Potential Roof Collapse Analysis of Tunnel Considering the Orthotropic Weak Interlayer on the Detaching Surface

Abstract

The evaluation of the roof collapse in tunnels or cavities remains one of the most complex issues in geotechnical engineering. Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, an analytical approach for determining the limit collapse range considering the arch effect of the tunnel is presented in this paper by the variation calculus. A discontinuity criterion moving from the anisotropic criterion proposed by the present authors is applied to the orthotropic interlayer. The phenomenon of sharp points in collapse blocks is further analyzed. Based on the proposed approach, illustrated examples are analyzed to investigate the effect of the strength parameters and the consideration of the collapse cusp, which show different influence laws on the range of collapse blocks. Those interesting conclusions can provide guidance for the prediction of the collapse mechanism of the tunnel.

Full text

Research Article
Potential Roof Collapse Analysis of Tunnel Considering the
Orthotropic Weak Interlayer on the Detaching Surface
Tong Xu ,
1
Dingli Zhang,
1
Zhenyu Sun,
1
Lin Yu,
1
Ran Li,
2
and Jiwei Luo
3
1
Key Laboratory of Urban Underground Engineering of Ministry of Education, Beijing Jiaotong University, Beijing 100044, China
2
China Tiesiju Civil Engineering Group Co., Ltd., Hefei 230023, China
3
Tianjin Research Institute for Water Transport Engineering, M.O.T, Tianjin 300456, China
Correspondence should be addressed to Tong Xu; 16115280@bjtu.edu.cn
Received 11 July 2022; Accepted 4 August 2022; Published 27 September 2022
Academic Editor: Pengjiao Jia
Copyright ©2022 Tong Xu et al. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
e evaluation of the roof collapse in tunnels or cavities remains one of the most complex issues in geotechnical engineering.
Taking the detaching surface of the tunnel roof collapse as an orthotropic weak interlayer, an analytical approach for determining
the limit collapse range considering the arch effect of the tunnel is presented in this paper by the variation calculus. A discontinuity
criterion moving from the anisotropic criterion proposed by the present authors is applied to the orthotropic interlayer. e
phenomenon of sharp points in collapse blocks is further analyzed. Based on the proposed approach, illustrated examples are
analyzed to investigate the effect of the strength parameters and the consideration of the collapse cusp, which show different
influence laws on the range of collapse blocks. ose interesting conclusions can provide guidance for the prediction of the
collapse mechanism of the tunnel.
1. Introduction
e stability problems of tunnels have always been of
overriding significance in geotechnical engineering. e
possible collapse of the tunnel remains one of the most
challenging problems. Due to the natural uncertainties of the
properties of the rock mass in situ, such as mechanical
parameters and the random variability of cracks or fractures
[1–8], the collapse mechanism of a cavity roof has yet to be
thoroughly grasped [9]. Because the limit analysis method
requires no elastic characterization and only refers to the
limit behavior, this approach can obtain more rigorous
results with fewer assumptions [10]. As a result, the limit
analysis method is very suitable for analyzing the collapse
mechanism of tunnel roofs and has been rapidly developed
in recent years.
Lippmann [11] firstly applied the limit analysis method
to the roof stability problems of tunnels considering the
Mohr–Coulomb (M-C) criterion. For many years, the roof
stability of tunnels is analyzed in this framework [10].
Guarracino and Guarracino [12] made encouraging progress
with the help of plasticity theory and calculus of variations,
and a closed-form solution of the collapsed outline was
obtained with the Hoek–Brown (H-B) criterion considered
instead of the M-C rule. Since then, many researchers
furthered their work by considering various cases of cavities
such as different excavation profiles [13], layered rock
masses or soils [9, 14–17], the presence of the karst cave [18],
the solutions for shallow tunnels [19–22] or progressive
collapse [23–26], consideration of the supporting pressure
[27–29], and the case considering the groundwater [30–33].
ese extending works moving from the approaches of
Guarracino and Guarracino [12] only focused on the H-B
rule expressed in the M-C form (nonlinear). In fact, a weak
interlayer may appear between the detaching surface when
roof collapse occurs [34]. e rock mass at the detaching
surface of the collapse zone can be taken as a weak interlayer
with thin thickness, which is related to the failure mecha-
nism of the surrounding rock [35, 36]. Under the influence
of the dislocation of the rock masses, the weak interlayer
exhibits different strength characteristics in the orthogonal
direction. For this reason, analysis considering the
Hindawi
Advances in Civil Engineering
Volume 2022, Article ID 3100011, 12 pages
https://doi.org/10.1155/2022/3100011

orthotropic characteristics of the weak interlayer between
the detaching surface can better describe the roof collapse
problems of tunnels or cavities. is consideration requires a
special criterion that can describe the failure behavior of the
orthotropic weak interlayer on the detaching surface.
In addition, we notice that most researchers obtained a
smooth collapse curve, which can be derivable at the axis of
symmetry [24]. In fact, a collapse cusp (not derivable at the
axis of symmetry) is usually observed in model tests or
numerical analysis [34, 37, 38], which means that the
condition at the axis of symmetry should be treated with
caution (Figure 1). To further explain this phenomenon, the
sharp point of the collapse curve is discussed in our study.
Once the assumption of a smooth curve (at the axis of
symmetry) is not applied to the analysis, it becomes more
difficult to get the collapse curve. As a result, we need to find
a reasonable restriction as an alternative to the smooth
assumption when considering the collapse cusp.
Based on the above considerations, a discontinuity yield
criterion for an orthotropic interlayer that moves from a
pressure-dependent, anisotropic criterion is applied in this
research. en, the theoretical formulas for the cases with
and without considering the collapse cusp are deduced to
figure up the collapse block. Finally, some examples are
analyzed, and the discrepancy between different cases is
further discussed in this paper. e results can help con-
stitute guidance for the prediction of the collapse range of
tunnels or cavities.
2. Problem Description
2.1. Orthotropic Criterion at the Velocity Discontinuity.
e orthotropic yield criterion can move from the aniso-
tropic criterion. Given the pressure-dependent of the rock
material, Caddell et al. [39] proposed an anisotropic yield
criterion in the following form:
%Ayz σy−σz
􏼐 􏼑2+Azx σz−σx
 􏼁2+Axy σx−σy
􏼐 􏼑2
+Byzτ2
yz +Bzxτ2
zx +Bxyτ2
xy +K1σx+K2σy+K3σz�1,
(1)
where the parameters A
yz
,A
zx
,A
xy
,B
yz
,B
zx
,B
xy
,K
1
,K
2,
and
K
3
characterize the properties of anisotropy. e subscript x,
y, and zdenote the reference axes of anisotropy. In con-
sideration of the orthotropic materials, these parameters
satisfy the following relations:
Ayz �Azx, Byz �Bzx , Bxy �2Ayz +2Axy
􏼐 􏼑, K1�K2.(2)
Because the detaching surface is consistent with the weak
layer, we take the normal direction of the detaching surface
as the zaxis (Figure 2). As a result, the failure on the
detaching surface only depends on σ
z
,τ
zy,
and τ
zx
, which
leads to an orthotropic yield criterion in the degenerative
form:
Bzx τ2
zx +τ2
zy
􏼐 􏼑+K3σz�1.(3)
For the plane strain problems, the equation (3) can be
further simplified as
Bτ2+Kσn�1,(4)
where σ
n
denotes the stress of normal direction (the com-
pressive stress is taken as positive in this paper), and the
parameters Band Kcan be determined according to the
shear (τ
0
) and tensile (σ
T
) strengths of the weak interlayer on
the detaching surface.
B�1
τ2
0
, K � − 1
σT
.(5)
On the basis of the above consideration, the disconti-
nuity yield criterion at the detaching surface of velocity can
be obtained as
f�τ2−τ2
0σ−1
Tσn−τ2
0�0.(6)
2.2. Collapse Mechanism of the Tunnel Roof. e key point
about the roof stability of tunnels or cavities is to determine
the shape and range of the potential collapsing blocks
(Figure 3). As it is usual, this paper considers the problem in
a plane and only makes reference to the cross section of a
long tunnel or cavity. e rock material is assumed to be
ideally plastic, and the plastic strain rate follows the asso-
ciated flow rule. Besides, strain within the collapsing body is
regarded as insignificant when the roof collapse occurs
(rigid-plastic behavior). Based on the above conditions, the
shape of the potential collapsing region can be given by using
the calculus of variations [12, 40].
In order to investigate the roof collapse on account of the
gravity field and refer to the upper bound principle [41], a
kinematically admissible field of vertical velocity, which
fulfills the compatibility with the strain rates, must be as-
sumed at first [42]. As shown in Figure 3, the collapse ve-
locity _
uvis in the negative direction of the y-axis, and the
symmetrical collapse curve is expressed as f(x). Moreover, as
shown in Figure 4, the value of the vertical velocity is
considered a variable that decreases from _
u(x�0) to zero
(x�R) linearly. As a result, the field of the variable vertical
velocity can be expressed as
_
uv�_
u1−x
R
􏼒 􏼓.(7)
According to the geometric conditions, the plastic strain
rate (the tensile strain rate is taken as negative) components
in the tangential (_
c) and normal (_
εn) directions can be
obtained as
_
c� − 1−x
R
􏼒 􏼓 _
u
w
􏼠 􏼡f′(x)1+f′(x)2
􏽨 􏽩−1
2,
_
εn� − 1−x
R
􏼒 􏼓(_
u/w)1+f′(x)2
􏽨 􏽩−1
2.
(8)
2Advances in Civil Engineering

(a)
Collapse cusp
(b)
Figure 1: Roof collapse in (a) an active trapdoor numerical test [34]; (b) a model test [38].
weak layer
Z
detaching surface
collapse curve
x
Figure 2: Orthotropic interlayer on the detaching surface.
Free boundary Sym
Sf (x)
c (x)
dx
dy
uv
nw
x
y
.
Figure 3: Possible collapsing area of the tunnel roof.
Advances in Civil Engineering 3

Coincident with the failure criterion mentioned in
Section 2.1 (obeying to the associated flow rule), the plastic
potential function ξcan be expressed as
ξ�τ2−τ2
0σ−1
Tσn−τ2
0.(9)
Further, the plastic strain rate can also be written in the
form:
_
c�λzξ
zτ�2λτ,
_
εn�λzξ
zσn
� −λτ2
0σ−1
T.
(10)
e association of equations (7) and (9) leads to the
following results:
λ�τ−2
0σT1−x
R
􏼒 􏼓 _
u
w
􏼠 􏼡 1+f′(x)2
􏽨 􏽩−1
2(11)
τ� −τ2
0σ−1
T
2f′(x).(12)
Finally, by substituting equations (10) into (6), we can
obtain
σn�τ2
0σ−1
T
4f′(x)2−σT.(13)
According to the equations (11) and (12), the tangential
and normal stress components are expressed by using the
derivative of the collapse function. Because a cusp (Figure 1)
can occur in roof collapse [34, 37, 38], the stress at the axis of
symmetry should be treated with caution (no derivative). In
particular, the shear stresses around the collapse cusp can be
described in Figure 5. Based on symmetry, the magnitude of
the shear stresses in the symmetrical tilt directions at the
cusp point must be equal. As a result, the inner horizontal
shear stress at the axis of symmetry naturally satisfies the
condition of being equal to zero, so long as the collapse
curves on both sides are symmetrical to each other.
3. Analysis without Considering Collapse Cusp
Because most researchers assumed a smooth collapse curve
in their studies [23, 24], the collapse curve is derivable at the
axis of symmetry, which must lead to zero of derivative
function f′(x). For comparison, we analyze the collapse
curve without considering collapse cusp in this section.
Meanwhile, a different criterion (i.e., the orthotropic yield
criterion proposed in section 2.1) is applied at the velocity
discontinuity.
Associating the equations (7), (11), and (12), the dissi-
pated power density of the internal stresses at the discon-
tinuity (
_
D) is expressed as
_
D�σn
_
εn+τ_
c
�τ2
0σ−1
T
4f′(x)2+kσT
􏼠 􏼡/w����������
1+f′(x)
􏼂 􏼃2
􏽱
􏼒 􏼓􏼨 􏼩 1−x
R
􏼒 􏼓_
u.
(14)
Besides, the power density of the applied loads is
_
We�c[f(x) − c(x)] 1−x
R
􏼒 􏼓_
u, (15)
where cdenotes the gravity per unit volume of the rock
mass.
Here, we consider the right half of the symmetrical block
(with respect to the y-axis). e total dissipated power of the
collapse system is further deduced as
_
U�􏽚L
0
_
Dw ����������
1+f′(x)
􏼂 􏼃2
􏽱dx −􏽚L
0
_
Wedx
�􏽚L
0
F f(x), f′(x), x
􏼂 􏼃_
udx,
(16)
where the F[f(x), f′(x), x]can be expressed as
F f(x), f′(x), x
􏼂 􏼃�
τ2
0σ−1
T
4f′(x)2+σT−c[f(x) − c(x)]
􏼨 􏼩 1−x
R
􏼒 􏼓_
u.
(17)
Because the effective collapse curve can be obtained
when the total dissipation power makes a minimum [13], the
problem can be solved by using the calculus of variations. In
order to obtain an extremum of the total dissipated power
_
U
over the interval of 0-L, the functional Fmust satisfy Euler’s
equation:
horiz
cusp
= 0
collapse cusp
collapse curve
collapse zone
axis of symmetry
Figure 5: e shear stresses around the collapse cusp.
R
ux=0 = u
..
ux=R =0
.
Figure 4: e field of the variable vertical velocity.
4Advances in Civil Engineering

δ
_
U�0⇒zF
zf(x)−d
dx
zF
zf′(x)
􏼠 􏼡�0.(18)
From equation (14), we can deduce that
zF
zf(x)� −c1−x
R
􏼒 􏼓_
u, zF
zf′(x)
�1
2τ2
0σ−1
Tf′(x)1−x
R
􏼒 􏼓_
u,
d
dx
zF
zf′(x)
􏼠 􏼡�1
2τ2
0σ−1
Tf″(x)1−x
R
􏼒 􏼓−1
Rf′(x)
􏼔 􏼕_
u.
(19)
By substituting equations (16) into (15), it is
−c1−x
R
􏼒 􏼓−1
2τ2
0σ−1
Tf″(x)1−x
R
􏼒 􏼓−1
Rf′(x)
􏼔 􏼕􏼚 􏼛_
u�0.
(20)
Integrating the equation (17), we can obtain the first
derivative of f(x) as follows:
f′(x) � cτ−2
0σTR−x−R2
R−x
􏼠 􏼡+C1R
R−x.(21)
Here, C
1
is an unknown parameter which needs to be
further determined. Similar to existing studies, f′(x�0)
should be equal to zero because a smooth symmetrical
collapse curve is assumed in this section, which results in
C
1
�0. en, the collapse curve f(x) can be deduced by
integrating the equation (18):
f(x) � cτ−2
0σTRx −1
2x2+R2ln (R−x)
􏼒 􏼓+C2,(22)
where C
2
is a pending parameter. Considering an implicit
constraint f(x�L)�0, we can obtain that
C2� −cτ−2
0σTRL −1
2L2+R2ln (R−L)
􏼒 􏼓,(23)
c(x) � ������
R2−x2
􏽱−������
R2−L2
􏽰.(24)
en, L can be further determined by equating the total
dissipated power to zero. Substituting equations (15) and
(20)–(23) into (14), we can obtain the equation which L
yields. It is
c2σTR3
4τ2
0
L
R+1
2
L
R
􏼒 􏼓2
−L
R
􏼒 􏼓3
+1
4
L
R
􏼒 􏼓4
−ln R
R−L
􏼒 􏼓􏼨 􏼩
−cR2
3−R2
2arcsin L
R
􏼒 􏼓+R
6
3L
R−L
R
􏼒 􏼓2
−2
􏼢 􏼣 R2−L2
􏼐 􏼑1
2
⎧
⎪
⎨
⎪
⎩⎫
⎪
⎬
⎪
⎭
+σTL
22−L
R
􏼒 􏼓�0
(25)
Note that equation (23) can be easily solved by using the
numerical method. After L is obtained, the collapse curve is
finally written as
f(x) �
cτ−2
0σTR(x−L) + 1
2L2−x2
􏼐 􏼑+R2ln R−x
R−L
􏼒 􏼓􏼒 􏼓,(x≥0)
cτ−2
0σTR(−x−L) + 1
2L2−x2
􏼐 􏼑+R2ln R+x
R−L
􏼒 􏼓􏼒 􏼓,(x<0)
⎧⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
.
(26)
4. Analysis Considering Collapse Cusp
When a possible collapse cusp (Figure 5) is considered in the
collapse analysis, the condition at the axis of symmetry
should be handled with care due to no derivative. As a result,
C
1
in equation (19) cannot be simply determined by equating
f′(x�0)with zero. So, f(x) coming from equation (18)
should be written as
f(x) � cτ−2
0σTRx −1
2x2+R2ln (R−x)
􏼒 􏼓
−C1Rln (R−x) + C2.
(27)
en, the pending parameter C
2
can be deduced by
considering f(x�L)�0. It results
C2� −cτ−2
0σTRL −1
2L2+R2ln (R−L)
􏼒 􏼓+C1Rln (R−L).
(28)
Substituting equations (17), (24), (27), and (28) into (14)
and equating the total dissipated power to zero, C
1
and L
yield
c2σTR3
4τ2
0
L
R+1
2
L
R
􏼒 􏼓2
−L
R
􏼒 􏼓3
+1
4
L
R
􏼒 􏼓4
−ln R
R−L
􏼒 􏼓􏼨 􏼩
−cR2
6
2−3 arcsin L
R
􏼒 􏼓
+3L
R−L
R
􏼒 􏼓2
−2
􏼢 􏼣 1−L
R
􏼒 􏼓2
􏼠 􏼡
1
2
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎭
+σTL
22−L
R
􏼒 􏼓+C2
1τ2
0R
4σT
ln R
R−L
􏼒 􏼓�0.
(29)
In addition, the fracturing azimuth is related to the
friction angle (φ) of the surrounding rock. Taking a stress
element at the collapse cusp of the tunnel and considering
the friction angle (Figure 6), we can get the angle between
directions of the fracture and maximum principal stress (the
horizontal and vertical shear stresses, i.e., τ
horiz
and τ
verti
, are
zero at the axis of symmetry) as (π/4 −φ/2). en, the one-
sided derivative f+
′(x)can be expressed as
Advances in Civil Engineering 5

f+
′(x�0) � C1� −tan π
4−φ
2
􏼒 􏼓.(30)
e friction angle corresponding to the M-C criterion at
the collapse cusp can be described in Figure 7. Considering
equation (30) and the approximate geometric conditions of
σ
n
and τ, the friction angle φyields
tan φtan2π
4−φ
2
􏼒 􏼓−2 tan π
4−φ
2
􏼒 􏼓
+4τ2
0σTτ0−σTtan φ
 􏼁�0.
(31)
Finally, by considering equations (29), (30), and (31)
together, we can get C
1
and Lnumerically. en, the collapse
curve considering a collapse cusp at the axis of symmetry
(x�0) can be written as
f(x) �
cτ−2
0σTR(x−L) + 1
2L2−x2
􏼐 􏼑+R2ln R−x
R−L
􏼒 􏼓􏼒 􏼓−C1Rln R−x
R−L
􏼒 􏼓,(x≥0)
cτ−2
0σTR(−x−L) + 1
2L2−x2
􏼐 􏼑+R2ln R+x
R−L
􏼒 􏼓􏼒 􏼓−C1Rln R+x
R−L
􏼒 􏼓,(x<0)
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
.(32)
5. Examples and Discussion
5.1. e Discrepancy between Different Cases. In the pre-
ceding sections, two different analytical results are obtained by
considering and not considering the collapse cusp, respectively.
e two results obtained under different conditions are
compared through an example. e parameters involved in the
example are τ
0
�20 kPa, σ
T
�22 kPa, c�25 kN/m
3
, and
R�3 m. As a result, the comparison between these two ana-
lytical results of the collapse curve is shown in Figure 8.
e results indicate that the collapse curve obtained by
considering the collapse cusp is higher than that obtained
without considering the collapse cusp. According to the
analytical results obtained in this paper, the height of the
collapse block can increase by 0.78 m when considering the
collapse cusp in the analysis. e span of the collapse curve
does not change whether the cusp is considered or not. Due
to the increase in the collapse height, the weight of the
collapse block will also increase. e gravity of the collapse
block can be calculated by using the following equation:
collapse curve
collapse zone
fracture direction
==0
42
3
axis of symmetry
verti horiz
ττ
πφ
σ1
σ
Figure 6: Microdescription of the fracture direction at the collapse
cusp.
M-C criterion
discontinuity criterion
σ
T
σ
3
σ
n
σ
1
σ
τ
n
τ
0
τ
φ
Figure 7: Acquisition of the friction angle by applying the M-C
criterion.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
H/m
X/m
no collapse cusp
with collapse cusp
variation of method by Fraldi and Guarracino
Figure 8: Comparison between cases considering and not con-
sidering the collapse cusp.
6Advances in Civil Engineering

P�2􏽚L
0
c[f(x) − c(x)]dx. (33)
Substituting equations (24) and (30) into (31), respec-
tively, we can easily calculate the gravity corresponding to
the collapse block for both cases. e results show that the
gravity obtained by considering the collapse cusp is in-
creased by 23% compared to the gravity obtained without
considering the collapse cusp. erefore, taking the collapse
cusp at the axis of symmetry into account when predicting
the collapse block can help ensure the safety of the tunnel
roof.
5.2. Comparison with Numerical Analysis. A numerical
analysis has been performed to further verify the above
analyses. e numerical parameters are consistent with those
mentioned in Section 5.1. e M-C friction angle can be
calculated from equation (29). e results for the example in
terms of vertical velocities have been obtained by FLAC3D.
e collapse block described the sudden change of vertical
velocities as shown in Figure 9.
By comparing the collapse block shapes as shown in
Figure 9, it is worth highlighting the similarity of the collapse
block shape from numerical analysis to that obtained by the
proposed analytical method. Both analytical results can
Velocity Z
-2.2500E-05
-2.2550E-05
-2.2600E-05
-2.2650E-05
-2.2700E-05
-2.2750E-05
-2.2800E-05
-2.2850E-05
-2.2900E-05
-2.2950E-05
-2.3000E-05
-2.3050E-05
-2.3100E-05
-2.3150E-05
-2.3200E-05
-2.3250E-05
-2.3300E-05
-2.3300E-05
No coll apse cusp
With collapse cusp
Sudden change of velocity
Figure 9: Comparison of collapse block shapes in terms of vertical velocities.
-2-10 1 2
2
3
4
5
6
7
Z (m)
X (m)
τ0=10 kPa
τ0=15 kPa
τ0=20 kPa
τ0=25 kPa
τ0=30 kPa
(a)
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
2
3
4
5
6
7
8
Z (m)
X (m)
σT=11 kPa
σT=16.5 kPa
σT=22 kPa
σT=27.5 kPa
σT=33 kPa
(b)
Figure 10: Effects of strength parameters involved in our analysis on the potential collapse of the tunnel roof without considering the
collapse cusp: (a) shear strength; (b) tensile strength.
Advances in Civil Engineering 7

change with
σ
T
change with
τ
0
H/H0
Variation (%)
50 75 100 125 150
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.7
1.5
1.3
1.1
0.9
0.7
0.5
0.3
1.5
H/H0
Var ati on of σT
Varation of τ0
1.25
1
0.75
0.5 0.5 0.75 11.25 1.5
(a)
Variation (%)
50 75 100 125 150
Var ati on of σT
Varation of τ
0
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
1.5
1.25
1
0.75
0.5 0.5 0.75 11.25 1.5
L/L0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
L/L0
change with
σ
T
change with
τ
0
(b)
Variation (%)
50 75 100 125 150
Var ati on of σT
Varation of τ
0
2.5
2
1.5
1
0.5
0
1.5 1.25
1
0.75
0.5 0.5 0.75 11.25 1.5
P/P0
P/P0
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
change with
σ
T
change with
τ
0
(c)
Figure 11: Comparison of the influences of shear and tensile strengths on (a) H/H
0
; (b) L/L
0
; (c) P/P.
8Advances in Civil Engineering

describe the collapse block shape well. As described in
Section 5.1, the analytical result will lead to a wider range of
collapse blocks when considering the phenomenon of sharp
points in roof collapse behavior.
5.3. Influence of Strength Parameters of the Weak Interlayer.
In the process of predicting the potential collapse of the
tunnel roof, the shear and tensile strengths of the weak
interlayer on the detaching surface are involved in our
analysis. In order to investigate the influence of these two
important parameters, we first discuss the different cases
without considering the collapse cusp. Figures 10(a) and
10(b) show the different results considering different values
of the shear and tensile strengths, respectively.
Figure 10(a) shows that the collapse curves obtained from
our proposed analytical result are significantly affected by the
shear strength of the weak interlayer on the detaching surface.
e width and the height of the collapse block increase as the
shear strength increases. Obviously, with the increase in shear
strength, a greater gravity of surrounding rock can be
maintained in the short term, but it also means that once the
collapse occurs, there will be a wider range of primary failures.
Figure 10(b) shows that the collapse curves obtained
from our proposed analytical result are also significantly
affected by the tensile strength of the weak interlayer on the
detaching surface. e height of the collapse block increases
as the tensile strength increases. However, the width of the
collapse block decreases as the tensile strength increases,
which is different from the effect of the shear strength.
Similar to the influence of shear strength, although the
increase in tensile strength may maintain a greater gravity of
surrounding rocks, there will be a wider range of primary
failure once the collapse occurs.
Variation (%)
50 75 100 125 150
Var ati on of σT
Varation of τ
0
1.3
1.1
0.9
0.7
0.5
1.5
1.25
1
0.75
0.5
ΔH/ ΔH0
ΔH/ ΔH0
0.7
0.8
0.9
1.0
1.1
0.5 0.75 11.25 1.5
change with
σ
T
change with
τ
0
(a)
Variation (%)
50 75 100 125 150
Var ati on of σT
Varation of τ
0
1.5
1.25
1
0.75
0.5
ΔP/ΔP0
ΔP/ΔP0
1.6
1.4
1.2
1
0.6
0.8
0.4
0.5 0.75 11.25 1.5 0.7
0.6
0.8
0.9
1.1
1.0
1.2
1.3
change with
σ
T
change with
τ
0
(b)
Figure 12: Comparison of the influences of shear and tensile strengths on the discrepancy between considering and not considering collapse
cusp: (a) ΔH/ΔH
0
; (b) ΔP/ΔP
0
.
Advances in Civil Engineering 9

Furthermore, in order to compare the different effects of
shear and tensile strengths on the collapse block, the changes
in height, width, and gravity of the collapse block are shown
in Figures 11(a), 11(b), and 11(c), respectively. e example
described in Section 5.1 is taken as the original case. By
comparing the change rates of each index with different
strength parameters, we can find that the height of the
collapse block is more sensitive to the change in tensile
strength. However, the width and the gravity of the collapse
block are more sensitive to the change in shear strength.
As described in Section 5.1, the height and weight of the
collapse block increase when considering the collapse cusp
compared to those without considering the collapse cusp,
while the span of the collapse curve does not change whether
the cusp is considered or not. As a result, when considering
the collapse cusp, only the changes in the collapse block
height (ΔH) and gravity (ΔP) relative to the results without
considering the collapse cusp are discussed. Figure 12 shows
the comparison of the influences of shear and tensile
strengths on the discrepancies of the collapse block height
and gravity.
According to Figure 12, the discrepancies in the collapse
block height and gravity between the two cases generally
decrease with the increase in the tensile strength. On the
contrary, the discrepancies in the collapse block height and
gravity between the two cases increase with the increase in
the shear strength. It is worth noting that these changes are
more sensitive to the change of shear strength than the
change of tensile strength, which is of directive functions
when considering the effect of the consideration of the
collapse cusp.
6. Conclusions
By using the orthotropic yield criterion which moves from
the anisotropic criterion proposed by Caddell et al. [39] for
the rock material, an exact solution to tunnel roof collapse
has been obtained with the help of the traditional plasticity
theory and the calculus of variations. In order to further
illustrate the impact of collapse cusps which have been
observed in previous studies [34, 37, 38], two different cases
according to whether the collapse cusp is considered are
analyzed in this paper. Our new theoretical results lead to the
following conclusions:
(1) Taking the detaching surface of the tunnel roof
collapse as an orthotropic weak interlayer, the the-
oretical formulas figuring up the collapse block are
obtained with and without considering the collapse
cusp, respectively. A case analysis shows that con-
sidering the collapse cusp can lead to a higher range
of collapse blocks.
(2) e strength parameters of the weak interlayer have a
significant impact on the range of collapse blocks.
e shear and tensile strength have similar effects on
the height of the collapse block, but their effects on
the width have the opposite trend. Moreover, be-
cause the increase in shear and tensile strengths may
maintain a greater gravity of surrounding rocks,
there will be a wider range of primary failures once
the collapse occurs. By sensitivity analysis, we can
find that the height of the collapse block is more
sensitive to the change in tensile strength, but the
width and the gravity of the collapse block are more
sensitive to the change of shear strength.
(3) e discrepancies between the two cases according to
whether the collapse cusp considered are related to
the strength parameters. e discrepancies between
the two cases generally decrease with the increase in
the tensile strength but increase with the increase in
the shear strength. ese changes are more sensitive
to the change of shear strength than the change of
tensile strength, which is of directive functions when
considering the effect of the consideration of the
collapse cusp.
Our theoretical results can provide guidance on the
collapse mechanism in tunnels or natural cavities, especially
they can explain the phenomenon of sharp points in collapse
blocks. Moreover, based on our proposed approach, many
extensions including various cases such as layered rock
masses and the presence of the karst cave can be further
studied in future research.
Data Availability
All data used to support the findings of this study are in-
cluded within the article.
Conflicts of Interest
e authors declare that there are no conflicts of interest
regarding the publication of this paper.
Acknowledgments
e authors gratefully acknowledge the National Natural
Science Foundation of China (Grant no. 51738002).
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