Blast resistance evaluation of urban utility tunnel reinforced with BFRP bars

Qing Zhou ,Hu-gung He ,Sn-feng Liu ,Xio-shuo Chen ,Ze-xun Tng ,Yng Liu ,Zhen-yu Qiu ,Sen-sen Li ,He Wng ,Yin-zhi Zhou ,Jin-nn Zhou ,**,Hu-lin Fn ,Feng-nin Jin ,***

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

b Research Center of Lightweight Structures and Intelligent Manufacturing,State Key Laboratory of Mechanics and Control of Mechanical Structures,Nanjing University of Aeronautics and Astronautics,Nanjing,210016,China

Keywords:Utility tunnel BFRP bars Explosion Dynamic response Crack Dynamic theory

ABSTRACT To improve corrosion-resistance of shallow-buried concrete urban utility tunnels(UUTs),basalt fiber reinforced polymer(BFRP)bars are applied to reinforce UUTs.As the UUT must have excellent survival capability under accidental explosions,a shallow-buried BFRP bars reinforced UUT(BBRU)was designed and constructed.Repetitive blast experiments were carried out on this BBRU.Dynamic responses,damage evolutions and failure styles of the BBRU under repetitive explosions were revealed.The tunnel roof is the most vulnerable component and longitudinal cracks develop along the tunnel.When the scaled distance is larger than 1.10 m/kg1/3,no cracks are observed in the experiments.When the BBRU is severely damaged,there are five cracks forming and developing along the roof.The roof is simplified as a clamped-supported one-way slab,proved by the observation that the maximum strain of the transverse bar is much larger than that of the longitudinal bar.Dynamic responses of the roof slab are predicted by dynamic Euler beam theory,which can consistently predict the roof displacement under large-scaleddistance explosion.Compared with the UUT reinforced with steel bars,the BBRU has advantages in blast resistance with smaller deflections and more evenly-distributed cracks when the scaled distance is smaller than 1.260 m/kg1/3 and the steel bars enter plastic state.Longer elastic defamation of the BFRP bars endows the UUT more excellent blast resistance under small-scaled-distance explosions.

1.Introduction

With the rapid development of economy and the acceleration of urbanization process,the city becomes increasingly congested.Ground power lines and communication lines seriously affect the appearance and the possibility of line damage is greatly increased.Water and gas pipelines are shallowed-buried and inconvenient for maintenance.All these problems can be solved by placing these pipelines in the shallowed-buried tunnel.These urban lifelines should have excellent protection ability of“disaster prevention in peacetime and air defense in wartime”.

Mussa et al.[1]numerically investigated the damage evolution of underground box-type tunnels caused by car bombs.Feldgun et al.[2]proposed a method to reveal the effect of an underground explosion on a neighboring buried structure.Through scaling model test and numerical simulation,De and Zimmie[3,4]found that polyurethane foam protective barriers outside the tunnel could effectively attenuate the blast load on the tunnel.Nan and Chuanbo[5]established the relationship between the dynamic stress of the tunnel and the peak velocity of surrounding rock.Yang et al.[6]analyzed the impact of ground explosion on subway tunnels and evaluated the structural safety.The common method to analyze the dynamic response of underground structures is suggested by TM5-855-1 manual[7].Ma et al.[8]proposed a more accurate model to study the dynamic response of shallow-buried box-type structures.Dancygier et al.[9]conducted explosion experiments on small scale buried structures subjected to surface explosions.Chen et al.[10-12]conducted explosion experiments on the shallow-buried reinforced concrete arches,analyzed the dynamic response and curvature effect of the structure,and proposed a dynamic soil-structure interaction model.

Table 1Tunnel structure size.

Fig.1.(a)Cross-section and(b)longitudinal section of the tunnel construction.

Fig.2.(a)Typical load-displacement curves and(b)elastic stress-strain curves of the BFRP bar and the steel bar.

Corrosion of steel bars will greatly reduce the durability of concrete structures.Replacing steel bars with BFRP bars is an effective method to prevent corrosion[13-15].BFRP bars have excellent corrosion resistance[16,17].Artemenko[18]conducted mechanical tests on different fiber reinforced plastics(FRP)and found that the BFRP is stronger than the glass fiber reinforced polymer but much cheaper than the carbon fiber reinforced polymer.Gao et al.[19],Zhou et al.[20],Liu et al.[21,22]and Feng et al.[13]applied FRP bars to reinforce concrete slabs and beams.The long elastic defamation of the FRP bars endows the concrete structures more excellent blast resistance compared with steel bars reinforced concrete beams and slabs[13,19-22].Then this technique can be extended to shallow-buried protective structures.

In this research,BFRP bars were applied to construct shallowburied UUT.Eleven repetitive explosions were loaded on the UUT.The defomations,the strains of the bars,the crack evolution and the damage pattern of the BBRU were revealed and compared with the referenced steel bar reinforced UUT(SBRU).The mechanism of the excellent performance of the BBRU in blast resistance is revealed.

Fig.3.Reinforcements of the tunnel reinforced with BFRP bars(Unit:mm).

Table 2Properties of BFRP bars.

Table 3Properties of steel bars.

2.Structural explosion experiments

2.1.Tunnel design

The dimensions of the tunnel are listed in Table 1.The experiment model is 1/3 of the prototype.As shown in Fig.1,the roof corners are strengthened by adding BFRP bars with spacing of 150 mm and diameter of 8 mm,and the width of the strengthened corner is 150 mm.The thickness of the protective layer is 20 mm.The cross section and the longitudinal section are shown in Fig.2 and Fig.3,respectively.The initial buried depth of tunnel is 2.0 m.As the reference,a SBRU of identical dimensions was also constructed and tested.The reinforcement ratio of the steel bars is identical to that of the BFRP bars.The tunnel is made of C40 concrete.The standard compressive strength of the concrete is 39.84 MPa.The tensile strength of the BFRP bars is 813.9 MPa,and the elastic modulus is 56.12 GPa,as listed in Table 2.Correspondingly,the yield strength of the steel bars is 416.79 MPa,and the elastic modulus is 183.3 GPa,as listed in Table 3.Compared with the steel bars,the BFRP bars have much higher tensile strength but much smaller stiffness.They also have much longer elastic deformation.The elastic strain of the BFRP bars is over 1.45%in average,while the steel bar fall into plastic deformation when the strain is over 0.227%,as shown in Fig.2.The fracture strain of the steel bar is 0.118 in average.

2.2.Tunnel construction

The diameter of the BFRP bars is 8 mm.The spacing between neighboring longitudinal and transverse bars is 150 mm.The reinforcement diagram is shown in Fig.3.The construction process of the tunnel generally includes:binding steel bars,supporting tunnel wall formwork,pouring concrete,demolition of the template and maintenance,as shown in Fig.4.In accordance with the reinforcement diagram,the transverse bars and longitudinal bars of the floor,the reinforcing bars of the bottom corners and the transverse bars of the side walls are bound firstly.After the binding is completed,the template is fixed and then the floor is poured.After the soleplate concrete reaches the early strength,the inner template is fixed,and then the longitudinal bars of the side walls,the transverse and longitudinal bars of the roof and the reinforcing bars of the top corners are bound.Next the template is fixed.After the template is completed,the side walls and the roof are poured.As the tunnel is poured in winter,after the concrete reaches the early strength,maintenance measures with thick insulation layers are taken for the tunnel,as shown in Fig.4.

Fig.4.Construction process of the BBRU.

2.3.Experimental scheme

The TNT block is placed on the surface of the soil cover above the center of the tunnel roof.After each explosion,backfill the soil to restore the soil surface to the original state,and then carry out the next explosion.The damage process of the experiment is cumulative.Compared with the experiment of intact components,it’s safer to use the result to evaluate the various indicators.In order to obtain the load distribution and dynamic response of the tunnel under different explosive equivalent and buried depth,eleven repetitive explosions from B1 to B11 with decreased scaled distance are carried out,as listed in Table 4.

Table 4Explosion experiment scheme.

To measure the blast load and reflect the load distribution on the tunnel roof,the pressure sensors are placed symmetrically along the axis of the roof,as shown in Fig.5(a).A total of 11 sensors are arranged.The maximum measuring range is 30 MPa,the response frequency is 200 kHz,and the accuracy is 1%.To measure the dynamic displacement of the tunnel roof and the side walls relative to the floor,three extensometers are arranged in each section,as shown in Fig.5(b).The measuring range is 100 mm and the accuracy is 0.5%.To measure the strains of the reinforcements,72 strain gauges are adhered to the surface of the bars,as shown in Fig.5(c).The resistance value of the sensors is 120.0±0.1Ωand the measuring range is 20,000με.To measure the accelerations,a total of 18 accelerometers are installed on the side walls,the roof and the floor,as shown in Fig.5(d).Two kinds of acceleration sensors with a range of 5000 g and 10,000 g are chosen and the response frequency is 10 kHz.After each explosion,the crack width and the evolution are recorded.

3.Experimental analys es

3.1.Pressure analyses

As shown in Fig.6,the time-history curves of the reflected pressures in the center of the tunnel roof are measured in tests B1,B2 and B3.The peak pressures are 0.016 MPa,0.036 MPa and 0.072 MPa,respectively.The measured data are compared with those calculated by the CONWEP,as shown in Fig.7.The parameters used for the CONWEP calculation are listed in Table 5.The measured reflected pressures are in good agreement with the CONWEP calculation,as compared in Table 6.

3.2.Strain analyses

Fig.8 shows the time-history curves of the longitudinal and transverse bars strains on the inner side of the tunnel roof center and the transverse bars strains on the inner and outer side of the side wall center.The data are summarized in Table 7.It is shown that the strain of the transverse bars is much larger than that of the longitudinal bars.The roof deformation exhibits strong unidirectional characteristic.The strain time-history curve of the outer transverse bars of the side wall has a small negative compression strain before the positive strain of outward bending,because when the blast load acts on the roof,the side wall is compressed first,and after reaching a certain force value,it bends outwards.Till test B9,the tensile strain of the outer transverse bars is greater than the compressive strain of the inner transverse bars,and the transverse strain of the side wall is at a small level,indicating that the backfill soil has a good constraint effect on the displacement of the side wall of the tunnel.

Fig.5.Distributions of measuring points for(a)pressures,(b)displacements,(c)strains and(d)accelerations.

Strain distribution along the tunnel roof is shown in Fig.9,where the negative data denote the upper surface while the positive data denote the lower surface.The distributions reveal that the BFRP bars work in elastic state and the roof is a typical flexural member under surface explosion.

3.3.Acceleration analyses

Fig.10 shows the time-history curves of accelerations in the central cross-section of the tunnel.The acceleration under test B9 is not displayed because of data loss caused by improper data collection.The data are also summarized in Table 8.

In tests B1 to B3,the vertical accelerations of the roof and the floor and the transverse accelerations of the side walls are equal,which are about twice of the vertical accelerations of the side wall.With the decrease of the scaled distance,the vertical acceleration of the roof is greater than that of the floor and the transverse acceleration of the side wall.The reason may be that the overall connection between the roof and the side wall is weak,so the bending degree of the side wall is small when the roof is bent.The small strain level of the side wall is also a good proof to this analysis.On the other hand,the side wall is under overall vertical compression.From the transverse and vertical accelerations of the side wall,the deformation of it is still dominated by bending.The overall movement of the tunnel is still well restrained by the soil and the floor cushion,so its value is the smallest.The vibration of the roof rigidly spreads to the bottom plate,so the acceleration of the floor is greater than that of the side wall.The center acceleration of the roof is larger,which shows the roof is easier to destroy.

3.4.Displacement analyses

The displacement time-history curves of the center position of the roof and the side walls are shown in Fig.11.The peak displacements of the center position of the tunnel roof from tests B1 to B11 are 0.28 mm,0.64 mm,1.35 mm,2.95 mm,5.72 mm,6.78 mm,11.17 mm,19.38 mm,23.08 mm,22.92 mm and 40.71 mm,respectively.The residual displacements are 0.02 mm,0.33 mm,0.56 mm,0.67 mm,1.24 mm,0.52 mm,1.69 mm,3.59 mm,1.22 mm,1.53 mm and 5.09 mm,respectively.The residual displacements are all at a small level.The outward peak displacements of the side wall are 0.19 mm,0.23 mm,1.43 mm,2.62 mm,1.70 mm,3.08 mm,4.21 mm,5.08 mm,5.23 mm and 7.81 mm,respectively.The inward peak displacements of the side wall are 0.21 mm,0.21 mm,0.66 mm,0.99 mm,1.75 mm,0.75 mm,1.51 mm,3.48 mm,4.00 mm,3.45 mm and 6.24 mm,respectively.The inside and outside peak displacements of the side wall are sizable.The residual displacement is small and the damage is small.

In Fig.12(b),the change rule of the displacement with the scaled distance is revealed.As no obvious spalling is observed,the damage of the BRRU is divide into three grades according to the displacement in this research.Only for test B11,the displacement doubles,reflecting the effect of accumulative damages in repetitive explosions.Before B11,the accumulative damage effect is not obvious.Fig.12(b)reveals the displacement distribution curves along the longitudinal axis of the roof from tests B4 to B11.As shown in Fig.12,the displacement attenuation is less in tests B4 to B7.In tests B8 to B11,the displacements from the center to both sides have a certain attenuation,and the proportional coefficient is gradually increasing,which is between 1.06 and 1.13.In test B11,the displacement increases sharply and the tunnel is damaged to some extent.

Fig.6.Pressure time-history curves.

Fig.7.CONWEP pressure time-history curves.

Table 5CONWEP computing parameters.

Table 6Pressure errors.

Fig.13 shows the displacement distribution curve of the center cross-section of the tunnel from tests B4 to B11.As shown in Fig.13,the maximum displacement of the cross section also presents a symmetrical distribution form,and the structure is symmetrically bent.With the decrease of the scaled distance,the ratio of the maximum displacement at the center of the roof and the maximum displacement at the center of the two side walls(taking the average displacement of the center of the two side walls)increase gradually,indicating that the constraint between the side wall and the roof is weakening.

3.5.Analyses of crack propagation and failure patterns

As shown in Fig.14,the crack in the BBRU is firstly observed in test B5.A micro-crack appears in the middle of the roof and the average crack width is 0.26 mm.In test B6,the number of the longitudinal cracks increases to 4,and the cracks are widened.The average width of the cracks is 0.38 mm.In test B7,the two microcracks in the middle of the tunnel are widened,and the average width of the cracks is 0.40 mm.In test B8,a new crack is added to the edge of the tunnel,and two oblique cracks are added along 45°on one side,with an average width of 0.52 mm.In test B9,two oblique cracks appear along 45°on the other side of the tunnel.The cracks are widened and the average width of the cracks is 0.57 mm.In test B10,a micro-crack is added to the longitudinal edge of the tunnel,and the cracks continue to widen.An oblique crack appears in the axillary part and the side walls on one side of the tunnel,with an average width of 0.84 mm.In test B11,the cracks continue to widen,the oblique cracks increase in the axillary part and the side walls,the concrete in the center of the roof is spalled by the shock wave,and the average width of the cracks is 0.86 mm.

When the BBRU is damaged,there are 5 evenly-distributed longitudinal main cracks crossing the roof.The damage forms mainly include bending cracks in the roof,shear cracks in the roof,shear cracks in the side walls and local spalling of the roof concrete.

According to the experiments,damage accumulation is not obvious in the first few explosions.But the damage of each explosion is accumulated in the last few tests,inducing abrupt increase of the displacement,the crack width and the strain.

4.Theoretical analyses

From the experiments,the BFRP bars work in elastic state although the concrete is crushed or stretched to form cracks.The dynamic behaviors of the BBRU can be analyzed by elastic dynamic theory.

4.1.Structural model

The length of the UUT is 5200 mm.As the corners are strengthened,the inner span is considered as 1500 mm.In Fig.12,the deflections along the tunnel are closely evenly-distributed.In Fig.14,five parallel cracks extend through the tunnel.From these experimental phenomena and the dimensions of the tunnel,the tunnel roof can be regarded as a clamped-supported one-way slab.The Euler beam theory with clamped ends can be used to analyze the dynamic responses.The bending rigidity of the beam,Dx,is given by

whereΨ=0.6,which is the stiffness reduction factor.Ecand Esrepresent the elasticity modulus of the concrete and the steel,respectively.Icand Isare the inertia moments contributed by the concrete and the steel in unit width,respectively.

4.2.Load model

The pressures and impulses calculated by the CONWEP are listed in Table 9,where Piand Prare the incident pressure and the reflect pressure on the roof center,respectively.Ptis the total pressure applied on the roof center.Iiis the impulse of the incident pressure.For the convenience of calculation,the explosive load acting on the roof can be equivalent to a uniform load.The equivalent uniform peak pressure P0is given by

whereαis the uniform coefficient[7].According to Fig.15,α=0.88,0.84,0.76 when the buried depth of the tunnel is 2 m,1.6 m,1.2 m,respectively.The load form is simplified as a triangular load with a pressure rising stage and given by

where P0is the uniform peak pressure calculated by Eq.(2);t is the time;t1is the time when the pressure reaches its peak which can be given by the CONWEP;and t2is the duration of the blast load which can be calculated by the equal impulse method as

Fig.8.Strain time-history curves.

Table 7Strain summary table(×10-6).

Fig.9.Strain distribution along the tunnel.

Fig.10.Acceleration time-history curves.

Table 8Acceleration summary table(m·s-2).

Fig.11.Displacement time-history curves.

Fig.12.(a)Displacement change rule with the scaled distance and(a)displacement distribution curves along the longitudinal axis of the roof.

where Iiand Piare listed in Table 9.Thus the parameters of the load can be given by Table 10.

4.3.Dynamic responses

The deflection of the beam is w,which can be assumed as

whereΦn（x）is the nth mode of the beam,and Zn（t）is generalized coordinate.Without considering damping,the equilibrium equations of a beam element can be given by

whereρrepresents the equivalent density of the beam;h represents the thickness beam;and P（t）is the explosive load given by Eq.(3).The bending moment,Mxis given by

Substituting Eq.(8)into Eq.(6)and Eq.(7),the governing equation of the beam is given by

Substituting Eq.(5)into the governing equation and according to the orthogonality of the characteristic modes,the governing equation for the nth mode of the beam is given by

and

The Roman numeral and dot superscripts denote the space and time derivatives,respectively.The solution of Eq.(10)is suggested in Appendix A in details.

4.4.Validations

The parameters of the blast load for the first four working conditions are listed in Table 10.So the displacements can be calculated by substituting these parameters into Eq.(A10),which are listed in Table 11.

The displacements curves are showed in Fig.16.The maximum displacements of the theory match those of the experiments with acceptable errors in the first three working conditions.But the experimental data are obviously larger than the theoretical values from the forth working condition,which means there were some damage in the tunnel roof.

As the damage accumulation is not considered in the model and the ends are assumed clamped,the prediction is smaller.Damping effect is also not included in the theory,so the predicted displacement curves have no attenuations.The blast load is also simplified.All these induce errors in prediction.

Fig.13.Displacement distribution curves of the central cross-section of the tunnel(Unit:mm).

Fig.14.Cracks evolutions of the BBRU.

Table 9Pressures and impulses calculated by CONWEP.

5.Comparisons between BBRU and SBRU

As a reference tunnel,blast responses of the SBRU is also tested from S1 to S11,corresponding to B1 to B11,which will be discussed in details in another paper.Comparisons between these two tunnels reveal interesting phenomena.

The strains are compared in Table 12 and Fig.17(a).When the scaled distance is large,the difference is not obvious.When the scaled distance is smaller than 0.881 m/kg1/3,the maximum strain of the steel bar is greater than that of the BFRP bar.When the scaled distance is smaller than 0.66 m/kg1/3,the maximum strain of the steel bar is several times of that of the BFRP bar.The steel bar is working in plastic state while the BFRP bar is still working in elastic state.

Table 10Parameters of the blast wave.

Table 11Displacements from the test and the theory(Unit:mm).

Fig.15.Uniform coefficient of the pressure for underground structure.

In this circumstance,the BBRU is much stiffer and the tunnel would have much smaller mid-span deflection,as compared in Table 13 and Fig.17(b).When the scaled distance is large,the difference is not obvious.When the scaled distance is smaller than 0.66 m/kg1/3,the maximum deflection of the SBRU are much larger than that of the BBRU.The SBRU has obvious residual deflection while the deformation of the BBRU nearly completely restored.

The crack information also strengthens the advantage of the BBRU in blast resistance,as shown in Fig.17(c)and(d)and compared in Table 14.From the crack number,the BBRU has more uniform crack distribution compared with the SBRU.As uniform crack distribution represents less damage,the damage degree of the BBRU is smaller than the SBRU when the scaled distance is smaller than 1.0 m/kg1/3,as shown in Fig.17(c).Another crack information to judge the damage is the crack width,as shown in Fig.17(d).When the scaled distance is large,the difference between the crack widths is not obvious.When the scaled distance is smaller than 0.743 m/kg1/3,the crack width of the SBRU is about twice of that of the BBRU.When the scaled distance is 0.557 m/kg1/3,the crack width of the SBRU(S11)is about 13 times of that of the BBRU(B11).As shown in Fig.18,the BBRU have five evenly-distributed longitudinal cracks while the SBRU has three concentrated longitudinal main cracks.The central crack is much wider than the other two.The crack width is 11.2 mm in test S11 and the crack penetrates the cross section and there is also a wide longitudinal crack outside of the tunnel roof,representing there is a fully-developed plastic hinge line at the mid-span of the roof and its rotation will greatly weaken the blast resistance of the SBRU.

The comparisons reveals that the long elastic deformation of the BFRP bars make the BBRU work in elastic in this research in which the minimum scaled distance is 0.557 m/kg1/3.The much smaller elastic deformation of the steel bars makes the SBRU works in plastic state when the scaled distance is smaller than 1.000 m/kg1/3.Plastic deformation of the steel bars greatly reduce the rigidity and strength of the SBRU,as well as the blast resistance.

In this research,the BBRU has much more excellent blast resistance than the SBRU through comparing the reinforcement strains,the mid-span roof deflections,the crack width and the crack distribution.

6.Conclusions

In this paper,BFRP bars are applied to construct concrete UUT.Explosion experiments are carried out to investigate the blast resistance of the BBRU.Based on the measurements,observations and analyses of the surface reflection pressures,the strains,the accelerations,the displacements,the crack information and the failure patterns,the blast performances of the BBRU are firstly revealed.Theoretical calculation is conducted on the center displacement of the tunnel roof under the first four working conditions.The following conclusions are concluded:

(1)The maximum strains of the central transverse bars at the bottom of the tunnel roof are much larger than those of the longitudinal bars,which shows an obvious one-way slab structural performance,and the stresses mainly distribute along the short span of the roof.It is suggested that the transverse bars should be strengthened in designing the BBRU.

(2)The main damage forms of the BBRU are the bending cracks in the roof and local spalling of the roof concrete.When the BBRU is damaged,5 cracks form and develop along the axis of the roof.The shear strength of the BFRP bars is not high,so shear cracks appear in the side wall.In designing the UUT,the hybrid application of steel bars and BFRP bars can be considered to enhance the shear strength of the structure.

(3)The displacements changing with the scaled distance reveal the damage degree and the accumulative damage effect of the BRRU.In this research,the BRRU has three damage degrees,including no damage,slight damage and medium damage.Only when the scaled distance is reduced to 0.557 m/kg1/3,the accumulative damage effect redoubles the displacement.In other cases,the accumulative damage effect is not so obvious.

(4)The tunnel roof is regarded as a clamped-supported one-way slab.And the Euler beam theory with clamped ends is applied to analyze the dynamic responses of the BRRU.The displacements of the theory are in agreement with the test in the first three working conditions.In test B4,the scaled distance is about 1.10 m/kg1/3and the error of the theory is large.It is considered that when the scaled distance is greater than 1.260 m/kg1/3in this research,the concrete in the tunnel is not damaged.As the damage accumulation is not considered in the model,it cannot predict dynamic responses in the experiment with small scaled distance.Also,clamped end assumption also underestimates the displacement.

(5)For the referenced SBRU,there are three main longitudinal cracks concentrating on the mid-span of the tunnel roof,forming a plastic-hinge mechanism.Compared with the SBRU,the BRRU has excellent elastic performance under intense explosion cases,which makes the BRRU much stiffer and stronger than the SBRU with smaller displacements,much smaller strains,much finer and more evenlydistributed longitudinal cracks.Combining with the corrosion resistance,the BRRU might be a better choice for coastal and protective UUTs.

Fig.16.Displacement curves of the BBRU.

Table 12Strain comparisons(Unit:με).

Fig.17.Comparisons between(a)strains,(b)displacements,(c)crack numbers and(d)crack widths.

Table 13Displacement comparisons(Unit:mm).

Table 14Crack width comparisons(Unit:mm).

Fig.18.Comparisons between typical crack developments of(a)the SBRU(S11)where the crack width is 11.2 mm and(b)the BBRU.

Declaration of competing interest

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

Acknowledgements

Supports from National Natural Science Foundation of China(51778622,11672130,and 11972184),Social Development Project of Science and Technology Department of Jiangsu Province(BE2017780),State Key Laboratory for Disaster Reduction in Civil Engineering(SLDRCE16-01)and State Key Laboratory of Mechanics and Control of Mechanical Structures(MCMS-0217G03)are gratefully acknowledged.

Appendix A.Solution of the dynamic response

Based on the clamped boundary conditions,Φn（x）is assumed as

with

where a is the span of the beam.When 0≤t≤t1,the initial conditions are Zn1（0）=0 and˙Zn1（0）=0 since the tunnel is static.According to the Duhamel integral,the solution of Eq.(10)is given by

When t1

According to Eq.(A5),Zn1（t1）and˙Zn1（t1）are given by

When t＞t2,the initial conditions are Zn2（t2）and˙Zn2（t2）in the stage of free vibration,and the solution of Eq.(10)is given by

According to Eq.(A6),Zn2（t2）and˙Zn2（t2）are given by

So the displacement of the beam is given by