Indan Geotechncal Journal, 41(1), 011, 48-53 Excavaton Induced Buldng Response by Lamnate Beam Method Kngshuk Dan 1 and Ramendu Bkas Sahu Key words Crackng of buldng, Dfferental settlement, Stran energy of shear, Lamnate beam, Permssble deflecton rato, Angular dstorton, Case study Abstract: Predcton of buldng damage or crackng due to ground settlement durng braced excavaton s mportant n urban areas. Reasonably good amount of work on numercal analyss and emprcal methods on the predcton of buldng damage potental have been reported n the lterature. However, theoretcal studes or mathematcal modelng have not been well addressed n the lterature. Here lamnate beam method, as avalable n the lterature for estmatng buldng response nduced by excavaton, s modfed usng stran energy concept. Buldng s consdered as smply supported beam wth load concentrated at centre of the beam. A deformaton profle of the ground surface near excavaton s generated whch s hoggng type n nature. Two equatons relatng bendng and shear stffness of a buldng to crtcal deflecton rato are derved. Deflecton of beam due to shear s calculated by usng the stran energy of shear. Conventonally deflecton due to shear s calculated consderng deflecton curve of beam where t s assumed that the beam s free to warp everywhere. Ths may not be vald for neghbourhood of plane mddle secton. But n stran energy such assumptons are not requred. The proposed method s used to estmate the response of three mult-stored buldngs adjacent to northern stretches of Kolkata Metro Constructon. Introducton Buldng damage adjacent to any knd of excavaton (Fgure 1) s a major desgn consderaton n urban areas. In spte of support system, excavaton leads to some ground movements and any buldng wthn the zone of nfluence s lkely to be affected. It s hence necessary to predct buldng damage for preventng any adverse effect due to excavaton nduced ground movement. A number of methods are used for calculatng buldng damage potental assocated wth ground movement. Most of these methods consst of estmatng crtcal dfferental settlement of a structure due to self weght (Skempton and McDonald, 1956; Polshn and Tolkar, 1957). Burland and Wroth (1975) proposed deep beam method and modeled a buldng as a deep sotropc beam to relate strans n the buldng to mposed deformatons. Boscardn and Cordng (1989) extended the deep beam model and consdered horzontal extenson strans (εh) for buldngs wth loadbearng brck walls caused by lateral ground movements due to adjacent excavaton and tunnellng. Boone (1996) presented another to evaluate buldng damage due to dfferental ground movement caused by adjacent excavaton consderng structure geometry & desgn, stran superposton and the crtcal strans of buldng materals. oss (00) extended Burland and Wroth (1975) equaton assumng buldng as a smply supported beam wth load concentrated at md pont and related lmtng deflecton rato wth bendng stran at the top and bottom of the beam. oss (003) and Fnno et. al. (005) used a complmentary vrtual work to determne the stran deflecton relatonshp of a lamnate beam n terms of bendng stran to deflecton rato (/L) and shear stran to deflecton rato (/L). In ths method deflecton due to shear stress s derved consderng general deflecton curve assumng that all cross sectons are free to warp. But from the condton of symmetry, mddle secton must reman plane whle adjacent sectons carryng a shear force P/. From contnuty of deformaton the abrupt change from plan mddle secton to warped adjacent secton s unlkely (Tmoshenko and Young, 1968). From ths consderaton neghbourhoods of the plane mddle secton cannot be free to warp and so the normal stress dstrbuton at plane mddle secton cannot be predcted by elementary beam theory. Excavaton sde Ground surface sedment profle Buldng restng on zone of nfluence Wall deflecton pattern Fg. 1 Buldng Restng on Deformed Profle Adjacent to the Excavaton 1 Research Scholar, Cvl Engneerng Department, Jadavpur Unversty, Kolkata 70003 Professor, Cvl Engneerng Department, Jadavpur Unversty, Kolkata 70003, Emal: rbsahu_1963@yahoo.co.n
Excavaton Induced Buldng Response by Lamnate Beam Method Kngshuk Dan and Ramendu Bkas Sahu 49 In ths paper, permssble deflecton rato of a buldng lyng wthn the nfluence zone of a braced excavaton s estmated usng lamnate beam method consderng ground movement profle suggested by Peck (1969) for large wall movements. Deflecton due to shear s calculated by the stran energy of shear whch does not requre the above mentoned assumptons. The calculated deflecton rato s compared wth the reported case studes for three buldngs (Som, 000) subjected to angular dstorton due to excavaton durng Kolkata Metro Constructon. General Consderatons Any structure or buldng located on the ground surface adjacent to an excavaton s tlted followng the deformed profle of the ground. But ths tlt of buldng has two components. These are rgd body rotaton and dfferental settlement. Rgd body rotaton of the structure causes no stress or stran n the buldng. So, the cracks n the buldng may develop only due to dfferental settlement. For sngle mode of deformaton (Fgure (a)) slope of the deformed profle s same as the rgd body rotaton. But f a settlement profle s such that a buldng experences multple mode of deformaton (Fgure (b)), then slope of each mode s not equal to rgd body rotaton and addtonal shearng stran may arse. The permssble deflecton rato s expressed n terms of crtcal bendng and shear stran. Ths crtcal bendng stran or shear stran vares from one materal to another. It manly depends on the materal propertes. Boone (1996) summarzed the crtcal stran that causes falure n common buldng materals. In lamnate beam method, buldng s consdered as a beam wth unt thckness. EI/GA s consdered as parameter to account for the varaton n bendng and shear stffness of structure. Here, deformaton due to bendng s proportonal to the bendng stffness EI, where I s the moment of nerta of the beam whle that due to shear s proportonal to the shear modulus tmes the area contrbutng to shear resstance GA. In the buldngs wth large area of floors or slabs provdes resstance aganst n plane deformaton or bendng deformaton and load bearng wall or column provde shear transfer from floor to wall. Dervaton of Deflecton Rato Deflecton Rato n terms of Bendng Stran Consderng buldng as a smply supported beam wth concentrated load at md secton (Fgure 3), then 3 PL maxmum deflecton (at centre) s 1 48EI P/ P L/ C L/ P P/ S = P/ P/ Fg. (a) Settlement Profle of Ground Surface wth Sngle Mode of Deformaton Sag Hog +ve -ve Shear Force Dagram PL/4 P/ S(sag) add(sag) add(hog) S(hog) Bendng Moment Dagram Fg. (b) Settlement Profle of Ground Surface wth Two Mode of Deformaton Fg. 3 Deflecton, Shear Force & Bendng Moment Dagram of a Smply Supported Beam wth Concentrated Load at Mddle
50 Indan Geotechncal Journal 41(), 011 From elementary theory of bendng for the cross secton at md span bendng stress () wll be PL h (For rectangular beam neutral axs s at md 4 I secton) 1 L b (1) L 1h where, λh = Dstance of neutral axs from bottom and equal to h/ for beam wth rectangular cross secton Now, addtonal deflecton occurs due to shear stress where there s non- unform bendng. These shear stresses are not unformly dstrbuted for a beam wth rectangular cross-secton. Slope of the deflecton curve due to shear at any cross secton s equal to shear stran at neutral axs. If s deflecton due to shear then, d max kx () dx G G But, here shear deformaton s calculated by usng the stran energy of shear as mentoned earler. The shear force at any secton of beam s P/, where, P s the concentrated load at the centre of the beam (Fgure 3). Shear stress at any element stuated at Q the dstance y from neutral surface s, where Q Ib s the statc moment at that secton. Now, h y I 4 Total stran energy n the entre beam s obtaned by ntegratng stran energy of any element, PLh U (For beam of unt thckness, b=1) 80GI Equatng the total stran energy to the work done, P/, (3) PLh (4) 40GI 3 PL 1.h E Now, t can be wrtten as 48EI LG 0.AE b and as the thckness of the beam s unt G so h = A, 0.EI 0.8EI b b (5) Gr GA h where, I= Ar and r = radus of gyraton, Av= Area contrbutng the shear resstance = A. So, total 3 deflecton rato, 1 L L L L 0.8EI b b 1h GA Lh (6) Deflecton Rato n Terms of Shear Stran From the pure bendng consderaton, deflecton 1 PL at centre. Now at the centre shear force s L 48EI = P/ or, 1 L L GA (7) L 4EI 4EI For the mult-stored buldng f s the shear stran at each storey and s the total shear stran of buldng then, for each storey 1 LGA L 4EI L ( GAv) 4 EI( / ) (8) Addtonal deflecton due to shear force PL 1.h E (9) L 48EI LG.5GA For mult-stored buldng addtonal deflecton ( ) rato for shear of each storey s GAv.5( / ) GA So, total deflecton rato due to combned bendng and shear for each storey 1 L ( GAv) ( GAv) (10) L L L 4 EI( / ).5( / ) GA Parameters Estmaton v In lamnate beam method parameters are estmated usng the procedure gven by Fnno et al (005) and oss (003). Frst, the dstance of the neutral axs (Fgure 4) from the bottom of the buldng (h) s estmated and then moment of nerta of whole lamnate beam s calculated. The moment of nerta of each floor slab about ts centrodal axs s gnored because thckness of each floor s small compared to overall heght of buldng. If the buldng s consdered as beam the equvalent shear stffness, GA n 1 H y n 0 1 y 1 h ( G A ) v v (11)
Excavaton Induced Buldng Response by Lamnate Beam Method Kngshuk Dan and Ramendu Bkas Sahu 51 A n Hoggng A n-1 GA vn y n Neutral Axs Locaton A n- A n-3 y n- h n-1 h n A B C D S AD A h A 1 A 0 GA v y h GA v1 y 1 h 1 Fg. 4 Neutral Axs Locaton for a Mult- Stored Buldng h n- Now these parameters are used to calculate the crtcal deflecton rato of a buldng. The mnmum value of /L as obtaned from the equatons (1) and () s the permssble deflecton rato and f the actual deflecton rato of the buldng s more than the permssble value, then cracks may develop n the buldng. Here ground surface deflecton profle (Fgure 5) s hoggng type n nature as gven by Peck (1969) for large wall movements of the braced excavaton. Any structure whch s stuated on ths surface s deformed followng the ground surface. The actual deflecton rato of a buldng s calculated by dvdng the maxmum devaton of the deformed profle from the straght lne jonng two extreme ponts of the buldng by ts length. L AD Fg. 5 Deflecton Quanttes for a Hoggng Type Deflecton Curve Comparson wth Case Study Deflecton rato s calculated for three buldngs of the northern stretches of Kolkata Metro Excavaton. These buldngs are 161, C. R. Avenue, 164, C.R. Avenue and 180A, C.R. Avenue. Detals of the buldngs are gven n Table 1. Cross-sectonal vews of three buldngs modeled as lamnate beam are shown n Fgure 6. Takng a secton of buldng the permssble deflecton rato for combned bendng and shear are calculated consderng the crtcal stran gven below. Crtcal bendng stran s taken as 0.067% (Burland and Wroth, 1975) and Crtcal shear stran s taken as 0.15% (Boone, 1996). Table 1 Deflecton Rato n terms of Bendng and Shear Stran Buldng 161, C.R. Avenue 164, C.R. Avenue 180A, C.R. Avenue A (m²) 0.90 0.90 0.75 h (m) 3.5 3.5 3.5 h (m) 10.50 10.50 8.75 n 6 5 5 0.5 0.5 0.5 Ibuldng (m 4 ) 308.7 19.9 160.8 Iwall (m 4 ) 1.786 1.786 1.786 (GA)wall (kn) 0.93 10 6 0.586 10 6 7.38 10 6 (GA)I (kn) 0.65 10 6 0.41 10 6 5.167 10 6 GA (kn) 3.9 10 6.064 10 6 5.796 10 6 EI/GA (m ) 1963.0 318. 154.6 /L(n terms of bendng) 6 10 3.6 10 3 1.608 10 3 /L(n terms of shear) 3.994 10 3 3.454 10 3 3.975 10 3 /L(permssble) 3.994 10 3.6 10 3 1.608 10 3 /L(observed) 3.69 10 4 1.174 10 3 6 10 5
5 Indan Geotechncal Journal 41(), 011 6.0m 6m 0.15m 5.0m 5m 0.15m 6.0m 6m 0.15m 0.5m 0.50m 0.50m 0.5m GFG.F GF G.F GFG.F cton Secton of 161,C.R.Avenue of Secton Secton of 180A,C.R.Avenue of Secton Secton of 164,C.R.Avenue of C.R. Avenue C.R. Avenue C.R. Avenue Fg. 6 Cross Sectonal ew of three Buldngs Modeled as Lamnate Beam The permssble deflecton rato s nearly 1/300 whch s the well accepted permssble value for a concrete buldng (McDonald and Skempton (1955)). Dscusson The permssble values of deflecton rato gven n Table 1 for three buldngs at Northern Stretches of Kolkata Metro Constructon are compared wth the actual deflecton rato obtaned from deformaton profle of observed data. Now, comparng the actual values wth the permssble deflecton rato gven n Table 1 t can be sad that the buldngs were safe n both bendng and shear durng excavaton. Further, t may be noted that the observed values of deflecton rato for those three buldngs are well wthn the permssble lmts. Relatonshp between Crtcal Deflecton Rato and L/H of the Buldng From the Fgure 7 (a) and 7(b) t s clear that when deflecton rato s expressed n terms of bendng stran, the lmtng value of /L s ntally drectly proportonal to L/H whch matches wth Burland s of deep beam method. Both the curves gve crtcal condton for lesser L/H. But when deflecton rato s expressed n terms of shear stran then such relatonshp s not observed. /(L b ) /(L) 14 1 10 0 0 0.5 1 1.5 L/H (a) Deflecton Rato n terms of Bendng Stran 3.5 3.5 1.5 1 0.5 8 6 4 Stran energy Conventonal Stran energy Conventonal 0 0 0.5 1 1.5 L/H (b) Deflecton Rato n terms of Shear Stran Fg. 7 Relaton between Deflecton Rato & L/H
Excavaton Induced Buldng Response by Lamnate Beam Method Kngshuk Dan and Ramendu Bkas Sahu 53 Conclusons 1. Calculaton of permssble deflecton rato from stran energy s more accurate as t does not requre any assumpton.. The assumpton of buldng as a smply supported beam s adequate for buldng of small length because t experences sngle mode of deformaton. 3. Permssble deflecton rato for three buldngs of Kolkata Metro Constructon s nearly equal to the conventonal permssble value for a concrete buldng. 4. A buldng wll be more crtcal f L/H rato s lesser. 5. The deflecton rato of the three buldngs n Kolkata Metro Constructon s wthn the permssble deflecton rato. So, proper predcton of buldng damage near any deep excavaton may be done. Symbols and Notatons s E G 1 h U Av EI GA Ground slope n each mode of deformaton Rgd body rotaton of the buldng near excavaton Young modulus of buldng component Shear modulus of buldng component Maxmum deflecton of the beam due to pure bendng Maxmum deflecton of the beam due to shear Dstance of neutral axs from the bottom of lamnate beam Stran energy of beam due to shear Area of the buldng contrbutng to shear resstance Equvalent bendng stffness when buldng modeled as lamnate beam Equvalent shear stffness when buldng modeled as lamnate beam / Percentage shear n storey by lamnate beam model y and Heght and shear force of th storey of buldng and Shear stran n th storey and total shear stran of buldng /L Permssble Deflecton rato of buldng References Boone, S.J.(1996): Ground-Movement-Related Buldng Damage, Journal of Geotechncal Engneerng, ASCE, ol. 1, No. 11, pp. 886-896. Boscardn, M.D. and Cordng, E.J. (1989): Buldng Response to Excavaton-Induced-Settlement, Journal of Geotechncal Engneerng ASCE, ol. 115 (1), pp. 1-1. Burland, J.B., and Wroth, C.P. (1975): Settlement of Buldngs and Assocated Damage, Proceedng of a conference on settlement of structures, Cambrdge pp. 611-654. Fnno, R.J., oss, F.T., Rossow, E., and Blackburn, J.T.(005): Evaluatng Damage Potental n Buldngs Affected by Excavatons Journal of Geotechncal and Geoenvronmental Engneerng, ol.131, No.10. McDonald, D.H. and Skempton, A.W. (1955): A Survey of Comparsons between Calculated and Observed Settlements of Structures on Clay, Insttuton of Cvl Engneers, London. Peck, R.B. (1969): Deep Excavaton and Tunnelng -State of the Art Report. Proc. 7 th ICSMFE, Mexco, ol.3 Som N. N. (1991): Performance study of Braced Cuts for Calcutta Metro Constructon. Proc. 9 th Asan Regonal Conf. on SMFE, Bangkok, ol., 387-394. Tmoshenko, S.P. and Young, D.H. (1968): Elements of strength of materals oss, F.T.(003): Evaluatng Damage Potental n Buldngs Affected by Excavatons MS thess, North Western Unversty, Evanston, IL. 166 p