Limit Analysis of Stability of Circular Foundation Pit

Transcription

1 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, Limit Analyi of Stability of Cicula Foundation Pit CUI XINZHUANG, YAO ZHANYONG, JIN QING, WU SHIMEI Scool of Civil Engineeing Sandong Univeity Scool of Civil Engineeing, Sout Campu of Sandong Univeity, Jingi Road 73#, Jinan, 2561 Cina ttp:// Abtact: Cicula foundation pit often appea in civil engineeing. In ode to obtain te citical dept of te non-uppoted cicula foundation pit, te uppe-bound metod in platicity mecanic wa employed. Te aumed lip uface in analyi wa te otational logpial uface. Te kinematically admiible velocity field wa obtained accoding to te aociated flow ule fo Coulomb mateial, and te optimization model of te citical dept wa etablied and olved wit SQP optimization algoitm. Te vaiation of te citical dept wit te lope angle, te atio of dept to adiu of pit and te intenal fiction angle of oil wee tudied. Te ac effect of te cicula foundation pit make te citical dept lage tan te citical eigt of te plane lope; oweve, wen te atio of dept to adiu of pit appoace zeo, te uppe-bound olution of te fome appoace tat of te latte. If te atio of dept to diamete of pit i le tan 1, te ac effect may be ignoed and te foundation pit can be analyzed a te plane lope wit te metod of lice. Compaion between uppe-bound olution and te olution fom limit equilibium metod owed tat te fome i cloe to te tue olution tan te latte. Key-Wod: cicula foundation pit; lope; citical dept; limit analyi; uppe-bound metod; ac effect 1 Intoduction Limit analyi teoy i an impotant banc of platic mecanic. It wa developed fom te metal platic teoy and a aleady been extended to ock and oil mecanic now. Limit analyi wa ued to olve ome engineeing poblem, uc a lope tability and limit load[1-4]. It contain two kind of baic metod, i.e. te uppe-bound metod and te lowe-bound metod. Baed on uppe-bound teoem, te uppe-bound metod need to etabli te kinematically admiible failue mecanim and velocity field in advance. Te velocity field mut meet motion bounday condition and aociated platic flow ule. Baed on lowe-bound teoem, te lowe-bound metod need to et up tatically admiible te field wic mut atify equilibium equation, te bounday condition and not diobey te failue citeion wic i Mo-Coulomb failue citeion fo ock and oil. Limit analyi can give te definite bound of ome poblem uc a te lope citical eigt and te pile beaing capacity[1-3]. Howeve, te olution fom igid limit equilibium metod, wic i anote analyi metod ued extenively in geomecanic, i difficult to tell tat it i a uppe-bound olution o lowe-bound olution. Fo Some failue mecanim ued in limit equilibium metod, te coeponding kinematically admiible velocity field fo limit analyi can be obtained accoding to vitual wok pinciple. So te limit equilibium olution fom tee failue mecanim ae uppe-bound olution, jut a te Sama metod wic i an impotant metod olving te afety facto of lope. But fo Some failue mecanim ued fequently in limit equilibium metod, te coeponding kinematically admiible velocity field can t be et up, o te limit equilibium olution aen t uppe-bound olution, jut a te vetical lice metod and te cicula lide metod ued fo te tability analyi of lope. Fo vetical lice metod, inte-lice foce can t atify Mo-Coulomb failue citeion. Fo Coulomb mateial obeying aociated flow ule, te angle between velocity jump vecto and te tangent of te lip uface ould be equal to te intenal fiction angle of mateial, but fo te cicula igid lide mecanim, we can t et up any velocity field atifying ti. In addition, te limit equilibium olution can t be ued a te lowe-bound olution becaue te te field in igid body i not known. Te uppe-bound metod i applied moe extenively tan te lowe-bound metod becaue te etabliment of tatically admiible te field i ate difficult. Wile olving te poblem wit te uppe-bound metod, a valid failue mecanim i aumed fitly, and ten te

2 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, intenal enegy diipation ate and te wok done by extenal load ae calculated epectively and equated wit eac ote. Tu a eial of uppe-bound olution coeponding to te pecific mecanim ae obtained, and finally, te optimum uppe-bound olution can be gotten by employing optimization tecnology[5]. Donald I and Cen Z Y tudied te tabilitie of plan tain and tee-dimenion lope uing igid block tanlational failue mecanim[1-2]. Muff obtained te lateal beaing capacity of pile wit tee-dimenional defoming mecanim[3]. Howeve, limit analyi an t be ued to olve te axiymetical poblem a yet. Axiymetical poblem appea fequently in geotcnical engineeing uc a te tabilitie of cicula foundation pit and mot in-itu tet uc a CPT and SPT. Te aim of ti pape i to olve te non-uppoted citical dept of te cicula foundation pit to demontate te application of uppe-bound metod to axiymetical poblem. Te cicula foundation pit i often analyzed a te plane tain poblem wen te atio of dept to adiu of pit i lage enoug. Wen te atio of dept to adiu of pit i mall, oweve, te plane tain olution i not accuate becaue te ac effect of oil ma lagely enance te tability of te pit. Uing axiymetical plit line teoy, Верезанцев В Г obtained te appoximate active eat peue on etaining wall of cicula foundation pit [6]. Toug integating te eat peue along te dept and equating te integal zeo, Ma S C [7] obtained te limit equilibium olution of citical dept of cicula foundation pit. In ti pape, limit analyi metod will be employed to olve te citical dept of te cicula foundation pit. In analyi, oil i aumed omogeneou and iotopic, and te failue i axiymetical. 2 Poblem Fomulation 2.1 Failue Mecanim A a kind of lope, te common failue uface of cicula foundation pit i logpial uface, jut a own in figue 1. Te otating cente of failue face i on te axi of foundation pit. Te function of logpial uface i: ( ) tanφ = e (1) Weeφ i intenal fiction angle of oil; i pola adiu of point b wee =. Te ape of foundation pit can be decibed wit te atio of dept to bottom adiu D / R and lope angle α of foundation pit, a own in figue 1. Once tey ae known, te dimenion of te wole failue zone can be detemined by two independent vaiable: te caacteitic angle and te dept of foundation pit D. Fig.1 Failue mecanim of cicula foundation pit In Fig.1, L i te lengt of ab ; β i defined a plit angle tat i te angle between lip uface and pit wall at lope toe. Let A = D / and B = L /. Ten equation (2)-(4) can be deived fom Fig.1: ( ) tanφ A = in e in (2) ( ) tan φ B = co co e A cotα (3) β = φ + α π / 2 (4) Te function of te line ab i: in = (5) in Te function of te line ac i: in + tanα (co B) = (6) in + tanα co Te pola angle of te point a i: co B a = ac cot (7) in Accoding to geometical elationip, can be obtained fom te following equation: ( ) tanφ D in e co = (8) ( ) tanφ R in e in 2.2 elocity Field, and ae adial, tangential and cicumfeential velocity component in peical

3 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, coodinate ytem (,, ), epectively. Accoding to te aociated platic flow ule, = on te logpial failue uface. Hee = we aume fute tat = in te wole = defoming egion. can be detemined accoding to te aociated platic flow ule. In peical coodinate, teno- tain-ate ae witten a: ε = 1 ε = cot ε = (9) 1 γ = ( ) 2 γ = γ = Becaue γ γ = ε i pincipal tain =, ate. Accoding to te nomality of ε, ε and ε, we know ε and ε ae alo pincipal tain ate. Subequently we can know: 1 γ = ( ) = 2 (1) Fo Coulomb mateial matcing aociated flow ule, Cen W F gave te following equation[5]: T ε + ε = t (11) 2 φ Wee T = tan ( π 4 2), ε t and ε c ae te ummation of pincipal tenional tain ate and te ummation of pincipal compeive tain ate, epectively. Note tat compeive tain ate ae poitive in ti pape. Becaue ε =, we know fom equation (11) tat tee i a maximum tain ate ε max and a minimum tain ateε min in ε and ε. In ode to obtain velocity field, we tudied te following two cae. ε = ε, ε = ε min Fom Eq.(9) and Eq.(1), we obtain: cot + T = (12) Fom Eq.(1) and Eq.(12), we obtain: T cot + = (13) (1) max c Te patial diffeential equation above can be olved uing te vaiable epaation metod. Let = f ( ) g( ), and ubtitute it into Eq.(13) and obtain: f ( ) g ( ) tan = = C1 (14) f ( ) g( ) T Wee C 1 i a contant. Solving te two odinay diffeential equation in Eq.(14), f () and g ( ) can be obtained. Te final expeion of i: ( ) ( 1 1 ) C in TC = f g = C (15) Wee C i a contant. Subtituting Eq.(15) into Eq.(1) o Eq.(12) lead to C 1, tu Eq.(15) can be witten a 1 = T = C in (16) Becaue = ε > max, we can know C > ε afte ubtituting Eq.(16) into Eq.(9). At te ame π time, < < 2. So we can know >. Ti indicate tat oil ma lide down and doen t diobey te tue pyical cae. ε = ε, ε = ε min Uing te ame metod, we obtain: (2) max T = C in 1 17 Becaue = ε < min, C < can be known ε afte ubtituting Eq.(17) into Eq.(9). At te ame π time, < < 2. o we can obtain <. Ti implie tat oil ma lide up and diobey te tue pyical condition. Now we know tat te maximum and minimum pincipal tain ate ae ε and ε, epectively, and i detemined uniquely by Eq. (16). 2.3 Intenal Enegy Diipation Rate and Wok by Extenal Foce Te intenal enegy diipation ate include toe on failue uface and in platicity defoming zone. Accoding to te liteatue [5], te enegy diipation ate of unit aea on failue uface i calculated wit te following equation: E = c (18) Wee c i coeion of oil; i tangential velocity jump aco te failue uface. Liteatue [5] alo gave te expeion of te intenal enegy diipation ate of unit volume in platicity defoming zone:

4 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, E 2 ε (19) = c T t Once velocity field i known, tain ate field can be obtained accoding to geometic equation. But it difficult to obtain te analytic expeion of pincipal tain ate and ti limit te application of Eq.(19). Cui et al deived te geneal olution of te enegy diipation ate of unit volume in platicity defoming egion in tem of bulk tain ate [8]: E = c cotφ ε (2) Te enegy diipation ate of unit volume in defoming egion can be obtained fom Eq.(9) and Eq.(2): T E = Cc(1 T )cotφ cot in (21) Fo te foundation pit tudied in ti pape, te gavity of oil i te only extenal foce. Te gavity wok ate of unit volume oil i expeed: w = γ co (22) Wee γ i oil bulk denity. Te total enegy diipation ate on failue uface Ḋ 1 i obtained by integating Eq.(18) along te uface tat i gotten by otating te logpial line bc about te axe of foundation pit, a own in Fig.1. Te total intenal enegy diipation ate in defoming egion Ḋ 2 and te gavity wok ate of oil ma W can be obtained by integating Eq.(21) and Eq.(22), epectively, in te annula domain tat i obtained by otating te aea abc about te axe of foundation pit, a own in Fig.1. Te integation above ave no analytic olution, o we need to employ numeical metod to olve tem. 2.4 Matematic Model Accoding to uppe-bound teoem, tee i D + D = W 1 2 (23) Te citical dept of foundation pit can be expeed a: c D ( ) γ f c = (24) In ode to obtain te minimum uppe-bound olution of citical dept, we need to olve te minimum of function f ). Let ( N = min f ( ), wee N i a dimenionle vaiable, independent of c andγ and only elated to φ, α and D / R. Te matematic model ued to olve te minimum uppe-bound olution of citical dept of foundation pit i a follow: N = min f ( ) B (25) Te containt condition in Eq.(25) make ue tat point b i on te igt of point a a own in Fig.1. Bae on SQP optimization algoitm, we ued matlab oftwae to olve te minimum uppe-bound of citical dept of cicula foundation pit. 3 Reult and Analyie Wen D / R i equal to.1 and 1., epectively, te vaiation of N witφ andα ae owed in Fig.2. N inceae wit te incement of φ. Moeove, te bigge φ i, te moe apidly N vaie. N deceae wit te incement of α, futemoe, te malle α i, te moe apidly N vaie. N N D/R = 1 φ / ( ) (a) D/R = 1 D/R =.1 φ / ( ) Fig.3 ow vaiation of N wit (b) D/R =.1 Fig.2 aiation of N wit φ and α D / R andφ wenα = 9. N inceae wit te incement of D / R. Moeove, wen D / R i vey mall, te vaiation of N i alo mall. Ti ow tat te ac effect of oil ma lagely enance te tability of te foundation pit.

5 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, α = 9 Failue uface N D/R β φ Fig.3 aiation of N wit D/R and φ Fig.5 Rigid block tanlational failue mecanim Fig.4 ow vaiation of caacteitic angle wit φ and α. inceae wit te incement of φ and α. Once and N ae known, ote caacteitic angle and and caacteitic dimenion uc a, L and o on can all be obtained. Tu te location and ape of te lip uface can be decided. /( ) D/R = 1 a Accoding to te eult computed wit te metod popoed in ti pape, we alo know tat wen D / R. Ti implie tat te otational logpial uface degeneate to te cicula tuncated uface. At te ame time, te citical lip angle β c coeponding to Dc i te ame a te citical value of β own in Fig.5 and bot ae equal to ( π / 4 φ / 2). But ti doen t imply tat te platicity defoming egion degeneate to igid egion becaue intenal enegy diipation ate expeed by Eq. (9) in t equal to zeo. φ /( ) Fig.4 aiation of wit φ and α Wen D / R i vey mall, te cicula foundation pit can be analyzed a te plane lope. In limit analyi of plane lope, two kind of failue mecanim ae uually ued, i.e. igid block tanlational and otational mecanim, a own in fig. 5 and fig. 6, epectively. In table 1, N of vetical cicula foundation pit wit D / R =.1 and vetical plane lope fom diffeent failue mecanim ae given, epectively. Wen D / R i vey mall, it can be found tat N of cicula foundation pit appoximate tat of plan lope fom igid block tanlational mecanim, and bot of tem ae ligtly geate tan tat fom igid block otational mecanim. Table 1 N obtained fom ome failue mecanim φ α Plane lope wit igid block tanlational failue mecanim N Plane lope wit igid block otational failue mecanim Logpial uface φ Fig.6 Rigid block otational failue mecanim Cicula foundation pit of D/R=.1 wit axiymetical failue mecanim Te tability of plane lope often be analyzed wit vetical lice metod, e.g. Janbu metod. Fo plan lope, wenφ = 2, N fom Janbu metod

6 5t WSEAS Int. Conf. on ENIRONMENT, ECOSYSTEMS and DEELOPMENT, Teneife, Spain, Decembe 14-16, i equal to 6.1. It i vey imila to te uppe limit olution of cicula foundation pit woe D / R i.2. So te ac effect of te foundation pit may be ignoed and it can be egaded a plan lope and analyzed wit te lice metod wen te atio of dept to diamete of pit i le tan 1. Compaion wa made between uppe-bound olution and limit equilibium olution obtained by Ma S C [7], a own in Fig 7. It can be een tat uppe-bound olution i ligtly geate tan limit equilibium olution wen φ =, oweve, in ote cae, te fome i alway le tan te latte. Ti implie tat te uppe-bound olution i bette tan limit equilibium olution and cloe to te tue olution tan te latte geneally. N Uppe-bound Limit equilibium φ /( ) Fig.7 Compaion between uppe-bound and limit equilibium olution of citical dept 4 Concluion Auming tat oil ma i omogeneou and iotopic and te failue model of cicula foundation pit i axiymetical, kinematically admiible failue mecanim and velocity field wee etablied accoding to aociated platic flow ule. Uing tem, te tability of te cicula foundation pit wa analyzed and te limit uppe-bound olution of citical eigt wa obtained. Te following ae ome impotant concluion: (1) Citical dept inceae wit te decement of lope angle of foundation pit and te incement of te atio of dept to adiu and intenal fiction angle of oil. (2) Te ac effect of cicula foundation pit make it citical dept geate tan te citical eigt of plane lope. Howeve, te ac effect i unconpicuou wen te atio of dept to diamete of pit i le tan 1. (3) Wen te atio of dept to adiu i vey mall, te uppe-bound olution of cicula foundation pit i vey cloe to te one of plane lope obtained fom igid block tanlational failue mecanim. Moeove, on te axiymetical uface, te otational logpial failue line degeneate to taigt line, and te citical lip angle i te ame a plane lope fom igid block tanlational mecanim, oweve, te platicity defoming zone in t igid yet. (4) By compaion between uppe-bound olution and limit equilibium olution, it wa found tat te fome i cloe to te tue olution tan te latte. 5 Acknowledgement Ti wok wa uppoted by Natual Science Foundation of Sandong Povince, Cina(No. Q26F2). Refeence: [1] Donald I, Cen Z Y. Slope tability analyi by te uppe-bound appoac: fundamental and metod. Canadian Geotecnical Jounal, ol.34, No.6, 1997, pp [2] Cen Z Y, Wang X G, Habefield C, et al. A tee-dimenional lope tability analyi metod uing te uppe-bound teoem Pat : teoy and metod. Intenational Jounal of Rock Mecanic and Mining Science, ol.38, No.6, 21, pp [3] Muff J D, Hamilton J M. P-ultimate fo undained analyi of lateally loaded pile. Jounal of Geotecnical Engineeing, ol.119, No.1, 1993, pp [4] Dece A, Detouany E. Limit load in tanlational failue mecanim fo aociative non-aociative mateial. Geotecnique, ol.43, No.3, 1993, pp [5] Cen W F. Limit analyi and oil platicity. Amtedam, te Neteland: Elevie Publiing Co., 1975 [6] Верезанцев В Г. Axial-ymmetical limit equilibium poblem of looe media. Beijing: Cina Acitectue and Building Pe, 1981 [7] Ma S C, Zou Y S, Wang Y S. Citical eigt calculation of upigt non-uppoted cicula foundation pit. Natual Science Jounal of Xiangtan Univeity, ol.24, No.2, 22, pp [8] Cui X Z, Jin Q, Sang Q S, et al. Geneal olving metod of enegy diipation ate of defoming egion in uppe-bound analyi fo Coulomb mateial. Key Engineeing Mateial, ol 36-38, 26, pp