed model
Outline
The proposed model is plotted in Fig. 4, where the strike-slip
displacement is Δf, the crossing angle is β (0◦ < β < 180◦ ), the tunnel
equivalent diameter is D (i.e., the diameter of a circular tunnel with the
same area as the prototype tunnel), the lining thickness is t, the burial
depth is C, the fault zone width is W (WF in the footwall and WH in the
hanging wall) and the tunnel length is L (LF in the footwall and LH in the
hanging wall). Note that the fault core is simplified to a fault plane, as
the width of the fault core is much smaller than that of the damage zone.
This simplification has been widely employed in previous works [17].
The left- and right-lateral strike-slip faults and crossing angles β are
considered in this work, as shown in Fig. 4(a) and (b).
As plotted in Fig. 4, the axial and transverse fault displacements Δfx
and Δfy acting on the tunnel can be calculated by Eq. (1) and Eq. (2). The
value of Δf is positive (+) for left-lateral strike-slip faults and negative (-)
for right-lateral strike-slip faults.
Fig. 1. Conceptual model of a fault zone illustrating the main architectural
elements: damage zone and fault core.
2
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Fig. 2. Daliang tunnel damaged due to strike-slip faulting in the 2022 Menyuan earthquake [7,8]: (a) circumferential crack; (b) collapsed lining; (c) bulging invert;
(d) spalling lining.
Fig. 3. Schematic diagram of (a) a shallow pipeline and (b) a deep tunnel crossing an active fault zone.
Δfx = Δf cosβ
(1)
Δfy = Δf sinβ
(2)
where N is the axial force caused by the combined action of axial and
transverse displacements, which will be discussed later.
Based on Eq. (3), the following equation of moment equilibrium is
obtained:
Derivation of the governing equation
Based on previous studies [31,40], a tunnel crossing an active fault
zone is simplified as a large-deformed beam acting on a nonlinear
foundation, as shown in Fig. 5. The tunnel-stratum interaction is
described by a series of nonlinear springs, and the fault displacements
are applied at the ends of the springs. The different mechanical char­
acteristics of the fault zone and host rock are reflected by different
stratum stiffnesses. Considering the different stratum parameters of the
fault zone and host rock, the parameter variable χ is introduced. The
parameters of the fault zone for χ = f and host rock for χ = h. For
−
(4)
where uy is the transverse displacement of the tunnel.
The bending moment of the tunnel is [41]:
M = − EI
d2 uy
dx2
(5)
where I and E are the inertia moment and Young’s modulus of the
tunnel, respectively.
Based on Eq. (4) and Eq. (5), yields:
f
instance, the nonlinear transverse stiffness of the fault zone is ky , and the
nonlinear stiffness of the host rock is khy . Under the action of fault dis­
EI
placements (Δfx and Δfy) and stratum stresses (Px and Py), the tunnel is
subjected to axial force N, shear force V and bending moment M, as
plotted in Fig. 5.
According to Fig. 5, the microelement equilibrium of moment yields:
Py
dM + M + Vdx + Ntanθdx = M + (dx)2
2
dM
duy
=N
+V
dx
dx
d3 uy
duy
=N
+V
dx3
dx
(6)
By taking the derivative from both sides of Eq. (6), yields:
EI
(3)
d4 uy dN duy
d2 uy dV
=
+N 2 +
dx dx
dx
dx4
dx
(7)
From the microelement equilibrium of forces in the y-direction,
3
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Fig. 4. (a) A tunnel crossing an active fault zone, (b) left-lateral strike-slip fault with 0◦ < β < 90◦ , (c) left-lateral strike-slip fault with 90◦ < β < 180◦ , (d) rightlateral strike-slip fault with 0◦ < β < 90◦ and (e) right-lateral strike-slip fault with 90◦ < β < 180◦ .
yields:
dV
= Py
dx
δy = Δfy − uy
(8)
Based on Eq. (4) ~ Eq. (10) yields the governing equation of trans­
verse tunnel displacement uy:
The nonlinear transverse stratum stress Py is proportional to the
value of the relative transverse tunnel-stratum displacements δy and
nonlinear stratum stiffnesses kχy :
Py = kyχ δy
(10)
EI d4 uy N d2 uy 1 dN duy
−
−
+ uy = Δfy
kyχ dx4 kxχ dx2 kxχ dx dx
(9)
(11)
Derivation of the axial force equation
Based on previous studies [31,40], the axial force N of a large
deformed tunnel consists of a frictional axial force Na (caused by fric­
tional axial tunnel-stratum interaction) and a membrane axial force Nb
The relative transverse tunnel-stratum displacements δy can be
calculated by
4
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Fig. 5. Schematic diagram of a tunnel in bending, tension and shearing.
(caused by large transverse deflection). Neglecting the term Nb leads to
erroneous results of tunnel response under faulting [27,31,40].
N = Na + Nb = EA(εa + εb )
The membrane axial strain εb of the tunnel owing to this large
transverse deflection is derived as [31,40]:
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
du
(18)
εb = 1 + ( y )2 − 1
dx
(12)
where A is the lining cross-sectional area; εa and εb are the axial
strains caused by frictional axial tunnel-stratum interaction and trans­
verse deflection, respectively.
Using the following series expansion yields:
√̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
du
1 du
1 du
εb = 1 + ( y )2 = 1 + ( y )2 − ( y )4 …
2 dx
8 dx
dx
(1) The frictional axial force Na
The higher-order terms in Eq. (19) are deleted under the premise of
ensuring the calculation accuracy. Therefore, the membrane axial strain
εb and axial force Nb can be calculated by
In the axial direction, the tunnel is subjected to the axial fault
displacement Δfx and frictional stress Px, resulting in axial tunnel
displacement ux and axial force Na, as shown in Fig. 6.
The axial force Na of the tunnel is [41]:
Na = EAεa = EA
dux
dx
1 duy 2
)
2 dx
(13)
Nb = EAεb =
by
(14)
Due to the nonlinear stratum stiffnesses kχx and kχy , it is difficult to
obtain the analytical solution of tunnel displacements ux and uy.
Therefore, the finite difference method is used to solve the governing
equations Eq. (11) and Eq. (17). As shown in Fig. 7, a tunnel with a
length of L is discretized along the axial direction into n + 5 node ele­
ments (including four virtual node elements), and the element length is l
= L/n.
(15)
(16)
Based on Eq. (14) ~ Eq. (16), the differential equation of axial tunnel
displacement ux is:
−
EA d2 ux
+ ux = Δfx
kxχ dx2
(21)
Solution of the governing equations
The relative axial tunnel-stratum displacements δx can be calculated
δx = Δfx − ux
( )2
EA duy
2 dx
Note that the key problem in analysing the tunnel response under
faulting is to solve the governing equations of the axial and transverse
tunnel displacements ux and uy, i.e., Eq. (11) and Eq. (17).
The nonlinear axial stratum stress Px is a function of the relative axial
tunnel-stratum displacement δx and nonlinear stratum stiffnesses kχx :
Px = kxχ δx
(20)
εb = (
According to Eq. (13), the axial tunnel displacement ux must be ob­
tained to calculate the axial force Na. Therefore, the governing equation
of the axial tunnel displacement ux is derived herein. From the free body
equilibrium of forces in the x-direction in Fig. 6, yields
Na = Na + Px dx + dNa
(19)
(17)
(2) Membrane axial force Nb
Fig. 7. Discretization of the tunnel.
Fig. 6. The frictional axial tunnel-stratum interaction.
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X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Solution of the axial tunnel displacement ux
According to the finite difference method, the difference expression
of Eq. (17) is:
)
EA (
− χ 2 ux,i+1 − 2ux,i + ux,i− 1 + ux,i = Δfx,i
kx,i l
(22)
where kχx,i , ux,i and Δfx,i are the axial stratum stiffness, tunnel
displacement and fault displacement of node i, respectively.
The frictional axial force Na at both ends of the tunnel is zero, i.e.,
Na,0 = Na,n = 0. According to the finite difference method and Eq. (13),
the difference expressions of the boundary conditions are:
Na,0 =
)
EA (
ux,1 − ux,− 1 = 0
2l
ux,n+1 = ux,n− 1
(26)
(39)
(41)
[
[A](n+1)×(n+1) = [A2 ](n+1)×(n+1) − [A1 ](n+1)×(n+1)
(28)
...
ux,n ]Tn+1
(29)
... Δfx,i
...
ux,i
]T
Δfx,n n+1
] [
Δfy = Δfy,0
(30)
Solution of the transverse tunnel displacement uy
According to the finite difference method, the difference expression
of Eq. (11) is:
kxχ =
)
)
EI (
Ni (
χ 4 uy,i+2 − 4uy,i+1 + 6uy,i − 4uy,i− 1 + uy,i− 2 −
χ 2 uy,i+1 − 2uy,i + uy,i− 1
ky,i
l
ky,i
l
kyχ =
(31)
where Ni, kχy,i , uy,i and Δfy,i are the axial force, transverse stratum
stiffness, transverse tunnel displacement and transverse fault displace­
ment of node i, respectively.
The axial force N, bending moment M and shear force V at both ends
of the tunnel are zero. According to the finite difference method and the
elastic beam theory, the difference expressions of the boundary condi­
tions are:
M0 = −
)
EI (
uy,1 − 2uy,0 + uy,− 1 = 0
l2
)
EI (
Mn = − 2 uy,n+1 − 2uy,n + uy,n− 1 = 0
l
(34)
Vn = −
)
EI (
uy,n+2 − 2uy,n+1 + 2uy,n− 1 − uy,n− 2 = 0
2l3
(35)
Δfy,1
Δfy,2
⎧P
x
⎪
⎪
⎨ |δ |
... uy,n ]Tn+1
]T
... Δfy,n n+1
... Δfy,i
Δx⩽|δx |
χ
⎪
⎪
⎩ Pxu
Δx
0⩽|δx | < Δx
(42)
(43)
⎧
Py
⎪
⎪
⎪ ⃒⃒δ ⃒⃒
⎨
⃒ ⃒
Δy⩽⃒δy ⃒
⎪
Pχ
⎪
⎪
⎩ yu
Δy
⃒ ⃒
0⩽⃒δy ⃒ < Δy
(44)
x
y
Pχxu = παχ cχ D + πCDγχ
(33)
)
EI (
uy,2 − 2uy,1 + 2uy,− 1 − uy,− 2 = 0
2l3
... uy,i
(45)
where Pχxu and Pχyu are the bearing capacities of the axial and trans­
verse stratum, respectively; Δx and Δy are the corresponding yield
displacements of the tunnel-stratum interaction; and the relative axial
and transverse tunnel-stratum displacements δx and δy can be calculated
by Eq. (16) and Eq. (10), respectively.
χ
χ
The axial and transverse stratum bearing capacity Pxu and Pyu can be
calculated by [42]:
(32)
V0 = −
uy,2
Nonlinear tunnel-stratum interaction
To solve the equations of tunnel displacements, Eq. (27) and Eq. (40),
χ
χ
the key point is to calculate the nonlinear stratum stiffnesses kx and ky .
The widely recognized nonlinear structure-stratum interaction curves
recommended by design guideline [42] are employed in this work as an
example, see Fig. 8. In addition, to assess the tunnel response under both
left- and right- lateral strike-slip faulting (see Fig. 4) with uniform
governing equations, it is assumed that these springs can withstand
compression and tension. Otherwise, it is necessary to establish different
calculation models for different situations. The force vector applied to
the tunnel by these springs is shown in Fig. 8.
According to Fig. 8, the axial and transverse stratum stiffnesses kχx
and kχy can be calculated by
The calculation of matrix [A] is detailed in Appendix A.
)
(
1
− χ 2 (Ni+1 − Ni− 1 ) uy,i+1 − uy,i− 1 + uy,i = Δfy,i
4ky,i l
uy,1
The calculation of matrix [B] is detailed in Appendix B.
where
Δfx,2
uy,n+2 = 4uy,n − 4uy,n− 1 + uy,n− 2
[ ]
uy = [ uy,0
(27)
[A][ux ] = [Δfx ]
Δfx,1
(38)
where
According to Eq. (22) ~ Eq. (26), the axial tunnel displacement ux
can be calculated by
[
[Δfx ] = Δfx,0
uy,n+1 = 2uy,n − uy,n− 1
(24)
(25)
...
(37)
[B](n+1)×(n+1) = [B1 ](n+1)×(n+1) − [B2 ](n+1)×(n+1) − [B3 ](n+1)×(n+1) + [B4 ](n+1)×(n+1)
ux,− 1 = ux,1
ux,2
uy,− 2 = 4uy,0 − 4uy,1 + uy,2
According to Eq. (31) ~ Eq. (39), the transverse tunnel displacement
uy can be calculated by
[ ] [ ]
(40)
[B] uy = Δfy
Therefore, the axial tunnel displacement ux of nodes − 1 and n + 1
can be obtained as follows:
ux,1
(36)
(23)
)
EA (
Na,n =
ux,n+1 − ux,n− 1 = 0
2l
[ux ] = [ ux,0
uy,− 1 = 2uy,0 − uy,1
1 + K0
tan(f1 φχ )
2
(46)
(47)
χ χ
χ χ
Pχyu = Nch
c D + Nqh
γ CD
where c is the stratum cohesion, γ is the stratum unit weight, K0 is
the lateral pressure coefficient, αχ is the adhesion factor, φχ is the stra­
tum friction angle, f1 is the coating dependent factor, and the bearing
χ
χ
capacity factors Nch
and Nqh
can be interpreted from design charts
[42].
χ
Therefore, the transverse tunnel displacement uy of nodes − 1, − 2, n
+ 1 and n + 2 can be obtained as follows:
6
χ
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Fig. 8. Nonlinear tunnel-stratum interaction curves [42] in the (a) axial and (b) transverse directions.
Calculation process
To obtain the values of the nonlinear stratum stiffnesses kχx and kχy , a
novel iterative method is proposed in this section. In short, the stratum
stiffnesses kχx and kχy are continuously decreased until the tunnel dis­
placements and axial force N converge. The specific calculation process
is as follows (see Fig. 9):
f
f
(2) The 1st iteration calculation.
• Calculate the initial tunnel displacements ux,i,1 and uy,i,1 by Eq. (27)
and Eq. (40).
• Calculate the initial relative tunnel-stratum displacements δx,i,1 and
δy,i,1by Eq. (16) and Eq. (10).
Ni Mi (D − t)
+
A
2I
(51)
Ni Mi (D − t)
−
2I
A
(52)
σ i,left
(53)
E
σ i,right
E
(54)
Note that the positive strain indicates the tensile state, and the
negative strain represents the compressive state.
f
Comparison with the model test
Parameters
f
the j-1th stratum stiffnesses khx,i,j− 1 , khy,i,j− 1 , kx,i,j− 1 and ky,i,j− 1 into Eq.
(27) and Eq. (40).
• Calculate the jth relative tunnel-stratum displacements δx,i,j and δy,i,j
by Eq. (16) and Eq. (10).
Wang et al. [43] conducted 1 g model tests to investigate the me­
chanical response of a tunnel subjected to right-lateral strike-slip fault­
ing, as shown in Fig. 10.
The detailed testing parameters are shown in Table 1.
In the theoretical model analysis, the tunnel-stratum interaction
properties are calculated by the guideline [42], as shown in Table 2.
Note that the coating-dependent factor f1 of the tunnel-stratum interface
is 0.7, as smooth gypsum is used to simulate the tunnel in the model
tests.
f
• Calculate the jth stratum stiffnesses khx,i,j , khy,i,j , kx,i,j and ky,i,j by Eq.
(44) and Eq. (45).
• Calculate the jth membrane axial forceNb,i,j by Eq. (21).
Equations of the tunnel forces and stresses
The tunnel axial force Ni, bending moment Mi and shear force Vi of
node i can be calculated according to the finite difference method in
difference form:
) EA(
)2
EA (
ux,i+1 − ux,i− 1 + 2 uy,i+1 − uy,i− 1
2l
8l
(50)
εi,right =
Eq. (44) and Eq. (45).
• Calculate the initial membrane axial force Nb,i,1 by Eq. (21).
(3) The j (j > 1)th iteration calculation.
• Calculate the jth tunnel displacements ux,i,j and uy,i,j by substituting
Ni =
) Ni (
)
EI (
uy,i+2 − 2uy,i+1 + 2uy,i− 1 − uy,i− 2 −
uy,n+1 − uy,n− 1
2l3
2l
εi,left =
• Calculate the initial stratum stiffnesses khx,i,1 , khy,i,1 , kx,i,1 and ky,i,1 by
f
Vi = −
σ i,right =
f
f
(49)
σ i,left =
kx,i,0 = Pxu /Δx and ky,i,0 = Pyu /Δy.
f
)
EI (
uy,i+1 − 2uy,i + uy,i− 1
l2
Under strike-slip faulting, the maximum stress occurs at the tunnel
hances, as shown in Fig. 4. The stress σi and strain εi at the left and right
tunnel hances of node i are respectively:
(1) Pretreatment
• Input parameters.
• Calculate Δfx,i and Δfy,i by Eq. (1) and Eq. (2).
• Assume the membrane axial force Nb,i,0=0..
• Assume the the stratum stiffnesses khx,i,0 = Phxu /Δx, khy,i,0 = Phyu /Δy,
f
Mi = −
Results
The longitudinal strain and transverse displacement at the right
hance obtained by the proposed model and model test are plotted in
Fig. 11. Note that, in the condition of only considering the nonlincar
(48)
7
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
tunnel-stratum interaction (i.e., analytical model II), the parameters of
the host rock are employed in the entire model; in the condition of only
considering the fault zone width (i.e., analytical model III), the linear
stratum stiffnesses are calculated by the equations listed in Appendix C
[44]. Specifically, the strain is symmetric about the fault plane, first
increases and then decreases as the fault distance increases and finally
reaches zero at both ends of the tunnel. The value of longitudinal strain
at the fault plane is zero, and the maximum value is located near the
fault. Under strike-slip faulting, the transverse displacement of the
tunnel shows an ‘S’ shape. In addition, the results of the proposed model
agree well with the 1 g model test, but ignoring the influence of fault
zone width or nonlinear tunnel-stratum interaction will result in obvi­
ously larger results.
Comparison with the 3D FEM numerical model
To comprehensively verify the proposed model, the calculation re­
sults of tunnel displacements and forces obtained by the proposed model
are compared with the results from the 3D FEM numerical model.
Establishment of the FEM numerical model
As shown in Fig. 12, an FEM numerical model with a geometry of
Table 1
Model testing parameters [43].
Parameter
Tunnel
Fault
Stratum
Value
D
t
C
L
LF
LH
β
E
ν
W
WF
WH
Δf
γf
cf
φf
Ef
νf
γh
ch
φh
Eh
Fig. 9. Calculation flow chart.
νh
Fig. 10. Model testing equipment [43].
8
Width
Thickness
Buried depth
Length
Length in the footwall
Length in the hanging wall
Crossing angle
Young’s modulus
Poisson’s ratio
Width
Width in the footwall
Width in the hanging wall
Fault displacement
Unit weight of the fault zone
Cohesion of the fault zone
Friction angle of the fault zone
Young’s modulus of the fault zone
Poisson’s ratio of the fault zone
Unit weight of the host rock
Cohesion of the host rock
Friction angle of the host rock
Young’s modulus of the host rock
Poisson’s ratio of the host rock
0.24 m
0.015 m
0.72 m
2.8 m
1.4 m
1.4 m
90◦
0.5 GPa
0.25
0.72 m
0.36 m
0.36 m
− 0.013 m
16.3 kN/m3
3.2 kPa
26◦
17 MPa
0.45
16.8 kN/m3
11.4 kPa
38◦
108 MPa
0.35
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Table 2
Tunnel-stratum interaction parameters[42].
Parameter
Phxu
Value
Axial bearing capacity of the host rock
αh = 0.608 − 0.123
ch
0.274
0.695
11.4
0.274
0.695
− (
+(
= 0.608 − 0.123 ×
+(
−
)
)
)3
100
100 ( ch )2
11.4 2
11.4 3
ch
+1
+1
+1
+1
100
100
100
100
= 1.018
)
(
1 + K0
Phxu = παh ch D + πDCγh
tan f1 φh
2
= 3.14 × 1.018 × 11.4 × 0.24 + 3.14 × 0.24 × 0.72 × 16.8 ×
f
Pxu
Axial bearing capacity of the fault zone
= 13.306kN/m
cf
0.274
0.695
3.2
0.274
0.695
− (
α = 0.608 − 0.123
−
+(
= 0.608 − 0.123 ×
+(
)
)
)3
100 ( cf )2
100
3.2 2
3.2 3
cf
+
1
+1
+1
+1
100
100
100
100
f
Pfxu = παf cf D + πDCγf
= 1.025
)
1 + K0 (
tan f1 φf
2
= 3.14 × 1.025 × 3.2 × 0.24 + 3.14 × 0.24 × 0.72 × 16.3 ×
Phyu
Transverse bearing capacity of the host rock
1+1
× tan(0.7 × 38◦ )
2
[
Nhch = min
= 5.380kN/m
1+1
× tan(0.7 × 26◦ )
2
0.72
11.063
7.119
− (
6.752 + 0.065 ×
)2 + (
)3 , 9
0.24
0.72
0.72
+1
+1
0.24
0.24
0.72
Nhqh = 6.816 + 2.019 ×
−
0.24
= 6.367
(
0.146 ×
]
)
)
)
(
(
0.72 2
0.72 3
0.72 4
+ 7.651 × 10− 3 ×
− 1.683 × 10− 4 ×
0.24
0.24
0.24
= 11.752
Phyu = Nhch ch D + Nhqh γh CD = 6.367 × 11.4 × 0.24 + 5.180 × 16.8 × 0.72 × 0.24
f
Pyu
Transverse bearing capacity of the fault zone
[
Nfch = min
= 51.536kN/m
0.72
11.063
7.119
− (
6.752 + 0.065 ×
)2 + (
)3 , 9
0.24
0.72
0.72
+1
+1
0.24
0.24
0.72
Nfqh = 3.332 + 0.839 ×
−
0.24
= 6.367
0.090 ×
(
]
)
)
)
(
(
0.72 2
0.72 3
0.72 4
+ 5.606 × 10− 3 ×
− 1.319 × 10− 4 ×
0.24
0.24
0.24
= 5.180
Pfyu = Nfch cf D + Nfqh γf CD = 6.367 × 3.2 × 0.24 + 5.180 × 16.3 × 0.72 × 0.24
Δx
Δy
K0
f1
Axial yield displacement
Transverse yield displacement
Lateral pressure coefficient
Coating dependent factor
= 17.479kN/m
0.01 m
0.015 m
1
0.7
Fig. 11. Tunnel response comparison with the 1 g model test [43]: (a) longitudinal strain and (b) vertical displacement.
tangential friction coefficients are tan(φf) and tan(0.6φχ) [42], respec­
tively, because there is usually a plastic waterproof plate between the
tunnel and the stratum.
100 m × 200 m × 85 m is established using ABAQUS to simulate the
tunnel’s response under strike-slip faulting. Both the stratum and the
tunnel are modelled by 3D solid continuum elements. The stratum is an
ideal elastic-plastic material conforming to the Mohr–Coulomb strength
criterion, and the elastic model is adopted for the tunnel. The normal
contact behavior between the hanging wall and footwall, as well as
between the stratum and tunnel, is defined as hard contact, while the
tangential behavior is set as penalty contact. The corresponding
Parameters
Note that a large strike-slip displacement of 3 m is considered, and
the calculation includes three simplified steps:
9
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Fig. 12. Geometry and mesh of the FEM numerical model.
(1) The initial ground stress balance.
(2) The installation of tunnel lining.
(3) The strike-slip displacement Δf is applied to the model bound­
aries to simulate fault dislocation, as shown in Fig. 13.
The detailed calculation parameters are listed in Table 3. Note that
the host rock has a larger unit weight than the fault zone, and this paper
takes 21 kN/m3 and 19 kN/m3 as an example.
The tunnel-stratum interaction parameters employed in the pro­
posed analysis model are obtained by Eq. (46) and Eq. (47), as listed in
Table 4. Note that since the normal and tangential contact behaviors
between the stratum and the tunnel are set as hard and penalty contact,
respectively, the value of αχ equals 0.
Table 3
Calculation parameters.
Parameter
Tunnel
Fault
Results
Stratum
As plotted in Fig. 14, the transverse tunnel displacement and forces
(bending moment, axial force and shear force) are symmetric about the
fault. The bending moment is almost zero at the tunnel-stratum crossing
point (at x = 0 m) and increases with increasing fault distance. The axial
and shear forces reach their peak value at the tunnel-stratum crossing
point and decrease with increasing fault distance. The maximum value
of the bending moment is reached approximately 25 m away from the
Value
W
t
H
C
L
LF
LH
β
E
ν
W
WF
WH
Ef
νf
γf
cf
φf
Eh
νh
γh
ch
φh
Width
Thickness
Height
Buried depth
Length
Length in the footwall
Length in the hanging wall
Crossing angle
Young’s modulus
Poisson’s ratio
Width
Width in the footwall
Width in the hanging wall
Young’s modulus of the fault zone
Poisson’s ratio of the fault zone
Unit weight of the fault zone
Cohesion of the fault zone
Friction angle of the fault zone
Young’s modulus of the host rock
Poisson’s ratio of the host rock
Unit weight of the host rock
Cohesion of the host rock
Friction angle of the host rock
10 m
0.35 m
10 m
30 m
200 m
100 m
100 m
90◦
31.5 GPa
0.2
50
25
25
0.5 GPa
0.4
19 kN/m3
100 kPa
18◦
1.2 GPa
0.33
21 kN/m3
400 kPa
28◦
fault plane and finally converges to zero at approximately 50 m away
from the fault plane. In addition, the comparisons also show that the
calculation results of the theoretical model (i.e., analytical model I)
considering the influence of fault zone width and nonlinear tunnelstratum interaction are in good agreement with the numerical simula­
tion results. Otherwise, the tunnel response will be highly enhanced by
approximately 40 %~110 % (i.e., analytical models II and III) under this
condition.
The peak axial force, shear force and bending moment calculated by
the proposed model and the 3D FEM model are plotted in Fig. 15.
Compared with the numerical model, the relative errors of the peak
bending moment, shear force and axial force calculated by analytical
model I (considering the fault zone width and nonlinear tunnel-stratum
Fig. 13. Numerical model boundary conditions.
10
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
Table 4
Tunnel-stratum interaction parameters.
Parameter
Phxu
value
Axial bearing capacity of the host rock
αh = 0
Phxu = παh ch D + πDCγh
f
Pxu
Axial bearing capacity of the fault zone
αf = 0
)
1 + K0 (
1+1
× tan(0.6 × 28◦ )
tan f1 φh = 0 + 3.14 × 10 × 30 × 21 ×
2
2
= 5972.538kN/m
)
1 + K0 (
1+1
× tan(0.6 × 18◦ )
tan f1 φf = 0 + 3.14 × 10 × 30 × 19 ×
2
2
= 3414.226kN/m
[
]
30
11.063
7.119
Nhch = min 6.752 + 0.065 ×
− (
)2 + (
)3 , 9
10
30
30
+1
+1
10
10
Pfxu = παf cf D + πDCγf
Phyu
Transverse bearing capacity of the host rock
Nhqh = 4.565 + 1.234 ×
= 6.340
( )2
( )3
( )4
30
30
30
30
− 0.089 ×
+ 4.275 × 10− 3 ×
− 9.159 × 10− 5 ×
10
10
10
10
= 7.574
Phyu = Nhch ch D + Nhqh γh CD = 6.340 × 400 × 10 + 7.574 × 21 × 30 × 10
f
Pyu
Transverse bearing capacity of the fault zone
[
= 73075.427kN/m
Nfch = min 6.752 + 0.065 ×
Nfqh = 2.399 + 0.439 ×
30
11.063
7.119
− (
)2 + (
)3 , 9
10
30
30
+1
+1
10
10
]
= 6.340
( )2
( )3
( )4
30
30
30
30
− 0.030 ×
+ 1.059 × 10− 3 ×
− 1.754 × 10− 5 ×
10
10
10
10
= 3.473
Pfyu = Nfch cf D + Nfqh γf CD = 6.340 × 100 × 10 + 3.473 × 19 × 30 × 10
Δx
Δy
K0
f1
Axial yield displacement
Transverse yield displacement
Lateral pressure coefficient
Coating dependent factor
0.01 m
0.1 m
1.0
0.6
= 26136.879kN/m
Fig. 14. Tunnel response comparison with the 3D FEM numerical model.
11
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
16000
2500
2000
12000
1500
8000
1000
4000
500
0
0
Fig. 15. Comparison of the tunnel peak forces with different analytical models.
interaction) are 6.37 %, 11.00 % and 9.91 %, respectively; the relative
errors of the peak bending moment, shear force and axial force calcu­
lated by analytical model II (only considering the nonlinear tunnelstratum interaction) are 31.67 %, 60.01 % and 31.32 %, respectively;
and the relative errors of the peak bending moment, shear force and
axial force calculated by analytical model III (only considering the fault
zone width) are 63.00 %, 199.92 % and 77.36 %, respectively. In
addition, the average errors of analytical model I, analytical model II
and analytical model III are 9.09 %, 41.02 % and 113.43 %, respectively.
Parameter analysis
In previous tunnel and pipeline analytical models [26–36], the ulti­
mate strain of the corresponding materials is usually employed as the
criterion of structural safety. Therefore, to assess the safety of tunnels
crossing active strike-slip fault zones, the strains at tunnel hances and
Fig. 16. Strain distribution of tunnel hances under left- and right- lateral strike-slip faulting.
12
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
failure scope with different parameters are analysed in this section. The
failure scope is determined by the ultimate tensile and compressive
strain of the concrete. Additionally, the ultimate compressive strain εcu
of C35 concrete is 3300 με, and the ultimate tensile strain εtu is 100 με
[45]. The tunnel length L is 1000 m, the fault displacement Δf is 2 m, the
crossing angle β is 80◦ or 100◦ , and the other parameters are the same as
in Table 3.
strike-slip faulting both linearly increase with increasing fault
displacement. The maximum strains are 6220, 8158, 9985, 11,730 and
13,356 με for 1.0, 1.5, 2.0, 2.5 and 3.0 m of fault displacement,
respectively. Additionally, the tunnel failure scope also linearly in­
creases as the fault displacement increases. Specifically, when the fault
displacement is 1.0–3.0 m, the tunnel failure scope is 208–352 m and
79–97 m under left- and right- lateral strike-slip faulting, respectively.
Longitudinal strain distribution along the tunnel
Effect of the fault zone width W
As shown in Fig. 16, the strain distribution of the left and right tunnel
hances is symmetrical about the fault plane under strike-slip faulting.
The maximum strain is located around the fault plane, and when the
fault plane distance is greater than 150 m, the tunnel strain tends to zero.
Under left-lateral strike-slip faulting with a crossing angle β = 80◦ , the
maximum tensile strain occurs at the right tunnel hance of the hanging
wall and left tunnel hance of the footwall, while the maximum
compressive strain occurs at the left tunnel hance of the hanging wall
and right tunnel hance of the footwall; under left-lateral strike-slip
faulting with β = 100◦ , the tunnel response is symmetric with that of leftlateral strike-slip faulting-. In addition, under left-lateral strike-slip
faulting with β = 80◦ , the failure scope is governed by the tensile strain,
which is 290 m; under left-lateral strike-slip faulting with β = 100◦ , the
failure scope is governed by the compressive strain, which is 86 m. In
addition, under right-lateral strike-slip faulting, the tunnel response is
symmetric with that of left-lateral strike-slip faulting. The results show
that the failure scope under left-lateral strike-slip faulting is obviously
larger than that under right-lateral strike-slip faulting under the condi­
tion of 0◦ < β < 90◦ because the ultimate tensile strain of concrete is
much smaller than its ultimate compressive strain. However, under the
condition of 90◦ < β < 180◦ , the opposite conclusion is obtained.
Then, the effect of fault displacement Δf, fault zone width W, fault
zone cohesion cf and crossing angle β on the maximum strain and failure
scope of the tunnel is discussed under the condition of 0◦ < β < 90◦ . The
tunnel response under the condition of 90◦ < β < 180◦ can be obtained
based on Fig. 16.
The effect of the fault zone width W on the tunnel response is
investigated by changing the fault zone width as W = 0, 10, 25, 50 and
100 m. In Fig. 18, the maximum tunnel compressive and tensile strains
under left- and right- lateral strike-slip faulting both decrease with
increasing fault zone width until W > 50 m, and the maximum strain is
14224–9148 με for different fault zone widths. The tunnel failure scope
increases with increasing fault zone width. For instance, when the fault
zone width is 0–100 m, the failure scope is 282–295 m and 79–93 m
under left- and right- lateral strike-slip faulting, respectively.
Effect of the fault displacement Δf
To investigate the influence of the crossing angle β on the mechanical
responses of the tunnel, a parametric analysis is performed where the
crossing angle β is 20◦ , 40◦ , 60◦ and 80◦ . As shown in Fig. 20, the
maximum compressive strain under left- lateral strike-slip faulting and
the maximum tensile strain under right- lateral strike-slip faulting in­
crease with increasing crossing angle, while the maximum tensile strain
Effect of the fault zone cohesion cf
To investigate the influence of the fault zone cohesion cf on the
tunnel responses, five cases with different fault zone cohesion cf are
investigated, with values increasing from 50 kPa to 400 kPa. Fig. 19
shows that the maximum tension and compressive strains monotonically
increase with increasing fault zone cohesion, and the maximum strain is
9748–11702 με for different fault zone cohesions. In contrast, the failure
scope decreases with increasing fault zone cohesion. For instance, when
the fault zone cohesion is 50–400 kPa, the failure scope is 292–281 m
and 89–78 m under left- and right- lateral strike-slip faulting, respec­
tively. This conclusion is also obtained by Zhong et al. [21], which is
because the stratum bearing capacity and stiffness increase with
increasing cohesion according to Eq. (44) ~ Eq. (47).
Effect of the crossing angle β
To investigate the influence of the fault displacement Δf on the
tunnel responses, a parametric analysis is performed where the fault
displacement Δf is 1.0, 1.5, 2.0, 2.5 and 3.0 m. In Fig. 17, the maximum
tunnel compressive and tensile strains under left- and right- lateral
400
130
120
300
200
110
100
90
100
0
Fig. 17. Tunnel response under strike-slip faulting with different fault displacements.
13
80
70
0.0
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
300
295
290
285
110
100
90
280
275
270
265
80
70
260
60
0.00
0.0
Fig. 18. Tunnel response under strike-slip faulting with different fault zone widths.
300
100
295
95
290
90
285
280
85
275
80
270
75
265
260
70
0.0
0.0
Fig. 19. Tunnel response under strike-slip faulting with different fault zone strengths.
700
350
600
300
500
250
400
200
300
150
200
100
100
50
0
0
Fig. 20. Tunnel response under strike-slip faulting with different crossing angles.
14
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
under left- lateral strike-slip faulting and the maximum compressive
strain under right- lateral strike-slip faulting first increase and then
decrease with increasing crossing angle. Note that under the condition of
90◦ < β < 180◦ , the opposite conclusion is obtained. Because the axial
component of the fault displacement Δfx decreases with increasing
crossing angle in the condition of 0◦ < β < 90◦ and decreasing crossing
angle in the condition of 0◦ < β < 90◦ , resulting in the reduction of axial
force N.
the maximum strain is located around the fault plane. Under leftlateral strike-slip faulting with 0◦ < β < 90◦ , the failure scope is
governed by the tensile strain. Under right-lateral strike-slip
faulting with 0◦ < β < 90◦ , the failure scope is governed by the
compressive strain. The failure scope under left-lateral strike-slip
faulting is obviously larger than that under right-lateral strikeslip faulting because the tensile strength of concrete is much
smaller than its compressive strength. However, under the con­
dition of 90◦ < β < 180◦ , the opposite conclusion is obtained.
(4) The value of the tunnel maximum strain is directly related to the
fault displacement Δf and fault zone cohesion cf and has an in­
verse relationship with the fault zone width Wf; the tunnel failure
scope is directly related to the fault displacement Δf, fault zone
width Wf, and crossing angle β (0◦ < β < 90◦ ) and has an inverse
relationship with the crossing angle β (90◦ < β < 180◦ ) and fault
zone cohesion cf.
Conclusion
In this work, a mechanical analysis model for tunnels under strikeslip faulting considering the fault zone width and nonlinear tunnelstratum interaction is first established. More specifically, tunnels
crossing active fault zones are simplified as an elastic beam acting on a
nonlinear foundation; the nonlinear tunnel-stratum interaction is
considered by a series of nonlinear axial and transverse nonlinear
springs; the different mechanical characteristics of the fault zone and
host rock are described by different spring stiffnesses; and the nonlinear
tunnel-stratum interaction is analysed by the finite difference method
and a novel iterative method for nonlinear spring stiffness. The con­
clusions are summarized as follows:
Additionally, the range of application of this proposed model and its
limitations should be noted as follows:
• The proposed model applies to strike-slip faults but does not extend
to dip-slip faults.
• In this model, the tunnel is assumed to possess elastic properties,
while disregarding the elastic-plastic characteristics of concrete.
• The proposed model only compares the strain and displacement of
the tunnel with the model tests. The stratum pressure and tunnel
failure scope should be considered in future works.
(1) The displacement, bending moment, shear force and axial force of
the proposed model agree well with the results obtained from
model tests and numerical simulations. Compared with the nu­
merical model, the relative errors of the peak bending moment,
shear force and axial force calculated by the proposed model are
6.37 %, 11.00 % and 9.91 %, respectively, and the average
relative error is 9.09 %.
(2) The proposed model significantly improves the accuracy of
existing analytical models. More specifically, neglecting the ef­
fect of the fault zone width or nonlinear tunnel-stratum interac­
tion, the tunnel response will be misestimated, e.g., when the
fault displacement is 3 m and the crossing angle is 90◦ , the
overestimated value is approximately 40 % to 110 %. The relative
average errors of analytical model I (considering the fault zone
width and nonlinear tunnel-stratum interaction), analytical
model II (only considering the nonlinear tunnel-stratum inter­
action) and analytical model III (only considering the fault zone
width) are 9.09 %, 41.02 % and 113.43 %, respectively.
(3) The strain distribution of the left and right tunnel hances is
symmetrical about the fault plane under strike-slip faulting, and
Declaration of competing interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgement
The authors are grateful for the support from the National Natural
Science Foundation of China (52378411 and 52208404).
Appendix A
⎡
− 2
⎢ χ
⎢ kx,0
⎢
⎢ 1
⎢
⎢ χ
⎢ kx,1
⎢
⎢
EA ⎢
⎢
[A1 ] = 2 ⎢ 0
l ⎢
⎢
⎢
⎢ 0
⎢
⎢
⎢ ...
⎢
⎢
⎣
0
⎤
2
χ
kx,0
0
0
...
− 2
χ
kx,1
1
χ
kx,1
0
...
1
χ
kx,2
− 2
χ
kx,2
− 1
χ
kx,2
...
...
1
χ
kx,3
...
− 2
χ
kx,3
...
0
0
0
0
...
...
...
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
... ⎥
⎥
⎥
− 2⎦
χ
kx,n
(A1)
(n+1)×(n+1)
15
X. Zhang et al.
⎡
1
⎢0
⎢
⎢0
[A2 ] = ⎢
⎢0
⎢
⎣ ...
0
Transportation Geotechnics 44 (2024) 101171
0 0
1 0
0 1
0 0
... ...
0 0
0
0
0
1
...
0
⎤
0
0⎥
⎥
0⎥
⎥
0⎥
⎥
... ⎦
1 (n+1)×(n+1)
...
...
...
...
...
...
(A2)
Appendix B
⎡
2
⎢ kχ
⎢ y,0
⎢
⎢− 2
⎢
⎢ χ
⎢ ky,1
⎢
⎢
1
EI ⎢
⎢
[B1 ] = 4 ⎢ kχ
l ⎢ y,2
⎢
⎢
⎢ 0
⎢
⎢
⎢ ...
⎢
⎢
⎣
0
2
χ
ky,0
0
...
5
χ
ky,1
− 4
χ
ky,1
1
χ
ky,1
...
− 4
χ
ky,2
6
χ
ky,2
4
χ
ky,2
...
1
χ
ky,3
...
− 4
χ
ky,3
...
6
χ
ky,3
...
0
0
0
⎡
− 2N0
⎢ kχ
⎢ y,0
⎢
⎢ N
⎢ 1
⎢ χ
⎢ ky,1
⎢
⎢
1⎢
⎢
[B2 ] = 2 ⎢ 0
l ⎢
⎢
⎢
⎢ 0
⎢
⎢
⎢ ...
⎢
⎢
⎣
0
⎤
− 4
χ
ky,0
−
...
...
...
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
... ⎥
⎥
⎥
2 ⎦
χ
ky,n
(B1)
(n+1)×(n+1)
⎤
2N0
χ
ky,0
0
0
...
− 2N1
χ
ky,1
N1
χ
ky,1
0
...
N2
χ
ky,2
− 2N2
χ
ky,2
N2
χ
ky,2
...
...
N3
χ
ky,3
...
− 2N3
χ
ky,3
...
0
0
0
0
...
...
...
0
⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
... ⎥
⎥
⎥
− 2Nn ⎦
χ
ky,n
(B2)
(n+1)×(n+1)
⎡
⎤
0
⎢
⎢ − (N2 − N0 )
⎢
χ
⎢
ky,1
⎢
⎢
⎢
0
1 ⎢
⎢
[B3 ] = 2 ⎢
4l ⎢
⎢
⎢
⎢
0
⎢
⎢
⎢
...
⎣
0
⎡
1 0
⎢0 1
⎢
⎢0 0
[B4 ] = ⎢
⎢0 0
⎢
⎣ ... ...
0 0
0 0
0 0
1 0
0 1
... ...
0 0
0
0
N2 − N0
χ
ky,1
0
− (N3 − N1 )
χ
ky,2
0
0
...
0
...
(N3 − N1 )
χ
ky,2
...
0
...
...
− (N4 − N2 )
χ
ky,3
...
...
...
0
0
0
...
0
0
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
⎥
⎥
⎥
0 ⎥
⎥
⎥
... ⎥
⎦
0
(B3)
(n+1)×(n+1)
⎤
... 0
... 0 ⎥
⎥
... 0 ⎥
⎥
... 0 ⎥
⎥
... ... ⎦
... 1 (n+1)×(n+1)
(B4)
Appendix C. [44]
√̅̅̅̅̅̅̅̅̅̅̅
Eχ D4
ky = [
]
EI
D 1 − (νχ )2
χ
1.3Eχ
(C1)
12
16
X. Zhang et al.
Transportation Geotechnics 44 (2024) 101171
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