Plate Bending Analysis by Two-dimensional Non-linear Partial Differential Equations
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1 Uversa Joura of Coputatoa Aass papersceces.co Pate Bedg Aass b To-desoa No-ear Parta Dffereta Equatos E.G. Ladopouos Iterpaper Research Orgazato 8 Da Str. Athes GR Greece eadopouos@terpaper.org Abstract A pate bedg aass s proposed ad vestgated b reducg the above structura aass probe to the souto of a to-desoa o-ear parta dffereta equato. The pate s subected to ufor dstrbuted oad ad the the trasverse dspaceets the bedg oets ad the trasverse shear forces are coputed. Ths coputatoa pate bedg ethod cossts to the aass of the oad to Fourer-doube seres. A appcato of pate bedg probes s fa gve to the deterato of the trasverse dspaceets the bedg oets ad the trasverse shear forces a orthogoa pate cosstg of reforced cocrete ad subected to ufor oad. Ke Word ad Phrases Pate Bedg No-ear Parta Dffereta Equato Structura Aass Trasverse Dspaceets Bedg Moets Trasverse Shear Forces Fourer-doube Seres. 1. Itroducto A ver terestg fed of apped echacs ad structura aass s the desg of pates of severa szes ad fors subected uder bedg oets. These pates are ade ether b sotropc or orthotropc sods or coud be sadch pates. The costtute basc uts of both budg frae structures ad udergroud structures. Speca atteto shoud be therefore gve he choosg the proper theor ad evauato ethod for the deterato of the pate bedg oets. Over the past ears the Boudar Eeet Method BEM ad the Sguar Itegra Equato Methods have bee successfu used for the uerca evauato of pate bedg probes. M.A.Jaso ad M.Mat [1] ere the frst scetsts ho pubshed a appcato of BEM to pate bedg probes he the reduced the pate bedg to the souto of a bharoc boudarvaue probe. Beod the above C.M.Seged ad D.G.A.Brce [] proposed a tegra ethod for the souto of pates th reetr corers. Soe te ater M.Mat ad S.K.Charabart [3] used a tegra equato ethod for the uerca evauato of sp supported pogoa pates. Furtherore H.Fabre [] proposed a uerca ethod of sguar tegra ad tegro-dffereta equatos for the souto of pate bedg probes. Aso N.J.Atero ad D.L.Sarse [5] apped the Boudar Itegra Equato Method to pates of arbtrar fors. O the other had G.P.Beze [6] [7] has further used the BEM for the souto of probes of pate feure th arbtrar boudar codtos. M.A.Ster [8] ade a stud o appg the boudar eeet ethod to the uerca evauato of pate bedg probes. Soe further studes of the boudar tegra equato ethod to the cassca pate theor ere ade b V.Sade ad J.Sade [9]. I ther oograph [10] C.A.Brebba J.C.F.Tees ad L.C.Wrobe ade a eteded research o the boudar tegra equato ethod for the souto of geera pate bedg probes. I addto J.Baas V.Sade ad J.Sade [11] used the BEM for the uerca souto of pates restg o eastc foudato. G. Burgess ad E.Mahaer [1] apped the boudar tegra equato ethod for the uerca evauato of atera oaded th pates. O the other had J.A.Costa ad C.A.Brebba [13] ade further proveets of the Boudar Eeet Method apped to pates o eastc foudato. 1
2 E.G. Ladopouos F.Pars ad S.Leó [1] [15] apped the Boudar Eeet Method to the souto of th bedg pates. Furtherore E.G.Ladopouos [16]-[18] proposed a ethod of o-ear tegrodffereta equatos for the uerca souto of bedg pates of severa fors. Aso B.Fater [19] proposed severa uerca ethods for the souto of bedg pates. G.A.Harte [0] ade further proveets to the deveopet of pate bedg eeets for frae aass. B the preset research the pate bedg probe be reduced to the souto of a todesoa o-ear parta dffereta equato. Thus the pate s subected to ufor dstrbuted oad ad the trasverse dspaceets the bedg oets ad the trasverse shear forces are coputed. The dervato of the proposed coputatoa pate bedg ethod cossts to the aass of the oad to Fourer-doube seres. Fa a appcato of pate bedg probes s gve to the deterato of the trasverse dspaceets the bedg oets ad the trasverse shear forces a orthogoa pate cosstg of reforced cocrete ad subected to a ufor oad.. Pate Bedg Aass b Parta Dffereta Equatos Cosder a pate bedg probe here the pae - cocdes th the ea surface of the pate ad the thcess of the pate s defed b t Fgure 1. The apped forces are oets ad trasverse shear forces. Fg. 1 Pate bedg probe uder ufor dstrbuted force p. The pate o-ear parta dffereta equato s gve b the foog forua th the trasverse dspaceet ad p the ufor dstrbuted oad: p.1 here Δ. deotes the Δ-operator:....
3 E.G. Ladopouos 3 ad s the bedg rgdt of the pate: 11 3 Et.3 th E the oduus of eastct ad ν Posso s rato. The dffereta equato.1 s vad for eastc ad sotropc pates. For a pate bedg probe as sho Fgure 1 the the oad p at a pot ca be gve b the foog Fourer-doube seres: a p 1 s s. th the coeffcets: d v b u p a s s s s 16.5 For the souto of the pate bedg probe the foog boudar codto s used: = Δ = 0.6 fro hch foos: 1 s s.7 ad because of.1: / / a.8 3. Bedg Moets ad Trasverse Shear Forces The bedg oets are gve b the reatos: [10] 3.1
4 E.G. Ladopouos 1 B usg 3.1 ad the foruas of prevous secto foos: s s 3. s s 3.3 cos cos 1 3. Furtherore the trasverse shear forces are gve b the reatos: q 3.5 q Thus fro 3. to 3.5 foos: q 3 s cos 3.6 q 3 cos s 3.7 Eqs 3. to 3. ad be used for the coputatos.
5 E.G. Ladopouos. Appcato of Pate Bedg Theor As a appcato of the pate bedg aass b to-desoa o-ear parta dffereta equatos the trasverse dspaceet the bedg oets ad the trasverse shear forces be coputed for a pate of szes 5. 0 ad. 0 subected uder a ufor oad p 1.0 N /. The pate cossts of reforced cocrete th eastct oduus E = 1 GPa ad Posso s rato v = 0.0. Furtherore the thcess of the pate s t 0. 0 ad the szes of the ufor oad are b d u ad v 3. 0 Fg.. Fg. Orthogoa pate of szes uder a ufor oad p=1.0 KN/. Thus b appg the ethod of the prevous paragraphs the the trasverse dspaceets the bedg oets ad the trasverse shear forces are coputed for severa teratos. Tabe 1 shos the resuts of the au trasverse dspaceets ad the bedg oets ad for severa teratos ad 100. Tabe 1 Trasverse Dspaceets Bedg Moets ad Y for the pate of Fg. Iteratos = M N/ M N/ As t ca be see fro Tabe 1 after 50 teratos the the vaues of the trasverse dspaceets ad the bedg oets cocde to ther fa vaues. 5
6 E.G. Ladopouos Furtherore Tabe shos the vaues of the trasverse dspaceets bedg oets ad ad trasverse shear forces q ad q for 100 teratos. Tabe Trasverse Dspaceets Bedg Moets Y ad trasverse shear forces q q for the pate of Fg. q q Fro Tabe foos that three teredate pots have bee tae b the coputer progra hch are approprate for the coputato of the au vaues. 6
7 E.G. Ladopouos 5. Cocusos A to-desoa pate bedg aass has bee proposed ad vestgated b usg the pate o-ear parta dffereta equato. Thus the pate as subected to ufor dstrbuted oad ad b usg a speca uerca ethod the trasverse dspaceets the bedg oets ad the trasverse shear forces ere coputed. Accordg to the proposed structura aass ethod the oad as aazed to Fourer-doube seres. The proposed ethod of reducg the pate bedg probe to the souto of to-desoa o-ear parta dffereta equatos has a advatages coparso to other uerca ethods e for eape the boudar eeet ethod as a cases s uch sper faster ad gvg ore accurate resuts. Furtherore the desg of pates of severa szes ad fors subected to bedg oets s ver portat for structura aass ad thus speca atteto shoud be gve b the egeer to choose the proper uerca ethod for the souto of the pate bedg probe. These tpes of pates are used ether budg frae structures or udergroud structures. A appcato of the pate bedg probe b usg to-desoa o-ear parta dffereta equatos as fa gve for the cacuato of the trasverse dspaceets the bedg oets ad the trasverse shear forces a orthotropc pate cosstg of reforced cocrete ad subected to a ufor oad. Refereces 1. Jaso M.A. ad Mat M. A tegra foruato of pate bedg probes J. Egg Math Seged C.M. ad Brce D.G.A. Itegra equato ethod for corer pates ASCE J. Struct. Egg Mat M. ad Charabart S.K. Itegra equato soutos for sp supported pogoa pates It. J. Egg Sc Fabre H. Nuerca souto of tegro-dffereta ad sguar tegra equatos for pate bedg probes J. Eastct Atero N.J. ad Sarse D.L. A boudar tegra ethod apped to pates of arbtrar pae for Cop. Struct Beze G.P. Boudar tegra foruato for pate feure th arbtrar boudar codtos Mech. Res. Cou Beze G.P. A boudar tegra equato ethod for pate feure th codtos sde the doa It. J. Nu. Meth. Egg Ster M.A. A geera boudar tegra foruato for the uerca souto of pate bedg probes It. J. Sods Struct Sade V. ad Sade J. O the Boudar Itegra Equato Method the cassca pate theor Mech. Res. Cou Brebba C.A. Tees J.C.F. ad Wrobe L.C. Boudar Eeet Techques: Theor ad Appcatos Egeerg Sprger Verag Ber Baas J. Sade V. ad Sade J. The boudar tegra equato ethod for pates restg o a to paraeter foudato ZAMM Burgess G. ad Mahaer E. A uerca ethod for atera oaded th pates Cop. Meth. App. Mech. Egg Costa J.A. ad Brebba C.A. O the reducto of doa tegras to the boudar for the BEM foruato of pates o eastc foudato Egg Aa Pars F. ad Leó S. Boudar eeet ethod apped to the aass of th pates Cop. Struct Pars F. ad Leó S. Th pates b the boudar eeet ethod b eas of to Posso equatos Egg Aa. Boud. Ee Ladopouos E.G. No-ear tegro-dffereta equatos used orthotropc spherca she aass Mech. Res. Cou
8 E.G. Ladopouos 17. Ladopouos E.G. No-ear tegro-dffereta equatos sadch pates stress aass Mech. Res. Cou Ladopouos E.G. Sguar Itegra Equatos Lear ad No-ear Theor ad ts Appcatos Scece ad Egeerg Sprger-Verag Ber Ne Yor B.Fater Statprograe für Persoacoputer Werer Verag Düssedorf Harte G.A. Deveopet of pate bedg eeets for frae aass Egg Aa. Boud. Ee
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