Limit-Equilibrium Stability Analysis of Spiling Soil Reinforcement in Tunneling

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1 58 Transportation Research Record 18 Limit-Eqilibrim Stability Analysis of Spiling Soil Reinforcement in Tnneling S. BANG ABSTRACT A limit-analysis procedre for a spiling reinforcement system in soft-grond tnneling is described. The system is composed of a series of radially installed reinforcing spiles along the perimeter of the tnnel opening before excavation. The reinforcing spile network is extended into the in sit mass both radially and longitdinally, The system ths forms a temporary spport with the advantage of not loosening the weak mass. It has been sccessflly sed in weak rock formations as well as in soft grond on limited occasions. The limitanalysis formlation incldes consideration of design parameters sch as soil type and geometry of the tnnel and the reinforcing spiles. The procedre developed can be sed to evalate the overall stability or to determine the design parameters of the system or both. In recent years ndergrond constrction has increased considerably as a logical part of the soltion to many rban problems. Effective and safe ndergrond excavation technology therefore has been soght to meet the challenge of this increasing demand. On several occasions a spiling reinforcement system has been sccessflly sed in tnneling to strengthen weak rock formations (1), On limited occasions the application also involved reinforcing tnnels in soft grond. The system typically consists of a series of radially installed reinforcing spiles 4.6 to 6.1 m long spaced between.6 and 1.5 m with an inclination angle of approximately 3 degrees to the tnnel axis. The reinforcing spiles are formed by inserting 2.5- to 3.B-cm diameter rebacs into predrilled holes with sbseqent grot. Figre 1 is a schematic representation of the system, the general principle of which is to stabilize a weak mass by installing an annlar spiling reinforcement network along the perimeter of the tnnel heading before excavation. The reinforcing network is extended into the in sit mass both radially and longitdinally. The tnneling operation is therefore always performed inside a tblar reinforced zone encircling the tnnel opening. Spile Set FIGURE I Spiling reinforcement system. A limit-analysis procedre for the spiling reinforcement system in soft-grond tnneling is presented. The formlation incldes consideration of design parameters sch as soil type, depth and diameter of the tnnel, and length, inclination, and spacing of the spiles. The analysis procedre can be sed to evalate the overall stability of the system and to determine the proper size, spacing, and length of the spiles for a given site condition. BACKGROUND The principle of the spiling reinforcement system is to strengthen the weak mass srronding the tnnel excavation. The prpose is to improve the grond in terms of stand-p time by preventing loosening ( immediate stabilization) and to contribte to permanent stabilization of the tnnel opening by restricting deformation. Korbin and Brekke () condcted small-scale laboratory model tests to investigate the mechanism of spiling reinforcement in weak-rock tnneling. Behavior of cylindrical models with and withot spiling reinforcement nder niform external pressres was examined. The stdy indicates that the system can stabilize a weak rock mass effectively by redcing deformations and thereby deterioration de to strain softening, providing increased available strength, and maintaining arching action, which allows for increased tangential stress in the immediate vicinity of the opening. Field instrmentation was also implemented at two sites (3), The investigation was designed to address qli' estions related primarily to the magnitde, distribtion, and time history of the deformation-indced tension and bending of the spiles as a reslt of excavation. Deformation-indced tension was fond to be the major mechanism of the spiling reinforcement. A generalized plane strain finite-element method of analysis was developed to investigate the be- 1\avior of the spiling reinforcement system in soft grond (.!l. The method of analysis developed was sed to compare the reslts of the model tests condcted by Korbin and Brekke and was fond very effective (.?_). The generalized plane strain condition assmes three nonzero displacement components, none of which is dependent on the ot-of-plane coordinate; ths the ot-of-plane strain remains zero instead of the ot-of-plane displacement, as is commonly the case in the conventional plane strain approach. The main advantage of this approach is that it calclates three-dimensional stresses and

2 Bang 59 displacements, as the finite-element grid remains in two dimensions. A comprehensive parametric stdy to identify the effects of the varios design parameters of the syst.em was also condcted (6) by sing the developed generalized plane strai;- f i nite-element method of analysis. LIMIT ANALYSIS AT EQUILIBRIUM To date there have been no prototype failre stdies of this system. Indirect methods therefore mst be sed to approximate the failre mechanism. As shown in Figre 2, contors of maximm shear strains were obtained from the previosly described generalized plane strain finite-element analysis <!> and ths a potential failre srface can be approximated. The potential failre srface passes more or less throgh the side wall of the tnnel opening and propagates pward to form a crved srface. In this analysis the crved failre srface is approximated by two planes, defined by angles a 1 and a 2, with a change in direction at the back face of the reinforced zone as shown in Figre 2. A potential failre srface can then be determined by finding a set of angles al and a2 that yields the lowest overall factor of safety. Note that it is always assmed that a 1.S. a 2.S. 9 degrees. Formlation Figre 3 shows the free-body diagrams of the assmed failre wedge. Becase of the symmetry, only half of the failre wedge is considered. Forces acting on element 1 are w 1, Ni, S1, Q1, N3 and s 3 : W 1 =rh 3 2 cota 2 N 1 =rka H3 2 S1 =/JN1 w body weight, Pot en tiol Failre Srface From Finite Element Analysis Appro1dmoted Failre Maximm Shear Strain Contor (xl- 4 ) (!) 2 1 IT, rl' -- r2 UL, //::_":). s N Ka ' ' -- / a, 2 L2 ELEMENT 2 N1 S1 (',:-f t r ELEMENT I FIGURE 3 Free-body diagrams of failre wedge. tangential force, normal force, and active earth pressre coefficient. H2 _J_ The coefficient e describes the ratio between S1 and N1. Normal and tangential forces (N3 and s 3 ) can be obtained from the eqilibrim: N3 = (W S1) cose<2 + N1 sina2 Q is the srcharge. The coefficient e can be obtained, on the assmption that the developed resisting force along the assmed failre srface is eqal to the driving forcei that is, S3 = C' (HJ/sina2) +NJ tan</>' =(W1 +1 +Si)sina2 -N1 cosa2 (3) C' is the developed cohesion and qi' is the developed friction angle. Sbstitting Eqations 1 and 2 into Eqation 3 yields /J=2{CW1 +1)sina2 2 -N1 cosa2 sina2 -C'HJ - [(W1+1) cos<>2 + N1 sina2] tanr/>'sina2} 7 Ka 'Y HJ 2 ( cosa2 tan</>' - sina2) sin<>2 Forces W2 and N4 acting on Element 2 are W2 = r [H(r + L2 cosa1) - (rrr 2 /4) - (H2L2/2) cosa1] N4 = 1 /zko rh1 2 r = radis of tnnel, L2 length of failre srface of Element 2, and Ko at-rest earth pressre coefficient. Note that the normal force acting on the plane of symmetry (N4) is assmed to be the at-rest condition and that no shear force exists on the p lane of symmetry. Force eqilibrim of Element 2 yields (2) (4) (5) N1 = (W S1) cos<>1 - (N 1 -N 4 ) sina1 (6) Reinforced Reoion The driving forces of Elements 1 and 2 are s 3 and S2, respectively, as the developed resisting forces of Elements 1 and 2 are FIGURE 2 Potential failre srface from contors of maximm shear strains. SR 1 = C'L1 +NJ tan</>' SR2 =C'L2 +N2 tan</>'+ LTTT (7) n=l

3 6 Transportation Research Record 18 m N; "' N2 + })TN, n=l m ITTN = resltant of normal component of n=l spile force TT, resltant of tangential component of spile force TT, = component of developed spile axial force Tn on X-Y plane, Li = length of failre srface of element i, and m = nmber of all spiles that cross assmed failre srface. Figre 4 is the schematic representation of the spile force components.,resltant Forces in Spiles It is assmed in this formlation that the spiles have the same length, inclination angle, and spacing. Neglecting the bending and shear as observed in field instrmentation (1_), the maximm developed axial force in each spile (Tnl can be calclated. Denoting stress tensors on X-Y-Z space and space as g and g', respectively, axis l is directed along the individal spile axis, one can obtain direction matrix is defined as follows: (8) Zn is the depth to the middle of the effective length of the nth spile. Therefore normal stresses acting on the spile (cr 2 and 3) can be calclated from Eqations 8 and 9. The mean normal stress acting on the spile is then (IO) The reslting force in the spile is the developed frictional resistance; that is, diameter of spile, effective length of nth spile, spile spacing, spile cross section, and yield stress of spile. (ll) The vales of n and Zn can be obtained from the geometry (Figre 5) The projection of the nth spile on the X-Y plane forms an angle en with the Y-axis, From triangle OAB in Figre 5, it is obvios that Rn 2 = r 2 + Ll + 2 r Li cose<1 Becase Rn cosan = L 2 sina 1, one can obtain Rn= r [-B + (B 2-4A)y,/2A] (12) (13) (14) [ COSQ = slnacoson slna sfn8n - slnasin On cos a sinacos -sina cos 11] -SinaslnOn cos a A= (cosn/sine<i) 2 - I B = 2(cosn/tane<i) Angles a and an are defined in Figre 4. Stresses acting along the effective length of the spile (portion of the spile otside the failre wedge) can be assmed as Uy ='Y Zn (9) Therefore the effective length of the nth ( nl and the depth to the middle of the tive length (Znl can be calclatedi that is, Rn= -[(Rn - r)/sine<] Zn = H -[(Rn + R)/2] coson i is the length of the spile. spile ef fec- (15) 2 y ASSUMED FAILURE SURFACE ---,., '',, - -,,, T TTN' Tylin(a,+8n-9"1 Tn TycotC 1 + 8n-9o"l Ty Tn tin a a Spilt inclination an9lt mta1rtd lrom tnnel axi9 Bn Spi11 lncllnollon on X- Y plane m11red lrom Y Glll 3 x F1GURE 4 Forces in reinforcing spiles.

4 Bang 61 y Section 1-1 FIGURE 5 Geometric parameters. EVALUATION OF OVERALL STABILITY On the basis of Eqations 3, 6, and 7, the overall stability of the system can be analyzed. At any stage, the driving force and the developed resisting force mst be in eqilibrim; that is, 53 = SRl for Element 1 and 52 = 5R2 for Element 2, or S = 5Rl + 5R2 The overall factor of safety with respect to shear strength (FS) is defined as a factor of safety when FSc = FS = FS (2), FSc is the factor of safety with respect to cohesion and FS is the factor of safety with respect to friction. The factor of safety with respect to cohesion or friction is the ratio between the available cohesion or friction and the developed cohesion or friction; that is, C' = C/FSc and tan ' = tan /FS U.J f/) I, "T'"" , L = 1.5 D Spile Len9th 1,4 L = 1.25 D L = D f.'3 L=.75D a: 1.2 I.I Tnnel D iometer ( D) = 3.5 m Tnnel Depth (H) = m The overall factor of safety (FS) can then be determined. Becase the expressions for driving and resisting forces contain an nknown FS, direct soltion is not possible. An iteration method is therefore sed. A compter program was developed to calclate the overall factor of safety. For a given set of geometric and strength parameters, this program calclated the minimm overall factor of safety by searching a series of potential failre srfaces with different angles of al and a 2 Typical reslts of this limit-eqilibrim analysis are shown in Figre 6. The soil is silty clayey sand with an ndrained strength of c = kpa and = 33 degrees (8). The angle of spile inclination is 3 degrees to the tnnel axis in all cases. The reslts indicate that 1. The improvement in the factor of safety with respect to spile spacing is greater for longer spiles, and 2. The rate of increase in the factor of safety for a given spile length is greater for small spile spacings, as can be seen from the changes in slopes. Figres 7 and 8 show the affects of tnnel depth and length and spacing of the spiles on the factor of safety. It is interesting to note that for a given spile length and spacing the factor of safety SPILE SPACING (m) FIGURE 6 Typical reslts of limit analysis U.J Ill 1.6 Tnnel Diameter ( D) 3.5 m Spile Spocin9(s). 91 m Len9th (L) = 3.81 m a: L = 3.5 m 1.4 L = 2.29m L = I. 52m ' TUNNEL DEPTH ( m) FIGURE 7 Effects of tnnel depth and spile length on factor of safety.

62 Transportation Research Record 18 2. Tnnel Diameter ( D) = 3 5 m UJ <[ CJ) a:: <[ 1.8 1.6 1.4 Spile Length (L) = 3.5m Spile Spacing (s) =.61 m s=.

5 62 Transportation Research Record Tnnel Diameter ( D) = 3 5 m UJ <[ CJ) a:: <[ Spile Length (L) = 3.5m Spile Spacing (s) =.61 m s=. 91m Potential Failre Srface From Finite Element Analysis Potential Failre Srface From Limit Analysis _... _. J TUNNEL DEPTH (m) F1GURE 8 Effects of tnnel depth and spile spacing on factor of safety. Tnnel Diameter= 3.5m Tnnel Depth= 6.1 m Spile Length = 2 29 m Spile Spacing=.91 m Splle Inclination= 3 Soil No. IC= kpa 4,, 33 y=l57n/m3 Reinforced Region decreases as tnnel depth increases ntil a critical depth is reached. The factor of safety increases slightly below the critical depth. The rate of the decreasing factor of safety above the critical depth is, however, mch greater than that of the increasing factor of safety below the critical depth. The critical depth is the location of the tnnel springline the sliding tendency of the assmed failre wedge is maximm. The analysis is based on the eqilibrim of driving and resisting forces along the assmed failre srfacei therefore the factor of safety indicates a ratio between available soil strength and reqired soil strength mobilization against the sliding. Indeed the sliding tendency wold be small at very shallow depths becase of the small overbrden and at greater depths becase of arching. DISCUSSION AND SUMMARY The stability analysis procedre of the spiling reinforcement system in soft-grond tnneling has been presented. A limiting stress eqilibrim analysis has been performed to establish a design method for the system. The potential failre srface is approximated by two planes, starting from the side of the tnnel with a transition at the interface between the reinforced and nreinforced regions. The parameters fond to be significant from the previos stdy are inclded in the formlation. Strength parameters, sch as cohesion and friction, are assmed to be only partially developed, nless failre is imminent. From the eqilibrim of the driving force and the developed resisting force, the overall factor of safety is obtained. A comparison of the pntent.ial failrp s11rf11r.p. predicted by the finite-element analysis with generalized plane strain conditions was made with that predicted by limiting stress eqilibrim analysis. Figre 9 shows one of the typical reslts. The agreement, thogh it has not yet been verified experimentally, between these two predicted crves is reasonably good. The proposed formlation for the design of the spiling reinforcement system in soft-grond tnneling is intended to inclde the most essential characteristics involved in the system. However, a great deal of additional research is needed to describe the behavior of the system completely. Frther research is necessary in areas sch as field instrmentation and model testing to verify the analytical reslts. F1GURE 9 Comparison of potential failre srfaces. ACKNOWLEDGMENT The research reported in this paper was spported by the.s. Department of Transportation. The athor is gratefl for this spport. REFERENCES 1. G.E. Korbin and T.L. Brekke. Model Stdy of Tnnel Reinforcement. Jornal of the Geotechnical Engineering Division, ASCE, Vol. 12, No. GT9, G.E. Korbin and T.L. Brekke. A Model Stdy of Spiling Reinforcement in Undergrond Openings. Technical Report MRD Missori River Division,.s. Army Corps of Engineers, G.E. Korbin and T.L. Brekke. Field Stdy of Tnnel Prereinforcement. Jornal of the Geotechnical Engineering Division, ASCE, Vol. 14, No. GT8, s. Bang. Analysis of a Spiling Reinforcement System in Soft Grond Tnneling. In Transportation Research Record 945, TRB, National Research Concil, Washington, D.C., 1984, pp s. Bang. Spiling Reinforcement System in Tnneling. Presented at 4th Astralia-New Zealand Conference on Geomechanics, Perth, Astralia, May s. Bang and C.K. Shen. Soil Reinforcement in Soft Grond Tnneling. Research Report DOT/RSPA/DNA- 5/83/5. U.S. Department of Transportation, Jan D.W. Taylor. Fndamentals of Soil Mechanics. John Wiley and Sons, Inc., New York, J.M. Dncan, P. Byrne, K.S. Wong, and P. Mabry. Strength, Stress-Strain and Blk Modls Parameters for Finite Element Analyses of Stresses and Movements in Soil Masses. Report UCB/GT/8-1. University of California, Berkeley, Ag Pblication of this paper sponsored by Committee on Sbsrface Soil-Strctre Interaction.

Lecture Notes: Finite Element Analysis, J.E. Akin, Rice University

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