Developing an Analytical Method to Study Vertical Stress Due to Soil Arching During Tunnel Construction

Transcription

1 Geotech Geol Eng (2016) 34: DOI /s x TECHNICAL NOTE Developing an Analytical Method to Study Vertical Stress Due to Soil Arching During Tunnel Construction Chunlin Li Received: 6 October 2015 / Accepted: 14 May 2016 / Published online: 27 May 2016 Springer International Publishing Switzerland 2016 Abstract Due to displacement constraints, the soil arching effect only plays a partial function during tunnel construction. When the shear stress of soil is less than anticipated relative to its shear strength, Terzaghi vertical stress on the soil arch is not reasonable. A calculation method of the lateral pressure coefficient was developed based on a previous research. The results show that the value of the lateral pressure coefficient varies between active and passive earth pressure coefficients. Subsequently, vertical stress was proposed by considering partially developed soil arching effect. On this basis, a simplified method to predict the arching effect zone associated with tunneling was drawn by studying the soil arch height. Finally, an analytical method for supporting force of the lining and tunnel vault displacement was developed considering the stiffness of the tunnel lining. Keywords Tunnel construction Soil arching effect Vertical stress Ground loss Tunnel lining C. Li (&) Institute of Civil and Architectural Engineering, Tongling University, Tongling, China lichunlin111@126.com 1 Introduction Several experimental, theoretical, and analytical studies have been conducted to evaluate the soil arching effect in the geotechnical engineering field (Tien 1996). Among these studies, the most widely known theoretical investigations of arching were conducted by Terzaghi, based on trapdoor experimental results (Terzaghi 1936). Terzaghi described the soil arching effect as stress transfer due to soil differential displacement and assumed that the shear stress is equal to the shear strength at the soil arching influencing zone. Soil arching theory is widely used in underground engineering such as tunnels but some issues still require deeper discussion. For example, the traditional soil arching theory is based on the assumption that the soil above the top of a tunnel is in an ultimately balanced condition and without considering soil arching it cannot fully function because of the limitations on vault displacement. However, the existence of tunnel linings can prevent the development of the differential settlement, which would consequently lead to the partial development of the soil arching effect. When the soil arching effect cannot fully take effect, the shear stress of the soil can be far less than its original shear strength. In view of this, a new analytical method for calculating the vertical stress of the soil arch is proposed considering a partially developed soil arching effect, which can then be adopted for tunnel design.

2 1248 Geotech Geol Eng (2016) 34: Terzaghi s Arching Theory According to the trapdoor test results and theoretical deduction, Terzaghi proposed a semi-empirical theoretical approach for the arching problems in sand under a plane strain condition (Terzaghi 1943). He defined the arching effect as the stress transfer between a yielding mass of soil and adjoining stationary parts. If the tunnel roof is located at a depth of h from the ground surface, the vertical stress on the roof is (Terzaghi 1943): r v ¼ r 0 v þ r00 v ¼ 1c c K tan uh 1 e 1 þ qe K tan u K tan uh 1 ð1þ where 2 1 = width of the yielding strip (m), h = depth, c = unit weight of soil, r v = vertical stress, K = lateral stress coefficient, q = surcharge at the soil surface, c = cohesion, u = friction angle. Figure 1 shows the vertical stress profile at the top of a tunnel. If the tunnel is very deep, the arching effect cannot extend beyond a certain elevation (D 1 ) above the tunnel crown (Fig. 1), and there is no arching effect and no shear resistance within D 2, the vertical stress on the roof then is expressed as: Fig. 1 Loosening earth pressure profile located above the tunnel r v ¼ r 0 v þ r00 v ¼ 1c c K tan ud 1 1 e 1 þðqþcd 2 Þe K tan ud 1 1 K tan u ð2þ ased on experiments, Terzaghi suggested that the value of D 1 is equal to 3 1. For a circular tunnel, the value of D 1 is advised to range from 1.5 times to 3 times the tunnel diameter (Kezdi 1986). Although the soil arch theory proposed by Terzaghi is still widely used today, some issues still need to be explored, which are listed as below: 1. In a real project, soil displacement is often limited (e.g., tunnel liner supporting) and the development of the soil arching effect is closely associated with the crown displacement of the tunnel. In this case, an analytical method to predict vertical stress of the soil arch requires further research. 2. Even if the soil displacement is not limited, the soil is not necessarily going to reach the ultimate strength. In other words, if the value of soil shear strength is so large that the sliding surface in soil cannot be formed, Terzaghi assumption can be unreasonable as its shear stress is less than the shear strength of soil and the assumed sliding surfaces are not true. For example, the vertical stress according to Terzaghi s theory is negative when geological engineering conditions are suitable and the soil is of high shearing strength, which is not obviously consistent with the fact. 3. To simplify the analysis, Terzaghi assumed that the normal stress is uniform across horizontal sections and the value of K is the constant value. However, the coefficient of lateral stress (K) depends on the development of the soil arching effect. K varies with the changing position of the calculated point according to experimental results and theoretical analysis. 4. Tunnel segment stiffness is closely related to tunnel crown vertical displacement, which further affects the loose earth pressure. However, in Terzaghi s soil arch theoretical analysis, the correlation between tunnel segment stiffness and segment supporting force has not been deeply explored. 5. Terzaghi proposed the concept of the equal settlement plane based on experimental results,

3 Geotech Geol Eng (2016) 34: where the thickness of the zone of the arching is assumed to be D 1. Terzaghi suggested that the thickness D 1 is roughly equal to 3 1, and the stress in the soil above the zone of arching is not affected by the tunnel operations. However, thickness D 1 is only obtained based on experimental data without theoretical support. (a) 3 Variation of Lateral Stress Coefficient Due to Arching Terzaghi assumed that the coefficient of lateral stress (K) is equal to 1 to simplify the analysis. Handy (1985), Giroud et al. (1990), and Chen et al. (2010) considered that the coefficient of lateral stress is a constant relative to the effective angle of internal friction. In reality, the coefficient of lateral stress is a changing parameter because of the soil arching effect. Partially developed soil arching is defined as a state in which shear stress does not fully develop compared to its shear strength (Wang and Miao 2009; Lv et al. 2012). Due to limitations on vault displacement, the soil arching effect only plays a partial function during tunnel construction. In such a case, the soil in the arching zone attains new equilibrium by segment support force and stress adjustment in the soil arch area. If the soil arching effect is partially developed, the coefficient of lateral stress of cohesionless soil can be derived about partially developed soil arching based on the Handy theory. Handy (1985) suggested a method to calculate the coefficient of lateral earth pressure in the soil arch when sufficient vertical settling of the soil occurs, as shown in Fig. 2. Minor principal planes drawn through the Mohr circle poles (Fig. 2a) showed radiating major principal stress directions, whereas the trajectory of the minor principal stress, r 3, defined a continuous compression arch that dips downward instead of upward (Fig. 2b). A force equilibrium equation is derived based on trigonometric functions as shown in Fig. 3: r h ¼ r 1 cos 2 h þ r 3 sin 2 h ð3þ (b) Fig. 2 a Krynine construction of Mohr circle to show arching stresses at rough wall; b continuous inverted arch defined by trajectory of minor principal stresses (Handy 1985) (a) (b) Fig. 3 Sketch of soil stress for soil arching effect

4 1250 Geotech Geol Eng (2016) 34: where r h = the horizontal stress of the soil arch (kpa), r 1 = the major principle stresses (kpa), r 3 = minor principle stresses (kpa), h = the angle between the major principle stresses direction and the horizontal direction. Similarly, r v ¼ r 1 sin 2 h þ r 3 cos 2 h ð4þ where r v is the vertical stress of the soil arch (kpa). When the soil arching effect is partially developed, as shown in Fig. 3, the shear stress of the soil arch is s ¼ðr h r 3 Þ tan h ¼ r h tan a ð5þ where s is the shear stress and the angle a is defined in Fig. 3, which depends on the developmental degree of the soil arch effect. Dividing Eq. (3)byr 1 and considering the soil arch in an active state, the following is obtained r h ¼ cos 2 h þ K a sin 2 h ð6þ r 1 where K a is the coefficient of active earth pressure K a ¼ r 3 =r 1 Dividing Eq. (4)byr 1 and considering the soil arch is in an active state, the following is obtained r v ¼ sin 2 h þ K a cos 2 h ð7þ r 1 Dividing Eq. (6) by Eq. (7) gives K ¼ r h ¼ cos2 h þ K a sin 2 h r v sin 2 h þ K a cos 2 h ¼ 1 þ K a tan 2 h tan 2 ð8þ h þ K a where K is the coefficient of lateral earth pressure in the soil arch and K a is the coefficient of lateral active earth pressure, passive lateral coefficients, and the coefficient of lateral earth pressure increases with the increase of the deflection degree of principal stress. Substituting Eq. (3) into Eq. (5) gives: r 1 cos 2 h þ r 3 sin 2 h tan a ¼ r1 cos 2 h þ r 3 sin 2 h r3 tan h r 1 cos 2 h þ r 3 sin 2 h tan a ¼ r1 cos 2 h r 3 cos 2 h tan h Therefore: ð tan a ¼ r 1 cos 2 h r 3 cos 2 hþtan h r 1 cos 2 h þ r 3 sin 2 ¼ ð1 K aþ tan h h 1 þ K a tan 2 h ð9þ Therefore the shear stress on the side of the soil arch can be calculated by combining Eqs. (5), (8), and (9): s ¼ Kr vð1 K a Þ tan h 1 þ K a tan 2 ð10þ h It can be seen from Eq. (10) that the shear stress (s)is directly bound with h, and h is closely dependent on the vault displacement of tunnel and stiffness of the tunnel segment. 4 Vertical Stress of the Soil Arch Under Partially Developed Soil Arching Effect The potential sliding surfaces are assumed to be vertical. The vertical sections ab and cd through the K a ¼ r 3 =r 1 ¼ tan 2 ð45 u=2þ: From this Eq. (8), K is a function of h. When the rotation of principal stress axes does not occur (h = 90), K is equal to K a. When the rotation of principal stress axes is equal to 90 (h = 0), K is equal to 1/K a, in other words, K is equal to K p, where K p ¼ tan 2 ð45 þ u=2þ is the coefficient of lateral passive earth pressure. The analysis indicates that the coefficient of lateral earth pressure in the soil arch varies between active and Fig. 4 Free body diagram for the loosening zone of cohesionlesss soil

5 Geotech Geol Eng (2016) 34: outer edges of the yielding strip represent surfaces of potential sliding in Fig. 4. The pressure on the horizontal strip is thus equal to the difference between the weight of the soil located above the tunnel and shearing stress along the vertical sections. The free body diagram for a slice of cohesionlesss soil in the loosening zone is shown in Fig. 4. Compared with Terzaghi s methods, the main difference in the presented method is that shear strength (s f ) is replaced with shear stress (s). For the equilibrium of the upward and downward vertical forces on the horizontal strip in Fig. 4, the following equation can be obtained dr v ¼ c 2s dh ¼ c s dh ð11þ 1 p=4 þ u=2 1 ¼ R cot ð12þ 2 In which, = width of the yielding strip (2 1 ), h = depth (m), c = unit weight of soil, r v = vertical stress (kpa), R = the radius of the tunnel (m). Substituting Eq. (10) in Eq. (11), and let ¼ Kð1 K aþ tan h 1 þ K a tan 2 h Equation (11) reduces to dr v ¼ c 2A 1r v dh ð13þ y separation of the variables, Eq. (13) can be rewritten as r v ¼ c þ C 3 e 2 h ð17þ 2 where C 3 is a constant, C 3 is equal to C 2 /(2 ). The boundary condition is r v = q at h = 0, solving Eq. (17), it can be concluded that C 3 is equal to q - c/ (2 ). Therefore the vertical stress r v on the roof is r v ¼ 1c 1 e h 1 þ qe h 1 ð18þ If a tunnel is located at a great depth from the surface, the vertical stress r v will reach a limit value r v(?), r v(?) is equal to 1 c/. In the special case in which q is zero, considering the boundary condition, r v = 0 at h = 0, solving Eq. (17), it can be concluded that C 3 is equal to -c/(2a), i.e., - 1 c/(a). In this case, the vertical stress r v on the roof is thus equal to r v ¼ 1c 1 e h 1 ð19þ Similarly, when h is very large, the vertical stress r v will reach a limit value r v(?), r v(?) is equal to 1 c/. For cohesion soil, the solution for the vertical stress can be obtained by making use of transformation of coordinate translation (Li et al. 2013), as shown in Fig. 5. r 0 ¼ r þ c cot u s 0 ¼ s ð20þ The coefficient of lateral earth pressure K is: K 0 ¼ r0 h r 0 ¼ 1 þ K a tan 2 h v tan 2 ð21þ h þ K a dr v c 2r v ¼ dh y indefinite integral, Eq. (14) becomes ln c 2A 1r v ¼ h þ C 1 2 where C 1 is constant. Equation (15) can be deduced as: ð14þ ð15þ c 2r v ¼ 2 e ðc1þhþ ¼ e 2 C 1 e 2 h ¼ C 2 e 2 h ð16þ where C 2 is a constant, C 2 is equal to e 2C 1 =. The solution for the vertical stress can be obtained by solving Eq. (16) Fig. 5 The analysis of Mohr s circle for cohesive soil

6 1252 Geotech Geol Eng (2016) 34: K ¼ r h ¼ r0 h c cot u r v r 0 v c cot u ¼ K0 r 0 v c cot u r 0 v c cot u ¼ K0 ðr 0 v c cot uþþðk0 1Þc cot u r 0 v c cot u ¼ K 0 þðk 0 1Þ c cot u r v Therefore the shear stress on the side of the soil arch is s ¼ K0 r 0 v ð1 K aþ tan h 1 þ K a tan 2 ¼ r 0 v ð22þ h Substituting Eqs. (22) (11), and the following equation is obtained dr v ¼ c 2r v 2A 1c cot u dh ð23þ Considering the boundary condition, r v = qath= 0, the vertical stress r v on the roof for cohesive soil is thus equal to r v ¼ 1c c cot u 1 e h 1 þ qe h 1 ð24þ 5 Vertical Stress of the Soil Arch Caused by Different Ground Loss For proposed method, is an important parameter, as the effect of soil arching depends on the number of. The greater the number of is, the more rapidly r v approaches limiting value with the increase of soil depth. The relation of and h can be expressed as follows: ¼ Kð1 K aþ tan h 1 þ K a tan 2 h ¼ 1 þ K a tan 2 h tan 2 ð1 K aþ tan h ð25þ h þ K a 1 þ K a tan 2 h ¼ ð1 K aþ tan h tan 2 h þ K a h is relevant to vault displacement of the tunnel. Figure 6 shows the relation between the angle h and tunnel crown vertical displacement S, in which the shape of the soil arch is assumed to be circular and L is the radius of the circular. Thus, the relation of L and S can be expressed as follows: L 2 ¼ðL SÞ 2 þ 2 1 L ¼ S 2 þ 2 ð26þ 1 =2S From the Fig. 6, the relation of h and S can be expressed as follows: tan h ¼ L S 1 ¼ S 2 þ 2 1 2S 1 S ¼ 2 1 S2 ¼ 1 2S 1 2S S 2 1 ð27þ Combining Eqs. (24), (25), and (27), the connection between S and vertical stress of the soil arch can be further established. Example Tunnel diameter = 6.28 m, h = 20 m, c = 20 kn/m 3, r cz = ch = 400 kpa, u = 15, The ratio of vertical stress of the soil arch to the weight of the soil located above tunnel φ=15º φ=30º φ=20º Vault displacement/mm Fig. 6 Calculating sketch of tanh Fig. 7 The ratios of vertical stress of the soil arch to the weight of the soil located above tunnel under different vault displacement of tunnel

7 Geotech Geol Eng (2016) 34: u = 20 or u = 30, K a = 0.59, 0.49 or 0.33, S = 0, 5 mm, 10 mm, 15 mm,, 80 mm. The ratios of vertical stress of the soil arch to gravity stress (r v /r cz ) under different vault displacement of tunnel are shown in Fig. 7. As shown in Fig. 7, soil pressure on the tunnel segment is decreased gradually with the increase of the tunnel vault displacement. Since the soil arching effect can play a greater role and the strength of soil can be more sufficiently utilized with the increase of the tunnel vault displacement. Furthermore, the soil arching effect is relevant to the internal friction angle of soil (u). The bigger internal friction angle (u) is, the more fully the soil arching effect plays and the smaller vertical stress is. 6 Comparison etween Terzaghi s Arching Theory and the Presented Method Terzaghi s arching theory r v ¼ 1c c K tan uh tan uh K 1 e 1 þ qe 1 ð28þ K tan u The presented method in this article: r v ¼ 1c c cot u 1 e h 1 þ qe h 1 ð29þ y comparing Eq. (29) with Eq. (28), it can be noted that both are exactly the same using to replace Ktanu and assuming K is equal to 1. In Terzaghi s arching theory, Ktanu is a constant; whereas for the proposed method, is not constant but rather a function of h, h depends on segment stiffness and construction factors, such as ground lose, time for injecting. Compared with Eq. (28), some limitations of Terzaghi s arching theory are modified to make the predictions closer to the physical conditions and some construction parameters may be taken into account in Eq. (29), so that it is more reasonable. Example In order to investigate the accuracy of the proposed method, the vertical stress on the tunnel using the proposed method were compared with those using the Terzaghi s arching theory. Tunnel diameter = 6.28 m, h = 20 m, c = 20 kn/ m 3, u = 20, C = 0, S = 60 mm. 6.1 The Proposed Method K a ¼ tan 2 ð45 u=2þ ¼0:49 p=4 þ u=2 1 ¼ R cot ¼ 3:14 cotð27:5 Þ¼6:03 2 tan h ¼ 1 2S S ¼ 6:03 0: :06 2 6:03 ¼ 50:3 ¼ 1 þ K a tan 2 h tan 2 ð1 K aþ tan h h þ K a 1 þ K a tan 2 h 1 þ 0:49 50:32 ð1 0:49Þ50:3 ¼ 50:3 2 þ 0:49 1 þ 0:49 50:3 2 ¼ 0:10 r v ¼ 1c c cot u 1 e h 1 ¼ 6: :10 1 e 0:1020 6: Terzaghi s Arching Theory r v ¼ 1c c K tan u ¼ 6: tanð20 Þ K tan uh ð1 e 1 Þ 1 e ¼ 339:7 kpa 1tanð20 Þ20 6:03 ¼ 150:9 kpa The results show that the calculated values of the proposed method is much larger than those obtained by Terzaghi s method when tunnel crown vertical displacement, S, is quite small and the soil arching cannot fully develop. The vertical stresses are compared for the proposed method and Terzaghi method when the tunnel crown vertical displacement S is equal to 60 mm in different internal friction angle of soil, as shown in Fig Discussion of Soil Arch Height and Range of Construction Disturbance When the buried depth of the tunnel is very large, the arching effect cannot extend beyond a certain elevation D 1 above the tunnel roof. Terzaghi suggested that D 1 was roughly equal to 3 1 (Tien 1996). The soil arch height, D 1, presented by Terzaghi, is established merely by experience and lacks systematic and theoretical analysis. In fact, D 1 is affected by soil conditions and construction factors. D 1 presented by

8 1254 Geotech Geol Eng (2016) 34: article, the soil arch height calculation formula is derived as follows 1 c c cot u 1 e D 1 1 ¼ 1c c cot u m ð30þ Equation (30) can be deduced as Fig. 8 The comparison between proposed method and Terzaghi method to calculate vertical stress in different internal friction angle of soil D 1 ¼ 1 lnð1 mþ ð31þ Assuming m is equal to 95 %, accordingly the soil arch height is equal to 3 1 / by Eq. (31). Essentially, there is no arching effect and soil disturbance outside the soil arch height range. Therefore, a simple method to determine disturbance scope is proposed based on Eq. (31). 8 Tunnel Lining Supporting Force and Vault Displacement Fig. 9 Variation of vertical stress in soil to the depth located and zone of arching above the tunnel Terzaghi needs further analysis based on the study of change law of vertical stress of the soil arch. Figure 9 shows the vertical stress varying with the depth at the top of a tunnel. When the buried depth of the tunnel, h, is large, the vertical stress r 0 v infinitely approaches the limiting value r vmax, and the arching effect only exists within a distance D 1 above the tunnel roof, as shown in Fig. 9. ased on this point, the soil arch height can be determined by the ratio of vertical stress r 0 v to the limiting stress value r vmax. Let us suppose that the vertical stress r 0 v of the soil arch reaches a certain percentage (m) of the limiting stress value r vmax (e.g., m = 95 %) at the soil arch height. According to the method proposed in this Tunnel lining supporting force is a very important parameter in the design of a tunnel because it is directly correlated with material selection, size, reinforcement, etc. ased on above analysis of soil arching, vault displacement of tunnel has great influence on the vertical stress of the soil arch; the greater the vault displacement, the more fully soil arching play and the smaller vertical stress of the soil arch is. The vertical stress of the soil arch will cause deformation of the tunnel lining. Conversely, the deformation will affect the vertical stress in the soil. This shows that the vertical stress of the soil arch and the deformation of the tunnel lining are mutually dependent, and eventually reach some kind of equilibrium. Therefore, to obtain correct vertical stress of the soil arch and vault displacement, it is necessary to use an iterative method. Assuming that the constitutive relation for support structure is linear elasticity, the supporting force of the segment is (Ward 1978) r a ¼ K s ðu a u 0 Þ ð32þ To keep a balance of forces on segment, the following equation is obtained r a ¼ r v

9 Geotech Geol Eng (2016) 34: According to the partially developed soil arching theory proposed in this article, lining supporting force size and vaults displacements are calculated as follows 1. Assuming the supporting force of segment r a. 2. Put r a into Eq. (32) to calculate the radial displacement u a. 3. Put u a into Eq. (27) to calculate tanh, and then put tanh into Eq. (25) to calculate. 4. Subsequently, put into Eq. (20) to calculate r v. 5. If r v is equal to r a, this indicates that the assumption is correct and the supporting force of segment is equal to r 0 a. If both are not equivalent, then substitute r v into Eq. (30) to calculate the radial displacement u 0 a. Then use u0 a in Eq. (27)to calculate tan h 0. Tanh 0 can then be used in Eq. (24) to calculate A 0 1. Subsequently, use A0 1 in Eq. (20) to calculate r 0 v.ifr0 v is equal to r a, the calculation is terminated, otherwise, the next cycle of computation is carried out until both are equal. Using the proposed method to calculate the supporting force of the segment, the factor, including segment stiffness and vault displacement, can be taken into account for the soil arching effect. Compared with Terzaghi s arching theory, the proposed method is more suitable for engineering applications when the soil arching effect only plays a partial function. The precise solution can be infinitesimally approached with the iterations. In general, accurate results with a tolerance of \5 % can be achieved through 3 5 iterative computations. 9 Conclusions 1. When there is a soil arching effect, the lateral pressure coefficient depends on the principal stress rotation angle, and its value varies between active and passive earth pressure coefficients. 2. After tunnel excavation, the vertical stress of the soil arch is less than the weight of the soil located above the tunnel. The more full the soil arching effect, the greater the difference between the supporting force of the segment and the weight of the covered soil. 3. According to the changing rule of the vertical stress of the soil arch with depth, the soil arch height can be derived. On this basis, a simplified calculation method of the tunneling construction disturbance was proposed. 4. Due to the reciprocal relationship existing between tunnel vault displacement and vertical stress of the soil arch, iterative calculation method for the support force of the tunnel lining and tunnel vault displacement can be proposed. Acknowledgments This paper was supported by Anhui Provincial Natural Science Foundation (Grant No ME98) and Anhui Provincial University Natural Science Foundation (Grant No. KJ2015A255). The author wishes to express his gratitude for the supports given to this work. esides, the author is also express heartfelt gratitude to his advisor Prof. Dr. Lin-Chang Miao for the valuable advice about the research idea. References Chen R, Zhu X, Chen YM et al (2010) Modified Terzaghi loozening earth pressure based on theory of main stress axes rotation. Rock Soil Mech 31(5): Giroud JP, onaparte R, eech JF et al (1990) Design of soil layer-geosynthetic systems overlying voids. Geotext Geomembr 9(1):11 50 Handy RL (1985) The arch in soil arching. J Geotech Eng 111(3): Kezdi A (1986) Lateral earth pressure. In: Winterkorn F, Fang HY (eds) Foundation engineering handbook. Van Nostrand Reinhold Co., New York Li L, Dubé JS, Aubertin M (2013) An extension of Marston s solution for the stresses in backfilled trenches with inclined walls. Geotech Geol Eng 31(4): Lv WH, Miao LC, Wang F (2012) Mechanism of geogrid reinforcement based on pratially developed soil arch effect and design method. Chin J Rock Mech Eng 31(3): Terzaghi K (1936) Stress distribution in dry and in saturated sand above a yielding trap-door. In: Proceedings of first international conference on soil mechanics and foundation engineering, Cambridge, pp Terzaghi K (1943) Theoretical soil mechanics. Wiley, New York, pp Tien HJ (1996) A literature study of the arching effect. National Taiwan University, Taipei Wang F, Miao LC (2009) New design method of geosyntheticreinforced embankment over sinkholes. J Southeast Univ Nat Sci 39(6): Ward WH (1978) Ground support for tunnels in weak rocks. Géotechnique 28(2):