S S symmetry
Article Stability Analysis of Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella
Kaihang Han 1,2 , Xuetao Wang 1,2,*, Beibei Hou 3, Cheng-yong Cao 1,2 and Xing-Tao Lin 1,2
1 College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China; [email protected] (K.H.); [email protected] (C.-y.C.); [email protected] (X.-T.L.) 2 Underground Polis Academy, Shenzhen University, Shenzhen 518060, China 3 China Jingye Engineering Company Limited, Beijing 100088, China; [email protected] * Correspondence: [email protected]
Received: 23 November 2020; Accepted: 11 December 2020; Published: 13 December 2020
Abstract: When tunnels are constructed under difficult geotechnical conditions in urban areas, tunnel face stability is one of the main issues to be addressed. To ensure tunnel face stability and reduce the impact of tunneling on adjacent structures, a few alternative procedures of ground reinforcement should be adopted, which includes reinforcing the soil ahead of the face using longitudinal fiberglass dowels alone or together with a steel pipe umbrella. It is of great academic value and engineering signification to reasonably determine the limit reinforcement density of these ground reinforcements. In this paper, an analytical prediction model is proposed by using the limit analysis method to analyze the tunnel face stability, and the favorable effects of longitudinal fiberglass dowels and steel pipe umbrella on tunnel face stability are investigated quantitatively. The analytical prediction model consists of a wedge ahead of the tunnel face, distributed force acting on the wedge exerted by overlying ground, and the support forces stem from the longitudinal fiberglass dowels. Moreover, sensitivity analysis is conducted to study the effect of the depth of cover, the tunnel shape, the reinforcement installation interval and the reduction factor on the required limit reinforcement density.
Keywords: face stability; shallow tunneling method; limit analysis; cohesive–frictional soils; reduction effect; reinforcement density
1. Introduction The stability of the tunnel face is one of the main problems to be solved when the tunnel passes through complicated geological conditions [1,2]. The shallow tunneling method is mainly applied to urban subways, municipal underground pipe networks and other shallow-buried underground structures. This method is mostly used in quaternary soft strata, and the excavation methods include the positive step method, single side wall method, middle wall method (also known as CD method and CRD method), double side wall method (eyeglasses method), etc. The shallow tunneling method has the advantages of flexibility; little influence on the ground building, road and underground pipe network; less land for demolition; no disturbance to nearby communities; no pollution to the urban environment, and so on. In order to ensure the face stability of the tunnels constructed using the shallow tunneling method, some flexible measures should be taken to improve the stability of the tunnel face under poor ground conditions. The commonly used pre-reinforcement techniques for tunnel construction are shown in Table1.
Symmetry 2020, 12, 2069; doi:10.3390/sym12122069 www.mdpi.com/journal/symmetry Symmetry 2020, 12, 2069 2 of 12
Table 1. Classification of pre-reinforcement techniques for tunnel construction.
Construction Safety Pre-Reinforcement Technology Stable Working Stability of Groundwater Stable Arch Face Arch Foot Control payments Advance Advance bolt √ √ Pipe shed √ √ Horizontal rotary jet pile √ √
xaae surfaces excavated of Reinforcement Core soil reserved for √ annular excavation Shotcrete on excavated √ surface Anchor of excavated surface √ Grouting of excavated √ surfaces fac foot arch of Reinforcement Anchor bolt reinforcement √ Reinforcement of locked pile √ Grouting reinforcement of √ arch foot Temporary inverted arch √ control Groundwater Surface Drainage √ √ √ drainage measures Drainage √ √ √ Waterproofing Grouting √ √ √ measures Freeze √ √ √ Formation reinforcement Contact grouting √ √ √ Full section grouting √ √ √ Joint grouting Surface pre-grouting √ √ √ √
Numerical, experimental and theoretical analytical methods have been developed to study the stability of tunnel faces reinforced with auxiliary methods. Based on the centrifugal test, it was found that the extrusion displacement of soil on the excavation surface decreases when the bolt is arranged in the outer area of the excavation surface, and the optimal bolt arrangement length is approximately twice the length from the excavation surface to the fracture surface [3]. A series of scale model tests and three-dimensional finite element simulations were carried out to study the constraints of reinforcement parameters such as density, length and stiffness on excavation face deformation. The results indicated that when the reinforcement element is installed, the reinforcement effect of the excavation face is obvious, and there is a critical value to exert the maximum effect of the reinforcement [4,5]. Centrifugal test equipment and discrete element simulation were used to investigate the influence of typical auxiliary bolt (front bolt, vertical pre-reinforcement bolt and advance support bolt) on the stability of the excavation face. The results showed that when the length exceeds 0.5 D, the front bolt will play a beneficial role, and the front bolt placed on the upper section of the tunnel is more effective than the lower section [6]. The effectiveness of the pre-reinforcement of the excavation face of deep buried tunnel was studied by numerical simulation methods, and the results were compared with the measured data [7]. The stability of the tunnel excavation face under the pipe umbrella was studied, and the support pressure needed to ensure the stability of the tunnel excavation face was deduced by upper limit theory and limit equilibrium theory. The results showed that under the condition of no water, the effect of multi-stage grouting on the supporting force needed to ensure the stability of the excavation face is not obvious. When seepage is considered, the steel pipe umbrella has a relatively strong influence on underwater tunnels. [8]. A series of centrifugal tests were carried out on the Symmetry 2020, 12, x FOR PEER REVIEW 3 of 14 the pipe umbrella was studied, and the support pressure needed to ensure the stability of the tunnel excavation face was deduced by upper limit theory and limit equilibrium theory. The results showed that under the condition of no water, the effect of multi-stage grouting on the supporting force needed Symmetryto ensure2020 the, 12 stability, 2069 of the excavation face is not obvious. When seepage is considered, the 3steel of 12 pipe umbrella has a relatively strong influence on underwater tunnels. [8]. A series of centrifugal testsexcavation were carried robot to out study on the excavation effect of tube robot shed to support study the and effect excavation of tube method shed support (full section and excavation excavation methodor ring excavation (full section with excavation reserved or core ring soil) excavation on the displacement with reserved above core thesoil) excavation on the displacement face of the tunnel.above Thethe excavation model test showedface of the that tunnel. the maximum The model settlement test show ofed the that full sectionthe maximum of the excavated settlement strata of the under full sectionthe action of ofthe the excavated pipe shed strata is one under quarter the of action that without of the pipe the action shed ofis theone pipe quarter umbrella of that [9 ].without A series the of actionlarge-scale of the model pipe tests umbrella are carried [9]. A out series to study of large the reinforcement-scale model tests mechanism are carried of the out tunnel to study roof tube the reinforcementshed [10]. The supportingmechanism system of the tunnel structure roof of tube the pipe shed shed [10] is. The analyzed, supporting and the system bending structure moment of andthe pipeshear shed force is ofanalyzed, the pipe and shed the are bending analyzed moment using and finite shear element force software. of the pipe Moreover, shed are theyanalyzed compared using finitethe results element with soft generalware. commercialMoreover, they software compared to verify the theresults reliability with general of the proposedcommercial finite software element to verifycalculation the reliability [11]. An analytical of the proposed study on finitethe stability element of the calculation tunnel excavation [11]. An faceanalytical strengthened study on by the stabilityauxiliary of method the tunnel includes excavation an equivalent face strengthened homogenized by material the auxiliary concept method [12–14 ]includes and the supportan equivalent forces homogenizedexerted by the material individual concept bolts [[1215–1914].] and the support forces exerted by the individual bolts [15–19]. In urban subway tunnel excavation, retaining core soil is widely used to partially realize the In urban subway tunnel excavation, retaining core soil is widely used to partially realize the retaining wall effect, which can control the stability of the tunnel face under the general stratum retaining wall effect, which can control the stability of the tunnel face under the general stratum conditions. However, in special stratum conditions, such as large water inflow and sand layers with conditions. However, in special stratum conditions, such as large water inflow and sand layers with relatively small water barrier thickness, the core soil of the tunnel face is often difficult to retain, and soil relatively small water barrier thickness, the core soil of the tunnel face is often difficult to retain, and collapse occurs frequently. Subsequent measures to deal with the collapse, such as grouting small soil collapse occurs frequently. Subsequent measures to deal with the collapse, such as grouting small pipes, have become the conventional means to maintain normal excavation of the working face of the pipes, have become the conventional means to maintain normal excavation of the working face of the tunnel under such stratum conditions. This cannot be said to be a passive face of soil pre-reinforcement. tunnel under such stratum conditions. This cannot be said to be a passive face of soil pre- A large number of engineering practices have also shown that the use of longitudinal fiberglass dowels reinforcement. A large number of engineering practices have also shown that the use of longitudinal in the tunnel face is an active and effective method to pre-reinforce the front soil (c.f., Figure1). It is of fiberglass dowels in the tunnel face is an active and effective method to pre-reinforce the front soil great academic value and engineering significance to reasonably determine the limit reinforcement (c.f., Figure 1). It is of great academic value and engineering significance to reasonably determine the limitdensity reinforcement of these ground density reinforcements. of these ground In this reinforcements. paper, based on In thethis limit paper, analysis based method, on the limit an analytical analysis predictionmethod, an model analytical is proposed prediction to analyzemodel is the propos favorableed to eanalyzeffects of the longitudinal favorable effects fiberglass of longitudinal dowels and fiberglasssteel pipe umbrelladowels and on steel face stability.pipe umbrella Moreover, on face sensitivity stability. analysis Moreover is carried, sensitivity out to analysis study the is ecarriedffect of outthe depthto study of cover,the effect the tunnelof the depth shape, of the cover, reinforcement the tunnel installation shape, the intervalreinforcement and the installation reduction factor interval on andthe requiredthe reduction reinforcement factor on density.the required reinforcement density.
Figure 1. 1. FaceFace failure failure reinforced reinforced with with longitudinal longitudinal fiberglass fiberglass dowels dowels together together with steel with pipe steel umbrella. pipe umbrella. 2. Stability Analysis of the Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together2. Stability with Analysis Steel Pipeof the Umbrella Tunnel Face Reinforced with Longitudinal Fiberglass Dowels Together with Steel Pipe Umbrella 2.1. The New Analytical Prediction Model 2.1. TheIn thisNew paper, Analytical a new Prediction analytical Model prediction model is proposed to investigate the stability of a tunnel face reinforced with auxiliary methods, as shown in Figure2a. The tunnel is a rigid cylinder with a In this paper, a new analytical prediction model is proposed to investigate the stability of a diameter of D, an underground cover depth of C and a surcharge applied on the ground surface of tunnel face reinforced with auxiliary methods, as shown in Figure 2a. The tunnel is a rigid cylinder σ . The analytical prediction model consists of a wedge ahead of the tunnel face, distributed force withs a diameter of D, an underground cover depth of C and a surcharge applied on the ground acting on the wedge exerted by overlying ground, and the support forces stem from the longitudinal
fiberglass dowels. On the one hand, the advanced pre-reinforcement structure of the steel pipe umbrella is deemed the beam on the Winkler elastic foundation, which shows that the existence of the steel pipe umbrella effectively reduces the vertical pressure exerted by overlaying ground. On the other Symmetry 2020, 12, x FOR PEER REVIEW 4 of 14 with a diameter of D, an underground cover depth of C and a surcharge applied on the ground surface of σs. The analytical prediction model consists of a wedge ahead of the tunnel face, distributed force acting on the wedge exerted by overlying ground, and the support forces stem from the longitudinal fiberglass dowels. On the one hand, the advanced pre-reinforcement structure of the steel pipe umbrella is deemed the beam on the Winkler elastic foundation, which shows that the existence of the steel pipe umbrella effectively reduces the vertical pressure exerted by overlaying ground.Symmetry 2020On, the12, 2069 other hand, the strengthening effect of the longitudinal fiberglass dowels depends4 of 12 primarily on the tensile bearing capacity of the bolt or the bond strength of the ground–bolt interface. The ground extrusion in front of the tunnel face is effectively reduced for the installation of hand, the strengthening effect of the longitudinal fiberglass dowels depends primarily on the tensile longitudinal fiberglass dowels. The mechanical force analysis of the new analytical prediction model bearing capacity of the bolt or the bond strength of the ground–bolt interface. The ground extrusion is shown in Figure 2b. in front of the tunnel face is effectively reduced for the installation of longitudinal fiberglass dowels. The mechanical force analysis of the new analytical prediction model is shown in Figure2b.
σs Ground surface
C Vertical pressure exerted by the prism upon the wedge H Pre-supports
Subgrade reaction
D Tunnel Longitudinal fiberglass dowels
(a)
σv
Ts G s H φ β
B
(b)
Figure 2. The new failure mechanism. (a) Concept map; (b) Mechanical force analysis. Figure 2. The new failure mechanism. (a) Concept map; (b) Mechanical force analysis.
2.2. Power of the External Loads Pe 2.2. Power of the External Loads Pe 2.2.1. Power of the Soil Unit Weight 2.2.1. Power of the Soil Unit Weight The power of self-weight of the wedge body is as follows: The power of self-weight of the wedge body is as follows: Pw = G[V cos(ϕ + β)] (1) =+= 1 H2 tan()ϕββBγ [V cos(ϕ + β)] = 1 γH2B tan β cos(ϕ + β)V PGVw 2 cos 2 (1) where G is the weight=+=+ of11 the22 wedge,βγV is the velocity ϕβ of the γ wedge, βB is the ϕβ width of the wedge, H is HBVtan cos() HB tan cos() V the height of the wedge,ϕ22is the angle of internal friction of the ground, and β is the inclination of the slope and the horizontal plane, as rendered in Figure2b. 2.2.2. Power Induced by the Friction on Both Sides The power of the friction on both sides is as follows:
R H nh i o P = c + (H z) + z [z ][V ] dz Ts 2 0 γ H σv tan ϕ tan β cos ϕ 2−σ +γH (2) = c + K tan ϕ v H2 tan β cos ϕV − 3 Symmetry 2020, 12, 2069 5 of 12 where c is the cohesion of the ground, z is the vertical coordinate, and K is the lateral pressure coefficient.
2.2.3. Power Induced by the Vertical Stress on the Sliding Wedge Body Basic ideas and assumptions of the mechanical model for advanced small pipes are as follows: (1) Both the beam theory and the theory of beam on Winkler elastic foundation are adopted to investigate the mechanical behavior and characteristics of advanced small pipes in tunnel construction. Beam theory is used to analyze the advanced small pipes that are embedded in soil in the front of the tunnel face, whereas the theory of beam on Winkler elastic foundation is used to analyze the advanced small pipes behind the tunnel face, as rendered in Figure3.
(2) It is assumed that the fixed end A has certain vertical displacement y0, which is a known value and is considered as the measured vault subsidence value. (3) The length of the advanced small pipes consists of two parts, which includes unsupported span (length in 1.5a, includes excavation footage 1.0a and the length 0.5a due to support delay effect) and the length of the wedge (length in l), as depicted in Figure3. The symbol a denotes the excavation footage—that is, the length of the tunnel for each excavation. (4) In order to simplify the analysis, the horizontal projection length of advanced small pipes is considered. The load q (uniform distribution) above the tunnel face will entirely act on the collapse slope surface of the tunnel face within length l. According to the analysis above, the stability analysis models of the upper-bound solution of the tunnel face with the following conditions are established, as follows:
(1) The remaining length le of the pipe in soil is longer than the length l of the wedge (Type I). The stability analysis model I is shown in Figure3a. The subgrade reaction force that acts on the wedge is p (triangular distribution).
(2) The remaining length le of the pipe in soil is shorter than the length l of the wedge (Type II). The stability analysis model II is shown in Figure3b. The load that acts on the wedge can be divided into two parts, one is the subgrade reaction force p (trapezium distribution) along the Symmetry 2020, 12, x FOR PEER REVIEW 6 of 14 pipe and the other is the uniform load q that acts on the wedge.
q(x) q(x) Joint with Joint with grid steel frame grid steel frame Pipe roof Pipe roof B C B x x A O A O p(x) p(x) 1.5a l 1.5a le≤l
le y y (a) (b)
Figure 3. Mechanical model of the steel pipe umbrella. (a) Type I; (b) Type II. Figure 3. Mechanical model of the steel pipe umbrella. (a) Type I; (b) Type II. (1) For Type I, the control differential equations of the reinforced foundation beam for different (1) Forsegments Type I, are the obtained control asdifferential follows: equations of the reinforced foundation beam for different segments are obtained as follows: d4 y bq(x) AO : = dx4 EI 4 4 bq() x dyd =y 4 bq(x) AO OB: : 4 + 4λ y = , (3) 4dx EI dxd4 y EI BC : + 4λ4 y = 0 4 dx4 bq() x dy+=λ 4 OB:44 y , (3) where y is the deflection of the beam,dxλ is the characteristic EI coefficient of the beam, E is the 4 elasticity modulus of the beam material,dy and+=Iλis4 the moment of inertia of the beam section. BC:404 y dx where y is the deflection of the beam, λ is the characteristic coefficient of the beam, E is the elasticity modulus of the beam material, and I is the moment of inertia of the beam section.
The deflection equations are obtained as follows:
=++++432() AO:/24 y11234 qbx EI C x C x C x C =+ OB: y23 y yt , (4) =+++λλxxλλ− λλ BCy:35 e ( C cos x C 6 sin x ) e ( C 7 cos x C 8 sin x ) where y1, y2 and y3 denote the deflections of the beams AO, OB and BC, respectively; C1, C2, C3, C4, C5, C6, C7 and C8 present the undetermined coefficients of differential equations; yt is a particular solution to a differential equation for beams OB.
Boundary conditions are as follows:
==θ y33→∞0, →∞ 0 xx yy33==,0θ 101xa=− xa =− 22, (5) ==θθ yy1212====, xxxx0000 MMQQ==, 1212xxxx====0000 where θ, M and Q denote the angle of rotation, bending moment and shear force of the beams, respectively.
According to the above boundary conditions, the following equations are obtained to calculate the unknown parameters in Equation (4):
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The deflection equations are obtained as follows:
4 3 2 AO : y1 = qbx /(24EI) + C1x + C2x + C3x + C4 OB : y = y + y , (4) 2 3 t λx λx BC : y3 = e (C5 cos λx + C6 sin λx) + e− (C7 cos λx + C8 sin λx) where y1, y2 and y3 denote the deflections of the beams AO, OB and BC, respectively; C1, C2, C3, C4, C5, C6, C7 and C8 present the undetermined coefficients of differential equations; yt is a particular solution to a differential equation for beams OB. Boundary conditions are as follows:
y = 0, θ = 0 3 x 3 x →∞ | →∞ y = y , θ 3 = 0 1 x= 3 a 0 1 x= a − 2 | − 2 , (5) y = y , θ = θ 1 x=0 2 x=0 1 x=0 2 x=0 | | M = = M = , Q = Q 1|x 0 2|x 0 1|x=0 2|x=0 where θ, M and Q denote the angle of rotation, bending moment and shear force of the beams, respectively. According to the above boundary conditions, the following equations are obtained to calculate the unknown parameters in Equation (4): A11 A12 A13 A14 A15 A16 C1 B1 A A A A A A C B 21 22 23 24 25 26 2 2 A31 A32 A33 A34 A35 A36 C3 B3 = , (6) A A A A A A C B 41 42 43 44 45 46 4 4 A51 A52 A53 A54 A55 A56 C7 B5 A61 A62 A63 A64 A65 A66 C8 B6
where 27qba4 A =27a3,A = 18a2,A =12a,A = 8,A =0,A =0,B = 8y , 11 12 − 13 14 − 15 16 1 16EI − 0 27qba3 A =27a2,A = 12a,A =4,A =0,A =0,A =0,B = , 21 22 − 23 24 25 26 2 12EI q A =0,A =0,A =0,A =1,A = 1,A =0,B = [1 cosh(λl) cos(λl)], 31 32 33 34 35 − 36 3 K − q (7) A =0,A =0,A =1,A =0,A =λ,A = λ,B = [sinh(λl) cos(λl) cosh(λl) sin(λl)], 41 42 43 44 45 46 − 4 K − q A = A = A = A = A = A = 2 B = [ ( l) ( l)] 51 0, 52 1, 53 0, 54 0, 55 0, 56 λ , 5 K sinh λ sin λ , q A = 3,A =0,A =0,A =0,A =λ3,A =λ3,B = [sinh(λl) cos(λl)+cosh(λl) sin(λl)], 61 − 62 63 64 65 66 6 K (2) For Type II, the control differential equations of the reinforced foundation beam for different segments are obtained as follows:
d4 y bq(x) AO = : dx4 EI d4 y bq(x) , (8) OB + 4 y = : dx4 4λ EI
The deflection equations are obtained as follows:
( 4 3 2 AO : y1 = qbx /(24EI) + C1x + C2x + C3x + C4 λx λx , (9) OB : y2 = e (C5 cos λx + C6 sin λx) + e− (C7 cos λx + C8 sin λx) + q/K Symmetry 2020, 12, 2069 7 of 12
Boundary conditions are as follows:
y = y , θ 3 = 0 1 x= 3 a 0 1 x= a − 2 | − 2 y = y , θ = θ 1 x=0 2 x=0 1 x=0 2 x=0 , (10) | | M1 x=0 = M2 x=0, Q1 x=0 = Q2 x=0 | | | | M = = y , Q = 0 2|x le 0 2|x=le According to the above boundary conditions, the following equations are obtained to calculate the unknown parameters in Equation (9): A11 A12 A13 A14 A15 A16 A17 A18 C1 B1 A A A A A A A A C B 21 22 23 24 25 26 27 28 2 2 A31 A32 A33 A34 A35 A36 A37 A38 C3 B3 A A A A A A A A C B 41 42 43 44 45 46 47 48 4 = 4 , (11) A51 A52 A53 A54 A55 A56 A57 A58 C5 B5 A A A A A A A A C B 16 62 63 64 65 66 67 68 6 6 A71 A72 A73 A74 A75 A76 A77 A78 C7 B7 A81 A82 A83 A84 A85 A86 A87 A88 C8 B8
where 27qba4 A =27a3,A = 18a2,A =12a,A = 8,A =0,A =0,A =0,A =0,B = 8y , 11 12 − 13 14 − 15 16 17 18 1 16EI − 0 27qba3 A =27a2,A = 12a,A =4,A =0,A =0,A =0,A =0,A =0,B = , 21 22 − 23 24 25 26 27 28 2 12EI q A31=0,A32=0,A33=0,A34=1,A35= 1,A36=0,A37= 1,A38=0,B3= , − − K A =0,A =0,A =1,A =0,A = λ,A = λ,A =λ,A = λ,B =0, 41 42 43 44 45 − 46 − 47 48 − 4 2 2 A51=0,A52=1,A53=0,A54=0,A55=0,A56= λ ,A57=0,A58=λ ,B5=0, − , (12) A =3,A =0,A =0,A =0,A =λ3,A = λ3,A = λ3,A = λ3,B =0, 61 62 63 64 65 66 − 67 − 68 − 6 λl λl λl A =0,A =0,A =0,A =0,A = e e sin(λle),A =e e cos(λle),A =e e sin(λle), 71 72 73 74 75 − 76 77 − λle A78= e cos(λle),B7=0, − − λle λle A81=0,A82=0,A83=0,A84=0,A85= e [cos(λle)+sin(λle)],A86=e [cos(λle) sin(λle)], − − λl λl A =e e [cos(λle) sin(λle)],A =e e [cos(λle)+sin(λle)],B =0. 87 − − 88 − 8 The average ground reaction forces are calculated as follows:
R min[l, l ] R l e p(x)dx y [l l ]dx 0 0 2min , e σv = = σv, (13) min[l, le] min[l, le] ≤
Power of the vertical stress on the sliding wedge body is calculated as follows:
Pσv = (σvHB tan β)[cos(ϕ + β)V], (14)
2.2.4. Power Induced by the Longitudinal Fiberglass Dowels According to the work by Anagnostou and Perazzelli [17], the reinforcement effect of longitudinal fiberglass dowels depends on the tensile axial force, and there are three distributions of support pressure induced by the longitudinal fiberglass dowels for different wedge angles β, as shown in Figure4. Symmetry 2020, 12, 2069 8 of 12 Symmetry 2020, 12, x FOR PEER REVIEW 9 of 14
(a) (b) (c)
Figure 4. Distribution of support pressure for different wedge angles β (Anagnostou and Perazzelli Figure 4. Distribution of support pressure for different wedge angles β (Anagnostou and Perazzelli 2015). (a) Condition I; (b) Condition II; (c) Condition III. 2015). (a) Condition I; (b) Condition II; (c) Condition III. Based on these assumptions, the tensile axial force s induced by the longitudinal fiberglass dowels Based on these assumptions, the tensile axial force s induced by the longitudinal fiberglass is as follows: dowels is as follows:
β β , s(z) = s (z) = nπdτmz tan β ≤ 1 ( 1 11,s z s 2(z ) = nn π dd τmm z( tanL0 z tan β), (z1 z H) β1 β β2, s(z) = − ≤ ≤ ≤ ≤ s1s(z) z, n d L ztan , ( z0 zz Hz1) 21 m ≤ ≤ , (15) 12 , sz s3(z) = 0, (z2 z H), s z, (0≤ z≤ z ) β β , s(z) = s (11z), (z z z ) 2 2 1 2 (15) ≥ ≤ ≤ s1s(32z), z 0 , (( z0 z Hz1 )) ≤ ≤ 2,,s z s 2 z z 1 z z 2 where n is the reinforcement density of the longitudinal fiberglass dowels, d is the diameter of the longitudinal fiberglass dowels, and τm is thes11 bond z strength of the soil–grout, (0 interface z z ) of the longitudinal fiberglass dowels. where n is the reinforcement density of the longitudinal fiberglass dowels, d is the diameter of the ( ! ! longitudinal fiberglass dowels,L and τm is the L bond strength0.5L of the soilL –grout interface of the β = arctan 0 , β = arctan 0 , z = 0 , z = 0 = 2z , (16) longitudinal fiberglass1 dowels.2 H 2 H 1 tan β 2 tan β 1
Power of the longitudinal fiberglassLLLL dowels reads for di0.5fferent conditions as follows: 1arctan , 2 arctan ,z 1 , z 2 2 z 1 , (16) 2HH tan2 tan s1(H)H H Ps = BV = nπdτm tan β BV, [β β1], (17) Power of the longitudinal− fiberglass2 dowels− reads for different2 conditions≤ as follows: s1(z1)z1 [s1(z1)+s2(H)](H z1) Ps = 2 + 2 − BV 2 − s1 H H H 2 Psmn πdτm tan βz BV[n πd τ nmtan d βz +tannπ dτm(L BVH tan , β)]( H z 1) , (17) = 221 + 1 0− − 1 BV − 2 2 2 , (18) nπdτm tan βz [nπdτm tan βz +nπdτm tan β(2z H)](H z ) = 1 + 1 1− − 1 BV s2 z s H H z 2 s111 z− z 1 1 2 1 2+[ ]( ) Ps z1 3z1 H H z1 BV 22= nπdτm tan β − − BV, [β β β ] − 2 1 ≤ ≤ 2 2 s1(z1n)z1 d stan2(z2)( z z2 z1 n) d L H tan H z n d m tan= z1 +mm11= [ ( ) ] Ps 2 2 − BV s1 z1 z1 BV BV 22− n o − , (19) 2 = nπdτm tan β z BV, [β β ] − 1 ≥ 2 , (18) 2 n d tan z n d tan 2 z H H z n d m tan z1 mm1 1 1 2.3. Dissipation Power on Discontinuity Surface Pv BV 22 Dissipation power on the discontinuity2 surface is obtained as follows: z13 z 1 H H z 1 nd m tan BV , 12 2 H Pv = c cos ϕBV, (20) cos β
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2.4. Critical Reinforcement Density of Longitudinal Fiberglass Dowels Based on the upper-bound limit theory, the power of external force is equal to the dissipation of internal force in the analytical prediction model, as follows:
Pw + PTs + Pσv + Ps = Pv, (21)
Since there are three distributions of support pressure induced by the longitudinal fiberglass dowels for different wedge angles β (c.f., Figure4), the critical reinforcement density ( n1, n2 and n3 corresponding to three distributions of support pressure) of longitudinal fiberglass dowels is calculated:
ncr = max(n1, n2, n3), (22)
where 1 2 2σv+γH 2 H γH B tan β cos(ϕ+β) c+K tan ϕ H tan β cos ϕ+σvHB tan β cos(ϕ+β) c cos ϕB 2 − 3 − cos β n = , 1 H2 πdτm tan β B 2 β β1 ; 1 ≤ 2 2σv+γH 2 H γH B tan β cos(ϕ+β) c+K tan ϕ H tan β cos ϕ+σvHB tan β cos(ϕ+β) c cos ϕB 2 − 3 − cos β n = " ( ) # , 2 z2+[3z H](H z ) 1 1− − 1 , (23) πdτm tan β 2 B β1 β β2 ; 1 ≤ 2 ≤ 2σv+γH 2 H γH B tan β cos(ϕ+β) c+K tan ϕ H tan β cos ϕ+σvHB tan β cos(ϕ+β) c cos ϕB 2 − 3 − cos β = h n o i n3 2 , πdτm tan β z B 1 β β . ≥ 2 3. Sensitivity Analysis
3.1. Longitudinal Fiberglass Dowels in the Excavation Face Alone
3.1.1. The Influence Rules of the Cover Depth on Limit Reinforcement Density
Figure5 renders the influence rules of the cover depth h on limit reinforcement density ncr with the variation of the cohesion c. The results indicate that when the cover depth is greater than the width of theSymmetry tunnel 2020 face,, 12, thex FOR limit PEER reinforcement REVIEW density does not increase significantly. 11 of 14
1.4
3 1.2 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m 1.0 h =
] 2 0.8 h = 8 m h = 4 m
0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
FigureFigure 5. The 5. The influence influence rules rules of of the the cover cover depth depthh hon on limitlimit reinforcement density density ncrn withcr with the the variation of thevariation cohesion of cthe. cohesion c.
3.1.2. The Influence Rules of the Tunnel Shape on Limit Reinforcement Density Figure 6 shows the influence rules of the different tunnel shapes on the limit reinforcement density ncr with the variation of the cohesion c. The results are markedly different between shapes A, B and C. Specifically, shape C needs a lower reinforcement density than shape A, though they have the same height, which agrees with common sense that the C-D tunneling method is more stable than the full-face tunneling method. Moreover, a comparison between the reinforcement required in the cases of tunnel shapes B and C indicates that the C-D tunneling method is more stable than the benching tunneling method.
1.4
1.2
1.0
3 = 25, = 20 kN/m , e = 0, h =
] 2 0.8 L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m
0.6 H = 8 m, B = 8 m (A) H = 4 m, B = 8 m (B)
[bolts/m
cr H = 8 m, B = 4 m (C) n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
Figure 6. The influence rules of the different tunnel shapes on limit reinforcement density ncr with the variation of the cohesion c.
3.1.3. The Influence Rules of the Reinforcement Installation Interval on Limit Reinforcement Density Figure 7 depicts the influence rules of the two installation intervals l on the limit reinforcement density ncr with the variation of the cohesion c. The results show that large installation intervals of bolts require greater reinforcement density ncr.
Symmetry 2020, 12, x FOR PEER REVIEW 11 of 14
1.4
3 1.2 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m 1.0 h =
] 2 0.8 h = 8 m h = 4 m
0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
SymmetryFigure2020 5., 12 The, 2069 influence rules of the cover depth h on limit reinforcement density ncr with the 10 of 12 variation of the cohesion c.
3.1.2. The Influence Influence Rules of the Tunnel ShapeShape onon LimitLimit ReinforcementReinforcement DensityDensity Figure6 6shows shows the the influence influence rules rules of the of di theff erent different tunnel tunnel shapes shapes on the on limit the reinforcement limit reinforcement density ndensitycr with n thecr with variation the variation of the cohesion of the cohesionc. The resultsc. The results are markedly are markedly different different between between shapes shapesA, B and A, CB. and Specifically, C. Specifically, shape shapeC needs C needs a lower a lower reinforcement reinforcement density density than than shape shapeA, though A, though they they have have the samethe same height, height, which which agrees agrees with with common common sense sense that that the C-Dthe Ctunneling-D tunneling method method is more is more stable stable than than the full-facethe full-face tunneling tunneling method. method. Moreover, Moreover, a comparison a comparison between between the reinforcement the reinforcement required required in the in cases the ofcases tunnel of tunnel shapes shapeB ands CB indicatesand C indicates that the thatC-D thetunneling C-D tunneling method is method more stable is more than stable the benching than the tunnelingbenching tunneling method. method.
1.4
1.2
1.0
3 = 25, = 20 kN/m , e = 0, h =
] 2 0.8 L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m
0.6 H = 8 m, B = 8 m (A) H = 4 m, B = 8 m (B)
[bolts/m
cr H = 8 m, B = 4 m (C) n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
Figure 6. The i influencenfluence rules of the different different tunnel shapesshapes onon limitlimit reinforcementreinforcement densitydensityn ncrcr with the variation of thethe cohesioncohesion cc..
3.1.3. The Influence Influence Rules of the Reinforcement InstallationInstallation IntervalInterval onon LimitLimit ReinforcementReinforcement DensityDensity Figure7 7 depicts depicts the the influenceinfluence rulesrules ofof thethe twotwo installationinstallation intervalsintervals l on the limit reinforcement density n with the variation of the cohesion c. The results show that large installation intervals of density ncr with the variation of the cohesion c. The results show that large installation intervals of bolts require greater reinforcement density n . bolts Symmetryrequire 2020 greater, 12, x FOR reinforcement PEER REVIEW density ncrcr. 12 of 14
1.4
3 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, 1.2 L = 12 m, = 150 kPa, d = 114 mm, h = . m
1.0 l = 9 m
] 2 0.8 l = 4.5 m
0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
FigureFigure 7. The 7. The influence influence rules rules of of the the two two installation installation intervals l lonon limit limit reinforcement reinforcement density density ncr withncr with the variationthe variation of the of the cohesion cohesionc. c.
3.2. Longitudinal Fiberglass Dowels in the Excavation Face Together with Pre-Supports The grouting pipe roof bears the vertical stress component induced by overlying soil pressure and seepage force that act on the tunnel crown, which leads to a decrease in the limit support pressure of the tunnel face. Figure 8 indicates the effect of the reduction factor (RF) on the required reinforcement density ncr.
1.4
3 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, h = 1.2 L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m
1.0 RF = 0 RF = 0.3
] 2 0.8 RF = 0.5 RF = 0.7
RF = 0.9 0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0
c [kPa]
Figure 8. The influence rules of the reduction factor (RF) on limit reinforcement density ncr with the variation of the cohesion c.
4. Conclusions This paper proposed an analytical prediction model by using the limit analysis method to analyze the tunnel face stability. Moreover, the sensitivity analysis is conducted to investigate the effect of the depth of cover, the tunnel shape, the reinforcement installation interval and the reduction factor on the limit reinforcement density. The main conclusions are as follows:
Symmetry 2020, 12, x FOR PEER REVIEW 12 of 14
1.4
3 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, 1.2 L = 12 m, = 150 kPa, d = 114 mm, h = . m
1.0 l = 9 m
] 2 0.8 l = 4.5 m
0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 c [kPa]
Symmetry 2020, 12, 2069 11 of 12 Figure 7. The influence rules of the two installation intervals l on limit reinforcement density ncr with the variation of the cohesion c. 3.2. Longitudinal Fiberglass Dowels in the Excavation Face Together with Pre-Supports 3.2. Longitudinal Fiberglass Dowels in the Excavation Face Together with Pre-Supports The grouting pipe roof bears the vertical stress component induced by overlying soil pressure and The grouting pipe roof bears the vertical stress component induced by overlying soil pressure seepage force that act on the tunnel crown, which leads to a decrease in the limit support pressure of and seepage force that act on the tunnel crown, which leads to a decrease in the limit support pressure the tunnelof the face. tunnel Figure face.8 indicatesFigure 8 indicates the e ffect the of the effect reduction of the reduction factor (RF factor) on the (RF required) on the reinforcement required densityreinforcementncr. density ncr.
1.4
3 H = 8 m, B = 8 m, = 25, = 20 kN/m , e = 0, h = 1.2 L = 12 m, l = 9 m, = 150 kPa, d = 114 mm. m
1.0 RF = 0 RF = 0.3
] 2 0.8 RF = 0.5 RF = 0.7
RF = 0.9 0.6
[bolts/m
cr
n 0.4
0.2
0.0 0.0 2.5 5.0 7.5 10.0 12.5 15.0
c [kPa]
FigureFigure 8. The 8. The influence influence rules rules of of the the reduction reduction factor (RF (RF) )on on limit limit reinforcement reinforcement density density ncr withncr thewith the variationvariation of the of cohesionthe cohesionc. c.
4. Conclusions4. Conclusions ThisT paperhis paper proposed proposed an analytical an analytical prediction prediction model model by using by using the thelimit limit analysis analysis method method to analyze to the tunnelanalyze face the stability. tunnel face Moreover, stability. theMoreover, sensitivity the sensitivity analysis is analysis conducted is conducted to investigate to investigate the effect the of the deptheffect of cover, of thethe depth tunnel of cover, shape, the thetunnel reinforcement shape, the reinforcement installation installation interval and interval the reduction and the reduction factor on the factor on the limit reinforcement density. The main conclusions are as follows: limit reinforcement density. The main conclusions are as follows: (1) The advanced pre-reinforcement structure of the steel pipe umbrella is considered as the beam on the Winkler elastic foundation, which shows that the existence of the steel pipe umbrella effectively reduces the vertical pressure exerted by overlaying ground. Under general conditions, with the increase in the length of the pre-reinforcement, its promoting effect on tunnel face stability is obvious. However, when the surplus length of the pre-reinforcement structure reaches the critical fracture length, the length of the pre-reinforcement structure on the stability of the tunnel face is no longer a key factor. (2) The strengthening effect of the longitudinal fiberglass dowels depends primarily on the tensile bearing capacity of the bolt or the bond strength of the ground–bolt interface. The ground extrusion in the front of the tunnel face is effectively reduced for the installation of longitudinal fiberglass dowels. Moreover, the limit reinforcement density of longitudinal fiberglass dowels is assessed under specific lengths with or without the consideration of the steel pipe umbrella. (3) The results indicate that the required reinforcement density does not increase significantly when the cover depth is greater than the width of the face. The C-D tunneling method is more stable than the full-face tunneling method and benching tunneling method. Moreover, the results show that large installation intervals of bolts require greater reinforcement density.
Author Contributions: Conceptualization, K.H. and X.W.; Data curation, B.H., C.-y.C. and X.-T.L.; Funding acquisition, K.H.; Investigation, K.H.; Methodology, K.H. and X.W.; Supervision, K.H.; Validation, X.W.; Writing—original draft, K.H., X.W., C.-y.C. and X.-T.L.; Writing—review and editing, K.H., X.W., B.H., C.-y.C. and X.-T.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the National Natural Science Foundation of China. (No. 51908371). Symmetry 2020, 12, 2069 12 of 12
Acknowledgments: The authors are deeply thankful to the reviewers and editor for their valuable suggestions to improve the quality of the paper. Conflicts of Interest: The authors declare no conflict of interest.
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