Effects of Forces Applied in the Middle Plane on Bending of Medium-Thickness Band

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1 MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig Effecs of Forces Applied i he Middle Plae o Bedig of Medium-Thickess Bad Adre Leoev * Moscow sae uiversi of civil egieerig Yaroslavskoe shosse 6 Moscow Russia 9337 Asrac. This paper eamies he prolem of calculaig a plae of medium hickess uploaded rasverse loads ad forces i he middle plae. We use varia equaios from he heor of medium hickess plaes as ipus as proposed B.F. Vlasov. The paper also cosiders he presece of esile or compressive forces i he plae of he plae. The paper offers a eample of calculaio for a semi-ifiie uploaded a he ed of he rasverse load ad pivoall suppored o he logiudial edges. I is demosraed ha i he middle plae he forces sigifical ifluece he plae s deflecios ad edig forces. There are maerial differeces ewee resuls oaied wih he heor of plaes of medium hickess ad hose of he classical he heor of edig of plaes. Iroducio A recagular plae wih various codiios o he coour isalled o a elasic ase is a widel used eleme i uildig srucures. I cerai cases whe calculaig plaes paricularl oes ha res o a elasic ase oe eeds o facor i some forces applied o he middle plae i addiio o he rasverse loads. Such era forces ca e caused seasoal ad dail flucuaios of emperaure pre-esed reiforceme seel impacs from producio equipme ad pressure of he eclosig walls o he foudaio sla. The sressed ad deformed sae of a medium hickess plae edig from he acio of a rasverse load is descried wih hese equaios as proposed B.F. Vlasov []: p h. Here is he clidrical rigidi of he plae h is is hickess. The deflecio w ad he roaio agles of he plae are associaed wih desired fucios as relaios: w where G he shear modulus Poisso's raio. We ca wrie he followig depedece: * Correspodig auhor: LeoievA@mgsu.ru The Auhors pulished EP Scieces. This is a ope access aricle disriued uder he erms of he Creaive Commos Ariuio Licese. hp://creaivecommos.org/liceses//./.

2 . 3 M M M Q Q. 6 Sae of he prolem If forces = Fig. eis i he middle plae of he plae he secod equaio remais uchaged ad i he firs equaio mus cosider projecios of such forces o ais Оz creaed he edig of he plae. d d d d d d Fig.. Effors i he middle plae of he plae oal such forces are ierrelaed wih equaios of equilirium:. 7 Projec forces = ad heir icremes o he verical ais. The i he ligh of he epressios 3 7 he firs equaio ca e wrie as: q. 8 3 Aalical mehod of solvig Le us assume ha he logiudial side edges of he plae have a higed suppor Fig.. OI:./ MATEC We of Cofereces maeccof/ XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig

3 MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig q O Fig.. Saeme of he prolem of calculaio of he plae The our desired fucios ca e represeed as: W cos 9 F si where. We susiue decomposiio 9 i equaio 8 due o he orhogoali of rigoomeric fucios for each of ge a ordiar differeial equaio of he fourh order which i he case of ol logiudial forces will appear as: IV II p W W W ad i he case of logiudial forces : IV II p W W W where p q cos d. Equaios ad ca e represeed usig he sadard form pical for a prolem of he edig of he eam locaed o a elasic ase: IV II p W r W sw..3 epedig o he relaio of he coefficie values r ad s we fid he pe of roos of he respecive characerisic equaios ad herefore he pe of paricular iegrals for is soluio. Firs eamie a case of he plae uder he acio of forces disriued evel over is logiudial edges Fig.. B comparig equaios ad 3 we ca see ha i his case he coefficies of equaio 3 will e values: r s relaios ewee which deped o he sig of forces : i he case of srechig plae s r i he case of compressio s r. For s r he roos k of he characerisic equaio is deermied he formula: k i where s r s r 6 while he geeral soluio of differeial equaio is wrie as: W C F C F C F C F W

4 MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig Here F... F are hperolo-rigoomeric fucios W is a paricular soluio ha depeds o he rasverse load p. For s r he roos of he characerisic equaio will appear as: k k r r s k3 k r r s 8 ad herefore he soluio o equaio will have his form: W C sh Cch C3sh Cch W. 9 If he plae is uder he acio of forces coefficies of equaio 3 appear: r s. From he relaios we ca see ha his ime i he case of srechig plae s r ad i he case of compressio plae s r he soluios o he prolem will e represeed respecivel as epressios 9 or 7. Us ow eamie he prolem of he edig of a semi-ifiie plae uder he acio of rasverse load q i is iiial secio ad logiudial compressig forces evel disriued over is logiudial edges Fig.. I his case if he rasverse load is represeed as q q cos where / he eve firs erm of decomposiio 9 prese a accurae soluio ad so ide ca e omied everwhere. Assumig he plae s ifiie legh soluio o equaio 3 ca e wrie as: W Ce Ce. To compleel solve he prolem of he edig of a medium-hickess plae we also eed o add o epressio 9 he iegral of he secod equaio from which i he ligh of decomposiio of ad he ifiie legh of he plae ca e wrie as: g F e where g h. h To fid he hree iegraio cosas C C a he iiial secio = oe ca se he followig oudar codiios: II h I M W W F cos I h II M W F F si 3 III I Q W W F cos q cos. Revealig hese codiios usig equaios ad we oai a ssem of hree algeraic equaios: h C C g C h C C g C q.

5 MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig Cosider he same prolem wih eisig esile forces of he soluio o 7 ca e wrie as: W C e sice cos. To fid he cosas of iegraio oudar codiios 3 ad epressios ad allow o oai he followig equaio: h C C g h C C g 6 q 3 C 3 C. Havig solved he equaio ssems ad 6 ad defiig he cosas C C we ca deermie our calculaed values usig he formulas of he heor of plae middlehickess []. For eample he deflecio w ad he edig mome M are deermie as: II w W W W h cos 7 h II I M W W F cos 8 where fucio W ha is par of 7 ad 8 appears as whe he plae is compressed ad as whe he plae is sreched. Eample of calculaio Fig. 3 has graphs represeig he depedec of dimesioless deflecio 3 w w / p i he middle of he plae s iiial secio o dimesioless logiudial forces. / w q 3 Fig. 3. iagrams of depedece of deflecio o logiudial forces

6 MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig I he diagram solid lies ad relae respecivel o he cases compressed ad sreched of he plae middle-hickess for h/ =. while doed lies 3 ad descrie he cases for he hi compressed ad sreched plae. Fig. shows depedecies of dimesioless edig momes M a he middle of he iiial secio o dimesioless logiudial forces. M q Fig.. iagrams of depedece of he edig momes o logiudial forces Coclusios Ca see ha a icrease i logiudial force causes maerial chages of deflecio ad edig momes: he are higher i he case of compressio ad lower i he case of srechig. Epressed as perceage deflecios var more ha edig momes. I should also e oed ha he effec of logiudial forces o he plae middle-hickess is somewha sroger ha o a hi plae ad i is greaer wih compressio ha wih srechig. Lies ad 3 earl sraigh a low values of ecome curves edig o rise arupl as icreases approachig heir respecive criical values. Refereces. B.F. Vlasov Equaio of edig plaes of average hickess. / Theoreical ad eperimeal sudies of sregh ad siffess of srucural elemes Moscow MSUCE 989. R.F. Gaasov V.V. Filaov Calculaio of compressed-e plaes i icomplee coac wih a elasic Foudaio Moscow MSUCE S.P. Timosheko S. Voovski-Kriger. / Plaes ad shells Moscow auka 966. A.T. Turgaaev Bedig of a recagular plae locaed o a elasic Wikler s Foudaio wih he ifluece of he logiudial force. / Bases foudaios ad soil mechaics