Determination of Support Pressure for Semi-Elliptical Tunnels in Anisotropic and Nonhomogeneous Soil by Lower Bound Limit Analysis

Sådhanå (2021) 46:19 Ó Indian Academy of Sciences

https://doi.org/10.1007/s12046-020-01540-w Sadhana(0123456789().,-volV)FT3](0123456789().,-volV)

Determination of support pressure for semi-elliptical tunnels in anisotropic and nonhomogeneous soil by lower bound limit analysis

PUJA DUTTA and PARAMITA BHATTACHARYA*

Civil Engineering Department, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, India e-mail: [email protected]; [email protected]

MS received 13 May 2020; revised 2 November 2020; accepted 5 November 2020

Abstract. This research work examines the stability of long semi-elliptical tunnels in purely cohesive and cohesionless soils. The support pressure (ri) required to apply along the tunnel periphery by means of lining and anchorage system has been evaluated using lower bound limit analysis with ﬁnite elements and second-order conic programming technique (SOCP). The results are presented in terms of normalized support pressure as (i) ri/cv0 in purely cohesive soil for different combinations of cD/cv0 and ac and (ii) ri/cD in cohesionless soil for different combinations of /v and a/ considering different values of 2h/D and H/D of tunnel where h, D and H are height of tunnel, width of tunnel and soil cover depth of the semi-elliptical tunnel, respectively; cv0, /v and c are soil undrained shear strength of cohesive soil in vertical direction at the ground level, peak vertical friction angle of cohesionless soil and unit weight of all type of soils, respectively; ac and a/ are anisotropy factors in cohesive and cohesionless soil, respectively. The value of ri/cv0 has been observed to increase with an increase in cD/cv0 and ac. The value of ri/cD increases with a decrease in /v and increase in a/. The increase in aspect ratio of tunnel proﬁle (2h/D) causes an increase in normalized support pressure in both purely cohesive and cohesionless soils. The effects of increase of undrained shear strength in isotropic purely cohesive soil on the stability of tunnel have also been studied. The support pressure is observed to decrease with increase in nonhomogeneity of cohesive soil. The combined effect of soil anisotropy and linearly increasing undrained shear strength may reduce the required magnitude of ri/cv0 in comparison with the value of ri/cv0 in isotropic homogeneous cohesive soil, especially for H/D [ 1.

Keywords. Stability; tunnels; limit analysis; optimization; soil anisotropy.

1. Introduction concluded that the tunnel stability is a function of its aspect ratio, soil cover depth ratio and shear strength parameters of The rapid growth of urbanization gives birth to challenges soil. Although circular tunnels are easier to construct in of creating space for fulfilling the increased infrastructural view of excavation, non-circular tunnels like rectangular/ demands. Therefore the usage of underground space square shaped have gained high interest due to maximum through tunnelling has become a popular alternate way to useable space. In the present research, support pressure meet requirement of land. Tunnels not only provide an required for peripheral stability of a semi-elliptical shaped efficient and smooth passage of traffic and railways but also tunnel has been investigated for a combination of soil cover create transportation infrastructures through mountainous depths and shear strength parameters. This type of tunnels terrain and are used for sewerage systems. Numerous may be adopted for multi-lane passage of traffic flow like studies were carried out to assess the stability of tunnels Figueroa Street Tunnels in Los Angeles, California, USA. based on (i) 1-g small-scale and large-scale centrifuge model tests [1–3], (ii) finite-difference analysis using FLAC2D [3], (iii) upper and lower bound theorems of limit analysis [4–21] and (iv) finite-element analysis [22–24]. 2. Problem definition Study on impact of tunnel on the capacity of single pile was also carried out by conducting small-scale model tests and A semi-elliptical tunnel of width D and crown height using PLAXIS 2D [25]. The aforesaid research works h measured from its floor is placed at a depth H below the ground surface in soil as shown in figure 1(a). The length of the tunnel is assumed to be very large compared with its *For correspondence width to maintain the plane-strain condition. The soil mass 19 Page 2 of 15 Sådhanå (2021) 46:19

Figure 1. (a) Schematic diagram of the problem with stress boundary conditions, (b) typical ﬁnite-element mesh for /v =0° and H/D = 3 and (c) a typical segment along tunnel periphery. Comparison between values of ri/cv0 of present analysis with available results in o literature for circular tunnel for /v = /h =0 for (d) cD/cv0 = 2.6, ac = 1 and (e) ac =1. is considered to follow Mohr–Coulomb yield criterion and as reported by Bishop [26]. Therefore the effect of non- an associated ﬂow rule. Since the shear strength of satu- homogeneity of saturated cohesive soil in undrained con- rated cohesive soil in undrained condition is lower and the dition is also studied in the present analysis. Finally the shear strength of dry cohesionless soil is higher among all peripheral support pressure is estimated for the semi-el- types of soils, two different types of soils, namely saturated liptical tunnel in anisotropic and nonhomogeneous clay in cohesive soils in undrained condition and dry cohesionless undrained condition. Table 1 presents all information about soil, are chosen in the present analysis to study the effect of the different analyses conducted with the different param- soil anisotropy on peripheral support pressure of the semi- eter values. elliptical tunnel. Moreover, shear strength of normally The stability analysis of tunnel in anisotropic soil has consolidated clay and slightly over-consolidated clay is been carried out following a systematic method as proposed found to increase with the depth below the ground surface by Veiskarami et al [27]. The approximate variation of Sådhanå (2021) 46:19 Page 3 of 15 19

shear strength parameters (c, tan /) in any direction x can be presented as

2 c ¼ ch þ ðÞcv ch sin x ð1aÞ

tan/ ¼ tan / þ ðÞtan / tan / sin2 x ð1bÞ 5 and 2 h v h : 1

¼ where (c , tan / ) and (c , tan / ) are shear strength

, with an interval of 5 v v h h /

a parameters of soil in vertical and horizontal directions, 45 / [ / [ x =0) respectively, tan v tan h, cv ch and is the angle c between major principal stress direction and the horizontal 20 B x B ° ¼ axis where 0 90 . v

/ The degree of anisotropy has been presented by anisotropy factors ac and a/ deﬁned as follows: / a cv a tan v c ¼ and / ¼ / ð1cÞ ch tan h 1 For normally consolidated isotropic clay, the soil cohe- ¼

/ sion increases with depth as a , with an interval of 5 cv0 45 ch ¼ ch0 þ qy ¼ þ qy ð1dÞ ac Isotropic (homogeneous) Anisotropic (homogeneous) 20 where ch0 and cv0 are deﬁned as soil cohesion at ground ¼

v surface in horizontal and vertical directions, respectively, q / is the rate of change of soil cohesion with depth and ch is 1

; soil cohesion in the horizontal direction at a depth y from 5 :

0 the ground surface. ; 3 and 5 5 and 2 : ; 25

1 For anisotropic and nonhomogeneous undrained cohe- 1 : 0 ¼ ¼ condition sive soil, ﬁrst the undrained cohesions (ch) in horizontal ¼ c 0 D v a c c D

0 direction at different depths are estimated using equation c v Non-homogenous c qa (1d) and then considering the degree of anisotropy of the soil (ac) the undrained cohesion (cv) in vertical direction is estimated at different depths using equation (1c). Thereafter 4 and 5 0 ; the undrained shear strength parameter c along the major 3 ; ¼ principal plane for undrained cohesive soil is calculated 2 1.5 and 2 D ; 0 c v 1 c ¼ condition using equation (1a), performing lower bound ﬁnite-element qa c Homogenous ¼ a

0 limit analysis and an iteration technique described in D v c c section 4. However, the study of the increase or decrease in shear 1

; strength of cohesionless soil with depth is not in the scope 5 : 0 1 of the current manuscript. ; 3 and 5 ¼ ; 25 1 : c 0 a ¼ condition ¼ 0 D v c c D

0 3. Domain, stress boundary conditions and ﬁnite- c v Non-homogenous c qa

Fully saturated clay under undrained condition Cohesionless soil ( element mesh Isotropic Anisotropic The problem domain with the boundary conditions is pre- 4 and 5 1 0 ; sented in ﬁgure 1(a). The entire soil domain is symmetric 3 ¼ ; ¼

2 about an axis passing through the centre of the domain and c D ; 0 a c v 1 c

Condition coinciding with vertical y-axis. Therefore, only one half of qa Homogenous ¼

0 the total domain, i.e. NJKM, has been chosen for present D v c c analysis. Size of the domain is chosen in such a way that

Details of the soil type and the analysis. yielded elements do not touch the boundaries KM and MN and increase in domain size cannot cause any further change in ri. The horizontal (Lr) and vertical (Di) dimen- Soil type Table 1. Soil parameters sions of the domain vary from (i) 8D to 24D and (ii) 6D to 19 Page 4 of 15 Sådhanå (2021) 46:19

18D, respectively, depending upon the values of H/D, cD/ linear equality constraints with the usage of shape functions cv0 and /v. The normal and shear stresses are zero at ground of triangular elements described in equation (2a). surface. The shear stresses are zero at all the edges along Statically admissible stress discontinuities are permitted the boundary JK. The domain is discretized into three- by ensuring continuity of normal (rn) and shear (s) stresses noded triangular elements. Typical finite-element mesh for along each common edge shared by two adjacent triangular /v = /h =0° and H/D = 3 is presented in figure 1(b) where elements. If E1 and E2 are two adjacent elements and node N, El and Dc are the number of nodes, elements and stress pairs (1,3) and (2,4) lie along the interface of elements E1 discontinuity edges, respectively. and E2, then the discontinuity equilibrium condition can be satisfied by employing following equation 4: r r ; s s r r ; s s 4. Analysis n1 ¼ n3 1 ¼ 3 and n2 ¼ n4 2 ¼ 4 ð4Þ

where (rn1,s1), (rn2,s2), (rn3,s3) and (rn4,s4) are normal and The present analysis is carried out using lower bound limit shear stresses at nodes 1, 2, 3 and 4, respectively. In this analysis with finite elements as proposed by Sloan [28] and case, nodes 1 and 2 belong to E1 and nodes 3 and 4 belong second-order conic programming (SOCP) following to E2. Makrodimopoulos and Martin [29]. Although different Additional equality constraints are imposed to ensure the researchers [30, 31] proposed nonlinear programming in normal stresses acting along the tunnel periphery to be numerical lower bound limit analysis till the beginning of uniform. In figure 1(c), the segment sq along the tunnel twenty first century, the usage of SOCP became popular periphery is shown comprising node 2q–1 and 2q. The r r s later [17–20]. The nodal stresses ( x, y, xy) are adopted as normal stresses acting on nodes 2q–1 and 2q can be basic unknown variables, which vary linearly throughout expressed in terms of (rx(2q–1), ry(2q–1), sxy(2q–1)) and (rx(2q), the elements according to ry(2q),sxy(2q))as

X3 X3 X3 2 2 j j j j j j r ¼ r sin h þ r cos h r ¼ N r ; r ¼ N r and s ¼ N s ð2aÞ nðÞ2q1 xðÞ2q1 1 yðÞ2q1 1 x x y y xy xy s h j¼1 j¼1 j¼1 xyðÞ2q1 sin2 1 and j th r r 2h r 2h s h : where N is the shape function at j node and can be nðÞ2q ¼ xðÞ2q sin 2 þ yðÞ2q cos 2 xyðÞ2q sin 2 2 ð5aÞ expressed as N j ¼ 1 a j þ b jx þ k jy , aj = Akl, b j = yk–y, kj = xl – xk The additional constraint imposed to ensure the unifor- 2jjA mity of normal stresses acting on all the edges along the and tunnel periphery can be defined as 2 3 1 x1 y1 r r r r ... r 4 5 n1 ¼ n2 ¼ nðÞ2q1 ¼ nðÞ2q ¼ ¼ nðÞ2Tp ð5bÞ A ¼ 1 x2 y2 : ð2bÞ 1 x3 y3 where q implies a particular edge number and Tp represents total segments present along the tunnel periphery. Akl is the area covered by the corner nodes k and l of a The Mohr–Coulomb yield criteria with tensile stress as triangular element and the origin of the Cartesian co-ordi- positive can be described as nate system (x–y) whereas j, k and l are the three nodes of a triangular element oriented in a counter-clockwise qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiÂÃ 2 2 direction. rx ry þ 2sxy 2c cos / rx þ ry sin / : The lower limit of normalized support pressure is ð6aÞ obtained by maximizing the collapse load under active collapse state in the statically admissible stress field, which : r r s Assuming conic variablessx ¼ 0 5 x y , sxy ¼ xy is constructed by satisfying element equilibrium equations, and sy ¼ c cos / 0:5 rx þ ry sin /, Mohr–Coulomb stress discontinuity equilibrium conditions along the edges yield criterion is converted into a second-order conic con- of discontinuity, stress boundary conditions along the straint as boundaries and yield criterion at every node. The following element equilibrium equations are satisfied in each element qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 : of the soil domain: sx þ sxy sy ð6bÞ or os os or x þ xy ¼ 0 and xy þ y ¼ c ð3Þ The relationship between basic stress variables and conic ox oy ox oy variables at ith node is where c is the unit weight of soil. The differential equations ÈÉ ÈÉÈÉ ri i i of equilibrium conditions can be converted into sets of ½ASOCP þ xSOCP ¼ bSOCP ð6cÞ Sådhanå (2021) 46:19 Page 5 of 15 19 where compressive in nature. Thus the minimum lining pressure 2 3 can be calculated as 0:50:50 4 5 ½¼ASOCP 0:5sin/ 0:5sin/ 0 fgg T fgx ri ¼ : ð9Þ 001 perimeter of the tunnel unit length perpendicular to xy plane 8 9 8 9 ri i Positive value of r implies that support pressure is < x = ÈÉ< sx = i , fgri ¼ ri , xi ¼ si and required to apply along the tunnel periphery whereas neg- : y ; SOCP : y ; r si si ative value of i implies that the tunnel is stable under 8 xy 9 xy active collapse state of failure. < 0 = ÈÉ The stability analysis for the tunnels embedded in ani- bi ¼ c cos / . SOCP : ; sotropic soil is performed using an iterative method [27]. 0 The shear strength parameters are updated at every iteration In this method the active collapse load (Qu) is calculated according to the direction of major principal stress obtained by numerically integrating the vertical components of dis- in the previous iteration at every node in the domain using crete forces as shown in equation 7: equations (1a)–(1b). The iteration procedure has been continued until the convergence criteria as stated in equa-

! XTp 2 2 2 2 rx 2q 1 sin ðÞþh1 ry 2q 1 cos ðÞh1 sxy 2q 1 sinðÞþ 2h1 rx 2q sin ðÞþh2 ry 2q cos ðÞh2 sxy 2q sinðÞ 2h2 Q ¼ ls ðÞ ðÞ ðÞ ðÞ ðÞ ðÞ u q 2 q¼1 ¼ fgg T fgx ð7Þ

where Tp is the total number of segments along the tunnel tions (10a)–(10b) are achieved. th periphery, sq is the length of q segment in ﬁgure 1(c) and l th is the out-of-plane thickness of the segment. The q seg- n n1 h Q Q ment has two end nodes (2q–1) and 2q inclined at angles 1 e ð10aÞ kkQt1 and h2 with horizontal axis, respectively; {g} is the vector of objective function coefﬁcients having units of area. After ln ln1 assembling the various equality and inequality constraints, ln ln1 / / c c e e the tensile load {g}T{x} is maximized as stated by the or ð10bÞ lt1 lt1 following canonical form: c /

n n-1 th maximizefgg T fgx or minimizefgg T fgx ð8aÞ where Q and Q are the collapse load at n and (n– th ln ln1 ÂÃÈÉ 1) iterations, respectively; c and c are vectors of th th subject to Aequality fgx ¼ Bequality ð8bÞ soil cohesion at all nodes at n and (n–1) iterations, n n1 respectively; l/ and l/ are vectors of friction coefﬁ- fxðÞ0 ð8cÞ cient (tan /) at all nodes at nth and (n–1)th iterations, e where respectively; the convergence tolerance is assumed to nobe 1%. For cohesive soil, convergence can be achieved T r1r1s1 r2r2s2 ...rN rN sN 1 1 1 2 2 2 ... N N N in a few iterations whereas for cohesionless soil a good fgx ¼ x y xy x y xy x y xysxsy sxysxsysxy sx sy sxy number of iterations are required to achieve conver-

,[Aequality] is the global coefﬁcient matrix of equality gence. Moreover, for cohesionless soil, although theo- constraints comprising element equilibrium, discontinuity retically the value of c is zero in each iteration a very equilibrium, uniformity of support pressure conditions and small value of c is chosen in such a way that c/cD = stress boundary conditions and global [ASOCP], {Bequality}is 0.00001. global vectors that appear on the right hand side of all A code in MATLAB [32] has been developed for the equality constraints and fxðÞ0 is Mohr–Coulomb yield present analysis. MOSEK [33] available as a toolbox in criterion as shown in equation 6b. MATLAB [32] is popular to solve SOCP [17–21]. Thus, for The minimum lining pressure should be equal to and the optimization purpose, MOSEK [33] available as a opposite of the maximum collapse pressure and thus toolbox in MATLAB [32] is used. 19 Page 6 of 15 Sådhanå (2021) 46:19

5. Results and comparison strength of soil increases with increase in soil friction angle the requirement of support pressure to resist collapse r c 5.1 Comparison of present result decreases. It has also been observed that the required i/ D increases with increase in H/D keeping /v unchanged and There is no study reported in literature on peripheral sta- the rate of increase in ri/cD is found to be higher at lower bility of semi-elliptical tunnels. Hence, for validation pur- o value of H/D. For an example, for 2h/D = 1 and /v =25 pose, the mesh of the semi-elliptical tunnel has been (i) the value of r /cD increases from 0.44 to 0.56 due to r i extended for circular opening to compute the values of i/ increase in H/D from 1 to 3 in figures 4(b) and 4(e), a q cv0 in purely cohesive soil with c = 1 and D/cv0 =0. respectively, whereas (ii) the value of r /cD increases from r i Variation of i/cv0 for circular tunnel has been compared 0.58 to 0.59 due to increase in H/D from 5 to 7 in fig- with the results provided by (i) Mair [2], Davis et al [4] and ures 5(b) and 5(e), respectively. However, for higher values Wilson et al [8] for cD/cv0 = 2.6 and H/D = 1–3 in fig- of /v, the increment in ri/cD is very small. Similarly, the ure 1(d) and (ii) Wilson et al [8] for cD/cv0 = 0–4 and H/D magnitude of ri/cD increases with an increase in 2h/D. For = 1–7 as presented in figure 1(e). Figure 1(d) shows that the an example, for tunnel at H/D = 5 in cohesionless soil with difference between experimental results [2] and present o /v =20, the value of ri/cD at (a) 2h/D = 0.5 is 0.90 (in analysis is very less at all H/D except at H/D = 2, which is figure 5(a)), (b) 2h/D = 1 is 0.94 (in figure 5(b)) and (c) 2h/ close to 15%. The results obtained from present analysis are D = 1.5 is 1.03 (in figure 5(c)). 4–12% lower than the solutions reported by Davis et al [4]. The present solutions are close to the lower bound solutions reported by Wilson et al [8]. 5.3 Variation of normalized peripheral support pressure for tunnels in anisotropic homogeneous soil 5.2 Variation of normalized peripheral support For undrained cohesive soil, the effect of soil anisotropy on pressure for tunnel in isotropic homogeneous soil the magnitude of ri/cv0 is studied for different values of ac

The normalized peripheral support pressure (ri/cv0) for = 1, 1.5 and 2 and presented in figures 2 and 3 for different c a stability of semi-elliptical tunnel in isotropic and homoge- combinations of H/D,2h/D and D/cv0. The increase in c r c neous cohesive soil is presented in figures 2 and 3 for can increase the required value of i/cv0 when D/cv0, H/D different combinations of (i) H/D = 1–7 with an interval of and 2h/D remain unchanged. It is noted that for a tunnel c r 2, (ii) 2h/D equal to 0.5, 1 and 1.5 and (iii) cD/cv0 = 1–5 with particular 2h/D and D/cv0, the rate of increase in i/ a with an interval of 1 where ac ¼ 1 and qDac=cv0 ¼ 0. The cv0 due to increase in c decreases with an increase in H/D. c magnitude of ri/cv0 increases with an increase in cD/cv0 for For an example, for a tunnel with 2h/D = 1 and D/cv0 =4, a r any combination of H/D and 2h/D. For an example, with an when c increases from 1 to 1.5 the increases in i/cv0 are increase in cD/cv0 from 4 to 5, the increases in the value of 22.76% and 4.65% for H/D = 1 (in figure 1(b)) and 7 (in ri/cv0 are (a) 52.84% and 48.82% for 2h/D = 0.5 (in fig- figure 3(e)), respectively. Therefore it can be concluded ure 2(a)) and 1.5 (figure 2(c)), respectively, at H/D = 1 and that the effect of anisotropy is very low for tunnels (b) 32.12% and 31.33% for 2h/D = 0.5 (in figure 3(d)) and embedded at higher depths. Similarly, for a tunnel with c r 1.5 (in figure 3(f)), respectively, at H/D = 7. The value of particular H/D and D/cv0, the rate of increase in i/cv0 due a ri/cv0 also increases with an increase in H/D when other to increase in c decreases with increase in 2h/D.Foran c parameters remain unchanged. It implies that higher sup- example, for H/D = 3 and D/cv0 = 5, due to an increase in a r port pressure is required with increase in self-weight of c from 1 to 1.5 the magnitude of i/cv0 increases by 7.6% cover soil. The increase in aspect ratio of tunnel profile (2h/ for 2h/D = 0.5 in figure 2(d) and 7.09% for 2h/D = 1.5 in r D) can cause an increase in the required value of ri/cv0. For figure 2(f). The rate of increase in i/cv0 due to an increase a c an example, for the tunnel in saturated clay with cD/cv0 =5 in c decreases with an increase in D/cv0. For an example, in undrained condition, due to an increase in (a) 2h/D from for the tunnel with 2h/D = 1 and H/D = 3, the increases in r 0.5 to 1, the increases in the value of ri/cv0 are 12.22% and i/cv0 as observed in figure 2(e) are 15.38% and 7.47% for c 3.62% for H/D = 1 (in figures 2(a) and 2(b)) and 7 (fig- D/cv0 = 3 and 5, respectively. Thus effect of soil aniso- ures 3(d) and 3(e)), respectively, and (b) 2h/D from 1 to tropy is prominent in stiffer clay. a 1.5, the increases in the value of ri/cv0 are 18.00% and For cohesionless soil, the effects of soil anisotropy, /, 3.77% for H/D = 1 (figures 2(b) and 2(c)) and 7 (fig- on the normalized support pressure ri/cD are studied for (i) ures 3(e) and 3(f)), respectively. H/D = 1–7 with an interval of 2, (ii) /v = 20°–45° with an ° a For cohesionless soil, the normalized support pressure ri/ interval of 5 , (iii) 2h/D = 0.5, 1 and 1.5 and (iv) / = 1, 1.5 cD is presented in figures 4 and 5 for (i) H/D = 1–7 with an and 2 and presented in figures 4 and 5. The results show a interval of 2, (ii) /v = 20°–45° with an interval of 5° and that the stability of tunnel decreases with an increase in / (iii) 2h/D = 0.5, 1 and 1.5. The figures show that the value keeping other parameters unchanged. For a tunnel with a of ri/cD decreases with an increase in /v. Since the shear particular 2h/D, due to increase in / the rate of increase in Sådhanå (2021) 46:19 Page 7 of 15 19

Figure 2. Variation of ri/cv0 with cD/cv0 for tunnel embedded in homogenous cohesive soil (qDac=cv0 =0)atH/D = 1 with (a)2h/D = 0.5, (b)2h/D = 1 and (c)2h/D = 1.5; and at H/D = 3 with (d)2h/D = 0.5, (e)2h/D =1 and (f)2h/D = 1.5. ri/cD increases with an increase in H/D. For an example, 5.4 Variation of normalized peripheral support / ° for the tunnel with 2h/D = 1 in soil with v =25, due to pressure (ri/cv0) for tunnels embedded in isotropic a r c increase in / from 1 to 1.5, i/ D increases by 24.58% for nonhomogeneous undrained cohesive soil H/D = 1 (in figure 4(b)) and 47.75% for H/D = 7 (in figure 5(e)). This shows that the effect of anisotropy is more The effects of linear increase of the undrained shear significant in case of tunnels embedded at higher H/D in strength for normally consolidated clay on the magnitude of r a q a = cohesionless soil. The increase in 2h/D can also increase i/cv0 for c = 1 and D c cv0 equal to 0.25, 0.5 and 1 are presented in Table 2. It is observed that for a particular 2h/ the value of ri/cD. For an example, for the tunnel in soil D, H/D and cD/c , r /c decreases with an increase in with /v =25° and a/ = 1.5 located at H/D = 5, the value of v0 i v0 qDac=cv0 due to increase in undrained shear strength of ri/cD is (i) 0.77 for 2h/D = 0.5 (in figure 5(a)), (ii) 0.83 for 2h/D = 1(in figure 5(b)) and (iii) 0.96 for 2h/D = 1.5 (in clay with depth. For an example, for the tunnel with 2h/D = a c r figure 5(c)). 1 located at H/D =3, c = 1 and D/cv0 =5, i/cv0 decreases 19 Page 8 of 15 Sådhanå (2021) 46:19

Figure 3. Variation of ri/cv0 with cD/cv0 for tunnel embedded in homogenous cohesive soil (qDac=cv0 =0)atH/D = 5 with (a)2h/D = 0.5, (b)2h/D = 1 and (c)2h/D = 1.5; and at H/D = 7 with (d)2h/D = 0.5, (e)2h/D = 1 and (f)2h/D = 1.5.

from 9.19 to 6.79 due to an increase in qDac/cv0 from 0.25 decrease in ri/cv0 due to increase in qDac/cv0 is found to to 0.5. In Table 2 the rate of change in ri/cv0 due to increase increase with an increase in H/D for any value of cD/cv0. in qDac/cv0 is noted to reduce with an increase in cD/cv0 for For an example, for the tunnel with 2h/D = 1 and soil with any combinations of H/D,2h/D and qDac/cv0.Foran cD/cv0 =5,ac= 1, the value of ri/cv0 decreases by 21.99% example, for the tunnel with 2h/D =1atH/D = 1 in iso- and 29.45% for H/D = 3 and 7, respectively, due to increase tropic cohesive soil with ac = 1, the value of ri/cv0 in qDac=cv0 from 0 to 0.25. It indicates that the increase in decreases by 47.94% and 15.89% for cD/cv0 = 3 and 5, nonhomogeneity of soil greatly reduces the requirement of respectively, due to increase in qDac=cv0 from 0 to 0.25. It support pressure for tunnels located at higher depths. For implies that the effect of linear increase in shear strength on any values of H/D, cD/cv0 and /v a semi-elliptical tunnel ri/cv0 is less prominent in very soft soil. The rate of (2h/D B 1.5) requires lesser support pressure (ri/cv0) than Sådhanå (2021) 46:19 Page 9 of 15 19

Figure 4. Variation of ri/cD with /v for tunnel embedded in cohesionless soil at H/D = 1 with (a)2h/D = 0.5, (b)2h/D = 1 and (c)2h/ D=1.5 and at H/D = 3 with (d)2h/D = 0.5, (e)2h/D = 1 and (f)2h/D = 1.5.

the support pressure required for circular and square tunnels 5.5 Variation of normalized peripheral support of same diameter/width, which may be due to the fact that pressure (ri/cv0) for tunnels embedded the amount of soil mass removed for the construction of a in anisotropic and nonhomogeneous undrained B semi-elliptical tunnel for 2h/D 1.5 is less in comparison cohesive soil with the circular and square shaped tunnels. For instance, for the tunnel embedded in isotropic homogeneous purely On one hand, due to the anisotropic nature of cohesive soil cohesive soil at H/D = 3 and cD/cv0 = 3, the magnitudes of the magnitude of required normalized peripheral support r ri/cv0 for semi-elliptical tunnel with 2h/D = 1.5, circular pressure ( i/cv0) increases with an increase in degree of a and square tunnels are 5.94 (present analysis), 6.1 [8] and anisotropy ( c) as illustrated in section 5.2. On the other 6.62 [10], respectively. hand the shear strength of normally consolidated clay and 19 Page 10 of 15 Sådhanå (2021) 46:19

Figure 5. Variation of ri/cD with /v for tunnel embedded in cohesionless soil at H/D = 5 with (a)2h/D = 0.5, (b)2h/D = 1 and (c)2h/ D=1.5 and at H/D = 7 with (d)2h/D = 0.5, (e)2h/D = 1 and (f)2h/D = 1.5.

slightly over-consolidated clay increases with depth [26] homogeneous clay (ac =1,qDac/cv0 = 0) to (a) 10.32 in and it reduces the required magnitude of ri/cv0 in isotropic anisotropic (ac = 2) and nonhomogeneous soil (qDac/cv0 = nonhomogeneous cohesive soil, which is discussed in sec- 0.25) and (b) 9.76 in anisotropic (ac = 1.5) and nonhomo- tion 5.3. In this section the change in ri/cv0 in anisotropic geneous soil (qDac/cv0 = 0.25). This trend is noted mainly and nonhomogeneous cohesive soil is studied to illustrate for (i) H/D C 3 and all values of 2h/D, cD/cv0, qDac/cv0 and the combined effect of soil anisotropy and nonhomogeneity ac and (ii) a few cases of H/D = 1 with higher value of on the required value of ri/cv0. The results of ri/cv0 in qDac/cv0 and lower values of ac. It can be inferred that anisotropic and nonhomogeneous cohesive soil are pre- although only soil anisotropy reduces the stability of tun- sented in Table 2 for different values of H/D, cD/cv0, qDac/ nels embedded at H/D C 3, the consideration of linear cv0 and ac. It is noted that in general, the magnitude of ri/ increase of undrained shear strength of soil can reduce the cv0 in anisotropic and nonhomogeneous clay is lower than effect of anisotropy on the required magnitude of support that in isotropic homogeneous clay. For an example, for the pressure. However, for H/D =1,ri/cv0 in isotropic homo- tunnel at H/D = 3 in undrained clay with cD/cv0 = 5, the geneous cohesive soil is observed to be lower than the value magnitude of ri/cv0 decreases from 10.91 in isotropic of ri/cv0 in anisotropic nonhomogeneous soil with (a) qDac/ S

r å

Table 2. Variation of i/cv0 for tunnel embedded in anisotropic nonhomogeneous purely cohesive soil. dhan r i/cv0 å

(2021) Nonhomogeneous undrained clay with different ac Isotropic and homogeneous undrained clay (ac =1,qD/cv0 =

0) 2h/D = 0.5 2h/D =1 2h/D = 1.5 46:19

2h/D ac ac ac

H/D cD/cvo 0.5 1 1.5qDac/cv0 1 1.5 2 1 1.5 2 1 1.5 2 11 - 1.18 - 1.25 - 1.20 0.25 - 1.57 - 1.09 - 0.86 - 1.77 - 1.16 - 0.86 - 1.82 - 1.11 - 0.77 0.5 - 1.95 - 1.41 - 1.16 - 2.25 - 1.55 - 1.21 - 2.4 - 1.58 - 1.16 1 - 2.68 - 2.03 - 1.72 - 3.2 - 2.34 - 1.89 - 3.49 - 2.48 - 1.93 3 0.97 1.12 1.44 0.25 0.58 1.07 1.31 0.59 1.22 1.52 0.78 1.52 1.87 0.5 0.19 0.74 1.02 0.06 0.80 1.16 0.14 1.02 1.45 1 - 0.55 0.12 0.44 - 0.93 0.00 0.46 - 1.07 0.07 0.65 5 3.14 3.53 4.16 0.25 2.73 3.24 3.50 2.97 3.62 3.93 3.43 4.21 4.58 0.5 2.34 2.91 3.19 2.43 3.19 3.56 2.76 3.68 4.12 1 1.58 2.27 2.62 1.39 2.36 2.84 1.48 2.69 3.27 31 - 1.42 - 1.12 - 0.74 0.25 - 3.77 - 2.55 - 2.00 - 3.54 - 2.21 - 1.58 - 3.20 - 1.82 - 1.16 0.5 - 6.09 - 4.44 - 3.70 - 5.91 - 4.10 - 3.28 - 5.60 - 3.70 - 2.80 1 - 10.7 - 8.20 - 7.09 - 10.63 - 7.84 - 6.59 - 10.37 - 7.43 - 6.05 3 4.73 5.33 5.94 0.25 2.38 3.61 4.16 2.84 4.17 4.79 3.41 4.82 5.49 0.5 0.05 1.71 2.45 0.45 2.27 3.11 0.98 2.91 3.83 1 - 4.56 - 2.05 - 0.95 - 4.27 - 1.49 2.62 - 3.81 - 0.84 0.53 5 10.91 11.79 12.83 0.25 8.53 9.76 10.32 9.19 10.57 11.21 10.07 11.51 12.20 0.5 6.20 7.86 8.61 6.79 8.65 9.48 7.60 9.56 10.49 1 1.59 4.10 5.21 2.05 4.87 6.20 2.77 5.77 7.20 51 - 0.55 - 0.17 0.25 0.25 - 5.82 - 3.92 - 3.03 - 5.33 - 3.32 - 2.42 - 4.74 - 2.70 - 1.76 0.5 - 10.92 - 8.13 - 6.83 - 10.36 - 7.37 - 6.01 - 9.63 - 6.56 - 5.10 ae1 f15 of 11 Page 1 - 21.07 - 16.49 - 14.40 - 20.36 - 15.43 - 13.18 - 19.33 - 14.25 - 11.82 3 9.7 10.38 11.09 0.25 4.32 6.24 7.13 5.03 7.05 7.97 5.86 7.94 8.89 0.5 - 0.78 2.02 3.33 - 0.01 2.99 4.36 0.95 4.05 5.52 1 - 10.93 - 6.35 - 4.25 - 10 - 5.08 - 2.79 - 8.77 - 3.66 - 1.26 5 19.98 21 22.12 0.25 14.47 16.39 17.31 15.37 17.44 18.42 16.50 18.62 19.61 0.5 9.36 12.17 13.49 10.32 13.37 14.73 11.56 14.69 16.17 1 - 0.79 3.80 5.91 0.32 5.28 7.56 1.81 6.94 9.41 19

7 1 0.84 1.22 1.65 0.25 - 7.91 - 5.31 - 4.08 - 7.21 - 4.49 - 3.26 - 6.39 - 3.66 - 2.40 0.5 - 16.28 - 12.24 - 10.34 - 15.34 - 11.06 - 9.13 - 14.19 - 9.89 - 7.85 1 - 32.97 - 26.03 - 22.83 - 31.51 - 24.17 - 20.81 - 29.72 - 22.27 - 18.77 3 15.12 15.8 16.57 0.25 6.24 8.85 10.08 7.17 9.88 11.15 8.21 10.97 12.27 0.5 - 2.14 1.91 3.80 - 0.96 3.30 5.23 0.39 4.72 6.77 1 - 18.83 - 11.89 - 8.67 - 17.14 - 9.82 - 6.45 - 15.16 - 7.68 - 4.18 5 29.41 30.48 31.63 0.25 20.40 23.00 24.22 21.50 24.29 25.61 22.84 25.65 27.02 19 Page 12 of 15 Sådhanå (2021) 46:19

cv0 = 0.25 for all values of 2h/D, cD/cv0, qDac/cv0 and ac and (b) qDac/cv0 = 0.5 for (i) all 2h/D and cD/cv0 = 1 and 3 and ac = 2, (ii) 2h/D = 0.5 and 1 for cD/cv0 = 5 and ac =2. For an example, for H/D = 1, due to increase in qDac/cv0 = 1.5 c

a from 0 to 0.25, the values of ri/cv0 increase from (i) 4.16 at

h/D ac = 1 to 4.58 at ac = 2 for 2h/D = 1.5 and cD/cv0 = 5 and

c (ii) 3.14 at a = 1 to 3.24 at a = 1.5 for 2h/D = 0.5 and cD/ a c c

0.58 6.93 10.47 cv0 =5. -

5.6 Proximity of the stress state to yield

For isotropic soil, i.e. cv = ch in purely cohesive soil and tan

=1 2 / /

c v = tan h in cohesionless soil, the proximity of the stress a state at a point with respect to shear failure is presented in h/D 2 2 2 terms of a ratio a/d where a =(rx–ry) ? (2sxy) and d =4c 2 in purely cohesive soil and d = {–(rx?ry)sin /} in 2.84 4.54 7.91 cohesionless soil. The plots of proximity of the stress state - to failure are presented in ﬁgure 6. The ratio of a/d becomes unity at any point where shear failure has occur- red. For all non-yielding points, a/d B 1. Figures 6(a, b)

Nonhomogeneous undrained clay with different present the stress contour plots, which illustrate that the size of the plastic failure zone decreases with an increase in qD/ = 0.5 2 c c as the shear strength of the soil increases with depth for

a v tunnels embedded in isotropic nonhomogeneous cohesive h/D v0 2 c /

i soil. Moreover, it is noted that in homogeneous clay soil r with uniform undrained shear strength, the failure zone has 4.69 2.26 5.49

- covered both the roof and ﬂoor of the tunnels whereas in clay soil where shear strength increases with depth the 0 v plastic failure zone can cover only the roof of the tunnel as /c c

a shown in ﬁgure 6(b). Figures 6(a, c) show that the size of D 0.51 12.00 16.06 17.98 13.35 17.67 19.62 14.99 19.35 21.41 q plastic zone increases with an increase in soil cover depth

= of tunnel. For cohesionless soil, the plastic failure zone 0 v covers only the roof of the tunnel as shown in ﬁgures 6(d)–

D/c (f). No failure occurs below the tunnel ﬂoor for any value of q soil friction angle. The size of the plastic failure zone

=1, increases with increase in aspect ratio of tunnel (2h/D)as c a indicated by ﬁgures 6(d, e) and it decreases with an increase in soil friction angle, /v, as shown in ﬁgures 6(e, f). 0) h/D 2 6. Conclusions

The normalized support pressures for different values of 2h/ D and H/D are obtained for tunnels in purely cohesive soil for different combinations of cD/cv0, ac and qDac/cv0 and

0.5 1 1.5for tunnels in cohesionless 1 1.5 soil for 2 different 1 combinations 1.5 2 1 1.5 2 of /v and a/. The increase in aspect ratio of tunnel (2h/D) can cause a decrease in the stability of tunnel located in Isotropic and homogeneous undrained clay ( both purely cohesive soil and cohesionless soil. The mag-

vo nitude of required ri/cv0 in cohesive soil increases with (i) continued

D/c an increase in H/D, cD/cv0, ac and (ii) a decrease in shear c strength parameter qDac/cv0. The magnitude of ri/cD in cohesionless soil is found to increase with (i) an increase in Table 2 H/D Sådhanå (2021) 46:19 Page 13 of 15 19

Figure 6. Proximity of the stress state to failure for smooth tunnel embedded in isotropic soil with (a)2h/D =1,H/D =3,/v = /h = 0°, cD/cv =3,qD/cv =0,(b)2h/D =1,H/D =3,/v = /h = 0°, cD/cv =3,qD/cv = 0.5, (c)2h/D =1,H/D =5,/v =/h = 0°, cD/cv =3,qD/cv = 0, (d)2h/D = 0.5, H/D =3,/ =20°, cv0 = ch0 =0,(e)2h/D = 1.5, H/D =3,/ =20°, cv0 =ch0 = 0 and (f)2h/D = 1.5, H/D =3,/ =35°, cv0 = ch0 =0.

H/D and a/ and (ii) a decrease in /v. However, the increase = 7 and 2h/D = 0.5 in anisotropic nonhomogeneous soil a q a of ri/cD is very less for /v C 35°. The increase in ac and a/ with c = 2 and D c/cv0 = 1. The semi-elliptical tunnel (2h/ implies that the shear strength in horizontal direction is D B 1.5) requires less support pressure in comparison with quite less than that in vertical direction, so the stability of the circular and square shaped tunnels of same diameter/ tunnel decreases and requirement of support pressure width, D, embedded in same soil keeping H/D unchanged increases. In anisotropic nonhomogeneous cohesive soil the in all the three cases. The solutions obtained in the present magnitude of required ri/cv0 is lower than that in isotropic work are expected to be useful to determine the support homogeneous cohesive soil for H/D C 3 and a few cases of pressure for the stability of semi-elliptical tunnel in wide H/D = 1. The maximum reduction in ri/cv0 is noted at H/D ranges of soils. 19 Page 14 of 15 Sådhanå (2021) 46:19

References

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