Présentation PowerPoint

Présentation PowerPoint

Cours MEF : Les structures treillis Etude des Treillis F F Premier thme du parcours Prsentation de llment barre tude des structures treillis Supports du cours Site web : https://pedagogie.ecnantes.fr/meefi/ Cours MEF : Les structures treillis Les Treillis F Poutres relies par des rotules Traction - compression F Modle barre f PFD On isole une tranche dx N dN N dx x 0, x x 0, dN fdx Sudx Comportement Su ESu, xx f ESu, x N d (t ) sur D u ud (t ) Du Les conditions aux limites N ESu,x sur "EDP" quations aux Drives Partielles Cours MEF : Les structures treillis

Les Treillis F Modle barre f Fo 0 u(M,t) F Poutres relies par des rotules Traction - compression F PTV Principe des Travaux Virtuels l A W xx xx o u xx u, x Sdx avec xx E xx Eu, x Su u dx ESu,x u,x dx + f o o o Ed Formulation variationnelle u dx + Fo uo + F u 2 2 Ed ES u,x dx o Cours MEF : Les structures treillis

Les Treillis F Poutres relies par des rotules Traction - compression F Equivalence des formulations "EDP 2u EL x 0, Su ES 2 f 0 " x P P ( Su ES 0 2u u P ES dx P ES x 2 x 0 0 P 2u x 2 re f ) dx 0 1 Forme intgrale Formulation forte P u ES x x dx 0

P u u P Sudx + x ES x dx P ES x 0 Pfdx 0 0 0 CL P u "PTV" Cours MEF : Les structures treillis Les Treillis Elment fini "Barre " i ui (e) xo x F j Poutres relies par des rotules Traction - compression F uj 2 variables nodales approximation linaire a1 (t ) u ( x, t ) a1 a2 x 1 , x a ( t ) 2 Maths Identification nodale u N Ue Approximation nodale x x ui ( t ) u ( x, t ) 1 , u

( t ) e e j x N 1 ( x ) 1 e 1 x N 2 ( x) e 1 N 1 0 0 x/ l e 1 N Physique 2 1 x/ l e Fonctions dinterpolation Cours MEF : Les structures treillis Les Treillis Poutres relies par des rotules Traction - compression F F Elment fini "Barre Matrice" raideur e lmentaire e Ed ESu,x u,x dx u,xT ES u,x dx o o u,x N , x Ed U e T

ui u j e T N , x ES N, x dx U e o e Ke N, x T ES N , x dx o Ke ES e 1 1 1 1 Matrice raideur lmentaire Cours MEF : Les structures treillis Les Treillis F Elment fini "Barre " Changement de base en 2DT vi x U e ui vi u j v j ui j (e) yo i xo

ui C u j 0 Poutres relies par des rotules Traction - compression F u u C S v ui S 0 0 vi T U P U e e 0 C S u j v j Pour lnergie 2 Ed U e T P T ES 1 1 T P U U Ke U e e e 1 1 ES e K e e Pour les ES ES contraintes N (u j ui ) C e

e u j ui S v v j i A A A A C2 avec A C S C S S2 Cours MEF : Les structures treillis Les Treillis Elment fini "Barre Analyse" dun treillis F a yo X1 xo F N nuds vecteur des dplacements T nodaux U a 2 Y1 F Poutres relies par des rotules Traction - compression Y2 u1 v1

u2 v2 u3 v3 6 DDL Conditions aux limites 6 inconnues Assemblage des matrices lmentaires Rsolution du systme matriciel Post traitement F T X1 Y1 0 Y2 Systme rduit sur U red T F u2 K U F u2 u3 0 u3 v3 6 quations v3 puis X1 Y1 Y2 ES Efforts sur les lments Lois de comportement N (u j ui ) e Equilibre de chaque lment (4 quations) Efforts aux noeuds Cours MEF : Les structures treillis A vous de jouer Traitez les exemples et exercices de cours Vous verrez Ltude des treillis par la MEF est plus simple que par et plusla la RDM structure est hyperstatique plus cest simple

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