DESIGNING AND CONSTRUCTION OF ROADS, SUBWAYS, AIRFIELDS, BRIDGES AND TRANSPORT TUNNELS

Transcription

1 Issue (), ISSN 5- DESIGNING AND CONSTRUCTION OF ROADS, SUBWAYS, AIRFIELDS, BRIDGES AND TRANSPORT TUNNELS UDC 59.. A. Kozlov THE DEFLECTED MODE OF MULTI COHERENT PRISMATIC CONSTRUCTIE ELEMENTS OF BRIDGE CONSTRUCTIONS oroezh Stte Uversty of Archtecture d Cvl Egeerg Russ, oroezh, tel.: ()---, e-ml: vkozlov@vgsu.vr.ru D. Sc. Physcs d Mthemtcs, Hed of Dept. of Theoretcl d Appled Mechcs Sttemet of the problem. The dstrbuto of pressure buldg elemets of brdge costructos the perod of stllto c dffer cosderbly from feld of pressure durg operto of complete object. I some cses brdge flghts durg ths perod represet the cosole wth oe fxed cross-secto. The clculto of the deflected mode of such elemets llows oe to provde sfe stllto durg the erecto of brdges. Results. The pressure d deformtos th-wlled mult coheret prsmtc costructo re defed t the rgd fxed oe bsc cotour d the other oe free. As dstct from kow works the vrble thckess of pels d supportg wlls-logeros log desg s cosdered. The bed from the dstrbuted lodg d cross-secto force, torso from the dstrbuted d cocetrted momets s cosdered. Coclusos. Through the use of the lw of vrto of thckess log costructo t s possble to receve redstrbuto of the pressure rsg uder the fluece of vrous power fctors. Keywords: deflected mode, buldg costructos, vrble rgdty. Itroducto Costructo of dfferet buldg structures requres eve dstrbuto of strs costructo elemets uder the effect of ppled lods d force to optmze ther performce durg ther operto d ssembly. I some cses t s by vryg geometrc prmeters, costructo form, chge of thckess, wys of fxg, etc. As for brght flghts, they re sometmes ot structure supported t both eds durg ssembly but cosole wth fxed logtudl secto where the dstrbuto of strs c dffer cosderbly from tht ssembled flght. Assembly from up to ts fl stge mght tke few yers whe there wll be forces d lods ctg o the costructo elemets s well s low bed d rollg rgd ty t dfferet stges of ssembly. Therefore t s ecessry to perform relevt clcultos to Kozlov. A.,

2 Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture provde sfety of the ssembly of costructo elemets desgg brdges d other buldg structures. The proposed pper dels wth the dstrbuto of str qudruply-coected prsmtc structure fxed wth logtudl set of regulr structure depedg o the lw of chge the thckess of the upper d lower pels d re of logtudl secto of logtudl wll grders. A structure s rgdly fxed log the support cotour wth free-ed secto. The geerl solutos obted the lytcl form usg the devce of specl fuctos for coe d prsmtc structures wth rdom cotour of logtudl secto re [, ], free osclltos re ddressed []. Obtg resolvg system of ordry dfferetl equtos. The vector of elstc dsplcemet of rdom pot s preseted s resoluto (, ) z s ( z) ( s), where z s logtudl, s s trsverse coordtes of the pot, re ukow geerlzed dsplcemets determg the dsplcemet of logtudl cotour z = cost, re correspodg specfed coordte vector fuctos wth,, re projectos of these fuctos oto ut vectors of movg trhedro of medum surfce of prsmtc structure. Accordg to [], the geerl scheme of resolvg dfferetl equtos s s follows () j b b c R G j j j j j j j j /,,...,, () for the followg geometrc d sttc boudry codtos t Here ulke [], j j j,,..., () z z l * j j bj j P zl j zl,,..., (), b, c re vrble coeffcets depedg o the chrcter of rge of chge the thckess h z of structure; *, P o free-ed secto d exterl lods respectvely. Resolvg the system () for the orgl geerlzed dsplcemets R correspod wth surfce d cocetrted d obeyg the obted solutos to the boudry codtos () d () usg the formuls

3 Issue (), ISSN 5-,,. E E,, E. (5) ( ) we c fd the deformtos d strs t rdom pot of the structure. There re sx geerlzed dsplcemets j determg the dsplcemet of logtudl cotour z=cost s sold body: three trsltol,, dsplcemets log the coordte xes Ох, Оу, Oz d three torso gles, 5, relto to these xes for the correspodg fuctos, j, j j, j=,,...,. However some cses the umber of wthdrw geerlzed dsplcemets c be reduced. Hece, e.g., for sectos symmetrcl to the xs Ох, durg bedg d torso of shell from sx geerlzed dsplcemets,..., oly three c be wthdrw:,,. The equlbrum equtos correspodg wth,, 5, c be met detclly. I order to obt lytcl soluto the resoluto () oly three members wll be wthdrw resposble for wrpg of the sectos. The frst oe z) ( ) ( S ccout for torso wrpg d the secod oe z) ( ) for bedg wrpg. The ( S compoets of pproxmtg vector fuctos ( correspodg to the dsplcemets of logtudl secto s sold body re s follows,, y( ), 5 x(, ; S x( ), y( ),,, S S 5 x( y( x( y( ); y( ), x( ), S S S, 5, x( x( y( y(. () Wrpg of logtudl sectos s pproxmted by two fuctos: d d x y, xy, Ky () where ccouts for torso wrpg, s bedg torso. Here K s the coeffcet of orthogolzto determed usg the codto ( ( h( z, ds. The compoets, =,,...,+ ccordg to the ccepted hypothess o wrpg of the cotour pots drecto to the geertg le.

4 Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture Besdes f logtudl secto hs two symmetry xes: relto to Ox d Oy, umber of the coeffcet of the system () become zero. I ths cse the coected system of fve dfferetl equtos s splt to two: the frst, secod d ffth equtos descrbe the bedg of structure d the thrd d fourth oe torso. These systems re s follows for symmetrcl cotour of logtudl secto b b b b c c b c c b R R R / G, () / G. / G, b b b c R R / G, / G. (9) For rgdly fxed structure loded t the ed secto z=l by cocetrted logtudl force Q y d М z, geometrc () d sttcl () boudry codtos respectvely re s follows: z,, ; b b Q / G,,. z zl z y zl zl b M /.,, ; z G z z zl zl () () Furthermore the soluto of groups of dfferetl equtos () d (9) for correspodg boudry codtos () d () s performed dvdully. Bedg torso. Let us solve the problem of torso of the structure usg the dstrbuted m z (z) d cocetrted M z the secto z=l momets (Fg.). h O A z x Вид ew А M z d y l h d y Fg.

5 Issue (), ISSN 5- I ths cse the system (9) R =m z (z), R =. Let the thckess of the structure h(z) d dstrbuted torque momet m z (z) chge lerly log the geertg le gog dow from h to h d from m to m respectvely s the free ed pproches h h( z) b z, z) z m, m z ( where b h, ( h h ) / l, ( m m) / ; l s the legth of the structure. l Followg the trsformtos of the system (9) we ed up wth heterogeeous Bessel dfferetl equto relto to the geerlzed torso wrpg A A A, () where b / z;, A, A, A re the coeffcets depedg o the geometrc prmeters of the structure. The geerl tegrl of heterogeeous dfferetl equto () s wrtte usg the modfed frst-order I d the secod-order Bessel equtos K s [5] s well s the Struve fucto L. Ultmtely the expressos for geerlzed dsplcemets re s follows ( A b dz Fz Rz C dz C, () C I C K A / ) L / ( A A ) /. Itegrto costts re detfed usg the boudry codtos () of the tsk. Kowg the expressos for the geerlzed dsplcemets t s ot dffcult to detfy the deformtos d strs t rdom pot of the structure usg the formuls (5). Accordg to the obted solutos for prsmtc structure of ler vrble thckess the umber of clcultos usg softwre ws performed. As exmple Fg. there s model of the structure, ts lodg scheme d grphs of the depedece of orml strs / E f ( ) depedg o z z / l. I the clcultos M z =. knm, m z =. knm, z d = - m, d =. - m, l = m, h = - m, h =.9 - d h =. - m. The curved les d descrbe the dstrbuto log the logtudl rb (x=-d, у=-d /) of the upper pel of the structure loded wth momet m z d the curved les d for combto of the torso wth the momets M z d m z. The grphs d re desged t h h / h =. d dt h 5. A comprtve lyss shows tht eve t h =. cocetrted momet M z cosderbly creses (curved le ) d t h 5 there s qulttve chge of the str of the 5

6 Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture structure (curved le ). For the ltter t z =. there s surge of strs cused by chge the re of orml secto. The bove grphs dcte tht mult-coected shell structures of costt d vrble rgdty wrpg strs do ot hve the edge effect s rods wth sold secto d gog dow depedg o the embeddg ple re dstrbuted log the etre shell. I the ed secto due to ts free wrpg these strs equl zero. Curved le Usg the rb: x = d, y = d / ; Torso m z, ; Torso m z d M z Fg.

7 Issue (), ISSN 5- Wrpg bed. Let us look t wrpg bed of qudruply-coected prsmtc structure wth symmetrcl squre cotour. From the exterl lods we ccept the dstrbuted lod q ( z)d cocetrted force Q y t the free ed s show Fg.. q(z) Вид ew АA h O A z x d y _ Q y h d _ Q y y Fg. Tkg to ccout the chrcter of lodg the equto system () descrbg the bed of the structure we get R ( z) q( z), R, R. For the ske of certty we ssume tht the dstrbuted lod q(z) obeys the prbolc lw for the ler lw of chge of thckess h (z) ( q l q z) q z q z, q ( q q l)/, where q d q determe the logtudl lod the secto z = d the chrcter of chges of the prbol log the xs z respectvely. As for the torso, the geerl tegrl of heterogeeous Bessel dfferetl equto relto to the geerlzed bedg wrpg s wrtte s sum of the solutos of homogeeous equto usg the modfed frst-order d secod-order Bessel fucto d prtculr solutos detfed usg the method of vrto of the Lgrg costts. Ultmtely the solted form of the obted solutos s the followg: CB RB FB HB b dz dz, C B C B RB FB / HB / /, / C5 C C A A / ) L / [ A A ( r z )]/ ( where A [ ( r z) dz ( r z) dz], z B ( z) rb ( z) (,,,), B l( b z) b, r,

8 Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture С D A R lf, C Qy / G l Hl C D C, D L r l A D ) ( r l A ) r l ( ) ( r l) ( ) ( ) ( A r l r z dz r l) ( r z) dz, zl ( zl C С C lc R F Hl l l, С B ) C B () RB ()/ FB () / HB () / /, 5 ( CB () RB () FB D () HB A ra r L () b dz z r A A dz A r ( r z) dz r ( r z) dz, D r r. z Hvg determed the dsplcemets ccordg to the formuls (5) deformtos d strs re foud t rdom pot. Fg. presets some of the results of umercl clcultos of qudruply-coected model structure wth the thckess of pels d logtudl wlls chgg ccordg to the ler lw. The depedece grphs f ( ) t the jot effect of the force Q y o the model z dstrbuted log the lodg rge q( z) qz q, where q = -5/l, q = 9. kn/m. For the clcultos l = m, h = - m, d = - m, d =, - m, h =. d 5. The z curved les d descrbe the dstrbuto f ( ) log the sde rb of the upper pel gve by the coordtes x = d, y= -d / o the ler lod q(z) d the curved les d t the jot operto of Q y =.9 kn d q(z). The grphs d re desged t h =. d d t h = 5. The ture of the curved les u ths cse s detcl d they re dfferet. A drop to the ed secto s full greemet wth the operto of the structure. Note tht the specfc trsto t h the clculto results re cosstet wth those kow from lterture [], whch proves the credblty of the results d umercl clcultos. z z,

9 Issue (), ISSN 5- Fg. Coclusos The lyss of the bove grphs shows tht for the ler lw of chge of the thckess the ture of dstrbuto the rge for h =. s the sme s for the structure of the costt thckess []. However s h creses, dffers the module from the strs t h cost. Hece for the bove lodg schemes the str-deformto of prsmtc th-wlled structures wth costt d vrble thckess s dfferet, whch should be cosdered clcultos of the elemets of ctul buldg structures. Hece by vryg the lw of chge of the thckess of the structure d thus ts rgdty redstrbutos of strs cused by dfferet forces c be detfed. 9

10 Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture Refereces. Bultov S. N., Kozlov. A. Reshee ekotorykh prkldykh zdch teor koche-skkh obolochek slozhoy geometr [The soluto of some ppled problems of the theory of cocl shells of complex geometry]. RAN. Problemy mshostroey dezhost msh,, o. 5, pp... Bultov S. N., Kozlov. A. Steseyy zgb s krucheem kosol'o zshchemleoy tsldrcheskoy obolochk s mogosvyzym koturom ekrugovogo ocherty [Costred bedg d torso of ctlever clmped cyldrcl shell wth mesh cotour ocrculr shpe]. estk oroezhskogo GASU. Dorozho-trsportoe strotel'stvo,, o., pp... Kozlov. A. Free vbrtos of cosole restred d prsmtc th-slb structures. Scetfc Herld of the oroezh Stte Uversty of Archtecture d Cvl Egeerg. Costructo d Archtecture,, o. (), pp. 9.. Obrztsov I. F., Oov G. G. Strotel'y mekhk skosheykh tokosteykh system [Structurl mechcs slopg th-wlled systems]. Moscow, Mshostroee Publ., 9, 59 p. 5. Kmke E. Sprvochk po obykoveym dfferetsl'ym urveym [Hdbook o ordry dfferetl equtos]. Nuk Publ., 9, pp., 9.. Obrztsov I. F. rtsoye metody rschet tokosteykh vtsoykh prostrstveykh kostruktsy [rtol methods of clculto of th wlled rcrft costructos]. Mshostroee Publ., 9. 9 p.. Sered P.. Spodl'yy rspd v eptksl'ykh tverdykh rstvorkh geterostruktur ALXG- XAS/GAAS() GAXIN-XP/GAAS(). Izvesty Smrskogo uchogo tsetr Rossyskoy kdem uk, 9, vol., o., pp. 5. 5