ESTIMATION OF DIVERGENCES IN PRECAST CONSTRUCTIONS USING GEODETIC CONTROL NETWORKS

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1 Proceedngs, 11 th FIG Symposum on Deformaton Measurements, Santorn, Greece, ESTIMATION OF DIVERGENCES IN PRECAST CONSTRUCTIONS USING GEODETIC CONTROL NETWORKS George D. Georgopoulos & Elsavet C. Telon Natonal Techncal Unversty of Athens, Rural & Surveyng Engneerng School, Department of Topography, Laboratory of General Geodesy, 9,Heroon Polytechnou, , Athens, Greece Abstract The dvergences of the geometrc features of the concrete elements are usually detected after the constructon s erecton. These dvergences are of promnent mportance n precast concrete constructon, snce they mght prove responsble for the deteroraton of ts bearng ablty. Usng geodetc methodology, they can be relably estmated and ther values compared aganst the permtted tolerances, set from the precast constructon rules. For ths purpose, a geodetc control network s establshed and ts elements are measured usng nstruments of hgh precson. After the adustment of the observatons, the vectors of dvergences are estmated and ther statstcal sgnfcance s tested. The specal requrements n the desgn and optmzaton of such a network are analyzed n ths paper, and a case study dealng wth the detecton of the dvergences of a two floor precast constructon s presented. The fnally estmated dvergences were all statstcally sgnfcant; ther values ranged up to 52 mm and exceeded the permtted tolerances. Snce the geodetc control network can also be used for the montorng of the knematc behavor of the constructon, the possblty of usng GPS measurements for ths purpose s examned through the results of an experment. 1. Introducton Durng the constructon of a concrete buldng, dfferences (named dvergences) between the nomnal values of ts geometrc features determned n the general constructon plans, and those fnally resultng after the buldng s erecton are nevtable. In case that these dvergences exceed the permtted tolerances, set n constructon rules, deteroraton of the buldng s bearng ablty s possble. Especally n precast concrete constructons, dvergences are of promnent mportance snce, f deformatons occur after the buldng s erecton, the redstrbuton of the stran forces s not possble (Andrans, Petsas, 1983). Dvergences n precast constructon may happen durng the fabrcaton, the storage and the transportaton of the concrete elements n the place of the buldng s constructon. These dvergences manly concern the elements dmensons and qualty and can be mnmzed f specal precautons are taken. The most sgnfcant and mportant dvergences, from the pont of vew of the bearng ablty of the buldng, are those occurrng durng the consecutve stages of the precast assemblng (.e. the buldng s erecton). They consst of dvergences due to ncorrect postonng of the foundaton-precast elements, declnatons from vertcal poston and dsplacements of the precast elements. The fnal total dvergences n precast constructon can be relably estmated usng geodetc methodology. For ths purpose, a geodetc control network s establshed, ts elements are measured usng nstruments of hgh precson, and the coordnates of the network s ponts are estmated. Fnally, the dvergences n the postonng of the precast elements, as well as those of the geometrc features of the constructon are determned and ther sgnfcance s tested by approprate statstcal tests (Georgopoulos 2000).

2 2. Use of geodetc methodology for the estmaton of relable dvergences n precast contsructon The geodetc control network establshed for the detecton of dvergences n precast constructon conssts of the reference and the control ponts. The answer to the zero order desgn problem of such a network s not necessary. Snce the horzontal dvergence or dsplacement vectors of the bearng frame of the constructon are determned by the comparson of the coordnates of the network s ponts, located on the same precast element, they are relatve and not absolute. Therefore the optmum reference system s a local geodetc coordnate system. The network s geodetc datum s defned by the mnmum constrants.e. fxng the coordnates of one of the network s ponts and the azmuth of the same fxed pont to another pont of the network, makng the assumpton that the coordnates system s centered to the fxed pont. Specal care must be taken for the selecton of the fxed pont. It should be located to the most stable area of the surroundngs of the constructon and ts stablty must be checked from tme to tme. The reference ponts, located n the vcnty of the constructon, are establshed durng the stage of the buldng s settng out. Ther number and poston depends on the sze and the geometrc characterstcs of the buldng and on the stablty of the surroundng area. Snce the reference ponts can be used from the begnnng (settng out) tll the end of the constructon works, ther monumentaton should be permanent. The control ponts are stuated on the constructon. Ther locaton depends on the knd of the precast element on whch they are establshed. For nstance, n case of lnear elements, such as columns, where the most sgnfcant dvergences are the columns declnatons from vertcal poston, two or even three ponts are establshed on the symmetry axs of at least one sde of each column. It has been proved n practce, that t would be very useful, f the markng of the control ponts s done at the stage of the elements fabrcaton n the factory. Therefore the answer to the frst order desgn problem s rather mposed by the specal geometrc characterstcs of the constructon under consderaton, than the optmum locaton of the network s ponts. The second order desgn problem of such a network concerns prmarly the accuracy of the observatons,.e. the selecton of the nstruments to be used. Accordng to the predefned accuracy of the ponts coordnates (usually of a few mm), the accuracy of the observatons s determned and the nstruments meetng these demands are chosen. The desgn of the geodetc control network s optmum f t satsfes the crtera adopted to defne the qualty of the network. These crtera are the requrements n precson and relablty of the network (Agatza Balodmou, 1999). Snce the geodetc control network s establshed for the estmaton of the dvergences of a precast constructon, t s also mportant to determne the mnmum horzontal dvergence vector that can be estmated by the network. That means that f the postulated dvergences occur, they can be detected wth a specfed confdence level. Ths s called the senstvty of the network (Krakwsky, 1991, Kuang 1991). The senstvty of the geodetc control network, concernng the horzontal dvergence vector between, control ponts, belongng to the same precast element, s determned through the followng equaton (Georgopoulos 2000): mnδr = 99 σδr, z (2.1) 0. max where: : the horzontal dvergence of the precast element, as determned through the coordnates δr, δr max dfferences between, control ponts ( 2 2,, δy δr = δx +, ), and σ : the maor sem axs of the error ellpse of the dvergence vector, determned through the a pror covarance submatrx. The adustment of the observatons leads to the estmaton of the ponts coordnates and ther standard errors. After the adustment, the well known global test on the a posteror and V δr, σ 0 v

3 Baarda s data snoopng (on the observatons) are appled n order to check the precson and relablty of the network (Krakwsky 1991, Kuang 1991). If the null hypothess H 0 (the mathematc and stochastc part of the model are both correct) s accepted, the dvergence vectors of the constructon are determned, and ther statstcal sgnfcance s tested for a specfed confdence level p% (usually p = 95%). The horzontal dvergence of each precast element s determned through the coordnates dfferences of the control ponts establshed for ths reason on the element:, D, ( ) ( ) 2 2, Y (2.2) The above computed dvergence s statstcally sgnfcant f the followng nequalty holds (Krakwsky 1991, Kuang 1991): > where: : the standard devaton of the dvergence computed through the errors law of propagaton, : the correspondng value of the normal dstrbuton for confdence level 95%. σ D, z 0.95 D = X - X + Y - D, D, z σ (2.3) A case study In order to estmate the dvergences of a two floor ndustral precast buldng, havng dmensons, the above descrbed geodetc methodology was appled. The bearng frame of the constructon conssts of 15 precast concrete columns, whle ts walls are precast elements too. Durng the stage of the network s desgn ts geometry was determned. The network conssts of 42 ponts: 8 reference ponts (K 1,, K 8 ) and 34 control ponts, two on each precast column. The control ponts are located on the symmetry axs of the column s external sde, except of three of the four columns at the corners of the constructon. At these columns four control ponts were establshed, two on each external sde. The observaton accuraces determned at the stage of the network s desgn were and 10 cc 20 m m ±3mm for the length and angle measurements respectvely. The trace of the a pror covarance matrx (tr ^ and the mnmum detectable horzontal dvergence ( ) for confdence level 99% were also determned at ths stage. The nstrument used for the observatons was the Total Staton TC1600 Wld, havng an accuracy of 0.3mgon n angle measurements and n length measurements. 110 length and 108 angular measurements formed the observatonal scheme. The shape of the network s depcted n Fg.1. The adustment of the observatons was performed wth the mnmum external constrants: the coordnates of pont K 1 together wth the azmuth of sde K 1 K 5 were kept fxed (X ( = Y m, g). From the adustment of the observatons, the estmates of the coordnates of the network s ponts were determned, together wth ther covarance ^ matrx V^. The null hypothess was accepted from the global test of the network, whle no observaton was reected when Baarda s data snoopng was appled. The heghts of the control ponts were determned by trgonometrc heght levelng from the reference ponts, whose heghts were determned by double geodetc levelng. In order to detect the possble vertcal dferental dsplacements of the upper parts of the columns, ther heghts were also determned by double geodetc levelng. For the determnaton of heghts of all the mentoned above ponts, the heght of pont K 1 was kept fxed ( H K1 = m). K 1 x V ) of the coordnates, the mean standard devaton ( σ ) of the coordnates estmaton x K 1 az K 1 K 5 = 100 mn D ( ) ± 3 mm ± 3ppm

4 K3 K4 K K6 K Y K1 X 0 5m 10m K8 K7 Fg. 1. The establshed geodetc control network for the detecton of the dvergences of a two-floor precast concrete buldng. For each column the coordnates of the upper control pont were compared to those of the lower one, and the correspondng vector of horzontal dvergence was calculated from equaton (2.2). Its statstcal sgnfcance was tested for sgnfcance level 95%. All the horzontal dvergence vectors were proved to be statstcally sgnfcant. They range from (or 0.001rad ) up to 52 mm (or rad ). It must be ponted out that the permtted tolerance of dvergences, accordng to precast constructon rules (Techncal Chamber of Greece, 1991), s D ± 20mm or D 0.003rad, therefore the estmated dvergences are 11mm D a sgnfcantly greater than the permtted ones. The determned horzontal dvergence vectors n ( mm) and ( rad ) together wth ther azmuths are depcted n Table 1. Column Control Lnear Dstance of Angular Azmuth Ponts dvergence control dvergence ( g ) ( mm ) ponts ( m ) ( rad ) Table 1. Horzontal dvergence vectors of the constructon s columns. (dvergences that exceed the permtted tolerances are gven wth bold characters).

5 Statstcally sgnfcant vertcal dvergences, concernng the co-horzontalty of the columns upper parts, were also determned, the maxmum one beng of 33 mm. In Fgure 2. the horzontal dvergence vectors of the constructon s columns together wth the vertcal ones are depcted. Dvergences from the nomnal values, as gven from the constructon plans, and concernng varous geometrc features of the constructon, such as dstances between the columns symmetry axs, the sze of the buldng s parts etc., were also determned and found greater than the preset tolerances Y X 0 5 m 10 m 0 5cm SCALE OF DISTANCES SCALE OF HORIZONTAL & VERTICAL DIVERGENCES Fg.2 Horzontal and vertcal dvergences of the columns of the two floor precast constructon. 4. Use of GPS measurements for the determnaton of the horzontal movement of the buldng s roof slab The use of GPS technology for the detecton of dvergences n precast constructon s lmted due to the poston of the control ponts. It could however be used for the determnaton of the sdes of the reference control network, and thus amelorate the overall accuracy of the network. Moreover, snce the geodetc control network establshed for the detecton of the vectors of dvergences n a precast constructon, can also be used for the montorng of ts knematc behavor, f deformatons occur after ts erecton, t s nterestng to nvestgate the possblty of usng GPS measurements for the roof slab s horzontal movements. For ths purpose an experment was carred out at the roof slab of the aforementoned buldng. Three control ponts A 1, A 2, A 3 were establshed on ths slab, formng a trangle wth sdes of the range of 12m. Thus a new control network was establshed, consstng of the 8 reference ponts and the 3 new control ones. Its sdes were determned by GPS measurements, and the coordnates of the ponts were estmated (Fg 3). Three more control ponts B 1, B 2, B 3 were, afterwards, establshed n such a way that the dstances A 1 B 1, A 2 B 2, A 3 B 3 are of some cm and the sdes of the trangle B 1 B 2 B 3 have equal lengths to those of the trangle A 1 A 2 A 3. Thus a pseudo shft and rotaton of the roof s slab occurred (Fg.3). The coordnates of the ponts B 1, B 2, B 3 were determned va GPS measurements and the dstances A 1 B 1, A 2 B 2, A 3 B 3, representng the slab s movement, were calculated. It must be ponted out here that the dfferences, between the calculated dstances and the truly mposed ones, were of the range of ±1mm, t s therefore concluded that GPS could be used for such a purpose.

6 K3 K4 K5 A1 B1 K2 B3 A3 A2 B2 BUILDINGS OF ROOF SLAB K6 Y K1 X K8 0 5m 10m SIDES CONNECTING POINTS WITHOUT K7 SIDES CONNECTING POINTS WITH RECIPROCAL VISIBILITY RECIPROCAL VISIBILITY A1A2 = A2A3 = A1A3 A1B1= A2B2= A3B3= m m = m 3.0 cm 5.0 cm 4.0 cm Fgure 3. The control network for the determnaton of the horzontal movement of the buldng s roof slab. The mean rotaton angle of the slab was computed from the dfferences of the azmuths of the sdes of the two trangles (A 1 A 2 A 3 ) and (B 1 B 2 B 3 ): φ φ = 3 φ 3 ( ) Σ = B 1 a - a A B A (4.1) From the pont of ntersecton of the perpendculars at the mddles of the three dsplacement vectors (A 1 B 1 ), (A 2 B 2 ), (A 3 B 3 ) the mean pole Π of the slab s shft was also determned. Thus the dsplacement of any other pont of the slab can be computed through the equaton: φ rad S,Π D D =., rad S Π where: : the rotaton angle determned through (4.1) n rad, and : the dstance of pont from the pole Π. 1 (4.2) 4. Conclusons From the methodology proposed for the detecton of the dvergences n precast constructon, as appled n the above case study, the followng conclusons are wthdrawn: The use of geodetc methodology s a relable tool for the detecton and estmaton of the dvergences occurrng durng the varous stages of precast constructon. These dvergences, estmated through the comparson of the coordnates of the control ponts of the geodetc network, are gven n the local reference system of the network. They can therefore be compared to each other as well as to other mportant geometrc characterstcs of the constructon.

7 At the stage of the desgn of the geodetc control network the approprate observaton scheme as well as the nstrumentaton s selected. Therefore the qualty measures of the network together wth the tme needed for the observatons can be pre-estmated so that the network s optmzed from the pont of vew of the maxmum relablty of the fnal results and of the mnmum total cost. The estmates of the adustment (coordnates of the network s ponts) are free from the possble gross errors of the observatons snce approprate statstcal tests concernng the precson and relablty of the results, are appled. Moreover, the statstcal testng of the sgnfcance of the estmated dvergences ensures ther relablty. Snce a large number of the control ponts of the network s naccessble, the use of reflectorless total statons of approprate accuracy, for the measurement of the network s elements, permts the measurement of all the sdes between the reference and the control ponts of the network. Therefore the network s scale wll be stronger and the qualty of the results better. The reference ponts of the network can be used for the settng out of the constructon n order to amelorate the accuracy n the postonng of the foundaton precast elements. They should therefore be establshed before the begnnng of the constructon works, n safe and stable locatons and ther monumentaton should be permanent. It would be very useful f the markng of the control ponts at the selected postons s done durng the precast elements fabrcaton n the ndustry. The geodetc control network establshed for the estmaton of the dvergences of a precast constructon, can also be used for the montorng of the constructon s knematc behavor, after ts erecton. Ths s especally mportant n areas where the sesmc hazard s great. Therefore the stablty of the reference ponts of the network should be ensured. For ths purpose GPS measurements can be used where possble. References Agatza Balodmou A..M., (1999). Theory of errors and Least Squares Adustment Vol. I & II, Student Notes, Rural & Surveyng Engneerng School, N.T.U.A.. Andrans S., Petsas H., (1983) The nfluence of dvergences n precast constructon, Dploma Thess, N.T.U.A. Georgopoulos G., Desgn - development of geodetc networks checkng the geometrcal features of buldngs, precastng especally. Ph. D. Thess, Rural & Surveyng Engneerng School, N.T.U.A.. Krakwsky E., (1991). The method of least squares: a synthess of advances, Dept. of Surveyng Engneerng, Unversty of Calgary. Kuang Shan-Long, (1991). Optmzaton and desgn of deformaton montorng schemes, Ph. D. Dssertaton, Dept. of Surveyng Engneerng, Unversty of New Brunswck, Canada.Techncal Chamber of Greece, (1991). Rules for the constructon of renforced concrete buldngs, Athens.