Author name used in this publication:  朱建军Author name used in this publication: 丁晓利, DING Xiao-liAuthor name used in this publication: 陈永奇Journal title in Traditional Chinese: 中國有色金屬學會會刊 (英文版)2000-2001 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Chen, YQ

Ding, XL

Zhu, JJ

Transactions of Nonferrous Metals Society of China

PolyU Institutional Repository

[ Article ID] 1003-6326(2001)02-0301-06Dynamic model for landsliding monitoring underrigid body assumption①ZHU Jian-jun(???)1 , DING Xiao-li(???)2 , CHEN Yong-qi(???)2(1.Department of Surveying Engineering , Central South Universi ty , Changsha 410083 , P.R.China;2.Department of Land Surveying and Geo-Informatics , Hong Kong Polytechnic University ,Kow loon , Hong Kong)[ Abstract] Based on the assumption tha t the slope bodies are rigid , the dynamic model of the landsiding(forw ard mod-el)was put fo rward.According to the dynamic model , the system equations of Kalman filter were constituted.The me-chanical status of a slope w as hence combined with the monitoring data by Kalman filter.The model uncertainties or mod-el errors could also be considered through some fictitious observa tion equations.Different from existed methods , the pre-sented me thod can make use fo r not only the statistic info rma tion contained in the data but also the info rma tion providedby the mechanical and geo logical aspect of slopes.At last a numerical example was given out to show the feasibility of themethod.[ Key words] dynamic model;Kalman filter;rigid body;landslide monito ring[ CLC number] U 416.1+4;P207　　　　　　　　　 　[ Document code] A1　INTRODUCTIONLandslide is a main cause of the natural disaster.In order to prevent the disaster caused by landslide , alot of slope monitoring were carried out , hence a lotof methods have been proposed to analyze the datacollected through slope monito ring[ 1] .But most ofthese methods w ere based on statistics , for example ,the reg ression-ty pe methods , the time series-ty pemethods w hich were based on the theory of time se-ries and the g rey theory-type methods w hich w ere de-veloped in the recent decades[ 2 ～ 4] .Only the statisticinformation is used in these methods.The mechanicaland geological aspect of slopes is not taken into con-sideration.No mat ter how the mechanical status andthe geological condit ions are , the same models w ill beused in these methods.These methods are effective attimes.But misleading results can also be obtained dueto the ignorance of the actual mechanical properties.Another kind of methods is that of determinantmodel[ 1 ,5] .This kind of methods w ere based on themechanics , but they w ork mainly in spatial domain.It is very difficult for them to be used in dynamic pro-cess(in time domain).This paper describes a method that can make useof bo th the information provided by monitoring dataas the statistic methods do and the information pro-vided by the mechanical status of the slope body .Themonitoring data and the mechanical status of the slopeare combined w ith Kalman filter.Based on the assumption that the slope bodies arerigid and the bodies keep touch w ith each other , thepaper at first establishes the dynamic model of thelandsliding (fo rw ard model).According to the dy-namic model , the sy stem equations of Kalman filterare const ituted.The mechanical status of a slope ishence combined w ith the monitoring data by Kalmanfilter.The model uncertainties o r model erro rs can al-so be considered through some fict itious observat ionequat ions.The concept is demonstrated through asimple slope sliding example.2　DYNAMICMODEL OF SLOPEThe slopes typically vary in the plane , wedge ,curve or toppling modes[ 6] .If only the plane problemis concerned w ith , the plane , wedge and curve modescan be described as a block system (Fig.1)[ 7] , thatis , the fallen portion of a slope can usually be dividedinto a number of blocks.Without loss of generality , only the system ofblocks like that showed in Fig.1 will be discussed inFig.1　Block failure systemVol.11　№.2　　　　　　　　Trans.Nonferrous Met.Soc.China　　　　　　　　Apr.2001① [ Foundation item] Project(49774209)supported by the National Natu ral Science Foundat ion of China[ Received date] 2000-05-26;[ Accepted date] 2000-10-09this paper.To simplify the problem , the blocks areconsidered as rigid bodies in the study .If the geometry and material propert ies (e.g .,coeff icient of f riction and unit w eight)of a block sys-tem are known , the motion of the system can be de-rived.From the New ton' s second law of motion , themotion of any block can be described by (Fig.2):Fig.2　Geometry and forces associatedw ith rigid block　　mia i x = N ix +R i x +P i ,(i-1)x +P i ,(i+1)x +　　　 Ti ,(i-1)x +Ti ,(i+1)xmia iz =N iz +R iz +P i ,(i-1)z +P i ,(i+1)z +　　　 Ti ,(i-1)z +T i ,(i+1)z -m ig (1)where　ax , az are accelerations of the mass centre ofthe i th sliding block , that isa i x = 2 xi t 2 , a iz = 2 z i t 2 (2)N is the normal force exerted on the surface of theblock by the stable part of the slope;R is the resis-tant force as a result of the f rict ion;N x , N z , R x ,Rz are the components of N and R in x and z direc-tions respectively;m is the mass of the block;g isthe acceleration due to g ravity;xi and z i are the dis-placements of the block;P and T are the co rrespond-ing forces f rom the adjacent blocks;and t is thetime.Eqn.(1)can also be w rit ten asM iai=A1iY i+Fi (3)whereA 1i =sinαi -cosαi sinβi , i-1 -sinβi , i-1 -cosβi , i-1 -cosβi , i+1cosαi sinαi -cosβi , i-1 cosβi , i+1 -sinβi , i-1 sinβi , i+1Y i=(N i R i P i-1 , i P i , i+1 T i-1 , iTi , i+1)TM i= m i 00 miai=(axi　azi)TFi=(0-m ig)T (4)For the entire system of blocks , a general equa-tion of Eqn.(3)can be formed:Ma=A1Y +G (5)whereY =(N1 , N 2 , ΛN iΛ, R1 , Λ, R 1 , Λ, P1 , 2 ,ΛP i , i+1 , ΛPn-1 , n , T 1 ,2 , Λ, T i , i+1 , Λ,T n-1 , n)Ta=(ax1 , az1 , Λax i , az i , Λ, axn , az , n)TG=(0 , m 1g , Λ0 , m ig , Λ, 0 , m ng)TM and A1 are formed according to M i and A1i re-spectively.Eqn.(5)describes the motion of a system ofrigid body blocks.Obviously , if only one block isconsidered , thenY = NR, G= 0mg, A1= sinα -cosαcosα sinα(6)Under the assumption of rigidity of the blocksand the constraints of the geometry , any block mustmove along the surface of discontinuities.This meansthat for any block , there must existaxi tanαi +az i =0 (7)If there are n blocks in the sy stem , there mustbe n such equations.Besides , the displacements , veloci ties and accel-erations of tw o adjacent blocks are assumed the samein directions normal to the surfaces of discontinuity.Therefore , for acceleration , there existsaxisinαi , i+1 -az icosαi , i+1 =　　ax(i+1)sinαi , i+1 -az(i+1)cosαi , i+1 (8)If there are n blocks in the sy stem , there mustexist n-1 such equations.As for the block system ,Eqns.(7)and (8)can be combined intoA2a=0 (9)where　A2 can be formed acco rding to Eqns.(7)and(8).The friction forces depend on the normal forces ,i.e.,R i = kN i +cT i = kP i +c (10)where　k is the coefficient of f riction;and c is thefo rce due to cohesion.For a system of n blocks ,there must exist 2 n -1 such equat ions.The equa-tions can be expressed asA3Y =C (11)Finally , the equations in(5), (9)and(11)canbe expressed asAX=L (12)whereA=-M A1A2 00 A3, L = -G0C, X= aYThere are 6n -2 unknowns and 6 n -2 equa-tions in Eqn.(12).Therefore the equat ions have u-nique solutions.Muliply ing Eqn.(12)by AT , onegetsATAX=AT L (13)andX=(AT A)-1AT L (14)·302· Trans.Nonferrous Met.Soc.China　　　　　　　　　Apr.2001Eqn.(14)can be easy to solve by many existedsof tw ares for survey ing adjustment.After the acceler-ations are found from the above , one can derive themotion veloci ty and displacement of the slope usingVxi = X i t =∫axidt ,Vzi = Z i t =∫az idt (15a)X i =∫axid td t , Zi =∫azid tdt (15b)In a small time step , one can get:Vxi = V xio +axi tVzi = Vzio +az itX i =X io +V xiot +t 2axi/2Z i =Z io +Vziot +t 2az i/2(16)Eqns.(12), (15)and (16)constitute the for-ward model of the slope.Based on the known geome-try and material properties , one can find the displace-ment of the slope.If Eqn.(12)is used directly , double precisionvariables are needed to use in programming since mostnumbers need at least 16 digits.In o rder to improvethe computation accuracy and make the numericalcomputation stable , we make a transformation of pa-rameters Y .LetY =Y 0+ΔYwhere　Y 0 is the value of Y when in the state of lim-iting equilibrium.Since the acceleration of any slopebody w ill be ax i = az i =0 in an equilibrium state ,from Eqns.(5)and(11), one can getA1Y0+G0=0A3Y0=C0 (17)G0 , C0 is the values of G and C in the limiting equi-librium state.Substi tuting Eqn.(17)into Eqn.(12),the equation becomes-M A1A2 00 A3aΔY =-ΔG0ΔC (18)whereΔY =Y -Y 0 , ΔC=C-C0ΔG=G -G0=(0 , Δm 1g , Λ, 0 , Δm ig , Λ)In order to keep symbols simple , we w ill still useX= aΔY , L=-ΔG0ΔCin the rest of the paper.Eqn.(18)can still be w rit tenasAX=L (19)It is obvious that ΔY describes the difference be-tween the present state and the stable state.3　KALMAN FILTERMODELThe state of a sliding slope can be described bythe following state vector ,X=(X T1 , V T , a T , ΔY T)T (20)whereX1=(x 1 , z 1 , Λ, xi , z iΛ)TV=X&1 =(v x1 , v z 1 , Λ, v xi , vz i , Λ)Ta=V&=(ax1 , az1 , Λ, axi , azi , Λ)TX1 is the vecto r of displacements , V is the vector ofvelocities.Four different state t ransit ion models can be de-rived:1)The first one is based on the rigid body mo-tion equation and on the fo rw ard analysis model.Ac-cording to the rigid body motion equation , when thestate t ransits f rom state k to state k+1 , the displace-ment and the veloci ty w ill change to:X1 , k+1=X1 , k+tVk+t 2 ak/2V1 , k+1=Vk+tak (21)In real si tuations , the external forces acting on aslope body usually will not change except when forexample the g round w ater level changes after rains org round shaking is induced by earthquakes.Therefo reit is reasonable to assume ΔG=0 and ΔYk+1=ΔY k.If there exists any change in the ex ternal forces , theycan be considered as state transi tion errors (sy stemnoises)if the changes are not very significant.Basedon this assumption and Eqn.(18), one can getΔYk+1=ΔYkak+1=M-1A1ΔYk+1+M-1ΔG　　=M-1A1ΔYk(22)Combining (21)with(22), we can get the fol-low ing state t ransition equation:Xk+1=ΥkXk=I t I t 2 I/2 00 I t I 00 0 0 M-1 A10 0 0 IX1 , kVkakΔYk(23)2)The second state transi tion model is based onthe forw ard analy sis model only.The time-dependentstate equation isX&=X&1V&a&ΔY &=Va00=VM-1A1ΔY +ΔG00=VM-1A1ΔY00=0 I 0 00 0 0 M-1A10 0 0 00 0 0 0X1VaΔY(24)whereX&= X t , X&1 = X1 t ,a&= a t , ΔY &= ΔY t , ΔG=0By discretising the above continuous-time stateequat ion using series expansions[ 8] , a discrete-timestate equation is obtained:·303·Vol.11　№.2　　　　　Dynamic model fo r landsliding monito ring under rigid body assumptionXk+1=ΥkXk=I tI 0 t 2M-1A1/20 I 0 tM -1 A10 0 I 00 0 0 IX1 , kVkakΔYk(25)3)The thi rd model is an improved form of thesecond.The model given in Eqn.(25)above can besubstituted byXk+1=ΥkXk=I t I t 2M -1A1/20 I tM-1A10 0 IX1 , kVkΔY k(26)that is , in this modelX=X1VΔYThis means that the third model is the same asthe second , but has less parameters in the state vec-tor.The model is therefore simpler.4)The last model is the one widely used in ki-netic systems.In this model , only the rigid mot ion e-quation is considered[ 9] ,Xk+1=ΥkXk=I t I t 2 I/20 I t 1 I0 0 IX1 , kVkak(27)The stochastic state t ransition equation is ob-tained if the state t ransition erro rs(the system nois-es)w are added to Eqns.(23), (25), (26)or(27):Xk+1=ΥkXk+w (28)Any of the above four models can be used as thestate equation in Kalman filter.In the following sec-tion w e will discuss which w ill be the best.In slope defo rmation monitoring , observation e-quations can be established fo r all the observations ,L1=B1X+ε1 (29)whereB1=(I　0　0)and L 1 denotes the observ ations vector , ε1 is the ob-servations error vector.Under the assumption thatthe slope bodies are rigid , one can establish some con-st raints.For example , according to the forw ard mod-el(18), we haveA2ak+1=0A3ΔY k+1=ΔC (30)These can be used as geometric const raints.Be-sides , the displacements and veloci ties of the twoblocks that are adjacent to each other must be thesame in the no rmal direction of the contact surface.Therefore , similar to Eqn.(9), the following shouldhold:A2X1 , k+1=0A2 Vk+1=0 (31)Eqns.(30)and(31)can be rew ritten asB2Xk+1=L 2 (32)whereB2=A2 0 0 00 A2 0 00 0 A2 00 0 0 A3, L2=000ΔGIn the real situations , Eqn.(32)may not becompletely satisfied due to the errors caused by theassumptions.For example , the blocks w ill alw ay smore or less defo rm and therefore are not exactlyrigid.Besides , the failure surfaces may also be knownonly approximately .One bet ter w ay to t reat the con-st raints is to consider them as observations w ith cer-tain uncertainties ,B2 Xk+1=L2+ε2 (33)where　ε2 is an error term.The above equations canbe used as observation equations.Based on Eqn.(28), we can get the followingKalman filter modelXk+1=ΥkXk+w (34)Lk+1=BXk+1+εPk+1 (35)B= B1B2, Lk+1= L1 , k+1L2,Pk+1= PL 00 Pε2, ε= ε1ε2The computation formulae of Kalman filterare[ 9]Xk+1=Υk XkQXk +1 =ΥkQ XkΥTk +Qw Xk+1=Xk+1+QXk +1BT(BQXk +1BT+　　 　Qk+1)-1(Lk+1-B Xk+1)Q Xk +1 =QXk +1 -QXk+1BT(BQXk +1 BT+　　 　Qk+1)-1B Xk+1(36)or[ 10] Xk+1=Xk+1+J(Lk-BXk+1)Q Xk +1 =(I-JB)QXk+1J=QXk+1BT(BQXk +1 BT+Qk+1)-1(37)whereQk+1=P-1k+1 , Qw =P -1wIn practical computations , tw o problems must beconsidered when the above formulae are used , i.e.,the initial values of X , and the variances of the sy s-tem errors w and the model errors ε2(o r the modeluncertainty).The initial values of X can be deter-mined by using the condition of limiting equilibrium.If a slope has already begun to slide , the displace-ments , velocities and the accelerations can be deter-mined by using initial tw o epochs of surveys.4　NUMERICAL EXAMPLESStimulated examples are given here to show thefeasibili ty of the Kalman filter model presentedabove.The block system used in the st imulated ex-·304· Trans.Nonferrous Met.Soc.China　　　　　　　　　Apr.2001amples is similar to that show n in Fig.1 and Fig.3.The parameters of the sy stem are taken as:m 1=798t , m 2=4kt , m 3=400 t , α1=60 , α2=45 , α3=30 ,β12=60 and β23 =75 , and the cohesion force c =0.The coefficient of f riction k in the state of limiting e-quilibrium is k =0.767 057 362 728 123 6.One ob-served point w as placed on each of the three blocks ,and observations of the horizontal and vertical dis-placements of the observed points w ere stimulated bycomputer.The observations w ere taken once every 5days for one year period.The observation erro rs areg iven by computer.The variance of the stimulatederrors is σ2=32 mm2=9mm2.Fig.3　Sliding block on plane　　After the observations were taken , the Kalmanfilter model(23), (25), (26)and (27)were usedrespectively to process the stimulated data.Figs.4 ,5 , 6 and 7 give the results f rom the dif ferent models.In these figures , the stimulated observations , the realdisplacements and values computed by the Kalman fil-ter are given.Table 2 lists the follow ing values fo reach of the Kalman filter models:δ∑=∑( xk -x k)2σk = δ∑n(38)where　 xk and xk are the filtered and the t rue dis-placements respectively , and n is the number ofepoches.It can be seen from Table 1 and the f igures.1)The results f rom the first three models aremuch more accurate than the original observed val-Fig.4　Displacement curves f rom model 1ues.The variances decreased by 6(32/1.2262)to 29(32/0.556 2).2)The results f rom Eqn.(25)are the same asthose from Eqn.(26).This means the two modelsare equiv alent , but the computations using Eqn.(26)are simpler.The results from both Eqn.(25) andEqn.(26)are bet ter than those from Eqn.(23).3)The results from Eqns.(23), (25)and(26)are all significantly better than those f rom Eqn.(27).This means that the Kalman f ilter based on the meFig.5　Displacement curves f rom model 2Fig.6　Displacement curves f rom model 3Fig.7　Displacement curves f rom model 4·305·Vol.11　№.2　　　　　Dynamic model fo r landsliding monito ring under rigid body assumptionTable 2　Filtering deviationsModel Parameter X1 Z1 X 2 Z2 X3 Z3Model(1) δ∑σ∑ 35.060.708 105.151.226 56.320.897 56.340.897 67.930.985 22.680.568Model(2) δ∑σ∑ 33.400.691 100.161.196 53.660.876 53.680.876 64.710.962 21.600.556Model(3) δ∑σ∑33.400.961 100.161.196 53.660.876 53.680.876 64.710.962 21.600.556Model(4) δ∑σ∑242.231.860 338.582.199 502.482.679 354.282.250 317.602.130 383.482.341chanical model of the slope is significant ly better.5　CONCLUSIONS1)Under the assumption that the slope bodiesare rigid , the dynamic model of slope sliding is de-rived.If we can get the accurate mechanical parame-ters , we can derive the dynamic process of a slope de-formation.2)According to the dynamic model , the systemequations of Kalman filter are constituted.The me-chanical status of a slope is hence combined w ith themonitoring data by Kalman filter.Based on the prin-ciple of mechanics and ex isting Kalman filter ap-proaches , four kinds of Kalman filter model are pre-sented.The results show that the Kalman filter mod-els based dynamic model is significantly bet ter.3)The uncertainty of the model or model erro rusually influence heavily on the results of analysis inmany fields.In this paper , it is take into considera-tion by the addi tional fictitious observation equations.4)As the mechanical propert ies of slopes varyconsiderably f rom one to ano ther , more research isstill required to take into account to study the specialcharacteristics of the various types of slopes.[ REFERENCES][ 1] 　CHEN Y Q.The Data P rocessing Approach for Defor-mation Observation [ M] .Beijing:Survey ing and Map-ping Publish House , 1988.1-350.[ 2] 　Chen Y Q and Tang Conrad.Application of the theory ofgrey theory sy stem in the analysis of defo rmation survey-ing s[ A] .7th International FIG Symposium on Deforma-tion Measurements [ C] .1993.303-309.[ 3] 　LIU Zu-qiang.P rediction o f landsliding by g rey theory[ J] .Site Investig ation Science and Technology , (in Chi-nese), 1991 , 5:1-5.[ 4] 　YIN Hui.An application of time series analysis and g reytheory in the defo rmation prediction at Lian Zi cliff [ A] .The 8th FIG International Symposium on DeformationMeasurements [ C] .25-28 , June , 1996 , Hong Kong ,1996 , 267-274.[ 5] 　Chrzanow ski.Combination of Geometrical Analysis withPhysical I nterpre tation for the Enhancement of Deforma-tion Modelling [ M ] .FIG XIX International Congress ,612(3):1-15.[ 6] 　Hoek E and Bray J W.Rock Slope Engineering [ M ] .London:Institution of Mining and Metallurgy , 1981.1-200.[ 7] 　SHI G H.Block System Modelling by Discontinuous De-fo rmation Analy sis [ M ] .Southampton UK and BostonUSA , Computational Mechanics Publications , 1993.1-200.[ 8] 　REN D and DING X.Dynamic defo rmation analysis ofopen pit slopes [ A] .The 8th Fig International Sympo-sium on Deformation Measurements [ C] .25-28 , June ,1996 , Hong Kong , 1996.157-163.[ 9] 　CHUI C K and CHEN G.Kalman Filtering with RealT ime Applications [ M] .Springer-Verlag , Berlin , NewYork , 1992.1-250.[ 10] 　Jia M , Ding X and Montg omery B.On reliability mea-sures for kinetic surveys [ J] .Geomatica , 1998 , 52(1):37-44.(Edited by HUANG Jin-song)·306· Trans.Nonferrous Met.Soc.China　　　　　　　　　Apr.2001

Dynamic model for landsliding monitoring under rigid body assumption

Abstract

Similar works

Full text

Available Versions

PolyU Institutional Repository