MEG 741 Energy and Variational Methods in Mechanics I

Transcription

1 MEG 74 Energ n Vrtonl Methos n Mechncs I Brenn J. O Toole, Ph.D. Assocte Professor of Mechncl Engneerng Howr R. Hghes College of Engneerng Unverst of ev Ls Vegs TBE B- (7) j@me.nlv.e Chpter : Prncples of Vrtl ork: Integrl orm of the Bsc Eqtons 5-

2 Vrtl ork n Vrtonl Methos Energ prncples prove n lterntve to ewtonn methos s mens of ervng n solvng governng eqtons. 5-

3 ork & Energ Apple forces, moments, n torqes o work on strctre, e.g.,. L s s, where s the component of force n the s recton s Ths work chnges the potentl energ stte of the nternl forces Ths s referre to s nternl energ or strn energ The strn energ cn e efne n terms of stresses & strns Apple ork (eternl) Chnge of nternl energ 5-3

4 ork n Energ (contne) The strn energ n n elstc o, U, s eql to n hs the opposte sgn of the work one the nternl forces,. U Let s s tht Π s the potentl energ store n -D strctre n t s gong to o some work,. An Π s the ect fferentl of Π. Π ( ) z z where n z re the forces n the n z rectons. The work one nternl energ s conservtve. The chnge n energ when force moves from pont A to pont B s nepenent of the pth tken. If force strts t pont A n moves ron n ens p ck t pont A, no net energ hs een store n the strctre. 5-4

5 ork n Potentl Energ of Internl orces Eternl forces,, re pple t the ens of the ro shown elow. ht s the work one rng the eformton of ths r? L rst emne the work one the nternl forces,, over the fferentl element of length,. The chnge n length of [( ) ] the fferentl element s The nternl force cn e wrtten n terms of σ A The net work over the fferentl element s : stress net work σ A σ A 5-5

6 Internl ork e to Al Lo The nternl work over the length of the r s fon ntegrtng the prevos epresson: L L L AE L VE σ A AE L E A Vσ Snce σ E If s constnt over L The cpct of the nternl forces to o work s clle strn energ, U. The strn energ s consere postve qntt. The work one the nternl forces s negtve: U U VE VE Vσ Vσ 5-6

7 5-7 Generl Long The nternl work over the length of the r s fon ntegrtng the prevos epresson: V V z z z z z z z z z γ γ γ γ τ γ τ γ τ σ σ σ V V V V V V U V V or E σ E σ T T T T The strn energ enst s efne s the strn energ per nt volme. E σ T T o o U V U U

8 Strn Energ Denst n Complementr Strn Energ Denst σ * U o U o U o strn energ enst * U o complementr strn energ enst 5-8

9 5-9 Emple.: n the Strn Energ n Bem wth n Internl Al orce n Benng Moment L A V V A I z M AI Mz A E U V I Mz A I Mz A E U E I Mz A U, σ σ σ A A I A z za tht recll L EI M EA U The Strn Energ for Bem Prolems:

10 In Some Cses, the Internl ork s not Smpl Eql to Potentl ncton If the nternl work epens on the long hstor (n not jst the en sttes) then t s not eql to potentl fncton. Emple: A slener r hs n pple l stress n tempertre chnge σ A σ A B B o ( ) fnl ( ) fnl C o C D D σ, T Cse I: Tempertre Chnge Apple rst T pple wth no constrnt (stress free epnson from A to B) Apple Stresses cse etween B n C Arrve t fnl stress n strn t D Cse II: Stress Apple rst Apple Stresses cse etween A n C T pple ner constnt stress from C to D Arrve t sme fnl stress n strn t D The nternl work s eql to the re ner the stress strn crves. It s not the sme n oth cses. 5-

11 5- ork of the Apple orces (Eternl ork) Ν Conser n l r wth n pple force e e e k k k ssme tht

12 Generl Eqtons for Eternl ork e S p p s V p V V or mterls wth lner response e S p p s V p V V 5-

13 Some nmentls of Vrtonl Clcls A revew of vrtonl clcls s presente n Appen I of the tetook n smmrze n the followng pges. Vrtonl clcls nvolves fnng etrem of fnctonls. A fncton s sll n epresson nvolvng nepenent vrles; for emple: f f() A fnctonl s fncton of fncton, or fncton of epenent vrles; for emple, f f(, ), where g() n / Vrtonl clcls s se to erve energ concepts sch s the prncpl of vrtl work. The prncple of vrtl work reqres n nerstnng of terms lke vrtl splcements vrtl forces, n vrtl work 5-3

14 Generl Vrtonl Clcls Prolem n the fncton () sch tht Π (, ), s renere sttonr (where ) In other wors, fn the () tht mkes Π n etreme vle ecessr Contons () n the ntervl s known fncton (lke strn energ enst for emple) Usll reqre tht () e twce fferentle wth respect to, ( n est) An e twce fferentle wth respect to,, n 5-4

15 Vrtl Dsplcements n the Vrtonl Opertor, Let () e fml of neghorng pths of the etremzng pth (), ˆ û( ) ˆ ( ) ( ) ( ) ( ) here () s the etremzng pth n s smll vrton w from the etremzng pth. ote tht: ˆ t the enponts otce tht there s g fference etween n s clle the vrtonl opertor n t hs smlr propertes s the fferentl opertor 5-5

16 5-6 Propertes of the Vrtonl Opertor The vrtonl opertor ehves lke the fferentl opertor n mn ws. or emple, ssme (, ), where /, Then: ( ) ( ) ( ) ( ) ( ) n n n

17 nctonls n A fnctonl s n ntegrl epresson whose ntegrn(s) re fnctons of epenent vrles n ther ervtves. ( ) (,, ), where Π The frst vrton of fnctonl cn e clclte s follows Π therefore Π 5-7

18 5-8 Etrem of nctonls Sppose we wsh to fn the etemm of Π, where The necessr conton for the fnctonl to hve n etremm s tht ts frst vrton e zero, or ( ) ( ) ( ) ( ) Π,,,, ( ) Π Π Π

19 5-9 Etrem of nctonls Use ntegrton prts to evlte the secon term n the prevos ntegrl. The generl procere for ntegrton prts: [ ] ( ) ( ) ecomes then,,, Defne t t s ts st st

20 5- Etrem of nctonls (contne) Ths eqton cn e smplfe The vrton of, (), t ponts n mst e zero snce s specfe t those ponts; therefore: < <, then vle, s n rtrr (non - zero) n f levng, Eler Lgrnge Eqton Ths s necessr conton for fncton () to etremze the fnctonl Π.

21 Emple: n the Pth of Mnmm Length Between Ponts n s s Π Π Π Π s ( ) In ths cse: 5-

22 5- Mnmm Pth Emple (Contne) Appl the Eler Lgrnge Eqton: Let for ths prolem: ( ) ( ) The Eler Lgrnge Eqton Becomes: ), (

23 5-3 Mnmm Pth Emple (Complete) Solve the Eler Lgrnge Eqton: ( ) B A A A C C C C C, Appl onr contons t ponts n to fn the nknown constnts A n B.

24 et Clss Dscss some Prolems from H Dscss Semester Projects Defne Vrtl ork Usng Vrtonl Clcls 5-4