The finite element method(I for three-diwnensional robles Potential enery applied to one eleet m=(4C4EME:d-(卡c-(t:处5 e 2 e Here is the plcture 七一 vre Introduce an abproximstton for the displacement field u wiMin the element =13k=1n D: number of nodes per cement What can be inferred about the appnokimretvon
② oC 几.7 oximation for stralns =4(+) 2(%4+A9) Procedre Is the sawe as before lace in potent ie I. minimize wilth respet to hodal displacements 取 s obtain finite element matrices stiffness I oroe vecor [R] Expressions look simpler if we Write in Matrix form
3 从→从≡风 e r 3×1 从L n)∈Rx E22 233 E13 En 。0;→=(sn)eRx V22 923 2 Gce C eRGXG =11 t→七=)∈Rx 七 1七
otential now reads ETCad-「.+d-utds e ve 4x66×66x1 4x233×4 333xL 4x1 1 olisplaevoue- He ue U: vedor of nodal displaced 8x4 What is the diMension of Ue And H? -{鸣噶呀…C" strains Ee- B Ue ‖d6x16x3n3nx1 Obviously He is obtained frow k and be from Hheir derivatives。 eplace in polential:
⑤ A(BCBU_「(严Hfd-(Hts 2 =AT(▲ BCBdy u_U(Hf([中ds 2 e K UT R e⊥ UT KU_ uTR where 2 ke BcB d3nx66x5×32 Ne nxn k-(Fd+「H+ds5as e 3n 1 lpon minimization ot re will resped to nodal displacements0° KUe=R° 3n×3n3nx13x1
Exawple: plane stress. linear square element Model thick ④|③ 七 ②2 4 三三三三 3 (4 3 4 Basls functions: g=4(1+2)()2 =4(1一)(+ etc 4=4(1-81(1)-4(1)(1-)
⑦ A={}=「。99Q01(0) o.b0亮 A」M 丿 5~y B u,y+U B-「4×O4kOAx,O7 o ry o yo piy bix ky oix,y osix puy p ×=4(+)y=4(什 x=-1(1+9)中y=4(一8 ×=-4(4-4y=-4(4-8 4,x=4(1-9)中 Hny
Replace in B, then in K with 4-yu10 See Mathemnahca file