ScienceDirect. About Verification of Discrete-Continual Finite Element Method of Structural Analysis. Part 2: Three-Dimensional Problems
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1 Avlle ole t wwwscecedrectcom SceceDrect Proced Egeerg 9 ( XXIII R-S-P semr heoretcl Foudto of Cvl Egeerg (RSP (FoCE 04 Aout Verfcto of Dscrete-Cotul Fte Elemet Method of Structurl Alyss Prt : hree-dmesol Prolems Pvel A Amov Mr L Mozglev Vldmr N Sdorov c Moscow Stte Uversty of Cvl Egeerg 6 Yroslvsoe Shosse Moscow 97 Russ Smr Stte Uversty Archtecture d Cvl Egeerg 94 Molodogvrdeysy Street Smr 4400 Russ c Kelce Uversty of echology Fculty of Cvl Egeerg d Archtecture l ysclec Pstw Polsego 7 Kelce 5-4 Pold Astrct hs pper s devoted to verfcto of so-clled dscrete-cotul fte elemet method (DCFEM of structurl lyss hreedmesol prolems of structurl lyss re uder cosderto Formulto of the prolem of three-dmesol theory of elstcty for structure wth pecewse costt physcl d geometrcl prmeters log so-clled ts sc drecto solutos oted y DCFEM d fte elemet method (FEM ther comprso re preseted DCFEM s more effectve the most crtcl vtl potetlly dgerous res of structure terms of frcture (res of edge effects where some compoets of soluto re rpdly chgg fuctos d ther rte of chge my cses c t e dequtely te to ccout y the stdrd FEM he he Authors Authors Pulshed Pulshed y Elsever y Elsever Ltd hs Ltd s ope ccess rtcle uder the CC BY-NC-ND lcese ( Peer-revew uder resposlty of orgzg commttee of the XXIII R-S-P semr heoretcl Foudto of Cvl Peer-revew Egeerg uder (RSP resposlty of orgzg commttee of the XXIII R-S-P semr heoretcl Foudto of Cvl Egeerg (RSP Keywords: dscrete-cotul fte elemet method structurl lyss multpot oudry prolem three-dmesol prolems verfcto Opertol formulto of multpot oudry prolem of three-dmesol structurl lyss Let x e sc drecto whle physcl d geometrcl prmeters of structure c e chged rtrrly log x d x Opertol formulto of resultt multpot oudry prolem of three-dmesol theory of elstcty t exteded dom emorderg cosderg structure wth DCFEM hs the form: U LU S x ( x x = B U ( x 0 B U ( x 0 g g = B ( 0 ( 0 U x B U x g g ( Pvel A Amov el: fx: E-ml ddress: pvelmov@gmlcom he Authors Pulshed y Elsever Ltd hs s ope ccess rtcle uder the CC BY-NC-ND lcese ( Peer-revew uder resposlty of orgzg commttee of the XXIII R-S-P semr heoretcl Foudto of Cvl Egeerg (RSP do:006/jproeg04004
2 Pvel A Amov et l / Proced Egeerg 9 ( E L L vv( L uu C L vvl S 0 uv L vvf u U v u U U ( v L 0 0 vv 0 0 L L L L L L uv uv uv vu vu uv L L vu uv ( c 0 0 L uu j j C ( 0 c 0 (4 j c F F f x ( x x ( ( x (5 0 x s the dom occuped y structure x ( x x x x x x re coordtes ( x correspods to sc dmeso x = re coordtes of oudry cross-sectos of structure ( prtculr coordtes of crosssectos wth dscotutes of the frst d of physcl d geometrcl prmeters of structure l s the legth of structure log sc dmeso x 0 ] = re correspodg frgmets of dom wth oudres [ l oted y seprto from dom y cross-sectos x x d x x = re exteded doms emorderg frgmets = x x s the chrcterstc fucto of dom ( x x x s the delt-fucto of order ( x [ ] s ut orml vector of dom oudry u = s the uow vector of dsplcemets dom B B = B B re mtrces (opertors of oudry codtos of the sxth order ( x -depedet g g = g g re rght-sde vectors of oudry codtos of the sxth order ( x -depedet F s the rght-sde vector dom dom f s the oudry trcto vector dom C s the mtrx of elstc prmeters of the supports (f y re Lme coeffcets of mterl dom c s the coeffcet of resstce the drec- to of the xs Ox / x / x F s the vector of ody forces v u u v v Dscrete-cotul formulto of multpot oudry prolem of three-dmesol structurl lyss DCFEM presupposes fte elemet pproxmto of exteded dom log drectos of structure perpedculr to the sc drecto whle log sc drecto prolem rem cotul (thus exteded dom s dvded to dscrete-cotul fte elemets Resultg multpot oudry prolem for the frst-order system of ordry dfferetl equtos wth pecewse-costt coeffcets wth DCFEM [ ] hs the form: ( y A y f x ( x x B y ( x 0 B y ( x 0 g g = B y ( x 0 B y ( x 0 g g (6 where A re mtrces of costt coeffcets of order 6NN (dscrete logs of opertors L f re vectors of sze 6NN (dscrete logs of vector fuctos S N s the umer of elemets log x N s the umer of elemets log x
3 6 Pvel A Amov et l / Proced Egeerg 9 ( y ] y ( x [ u ( x v ( x (7 ( u u ( x [ ( u ( ( u ( v v ( x [ ( u v ( p q ( p q ( p q ( ( u ( ( u ( ( ( p q ( p q ( N ( u ( u ( p q ( N ( N ( N ( u ( N ( N ( u ( N ( N ( u ( ( N N N N (8 ] (9 ] u ( x [ u u u ] p N q N (0 v ( x [ v v v ] p N q N ( ( p q ( p q ( p q ( p q ( q ( q u p u p x p N q N re fuctos whch defe compoet of ( p q dsplcemet u the ode wth coordte ( x x x the tervl x ( x x Soluto of prolem (6 s ccetuted y umerous fctors hey clude oudry effects (stff systems d cosderle umer of dfferetl equtos (severl thousds Moreover mtrces of coeffcets of system ormlly hve egevlues of opposte sgs d correspodg Jord mtrces re ot dgol Specl method of soluto of multpot oudry prolems for systems of ordry dfferetl equtos wth pecewse costt coeffcets structurl lyss hs ee developed [] Not oly does t overcome ll dffcultes metoed ove ut ts mjor peculrtes lso clude uverslty computer-oreted lgorthm computtol stlty optml codtolty of resultt systems d prtl Jord decomposto of mtrx of coeffcet elmtg ecessty of clculto of root (prcpl vectors [] Numercl Smple DCFEM cosderg the dstctve pper hs ee relzed softwre DCFEMDpc Progrmmg evromet s Mcrosoft Vsul Studo 0 Professol d Itel Prllel Studo XE 0 (Itel Vsul Fortr Composer XE 0 est model d prctclly mportt prolems of structurl lyss hve ee solved wth the use of DCFEMDpc Let s cosder three-dmesol structure wth rectgulr cross sectos fxed o two sdes ( x 0 d x L wth zero dsplcemets u u u 0 Legth of structure ( L s equl to 600 cm (Fgure Heght of structure ( h s equl to 50 cm wdth ( h s equl to 50 cm Addtol geometrcl prmeters (Fg : 50 cm 450 cm Elstc modules of mterl for the frst ( x (0 L / d the secod ( x ( L / L prts of structure ( E E re equl to 000 N/cm d to 500 N/cm respectvely Posso's rtos of mterl for the frst d the secod prts of structure ( re equl to 06 d to 04 respectvely Structure s loded y cocetrted forces P 00 N d P 00 N P P h / h E E h L/ L/ L x x x Fg Desg model of structure
4 Pvel A Amov et l / Proced Egeerg 9 ( ANSYS Mechcl (ANSYS 50 smulto softwre hs ee used for soluto of prolem terms of FEM Crtes coordte system ( x y z hs ee used We hve x x y x z x Uform squre mesh 60x00 ( ANSYS 50 hve ee costructed from SOLID85 fte elemet [] DCFEMDpc smulto softwre hs ee used for soluto of prolem terms of DCFEM Uform pproxmtg mesh log x d x cludes 00 dscrete-cotul fte elemets Dstrutos of dsplcemets u u u d stress t cross secto x 40 sm re preseted t Fg Comprso of stresses d dsplcemets oted y ANSYS Mechcl d DCFEMDpc t severl cross-sectos of deep em re preseted t Fg hus we c coclude tht the results of lyss oted y the ANSYS Mechcl (ANSYS 50 d DCFEMDpc smulto softwre geerlly gree well wth ech other
![effects where some compoets of soluto re rpdly chgg fuctos d ther rte of chge my cses c t e dequtely te to ccout y the stdrd fte elemet method [ 4]](/docs-images/88/115113550/images/5-1.jpg "Acowledgemets hs wor ws fclly supported y the Grts of Russ Acdemy of Archtecture d Costructo Sceces (77 78 d y the Mstry of educto d scece of Russ uder grt")
5 8 Pvel A Amov et l / Proced Egeerg 9 ( c d Fg DCFEM dstruto of prmeters t cross secto x 40 sm: ( dsplcemets u (cm dsplcemets u (cm (c dsplcemets u (cm (d stresses (N/cm It ws cofrmed tht DCFEM s more effectve the most crtcl vtl potetlly dgerous res of structure terms of frcture (res of the so-clled edge effects where some compoets of soluto re rpdly chgg fuctos d ther rte of chge my cses c t e dequtely te to ccout y the stdrd fte elemet method [ 4] Acowledgemets hs wor ws fclly supported y the Grts of Russ Acdemy of Archtecture d Costructo Sceces (77 78 d y the Mstry of educto d scece of Russ uder grt umer No 04/07
6 Pvel A Amov et l / Proced Egeerg 9 ( c Fg Comprso of results oted y ANSYS d DCFEMDpc: ( dstruto of dsplcemets u log x ( x x 0 cm ( dstruto of dsplcemets u log x ( x 0 cm x 0 cm (c dstruto of stresses log x ( x 5 cm x 5 cm Refereces [] Amov PA: Correct Dscrete-Cotul Fte Elemet Method of Structurl Alyss Bsed o Precse Alytcl Solutos of Resultg Multpot Boudry Prolems for Systems of Ordry Dfferetl Equtos // Appled Mechcs d Mterls Vols (0 pp [] Amov PA Sdorov VN: Correct Method of Alytcl Soluto of Multpot Boudry Prolems of Structurl Alyss for Systems of Ordry Dfferetl Equtos wth Pecewse Costt Coeffcets // Advced Mterls Reserch Vols pp [] Brero EJ: Fte Elemet Alyss of Composte Mterls Usg ANSYS CRC Press 0 66 pges
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