MEG 741 Energy and Variational Methods in Mechanics I

Transcription

1 MEG 741 Energy and aratonal Methods n Mechancs I Brendan J. O Toole, Ph.D. Assocate Professor of Mechancal Engneerng Howard R. Hughes College of Engneerng Unversty of Nevada Las egas TBE B-122 (702) bj@me.unlv.edu Chater 2: Prncles of rtual Work: Integral Form of the Basc Equatons 6-1

2 Revew Problems from HW 1 6-2

3 Project Tocs Name Elumala Govndaraj Raghunandan A. Karamchet Subhra Bandyoadhyay Unnkrshnan allyl enkat Sla Budugur Suresh Shahdur Rahman Phan P. Gudat Debajyot Matra Umakanth Sakaray jagadesh k yelavarth Kaarambl, Ancla Radhakrshnan Santhanakrshnan Jagannadha Rao Naraaraju KARTHIK DOPPALA Srujanbabu Srdharala krshna.j.v.kuncham Kumar SURYA KIRAN PARIMI rashanth reddy D.K jayasarathy Subramanan Jaml M. Renno INOD K CHAKKA rama s korell srnvas chanda Jagadee Thota John Motaka Jayant Patl Robn Jenkns Erk Wolf Balaj Sadasvam Project Toc Materal Performance Delayed Hydrogen Crackng of zrconum alloys Hgh Tem roertes of Nckel based alloys C22 Postron Annhlaton method for resdual stress measurement stress corroson crackng for zrconum alloys Stress corroson crackng on self-loaded samles Resdual Stress measurement wth Neutron dffracton Falure Behavor of Syntactc Foams Hydrogen embrttlement of martenstc alloys Electrcal roertes Coer Tungsten comostes Damng Proertes of Materals Hgh Seed Photograhy for materal characterzaton (Seckle nterferometry) Rear methods for olymer comostes mechancs of nanoscale materals or structures Structural Dynamcs/Shock Proagaton/Imact Dynamc analyss of bolted jonts Dynamcs of adhesve jonts Dynamcs of comoste lates Dynamcs of lates Imact of honeycomb materals Flexble beam actuators onc olymer actuators MR olymer shock solaton Stran based control of flexble manulators Structural Damng Slt Hoknson Pressure Bar for dynamc materal characterzaton ALE methods for flud-structure nteracton Extreme Loadng of Structures Internal blast loadng on vessels blast loadng of vehcle structures Machne Desgn ehcle Structures Shock/braton of ehcle Seat Structures blast loadng of buldng structures 6-3

4 Defne rtual Work Usng aratonal Calculus 6-4

5 rtual Dslacements A gven body can take many ossble confguratons consstent wth the geometrc constrants of the system. Only 1 confguraton (the actual one) satsfes equlbrum condtons. The set of confguratons that satsfy geometrc constrants but not necessarly equlbrum s called the set of admssble confguratons. If these admssble confguratons are only nfntesmally dfferent than the actual confguraton, they are know as vrtual dslacements 6-5

6 rtual Forces A set of forces that are n equlbrum amongst themselves. Can be nternal or external Not necessarly related to actual forces Examle A vrtual ont load on a cantlever beam must be balanced by a vrtual reacton force and moment at the wall. δm=l* δp δp F (actual force) L δp 6-6

7 rtual Work rtual work done by actual forces movng through a vrtual dslacement: δ W = F δud Comlementary rtual work done by vrtual forces movng through an actual dslacement: δ W * = δf ud 6-7

8 External rtual Work Gven a sold body,, subjected to body forces,, and surface tractons,, on the boundary, S The external vrtual work s defned as: δ W e = δu ds + S where ds s a surface element and S s the orton of the boundary for whch stresses are secfed The comlementary external vrtual work due to vrtual body forces and vrtual surface tractons s defned as: δ u d δ W * e = δ u ds + S δ u d 6-8

9 Internal rtual Work Forces on a body cause t to deform resultng n stresses and strans The forces assocated wth the stresses cause dslacements corresondng to the strans whch means work s beng done The nternal vrtual work due to vrtual dslacements (whch nduce vrtual strans): δ W = σ δε j j d 6-9

10 Prncle of rtual Work If a contnuous body s n equlbrum, the vrtual work of all actual forces n movng through a vrtual dslacement s zero. The vrtual dslacements, δu, must satsfy knematc equatons and boundary condtons σ j δε δε d T j σd δw = δw δu T + δw δu d d e S = S 0 δu ds δu T ds = = 0 0 where are the aled body forces, are the aled external forces, s the volume, and S s the orton of the surface havng aled loads. δε j = δ ( u ) 1 2 u, j + j, 6-10

11 Next Class Aly Prncle of rtual Work to Truss Problems Beam Problems Revew Other Energy Prncles Statonary Potental Energy Comlementary rtual Work Statonary Comlementary Energy Generalzed aratonal Prncles 6-11