research article analysis of seismic damage of underground...

Research ArticleAnalysis of Seismic Damage of UndergroundPowerhouse Structure of Hydropower Plants Based onDynamic Contact Force Method

Yang Yang12 Juntao Chen12 and Ming Xiao12

1 State Key Laboratory of Water Resources and Hydropower Engineering Science Wuhan University Wuhan 430072 China2 Key Laboratory of Rock Mechanics in Hydraulic Structural Engineering Wuhan University Ministry of EducationWuhan 430072 China

Correspondence should be addressed to Yang Yang yangyang2012whueducn

Received 10 March 2014 Revised 4 August 2014 Accepted 16 August 2014 Published 2 September 2014

Academic Editor Longjun Dong

Copyright copy 2014 Yang Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on the characteristics of the dynamic interaction between an underground powerhouse concrete structure and itssurrounding rock in a hydropower plant an algorithm of dynamic contact force was proposed This algorithm enables thesimulation of three states of contact surface under dynamic loads namely cohesive contact sliding contact and separation Itis suitable for the numerical analysis of the dynamic response of the large and complex contact system consisting of undergroundpowerhouse concrete structure and the surrounding rock This algorithm and a 3D plastic-damage model were implemented ina dynamic computing platform SUCED to analyze the dynamic characteristics of the underground powerhouse structure ofYingxiuwan Hydropower Plant By comparing the numerical results and postearthquake investigations it was concluded thatthe amplitude and duration of seismic waves were the external factors causing seismic damage of the underground powerhousestructure and the spatial variations in structural properties were the internal factors The existence of rock mass surrounding theunderground powerhouse was vital to the seismic stability of the structure This work provides the theoretical basis for the anti-seismic design of underground powerhouse structures

1 Introduction

The southwest region is the key area for hydropower devel-opment in China during the past few decades A number oflarge-scale underground powerhouses of hydropower plantshave been built in this region It is also an earthquake-prone zone with the seismic intensity usually above VIITherefore the underground powerhouses should have suffi-cient earthquake resistance capability which is vital to ensurenormal operation of powerhouse and the safety of power-house personnel Postearthquake investigations of the under-ground powerhouses in the epicentral region of Wenchuanearthquake such as in Yingxiuwan Yuzixi and Gengdahydropower plants reveal that underground powerhouseshave in general stronger earthquake-resistant capabilitycompared to their surface counterparts Surrounding rock

of the three underground powerhouses was generally stablewhile the powerhouse concrete structure experienced obvi-ous local cracking and failure Concrete structure behaved asa weak link to affect the earthquake resistance capability ofthe underground powerhouses Thus it is very important tostudy the characteristics of seismic damage of undergroundpowerhouse concrete structure

It has been recognized that numerical method is anincreasingly popular and effective strategy to solve the prob-lem of structural dynamic response A variety of works [1ndash5] have been devoted to the study of the dynamic responseof underground powerhouse using the numerical methodEven though great achievement has been made in the fieldthe research on dynamic characteristic of powerhouse con-crete structure is still lacking From the microscopic pointof view the seismic damage and failure of underground

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 859648 13 pageshttpdxdoiorg1011552014859648

2 Shock and Vibration

powerhouse concrete structure are governed microscopiccrack generation and propagation In this way the continuummethod [6ndash8] (such as the finite element method the finitedifference method and the boundary element method) andthe discrete particle simulation method [9ndash22] (such asthe discrete element method the lattice-solid method andthe contact dynamics method) are the natural choice tosimulate the damage and failure process Among them thefinite element method is by far the most common anduseful numerical methodThe integration scheme it employsmakes it appropriate for the widest variety of geologic andstructural problems and it can handle the most sophisticatedconstitutive relationships [23ndash25]

Underground powerhouse could be regarded as a systemcomposed of powerhouse concrete structure and surround-ing rock Surrounding rock provides the external supportfor powerhouse concrete structure As the concrete structurecomes into direct contact with surrounding rock seismicwave is propagated through rock mass to the powerhouseconcrete structure Therefore the complex consisting ofpowerhouse concrete structure and surrounding rock under-goes a forced vibration The simulation and analysis of thedynamic contact surface between surrounding rock andconcrete structure is the key to the dynamic calculation ofunderground powerhouse concrete structure using the finiteelement method

At present the numerical calculation methods fordynamic contact problems are primarily the Lagrange mul-tiplier method [26] penalty method [27 28] and theirimproved versions [29 30] However these methods tendto either increase the degree of freedom of the system orinfluence the time step of integration [31 32] These methodsadversely affect the precision and speed of calculation whenapplied to the analysis of an underground powerhouseconcrete structure which involves a large number of contactelements and complex contact states Liu et al [33 34]put forward the dynamic contact force method targeted atthe dynamic response problem of the contact crack Theconvergence and stability of this algorithmwere easy tomeetmaking it suitable for large and complex contact systemsBut it cannot reflect the bond-slip properties of the contactsurface

In this paper a new dynamic contact force methodconsidering the bond-slip properties of the contact surfaceis suggested using the fundamental integration formulationof the dynamic contact force method The algorithm of themethod considers the cohesive effect of the contact surfacebetween concrete structure and the surrounding rock Itis capable of simulating the large slip phenomenon of thecontact surface under dynamic loads Based on the pro-posed algorithm a finite elementmodel considering dynamicconstitutive properties of materials is built for the dynamicnumerical analysis of the underground powerhouse structureof Yingxiuwan Hydropower Plant Then the characteristicsof seismic damage of the underground powerhouse structureunder dynamic loads are studied by means of the numericalanalysis and postearthquake investigations The results willprovide a theoretical basis for the antiseismic design ofunderground powerhouse

2 Dynamic Contact Force Method consideringthe Bond-Slip Properties of Contact Surface

The contact model of underground powerhouse concretestructure and surrounding rock is shown in Figure 1 Beforeapplication of the dynamic loading the nodes on the contactsurface between concrete and surrounding rock belong topoint-to-point contact A certain amount of cohesive forceexists between the nodes which are in cohesive contact stateDuring dynamic loading process the stress in some contactnodeswould exceed the cohesive force and enter into the stateof sliding contact or even separation When a large relativesliding occurs between these contact nodes they would comeinto contact with surfaces of the adjacent elements and thenbelong to point-to-surface contact

21 Fundamental Integration Formulation of the DynamicContact Force Method According to the dynamic contactforce method proposed by Liu et al [33 34] the problemof dynamic response of structure containing interface isdiscretized using the finite element method Then the differ-ential equations of the system could be obtained as follows

M U + C U + KU = F + R (1)

where M C and K are the mass damping and stiffnessmatrix respectively U is the displacement vector U is thevelocity vector U is the acceleration vector F is the knownvector of external forces R is the dynamic contact forcevector

The central difference method is used to solve the differ-ential equation at a time 119905The time domain integral equationof the displacement and velocity of contact nodes containingthe term of dynamic contact force could be obtained asfollows

U119905+Δ119905 = U119905+Δ119905 + Δ1199052

2

Mminus1R119905 (2)

U119905+Δ119905 =2 (U119905+Δ119905 minus U119905)

Δ119905

minus

U119905 (3)

U119905+Δ119905 = (1 minus Δ1199052

2

Mminus1K)U119905

+ (Δ119905 minus

Δ119905

2

2

Mminus1C) U119905 + Δ1199052

2

Mminus1F119905

(4)

where Δ119905 is the time stepR119905 depends on the state of motion not only at time 119905

but also at time 119905 + Δ119905 Therefore U119905+Δ119905 and U119905+Δ119905cannotbe obtained directly by using (2)sim(4) In order to obtain themotion state of the contact node at 119905 + Δ119905 the contact forceshould be solved according to the contact conditions

22 The Solution of Dynamic Contact Force under Point-to-Point Contact Condition If relative sliding does not occurbetween contact nodes or the relative sliding is small thecontact nodes are in or approximately in point-to-point

Shock and Vibration 3

Element of rock mass

Contact surface

Contact node

Element of concrete structure

Figure 1 Contact model of concrete structure and surroundingrock

i

Rii998400

Ri998400

Figure 2 Relationship of point-to-point contact

contact condition As shown in Figure 2 the node (119894) on thecontact surface of concrete and the node (1198941015840) on the contactsurface of the surrounding rock are in point-to-point contactat time 119905

Suppose the contact node pair at time 119905+Δ119905 is in cohesivecontact state then in the normal and tangential direction thecontact nodes pair should meet the nonintrusive conditionand the displacement compatibility conditionwith no relativesliding respectively as follows

[(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894)n119894]n119894= 0

(U119905+Δ119905119894

minus U119905+Δ1199051198941015840 ) minus [(U119905+Δ119905

119894minus U119905+Δ1199051198941015840 )n

119894]n119894

= (U119905119894minus U1199051198941015840) minus [(U119905

119894minus U1199051198941015840)n119894]n119894

(5)

where n119894is the unit normal vector of contact node

Equation (2) is substituted into (5) According to theprinciple that a pair of dynamic contact force is equal inmagnitude but opposite in direction that is R119905

1198941015840 = minusR119905

119894 we

have

N119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ1 (6)

T119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ2 (7)

where N119905119894 T119905119894are the normal and tangential components of

R119905119894 and Δ

1 Δ2 N119905119894 and T119905

119894satisfy the following equations

respectively

Δ1= [(U119905+Δ119905

1198941015840 minus U119905+Δ119905

119894)n119894]n119894 (8)

Δ2= U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840

minus [(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840)n119894]n119894

(9)

N119905119894= (R119905119894n119894)n119894 (10)

T119905119894= R119905119894minus (R119905119894n119894)n119894 (11)

In the above equations T119905119894and N119905

119894were obtained from

the analysis of motion of the node pair Therefore they mustsatisfy the following inequalities

(a)

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

1003817

le 120583

119904

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

+ 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

ge 0 (12)

If the value of T119905119894does not satisfy (12) then the node pair

would enter into the state of sliding contact We have

T119905119894= 120583

119889

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

(13)

(b)

radic

(T119905119894)

2

+ (N119905119894)

2

le 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

lt 0

(14)


entered into the separation state We have

T119905119894= 0 N119905

119894= 0 (15)

where 120583119904and 120583

119889are the coefficients of static friction and

kinetic friction respectively 119860 is the control area of thecontact node to be calculated 119888 is the cohesive force betweenthe contact surface Before time 119905+Δ119905 if sliding or separationhas not occurred between the node pair 119888 gt 0 otherwise119888 = 0


ni

i(i998400)

Figure 3 Relationship of point-to-surface contact

23 The Solution of Dynamic Contact Force under Point-to-Surface Contact Condition If larger relative sliding hasoccurred between contact nodes under dynamic loads thecontact nodes will be in the state of point-to-surface contactThen there are no cohesive forces between the contact nodesand surfaces As shown in Figure 3 at time 119905 one node 119894 onthe contact surface of concrete structure (or the surroundingrock) comes into contact with the surface of the surroundingrock (or concrete structure) Such a contact point on thecontact surface is denoted as 1198941015840

It is assumed that node 119894 is in the state of cohesive contactwith the corresponding contact surface at time 119905 + Δ119905 In thenormal and tangential direction the node should meet thenonintrusion condition and the displacement compatibilitycondition with no relative sliding respectively representedby (5) The displacements of the contact point 1198941015840 at time 119905 andtime 119905 + Δ119905 are respectively given by

U1199051198941015840 = sum

119895

120601

119895U119905119895 U119905+Δ119905

1198941015840 = sum

119895

120601

119895U119905+Δ119905119895

(16)

where 120601119895is the shape function 119895 is the node number of the

contact surfaceSubstituting (16) and (2) into (5) we have

Δ119905

2

2119872

119894

N119905119894minussum

119895

Δ119905

2120601

119895

2119872

119895

N119905119895= Δ3 (17)

Δ119905

2

2119872

119894

T119905119894minussum

119895

Δ119905

2120601

119895

2119872

119895

T119905119895= Δ4 (18)

where

Δ3=

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

)n119894]

]

n119894

Δ4= sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895

minus

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895)n119894]

]

n119894

(19)

In the above equations two conditions should be dis-cussed

(a) If Δ3 lt 0 node 119894 is separated from the correspond-

ing contact surfaceN119905119894T119905119894can be computed from (15)

(b) If Δ3 ge 0 node 119894 is in contact with the correspond-

ing contact surface It is assumed that (17) will resultin Δ3 ge 0 for 119898-number of nodes Thus (17) and

(18) of the119898 nodes can be represented as follows

[H]119898times119898

[N119905]1times119898

= [Δ3]

1times119898 (20)

[H]119898times119898

[T119905]1times119898

= [Δ4]

1times119898 (21)

where [H]119898times119898

is the coefficientmatrix relatingΔ119905119872and 120601

Solving (20) and (21) the contact forcesN119905 and T119905 can beobtained for the nodes in point-to-surface contact conditionIf one node has T119905 gt 120583

119904N119905 then the node enters into

the state of sliding contact For this particular node T119905can becomputed from (13) and (18) of the node should be removedfrom (21) Solving the new equation (21) the tangentialcontact force of other nodes can be known

After solving N119905 and T119905 of all the contact nodes from theabove equations the total dynamic contact force R119905 can beobtained as follows

R119905 = N119905 + T119905 (22)

Then the displacement and velocity of each contact nodefor the next time step can be computed from (2) and (3)Figure 4 presents the flow chart for the proposed method

3 Dynamic Constitutive Model ofConcrete and Rock Mass

Under cyclic loading the unloading stiffness of concrete androck mass at the later yielding stage is lower than the stiffnessat the initial linear stageTheplastic-damagemodel proposedby Lubliner et al [35] improved by Lee and Fenves [36 37]and generalized for 3D model by Omidi and Lotfi [38]could effectively simulate such a phenomenon The model issuitable for the dynamic analysis of the quasibrittle materialssuch as concrete and rock mass [3]


Figure 4 The flow chart for the dynamic contact force method

According to the basic theory of plastic-damage modelthe plastic-damage stress-strain relationship of rock mass orconcrete can be expressed as follows

120590 = (1 minus 119863)120590

120590 = E0 (120576 minus 120576

119901)

(23)

where 120590 is the stress tensor 120590 is the effective stress tensor E0

is the initial stiffness of the material 120576 is the strain tensor 120576119901is the plastic strain tensor119863 is the damage coefficient

The structural damage is the result of the microcracks ofthematerial Under cyclic loading the opening and closure ofmicrocracks may happen making the damage as a complexmechanism When the state of stress especially changes from

tensile to compressive the stiffness weakened by the damagebegins to recover In order to simulate this phenomenon thedamage coefficient can be written as follows

119863 = 1 minus (1 minus 119863

119888) (1 minus 119904119863

119905)

119863

119905= 1 minus exp (minus119889

119905120576119901)

119863


119888120576119901)

(24)

where 119863119905and 119863

119888are tensile and compressive damage coeffi-

cients respectively 119889119905and 119889

119888are the dimensionless constants

as the functions of plastic strain 119904 (0 le 119904 le 1) is the coefficientof restitution when the material shifts from tensile state tocompressive state


The yield function of the model in form of effective stressis given as follow

119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨

120590max⟩ minus 120574 ⟨minus

120590max⟩]

minus 120590119888(120576119901)

(25)

where 120572 and 120574 are the dimensionless constants and 120573 is aconstant variable For more details one can consult Omidiand Lotfi [38] 119868

1and 119869

2are the first and second invariants

of the effective stress tensor 120590max is the maximum effectiveprincipal stress

Concrete and the surrounding rock are used as frictionmaterial The nonassociated flow rule can simulate thevolume expansion properties under the compressive stateTherefore the plastic-damage model employs nonassociatedflow rule The plastic potential function follows the Drucker-Prager hyperbolic function as follows

Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)

where 120585 is the parameter which controls the potential func-tion to approach the asymptote 120572

119901is the dilatancy parameter

and 119891

1199050is the maximum uniaxial tensile strength of the

materialAn implicit cylindric anchor bar element method is

adopted for the finite element implementation of the steelbars embedded in rock or concrete structuresThe embeddedsteel bars are considered to improve the stiffness of the rockor concrete structures in this model Therefore the stiffnessof the steel bars can be superimposed onto the stiffness ofrock or concrete element during numerical simulation Thismethod is detailed in [39]

4 Seismic Analysis of UndergroundPowerhouse Structure of YingxiuwanHydropower Plant

The dynamic analysis of underground powerhouse structurewas performed with an in-house FEM numerical simulationplatform SUCED [40] which was designed for estimatingseismic damage of large underground caverns group Theplatform is based on the dynamic time-history method anduses explicit central difference method to solve the finiteelement equation

41 Postearthquake Investigations of Underground PowerhouseStructure of Yingxiuwan Hydropower Plant YingxiuwanHydropower Plant is one of the nine cascade hydropowerplants ofMinjiang RiverThis plant has an underground pow-erhouse with three generating units The seismic intensityof Yingxiuwan Hydropower Plant is up to XI according tothe distribution of seismic intensity ofWenchuan earthquakeLocated just 8 km away from the epicenter Yingxiuwan is one

of the hydropower plants closest to the epicenter A surveyperformed after the earthquake revealed some importantfindings (a) Existing cracks on the arch of the undergroundpowerhouse structure had propagated A number of cracksappeared along the river and along the axis of the powerhouseafter the earthquake (as shown in Figure 5(a)) (b) Partof sidewall lining cracked but the crack width and thelength were limited (as shown in Figure 5(b)) (c) The floorof turbine and generator layer showed closed cracks Theoverlying floor was uplifted (as shown in Figure 5(c)) (d)Theconcrete of the inside wall of generator pedestal experienceda large number of cracks (as shown in Figure 5(d))

42 Numerical Calculation Model Three-dimensional finiteelement model including surrounding rock and concretestructure of the powerhouse was established The modelconsisted of 189187 nodes and 171432 hexahedron elements(eight nodes) The powerhouse is 528m long 170m wideand 372m high The model ranges along 119909 119910 and 119911-axes are960 882 and 1477m respectively In order to guarantee theaccuracy of dynamic calculation results the maximummeshsize should be less than 75m based on the cut-off frequencyof seismic waves As the maximum mesh size of the modelis 70m the demand of FEM dynamic simulation can besatisfied The profile of the model is shown in Figure 6

43 Mechanical Parameters of Material The mechanicalparameters of the rock mass and concrete material areshown in Table 1 The 3D plastic-damage model discussed inSection 3 was used for the numerical simulation of thesurrounding rock and concrete structure

44 Boundary Conditions Free field boundary conditionwasapplied to absorb the reflection waves along the four verticalboundaries Viscoelastic artificial boundary condition wasused to absorb the incident waves from the bottom of themodel [41]

45 Dynamic Loads Wolong was the nearest strong motionstation to the epicenter to record strong ground motion fromthe Wenchuan earthquake Its epicentral distance was 19 kmThe seismic waves recorded inWolong were concluded morerepresentative than those recorded in other stations The 20 sto 80 s sections ofWolongmonitored acceleration data whichhave high intensity and amplitude were used as the seismicinput load The input acceleration time-histories used in thecalculation model are shown in Figure 7

46 Numerical Calculation Results Static analysis of thesurrounding rock excavation and concrete structure of theunderground powerhouse was performed before performinga dynamic analysisThe results provided the initial conditionsfor a dynamic analysis


Arch

(a)

Sidewall

(b)

Floor of generator layer

(c)

Inside wall of generator pedestal

(d)

Figure 5 Destruction of powerhouse concrete structure

Table 1 Mechanical parameter of rock mass and concrete

Material Deformationmodulus (GPa) Poisson ratio Cohesion

(MPa)Friction angle

(o)Tensile strength

(MPa)Rock 10 025 218 416 197Concrete (liningand innerstructure)

25 017 20 46 130

A formula as introduced in (27) was used to calculatethe damage volume of all damaged elements of undergroundpowerhouse concrete structure

119881 =

119899

sum

119894=1

V119863119894 (27)

where 119881 is the total damaged volume of the concrete struc-ture V119863

119894is the tensile or compressive damaged volume of each

single element and 119899 is the number of damaged elements

461 Analysis of Evolving Process of Seismic Damage Thetime history of the total damaged volume of powerhousestructure was plotted as shown in Figure 8 As can be seenthe total damaged volumewas 632m3 before seismic loadingSeismic damage on the powerhouse structure initiated when119905 = 26 s From 119905 = 26 s to 20 s and from 119905 = 30 s to 40 s theseismic waves entered into the two peaks The total damaged

volume increased sharply At other times the amplitudes ofthe seismic waves were small So the total damaged volumeincreased slowly When 119905 = 60 s the total damaged volumewas about 4603m3

Figures 9 and 10 show the plot of the damage coefficientwhen 119905 = 0 s and 119905 = 60 s respectively Before seismicloading a small amount of damage zones distributed inthe upper part of the lining The damaged coefficient didnot exceed 03 When the seismic loading was completed alarge amount of damage zones appeared in the lining Thedamaged areas were mainly distributed in the arch and theupper sidewall The damage coefficient at some locations wasobserved as high as 10 The concrete showed the risk ofcracking failure

The damage coefficient distribution of the inner concretestructure of the powerhouse when 119905 = 0 s and 119905 = 60 swas plotted (Figures 11 and 12) As can be seen when theseismic loading was completed a wide range of damaged


Surrounding rockLiningInner concrete structureof powerhousexy

z

Figure 6 Profile of finite element model

areas appeared at the floor of turbine and generator layer andthe inside wall of generator pedestal The damage coefficientwas close to 10 at most of the damaged area The concreteshowed the risk of cracking failure The damaged areas werealso noticed at the crane beam corbel and the column ofturbine layer However the damage coefficient was not large

462 Analysis of the Sliding and Separation of Contact SurfaceThe distribution of the sliding and separation zone of thecontact surface between concrete structure and surroundingrock when 119905 = 0 s and 119905 = 60 s was plotted as shown inFigures 13 and 14 When 119905 = 0 s a small amount of sliding orseparation zones occurred on the lining When 119905 = 60 s thesliding or separation zones of the contact surface were greatlyexpanded which mainly occurred on the arch the junctionof the sidewall with the arch and the junction of cavernsThisis consistent with the distribution of the damaged areas of thelining structure

5 Analysis of Seismic Damage Characteristicsof Underground Powerhouse Structure

51 Comparison between Numerical Calculation and Post-earthquake Investigation The numerical calculation resultsshowed that the damaged areas of underground powerhouseconcrete structure mainly occurred on the arch the uppersidewall the floor of generator and turbine layers the insidewall of generator pedestal and the junction of cavernsThis is basically consistent with the cracking failure areasnoticed in the postearthquake investigations Therefore thedynamic calculation method for underground powerhouseconcrete structure proposed in this paper is proved to bereasonable and effective The results could truly reflect thedamage characteristics of powerhouse structure and couldprovide theoretical basis for antiseismic design of suchstructures

52 Influence of Seismic Waves on Seismic Damage of theStructure The degree of damage of the underground pow-erhouse structure was closely related to the amplitude andduration time of seismic waves The larger the amplitudeand the longer the duration of seismic waves were the moreobvious the damage of the underground powerhouse struc-ture was In the numerical simulation of the undergroundpowerhouse structure of Yingxiuwan Hydropower Plant thedistribution range of the damaged areas and the damagecoefficient increased significantly at the time of the twoobvious vibration processes of seismic waves When the twoobvious vibration processes were over the amplitude of thesubsequent seismic waves was smaller However due to thecontinued input of seismicwaves the extent of damaged areasand the damage coefficient increased to a certain degreeThis shows that the amplitude and duration time of seismicwaves determine the degree of the damage of an undergroundpowerhouse structure These are indeed the external factorscausing the seismic damage of the underground powerhousestructure

53 Influence of Structural Properties on Seismic Damage ofthe Structure The structure of the underground powerhousepossesses significant spatial variation The structure abovethe generator layer is mainly the lining which is subjectedonly to the unidirectional constraint of the surroundingrock The upper structure is relatively weak Under seismicloads the damaged areas were distributed extensively Thestructure below the generator layer includes beams platescolumns and other massive concrete structures which aresubject to the constraints of surrounding rock on all sidesThe proportion of the damaged areas was relatively smallercompared with that of the upper structure Neverthelessa large number of damaged areas occurred at the beamsplates columns and the inner wall of the generator pedestalwhich have lower structural strength It is apparent that thespatial variation of underground powerhouse structure ledto the varying degrees of damage This is an internal factorcausing the seismic damage of the underground powerhousestructure

54 Influence of Surrounding Rock on Seismic Damage ofthe Structure The postearthquake investigations in manyother plants showed that the characteristic seismic damageof surface powerhouses was predominantly the horizontalshear failure These damages were serious and difficult torepair On the other hand the damage type of undergroundpowerhouse structure was mainly the surface shedding andclosed cracks The degree of damage was generally lighterThe results of the numerical analysis showed that the damageof the powerhouse structure in the areas which could sliderelative to or separate from the surrounding rock was moreserious while that in the areas having good contact withthe surrounding rock was lighter Hence the constraint ofthe surrounding rock remained influential to maintain thestability of the underground powerhouse structure underseismic loads This influence of the surrounding rock is


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)

(a) 119883-direction

0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)

(b) 119884-direction

0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)

(c) 119885-direction

Figure 7 Input acceleration time-histories of calculation model

0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)

Figure 8 Time history of the variation of the seismic damagedvolume of the powerhouse structure

the main reason for the lesser damage of an undergroundpowerhouse structure compared to its surface counterpart

Under seismic loads the surrounding rock deformedinwardThe concrete structure was subjected to compressioncaused by the deformation of the surrounding rock It wasmore pronounced in the lower part of underground pow-erhouse structure which would be subjected to constraintson all sides For the underground powerhouse structure of

00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z

Figure 9 Damage coefficient distribution of the lining when 119905 = 0 s

Yingxiuwan Hydropower Plant the floor of the generatorlayer was uplifted under the action of compression imposedby the surrounding rock Therefore the surrounding rockmay have both advantages and disadvantages for under-ground powerhouse structure However in general theadvantages outweigh the disadvantages


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z

Figure 10 Damage coefficient distribution of the lining when 119905 =60 s

00

01

02

03

04

06

07

08

09

10

X

Y

Z

Figure 11 Damage coefficient distribution of the inner concretestructure when 119905 = 60 s

6 Discussion on the Antiseismic Measures ofUnderground Powerhouse Structure

According to the seismic damage characteristics the con-straint imposed by the surrounding rock remained as animportant factor to the antiseismic stability of the under-ground structure The powerhouse structure above the gen-erator layer suffered from the problem of ldquounderconstraintrdquoresulting in a large area of damage in the upper structureunder seismic loads For the structures below the generatorlayer the problem of ldquooverconstraintrdquo made the action ofcompression caused by the surrounding rock deformationmore obvious

To ensure the antiseismic design surrounding rock isrecommended to be reinforced to improve its stability In

00

01

02

03

04

06

07

08

09

10

X

Y

Z


Sliding and separation zone

XY

Z

XY

Z

Figure 13 Distribution of sliding and separation zone when 119905 = 0 s

this way the favorable constraint of the surrounding rockon the internal structure can be increased while the unfa-vorable impact is minimized Secondly anchorage supportor other support measures are recommended to strengthenthe constraint of surrounding rock for the structure above thegenerator layer so as to promote the antiseismic capacity ofthe upper structure Finally soft-seismic isolation layer canbe added between the lower parts of the concrete structureand the surrounding rockThe soft-seismic isolation layer cannot only weaken the compression caused by the surroundingrock deformation but also reduce the seismic response ofthe structure to a certain extent The enhancement in theantiseismic capability of the structure after incorporating theabovementioned measures was evident from the numericalsimulation For instance Figures 15 and 16 show the distribu-tion of the damaged areas in the powerhouse structure after



XY

Z

XY

Z

Figure 14Distribution of sliding and separation zonewhen 119905 = 60 s

00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


adopting the abovementioned antiseismic measures Signifi-cant improvement is seen in the structural performancewhencompared to the damage in Figures 10 and 12Thedistributionrange and the damage coefficient were greatly reduced Thestability of the powerhouse concrete structure was enhancedconsiderably

7 Conclusion

Based on the dynamic contact forcemethod which considersthe bond-slip properties of the contact surface a finiteelement model considering dynamic constitutive propertiesof materials is built for the dynamic numerical analysisof the underground powerhouse structure of Yingxiuwan

00

01

02

03

04

06

07

08

09

10

X

Y

Z


Hydropower Plant The numerical analysis results were com-pared against the findings from postearthquake investiga-tionsThe following conclusions can bemade from this study

(1) The dynamic contact force method fully consid-ered the dynamic contact characteristics between theunderground powerhouse concrete structure and thesurrounding rock It could simulate three contactstates namely the cohesive contact sliding contactand separation of contact surface It could be appliedeffectively in the dynamic calculation of large com-plex contact systems

(2) The numerical analysis and the postearthquake inves-tigations revealed the evolution process and char-acteristics of the seismic damage of undergroundpowerhouse structure The amplitude and durationof seismic waves determined the degree of seismicdamage These are the external factors causing thedamage of the underground powerhouse structureThe spatial variations of the structural properties ledto the variation in the degree of damage This is aninternal factor causing the damage of the structureThe surrounding rock not only imposed favorableconstraint but also caused unfavorable compressionon the concrete structure However in general theadvantages outweigh the disadvantages

(3) In antiseismic design of the underground power-house structure reinforced support system should beadopted to improve the stiffness of the surroundingrock and reduce the deformation The upper partof the structure could be strengthened by anchoragesupport or other support measures to increase theconstraining effect of the surrounding rock Theinterstitial spaces between the lower parts of theconcrete structure and the surrounding rock arerecommended to be filled with soft-seismic isolation


layer so as to weaken the unfavorable compressiondue to surrounding rock and to reduce the seismicdamage of the structure

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was supported by the National Key Basic ResearchProgram of China (2010CB732005 2015CB057900) theMajor Program of the National Natural Science Foundationof China (91215301) the National Natural Science Founda-tion of China (51279136 51209164) and the Research Fundfor the Doctoral Program of Higher Education of China(20130141110015) These supports are greatly acknowledgedand appreciated

References

[1] Z Z Shen and H C Ren ldquoDynamic response characteristics ofunderground powerhouse caverns for Sandaowan hydropowerstationrdquo Advanced Materials Research vol 382 pp 80ndash83 2012

[2] Y Zhang M Xiao and J Chen ldquoSeismic damage analysisof underground caverns subjected to strong earthquake andassessment of post-earthquake reinforcement effectrdquo DisasterAdvances vol 3 no 4 pp 127ndash132 2010

[3] B-Y Zhao and Z-Y Ma ldquoInfluence of cavern spacing on thestability of large cavern groups in a hydraulic power stationrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 46 no 3 pp 506ndash513 2009

[4] W Q Sun Z Y Ma X Yan J Qi and X Du ldquoIntelligentidentification of underground powerhouse of pumped-storagepower plantrdquo Acta Mechanica Sinica vol 21 no 2 pp 187ndash1912005

[5] J Chen Z Zhang and M Xiao ldquoSeismic response analysis ofsurrounding rock of underground powerhouse caverns underobliquely incident seismic wavesrdquo Disaster Advances vol 5 no4 pp 1160ndash1166 2012

[6] O C Zienkiewicz R L Taylor and J Z ZhuTheFinite ElementMethod Its Basis and Fundamentals Butterworth-Heinemann2005

[7] G Gonzalez M Gerbault J Martinod et al ldquoCrack formationon top of propagating reverse faults of the Chuculay FaultSystem Northern Chile insights from field data and numericalmodellingrdquo Journal of Structural Geology vol 30 no 6 pp 791ndash808 2008

[8] G E Hilley I Mynatt and D D Pollard ldquoStructural geometryof Raplee Ridge monocline and thrust fault imaged usinginverse Boundary Element Modeling and ALSM datardquo Journalof Structural Geology vol 32 no 1 pp 45ndash58 2010

[9] D O Potyondy and P A Cundall ldquoA bonded-particle modelfor rockrdquo International Journal of Rock Mechanics and MiningSciences vol 41 no 8 pp 1329ndash1364 2004

[10] M Xia and K-P Zhou ldquoParticle simulation of the failureprocess of brittle rock under triaxial compressionrdquo InternationalJournal of Minerals Metallurgy and Materials vol 17 no 5 pp507ndash513 2010

[11] M Xia and C B Zhao ldquoSimulation of rock deformation andmechanical characteristics using clump parallel-bond modelsrdquoJournal of Central South University vol 21 no 7 pp 2885ndash28932014

[12] M Xia C B Zhao and B E Hobbs ldquoParticle simula-tion of thermally-induced rock damage with consideration oftemperature-dependent elastic modulus and strengthrdquo Com-puters and Geotechnics vol 55 pp 461ndash473 2014

[13] J S Yoon A Zang and O Stephansson ldquoSimulating frac-ture and friction of Aue granite under confined asymmetriccompressive test using clumped particle modelrdquo InternationalJournal of Rock Mechanics and Mining Sciences vol 49 pp 68ndash83 2012

[14] Z Zhao L Jing and I Neretnieks ldquoParticle mechanics modelfor the effects of shear on solute retardation coefficient in rockfracturesrdquo International Journal of Rock Mechanics and MiningSciences vol 52 pp 92ndash102 2012

[15] P A Cundall and O D L Strack ldquoA discrete numerical modelfor granular assembliesrdquo Geotechnique vol 29 no 1 pp 47ndash651979

[16] Z H Zhao ldquoGouge particle evolution in a rock fractureundergoing shear a microscopic DEM studyrdquo Rock Mechanicsand Rock Engineering vol 46 no 6 pp 1461ndash1479 2013

[17] P Mora and D Place ldquoA lattice solid model for the non-lineardynamics of earthquakesrdquo International Journal of ModernPhysics vol 6 pp 1059ndash1074 1993

[18] F Radjaı and F Dubois Discrete-Element Modeling of GranularMaterials Wiley-I STE New York NY USA 2011

[19] L-J Dong and X-B Li ldquoThree-dimensional analytical solutionof acoustic emission or microseismic source location undercube monitoring networkrdquo Transactions of Nonferrous MetalsSociety of China vol 22 no 12 pp 3087ndash3094 2012

[20] L J Dong and X B Li ldquoA microseismicacoustic emissionsource location method using arrival times of PS waves forunknown velocity systemrdquo International Journal of DistributedSensor Networks vol 2013 Article ID 307489 8 pages 2013

[21] L J Dong X B Li and G Xie ldquoAn analytical solution foracoustic emission source location for known P wave velocitysystemrdquoMathematical Problems in Engineering vol 2014 Arti-cle ID 290686 6 pages 2014

[22] X B Li and L J Dong ldquoAn efficient closed-form solutionfor acoustic emission source location in three-dimensionalstructuresrdquo AIP Advances vol 4 no 2 Article ID 027110 9pages 2014

[23] D M Potts and L Zdravkovic Finite Element Analysis inGeotechnical Engineering Thomas Telford 1999

[24] M Sharafisafa and M Nazem ldquoApplication of the distinctelement method and the extended finite element method inmodelling cracks and coalescence in brittle materialsrdquo Compu-tational Materials Science vol 91 pp 102ndash121 2014

[25] G G Gray J K Morgan and P F Sanz ldquoOverview of contin-uum and particle dynamics methods for mechanical modelingof contractional geologic structuresrdquo Journal of Structural Geol-ogy vol 59 pp 19ndash36 2014

[26] N Hu ldquoA solution method for dynamic contact problemsrdquoComputers and Structures vol 63 no 6 pp 1053ndash1063 1997

[27] D Peric and D R J Owen ldquoComputational model for 3-Dcontact problems with friction based on the penalty methodrdquoInternational Journal for Numerical Methods in Engineering vol35 pp 1289ndash1309 1992


[28] Y Kanto and G Yagawa ldquoDynamic contact buckling analysisby the penalty finite element methodrdquo International Journal forNumerical Methods in Engineering vol 29 no 4 pp 755ndash7741990

[29] J C Simo and T A Laursen ldquoAn augmented Lagrangiantreatment of contact problems involving frictionrdquo Computers ampStructures vol 42 no 1 pp 97ndash116 1992

[30] T A Laursen and V Chawla ldquoDesign of energy conservingalgorithms for frictionless dynamic contact problemsrdquo Interna-tional Journal for Numerical Methods in Engineering vol 40 no5 pp 863ndash886 1997

[31] G D Pollock and A K Noor ldquoSensitivity analysis of thecontactimpact response of composite structuresrdquo Computersand Structures vol 61 no 2 pp 251ndash269 1996

[32] T Belytschko W K Liu and B Moran Nonlinear FiniteElements for Continua and Structures John Wiley amp SonsChichester UK 2000

[33] J Liu and S K Sharan ldquoAnalysis of dynamic contact of cracksin viscoelastic mediardquo Computer Methods in Applied Mechanicsand Engineering vol 121 no 1ndash4 pp 187ndash200 1995

[34] J Liu S Liu and X Du ldquoA method for the analysis of dynamicresponse of structure containing non-smooth contactable inter-facesrdquo Acta Mechanica Sinica vol 15 no 1 pp 63ndash72 1999

[35] J Lubliner J Oliver S Oller and E Onate ldquoA plastic-damagemodel for concreterdquo International Journal of Solids and Struc-tures vol 25 no 3 pp 299ndash326 1989

[36] J Lee andG L Fenves ldquoPlastic-damagemodel for cyclic loadingof concrete structuresrdquo Journal of Engineering Mechanics vol124 no 8 pp 892ndash900 1998

[37] J Lee and G L Fenves ldquoA plastic-damage concrete modelfor earthquake analysis of damsrdquo Earthquake Engineering ampStructural Dynamics vol 27 pp 937ndash956 1998

[38] O Omidi and V Lotfi ldquoFinite element analysis of concretestructures using plastic-damage model in 3-d implementationrdquoInternational Journal of Civil Engineering vol 8 no 3 pp 187ndash203 2010

[39] M Xiao ldquo3-D elastoplastic FEM analysis of implicit cylindricanchor bar element for underground openingrdquo Chinese Journalof Geotechnical Engineering vol 14 no 5 pp 19ndash26 1992

[40] Z G Zhang M X Xiao and J T C Chen ldquoSimulation ofearthquake disaster process of large-scale underground cavernsusing three-dimensional dynamic finite element methodrdquo Chi-nese Journal of Rock Mechanics and Engineering vol 30 no 3pp 509ndash523 2011

[41] Z Zhang J Chen and M Xiao ldquoArtificial boundary settingfor dynamic time-history analysis of deep buried undergroundcaverns in earthquake di sasterrdquoDisaster Advances vol 5 no 4pp 1136ndash1142 2012

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



powerhouse concrete structure are governed microscopiccrack generation and propagation In this way the continuummethod [6ndash8] (such as the finite element method the finitedifference method and the boundary element method) andthe discrete particle simulation method [9ndash22] (such asthe discrete element method the lattice-solid method andthe contact dynamics method) are the natural choice tosimulate the damage and failure process Among them thefinite element method is by far the most common anduseful numerical methodThe integration scheme it employsmakes it appropriate for the widest variety of geologic andstructural problems and it can handle the most sophisticatedconstitutive relationships [23ndash25]

Underground powerhouse could be regarded as a systemcomposed of powerhouse concrete structure and surround-ing rock Surrounding rock provides the external supportfor powerhouse concrete structure As the concrete structurecomes into direct contact with surrounding rock seismicwave is propagated through rock mass to the powerhouseconcrete structure Therefore the complex consisting ofpowerhouse concrete structure and surrounding rock under-goes a forced vibration The simulation and analysis of thedynamic contact surface between surrounding rock andconcrete structure is the key to the dynamic calculation ofunderground powerhouse concrete structure using the finiteelement method

At present the numerical calculation methods fordynamic contact problems are primarily the Lagrange mul-tiplier method [26] penalty method [27 28] and theirimproved versions [29 30] However these methods tendto either increase the degree of freedom of the system orinfluence the time step of integration [31 32] These methodsadversely affect the precision and speed of calculation whenapplied to the analysis of an underground powerhouseconcrete structure which involves a large number of contactelements and complex contact states Liu et al [33 34]put forward the dynamic contact force method targeted atthe dynamic response problem of the contact crack Theconvergence and stability of this algorithmwere easy tomeetmaking it suitable for large and complex contact systemsBut it cannot reflect the bond-slip properties of the contactsurface

In this paper a new dynamic contact force methodconsidering the bond-slip properties of the contact surfaceis suggested using the fundamental integration formulationof the dynamic contact force method The algorithm of themethod considers the cohesive effect of the contact surfacebetween concrete structure and the surrounding rock Itis capable of simulating the large slip phenomenon of thecontact surface under dynamic loads Based on the pro-posed algorithm a finite elementmodel considering dynamicconstitutive properties of materials is built for the dynamicnumerical analysis of the underground powerhouse structureof Yingxiuwan Hydropower Plant Then the characteristicsof seismic damage of the underground powerhouse structureunder dynamic loads are studied by means of the numericalanalysis and postearthquake investigations The results willprovide a theoretical basis for the antiseismic design ofunderground powerhouse

2 Dynamic Contact Force Method consideringthe Bond-Slip Properties of Contact Surface

The contact model of underground powerhouse concretestructure and surrounding rock is shown in Figure 1 Beforeapplication of the dynamic loading the nodes on the contactsurface between concrete and surrounding rock belong topoint-to-point contact A certain amount of cohesive forceexists between the nodes which are in cohesive contact stateDuring dynamic loading process the stress in some contactnodeswould exceed the cohesive force and enter into the stateof sliding contact or even separation When a large relativesliding occurs between these contact nodes they would comeinto contact with surfaces of the adjacent elements and thenbelong to point-to-surface contact

21 Fundamental Integration Formulation of the DynamicContact Force Method According to the dynamic contactforce method proposed by Liu et al [33 34] the problemof dynamic response of structure containing interface isdiscretized using the finite element method Then the differ-ential equations of the system could be obtained as follows

M U + C U + KU = F + R (1)

where M C and K are the mass damping and stiffnessmatrix respectively U is the displacement vector U is thevelocity vector U is the acceleration vector F is the knownvector of external forces R is the dynamic contact forcevector

The central difference method is used to solve the differ-ential equation at a time 119905The time domain integral equationof the displacement and velocity of contact nodes containingthe term of dynamic contact force could be obtained asfollows

U119905+Δ119905 = U119905+Δ119905 + Δ1199052

2

Mminus1R119905 (2)

U119905+Δ119905 =2 (U119905+Δ119905 minus U119905)

Δ119905

minus

U119905 (3)

U119905+Δ119905 = (1 minus Δ1199052

2

Mminus1K)U119905

+ (Δ119905 minus

Δ119905

2

2

Mminus1C) U119905 + Δ1199052

2

Mminus1F119905

(4)

where Δ119905 is the time stepR119905 depends on the state of motion not only at time 119905

but also at time 119905 + Δ119905 Therefore U119905+Δ119905 and U119905+Δ119905cannotbe obtained directly by using (2)sim(4) In order to obtain themotion state of the contact node at 119905 + Δ119905 the contact forceshould be solved according to the contact conditions

22 The Solution of Dynamic Contact Force under Point-to-Point Contact Condition If relative sliding does not occurbetween contact nodes or the relative sliding is small thecontact nodes are in or approximately in point-to-point



Contact surface

Contact node



i

Rii998400

Ri998400




[(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894)n119894]n119894= 0

(U119905+Δ119905119894

minus U119905+Δ1199051198941015840 ) minus [(U119905+Δ119905

119894minus U119905+Δ1199051198941015840 )n

119894]n119894

= (U119905119894minus U1199051198941015840) minus [(U119905

119894minus U1199051198941015840)n119894]n119894

(5)



1198941015840 = minusR119905

119894 we

have

N119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ1 (6)

T119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ2 (7)


R119905119894 and Δ

1 Δ2 N119905119894 and T119905


respectively

Δ1= [(U119905+Δ119905

1198941015840 minus U119905+Δ119905

119894)n119894]n119894 (8)

Δ2= U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840

minus [(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840)n119894]n119894

(9)

N119905119894= (R119905119894n119894)n119894 (10)

T119905119894= R119905119894minus (R119905119894n119894)n119894 (11)




(a)

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

1003817

le 120583

119904

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

+ 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

ge 0 (12)



T119905119894= 120583

119889

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

(13)

(b)

radic

(T119905119894)

2

+ (N119905119894)

2

le 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

lt 0

(14)



T119905119894= 0 N119905

119894= 0 (15)

where 120583119904and 120583




ni

i(i998400)




U1199051198941015840 = sum

119895

120601

119895U119905119895 U119905+Δ119905

1198941015840 = sum

119895

120601

119895U119905+Δ119905119895

(16)



Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

N119905119895= Δ3 (17)

Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

T119905119895= Δ4 (18)

where

Δ3=

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

)n119894]

]

n119894

Δ4= sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895

minus

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895)n119894]

]

n119894

(19)







[H]119898times119898

[N119905]1times119898

= [Δ3]

1times119898 (20)

[H]119898times119898

[T119905]1times119898

= [Δ4]

1times119898 (21)







R119905 = N119905 + T119905 (22)







120590 = (1 minus 119863)120590

120590 = E0 (120576 minus 120576

119901)

(23)





119863 = 1 minus (1 minus 119863

119888) (1 minus 119904119863

119905)

119863


119905120576119901)

119863


119888120576119901)

(24)

where 119863119905and 119863







119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨


120590max⟩]

minus 120590119888(120576119901)

(25)


1and 119869




Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)



and 119891














Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Contact surface

Contact node



i

Rii998400

Ri998400




[(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894)n119894]n119894= 0

(U119905+Δ119905119894

minus U119905+Δ1199051198941015840 ) minus [(U119905+Δ119905

119894minus U119905+Δ1199051198941015840 )n

119894]n119894

= (U119905119894minus U1199051198941015840) minus [(U119905

119894minus U1199051198941015840)n119894]n119894

(5)



1198941015840 = minusR119905

119894 we

have

N119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ1 (6)

T119905119894=

2119872

119894119872

1198941015840

(119872

119894+119872

1198941015840) Δ1199052Δ2 (7)


R119905119894 and Δ

1 Δ2 N119905119894 and T119905


respectively

Δ1= [(U119905+Δ119905

1198941015840 minus U119905+Δ119905

119894)n119894]n119894 (8)

Δ2= U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840

minus [(U119905+Δ1199051198941015840 minus U119905+Δ119905

119894+ U119905119894minus U1199051198941015840)n119894]n119894

(9)

N119905119894= (R119905119894n119894)n119894 (10)

T119905119894= R119905119894minus (R119905119894n119894)n119894 (11)




(a)

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

1003817

le 120583

119904

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

+ 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

ge 0 (12)



T119905119894= 120583

119889

1003817

1003817

1003817

1003817

1003817

N119905119894

1003817

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

T119905119894

1003817

1003817

1003817

1003817

(13)

(b)

radic

(T119905119894)

2

+ (N119905119894)

2

le 119888119860

1003817

1003817

1003817

1003817

Δ1

1003817

1003817

1003817

1003817

lt 0

(14)



T119905119894= 0 N119905

119894= 0 (15)

where 120583119904and 120583




ni

i(i998400)




U1199051198941015840 = sum

119895

120601

119895U119905119895 U119905+Δ119905

1198941015840 = sum

119895

120601

119895U119905+Δ119905119895

(16)



Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

N119905119895= Δ3 (17)

Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

T119905119895= Δ4 (18)

where

Δ3=

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

)n119894]

]

n119894

Δ4= sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895

minus

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895)n119894]

]

n119894

(19)







[H]119898times119898

[N119905]1times119898

= [Δ3]

1times119898 (20)

[H]119898times119898

[T119905]1times119898

= [Δ4]

1times119898 (21)







R119905 = N119905 + T119905 (22)







120590 = (1 minus 119863)120590

120590 = E0 (120576 minus 120576

119901)

(23)





119863 = 1 minus (1 minus 119863

119888) (1 minus 119904119863

119905)

119863


119905120576119901)

119863


119888120576119901)

(24)

where 119863119905and 119863







119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨


120590max⟩]

minus 120590119888(120576119901)

(25)


1and 119869




Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)



and 119891














Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ni

i(i998400)




U1199051198941015840 = sum

119895

120601

119895U119905119895 U119905+Δ119905

1198941015840 = sum

119895

120601

119895U119905+Δ119905119895

(16)



Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

N119905119895= Δ3 (17)

Δ119905

2

2119872

119894


119895

Δ119905

2120601

119895

2119872

119895

T119905119895= Δ4 (18)

where

Δ3=

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

)n119894]

]

n119894

Δ4= sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895

minus

[

[

(sum

119895

120601

119895U119905+Δ119905119895

minus U119905+Δ119905119894

+ U119905119894minussum

119895

120601

119895U119905119895)n119894]

]

n119894

(19)







[H]119898times119898

[N119905]1times119898

= [Δ3]

1times119898 (20)

[H]119898times119898

[T119905]1times119898

= [Δ4]

1times119898 (21)







R119905 = N119905 + T119905 (22)







120590 = (1 minus 119863)120590

120590 = E0 (120576 minus 120576

119901)

(23)





119863 = 1 minus (1 minus 119863

119888) (1 minus 119904119863

119905)

119863


119905120576119901)

119863


119888120576119901)

(24)

where 119863119905and 119863







119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨


120590max⟩]

minus 120590119888(120576119901)

(25)


1and 119869




Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)



and 119891














Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












120590 = (1 minus 119863)120590

120590 = E0 (120576 minus 120576

119901)

(23)





119863 = 1 minus (1 minus 119863

119888) (1 minus 119904119863

119905)

119863


119905120576119901)

119863


119888120576119901)

(24)

where 119863119905and 119863







119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨


120590max⟩]

minus 120590119888(120576119901)

(25)


1and 119869




Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)



and 119891














Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119865 (120590 120576119901)

=

1

1 minus 120572

[120572119868

1+

radic

3119869

2+ 120573 (120576

119901) ⟨


120590max⟩]

minus 120590119888(120576119901)

(25)


1and 119869




Φ (120590) = radic(120585120572119901119891

1199050)

2

+ 2119869

2+ 120572

119901119868

1

(26)



and 119891














Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Arch

(a)

Sidewall

(b)


(c)


(d)




(MPa)Friction angle

(o)Tensile strength


25 017 20 46 130


119881 =

119899

sum

119894=1

V119863119894 (27)










z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











z










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)


0

2

4

0 10 20 30 40 50 60

Acce

lera

tion

(ms

2 )

minus2

minus4

Time (s)



0

100

200

300

400

500

0 10 20 30 40 50 60Time (s)

Dam

age v

olum

e (m

3)




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z




00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

X

Y

Z





00

01

02

03

04

06

07

08

09

10

X

Y

Z



XY

Z

XY

Z





XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











XY

Z

XY

Z


00

01

02

03

04

06

07

08

09

10

XY

Z

XY

Z



7 Conclusion


00

01

02

03

04

06

07

08

09

10

X

Y

Z










Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References













































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation


























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article analysis of seismic damage of underground...

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