MEG 741 Energy and Variational Methods in Mechanics I
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1 ME 7 Energy n rition Methos in Mechnics I Brenn J. O ooe, Ph.D. Associte Professor of Mechnic Engineering Hor R. Hghes Coege of Engineering Uniersity of e Ls egs BE B- (7) j@me.n.e Chter : Strctr Anysis Methos I: Bem Proems -
2 Css Otine Stiffness Mtrices se on ri Fnctions -
3 Stiffness Mtri Bse on Poynomi ri Fnctions For em eements, it s ossie to etermine stiffness mtri firy simy from three ifferent nysis techniqes: mechnics of mteris, theory of esticity, energy methos. For some eements, it is iffict to etermine the ect sotion neee to formte the stiffness mtri. In these cses, it is ossie to etermine stiffness mtri y ssming the iscements re some oynomi fnction of osition. his oynomi tri fnction metho cn e se to erie the sme stiffness mtri efine reiosy for em eements. -
4 - Poynomi ri Fnctions for Bem Eement [ ] interotion fnction she fnction or the is here oynomi fnction cn e ritten in mtri form : his re nnon constnts,n,, n the tie inictes n roimte eqtion here Assme
5 -5 Poynomi ri Fnctions for Bem Eement [ ] ( ) ( ) () () therefore, reresentssoe s fnction of n reresentsiscement s fnction of
6 - Poynomi ri Fnctions for Bem Eement (cont. ) : Soefor here
7 -7 Poynomi ri Fnctions for Bem Eement (cont. ) [ ] [ ] or osition rie, nonimension Define n
8 -8 Poynomi ri Fnctions for Bem Eement (cont. ) [ ] [ ] ( ) ( ) ( ) ( ) or or
9 Reie of irt Wor in erms of Discement (o Be Use in Deriing Stiffness Mtrices) ener irt Wor Eqtion : σ ε ij ij i i S or in mtri form : S i i ε σ S S Eress stress n strin in terms of iscement σ Eε ( ) ED E D -9
10 σ Eε D L y ( ) E D here y L ED L y Differenti Oertor tht the ε ( D) ( D) ε hesscrit in front ifferentioertor oertor is of inictes ie to thereceingterm. D the he rti erities in the ifferenti oertor mtri cn e rece y orinry erities if the eformtion is not fnction of time. -
11 irt Wor in erms of Discement σ Eε ( ) ED E D ( D) D ε ener irt Wor Eqtion : ε D σ ED S S S S is efine D D D ED s symmetric oertor mtri, then the gener rincie of irt or in terms of iscement. S S -
12 - Smmry of Engineering Bem heory Discements κ ε D ε Strin-Discement Retion κ ε EA M Eε s Mteri L
13 - Smmry of Engineering Bem heory With Sher Deformtion Effects Discements (incing sher eformtion) κ γ ε D ε Strin-Discement Retion (incing sher eformtion) κ γ ε s A EA M Eε s Mteri L (incing sher eformtion)
14 Princie of irt Wor in erms of Discements for Engineering Bem heory irt or in terms of iscements : D D EA here D ED For em negecting the effects of sher eformtion. he first term hs chnge from ome integr (in the gener form of the irt or eqtion) to ine integr (in the em theory form of the irt or eqtion). he re ortion of the ome integr hs een cconte for y the 'A' n 'I' terms in the 'E' mtri. S S -
15 Etern irt Wor erms for Engineering Bem heory Boy Force erm :, here m n re force er nit ength n m is moment er nit ength. [ ] L Srfce Lo erm : S S, here s s S S M here re istrite srfce os n s re oint os ie t the ens of the em here or L. -5
16 - irt Wor for A Bem Loe in Bening Ony ( ) ( ) ( ) :, iscement, in termsof irtor Discement in erms of ri Fnction ( ) ( ) [ ]
17 -7 Stiffness Mtri from ri Fnction n irt Wor ( ) ( ) ( ) ( ) ( ) the tri fnction in termsof o t here
18 -8 Ete he Stiffness Mtri [ ] [ ] n here /
19 -9 Contine Eting he Stiffness Mtri 8
20 - Stiffness Mtri from ri Fnction n irt Wor 8 Sme stiffness mtri fon erier in chter.
21 - Lo ector from ri Fnction n irt Wor ( ) ( ) ( ) ( ) ( ) the tri fnction in termsof o t here
22 - Lo ector from ri Fnction n irt Wor ( ) ( ) ( ) then, the ieistrite o is constnt, If M M B B A A
23 -
24 et Css Emes from Chter -
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