Why symmetry? Symmetry is oten rgued rom the requirement tht the strin energy must be positie. (e.g. Generlized -D Hooke s lw) One o the derities o energy principles is the Betti- Mxwell reciprocity theorem. (Very useul in structurl mechnics) Betti-Mxwell Reciprocl theorem: I two lod sets ct on linerly elstic structure, work done by the irst set o lods in cting through the displcements produced by the second set o lods is equl to the work done by the second set o lods in cting through displcements produced by the irst set. R.. Htk EML556 Finite Element Anlysis Uniersity o Florid Exmple or Betti-Mxwell Reciprocl heorem x L=+b b P P For this system the Betti_Mxwell Reciprocl theorem proides the condition P P he displcement nd rottion o n end loded cntileer bem is gien by P P EI Lx x 6 P EI Lx x P Using these we cn write the displcements s P b EI P EI P 6EI P EI b P b EI R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Betti-Mxwell theorem pplied to inite element equtions he theorem is sme s beore, except now we use system o equtions R D R D he terms boe re sclr quntities (representing work done) I we cn expnd the equtions by substituting or the orce ectors R in terms o the stiness mtrix nd displcement ector D D D D D D D D D Since they re sclr terms the trnspose should be the sme D D D D R.. Htk EML556 Finite Element Anlysis Uniersity o Florid 4 Sprsity Sprsity is term used to quntiy the number o zeros in stiness mtrix. In ery lrge models only ew nodes re connected to ech other. his cretes lot o zeros. o economize storge (computer memory) commercil progrms oten use dierent wys to oid storing the zero loctions. Node numbering in FE model ects the topology o the stiness mtrix For liner ssembly o br or bem elements you obtin bnded mtrix. Howeer or most -D complex structures the bndwidth increses nd cn oten result in dense mtrix. We need to ind topology tht ors storge nd eicient soling o the eqution R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Skyline o Mtrix 5 y k k u u u u x k 4 k 4 u 4 4 5 6 u 4 5 6 5 u 4 5 5 5 55 56 u 6 6 65 66 68 u 4 68 88 4 Due to symmetry only terms on the digonl nd boe need to be stored. Skyline o mtrix encloses the uppermost nonzero coeicients in ech column he coeicients re then stored in one-dimensionl rry. he rry tht describes the proile o the mtrix is needed to deelop -D storge or the stiness mtrix Zeros under the skyline need to be stored s they will become illed in the solution process x y x y x y x4 y4 R.. Htk EML556 Finite Element Anlysis Uniersity o Florid 6 Proile o mtrix 5 6 u 4 5 6 x y 5 u 4 x y 5 5 5 55 56 u x 6 6 65 66 68 y u 4 x4 68 88 4 y4 x he uxiliry rry tht describes the skyline is 5 6 he sum o the entries is the proile description is the number o coeicients tht must be stored nd is reerred to s the proile o the mtrix he proile or the boe cse is Note: See textbook exmple in Figure.8- on how you cn chnge the proile R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Bndwidth o the mtrix 7 6 5 6 u 5 4 5 6 5 u 4 5 5 5 55 56 u 6 6 65 66 68 u 4 68 88 4 Semi-bndwith b i o ny row i is equl to the number o columns rom the digonl to the rightmost non-zero term A mtrix o lrge proile lso hs lrge bndwidth he root men squre o the semi-bndwidths o the rows is used s mesure o the bndwidth o the mtrix For our exmple the bndwidth is.8 he solution procedure hs to crete ewer ills in the mtrix when the coeicients re tightly clustered round the digonl x y x y x y x4 y4 R.. Htk EML556 Finite Element Anlysis Uniersity o Florid 8 Solution o Equtions We he deeloped FE structurl equtions o the orm D R nd indicted tht this cn be soled (when the system is nonsingulr) s D R his must be simply interpreted s solution o the eqution set. Seldom do we need to obtin the inerse mtrix nd multiply them s shown boe. Solution procedure re o two types: Direct solers (e.g. Guss elimintion) Itertie solers (e.g. Guss-Siedel itertion) R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Direct Solers 9 Direct solers use methods to trnsorm the equtions into n upper or lower tringle mtrix tht cilittes the solution (Guss elimintion), or decomposes the mtrix into product o upper nd lower tringle (LU decomposition) he eort o soling system o equtions using the direct solers is unction o its sprsity nd proile (or bndwidth) ypiclly the number o opertion required to sole nxn mtrix system o equtions is nb when the mtrix bndwidth is b. (For dense mtrix this is n / ) Reiew bsics o Guss elimintion For ery lrge structures, the mtrix equtions cn be soled een beore the entire eqution cn be ssembled. hese methods re clled rontl solers. R.. Htk EML556 Finite Element Anlysis Uniersity o Florid Itertie solers hese methods strt with guess nd iterte till it conerges to solution (e.g Newton-Rphson method or soling lgebric equtions) For mtrices reiew Guss-Siedel Itertion he eort required oten is diicult to predict in itertie solers he conergence depends on the condition number o the stiness mtrix Condition number is the rtio o the lrgest nd smllest eigenlues o the stiness mtrix In structures the eigenlues relte to the nturl requencies o the structure Pre-conditioned itertie solers trnsorm the system o equtions to improe its conditioning beore soling it. R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Wht i the pplied lod is not concentrted lod? W / x Grity cting down W / g W / he nodl lods cn be clculted using equilent physicl deormtions or energy Note: he weight o ech element is ssigned to the nodes In the ctul structure the stress/strin is zero t the ree end where s there is some strin t the element hing the ree end his error is reduced i the elements re mde smller W / R.. Htk EML556 Finite Element Anlysis Uniersity o Florid Lods on Br Elements Uniormly loded br Force/length = q ql/ ql/ L he strin rition in uniormly loded br is liner. he br element only llows constnt rition. For ccurcy more elements re needed. Br loded between nodes P Pb/(+b) P/(+b) b Oten required becuse mesh does not include node t the point o orce ppliction R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Boundry Conditions Lets us consider model with 6 DOF. he ssembled stiness mtrix must pper s ollows, where ij = ji. 4 5 6 4 5 6 4 5 6 4 4 4 54 64 5 5 5 45 55 65 6 6 6 46 56 66 u u u x y x y x y Let us ssume tht the DOFs u, u nd re known he equtions cn be reorgnized s ollows 4 5 6 4 5 6 4 4 4 54 64 4 5 6 5 5 45 5 55 65 6 6 46 6 56 66 u u u y x y x x y R.. Htk EML556 Finite Element Anlysis Uniersity o Florid Prtitioning o FE equtions he prtitioned set o equtions nd be expnded nd denoted by the ollowing mtrix orm 4 5 6 4 5 6 4 4 54 64 u u 4 5 6 5 5 45 5 55 65 6 6 46 6 56 66 u u u u y x y x x y 4 he equtions or the unknown displcements re then y 5 6u 4u x 5 6u 4 4 y 4 45 46 Note: When ll the speciied DOF tke zero lues then the boe orm is the sme s obtined by the row-column elimintion procedure R.. Htk EML556 Finite Element Anlysis Uniersity o Florid
Compct representtion o the prtitioning method 5 he prtitioned stiness mtrix nd orce/displcement ectors cn be denoted s ollows. D R D, R re the unknown displcements nd orces.,, re the unknown displcements nd orces. x c x x Dc Rx Dc Rc Note: In boe equtions ij, D x, D c, R x, R c denote mtrices nd ectors nd not entries o mtrix or ector s beore (hence the bold lettering) When the displcements re speciied, the rections orces re unknown When Forces re speciied the displcements re unknown he solution cn now be written s R c D c D D Dx R x x c he irst set is used to get displcements he second set is used to clculte rection orces R.. Htk EML556 Finite Element Anlysis Uniersity o Florid