Shao-liu Liu,Yue-tang Zhao,Kang Hu,Shi-hao Wang

State Key Laboratory of Disaster Prevention&Mitigation of Explosion&Impact,Army Engineering University of PLA,Nanjing,Jiangsu,210007,China

Keywords: Intersecting cavern Ground shock Initial ground stress Dynamic response

ABSTRACT An intersecting cavern is a common structural form used in underground engineering,and its safety and stability performance directly control the service performance of the whole project.The dynamic responses of the three kinds of crossing type(＋-shaped,T-shaped,L-shaped)caverns subjected to ground shock were studied by numerical simulation.The velocity plus force mode boundary setting method was proposed in the coupled static and dynamic analysis of a deep underground cavern.The results show that,among the three types of crossing caverns,the＋-shaped cavern is the most signi ficantly affected by the dynamic action,followed by T-shaped,and then L-shaped caverns.The vault settlement,straight wall deformation,vault peak particle velocity,effective plastic strain of surrounding rock,and maximum principal stress and strain at the bottom of the lining of the straight wall increase with the increase of cavern span.The vault settlement,straight wall deformation,effective plastic strain of surrounding rock,and the maximum principal stress and strain at the bottom of lining to the straight wall decrease with the increase of lateral pressure coef ficient,and the peak particle velocity at the vault increases.The variation is small compared with the change of cavern span.The in fluence range of the underground cavern intersection is two cavern diameters from the intersection centre.The bottom of the straight wall at the intersection is the weak part.It is suggested to thicken the support locally to improve the stability of the cavern.

1.Introduction

To meet the needs of construction and use and to reduce the impact of external explosion shock wave on the main structure of underground caverns,a variety of different forms of cross structures will be established,including＋-shaped,T-shaped,and Lshaped intersections.Compared with the single cavern,the stress concentration at the intersection is more signi ficant and the lining deformation is more severe.It is the weak part of the underground structure,and even affects the service life of the whole cavern.

Much previous research into on tunnel behaviour in the intersection area is based on static analysis:Riley[1]measured the stress distributions of three common types(tee,cross,and right angle)of cylindrical tunnel intersections using a three-dimensional photoelastic method.It was found that the maximum compressive stresses at the intersection were 60%greater than those far fromthe intersection,and the maximum compressive stress on the wall of the tunnel opposite the corner(tee and right angle intersections)was not signi ficantly in fluenced by the intersection.Gercek[2]summarised the in fluences of different intersection angles on the tangential stress concentration factor of X and Y-junctions,the extent may reach several tunnel diameters at the acute angle side of the intersection.Li et al.[3]used numerical simulations to investigate the deformation,stress and plastic zone of the surrounding material and the internal forces acting on the support lining of a tunnel junction between a subway station tunnel and a construction tunnel in Chongqing in China.The calculated extent of the in fluence zone is either 2.4Dof 1.6D,whereDis the width of the junction tunnel,regarding as reference the deformation or stress,respectively.Liu et al.[4]studied the plastic zone distribution,the extrusion displacement of working face,and the sidewall deformation of T-shaped crossing tunnels through in-situ monitoring and numerical simulation.Chortis and Kavvadas[5]investigated the effect of constructing a junction tunnel,intersecting an existing main tunnel at a normal angle by 3D Finite Element analyses,they focused on calculating the axial forces acting on the primary support at the intersection area before,during and after the construction of the junction tunnel.Dynamic research mainly focuses on seismic load and train vibration load.Hassani et al.[6]studied axial force and bending moment changes of T-shaped crossing tunnel under operational design earthquake and maximum design earthquake loading conditions.In this paper,the method of statically indeterminate reaction is used to design a tunnel lining.Yan et al.[7]used numerical simulation and model tests to study the vibration response characteristics and vibration variation of the crossing part of twin shield tunnels under moving train load.Yang et al.[8]analysed the initial stress state of the intersection and the dynamic response characteristics of a structure under train load through numerical calculation and indoor fatigue test and a method to determine the in fluence zoning of vibration.Due to the strong dynamic load effects(i.e.,under explosion load regimes)being mainly applied by the military,there is less public information available.Bagge[9]studied the dynamic and static coupling problem of T-shaped cylindrical intersections subjected to a nuclear impact explosion under the condition of deep high ground stress and provided suggestions for the design of protective works.Lampman et al.[10]studied the damage of＋-shaped caverns under different nuclear explosion overpressures by using ADINA.Heuze and Morris[11]identi fied the in fluences of rock joint direction,spacing,and durability on the structure of large underground caverns under ground-impact load,and conducted indepth analysis on structural response of the intersections.

There remain certain shortcomings:first,much research on Tshaped caverns in the form of tunnel intersections,mainly involving the excavation of a construction adit or tunnel transverse passage is available,but the research on＋-shaped,especially Lshaped caverns,is sparse.For large underground caverns or tunnels,three kinds of crossing forms exist,whether the failure mode is the same need to study.Second,lateral pressure coef ficient,as one of the important parameters which characterise the complexity of strata,directly determines the load distribution and its magnitude of tunnel lining[12],and whether it has any in fluence on T-shaped and L-shaped asymmetric structures has rarely been studied.

This paper focuses on these two problems:by selecting the caverns with typical span and buried depth as the research object,the numerical analysis method is used to study the in fluences of three crossing modes(＋-shaped,T-shaped,and L-shaped)on the dynamic response of an underground cavern subjected to ground shock.The vault settlement,straight wall deformation,vault peak particle velocity,effective plastic strain of surrounding rock,and maximum principal stress and strain are used.The in fluences of different lateral pressure coef ficients on the failure law of three kinds of crossing caverns are discussed.The results can provide theoretical basis for the design and construction of underground caverns.

2.Case background

Two parallel tunnels located in Southwest China have a total length of 8.72 km and their section are arched straight wall.The tunnels mainly pass through the mountain with the ground elevation of 1224-1259 m,and the relative height difference of 35 m.The ground vegetation is mainly low shrubs and herbs.The tunnels were constructed in a rock mass consisting of mostly marble with some granitic intrusions.Fresh intact rock has uniaxial compressive strengths of 300-350 MPa.The rock mass quality is considered “good’’with averageQvalues of 35-40.According to the national standard for engineering classi fication of rock mass[13],the surrounding rock is classi fied as Level III.The average rock cover over the project is about 80 m.There are several transverse passages between the two parallel tunnels with T-shaped and Lshaped crossing intersections.The main tunnel has a width of 8 m and a height of 6 m.The transverse passage is 4 m wide and has the same height of the main tunnel.The tunnels are used as traf fic tunnels in peacetime but as a shelter in wartime to resist the attack of nuclear weapons.

To study the in fluences of different crossing types on caverns with different spans,the spans of some tunnels and caverns used for protection testing are shown in Table 1:4-8 m span caverns are widely used in various underground operations.In this paper,three typical spans of 4 m,6 m,and 8 m are selected to study the influences of different crossing types on the dynamic response of caverns with different spans.

Table 1 Cavern span list.

The deep-buried cavern is subjected to vertical and horizontal stresses.The vertical stress is caused by gravity,which is generally the weight of overlying rock mass,while the horizontal stress is formed by both gravity and tectonic stress.The ratio of the average horizontal stress to the vertical stress isK.According to in-situ stress measurement data in China,when the depth is less than 465 m,K>1,and the horizontal stress plays a leading role,while when the depth exceeds 465 m,the vertical stress plays a leading role,and the lateral pressure coef ficient rapidly converges to 0.68[23].Fig.1 shows the ratio of horizontal stress to vertical stress changing with the burial depth,which based on 1357 in-situ stress datapoints collected by Kang et al.[24].The fitting curve is obtained through regression analysis,and the regression formula is as follows:

Fig.1.The ratio of horizontal stress to vertical stress versus burial depth.

where,KH,Kav,Khare the ratios of maximum,average and minimum horizontal principal stress to vertical stress respectively,His the burial depth of the cavern.For an underground cavern with a depth of 80 m,the range of the lateral pressure coef ficientKis 1.65-2.81,so hereK=2 for convenience of calculation and analysis.

According to the research on the stress and deformation characteristics of surrounding rock of circular and rectangular caverns under different lateral pressure coef ficient conditions by Dong et al.[25],it is found that the distribution of maximum principal stress,shear strain increment,and maximum displacement of surrounding rock differ when the lateral pressure coef ficient is either greater than 1 or less than 1.Therefore,in this work,lateral pressure coef ficients of 0.5,1,and 2 are selected to study the in fluences of different crossing types on the dynamic response of caverns with different lateral pressure coef ficients.

3.Numerical study

3.1.Model and parameters

To understand the dynamic response of the underground cavern intersection in details,LS-DYNA finite element analysis software was used to simulate actual projects.The damage at an underground intersection arises from local damage to the structure,so the element should be as small as possible when establishing the finite element model.The grid size is between 0.1 m and 1 m,which near the cavern is 0.1 m and far away from the cavern is 1 m.To facilitate calculation,1/4 of the model was taken for the＋-shaped cavern,1/2 for the T-shaped cavern,the whole system for the Lshaped cavern,and 1/4 of the model for the single cavern.To reduce the in fluence of boundary effects on the accuracy of the numerical results,the distance between the side of the cavern and the boundary of the numerical model was set to three times the width of the cavern.The length,width,and height of the whole model were 36 m,36 m,and 37 m,respectively.The caverns were arched straight wall with a height of 6 m and a span of 4 m,6 m,8 m,respectively.The concrete lining with a thickness of 0.2 m was poured into the cavern.The caverns with different crossing types are shown in Fig.2.The rock mass behaviour was assumed to be elasto-plastic with yield governed by the Mohr-Coulomb criterion(see Table 2).The lining structure was made of C30 concrete and we used Johnson-Holmquist-concrete criterion(see Table 3).The physical meaning of each parameter was shown elsewhere[26,27].

Table 2 Physical and mechanical parameters of surrounding rock.

Table 3 Physical and mechanical parameters of lining.

Fig.2.Numerical model.

3.2.Loading

The underground caverns are subjected to the initialin-situstress and the ground impact load caused by the nuclear explosion.The depth of the caverns is 80 m,which can be converted into the vertical constant load on the surface.The vertical stress of the initial stress field is:P=σy= -ρgh= -1.96MPa.To study the in fluences of different lateral pressure coef ficients on the caverns,the horizontal stress is taken as 0.5,1,and 2 times the vertical stress,respectively.

The formula for calculating the radial peak stress in rock mass caused by nuclear explosion is[28]:

when

where,pvis the radial peak stress(kPa),rrepresents the distance from the observation point to the explosion centre(m),the burial depth of the tunnel is 80 m,and the equivalent chargeQ(kt)of the nuclear explosion is taken as 100 kt.The calculated ground impact load is about 15 MPa.

The ground impact load caused by nuclear explosion can be simpli fied as a triangular load,and the rising time and positive pressure time can be approximately estimated by the following formula[28]:

Fig.3.Impact load waveform.

where,ris the distance from the observation point to the blasting centre(m),cpis the longitudinal wave velocity in rock(m/s),and the calculated rising time is 5 ms,and the positive pressure time is 15 ms(see Fig.3).

3.3.Working condition

In this paper,20 working conditions are selected to study the in fluences of different crossing types on the dynamic response of caverns with different spans and different lateral pressure coef ficients(see Table 4).

Table 4 List of calculation conditions.

3.4.Boundary setting of coupled static and dynamic response

3.4.1.Boundary setting considering initial stress

The initial stress and strain state of deep underground cavern is very complex due to the combined action of surrounding rock and the tectonic stress field.In LS-DYNA,only a viscous boundary is provided to simulate wave propagation in an in finite region.The viscous boundary[29]does not converge subjected to static load due to us only considering the in fluence of velocity.Deek and Randolph[30]proposed a viscoelastic arti ficial boundary based on a viscous boundary.A series of simple mechanical models composed of linear springs and dampers are set at the arti ficial boundary to absorb the wave energy emitted to the arti ficial boundary and simulate the elastic restoring force of in finite foundation;however,a large number of springs and dampers are needed to establish the viscoelastic arti ficial boundary,and the initial stress and strain field will cause spring deformation at the arti ficial boundary due to the interaction between the region and the spring system,the stress and strain in the calculation area will also change,which is not consistent with the actual situation.

Before dynamic analysis,if there is initial stress in the calculation model,there must be a group of forces to balance on the boundary of the calculation model.If the initial static equilibrium state can be transformed into a dynamic equilibrium state,as a part of the dynamic load embedded in the process of dynamic analysis,the calculation can be realised by the interaction of the initial static equilibrium state and the subsequent dynamic load.In this way,the interaction process of static and dynamic coupling can be completely simulated by conventional dynamic analysis.

The static load on the boundary of the calculation model can be described as a constant value load with in finite time of action.In mathematical terms,it can be expressed by step loadH(0)σ,whichH(0)is a step function.To calculate the static load on the boundary,the load acts from the moment when the dynamic load begins to act.The step loadH(0)σon the boundary and the initial stress in the model will form a balanced force system,which will automatically meet dynamic equilibrium conditions.In the dynamic calculation,a composite boundary with viscous dampers and powerful functions will be formed at the boundary where the transmitting boundary needs to be set.At the boundary where dynamic loads need to be applied,there will be a certain superposition of loads.The solution of the initial static equilibrium state can avoid the possible “overshoot” phenomenon caused by applying the initial stress to the structure directly as a dynamic force.

Considering the in fluence of initial ground stress in the calculation,there are two steps,in the first step,the dynamic relaxation method in LS-DYNA is used to obtain the stress and strain state of surrounding rock under the action of the initial in-situ stress,and then the stress and strain state is treated as the initial condition from which to solve the dynamic response of underground cavern subjected to ground shock.In the second step,the initial stress and strain are applied as usual to balance the calculated stress and strain in the previous step.The two boundaries of the model under horizontal load are changed into normal constraint boundaries.The transmission boundary is set at the top and bottom of the model,and the original normal constraint condition at the bottom is changed to a load boundary condition.The load is calculated according to the calculated initial stress and strain,and then the dynamic response analysis is completed by applying the ground impact load.In Fig.4,taking the XY-plane as an example,the symbol(0)indicates that the element is only used in the calculation of the initial stress field,symbol(1)represents that the element is only used in the subsequent dynamic calculation,symbol(0,1)indicates that the element is used in both initial stress field calculation and dynamic calculation.

3.4.2.Numerical simulation verification of boundary setting

To simplify the calculation,the single cavern model was adopted without considering the in fluence of gravity.The cavern span is 6 m,and the lateral pressure coef ficientKis 2.0.The model size,material parameters,and impact loads are the same as those in Section 3.The boundary setting is shown in Fig.4.In the dynamic calculation,only the initial stress and strain are considered,and the displacement returns to zero,which is consistent with the actual situation.The selection of elements and nodes is shown in Fig.4.

Fig.4.Model boundary setting.

Fig.5 shows the time history curve of vertical stress for E7311 in the elastic region and E4631 in the plastic region.It can be seen fromthe figure that the difference between that case without initialin-situstress and that considering initialin-situstress in the elastic region is 1.96 MPa,which is consistent with the superposition principle invoked in elastic analyses.The stress calculation results of other positions in elastic region are identical in principle.however,the stress variation in the plastic region differs from that in the elastic region,and the difference between the two is not equal to the initialin-situstress,so the superposition principle is no longer satis fied.

Fig.6 shows the time history curve of vertical displacement for N14938 node in the elastic region and N7874 node in the plastic region.Since the displacement at the initial moment is not considered in the dynamic calculation process,the initialin-situstress has no effect on the displacement calculation results,that is,the calculation results of the two are completely coincident,and the displacement variation within the plastic region is the same as that in the elastic region.The results of displacement calculation in other positions obey the same law.

The calculated results of stress and displacement are in good agreement with the theoretical solution,which shows that the boundary setting method is reasonable.

3.5.Numerical simulation verification

In order to ensure the accuracy of the numerical calculation method,the model test conducted by Chen et al.[31]was selected to perform numerical simulation analysis on the stress and deformation characteristics of the cavern under plane charge explosion.The anti-explosion model test device and arrangement of measure points were shown in Fig.7.The model test required that plane stress wave be generated on the top of the cavern and the wave elimination technology of the model boundary was considered,which was similar to the situation studied in this paper.The arched straight wall cavern had a buried depth of 100 m and a span of 5.5 m.There were two conditions:an unlined cavern and a reinforced concrete lined cavern,with a lining thickness of 1.0 m.Other parameters and model tests were shown in Ref.[31].The stress and deformation of surrounding rock of the vault and straight wall of the cavern were compared with each other,the results were shown in Fig.8,in which the left side was the unlined cavern,and the right side was the lined cavern.In order to compare the data size conveniently,the peak stress of numerical simulation had been reduced according to the stress scale.r/Din the figure represents the ratio of the distance r from the observation point to the spanDof the cavern.

Fig.5.Time history curve of vertical stress for E7311 and E4631.

Fig.6.Time history curve of vertical displacement for N14938 and N7874.

4.Results of analysis and assessments

4.1.In fluences of different crossing types on different span caverns

4.1.1.Axial vault settlement of surrounding rock

Due to the signi ficant difference between the vault settlement of different crossing types,the incremental percentage change in settlement is used to measure the differences,that is,compared with the vault settlement of single cavern under the same load,the percentage increase in vault settlement at different positions from the intersection centre is used:

where,dis the vault settlement at different distances from the intersection centre(m),d0is the vault settlement of a single cavern(m),and ddis the increment of vault settlement of a crossing cavern compared with that of the single cavern(m).

Fig.8.Comparison of model test and numerical simulation of surrounding rock stress and strain.

Fig.9.The vault settlement incremental distributions of different span caverns with different crossing types.

Fig.10.Arch displacement nephogram:L-shaped cavern.

The vault settlement curves of different span caverns with different crossing types are shown in Fig.9,and the variation trend of the curves are similar.The vault settlement at the intersection centre reaches the maximum,and the settlement decreases with the increase of the distance from the intersection centre.The increment of vault settlement of a＋-shaped cavern with an 8-m span is the largest,and the vault even reaches 30.93%strain.For the same span of cavern,the vault settlement of＋-shaped caverns is the largest,followed by T-shaped,and L-shaped caverns,which first increases,then decreases.This is because the outside of the intersection of L-shaped caverns is restricted by the sidewall,which slows the rate of settlement of the vault subjected to impact and changes the stress pattern from one of centre symmetry.The maximum settlement along the axial direction of the L-shaped cavern is not in the vault.It can be seen from the settlement plot(see Fig.10)that the deeper the blue is,the greater the settlement.With the increase of the distance from the intersection centre,the vault settlement will first increase,then decrease.According to the suggestion proposed by Hsiao et al.[32],the area with 10%increase in tunnel deformation should be reinforced with additional support.It can be seen from the figure that the deformation increase of an L-shaped cavern is within 10%,and in T-shaped caverns,within the distance of 0.6 times the cavern diameter from the intersection centre should be reinforced only for cavern with an 8-m span.＋-shaped caverns are greatly affected:the area of in fluence of the 4-m span cavern is 0.6 times the cavern diameter from the intersection centre,while the 6-m and 8-m span caverns reach 1.7 times that from the intersection centre.

4.1.2.Circumferential arch and straight wall deformation of surrounding rock

For the two symmetrical structures of the＋-shaped cavern and single cavern,the settlement of the surrounding rock vault is the largest,but for the asymmetric structures such as T-shaped and Lshaped caverns,the maximum settlement of the arch is often not within the vault.It can be seen from Fig.10 that the maximum settlement occurs close to the inner side of the intersection,therefore,to understand the settlement of such a circular arch more comprehensively,the arch of the intersection is analysed.The observation angle is anticlockwise(see Fig.11).

Fig.11.Arch schematic diagram of surrounding rock intersection.

Fig.12 shows the circumferential arch settlement distributions of different span caverns with different crossing types:the arch settlement patterns of cavern with three different spans are similar.For the same span,the arch settlement of a＋-shaped cavern is the largest,followed by that of the T-shaped cavern.In the range of 30°-90°(close to the inner side)of the intersection,the settlement from arch foot to vault of L-shaped cavern is larger than that of a single cavern,while from 90°to 150°(close to the outside)of intersection the settlement from arch foot to vault is less than that of a single cavern.The maximum settlement of a＋-shaped cavern and single cavern occur at the vault,and the maximum settlement of T-shape and L-shaped caverns occur close to the inner side of the intersection.The maximum settlement of an L-shaped cavern is closer to the inner side of intersection than that of a T-shaped cavern:the maximum settlement of the L-shaped cavern occurs at nearly 48°and the maximum settlement of the T-shaped cavern at nearly 66°.This is because,when the stress wave propagates from the top of the cavern,it gathers on the inside of L-shaped and Tshaped intersection and is dispersed on the outside,which makes the pressure on the inside greater than that outside.The maximum settlement position of the arch is shifted from the vault to the inner side.Compared with the L-shaped intersection,both sides of the Tshaped intersection share some of the pressure symmetrically,while all the stress waves propagated from the vault gather on the inner side of the L-shape,so the maximum settlement of the arch will be closer to the inner side for an L-shaped intersection.For different span caverns,with the increase of cavern span,the arch settlement of the same crossing type increases.The largest increase is 4 mm for a＋-shaped cavern,followed by that of a T-shaped cavern at 2.8 mm,then that of an L-shaped cavern at 2.4 mm,with the smallest increase being 1.8 mm for a single cavern.This shows that a＋-shaped intersection should be paid much attention when constructing caverns under the same geological conditions.

In addition to the settlement deformation of the arch(as directly affected by ground impact),the straight wall will also deform towards the free surface due to the diffraction of shock waves.Fig.13 shows the deformation distribution on the circumferential straight wall of different span caverns with different crossing types.The horizontal coordinate is the deformation offset of each point to the straight wall,the vertical coordinate on the right is the inside of the intersection,and the vertical coordinate on the left is the outside thereof(see Fig.11).

It can be seen from the figure that the deformation of caverns with different spans is similar.The maximum deformation occurs in the middle of the straight wall,and the deformation decreases from the middle to both sides.The straight wall deformation of a single cavern is the largest,and the deformation of the L-shaped cavern is the smallest.To the inside of the intersection,there is little difference between the deformation of T-shaped and L-shaped caverns,and the straight wall deformation of the＋-shaped cavern is greater than that of the T-shaped and L-shaped caverns,while,to the outside of the intersection,the straight wall deformation of the L-shaped and the single cavern are close,while being much greater than that of the＋-shaped and L-shaped caverns.This is because the single cavern and＋-shaped cavern are symmetric structures,the straight wall deformation of three crossing types is much less than that of the single cavern due to the constraint imposed by the inside of the intersections.There is no constraint on the outside of the T-shaped intersection,so the stress state and deformation are similar to those around a single cavern.The deformation of L-shaped intersections is the smallest due to the crossing constraint.For different span caverns,with the increase of cavern span,the straight wall deformation of different crossing types increases.The deformation of single caverns increases the most,and the maximum deformations of 4 m,6 m,and 8 m span caverns are 3.69 mm,5.56 mm,and 7.12 mm,respectively,with the increase in range of 50.68%and 28.06%,followed by＋-shaped caverns,Tshaped caverns,and L-shaped caverns increasing the least.

4.1.3.Axial vault peak particle velocity of surrounding rock

According to the underground explosion test research conducted by the US Army,Henderson proposed that,when the peak particle velocity(ppv)exceeded 4 m/s,severe spalling and even collapse would occur in the cavern.A similar calculation has been done for rock of the US Army’s Underground Explosion Tests(UET)and shown in Table 5,the number of spalls corresponding to a ppv of 4 m/s for Zone 3 damage is 4,which would seem to fit the physical description of tunnel damage quite well[17,33].

Table 5 Comparison of UET(Hendron)and I-D calculation(Damage Zones are shown in Fig.14).

When the ground impact load is 15 MPa,the maximum peak particle velocity of each point in the arch of different caverns is 3.86 m/s,which is not in a severe spalling state.Fig.15 shows the effects of different values of velocity incrementδv to compare the difference in various crossing types of caverns with different spans,that is,compared with the peak particle velocity of single cavern vault under the same load,the percentage increase of the peak particle velocity at the vault at different locations from the intersection centre is calculated as follows:

where,vis the peak particle velocity(m/s)at the vault at different distances from the intersection centre,v0is the peak particle velocity(m/s)at the vault of the single cavern,and dvrepresents the increment of the peak particle velocity(m/s)at the vault of the crossing cavern compared with that of the single cavern.

It can be seen from the figure that the variation across different crossing types of caverns with different spans is similar.The peak particle velocity of＋-shaped caverns is the largest,followed by that for T-shaped,and L-shaped caverns:these are all reduced to the peak particle velocity value of a single cavern at the same position.The range of in fluence of the cavern with a 4 m span is 2.25 times the cavern diameter,that of a 6 m span cavern is 1.5 times the cavern diameter,and for an 8 m span cavern it is 1 times the cavern diameter.The peak particle velocities of＋-shaped,T-shaped,and L-shaped caverns with a 4 m span are 3.19 m/s,2.93 m/s,and 2.79 m/s,respectively.The peak particle velocities of＋-shaped,Tshaped,and L-shaped caverns with a 6 m span are 3.78 m/s,3.45 m/s,and 3.32 m/s,respectively.The peak particle velocities of＋-shaped,T-shaped,and L-shaped caverns with an 8-m span are 3.86 m/s,3.62 m/s,and 3.49 m/s,respectively.The peak particle velocity of caverns with a 6-m span and 8-m span is similar,and is much larger than that under a 4-m span.This is due to the different failure mechanisms of the two types of caverns.Through many tests and dimensional analysis,Person[34]shows that the ratio of radiusRto wavelengthLis an important parameter affecting the collapse failure of caverns.WhenR/L?1,the stress wave quickly surrounds the cavern and completes the stress redistribution,which can be simpli fied to the problem of stress concentration with a circular hole under plane stress conditions.WhenR/L?1,it is a dynamic problem.When the cavern span is 4 m,L/R=10,and the stress wavelength is much greater than the radius of the cavern,making it a quasi-static problem.When the cavern span is 6 m and 8 m,L/Ris 6.67 and 5,respectively,the fluctuations in the stress wave are weak,and the diffraction of wave is insigni ficant.The stress wave mainly acts on the vault of the cavern,imposing simultaneous static and dynamic effects,so the peak particle velocities of different crossing types in caverns with spans of 6 m and 8 m are much larger than that in caverns with a span of 4 m.

4.1.4.Plastic zone distribution in the surrounding rock

Fig.12.Circumferential arch settlement distributions of different span caverns with different crossing types.

Fig.13.Circumferential straight wall deformation distributions of different span caverns with different crossing types.

Fig.14.Tunnel damage zones.

The distribution of the plastic zone plays a key role in the stability of the cavern,and the position of maximum plastic strain is the position where the surrounding rock is about to be destroyed.Taking the single cavern as an example,if the combined tensile stress generated by superposition of re flected tensile waves and incident waves exceeds the threshold for dynamic fracture of the rock after local impact load acts on the surrounding rock,the fracture will expand into the surrounding rock,and the block between each of the two tensile cracks will be sheared and separated at the root under impact and compression.The surrounding rock collapses toward the centre of the cavern,forming an inclined shear slip line(see Fig.16).When the slip line runs through the whole cavern,a distinct plastic zone is formed.

Fig.15.Axial vault peak particle velocity distributions of different span caverns with different crossing types.

Fig.16.Effective plastic strain distribution around a single cavern.

The effective plastic strains at the intersection of different span caverns with different crossing types are shown in Fig.17.Similar to the case of a single cavern,multiple shear slip lines appear near the straight wall,and the plastic zone is much larger than that of single cavern due to the in fluence of the cavern crossing.The results show that the effective plastic strain in the＋-shaped cavern is the largest,followed by that in T-shaped and L-shaped caverns.The effective plastic strain at the inside is larger than that at the outside of the T-shaped intersection.Only the inside of the L-shaped intersection undergoes plastic deformation,while the outside has none,which indicates that the stress wave propagates to the Lshaped intersection,the stress on the inside increases due to the accumulation of stress waves,and the stress wave on the outside is relatively divergent,so the stress is decreased thereat.With the increase of cavern span,the effective plastic strain in caverns with the same crossing type increases.When the span is 4 m,the effective plastic strain near the straight wall is the largest,presenting a wedge-shaped distribution.When the span is 6 m,the maximum effective plastic strain distribution comprises two distinct intersecting plastic zones,and the area of the maximum strain is further reduced when the cavern span is 8 m.With the increase of the cavern span,the position of this effective plastic zone gradually develops from the straight wall to the arch and the bottom.Plastic deformation has already occurred at the vault of＋-shaped intersection with a span of 8 m.If the load continues to increase,the plastic strain in the vault will increase,thus leading to collapse.

4.1.5.Deformation and stress response of the tunnel lining

The intersection of the lining straight wall and the floor is an unfavourable position with regard to stress concentration,which will lead to the fracture and falling of the supporting structure(i.e.,sprayed concrete support),further affecting the stability of the straight wall and arch,thus leading to the overall failure of the lining structure.According to the analysis,the maximum principal stress and maximum principal strain at the bottom of the straight wall are the largest.Table 6 lists the maximum principal stress and maximum principal strain at the bottom of the straight wall of different crossing types with different spans.

Table 6 1 The maximum principal stress and maximum principal strain at the bottom of the straight wall:4 m span cavern,2 The maximum principal stress and maximum principal strain at the bottom of the straight wall:6 m span cavern,3 The maximum principal stress and maximum principal strain at the bottom of the straight wall:8 m span cavern.

It can be seen from the table that,when the cavern span is 4 m,the maximum principal stress and maximum principal strain at the intersection centre of the＋-shaped cavern are the largest,followed by those of the T-shaped and L-shaped caverns.When the cavern span is 6 m or 8 m,the trend is reversed:the maximum principal stress and maximum principal strain at the intersection centre of the＋-shaped cavern are the smallest,followed by those of the T-shaped and L-shaped caverns.With the increase of the cavern span,the maximum principal stress and maximum principal strain increase,and the maximum principal strain at the intersection centre of caverns with spans of 6 m and 8 m are signi ficantly larger than that of caverns with a span of 4 m.With the increase of the distance from the intersection centre,the maximum principal stress and maximum principal strain decrease rapidly.For T-shaped and L-shaped caverns,the maximum principal stress and maximum principal strain inside the intersection centre are signi ficantly greater than those outside it.The maximum principal stress inside the intersection centre is 1.23-1.62 times that outside the intersection centre.The maximum principal strain inside the intersection centre of the L-shaped cavern with a span of 8 m reaches nearly 40 times that of the outside the intersection centre.The maximum principal stress and maximum principal strain outside the intersection centre of T-shaped and L-shaped caverns are close to that of the single cavern,which shows that the outside of the intersection centre is almost unaffected by the crossing part,and the stress and deformation are the same as that in the single cavern.This phenomenon is akin to the findings of Riley[1],who used a three-dimensional photoelastic method to measure the stress distribution around three common types (T-shaped,＋-shaped,and L-shaped)cylindrical tunnel intersections.

Fig.17.Effective plastic strain distribution at the intersection of different span caverns with different crossing types.

4.1.6.Failure mechanism analysis

The rock mass at the edge of the intersection centre of the underground cavern has the highest degree of stress concentration and is most likely to reach the plastic deformation state and even to be crushed,thus losing the effective support for the arch of the intersection.This kind of deformation and failure phenomenon essentially expands the open roof area of the intersection,which will inevitably affect the stability of the intersection.

In view of this intersection failure,Professor Ping proposed the concept of “equivalent span” ,and took it as a speci fic parameter to analyse the failure and support of the cavern at the intersection[35].For the intersection,there are two main characteristic spans:the maximum equivalent span(Lmax)and the minimum equivalent span(Lmin).The schematic diagram of equivalent span at the intersection of three different intersecting caverns is shown in Fig.18.Where,Dis the span of the cavern,Δris the maximum failure depth,d1,d2are the failure depth of crushing zone,andd1>d2,Lmaxis the maximum equivalent span,andLminis the minimum equivalent span,the calculation formula are as follows:

Fig.18.Schematic diagram of equivalent span.

Since the maximum crushing depthΔris much greater thand1andd2,andd1>d2,for the three intersecting caverns,Lmin+>LminT>LminLandLmax+>LmaxT>LmaxL,the vault at the intersection centre of＋-shaped cavern is more vulnerable to damage.So for the＋-shaped cavern,the value of vault settlement,straight wall deformation,vault peak particle velocity,effective plastic strain in the surrounding rock,and the maximum principal stress and strain at the bottom of the lining straight wall are the largest,followed by those of the T-shaped cavern and L-shaped cavern,and the value of T-shaped cavern and L-shaped cavern are close to each other.The＋-shaped cavern is the most signi ficantly affected by dynamic action.

4.2.In fluences of different crossing types on caverns under different lateral pressures

4.2.1.Axial vault settlement of surrounding rock

The incremental percentage ddof vault settlement is still used to measure the change in vault settlement under different lateral pressures.

The vault settlement incremental distributions of caverns with different crossing types under different lateral pressures are shown in Fig.19.The variation trend of vault settlement curve under three kinds of lateral pressures is the same.The settlement at the vault of the intersection centre reaches a maximum,the vault settlement of＋-shaped and T-shaped caverns decreases with increasing distance from the intersection centre,and the vault settlement of Lshaped caverns first increases,then decreases.At a distance of three cavern diameters,the vault increment of settlement decreases to near 0,which is no different from that of the single cavern.With the increase of lateral pressure coef ficient,the vault settlement of the same crossing type increases,but the rate of increase is only 3%.There is no obvious increase compared with that seen when changing the span of such caverns,which indicates that the change of lateral pressure coef ficient exerts little in fluence on the vault settlement.In the three cases,only the＋-shaped caverns need to be reinforced at the intersection,and the reinforced ranges are 1.5 times,1.6 times,and 1.7 times the diameter from the intersection centre,respectively.

4.2.2.Circumferential arch and straight wall deformation of surrounding rock

Fig.20 shows the circumferential arch settlement distributions of caverns with different crossing types under different lateral pressures,and the observation angle is shown in Fig.11.

The variation in circumferential arch settlement of three different types of crossing caverns under different lateral pressure coef ficients is the same,akin to that at different spans.The arch settlement of the＋-shaped cavern is the largest,followed by that of the T-shaped cavern,in the range of 30°-90°(close to the inner side)of the intersection,the settlement from the arch foot to the vault of the L-shaped cavern is larger than that of a single cavern,while between 90°and 150°(close to the outside)of the intersection,the settlement from arch foot to vault is less than that in a single cavern.The maximum settlement of T-shaped and L-shaped caverns occur close to the inner side of the intersection,not at the vault,and the maximum settlement of the L-shaped cavern occurs closer to the inner side of the intersection than in the T-shaped cavern.It can be seen from the figure that,with the increase of lateral pressure coef ficient,the circumferential arch settlement of the same type of crossing cavern decreases,and the maximum settlement decreases to a signi ficant extent,while the rate of settlement on both sides of the arch decreases slowly.This is due to the restriction imposed on the cavern increase by the increase of lateral pressure coef ficient,and the structural bearing capacity of lining being enhanced,which limits the arch settlement.The settlement of the arch foot is small due to the support provided by the straight wall,and the maximum settlement position occurs on the free surface as this is subject to fewer constraints,so the magnitude of the reduction in settlement thereat is large.

Fig.19.Vault settlement incremental distributions of caverns with different crossing types under different lateral pressures.

Fig.20.Circumferential arch settlement distributions of caverns with different crossing types under different lateral pressures.

Fig.21 shows the circumferential straight wall settlement distribution of caverns with different crossing types under different lateral pressures.

Fig.21.Circumferential straight wall deformation distributions of caverns with different crossing types under different lateral pressures.

The results show that the straight wall deformation trends in the three crossing caverns under the three lateral pressure regimes are similar.The straight wall deformation of the single cavern is the largest.For the inner side of the intersection,the deformation of the＋-shaped cavern is second largest,and the deformation of the Lshaped and T-shaped caverns are similar(and are the smallest due to the constraint imposed by the intersection).For the outer side of the intersection,the stress conditions of the T-shaped cavern and the single cavern are the same,so the straight wall deformations are similar,and are much larger than those of＋-shaped and Lshaped caverns.Under different lateral pressures,the straight wall deformation of the same type of cavern is insigni ficant at no more than 5%.The largest deformation range of a single cavern is 4.98%,followed by that of the＋-shaped cavern at 3.64%,the L-shaped cavern at 2.99%,and the T-shaped cavern at 1.42%.With the increase of lateral pressure coef ficient,the straight wall deformation at the inner side of the intersection decreases slightly,while at the outside it increases slightly.

4.2.3.Axial vault peak particle velocity of the surrounding rock

The velocity increment dvis selected to compare the difference of different types of caverns under different lateral pressures(see Fig.22).

Fig.22.Axial vault peak particle velocity distributions of caverns with different crossing types under different lateral pressures.

It can be seen from the figure that the variation trends of all types of caverns are the same under different lateral pressures.The peak particle velocity of the＋-shaped cavern is the largest,followed by that of the T-shaped cavern,then the smallest is that of the L-shaped cavern,which all drop to the vault peak particle velocity of a single cavern at the position some 1.5 times the cavern diameter from the intersection centre.The difference of vault peak particle velocity increment is small,and the largest increment is seen in the＋-shaped intersection.When the lateral pressure coef ficient is 0.5,1.0,and 2.0,the maximum particle velocity is 3.74 m/s,3.75 m/s,and 3.78 m/s,respectively(increments of 19.72%,19.78%,and 19.82%,respectively).Compared with the in fluence of the change in span on the peak particle velocity,the in fluence of lateral pressure coef ficient is very small.When the lateral pressure coef ficient increases from 0.5 to 2.0,the peak particle velocity only increases by 1%.This is because the deeply buried crossing cavern is always in a three-dimensional state of stress,and it cannot be regarded as a plane strain case as with a single cavern.When the lateral pressure coef ficient increases from 0.5 to 2.0,only the principal stress direction changes fromvertical to horizontal,which does not change the stresses imposed on the crossing cavern,therefore,the value of peak particle velocity changes little.

4.2.4.Plastic zone distribution in the surrounding rock

The effective plastic strain at the intersection of caverns with different crossing types under different lateral pressures is shown in Fig.23.

Fig.23.Effective plastic strain distribution at the intersection of caverns with different crossing types under different lateral pressures.

It can be seen from the figure that,under the same lateral pressure coef ficient,the effective plastic strain distributions of caverns with different crossing types are the same.The effective plastic strain value and plastic zone at the intersection of＋-shaped cavern are the largest,followed by those around the T-shaped and L-shaped caverns:the maximum effective plastic strains in T-shaped and L-shaped caverns are similar and much less than that in a＋-shaped cavern.The effective plastic strain at the inside is larger than that at the outside of the T-shaped intersection.There is no plastic zone at the outside of an L-shaped intersection.The results show that the larger effective plastic strain of the＋-shaped cavern appears at the bottom,middle,and top of the straight wall,while the larger effective plastic strain of the T-shaped and L-shaped caverns appears near the straight wall,showing two intersecting plastic bands.The plastic deformation on the floor of an L-shaped cavern is the largest,and the area close to the outside of the intersection is larger than that inside.With the increase of lateral pressure coef ficient,the maximum effective plastic strain of the same type of crossing cavern decreases,and the＋-shaped cavern is the most affected.WhenK=0.5,there is a large area of plastic deformation at the vault,the area of plastic deformation decreases with the increase of the lateral pressure coef ficient.WhenK=2.0,no plastic deformation occurs at the vault.This is because,when the lateral pressure coef ficient is small,the stress on the cavern acts mainly in the vertical direction,and the stress wave acts on the vault,which readily undergoes plastic deformation.With the increase of the lateral pressure coef ficient,the stress state of the cavern changes from vertical to horizontal,the stress on the vault decreases,and the plastic deformation also decreases.

4.2.5.Deformation and stress response of the tunnel lining

Table 7 lists the maximum principal stress and maximum principal strain at the bottom of straight walls with different crossing types under different lateral pressure coef ficients.

Table 7 1 The maximum principal stress and maximum principal strain at the bottom of straight wall when K=0.5,2 The maximum principal stress and maximum principal strain at the bottom of straight wall when K=1.0,3 The maximum principal stress and maximum principal strain at the bottom of straight wall when K=2.0

Table 8 Range of in fluence of caverns with different crossing types.

It can be seen from the table that,with the increase of the lateral pressure coef ficient,the maximum principal stress and maximum principal strain at the bottom of the straight walls of different types of cavern decrease.For the same lateral pressure coef ficient,the maximum principal stress at the intersection centre of different crossing types is similar,the＋-shaped cavern bears the smallest maximum principal stress,and the L-shaped the largest.The maximum principal stress and maximum principal strain inside the intersection centre of T-shaped and L-shaped caverns are signi ficantly larger than those outside the intersection centre.The maximum principal stress inside the intersection centre is 1.34-1.71 times that outside the intersection centre.The maximum principal strain inside the intersection centre is 33.34-38.2 times that outside the intersection centre.The maximum principal stress and maximum principal strain outside the intersection centre are similar to those on a single cavern,and are unaffected by the intersection.With increasing distance from the intersection centre,the maximum principal stress and maximum principal strain on Tshaped and L-shaped caverns decrease rapidly,and decrease to the corresponding values of single cavern within the range of two times the cavern diameter.

4.3.In fluence range of the intersection

The stress,deformation,and failure characteristics of the intersection of underground caverns differ from those of ordinary single caverns.It is necessary to determine the in fluence range of intersection for engineering design and construction purposes.According to the analysis above,the in fluence range of caverns with different crossing types differs depending on parameter measured(see Table 8,where the in fluence range refers to the distance from the intersection centre in the axial direction).The in fluence range of caverns with spans of 6 m and 8 m are similar,yet different from that of caverns with a span of 4 m.To consider safety in engineering practice,the in fluence range of such intersections is determined to be two times the cavern diameter.The supporting structures in this range need to be thickened to improve the stability of the cavern.

5.Conclusion

Through the numerical simulation of the dynamic response of three kinds of crossing-type caverns with different spans under different lateral pressures,the following conclusions may be drawn:

(1)The proposed velocity plus force mode boundary setting method is simple to implement and does not require additional parameters.It can not only meet the transfer of incident stress wave load,but also meet the transmission of external radiation stress wave,and can accurately consider the in fluence of initial stress.The simulation results for the initial stress field are accurate,and meet the requirements of boundary condition setting for deformation and failure analysis of deeply buried structures under coupled static and dynamic load.

(2)Among the three types of crossing caverns,the＋-shaped cavern is the most signi ficantly affected by dynamic action.The vault settlement,straight wall deformation,vault peak particle velocity,effective plastic strain in the surrounding rock,and the maximum principal stress and strain at the bottom of the lining straight wall are the largest,followed by those of the T-shaped cavern and L-shaped cavern.It is suggested that the＋-shaped intersection should be used less in the design of underground protective engineering works.

(3)The vault settlement,straight wall deformation,vault peak particle velocity,effective plastic strain in the surrounding rock,and maximum principal stress and strain at the bottom of the lining straight wall increase with increasing cavern span.The values in caverns with spans of 6 m and 8 m are similar,and much greater than the values in the corresponding 4-m span cavern.When the cavern span is 4 m,the problem is one of quasi-static stress:the stress wave rapidly surrounds the cavern,and the damage is mainly concentrated near the straight wall.When the span of the cavern is 6 m or 8 m,dynamic failure characteristics are manifest:the stress wave mainly acts on the vault of the cavern,and damage is concentrated at the vault.

(4)The vault settlement,straight wall deformation,effective plastic strain of surrounding rock,and the maximum principal stress and strain at the bottomof the lining straight wall decrease with the increase of lateral pressure coef ficient,and the peak particle velocity at the vault increases.The variation is not signi ficant compared with that caused by changes to the span of such caverns,and the in fluence of lateral pressure coef ficient on cavern with different crossing types is thus small.

(5)The in fluence range of the underground cavern intersection is two times the cavern diameter from the intersection centre.The bottom of the straight wall at the intersection is the weakest part.It is suggested to thicken the support locally thereat to improve the stability of the cavern.

Declaration of competing interest

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.

Acknowledgment

This research is funded by the National Natural Science Foundation of China(No.51478469).