MEG 741 Energy and Variational Methods in Mechanics I
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1 EG 74 Energy nd rton ethod n echnc I Brendn J. O Tooe, Ph.D. Aocte Profeor of echnc Engneerng Hord R. Hughe Coege of Engneerng Unerty of Ned eg TBE B- (7) j@me.un.edu Chter 4: Structur Any ethod I: Bem Proem -
2 C Outne Sgn Conenton Fundment Reton Eement trce Stffne trce -
3 Sgn Conenton for Bem Eement N z x + + N Sgn Conenton N z x + + N Sgn Conenton -
4 Sgn Conenton for Bem Proem Sgn Conenton : Uuy ued hen nyzng em. The force on the end of the dgrm rereent the ntern recton Sgn Conenton : Uuy ued for Bem Sytem. Dcement nd nge re the me for oth conenton. -4
5 Stchng Beteen Sgn Conenton Ue to rereent force nd moment hen ung gn conenton Ue to rereent force nd moment hen ung gn conenton Equton for conertng gn conenton to gn conenton. -5
6 - the em for Sgn Conenton. the force (nd moment) of degnte the em for Sgn Conenton. the force (nd moment) of degnte K K Summry of Force ector Notton
7 Fundment Reton for Bem Eement Sgn Conenton Equrum Reton Wrte Force nd oment equrum equton: + N Geometry of Deformton Shon ter ter Shon ter N -7
8 Geometry of Deformton -8
9 ter -9
10 Summry of Fundment Equrum Reton Geometry of Deformton ter -
11 - Trnfer trx for Bem The mtrx U trnfer the re,,, nd from xx to xx. The ector z of dcement nd force ced the tte ector ecue t fuy decre the reone or tte of the em. z U z
12 - Trnfer trx (contnued) ( ) + + U U U U U U The ector rereent dcement nd oe. The ector rereent force nd moment (th Sgn Conenton ). U rereent rgd ody dcement. U rereent mter. U (n th ce), rereent nfuence of rng, foundton, etc. U rereent equrum.
13 - Stffne trx The tffne mtrx for eement rete the dcement (nd oe) to the force (nd moment) t nd. The tffne mtrx defned ung force ector of Sgn Conenton
14 -4 Stffne trx (contnued) To comete the defnton of n eement, ector, rereentng ed od houd e dded to Ech term n the tffne mtrx, j, cn e condered the force deeoed t coordnte due to unt dcement t coordnte j, th other dcement equ to zero.
15 Fexty trx The fexty mtrx for eement rete the force (nd moment) to the dcement (nd oe) t nd. The fexty mtrx f the nere of the tffne mtrx. The reou tffne mtrx rete of the dcement to of the force, thout regrd to ho the eement uorted or contrned. The rgd ody dcement term cue ome of the ro or coumn of the tffne mtrx to e nery deendent. A fexty mtrx cnnot e found for the reou tffne mtrx ecue the nery deendent term me the mtrx ngur nd t cn not e nerted. A fexty mtrx cn e defned for reduced tffne mtrx tht defned for em th ecfc contrnt or oundry condton. -5
16 - Reduced Stffne trx nd Fexty trx for Smy Suorted Bem 4 4 equton. reducedytemof Exmnetheecondnd fourth ro 4 4 GenerStffne trx for Bem Eement: R R R R R R f
17 Exme: Fnd the reduced tffne mtrx retng force t the end of cnteer em to the dcement t the end. nd f f f f It uuy eer to determne the term n the fexty mtrx. Ech term, f j, the dcement t degree-of-freedom (DOF) due to unt force ed t DOF j. The degree of freedom for th roem re nd. f dcement due to unt her force. f dcement due to unt endng moment. f oe (rotton) due to unt her force. f oe (rotton) due to unt endng moment. -7
18 Exme: Dcement nd oe t the end of cnteer th n ed unt her force nd n ed unt endng moment. Deformed he f f Defecton nd oe for thee roem cn e found ung ny method from: Deformed he f f echnc of ter, Etcty theory, or Energy ethod -8
19 -9 Exme (Contnued): Fnd the reduced tffne mtrx retng force t the end of cnteer em to the dcement t the end. The fexty mtrx emed from the term found on the reou ge. f f f f The tffne mtrx found y nertng the fexty mtrx. 4
20 Stffne trx Bed on Poynom Tr Functon For em eement, t oe to determne tffne mtrx fry my from three dfferent ny technque: mechnc of mter, theory of etcty, energy method. For ome eement, t dffcut to determne the exct outon needed to formute the tffne mtrx. In thee ce, t oe to determne tffne mtrx y umng the dcement re ome oynom functon of oton. Th oynom tr functon method cn e ued to dere the me tffne mtrx defned reouy for em eement. -
21 Next C Stffne trx Bed on Poynom Tr Functon -
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