Available online at ScienceDirect. Procedia Engineering 91 (2014 ) 32 36

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1 Aville online t wwwsciencediectcom ScienceDiect Pocedi Engineeing 91 (014 ) 3 36 XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) (TFoCE 014) Stess Stte of Rdil Inhomogeneous Semi Sphee unde the Veticl Unifom Lod Vldimi I Andeev * Dniil Kpliy Moscow Stte Univesity of Civil Engineeing Yoslvl highwy 6 Moscow Russi Astct Thee is solution of the polem of stess-stin stte detemining in the dilly inhomogeneous concete semi sphee The inhomogeneity is induced y the tempetue field The polem educes to diffeentil eqution system with vile coefficients The llownce fo the vile Young s module lets to ive to moe ccute solution 014 The The Authos Authos Pulished Pulished y Elsevie y Elsevie Ltd Ltd This is n open ccess ticle unde the CC BY-NC-ND license Selection ( nd pee-eview unde esponsiility of the ognizing nd eview committee of 3RSP Pee-eview unde esponsiility of ognizing committee of the XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) Keywods: sphee Fouie seies; Legende polynomil; stess-stin stte; inhomogeneity 1 Intoduction In most constuctions which e used nowdys elements hve unchnged geometicl shpe long the whole length s well s constnt mechnicl-nd-physicl popeties Stesses in such constuctions e unevenly sped the limit stte cn occu only in insignificnt es the esouce of the mteil is not completely used which leds to its ovespending One of the pomising es of stuctul mechnics is to develop methods tht llow you to moe fully use the stength of mteil esouce Such methods cn educe the defomtion chcteistics of mteils ie educe the stess tht cn educe the thickness of the einfoced concete shell moe usefully sped einfocement in section incese pek lod In woks [1-3 nd othes] wee developed methods to solve polems of elsticity theoy fo odies with continuous inhomogeneity of defomtion chcteistics (elstic modulus nd Poisson's tio ) including ones which e due to tempetue field Unlike pevious woks in this ppe we conside the polem fo the cse when n inhomogeneous semi sphee stnds unde the veticl unifom lod Fig 1 The model * Coesponding utho Tel: ; fx: E-mil ddess: sv@mgsuu The Authos Pulished y Elsevie Ltd This is n open ccess ticle unde the CC BY-NC-ND license ( Pee-eview unde esponsiility of ognizing committee of the XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) doi:101016/jpoeng

2 Vldimi I Andeev nd Dniil А Kpliy / Pocedi Engineeing 91 ( 014 ) Sttement of the polem As mechnicl model we chose thick-wlled einfoced shell (semi sphee) the inne dius is the oute dius is > The shell pmetes: = 33 m =45 m; T = 500 C tempetue t the inne fce of the sphee; 0 tempetue t the oute fce of the sphee; f 1MP veticl lod distiuted ove the entie oute fce (Fig 1) 3 Solution of the polem In the cse of stedy stte of the tempetue distiution the solution of the het eqution in spheicl shell hs the fom: 1 T ( ) T T T T (1) MP Fig Expeimentl dt ( ) of the Young's modulus depending on the tempetue of concete nd the gph of the ppoximting function ET ( ) Foced tempetue defomtions t constnt coefficient of line theml expnsion coefficient equl: T ( ) 4 1 In the clcultions it ws ssumed The dependence of the modulus of elsticity on tempetue [3] cn e ppoximted y the polynomil (Fig ): 0 i i E ( ) E T( ) N i 1 Displcement equtions of equiliium coesponding to the xisymmetic tosion-fee polem: m v u 1 1 u v v m T u 3 u vctg 3 3 K R 0 3 m u v 3 1 u v v v 3 v m u KT sin 0 whee 1 1 u 1 v u v sin ; 3 ctg m sin () In this polem Young s module is function of one vile (dius) If in the xisymmetic polem in spheicl coodintes T is function of one vile nd the inhomogeneity is dil then equtions () (3) cn e simplified: m v u m T u 3 u vcot 3 3 K R 0 3 m u v 1 u v v 0 sin v The stess oundy conditions fo the xisymmetic polem cn e fomulted in the following wy: (3) (4) (5) p q ; p q (6)

3 34 Vldimi I Andeev nd Dniil А Kpliy / Pocedi Engineeing 91 ( 014 ) 3 36 Let us wite expessions fo stesses in tems of defomtions: u 1 v u v u ctg 3 KT u 1 v u v 1 v u ctg 3 KT u 1 v u v u v ctg cot 3 KT v v 1 u (7) We will find the solution of equtions (4) (5) in tems of Legende polynomils n ncos u u P n0 dpn cos v vn d n1 (8) P cos is Legende polynomil of degee n which is the solution of the eqution: n d Pn cos dpn cos d d Pn cot n n 1 cos 0 (9) Also the sufce lod must e expessed in tems of seies: p pn 0 0 Pn cos P0cos P1cos ; p p 0 f n0 n n1 n cos 0 cos q qn dpn dp1 q q d f d ; (10) whee P cos 1 Pcos cos 0 1 Bsed on the nlysis of the expnsion of the sufce lod in Fouie seies we cn choose the nume of tems of the seies In this polem we need just two tems Hving inseted the eltion (8) in (4) nd (5) we come to diffeentil eqution of second ode fo u 0 : nd we lso come to two diffeentil equtions of second ode fo u1 v 1: u 0 u 0 u 03KT 0; (11)

4 Vldimi I Andeev nd Dniil А Kpliy / Pocedi Engineeing 91 ( 014 ) u1 u1 u1 u 1 v1 v 1 0; v1 v1 v 1 u1 u 1 0 (1) Hving used the eltions (7) we otin the oundy conditions fo the equied functions fom the oundy conditions (6): u 0 u03kt 0; u 0 u03kt 0 u 1 u1 v10; u 1 u1 v1 f; v1 u1 v 1 0; v1 u1 v 1 f (13) (14) Hving solved the oundy vlue polem with equtions (11) (1) nd oundy conditions (13) (14) in the compute lge system Mple we otin displcements which we will sustitute in eltions (7) Fig 3-6 show digms of the dil nd cicumfeentil stesses whee dshed lines show stesses fo the homogeneous mteil 3 Digms of dil stesses when 4 Digms of dil stesses when

5 36 Vldimi I Andeev nd Dniil А Kpliy / Pocedi Engineeing 91 ( 014 ) Digms of cicumfeentil stesses when 6 Digms of cicumfeentil stesses when As we cn see fom these figues the digms fo nd the digms fo e vey simil This is due to the smll influence of the sufce lod on the stess-stin stte Fig 7 4 Conclusions Mximl compessive stesses with llownce fo inhomogeneity e educed y 30% comped with the cse when the inhomogeneity is ignoed But it is not so impotnt comped with 15 times decese in the tensile stess on the oute oundy of the semi sphee s concetes genelly hve tensile stength which is sustntilly less thn the compessive stength Acknowledgements This wok ws suppoted y the Ministy of eduction nd science of Russi unde gnt nume 71014/ Refeences 7 Digms of dil nd cicumfeentil stesses when the influence of tempetue is ignoed nd when [1] Andeev VI Some Polems nd Methods of Mechnics of Inhomogeneous odies Pul house ASV Moscow (00) 86 p [] Andeev VI Duovskiy IA Stess stte of the hemispheicl shell t font movement diting field Applied Mechnics nd Mteils Tns Tech Pulictions Switzelnd Vols (013) pp [3] Andeev VI Minyev S The modeling of equl stessed cylinde unde ction foced nd tempetue lods Intentionl Jounl fo Computtionl Civil nd Stuctul Engineeing Volume 7 Issue 1 (011) pp [4] Bonshteyn IN Semendyyev KA Spvochnik po mtemtike dly inzhineov i uchschihsi vtuzov (Hndook of mthemtics fo enginees nd students of technicl univesities) Moscow Nuk (1986) 544 p

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Fourier-Bessel Expansions with Arbitrary Radial Boundaries Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk