u 1 Figure 1 3D Solid Finite Elements

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1 Sold Elemets he Fte Elemet Lbrary of the MIDAS Famly Programs cludes the follog Sold Elemets: - ode tetrahedro, -ode petahedro, ad -ode hexahedro sho Fg.. he fte elemet formulato of all elemet types s based o the soparametrc procedure (the elemet geometry ad dsplacemets are terpolated the same ay). Z Y X u u u u u u u u u u u u u u u u u u Fgure D Sold Fte Elemets he odal degrees of freedom (DOF s) are llustrated Fg.. he elemet geometry ad dsplacemet feld are defed terms of odal coordated ad DOF s by the follog fuctos: here u,, = dsplacemets drecto of global X, Y, Z axes, respectely, at th ode x, y, z = x, yz, coordates at x= f(,, ) x u = f(,, ) u = = y = f(,, ) y = f(,, ) u= fq = = z = f(,, ) z = f(,, ) = = th ode

2 f (, ), = terpolato fucto related to system of the elemet th arables,, + ),, = atural coordates of the th ode of the elemet = umber of odes elemet he odal terpolato fuctos f (,, ) are of the follog form: th ode ad defed the atural coordate ( +, +, ad For -ode tetrahedro elemet f ( + ) for =,, = ( + )( ) for= ( + )( )( ) for =, ( ) For -ode petahedro (edge) elemet f (,, ) ( + )( + ) for =, = ( + )( + )( + ) for =,,, For -ode hexahedro (brck) elemet f,, = for =,,,..., ( ) ( )( )( )

3 No, the stras at ay pot th elemet doma are expressed terms of odal dsplacemets as, u 0 0 x x 0 0 ε x y y ε y 0 0 u ε z z z = = ε γ u = du = dfq = Bq xy + 0 γ y x y x yz γ zx + 0 z y z y u + 0 x z z x here B = stra-dsplacemet matrx q = ector of odal dsplacemets he, the stress-stra relato become, σ = Dε = DBq here, D s the elastcty matrx defg mechacal propertes of the materal. For a lear sotropc materal D matrx takes the follog form: ν ν ν ν ν ν E ν 0 0 D = ( + ν)( ν ) ν Sym. 0 ν hch E = Youg modulus ν = Posso s rato

4 Accordgly, the stffess matrx ad force ectors for a typcal soparametrc D sold elemet are defed by the follog tegrals: K = B DB V dv p b = f bdω Ω here K V = stffess matrx = olume of the elemet p = B Dε dω 0 0 Ω A = surface area of the elemet p b = odal force ector due to dstrbuted body forces b p0 = odal force ector due to tal stra ε 0 Ω = rage of the tegrato So far e hae demostrated the stadard soparametrc formulato procedure, hch s detcal for all D solds elemets. It should be oted the -ode tetrahedro ad -ode edge elemets preseted here are fact the degeerated forms of the -ode hexahedro (brck). Formulato of these elemets by collapsg the -ode hexahedra allos ealuatg the aboe tegrals for all types of D sold elemets by use of the same stadard D umercal tegrato procedure, based o the Gauss-Legedre quadrature. hus the umercal tegrato formulas used for D elemets are: here: W l m = j= k= (,, ) (,, ) (,, ) K WW W B DB J = j k k k k k k k l m b = j k k k k k = j= k= (,, ) (,, ) (,, ) p WW W f b J k k l m 0 = WW jwk k k ε0 k k k k = j= k= (,, ) (,, ) (,, ) p B D J = eghtg factor of -th tegrato pot,, = atural coordates of -th tegrato pot (, j, k) J = determat of the Jacoba matrx l, m, = umber of tegrato pots drecto of,, ad, respectely. he approprate order of umercal tegrato ad correspodg locatos of tegrato pots used the D sold elemets are sho able bello.

5 able Gauss-Legedre Itegrato for D Sold Elemets Itegrato order Locato of tegrato pots 9 0 Here t should be oted that the compatble -ode hexahedro elemet based o stadard soparametrc formulato does ot produce accurate results may cases for both dsplacemets ad stresses. I order to mproe the performace of ths elemet are added so called the compatble dsplacemets modes. akg to accout the compatble modes, the dsplacemet feld s defed as, here, P, P, P u = f(,, ) u + αp + αp+ αp = = f(,, ) + αp + αp+ αp = = f(,, ) + αp + αp+ α9p = are the extra shape fuctos related to the compatble modes expressed as, ( ) ( ) ( ) P,, =, P,, = ad P,, = ad α ( =,,,...,9) are so called odeless extra DOF s. Note that the extra shape fuctos permt a parabolc deformato alog the elemet edge ad mproe the elemet stffess performace. Here t should be oted that the fal stffess matrx s obtaed by statc codesato of the

6 compatble modes the same ay as case of -ode quadrlateral D plae elemet.