MEG 741 Energy and Variational Methods in Mechanics I

Transcription

1 MEG 741 Energy nd rtonl Methods n Mechncs I Brendn J. O oole, Ph.D. Assocte Professor of Mechncl Engneerng Howrd R. Hghes College of Engneerng Unversty of Nevd s egs BE B-1 (7) bj@me.nlv.ed Chter : Prncles of rtl Work: Integrl orm of the Bsc Eqtons 7-1

2 Project ocs Nme Elml Govndrj Rghnndn A. Krmchet bhr Bndyodhyy Unnkrshnn llyl enkt l Bdgr resh hhdr Rhmn Phn P. Gdt Debjyot Mtr Umknth kry jgdesh k yelvrth Krmbl, Ancl Rdhkrshnn nthnkrshnn Jgnndh Ro Nrrj KARHIK DOPPAA rjnb rdhrl krshn.j.v.knchm Kmr URYA KIRAN PARIMI rshnth reddy D.K jysrthy brmnn Jml M. Renno INOD K CHAKKA rm s korell srnvs chnd Jgdee hot John Motk Jynt Ptl Robn Jenkns Erk Wolf Blj dsvm Project oc Mterl Performnce Delyed Hydrogen Crckng of zrconm lloys Hgh em roertes of Nckel bsed lloys C Postron Annhlton method for resdl stress mesrement stress corroson crckng for zrconm lloys tress corroson crckng on self-loded smles Resdl tress mesrement wth Netron dffrcton lre Behvor of yntctc oms Hydrogen embrttlement of mrtenstc lloys Electrcl roertes Coer ngsten comostes Dmng Proertes of Mterls Hgh eed Photogrhy for mterl chrcterzton (eckle nterferometry) Rer methods for olymer comostes mechncs of nnoscle mterls or strctres trctrl Dynmcs/hock Progton/Imct Dynmc nlyss of bolted jonts Dynmcs of dhesve jonts Dynmcs of comoste ltes Dynmcs of ltes Imct of honeycomb mterls leble bem cttors onc olymer cttors MR olymer shock solton trn bsed control of fleble mnltors trctrl Dmng lt Hoknson Pressre Br for dynmc mterl chrcterzton AE methods for fld-strctre ntercton Etreme odng of trctres Internl blst lodng on vessels blst lodng of vehcle strctres Mchne Desgn ehcle trctres hock/brton of ehcle et trctres blst lodng of bldng strctres 7-

3 Clss Otlne Aly Prncle of rtl Work to rss Problems Bem Problems Revew rss Dslcement Problem 7-3

4 Prncle of rtl Work If contnos body s n eqlbrm, the vrtl work of ll ctl forces n movng throgh vrtl dslcement s zero. he vrtl dslcements, δ, mst stsfy knemtc eqtons nd bondry condtons σ j δε δε d j σd δw δw δ + δw δ d d e δ d δ d where re the led body forces, re the led eternl forces, s the volme, nd s the orton of the srfce hvng led lods. δε j δ ( ) 1, j + j, 7-4

5 σ Prncle of rtl Work j δε δε d j σd δ δ d d δ d δ d We wnt to mke the solton rocedre s smle s ossble. We do not wnt to erform comle ntegrls for every roblem. he generl ln of ttck s to: Convert ll terms to volme ntegrls hen combne ll the ntegrnds nder one volme ntegrl hen fctor ot the vrtonl oertor nd other common constnt or vrles n ech term hen set the ntegrnd eql to zero (slly smle eresson) 7-5

6 Aly Gss (Dvergence) Integrl heorem It s often necessry to trnsform srfce ntegrls nto volme ntegrls or volme ntegrls nto srfce ntegrls. hs s ccomlshed sng Gss Integrl heorem: v d dv vd where v s n rbtrry vector nd s the otwrd norml vector t ont on. dvv v, v1,1 + v, + v3, 3 v + v y y + v z z 7-6

7 Bondry Condtons orce Bondry Condtons long ths edge. Dslcement Bondry Condtons long ths edge. trctrl Body he entre bondry of the strctre mst be defned s OR. on orce bondry condton my be ressre, moment, ont lod, or zero lod. Where re the rescrbed dslcements. Aled srfce forces (er nt re) re referred to s srfce trctons. on How re srfce forces relted to srfce stresses on the bondry? 7-7

8 rfce orce-tress Reltonsh orce Blnce n the -drecton y ds ( ds) + τ ( ds) σ, or σ y + τ y y y ds σ τ y dy ds θ y θ A smlr eqton s fond for the force n the y-drecton. A smlr eqton cold be derved n 3-D. he generl 3-D reltons cn be wrtten s: y z y z y z σ σ y σ z z τ y y τ z τ yz τ y d σ y y ds he generl Eqtons n vector nd nde notton: A σ σ j 7-8

9 Aly Gss heorem to the Eternl rtl Work rfce Integrl erm δ d bsttte defnton of σ δ d j j Rerrnge order of oertons ( σ δ ) j j d Aly Gss s heorem δ d ( σ δ ) j, j d End rtl dervtve oerton δ d ( σ δ + σ δ )d j, j j, j 7-9

10 Prncle of rtl Work Cn be frly esy to ly for some roblems. Recll tht we derved the strn energy for bem member erler: U N EA + M EI d If the strctre of nterest s comosed of trss members (constnt l forces only), then ths eresson s even smler: U N d EA N EA 7-1

11 7-11 Eternl Work De to Pont ods e e e W k d k W k d W 1 1 ssme tht P v A smlr nlyss cn be condcted for the work done by the vertcl force so tht the totl eternl work becomes: e v W P 1 1 +

12 c 4 3 rss Emle (Work Eqlbrm Method) 3 P 4 l δ b b ttcs: Geometry: 4 l δ nd the vertcl deflecton t ont b de to the led force P. W Assme tht both members nd hve the sme cross-sectonl re, A, nd modls, E. 1 P e v N W U 4 P N N EA P ( P) ( ) ( P) ( ) 5 4 EA 5 4 v f + EA Pl.78 AE N EA Pv f 7-1

13 7-13 rss Emle (rtl Work Method) Assmng no body forces nd only sngle led force P c b ( ) W W W + P A d d d d d e δ δε σ δ δ δ δ δ δ δ δ σ ε σ ε Where 1,, dentfes the trss member

14 rss Emle (rtl Work Method) he ertcl orce, P, wll cse horzontl (H) nd vertcl () dslcement. Consder these dslcements sertely. c c 53.1 P 53.1 P l Geometry: 4 l b 53.1 b 36.9 H H H.8. 6 H.6H H. 8H otl Chnge n ength of Brs: + H.8 +.6H + H H 7-14

15 rss Emle (rtl Work Method) c 53.1 A Horzontl dmmy force, Q, wll be necessry for some of the vrtl work clcltons. hs force wll be set to zero lter n the roblem. P Wrte the chnge n length, strn, nd stress n the brs s fncton of the ctl dslcements, nd H. l Q Geometry: 4 l b H b otl Chnge n ength of Brs: ε + + H H.8 trn n ech br:.6 +.8H.8 +.6H.8 +.6H 36.9 σ σ ε Eε Eε E.6 +.8H.6 tress n ech br: E.8 +.6H (.8 ).6 +.8H ( )

16 c 53.1 l Geometry: 4 l Q rss Emle (rtl Work Method) P b H δh b Aly vrtl dslcement n the vertcl drecton nd ly the rncle of vrtl work. δ Pδ σ δε A + σ Where Where σ nd σ re the stresses csed by the ctl dslcements nd H, ( Eq.1) δε nd δε re the vrtl strns csed by the vrtl dslcement δ only. δε σ nd σ re the stresses csed by the ctl dslcements nd H, δε nd δε re the vrtl strns csed by the vrtl dslcement δh only. A Aly the rncle of vrtl work n the horzontl drecton. Qδ H σ δε A + σ δε A ( Eq. ) 7-16

17 σ δε δε δε σ δε δ δ δ Eε Eε rles:.8.6 rtl trns de to δ: δ.8δ.8.6δh.8.8δh.6 E E δ rss Emle (rtl Work Method).8 +.6H (.8 ).6 +.8H ( ).6 δ rtl trns de to δh: 3δH 4 4δH 3 QδH QδH Pδ Pδ Q AE P AE P AE P AE bsttte vrles nto eqtons 1: δε A + σ δε A ( Eq. ).8 +.6H 3δH E(.8 )( 4 )(.8) A H 4δH E(.6 )( 3 )(.6) A 3 4 (.8 +.6H )( 4 ) + (.6 +.8H )( 3 ) H + (.6) + (.8) H σ δε A + σ δε A ( Eq.1).8 +.6H δ E(.8 )( )(.8) A H δ E(.6 )( )(.6) A.8 +.6H.6 +.8H ( )(.8) + ( )(.6) σ H +.H (Eq 1) +.H.6 +.8H bsttte vrles nto eqtons : 9 (Eq ) 7-17

18 rss Emle (rtl Work Method) H H (Eq ) P AE P AE P AE +. +.H ( ) P.78 AE P H.96 AE (Eq 1) 7-18

19 Comrson of rtl Work Method nd Drect Energy Blnce Method Note tht: We rrve t the sme solton for sng ether method. We lso get solton for H wth the vrtl work method. he vrtl work method s more cmbersome thn smle energy blnce. he energy blnce method wll only work f yo cn solve for nternl forces n ech member. he vrtl work method cn work for sttclly ndedermnnt roblems lso. We dd not hve to know the ect fnl oston. We gessed tht H wold move to the rght. We obtned H so t ctlly moves to the left. Eternl Work from Pont ods: Eternl Work of ont force, P, movng throgh dslcement, : Eternl rtl Work of ont force, P, beng moved throgh vrtl dslcement, δ: 1 W e P δw e Pδ 7-19

20 Another rss Emle ee emle.4 for nother trss emle 7-

21 Net Clss mmrze Chter 7-1