Numerical analysis of buried pipes subjected to lateral soil movements

Transcription

1 Numercal analss of bured ppes subjected to lateral sol movements P. Casamchele, M. Mauger & E. Motta Department of Cvl and Envronmental Engneerng, Catana Unverst, Ital Abstract he response of bured ppes subjected to lateral ground movements n falng slopes s nvestgated wth the am to establsh ppe falure rsk. In ths paper, the analss s carred out usng the dscrete element method. he ppe, crossng the stable and the unstable zones of the slope, s dvded nto a number of dscrete elements. he sol-ppe nteracton s modeled b sprngs wth a non-lnear stress-stran behavour, to consder the eldng of the sol. he soluton s determned wth an ncremental procedure, ncreasng at each step the slope movement and updatng the correspondng stffness related to the sol-ppe nteracton. Dsplacement, shear force and bendng moment along the ppe are calculated for the prevson of unacceptable condtons for ppelnes and to prevent sesmc hazard n a rsk analss. ewords: ground falure, non-lnear analss, ppelne. 1 Introducton One of the most common causes of collapse of ppelnes crossng unstable slopes s the large deformaton nduced b landsldes. Some of the ppe falures occur n slopes durng a sesmc loadng where earthquake-nduced movements take place. he sesmc response of bured ppelnes depends on varous factors, such as the drecton and the amount of the ground movement, the geometrcal characterstcs and the stffness of the ppe, the sol characterstcs. Man studes have been carred out to evaluate the sol-ppe response due to sol moton. Smlar nteractons are present n a varet of stuatons such as a ppelne subjected to fault or landslde movements or a laterall loaded ple embedded n a stff sol Rajan and Mongenstern [1], Motta [2]. Analtcal soluton n form Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

2 142 Rsk Analss IV of dmensonless charts for a slope movng perpendcularl to the ppe as, as shown n fg. 1,were gven b Casamchele et al [3]. Fgure 1: Schematc representaton of a bured ppelne crossng an unstable slope. In recent ears numercal methods have been ncreasngl used to nvestgate the behavour of bured ppelnes subject to sol movement. he nteracton between bured ppelnes and the surroundng sol, was studed b Brusch et al [4] and [5], usng the fnte element method, as well as feld and laborator tests. In the most general case the ground around the ppe can be modelled, wth sol-ppe nterfaces n the three drectons fg. 2. Fgure 2: Sol-ppelne nteracton: ground modellng wth three space drectons sprngs. In ths stud the soluton has been obtaned usng a dscrete element approach wth an ncremental procedure, updatng at each step the tangent stffness of the sol-ppe consttutve law and solvng the equlbrum equatons n the drecton of the sol movement. Lateral force-dsplacement response of the bured ppe s Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

3 Rsk Analss IV 143 modelled wth load transfer functons epressng the relatonshp between the appled load on the ppe and the relatve dsplacement between sol and ppe. hs approach has been used b Mauger and Motta [6] and b Mauger et al [7] for the numercal analss of passve ples n sldng sols. 2 Numercal analss 2.1 Sesmc nduced slope movement prevson Snce the entt of stress and stran on the ppelne depends on the amount of the permanent sol dsplacement, the stud of the sol-ppelne nteracton mples the prevous evaluaton of the slope dsplacement caused b an earthquake. For the prevson of the slope permanent dsplacement the Newmark approach can be used, however snce the sldng mass usuall has a lmted cross-sectonal etenson d fg. 2, the boundar effects can be sgnfcant and the slope stablt as well as the slope movements, should be evaluated wth a three dmensonal analss. A smple epresson of the factor of safet for a three dmensonal 3D analss of an ndefnte slope can be deduced usng a smple 3D bo element model. It s possble to show that the 3D safet factor for an nfnte slope wth a wdth d s Casamchele et al [3]: F S3 D Where: [ 1 k cos k γh sn ] [ 1 k sn k cos ] c tanφ γ H cos [ 1 k sn k cos ] 1 2c 0H cos tanφ γd [ 1 k sn k cos ] d[ 1 k sn k cos ] c coheson φ angle of frcton angle of the sldng surface k sesmc coeffcent n the horzontal drecton k sesmc coeffcent n the vertcal drecton 0 lateral coeffcent of earth pressure γ unt weght of the sol H depth of the sldng surface d wdth of the sldng mass from eq. 1, assumng a ground acceleraton parallel to the slope and a cohesonless sol, one obtans for the crtcal acceleraton: a c 0 H g 1 cos tan φ sn 2 d Once the value of a c s determned, the dsplacement analss for a 3D slope Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

4 144 Rsk Analss IV can be carred out n the tme doman assumng a desgn accelerogram and a prevson of the total amount of the slope movement can be made. s n m m n s a M M 1 s N V O Dsplacement of th ppe element 1 V N 1 M P s Sol movement Sol load b Fgure 3: Mathematcal model used n the analss: a ppe dscretzaton; b Actons actng on a ppe element. 2.2 Method of soluton he dscrete element method adopted s based on a dscretzaton of the ppe nto n blocks jonted each other wth elastc sprngs and connected to the surroundng sol through non lnear sprngs. hs approach was prevousl used b Mauger and Motta [8] and b Chang [9] for slope stablt analss. Snce the sol-ppe nterface sprngs have non lnear behavour, the load transmtted to the ppe b the sol movement ncrease untl an ultmate value for the load s acheved. Referrng to fg. 3, the load transmtted to the ppe b the movng sol and the actons between the ppe elements can be evaluated, once the ppe dsplacents are known. Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

5 he followng epressons for the th ppe element can be wrtten respectvel: 1 N 3 1 V 4 1 m M 5 S n P 6 S s 7 M 8 where: N, V and M are the aal load, the shear force and the bendng moment transmtted at each face of the dscrete ppe elements; P, and M are the normal load, the shear load and the bendng moment transmtted b the sol to the dscrete ppe element;,, m, n, s and are the stffness of the load transfer curves n fg. 4; s and s are the sol movement components n and drecton;, and are the dsplacements n, and rotaton of the th element respectvel. B mposng the force equlbrum n the drecton, for each dscrete element of the ppelne, one has: 0 1 P V V S n 10 B mposng the force equlbrum n the drecton and the moment equlbrum around the pont O, for each dscrete element of the ppe, one has: 0 1 E E S s V V M M M 13 { } m m 14 where s the wdth of the element n the drecton WI Press, ISBN Rsk Analss IV, C. A. Brebba Edtor Rsk Analss IV 145

6 146 Rsk Analss IV In a matr form we have: { } { c} 15 where [] s the stffness matr, {} s the vector of the unknown dscrete element dsplacements and {c} s the vector of the known terms. 2.3 Incremental procedure he soluton s acheved wth a step b step ncremental procedure. Frstl the sesmc nduced slope movement determned b the Newmark analss s dvded nto a number of small ncrements. At each step the ncremental ppe dsplacement s determned b solvng the smultaneous equatons n 15: the tangent stffness of the sol-ppe nterface s then updated as a functon of the sol-ppe relatve dsplacement. he ppe s assumed elastc whle for the solppe response an hperbolc load transfer functon was chosen. he normal and the shear loads transferred to the ppelne b the sol are related to the relatve sol-ppe dsplacements accordng to the followng epressons: P s 1 S P n, 1 s, LIM s S LIM where n, and s, are the ntal stffness of the sol sub-grade reacton; P LIM and LIM are the ultmate resstances of the sol n and drectons fg. 4. P P LIM LIM n, s, s - s - a b Fgure 4: Load transfer functons for the sol-ppelne nteracton; a normal load; b shear load. Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

7 Rsk Analss IV 147 Indcatons on how to evaluate the modulus of sub-grade reacton and the lmt load of bured conducts are gven b Audbert and Nman [10], rautmann et al [11] and rautmann and O Rourke [12]. A procedure proposal for the numercal analss of bured ppes subjected to lateral sol movements s reported n the flow chart n fg. 5. Start Input data ac, s, s, P LIM, LIM,, n,, s, Ns/ s, J0 0 JJ1 Ppe element dsplacement evaluaton solvng equatons 10, 12, 14,J,J Updatng sol stffness, n s NO JN ES End Fgure 5: Flow chart for response of ppes subjected to sol movements. 3 Numercal results Even f the method proposed has a general valdt and can be utlzed for ever drecton of the sol movement respect to the ppelne as, n ths paper onl the case where the ground moton s perpendcular to the ppe was consdered. Some plots for a cohesonless sol are presented. he stresses on the ppelne are determned b numerc dervaton, once the ppe dsplacement at each element s known. Fgures 6, 7 and 8 show some eamples of numercal results n terms of ppe dsplacement, shear force and bendng moment for a 1 m dameter steel ppe subjected to dfferent values of the sol movement s. he ppe thckness s 0,015 m and the etenson of the unstable zone s 240 m. he frcton angle of Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

8 148 Rsk Analss IV the sol used n the analss s 35 both for the stable and the unstable zones; the stffness and the ultmate resstant for the sol were evaluated usng the approach suggested b rautmann and O Rourke [12]. he analss shows the non lnear response of the ppe dsplacement due to eldng of the sol. 0,40 Ppe dsplacement [m] 0,35 0,30 0,25 0,20 0,15 0,10 Z Unstable Zone s 0,20 m s 0,10 m s 0,50 m Z 0,05 s 0,05 m s 0,01 m 0, Ppelne abscssa [m] Fgure 6: Ppe dsplacement n drecton for dfferent value of s. 150 s 0.01 m s 0.05 m s 0.1 m s 0.2 m s 0.5 m 100 Unstable Zone Shear force [kn] Ppelne abscssa [m] Fgure 7: Shear force on the ppe for dfferent value of s. Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

9 Rsk Analss IV s 0.01 m s 0.05 m s 0.1 m s 0.2 m s 0.5 m Bendng moment [knm] Unstable Zone Ppelne abscssa [m] Fgure 8: Bendng moment on the ppe for dfferent value of s. 4 Conclusons he behavour of bured ppes subjected to lateral sol movements has been nvestgated wth a numercal analss based on the dscrete element approach. hs s the case of a bured ppe crossng an unstable slope that can eperence large movements durng a sesmc loadng. For the prevson of the sesmc nduced slope dsplacement one can appl the Newmark method, however because the three dmensonal effects can be sgnfcant, the crtcal acceleraton should be evaluated usng a 3D approach. Indcatons on the value of the crtcal acceleraton n a three-dmensonal analss are then gven based on a smple nfnte slope model. For the sol-ppelne nteracton, non-lnear transfer loads functons have been adopted. Partcularl, an hperbolc functon has been used. Wth an ncremental procedure step b step the soluton based on equlbrum condtons has been evaluated. At each step the slope movement and the correspondng stffness related to the sol-ppe nteracton were updated. As an eample, the case of the sol movement perpendcular to the ppelne as was analzed wth dfferent values of the sol dsplacement. he results show that the mamum ppelne dsplacement occurs at the mddle of the unstable zone and t s dependent on the amount of the sol movement as well as on the eldng of the movng sol n the unstable zone. References [1] Rajan B, Morgenstern N. Ppelnes and Laterall Loaded ples n Elastoplastc Medum, ASCE Journal of Geotechncal Engneerng, 1993; 1199: Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN

10 150 Rsk Analss IV [2] Motta E. Dscusson on Ppelnes and Laterall Loaded ples n Elastoplastc Medum b Rajan B, Morgenstern N, 1993; 1199: ASCE Journal of Geotechncal Engneerng, 1995, 1211: [3] Casamchele P, Mauger M & Motta E. Non-lnear analss of solppelne nteracton n unstable slopes. III World Conference o Earthquake Engneerng, Vancouver, Canada, 2004 In press. [4] Brusch R, Glavna S, Spnazzè M, omassn D, Bonann S, Cuscunà S. Ppelnes subject to slow landslde movements: structural modellng vs feld measurement. Proceedng of the 15th Internatonal Conference on Offshore Mechancs and Arctc Engneerng, ASME, 1996, Vol. V. [5] Brusch R, Spnazzè M, omassn D, Cuscunà S, Venz S. Falure modes for ppelnes n landsldes areas. Proceedngs of the 14th Internatonal Conference on Offshore Mechancs and Arctc Engneerng, ASME, 1995, Vol. V. [6] Mauger M & Motta E. Stresses on ples used to stablze landsldes. Proc. 6 th Internatonal Smposum on Landsldes, Chrstchurch, New Zealand, 1992: [7] Mauger M, Castell F & Motta E. Analss of ples n sldng sol. 3 rd Internatonal Conference on Deep Foundaton Practce, Sngapore, 1994: [8] Mauger M, Motta E Un modello per l'anals d stabltà de pend n campo non lneare. Proc., Deformazone de terren ed nterazone terreno- struttura n condzon d eserczo, Monselce PD, October 5-6, 1988, pp. II.241-II.257. [9] Chang C.S. Dscrete element method for slope stablt analss. ASCE Journal of the Geotechncal Engneerng, 1992; 11812: [10] Audbert JME, Nman J. Sol restrant aganst horzontal moton of ppes. ASCE Journal of the Geotechncal Engneerng Dvson, 1977; 10310: [11] rautmann CH, O Rourke D, ulhaw FH. Uplft force-dsplacement response of bured ppe. ASCE Journal of Geotechncal Engneerng, 1985; 1119: [12] rautmann CH, O Rourke D. Lateral force-dsplacement response of bured ppe. ASCE Journal of Geotechncal Engneerng, 1985; 1119: Rsk Analss IV, C. A. Brebba Edtor 2004 WI Press, ISBN