1.050 Engineering Mechanics I. Summary of variables/concepts. Lecture 27-37

Transcription

1 .5 Engneeng Mechancs I Summa of vaabes/concepts Lectue 7-37

2 Vaabe Defnton Notes & ments f secant f tangent f a b a f b f a Convet of a functon a b W v W F v R Etena wok N N δ δ N Fee eneg an pementa fee eneg Compementa fee eneg 3 N N N 3 Fee eneg δ N N δ δ δ δ δ 3 Lectues 7 an 8: Basc concepts: Convet, etena wok, fee eneg, pementa fee eneg, ntouce nta fo tuss stuctues see schematc show n the owe ght pat.

3 Vaabe Defnton Notes & ments Tuss pobems v! v R F Compementa otenta eneg eneg : : pot pot At eastc souton: otenta eneg s equa to negatve of pementa eneg ' ' ma N, R ' N S.A. ' ' ' ' N, R s equa to pot δ, mn δ ', ' Lowe boun ' pot δ K.A. Uppe boun Uppe/owe boun At the souton to the eastct pobem, the uppe an owe boun conce Consequence of convet of eastc potentas, Lectues 7 an 8: Intoucton to potenta eneg an pementa eneg, efnton at the eastc souton, uppe/owe boun, eampe of eneg bouns fo tuss stuctues. The uppe/owe bouns of the epessons ae a consequence of the convet of the eastc potentas see pevous se. 3

4 Vaabe Defnton Notes & ments Compementa fee eneg -D Fee eneg -D W F v Contbutons fom etena W R W,W..N..N wok W W Capeon s fomuas Sgnfcance: Enabes one W W cacuate fee eneg, pementa fee eneg, potenta eneg an pot W W pementa eneg ect fom the bouna W W contons etena wok, at the souton taget! Lectues 7-9: The equatons fo fee eneg an pementa fee eneg fo tuss stuctues ae summaze. Lowe pat: Capeon s fomuas, use to cacuate the taget souton, that s, the esuts at the souton. These equatons ae genea va, not on fo tuss stuctues but the epessons of how to cacuate the nvua tems that appea n these equatons ae ffeent. 4

5 Vaabe Defnton Notes & ments ma σ ' σ ' S.A. σ ' s equa to pot ' σ ' S.A. ' K.A. mn pot ' ' K.A. Lowe boun Compementa eneg appoach σ ' σ ' W T ' ' W pot Souton Voume foce contbuton Uppe boun otenta eneg appoach Dspacement contbuton Stess vecto contbuton s σ m Ω Ω K G σ tace σ s σ : σ 3σ m 3 m tace v Ω K G Ω v : v 3 Uppe/owe boun fo 3D eastct pobems Compementa eneg an potenta eneg Etena wok contbutons Compementa fee eneg 3-D, sotopc matea Fee eneg 3-D, sotopc matea Lectue 3: Eneg bouns fo 3D sotopc eastct. Note that the etena wok contbuton une foce stess bouna contons nvoves a voume ntega ue to the voume foces gavt. The owe pat summazes the equatons use to cacuate the fee eneg an pementa fee eneg, as we as the etena wok contbutons etena wok contbuton pat. 5

6 Vaabe Defnton Notes & ments EI M ES N.. Compementa fee eneg fo beams EI ES.. ϑ Fee eneg fo beams Note : Fo D, the on contbutons ae aa foces & moments an aa stans an cuvatues Note : Taget souton usng Capeon s fomuas / / δ δ unknown spacement at pont of oa appcaton Taget souton δ [ ] [ ] R z z R M R R M R W,, ω ω Etena wok b pescbe spacements [ ] [ ] [ ] z z z z M F F f f M F f W.... ω ω Etena wok b pescbe foce enstes/foces/moments Lectue 3: How to cacuate fee eneg, pementa eneg an etena wok fo beam stuctues. 6

7 Vaabe Defnton Notes & ments ma F ', M ' F ',M 'S.A. F ', M ' s equa to pot ',ω ' F ',M 'S.A. pot ',ω ' K.A. ',,ω 'K.A. mn ',ω ' Lowe boun Souton Uppe boun F ', M ' Compementa otenta eneg that pove eneg appoach absoute appoach ma of Dspacement Stess appoach appoach ',ω ' Wok wth unknown that pove Wok wth unknown but S.A. moments an absoute but K.A. foces mn of pot spacements Step : Epess taget souton Capeon s fomuas cacuate pementa eneg AT souton Step : Detemne eacton foces an eacton moments Step 3: Detemne foce an moment stbuton, as a functon of eacton foces an eacton moments nee M an N Step 4: Epess pementa eneg as functon of eacton foces an eacton moments ntegate Step 5: Mnmze pementa eneg take pata evatves w..t. a unknown eacton foces an eacton moments an set to zeo; esut: set of unknown eacton foces an moments that mnmze the pementa eneg Step 6: Cacuate pementa eneg at the mnmum base on esutng foces an moments obtane n step 5 Step 7: Make pason wth taget souton fn souton spacement Step-b-step poceue how to sove beam pobems wth pementa eneg appoach Lectues 3-3: How to sove beam pobems usng the pementa appoach. Ths se shows the ovevew ove the uppe/owe bouns. The owe pat summazes a step b step poceue of how to sove statca netemnate beam pobems wth a pementa eneg appoach. 7

8 Vaabe Defnton Notes & ments Fo an homogeneous beam pobem, the mnmzaton of the pementa eneg wth espect to a hpestatc foces an moments X { R, M, ; } es the souton of the nea eastc beam pobem: X X W W mn X X Eampe: R' R' R' EI M M R Hpestatc foce R' R ' R ' R ' EI 5 7 δ R ' EI δ EI 3 3 Lectues 3-3: Cooa, how to sove statca netemnate beam pobems usng the pementa appoach. Summa of the concept that the mnmzaton of the pementa eneg wth espect to hpestatc foces an moments poves the eact souton of the nea eastc beam pobem. 8

9 Vaabe Defnton Notes & ments Eue beam buckng Dffeent bouna contons Eampe: Eue buckng of a fame stuctue Lectues 33: Buckng of beam stuctues une pessve oa. The owe pat summazes the epement pesente n cass. 9

10 Vaabe Defnton Notes & ments opetes an chaactestc of nstabt phenomenon Images emove ue to copght estctons: photogaph of faut ne, Wo Tae Cente towes, shattee wne gass, X-a of boken bone. Intoucton: Factue appcaton an phenomena Lectues 34: Summa chaactestcs of buckng phenomenon equvaenc of vegence of sees, nonestence of souton/bfucaton pont/oss of convet. Intoucton to factue.

11 Vaabe Defnton Notes & ments γ bei s ma Out-of-pane thckness: b Usefu scang aws b Γ pot G γ G unt s b cack aea Gffth conton fo cack ntaton Lectues 34 an 35: Factue mechancs. The most mpotant concept s the Gffth conton. The eampe on the top summazes the evaton one n cass, epesentng two beams that ae pue awa fom each othe. Ths

12 Vaabe Defnton Notes & ments σ π aσ G. γ E γe. π a σ a Factue n a contnuum Inta suface cack of ength a σ Lectues 35: Factue n contnuum. The equatons summaze n the eft se pove the eneg eease ate G fo the geomet shown on the ght. At the pont of factue, the eneg eease ate must equa the suface eneg. Ths conton can then be use to etemne the ctca stess at whch the stuctue begns to fa.