Analytical computation of support characteristic curve for circumferential yielding lining in tunnel design

Kui Wu ,Zhushan Shao ,*,Mostafa Sharifzadeh ,Siyuan Hong ,Su Qin

aSchool of Science,Xi’an University of Architecture and Technology,Xi’an,710055,China

b Shaanxi Key Laboratory of Geotechnical and Underground Space Engineering(XAUAT),Xi’an University of Architecture and Technology,Xi’an,710055,China c Department of Mining and Metallurgical Engineering,Western Australian School of Mines,Curtin University,Kalgoorlie,6430,Australia

Keywords:Squeezing ground Circumferential yielding lining Highly deformable element Support characteristic curve(SCC)Analytical method

ABSTRACT Circumferential yielding lining is able to tolerate controlled displacements without failure,which has been proven to be an effective solution to large deformation problem in squeezing tunnels.However,up to now,there has not been a well-established design method for it.This paper aims to present a detailed analytical computation of support characteristic curve(SCC)for circumferential yielding lining,which is a signi ficant aspect of the implementation of convergence-con finement method(CCM)in tunnel support design.Circumferential yielding lining consists of segmental shotcrete linings and highly deformable elements,and its superior performance mainly depends on the mechanical characteristic of highly deformable element.The deformation behavior of highly deformable element is firstly investigated.Its whole deforming process can be divided into three stages including elastic,yielding and compaction stages.Especially in the compaction stage of highly deformable element,a nonlinear stress-strain relationship can be observed.For mathematical convenience,the stress-strain curve in this period is processed as several linear sub-curves.Then,the reasons for closure of circumferential yielding lining in different stages are explained,and the corresponding accurate equations required for constructing the SCC are provided.Furthermore,this paper carries out two case studies illustrating the application of all equations needed to construct the SCC for circumferential yielding lining,where the reliability and feasibility of theoretical derivation are also well veri fied.Finally,this paper discusses the sensitivity of sub-division in element compaction stage and the in fluence of element length on SCC.The outcome of this paper could be used in the design of proper circumferential yielding lining.

1.Introduction

Deep excavation in squeezing ground normally contributes to very high rock displacements,and if a stiff support concept is applied,very high bearing capacity is required.This has been proved to be both an uneconomic and an unsafe practice,resulting in extremely high lining thickness and lining brittle-violent failure(Radonˇciˊc et al.,2009;Barla,2016;Langford et al.,2016;Wu and Shao,2019;Aygar,2020;Wu et al.,2020a).The only feasible solution in heavily squeezing ground is a tunnel support that is able to tolerate controlled rock displacements without failure(Cantieni and Anagnostou,2009;Wu et al.,2019,2020b).In order to achieve such purpose,a novel support form,i.e.shotcrete lining embedded by highly deformable elements,was proposed and first applied in Galgenberg tunnel,Austria in 1994(Schubert et al.,2018).In this support system,those elements show stronger deformability compared with shotcrete,and a reduction in the lining circumference is allowed upon loading due to considerable shortening of highly deformable elements,thus avoiding shotcrete failure-this is the so-called“circumferential yielding lining”.

Due to signi ficant advantages of circumferential yielding lining in the field of squeezing tunnel support,up to now,several different types of highly deformable elements have been sequentially developed and improved,for instance lining stress controller(LSC)(Moritz,2011),telescope yielding element(Verient et al.,2015)and HidCon element(Barla et al.,2011).Based on the type of production material,these highly deformable elements can be categorized into two basic groups:porous elements based on binders and steel elements(Moritz,2011).The concept of circumferential yielding lining has been successfully applied in many squeezing tunnels throughout the world(Thut et al.,2006;Kovári,2009;Deng et al.,2020;Lu et al.,2021),and some application examples are shown in Fig.1.The use of highly deformable element in shotcrete lining seems a simple task.In practice,it is rather challenging due to the time-dependent properties of shotcrete and time-and facedependent development of tunnel closure(Paraskevopoulou and Diederichs,2018;Fan et al.,2020;He et al.,2021;Sun et al.,2021).Even,it is an approach that can fail dramatically if used incorrectly(Asef et al.,2000).In Saint Martin La Porte access adit of Base tunnel connecting Lyon and Torino,the number and position of highly deformable elements in shotcrete lining were changed several times to ful fill the requirements of tunnel design and lining bearing capacity(Bonini and Barla,2012).Many researchers attempted to predict the response of tunnels using the system of circumferential yielding lining.Ramoni and Anagnostou(2011)reported the tunnel response in different systems of circumferential yielding lining based on the numerical method.Tian et al.(2016,2018)carried out a series of numerical investigations and discussed the in fluence of design parameters of highly deformable element on tunnel performance.Hasanpour et al.(2018)studied the interaction between circumferential yielding lining and squeezing ground by means of two different numerical approaches.Although it has been recognized as an effective measure for tunneling in squeezing ground,there is still no well-established design method for circumferential yielding lining(Hammer et al.,2018).

The convergence-con finement method(CCM)of support design is a convenient analytical-graphical procedure for estimating the required bearing capacity of tunnel support(Carranza-Torres and Engen,2017;De La Fuente et al.,2019;Wang et al.,2021),particularly in the early stage of support design.This method consists in constructing ground reaction curve(GRC)based on the planned tunnel diameter,ground stress and rock mass quality,together with support characteristic curve(SCC)based on the support mechanical properties.Numerous studies on constructing GRC have been carried out by Brown et al.(1983),Wang(1996),Carranza-Torres(2003),Oreste(2003a),Kabwe et al.(2020a),and Xu and Xia(2021),among others.In their solutions,many different conditions were considered.Now,it is suf ficient to make an easy and quick determination of the GRC based on the existing methods.In addition,because conventional stiff lining experienced a lengthy developing process,there have been many approaches to conveniently construct its SCC(Kumasaka,2007;Oreste,2003b;Oke et al.,2018;Yan et al.,2018;Lu and Sun,2020;Kabwe et al.,2020b).Regarding some special support structures,Rodríguez and Díaz-Aguado(2013)attempted to propose accurate equations of the characteristic curve for yielding steel ribs.Carranza-Torres and Engen(2017)provided the analytical expression of the SCC for blocked steel sets.Due to the short time of development,few researches focused on the construction of SCC for circumferential lining,which seriously limits its wide application in squeezing tunnel engineering.

This paper pays attention to the analytical computation of SCC for circumferential lining.The mechanical behavior of highly deformable elements is fully analyzed.The reasons for closure of circumferential yielding lining in different stages are explained,and the corresponding accurate equations required for constructing the SCC are then provided.Furthermore,two case studies are carried out,illustrating the application of all proposed equations.Finally,a parametric investigation on highly deformable element is performed.

2.Ground reaction curve

Fig.1.Applications of circumferential yielding lining in tunnels:(a)Lining stress controller(LSC)(Moritz,2011);(b)Wabe element(Radonˇciˊc et al.,2009);(c)HidCon element(Barla et al.,2012);and(d)Support resistant limiting damper(SRLD)(Qiu et al.,2018).

As previously mentioned,the implementation of the CCM consists in constructing the GRC and SCC.The GRC actually exhibits the relationship between support pressure and tunnel radial displacement.In a linear elastic ground,the expression of GRC is usually given by

whereλ(x)is the displacement release coef fi cient closely associated with the rock property and tunnel face advancement,and its expression has been provided by many researchers(Chu et al.,2019,2021);u(∞)ulrepresents the tunnel radial displacement far from the tunnel face in elastic geomaterial,which can be calculated as

wherep0denotes the initial ground stress,r0is the tunnel radius,andGis the rock shear modulus.

When the Mohr-Coulomb yield criterion is satis fi ed,the computing formula of GRC for an elastoplastic ground is given as follows(Panet,1995):

whereEandνare the elastic modulus and Poisson’s ratio of geomaterial,respectively;andc,φandψdenote the cohesion,friction angle and dilatancy angle,respectively.

3.Support characteristic curve for circumferential yielding lining

3.1.Highly deformable element behavior

Due to its low elastic modulus and strong deformability compared with shotcrete,major rock displacement can be absorbed by the highly deformable elements integrated into shotcrete lining.Fig.2 shows the general stress-strain curves of highly deformable element,and its whole deforming process can be basically divided into three stages:elastic stage(I),yielding stage(II)and compaction stage(III).In the elastic stage(I),the strain of highly deformable element generally linearly increases with stress.This stage only lasts for a short time owing to its low yield stress.Once the yield stress of highly deformable element is achieved,plastic strain occurs without stress increase.This is the yielding stage(II).As the strain continuously accumulates,highly deformable element subsequently enters its compaction stage,and in general a nonlinear stress-strain relationship can be observed in this stage,which indicates that the elastic modulus of highly deformable element is not a constant.When it experiences the maximum(limit)strain,highly deformable element eventually can be regarded as a practically rigid one.

Fig.2.Actual and assumed stress-strain curves for highly deformable elements.

As highly deformable element shows a nonlinear stress-strain relationship in compaction stage(III),its elastic modulus is not a constant,leading to the mathematical dif fi culty of analytical computation.According to Sakai and Schubert(2019),it is practical and feasible to divide the nonlinear curve in this period intoilinear sub-curves,and then the simpli fi ed sub-curves are plotted in red dashed lines,as shown in Fig.2.Accordingly,the elastic modulus of highly deformable element in this stage can be easily calculated.

3.2.Equation derivation for constructing support characteristic curve

Based on the theory of elasticity,the mean circumferential stress in a tunnel lining can be expressed in the form of external load as follows:

whereσθdenotes the mean circumferential stress in the lining,piis the rock pressure(support pressure),andtsis the lining thickness.For a circumferential lining subjected to rock pressure,its closure can be induced by two parts:shortenings of highly deformable elements and segmental shotcrete linings.The mechanical model of a circumferential yielding lining is illustrated in Fig.3,wheremis the number of highly deformable element.The radial displacement of this circumferential yielding lining can be given by

whereurdenotes the radial displacement;andεθ,trepresents the total circumferential strain of lining(including strains of highly deformable element and segmental shotcrete lining),which can be calculated as

Fig.3.Mechanical model of a circumferential yielding lining:(a)before and(b)after deformation.

whereLandΔLare the lining length and its reduction,respectively;andΔLsandΔLhrepresent the reductions in the length of segmental shotcrete lining and highly deformable element,respectively.

The reduction in the length of segmental shotcrete liningΔLscan be expressed by

whereεθ,sis the strain of shotcrete lining,lhrepresents the length of each highly deformable element,andEsis the elastic modulus of shotcrete.In practice,the reduced elastic modulus is usually used for analysis due to the time-dependency of shotcrete.

The reduction in the length of highly deformable elementΔLhcan be calculated as

whereεθ,hstands for the strain of highly deformable element.

On account of special deformation behavior of highly deformable element,its strain depends on the circumferential stress in the lining,which is expressed as

Whenσθ＜σy(the yield stress of highly deformable element),elastic deformation occurs,and the stress-strain relationship of highly deformable element satis fies the following equation:

whereEh1denotes the elastic modulus of highly deformable element in the elastic stage.

Substituting Eq.(17)into Eq.(15)provides the shortening of highly deformable elements in the elastic stage as follows:

Combining Eqs.(11),(13),(14)and(18),the total lining circumferential strain in this stage can be given by

Furthermore,the expression of SCC for circumferential lining in the elastic stage,using Eqs.(12)and(19),can be obtained as

If the circumferential stress in the liningσθreaches the yield stressσy,this indicates that the yielding stage starts for highly deformable elements.In this stage,the closure of lining is totally induced by plastic shortening of highly deformable elements,and the lining circumferential stress therefore remains unchanged.The reduction in the length of shotcrete lining in the yielding stage keeps the value corresponding to that atσθ=σy,as shown in Eq.(21).The shortening of highly deformable elements varies in a certain range,as shown in Eq.(22).

whereεhyrepresents the strain of element when the yielding stage completes.

Subsequently,the total circumferential strain of lining in this stage varies within a range as follows:

By using Eqs.(11),(12)and(23),the expression of SCC for circumferential yielding lining in the yielding stage can be written as

As previously mentioned,for mathematical convenience,the nonlinear stress-strain curve of highly deformable elements in the compaction stage can be processed asilinear sub-curves,and the elastic modulus corresponding to each linear stress-strain subcurve can be considered to be equal toEh3,i.Then,the shortening of highly deformable elements in the compaction stage can be calculated as follows:

whereσθ3,idenotes the circumferential stress in the lining in theith step of compaction stage.The range of element length reduction in this stage is

whereεhlimstands for the limit strain of highly deformable element.

The corresponding reduction in the length of shotcrete lining in the compaction stage is

The total circumferential strain of lining in the compaction stage can be subsequently given by

Substituting Eqs.(11)and(28)into Eq.(12)provides the expression of SCC for circumferential yielding lining in the compaction stage as follows:

When the highly deformable elements integrated into shotcrete lining experience their limit strain,they will become practically rigid and the gaps filled with these elements can be considered to be closed.Thereafter,the shortening of highly deformable elements keeps constant:

The reduction in the length of shotcrete lining after gaps closing still can be calculated using Eq.(14).The total circumferential strain of lining is formulated in Eq.(31)and then the expression of SCC is shown in Eq.(32).

4.Case studies

In this section,two case studies illustrating the application of all equations needed to construct the SCC for the circumferential yielding lining are carried out.Generally,the computation of SCC for the circumferential yielding lining involves the following parameters:tunnel radiusr0,lining thicknessts,shotcrete elastic modulusEs,the number(m)and length(lh)of highly deformable element,and its mechanical parameters(Eh1,σy,εhy,Eh3,i,εh3,iand εhlim).These mechanical parameters of highly deformable element can be easily determined based on its load-strain curve.

The first case study involves a circular tunnel of radiusr0=6 m.The reduced elastic modulus of shotcreteEsis equal to 23 GPa.The thickness of shotcrete liningtsis 20 cm.The number(m)and length(lh)of highly deformable element integrated into shotcrete lining are 9 and 40 cm,respectively.The elastic modulus(Eh1)and yield stress(σy)of highly deformable element are 510 MPa and 8.5 MPa,respectively.In addition,the maximum strainεhlimof highly deformable element is equal to 50%.The stress-strain relationship of highly deformable element is shown in Fig.4.According to Wu et al.(2020c),the whole deforming process of highly deformable element was divided into two stages including elastic stage(I)and yielding stage(II),as shown in red dashed lines in Fig.4,where the compaction stage(III)has not been considered.Herein,in order to perform a comparison with Wu et al.(2020c)’s result,two deforming stages of highly deformable element are assumed as well.

Based on the solution proposed in this study,SCC for the circumferential yielding lining is shown in Fig.5.It can be seen that a good agreement is achieved between the proposed solution and Wu et al.(2020c)’s result.As shown in Fig.5,the whole loaddisplacement curve of circumferential yielding lining can be divided into three parts due to the existence of highly deformable element.The highly deformable elements begin to yield when the lining pressurepi=0.283 MPa corresponding to the circumferential stress of liningσθ=8.5 MPa.The displacement of lining is about 124 mmwithout an increase of lining pressure in the yielding stage.However,after highly deformable elements become practically rigid,the lining pressure rapidly increases to 0.84 MPa(the bearing capacity of C25 shotcrete lining)with the lining radial displacement increasing by approximately 4 mm.

Fig.4.Actual and assumed stress-strain curves for HidCon element(Barla et al.,2012;Wu et al.,2020c).

Fig.5.Comparison of support characteristic curve between results of Wu et al.(2020c)and this study.

Fig.6.Actual(Solexperts,2007)and assumed stress-strain curves for high ductility concrete element.

In Wu et al.(2020c)’s solution,based on the principle of equivalent deformation,the circumferential yielding lining was processed into a“homogenized lining”and the corresponding support stiffness equations were provided.In order to obtain the SCC for the circumferential yielding lining,a further calculation based on the support stiffness equations is necessary in this study.Because in Wu et al.(2020c)’s solution,the Poisson’s ratios of shotcrete and highly deformable elements were involved and they only used the Poisson’s ratio of shotcrete to represent the homogenized lining for simpli fication.This may lead to slight differences in SCCs in Fig.5.In addition,as previously mentioned,Wu et al.(2020c)have not taken the compaction stage of highly deformable element into consideration and they assumed that the highly deformable element only experienced elastic and yielding stages.However,it can be observed from Fig.5 that the nonlinear compaction stage for the highly deformable element has started at the strain of approximately 30%and lasted till the strain of 50%.The maximum stress value of highly deformable element was up to about 20 MPa,which was highly larger than its yield stress of 8.5 MPa.Obviously,if the compaction stage of highly deformable element is ignored,it will result in the waste of material’s actual strength.

The other case study involves a circular tunnel of radiusr0=4.75 m.The shotcrete elastic modulusEsis reduced to 10 GPa and the thicknesstsof shotcrete lining is 25 cm.The number of highly deformable element(m)equals 4 and the length of each element islh=15 cm.In this case study,the compaction stage of highly deformable element is considered and the processing scheme of its stress-strain curve is shown in Fig.6.It can been seen from Fig.6 that in the element compaction stage,it is suf ficient to process this nonlinear stress-strain curve as three linear sub-curves in order to well describe its deforming behavior.The yield stress(σy)and maximum strain(εhlim)of highly deformable element are 7 MPa and 50%,respectively.Based on the element stress-strain curve,its other mechanical parameters involved in the SCC computation can be determined,that isEh1=514 MPa,εhy=20%,Eh3,1=13.6 MPa,εh3,1=30%,Eh3,2=42 MPa,εh3,1=40%andEh3,3=124 MPa.

The SCC for the circumferential yielding lining in this case is shown in Fig.7,and that based on the proposed solution in this study is also potted in red line.Using the same parameters,Ramoni and Anagnostou(2011)carried out a numerical investigation and obtained the SCC indicated as blue dashed line in Fig.7.It can be observed from Fig.7 that the SCC obtained in this study well matches the numerical result obtained by Ramoni and Anagnostou(2011).The highly deformable element yields at the lining pressurepi=0.368 MPa.Besides,it can be found that even in the compaction stage,the elastic modulus of highly deformable element is still signi ficantly smaller than that of shotcrete.This indicates that in the compaction stage,highly deformable elements can play a good role in releasing surrounding rock deformation as well.Based on the results in Fig.7,the lining is able to radially displace approximately 40 mm with pressure increasing by 0.93 MPa when the highly deformable elements enter compaction stage.

Ramoni and Anagnostou(2011),based on numerical approach,investigated the mechanical response of circumferential yielding lining in a squeezing tunnel.As accepted,the tunnel performance could be well evaluated after the correct input of tunnel geometry parameters,and rock and support parameters used in numerical analysis.However,Ramoni and Anagnostou(2011)have not established a clear design method for circumferential yielding lining,which may be unable to greatly promote this novel support system in squeezing tunnels.In this study,a detailed analytical computation of SCC for circumferential yielding lining is provided,where few assumptions are made and the actual deformation behavior of highly deformable element is fully considered.The proposal of equations for determining the SCC of circumferential yielding lining can accelerate the implementation of CCM in preliminary tunnel support design.Certainly,it is also very helpful for the other sophisticated tunnel design and analysis.

In summary,the good applications of proposed equations for the SCC of circumferential yielding lining in the above two cases indicate the effectiveness and reliability of theoretical derivation in this study.

5.Parametric investigation and discussion

Based on the proposed equations for the SCC of circumferential yielding lining,it is very convenient to carry out corresponding parametric investigation.For the circumferential yielding lining,its distinguished support performance largely depends on the characteristic of highly deformable element.In this section,the sensitivity of sub-division in the element compaction stage and the in fluence of element length are investigated,respectively.

Fig.7.Comparison of support characteristic curve between results of Ramoni and Anagnostou(2011)and this study.

5.1.Sensitivity of sub-division in the compaction stage

How does the sub-division of the element compaction stage affect the SCC and support effect?As shown in Fig.8,the SCCs are plotted with sub-divisionsi=0,1 and 3,respectively,wherei=0 indicates that the compaction stage of highly deformable element is not considered and the element yielding stage ends at its maximum strain achieved.In addition,other parameters involved in this analysis are all taken from the second case in Section 4.

Fig.8.In fluence of sub-division of compaction stage of highly deformable element.

Fig.9.In fluence of highly deformable element length.

As illustrated in Fig.8,the sub-division of the element compaction stage has a signi ficant in fluence on the SCC for circumferential yielding lining.Assuming that an equilibrium point between ground and support is achieved during the compaction period,the implementation of CCM in the above-mentioned three cases is shown in Fig.8.Noting that in the case without considering the element compaction stage,the equilibrium point is achieved after the element yielding stage is completed.Obviously,support pressurepiis reduced to the minimum value when the element compaction stage is not considered.In comparison,the maximum support pressure appears under the condition ofi=1,and the support pressure withi=3 is in between those of the other two cases.Based on the results in Fig.8,it is safe to conclude that wheni＞1,the support pressure at the equilibrium point should range between those obtained in the cases ofi=1 and 0(without considering the compaction stage).Of course,when an equilibrium point between ground and support is obtained during the element compaction period,it is very necessary to make the sub-curves describe the element deforming behavior as real as possible.

Interestingly,from the results in Fig.8,if compaction deformation of highly deformable elements has been completed and an equilibrium point between ground and support has not been achieved yet,the following SCC in three cases will become the same.This indicates that it is not very important to carry out careful subdivision of element compaction stage and a same support equilibrium point will be obtained in all cases.

5.2.In fluence of length of highly deformable elements

Herein,the in fluence of element length on the SCC is investigated.As shown in Fig.9,these SCCs are drawn for element lengthmlh=0 m,0.3 m and 0.6 m,respectively.Note that the element compaction stage is not considered and other related parameters are also taken from the second case in Section 4.

Fromthe results in Fig.9,the SCC is greatly affected by the highly deformable element length.It can be seen that the lining without installing highly deformable element(mlh=0 m,that is the socalled“stiff lining”)is unable to provide suf ficient support resistance to restrain ground deformations without failure.As the length of highly deformable element increases,an equilibrium point between ground and support is achieved with smaller support pressure and larger displacement.The results in Fig.9 show that the use of highly deformable element can well improve the performance of shotcrete lining,and it is a feasible and effective solution to deal with excessive ground deformations by increasing the length of highly deformable element.Based on experience,it should be highlighted that there is a high risk of sudden change of ground from the loosening state to instability during the deforming period of highly deformable element due to relatively small support resistance.Practically,tunnel engineers are required to determine an optimal element length instead of continuously increasing its length for the purpose of smaller support pressure.

6.Conclusions

Circumferential yielding lining uses the strong deformability of highly deformable elements to release surrounding rock deformation without damaging shotcrete lining.The deforming process of highly deformable element is divided into three stages including elastic,yielding and compaction stages.Specially,in its compaction stage,the stress-strain curve is processed as several linear subcurves due to obvious nonlinear characteristic.The closure of circumferential yielding lining can be then divided into four stages.The first is induced by elastic deformations of shotcrete lining and highly deformable elements.The second is totally caused by element plastic shortening as the stress in the lining reaches the element yield stress.In the third stage,lining closure is triggered by element compaction shortening and elastic deformation of shotcrete lining.Finally,after experiencing the limit strain,the shotcrete lining continues to work until failure.

The analytical computation of SCC for circumferential yielding lining is presented,and the accurate equations for constructing SCC in different stages are provided.The use of SCC for circumferential yielding lining involves the parameters of tunnel radius,lining thickness,shotcrete elastic modulus,and the number,length and mechanical parameters of highly deformable element.The solution derived can be easily reduced to the previous simple case that does not consider the element compaction stage,which possibly causes the waste of material’s actual strength.

Based on the construction of SCC for circumferential yielding lining,engineers can find that it is important to consider the sensitivity of sub-division of element in compaction stage if an equilibrium point between ground and support is achieved during compaction period.As the length of highly deformable element increases,that equilibrium point can be achieved with lower support pressure.

Declarationofcompetinginterest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to in fluence the work reported in this paper.

Acknowledgments

This work is supported by National Natural Science Foundation of China(Grant Nos.11872287 and 51908431),and Fund of Shaanxi Key Research and Development Program(Grant No.2019ZDLGY01-10).