element k Using FEM to Solve Truss Problems
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1 sng EM t Slve Truss Prblems A truss s an engneerng structure cmpsed straght members, a certan materal, that are tpcall pn-ned at ther ends. Such members are als called tw-rce members snce the can nl transmt r supprt rce alng ther length r as, whether n tensn r cmpressn. A plane truss s ne where all the members and lads le n ne spatal plane ppste t a space r -D truss. Tw-rce members als derm alng ther length r as and nt transverse t t. In the EM, each truss member s cnsdered as an element n the mesh as llws: L element k Each element s essentall cnsdered an elastc sprng the dermatn the member s small. Ths analg actuall cmes naturall ut the dentn stress, stran and Hke s law that cuples the tw: σ Ee Here, the stress σ s dened as usual: σ A, where s the rce actng n the element and A s ts area. Als, stran s dened as bere: L e L, where L s the undermed length the element and L s the stretch r rather the change n length the element. Nte that the last equatns can be re-arranged t wrte: AE L keq L L rm the last equatn we can clearl see that there s a lnear relatnshp between rce actng n an element and ts change length. Ths s eactl lke the rce-etensn relatnshp that ests r a lnear sprng and hence an analg between the tw ests. Therere, the nte elements here can smpl be replaced b lnear sprngs such that the equvalent stness (r sprng cnstant each sprng s gven b: AE k eq L
2 Nte that L s equal t the derence n the aal dsplacement the tw end r ndal pnts an element/member. Hence r an element k wth ndal pnts and, ne can re-sketch the last sketch as: k eq (u u k eq (u u u element k u element k u k eq (u u u k eq (u u r smplct rm nw n, we wll drp the eq subscrpt rm k eq and smpl use k nstead. Mrever, nte that mre apprpratel, u shuld be dented as u and u shuld be dented as u snce bth are ndal dsplacements alng the -drectn and nt the - drectn, where the -crdnate sstem s attached t the member such that the -as pnts rm nde t n the element and the -as curse s perpendcular t that. Such a crdnate sstem s als called a lcal crdnate sstem r lcal rame reerence. In addtn, shuld reall be labeled and shuld be labeled as. Nte als that r a tw-rce member, u and u shuld be equal t zer ust as and shuld be. r nw, hwever, we wll cntnue t nclude them bth n the equatns belw ust r cmpleteness sake and easness the descrptn. Hence, at nde, the transmtted rce n the pstve -drectn s: k (u u Als, at nde, the transmtted rce n the pstve -drectn s: k (u u Alng the -as, hwever, we smpl have: (u u And (u u The last equatns can be wrtten n matr rm as llws: k k u u k k u u The last equatn can be smpl wrtten as: {} {u}
3 T be mre general and precse n ur descrptn, each element n a planar truss wuld n general be rented at sme angle θ rm the -as (glbal -as whch s ed n space and each wuld then lk smethng lke: u u u u θ lbal crdnate sstem θ Mrever, snce each element s rented derentl wth respect t a glbal crdnate sstem, t s necessar t epress all the rces and dsplacements n terms ths glbal crdnate sstem n rder t use ne reerence rame r all the equatns. In ther wrds, we need t wrte each the rce cmpnents and dsplacement cmpnents n terms ths glbal crdnate sstem. The relatnshp between the lcal and glbal dsplacement cmpnents s gven b: u + u u u + + +
4 Alternatvel, we can wrte the glbal dsplacement cmpnents n terms the lcal dsplacement cmpnents as: u u u u u + u u + u Smlar equatns can be wrtten t relate the glbal and lcal descrptns the rce cmpnents: + + raphcall, the relatnshp between the glbal and lcal cmpnents can be llustrated: u u u u θ lbal crdnate sstem θ
5 The last tw sets equatns can be wrtten mre cncsel n matr rms as llws: {} T{u} Where u { }, u T, { u } u u And {} T{} Where {}, { } Nw, startng wth { } { u}, we can nstead wrte: T { } T { } { } { u} Where T Lastl, multpl the abve equatn b T, we get: { } T T { } Snce TT - I In epanded rm, the last equatn s: cs θ cs θ sn θ sn θ k cs θ cs θ sn θ sn θ where s the appled glbal rce vectr r an element wth ndal pnt and,
6 s the glbal ndal dsplacement/delectn vectr r an element wth ndal pnt and, and cs θ cs θ e sn θ sn θ k cs θ cs θ sn θ sn θ s the element stness matr that relates the rces t the dsplacements. Net, the last three matrces need t be epanded t t nt ther respectve pstn n the glbal stness matr, glbal rce vectr and glbal dsplacement vectr. Once ths s accmplshed, the dsplacements can be slved r llwng the applcatn bundar cndtns. Pstprcessng can then be perrmed t nd, r eample, the stress n each element. Eample: Slve the llwng truss prblem b determnng the dsplacements/delectns each ndal pnt and then the stress n each member. Here, E ung s mdulus (Duglas-r wd.9e6 b/n, and A crss-sectnal area a member 8 n. t b b t t ( (6 ( ( ( ( Slutn: Preprcessng: The rst thng t d s t cnstruct a mesh elements and ndes. Each member s cnsdered an element and each member end/nt s a nde. The llwng table lsts the elemental/ndal data: 6
7 Element Nde Nde θ rm sketch ( ( ( ( 9 ( (6 The equvalent stness k r elements (, (, (, and (6 s gven b: AE (8n (.9 E6b/n k eq k.eb/n L 6n r elements ( and (, k s gven b: AE (8n (.9 E6b/n k eq k.98eb/n L.9n Nw, r elements (, (, and (6, θ, and the element stness matrces r these elements becme: cs θ cs θ e sn θ sn θ k cs θ cs θ sn θ sn θ cs sn cs cs sn cs ( sn cs sn sn cs sn.e cs sn cs cs sn cs sn cs sn sn cs sn (.E 7
8 8 The pstn the last matr n the glbal stness matr (as part the assembl prcess s: (.... Smlarl, E (. (.... E (6.
9 9 (6.... Nw, r element (, θ 9, and the stness matr r ths element becmes: E E ( (. 9 sn cs9 sn 9 9 sn cs9 sn 9 cs9 sn 9 9 cs cs9 sn 9 9 cs 9 sn cs9 sn 9 9 sn cs9 sn 9 cs9 sn 9 9 cs cs9 sn 9 9 cs. The pstn the last matr n the glbal stness matr (as part the assembl prcess s: (....
10 Nw, r element (, θ, and the stness matr r ths element becmes: cs sn cs cs sn cs ( sn cs sn sn cs sn.98e cs sn cs cs sn cs sn cs sn sn cs sn ( E The pstn the last matr n the glbal stness matr (as part the assembl prcess s: ( Nw, r element (, θ, and the stness matr r ths element becmes: cs sn cs cs sn cs ( sn cs sn sn cs sn.98e cs sn cs cs sn cs sn cs sn sn cs sn.... ( E
11 The pstn the last matr n the glbal stness matr (as part the assembl prcess s: ( The glbal stness matr r the whle prblem s smpl the sum the glbal stness matrces r the derent elements,.e. the sum the elemental glbal stness matrces. Ths prcess s termed assembl the glbal stness matr: ( (6 ( ( ( ( ( ( Appl bundar cndtns and lads: What s let nw s t slve r the ndal dsplacements usng the equatn that we ve seen bere: } { } }{ { matr lad matr dsplacement matr stness Once ths s dne, we can then resrt back t } { } }{ { } { u R t nd the reactn rces.
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