Chao Ma,Shenghui Zhou,Jingwei Chi

School of Civil and Transportation Engineering,Beijing University of Civil Engineering and Architecture,Beijing,102616,China

Keywords: Underground structures Random field Seismic performance FEM analysis Variable coefficient

ABSTRACT Soils with spatial variability are the product of natural history.The mechanical properties tested by soil samples from boreholes in the same soil layer may be different.Underground structure service in surrounding soils,their seismic response is controlled by the deformation of the surrounding soils.The variability of soil mechanical parameters was not considered in the current research on the seismic response of underground structures.Therefore,a random field model was established to describe the spatial variability of surrounding soils based on the random field theory.Then the seismic response of underground structures in the random field was simulated based on the time-domain explicit global FEM analysis,and the soil mechanical parameters and earthquake intensity influencing the seismic response of surrounding soils and underground structures were studied.Numerical results presented that,the randomness of soil parameters does not change the plastic deformation mode of surrounding soils significantly.The variation coefficients of inter-story deformation of structures and lateral deformation of columns are much smaller than that of mechanical parameters,and the randomness of soil parameters has no obvious effect on the structural deformation response.

The seismic response of underground structures is mainly constrained by the deformation of surrounding soils (Iida et al.,1996;Nakamura et al.,2000;Hashash et al.,2001;Chen et al.,2016;Sawamura and KishidaKimura,2016;Ma et al.,2018b;Dong et al.,2020).The deformation mode and size of the surrounding soils determine the distribution and size of earthquake-induced soil pressure on the underground structures (Ma et al.,2019).Natural variability is one of the basic characteristics of soils,in this sense,the mechanical properties or parameters of soil samples from the same soil stratum might be different (Duncan,2000).Moreover,the mechanical parameters tested on the same soil sample may also be different due to the difference in equipment or experience of the tester.Therefore,the influence of soil mechanical parameter variation on the seismic response of underground structures is worth exploring.

The influence of soil parameter variability on geotechnical engineering has gained extensive attention and has been studied (Zhu et al.,2013;Wang et al.,2018;Hamrouni et al.,2020;Stuedlein et al.,2021).Such as,the variability of soil parameters influencing earthquake-induced slope safety was discussed based on the random field theory (Vanmarcke,1977;Srivastava and Babu,2009;Kim and Sitar,2013).It presented that,the slope safety factor obtained from the random field is significantly lower than that obtained from the homogeneous field.On the other hand,soil parameter variability influencing the seismic performance of underground structures was also studied,but mainly focused on tunnels,for example,the transverse seismic response(Yue et al.,2016)and longitudinal seismic response(Yu et al.,2020)of shield tunnels,and the seismic response of immersed tunnels(Chen et al.,2017).It presented that the random field had a significant influence on the force response of tunnels.

Fig.1.The basic flow of random field-underground structure seismic response analysis.

Fig.2.Subway station cross section (Parra-Montesinos et al.,2006).

Fig.3.Random field-underground structure FEM model.

During previous studies,authors simulated the earthquake-induced failure mode of rectangular underground structures and revealed their failure mechanism (Du et al.,2017;Ma et al.,2018a,2018b,2021).However,the influence of the variability of soil mechanical parameters was not considered.Therefore,this study presents a numerical study on random fields influencing the seismic performance of rectangular underground structures based on the random field theory.A 2Dtime-domain explicit global FEM model of random field and underground structures was established first.Then the seismic response of underground structures in the random field was simulated,and the soil mechanical parameters and earthquake intensity influencing the seismic response of surrounding soils and underground structures were studied.Numerical results presented that the randomness of soil parameters has no obvious effect on the seismic performance of rectangular underground structures.

2.1.Description of the analysis method

The basic flow of random field-underground structure seismic response analysis is shown in Fig.1 and described as follows.

1.A geometric model of surrounding soils and underground structures is established based on FEM software (e.g.ABAQUS),and the model will have meshed.The detailed information of nodes and elements about the geometric model will be recorded,herein,the quantity of the soil elements is recorded as n.

2.A random field generation program is compiled based on MATLAB software with the Monte Carlo method.After that,n groups of soil parameters will be generated.These parameters include elastic modulus,cohesion,internal frictional angle,and obey normal distribution.During the generation of soil parameters,their mean and square deviation will be determined.Then the generated n groups of soil parameters will be assigned to the surrounding soils.

3.The seismic analysis model of surrounding soils and underground structures will be established by considering the reasonable boundary conditions,earthquake input and dynamic interaction between soils and structures.Afterwards,the seismic response of underground structures in the random field will be simulated.

4.Finally,the seismic performance of underground structures will be explored based on the numerical results.

Fig.4.Acceleration time history and frequency spectrum of ground motion.

Fig.5.Equivalent plastic strain nephogram of surrounding soils when PGA ＝ 0.3 g (Unit: %).

2.2.Information about the target structure

A typical single-story double-span subway station and a single stratum were selected as a target case in this study.The subway station has a buried depth of 4.80 m,which cross-sectional dimensions are shown in Fig.2.The outer dimension of this station is 17.00 m in width and 7.17 m in height.The thicknesses of the ceiling and bottom slabs are 0.80 m and 0.85 m,respectively.The sidewalls have a thickness of 0.70m.The cross-section of the central columns is 0.40 m×1.00 m,and the net height and clear spacing of the columns are 3.82 m and 2.5 m,respectively.There are bedrock 50 m under the ground surface.

A 2D FEM model of the station and the surrounding soil was built and shown in Fig.3.In order to eliminate the influence of the boundary effect on the seismic response of underground structures,the soil on each side of the structure should be at least 5 times larger than its width(Liu and Li,2005),therefore,the horizontal size of the surrounding soil was selected to be 120 m.Previous studies on the seismic response of underground structures(Du et al.,2016;Ma et al.,2022)present that,the deformation of surrounding soils is mainly nearby the structures.Then the randomness of surrounding soils around the structure is considered to improve the numerical efficiency during simulations.The cross-sectional dimension of the random field was selected as 51 m × 19.14 m,as shown in Fig.3.Namely,the horizontal dimension of the random field is three times the structure width.The vertical dimension of the random field is the sum of the buried depth and twice the size of the structure height.Another part of the surrounding soil is considered as the homogeneous field.Furthermore,quadrilateral solid elements were used to discretize the surrounding soil and underground structure.According to the study of Hatzigeorgiou and Beskos (2010),N (6–12) finite elements per wavelength (λW) are adequate to describe elastic wave propagation in solid media accurately.The maximum vertical size of the soil element in Fig.3 is 1.0 m,i.e.,LE ¼ 1.0 m,and the shear wave velocity (vs) of the numerical field is about 170 m/s.Therefore,the maximum frequency (fmax) of earthquakes which can be accurately transmitted by the FEM models could be calculated as:

Thus,frequencies up to 14.2 Hz–28.3 Hz can be accurately transmitted.

Mohr-Coulomb model was used to simulate the strength and deformation behavior of the surrounding soil.During the generation of the parameters of the random field,the coefficient of variation δ1of parameters is defined as:

Fig.6.Lateral deformation of central columns when PGA ＝ 0.2 g.

where X and x are the mean and square deviation of soil parameters.The coefficient of variation of the parameters is shown in Table 1.The density of the soil is determined as 2000 kg·m-3,and Poisson"s ratio is 0.3.The linear elastic model was selected to simulate the deformation behavior of concrete,which material parameters are the density of 2500 kg·m-3,the elastic modulus of 24 GPa and Poisson"s ratio of 0.15.Additional viscous damping was involved separately assuming the Rayleigh damping.

Table 1 Mean and coefficient of variation of soil mechanical parameters.

where [K] and [M] are stiffness and mass matrixes.ζ is the damping ratio of the fundamental mode,taken as 5% in this study.ω1and ω2are the first and second natural vibration frequencies of the system.The calculated α and β are 0.0508 and 0.0492,respectively.Moreover,hard contact was used to describe the normal interaction between the surrounding soil and underground structure.In other words,the elements between surrounding soils and underground structures can be separated but not intruded into each other.The Coulomb frictional law was used to describe the tangential interaction between the surrounding soil and underground structure,in which the friction coefficient is set as 0.4(Huo et al.,2005;Zhuang et al.,2020).

2.3.Boundary conditions and earthquake input

According to the study by Li et al.(2018),the tied degree of the freedom boundary is selected to avoid the possible wave reflection issue with the truncated boundaries,because there is underlying bedrock under the soil.In the freedom boundary,the degrees of the freedom of the lateral boundary nodes of the field are coupled.The horizontal ground motion recorded by the Kobe Marine Observatory in the Hanshin earthquake was selected as the input ground motion.The acceleration time history and Frequency spectrum of the ground motion is shown in Fig.4.The ground motion was applied in the horizontal direction,and the movement of the soil stratum in the vertical direction was restricted.During the numerical simulations,the PGA of the earthquake was adjusted to 0.1 g,0.2 g and 0.3 g,in order to explore the influence of earthquake intensity on the seismic response of underground structures.

It is proved that the seismic response of underground structures is constrained by the deformation of surrounding soils.Therefore,the deformation mode of surrounding soils can be used to reflect the deformation behavior of underground structures.Then 100 random fields were generated to simulate the seismic response of the underground structure.And during the simulations,the PGA of the ground motion was adjusted to 0.3 g.

Fig.5 shows the equivalent plastic strain nephogram of the surrounding soil,which was randomly selected for the numerical results.It presents that the earthquake-induced plastic deformation of surrounding soils is around the structure,this is the same as the previous studies.The plastic deformation modes of the surrounding soils are almost the same.The largest plastic deformation of the surrounding soil is mainly on the upper and lateral sides of the structure.The plastic deformation of the soil on the left of the structure extends from the wall-slab junction to the ground surface obliquely.The plastic deformation of the soil on the right of the structure extends along the outer side of the sidewalls.Moreover,slight plastic deformation of the surrounding soil also occurs under the structure.However,the distribution of the plastic deformation area of the soil under and upper the structure is relatively random,and the value of the deformation is smaller.In this sense,the deformation mode of the surrounding soil,especially the plastic deformation,is not influenced significantly by the randomness of surrounding soils when simulating the seismic response of underground structures.

The lateral deformation of central columns and inter-layer deformation of structures were selected as indicators to evaluate the seismic response of underground structures.Then the randomness of soil mechanical parameters and earthquake intensity influencing the seismic response of underground structures were studied.To discuss the influence of soil parameters,the randomness of a single parameter and multiparameter was considered respectively.When discussing the influence of earthquake intensity,the randomness of all the parameters was considered.Note that,the fields considering the randomness of internal friction angle,cohesion,elastic modulus and all parameters are referred to as F1,F2,F3and F4for short in the following.Moreover,100 random fields were generated when studying each parameter influencing the seismic response of the underground structure.

4.1.Influence of soil mechanical parameters

The seismic response of the underground structure under the earthquake with the PGA of 0.2 g was simulated.The lateral deformation of the central columns and inter-layer deformation of the structure obtained from the numerical results are shown in Fig.6 and Fig.7,respectively.As presented,the maximum and minimum lateral deformations of the central columns are 8.24 mm and 7.03 mm,respectively.The maximum and minimum inter-layer deformation of the structure is 5.81 mm and 6.84 mm,respectively.In other words,the central column is the key vertical supporting component for seismic resistance (Huo et al.,2005;Wu,2007;Ma et al.,2019,2022),because the earthquake-induced deformation of columns is larger than that of the inter-layer deformation of the structure.Furthermore,the mean lateral deformation of the central columns and inter-layer deformation of the structure is about 8.04 mm and 6.66 mm,respectively.Namely,the deformation of the underground structure is relatively concentrated in Cases F1 and F2.In this sense,the seismic response of the underground structure is not influenced significantly by the randomness of the internal friction angle and cohesion.Therefore,the variability of the internal friction angle and cohesion and the experimental deviation has no obvious effect on the earthquake-induced deformation of underground structures.

Fig.7.Inter-layer deformation of structures when PGA ＝ 0.2 g.

Fig.8.Lateral deformation of central columns when under different earthquake intensities.

Figs.6 and 7 present that,the lateral deformation of the central columns and inter-layer deformation of the structure are relatively divergent,but still normally distributed in F3.The differences between the maximum and minimum deformation of the central columns and interlayer are about 0.3 mm.However,the differences are not obvious compared with the value of the deformation.The lateral deformation of the central columns and inter-layer deformation of the structure in F4 are more divergent than those in F3,whereas,the deformations are also normally distributed.The differences between the maximum and minimum deformation of the central columns and inter-layer are more than 1.0 mm.And the differences are more than 15% of the value of the structural deformation.However,the quantity of numerical cases with the lateral deformation of the central columns between 8.04 mm and 8.17 mm is about 66%of the total cases.The quantity of numerical cases with the inter-layer deformation between 6.57 mm and 6.79 mm is about 94 % of the total cases.Herein,define the coefficient of variation δ2of structural deformation as:

where Δ and σΔare the mean and square deviation of structural deformation.The coefficient of variation of structural deformation is shown in Figs.6 and 7.As presented,the coefficient of variation of structural deformation is less than 1%.In this sense,the structural deformation is not influenced significantly by the randomness of soil mechanical parameters.

4.2.Influence of earthquake intensity

The seismic response of the underground structure in the random field whenunder the earthquake withPGAof 0.1g,0.2 g and0.3g was simulated.The lateral deformation of the central columns and inter-layer deformation of the structure obtained from the numerical results are shown in Fig.8 and Fig.9,respectively.As presented,the structural deformation is normally distributed.There is no significant difference between structural deformations when under the same earthquake intensity.When the PGA of earthquakes are 0.1 g,0.2 g and 0.3 g,the lateral deformation of the central columnsisgenerallybetween3.93mmand4.05mm(about 85%ofthetotal numerical cases),7.91 mm–8.17 mm (about 66% of the total numerical cases)and 10.90 mm–11.15 mm(about 90%of the total numerical cases),respectively.The inter-layer deformation of the structure is generally between 3.28 mmand 3.38 mm(about 91%of the total numerical cases),6.57 mm–6.79 mm(about 92%of the total numerical cases)and 9.11 mm–9.31 mm(about 79%of the total numerical cases),respectively.Therefore,the difference in the structural deformation is not significant under each earthquake intensity.Furthermore,the coefficient of variation of the structural deformation is also less than 1%.

Fig.10 shows the relationship between the structural deformation and PGA of the ground motion.It presents that the lateral deformation of the central columns and inter-layer deformation of the structure almost satisfy the nonlinear increase trend with the increase of PGA.Moreover,the difference in the structural deformation is not influenced significantly by the randomness of surrounding soils.In this sense,the structural deformation obtained by homogeneous fields can be used to evaluate the seismic response of underground structures within the variation range of soil parameters.

Fig.9.Inter-layer deformation of structures when under different earthquake intensities.

Fig.10.Relationship between structural deformation and PGA.

This study presents a numerical study on the seismic response of underground structures influenced by the spatial variability of the field.Firstly,the random field model was generated to describe the spatial variability of surrounding soils based on the random field theory and Monte Carlo method.Then the 2D FEM model of random field and underground structures was established.The seismic response of underground structures in the random field was simulated.Finally,the soil mechanical parameters and earthquake intensity influencing the seismic response of surrounding soils and underground structures were discussed.The main achievements of this study could be summarized as follows.

(1) The randomness of soil parameters does not change the plastic deformation mode of the surrounding soil significantly.Meanwhile,the plastic deformation of the surrounding soil is mainly on the upper and lateral sides of the structure and gradually extends to the ground surface.

(2) The central column is the key vertical supporting component of underground structures for seismic resistance.The coefficient of variation of the structural deformation is much smaller than that of mechanical parameters when under different earthquake intensities.Therefore,the deformation of underground structures is not influenced significantly by the randomness of soil parameters.

Declaration of competing interest

No conflict of interest exits in the submission of this manuscript,and the manuscript is approved by all authors for publication.I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously,and is not under consideration for publication elsewhere,in whole or in part.

Acknowledgements

This paper was supported by the Beijing Natural Science Foundation(8212007)and the Pyramid Talent Training Project of Beijing University of Civil Engineering and Architecture(JDYC20200311).