Finite Elements Formulations

Transcription

1 Chapter Fnte Elements Formulatons ( ) (In collaboraton wth Dr. Zhao Cheng, Dr. Nma Tafazzol, Prof. José Antono Abell Mena), Mr, Yuan Feng, Mr. Sumeet Kumar Snha 69

2 Jeremć et al... CHAPTER SUMMARY AND HIGHIGHTS 7 of. Chapter Summary and Hghlghts. Formulaton of the Contnuum Mechancs Incremental Equatons of Moton Consder the moton of a general body n a statonary Cartesan coordnate system, as shown n Fgure (.), and assume that the body can experence large dsplacements, large strans, and nonlnear consttutve response. The am s to evaluate the equlbrum postons of the complete body at dscrete tme ponts, t, t,..., where t s an ncrement n tme. To develop the soluton strategy, assume that the solutons for the statc and knematc varables for all tme steps from to tme t nclusve, have been obtaned. Then the soluton process for the next requred equlbrum poston correspondng to tme t+ t s typcal and would be appled repettvely untl a complete soluton path has been found. Hence, n the analyss one follows all partcles of the body n ther moton, from the orgnal to the fnal confguraton of the body. In so dong, we have adopted a agrangan ( or materal ) formulaton of the problem. t t+ t x3, x3, t t+ t x x,, x t P( x,, ) x 3 V x 3 P( P( x, t t x, ) x x 3 Confguraton at tme A t+ t x t x t+ t x t+ t t+ t t+ t x ), x x, 3 t V Confguraton at tme = = = t A x x t x t n t+ t u t u u t+ t t+ t A V Confguraton at tme t+ t =,,3 t x, x, n+ t+ t x Fgure.: Moton of body n statonary Cartesan coordnate system detaled dervatons and explanatons are gven n Bathe (98) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

3 Jeremć et al... FORMUATION OF THE CONTINUUM M... 7 of In the agrangan ncremental analyss approach we express the equlbrum of the body at tme t+ t usng the prncple of vrtual dsplacements. Usng tensoral notaton ths prncple requres that: t+ t V t+ t σ j δ t+ t ǫ j t+ t dv = t+ t R (.) where the t+ t σ j are Cartesan components of the Cauchy stress tensor, the t+ t ǫ j are the Cartesan components of an nfntesmal stran tensor, and the δ means varaton n.e.: δ t+ t ǫ j = δ ( u t+ t + u ) j x j t+ t = ( δu x t+ t + δu ) j x j t+ t x (.) It should be noted that Cauchy stresses are body forces per unt area n the confguraton at tme t + t, and the nfntesmal stran components are also referred to ths as yet unknown confguraton. The rght hand sde of equaton (.),.e. t+ t R s the vrtual work performed when the body s subjected to a vrtual dsplacement at tme t+ t: t+ t R = t+ t V ( ) t+ t f B ρü t+ t δu t+ t t+ t dv + t+ t f S δu t+ t t+ t ds (.3) t+ t S where t+ t f B and t+ t f S are the components of the externally appled body and surface force vectors, respectvely, and ρü t+ t s the nertal body force that s present f acceleratons are present 3, δu s the th component of the vrtual dsplacement vector. A fundamental dffculty n the general applcaton of equaton (.) s that the confguraton of the body at tme t+ t s unknown. The contnuous change n the confguraton of the body entals some mportant consequences for the development of an ncremental analyss procedure. For example, an mportant consderaton must be that the Cauchy stress at tme t+ t cannot be obtaned by smply addng to the Cauchy stresses at tme t a stress ncrement whch s due only to the stranng of the materal. Namely, the calculaton of the Cauchy stress at tme t+ t must also take nto account the rgd body rotaton of the materal, because the components of the Cauchy stress tensor are not nvarant wth respect to the rgd body rotatons. The fact that the confguraton of the body changes contnuously n a large deformaton analyss s dealt wth n an elegant manner by usng approprate stress and stran measures and consttutve relatons. Ensten s summaton rule s mpled unless stated dfferently, all lower case ndces (,j,p,q,m,n,o,r,s,t,... ) can have values of,, 3, and values for captal letter ndces wll be specfed where need be. 3 Ths s based on D Alembert s prncple. However, that problem wll not be addressed n ths work snce ths work deals wth Materal Nonlnear Only analyss of solds, thus excludng large dsplacement and large stran effects. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

4 Jeremć et al... FORMUATION OF THE CONTINUUM M... 7 of When solvng the general problem 5 one possble approach 6 s gven n Smo (988). Prevous dscusson was orented toward small deformaton, small dsplacement analyss leadng to the use of Cauchy stress tensor σ j and small stran tensor ǫ j. In the followng, we wll brefly cover some other stress and stran measures partcularly useful n large stran and large dsplacement analyss. The basc equaton that we want to solve s relaton (.), whch expresses the equlbrum and compatblty requrements of the general body consdered n the confguraton correspondng to tme t + t. The consttutve equatons also enter (.) through the calculaton of stresses. Snce n general the body can undergo large dsplacements and large strans, and the consttutve relatons are nonlnear, the relaton n (.) cannot be solved drectly. However, an approxmate soluton can be obtaned by referrng all varables to a prevously calculated known equlbrum confguraton, and lnearzng the resultng equatons. Ths soluton can then be mproved by teratons. To develop the governng equatons for the approxmate soluton obtaned by lnearzaton we recall that the solutons for tme, t, t,...,t have already been calculated and that we can employ the fact that the nd Pola Krchhoff stress tensor s energy conjugate to the Green agrange stran tensor: V ( ) t S j δ t ǫ ρ (t j dv = V t ρ tx t,m σ mn t x j,n x k, δ t ) t tǫ kl x l,j dv = V because: and snce: we have: t x k,l t x l,m = δ km ρ dv = t ρ t dv V ρ t ρ t σ mn δ t ǫ mn dv(.) t S j δ t ǫ j dv = t σ mn δ t tǫ t mn dv (.5) V 5 That s, large dsplacements, large deformatons and nonlnear consttutve relatons. 6 Ths s stll a hot research topc! Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

5 Jeremć et al... FORMUATION OF THE CONTINUUM M of where nd Pola Krchhoff stress tensor s defned as: t S j = ρ t ρ tx t,m σ mn t x j,n (.6) where tx j,n = x t x m and ρ t ρ represents the rato of the mass densty at tme and tme t, and the Green agrange stran s defned as: t ǫ j = ( t u,j + t u j, + t ) t u k, u k,j (.7) Then, by employng (.5) we refer the stresses and strans to one of these known equlbrum confguratons. The choce les between two formulatons whch have been termed total agrangan and updated agrangan formulatons. In the total agrangan formulatons, also termed agrangan formulaton, all statc and knematc varables are referred to the ntal confguraton at tme, whle n the updated agrangan formulaton all statc and knematc varables are referred to the confguraton at tme t. Both the total agrangan and updated agrangan formulatons nclude all knematc nonlnear effects due to large dsplacement, large rotatons and large strans, but whether the large stran behavor s modeled approprately depends on the consttutve relatons specfed. The only advantage of usng one formulaton rather than the other les n ts greater numercal effcency. Usng (.5) n the total agrangan formulatons we consder ths basc equaton: V t+ t S j δ t+ t ǫ j dv = t+ t R (.8) whle n the updated agrangan formulatons we consder: t V t+ t t S j δ t+ t t ǫ t j dv = t+ t R (.9) n whch t+ t R s the external vrtual work as defned n (.3). Approxmate soluton to the (.8) and (.9) can be obtaned by lnearzng these relatons. By comparng the total agrangan and updated agrangan formulatons we can observe that they qute analogous and that, n fact, the only theoretcal dfference between the two formulatons les n the choce of dfferent reference confguratons for the knematc and statc varables. If n the numercal soluton the approprate consttutve tensors are employed, dentcal results are obtaned. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

6 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 7 of Couplng of large deformaton, large dsplacement and materal nonlnear analyss s stll the topc of research n the research communty. Possble drecton may be the use of both agrangan and Euleran formulaton ntermxed n one scheme..3 Fnte Element Dscretzaton Consder the equlbrum of a general three dmensonal body such as n Fgure (.) (Bathe, 996). The external forces actng on a body are surface tractons f S and body forces f B. Dsplacements are u and stran tensor 7 s ǫ j and the stress tensor correspondng to stran tensor s σ j. x 3 f B 3 f S 3 f S f S r 3 f B f B r x r x Fgure.: General three dmensonal body Dynamc equlbrum equaton s gven as σ j,j = f ρü (.) where σ j,j s a small deformaton (Cauchy) stress tensor, f are external (body (f B) and surface (fs ) ) forces, ρ s materal densty and ü are acceleratons. Inertal forces ρü follow from D Alembert s prncple (D Alembert, 758). Above equaton can be premultpled wth vrtual dsplacements δu and then ntegrated by parts to obtan the weak form, as further elaborated below. 7 small stran tensor as defned n equaton: ǫ j = (u,j +uj,). Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

7 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 75 of Assume that the externally appled forces are gven and that we want to solve for the resultng dsplacements, strans and stresses. One possble approach to express the equlbrum of the body s to use the prncple of vrtual dsplacements. Ths prncple states that the equlbrum of the body requres that for any compatble, small vrtual dsplacements 8 mposed onto the body, the total nternal vrtual work s equal to the total external vrtual work. Ths statement can be mathematcally expressed usng equaton (.) for the body at tme t + t, and snce we are usng the ncremental approach let us drop the tme dmenson, so that all the equatons are mposed for the gven ncrement 9, at tme t+ t. The equaton s now, usng tensoral notaton : V σ j δǫ j dv = V ( ) f B ρü δu dv + f S δu ds (.) S The nternal work gven on the left sde of (.) s equal to the actual stresses σ j gong through the vrtual strans δǫ j that corresponds to the mposed vrtual dsplacements. The external work s on the rght sde of (.) and s equal to the actual (surface) forces f S and (body) forces f B ρü gong through the vrtual dsplacements δu. It should be emphaszed that the vrtual strans used n (.) are those correspondng to the mposed body and surface vrtual dsplacements, and that these dsplacements can be any compatble set of dsplacements that satsfy the geometrc boundary condtons. The equaton n (.) s an expresson of equlbrum, and for dfferent vrtual dsplacements, correspondngly dfferent equatons of equlbrum are obtaned. However, equaton (.) also contans the compatblty and consttutve requrements f the prncple s used n the approprate manner. Namely, the dsplacements consdered should be contnuous and compatble and should satsfy the dsplacement boundary condtons, and the stresses should be evaluated from the strans usng approprate consttutve relatons. Thus, the prncple of vrtual dsplacements embodes all requrements that need be fulflled n the analyss of a problem n sold and structural mechancs. The prncple of vrtual dsplacements can be drectly related to the prncple that total potental Π of the system must be statonary. In the fnte element analyss we approxmate the body n Fgure (.) as an assemblage of dscrete fnte elements wth the elements beng nterconnected at nodal ponts on the element boundares. The dsplacements measured n a local coordnate system r, r and r 3 wthn each element are assumed to 8 whch satsfy the essental boundary condtons. 9 t+ t wll be dropped from now one n ths chapter. Ensten s summaton rule s mpled unless stated dfferently, all lower case ndces (,j,p,q,m,n,o,r,s,t,... ) can have values of,, 3, and values for captal letter ndces wll be specfed where need be. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

8 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 76 of be a functon of the dsplacements at the N fnte element nodal ponts: u û a = H I ū Ia (.) where I =,, 3,..., n and n s number of nodes n a specfc element, a =,, 3 represents a number of dmensons (can be or or 3), H I represents dsplacement nterpolaton vector, ū Ia s the tensor of global generalzed dsplacement components at all element nodes. The use of the term generalzed dsplacements means that both translatons, rotatons, or any other nodal unknown are modeled ndependently. Here specfcally only translatonal degrees of freedom are consdered. The stran tensor s defned as: ǫ ab = (u a,b +u b,a ) (.3) and the by usng (.) we can defne the approxmate stran tensor: ǫ ab ê ab = (û a,b +û b,a ) = = ) ((H I ū Ia ),b +(H I ū Ib ),a = = ((H I,b ū Ia )+(H I,a ū Ib )) (.) The most general stress stran relatonshp for an sotropc materal s: ˆσ ab = E abcd (êcd ǫ cd) +σ ab (.5) where ˆσ ab s the approxmate Cauchy stress tensor, E abcd s the consttutve tensor, ê cd s the nfntesmal approxmate stran tensor, ǫ cd s the nfntesmal ntal stran tensor and σ ab s the ntal Cauchy stress tensor. Usng the assumpton of the dsplacements wthn each fnte element, as expressed n (.), we can now derve equlbrum equatons that corresponds to the nodal pont dsplacements of the assemblage of fnte elements. We can rewrte (.) as a sum 3 of ntegratons over the volume and areas of all fnte elements: ˆσ ab δê ab dv m = ( ) f B a ρü a δûa dv m + fa S δû S a ds m (.6) V m m V m m S m m n terms of exact stress and stran felds but t holds for approxmate felds as well. Ths tensor can be elastc or elastoplastc consttutve tensor. 3 Or, more correctly as a unon m snce we are ntegratng over the unon of elements. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

9 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 77 of where m =,,3,...,k and k s the number of elements. It s mportant to note that the ntegratons n (.6) are performed over the element volumes and surfaces, and that for convenence we may use dfferent element coordnate systems n the calculatons. If we substtute equatons (.), (.3), (.) and (.5) n (.6) t follows: m m V m f B a δ(h I ū Ia ) dv m m ( Eabcd (êcd ǫ cd V m ( ) ) ) +σ ab δ (H I,b ū Ia +H I,a ū Ib ) V m H J ū Ja ρ δ(h I ū Ia ) dv m + m dv m = S m f S a δ(h I ū Ia ) ds m (.7) or: m V m ( (( ) ) ) E abcd (H J,d ū Jc +H J,c ū Jd ) ǫ cd +σab ( δ (H I,b ū Ia +H I,a ū Ib ) ) dv m = = m V m f B a δ(h I ū Ia ) dv m m V m H J ū Ja ρ δ(h I ū Ia ) dv m + m S m f S a δ(h I ū Ia ) ds m (.8) We can observe that δ n the prevous equatons represents a vrtual quantty but the rules for δ are qute smlar to regular dfferentaton so that δ can enter the brackets and vrtualze the nodal dsplacement. It thus follows: m V m = m ( (( ) ) ) ( ) E abcd (H J,d ū Jc +H J,c ū Jd ) ǫ cd +σab (H I,b δū Ia +H I,a δū Ib ) dv m = V m f B a (H I δū Ia ) dv m m V m H J ū Ja ρ (H I δū Ia ) dv m + m S m f S a (H I δū Ia ) ds m (.9) et us now work out some algebra n the left hand sde of equaton (.9): ( ( ) ) ( ) (HJ,d ū Jc +H J,c ū Jd ) E abcd E abcd ǫ (HI,b δū Ia +H I,a δū Ib ) cd m V m +σ ab dv m = = fa B (H I δū Ia ) dv m H J ū Ja ρ H I δū Ia dv m + fa S (H I δū Ia ) ds m m V m m V m m S m (.) snce they are drvng varables that defne overall dsplacement feld through nterpolaton functons Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

10 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 78 of and further: m V m (( ) (H J,d ū Jc +H J,c ū Jd ) + m V m ( )) E abcd (H I,b δū Ia +H I,a δū Ib ) dv m + ( ( )) E abcd ǫ cd (H I,b δū Ia +H I,a δū Ib ) dv m + + ( ) ( ) σ ab m V m (H I,b δū Ia +H I,a δū Ib ) dv m = fa B (H I δū Ia ) dv m m V m H J ū Ja ρ H I δū Ia dv m m V m + fa S (H I δū Ia ) ds m (.) m S m Several thngs should be observed n the equaton (.). Namely, the frst three lnes n the equaton can be smplfed f one takes nto account symmetres of E jkl and σ j. In the case of the elastc stffness tensor E jkl major and both mnor symmetres exst. In the case of the elastoplastc stffness tensor, such symmetres exsts f a flow a rule s assocated. If flow rule s non assocated, only mnor symmetres exst whle major symmetry s destroyed 5. As a matter of fact, both mnor symmetres n E jkl are the only symmetres we need, and the frst lne of (.) can be rewrtten as: m V m (( ) (H J,d ū Jc +H J,c ū jd ) ( )) E abcd (H I,b δū Ia +H I,a δū Ib ) dv m = = (H J,d ū Jc ) E abcd (H I,b δū Ia ) dv m = m V m = (H I,b δū Ia ) E abcd (H J,d ū Jc ) dv m (.) m V m Smlar smplfcatons are possble n second and thrd lne of equaton (.). Namely, n the second lne we can use both mnor symmetres of E jkl so that: 5 for more on stffness tensor symmetres see sectons (.6.,.3 and.) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

11 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 79 of m = m V m ( E abcd ǫ cd ( )) (H I,b δū Ia +H I,a δū Ib ) dv m = V m ( Eabcd ǫ cd (H I,b δū Ia ) ) dv m (.3) and the thrd lne can be smplfed due to the symmetry n Cauchy stress tensor σ j as: m = m ( ) σ ab V m ( ) (H I,b δū Ia +H I,a δū Ib ) dv m = V m ( σ ab ) (HI,b δū Ia ) dv m (.) After these smplfcatons, equaton (.) looks lke ths: (H I,b δū Ia ) E abcd (H J,d ū Jc ) dv m + m V m + ( Eabcd ǫ cd (H I,b δū Ia ) ) dv m + ( ) σ ab (HI,b δū Ia ) dv m = m V m m V m = fa B (H I δū Ia ) dv m H J ū Ja ρ H I δū Ia dv m + fa S (H I δū Ia )(.5) ds m m V m m V m m S m or f we leave the unknown nodal acceleratons 6 ū Jc and dsplacements ū Jc on the left hand sde and move all the known quanttes on to the rght hand sde: H J δ ac ū Jc ρ H I δū Ia dv m + (H I,b δū Ia ) E abcd (H J,d ū Jc ) dv m = m V m m V m = fa B (H I δū Ia ) dv m + fa S (H I δū Ia ) ds m + m V m m S m + ( Eabcd ǫ cd (H I,b δū Ia ) ) dv m ( ) σ ab (HI,b δū Ia ) dv m (.6) m V m m V m To obtan the equaton for the unknown nodal generalzed dsplacements from (.6), we nvoke the vrtual dsplacement theorem whch states that vrtual dsplacements are any, non zero, knematcally 6 It s noted that ū Jc = δ ac ū Ja relatonshp was used here, where δ ac s the Kronecker delta. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

12 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 8 of admssble dsplacements. In that case we can factor out nodal vrtual dsplacements δū Ia so that equaton (.6) becomes: [ H J δ ac ū Jc ρ H I dv m + ] (H I,b ) E abcd (H J,d ū Jc ) dv m δū Ia = m V m m V m = [ ] fa B H I dv m δū Ia + [ ] fa S H I ds m δū Ia + m V m m S m + [ ] ( Eabcd ǫ cd H ) I,b dv m δū Ia [ ] ( ) σ ab HI,b dv m δū Ia (.7) m V m m V m and now we can just cancel δū Ia on both sdes: H J δ ac ρ H I ū Jc dv m + m V m (H I,b ) E abcd (H J,d ū Jc ) dv m = m V m = fa B H I dv m + fa S H I ds m + m V m m S m + ( Eabcd ǫ cd H I,b) dv m ( ) σ ab HI,b dv m (.8) m V m m V m One should also observe that n the frst lne of equaton (.8) generalzed nodal acceleratons ū Jc and generalzed nodal dsplacements ū Jc are unknowns that are not subjected to ntegraton so they can be factored out of the ntegral: H J δ ac ρ H I dv m ū Jc m V m + H I,b E abcd H J,d dv m ū Jc m V m = fa B H I dv m + fa S H I ds m + m V m m S m + ( Eabcd ǫ cd H I,b) dv m ( ) σ ab HI,b dv m (.9) m V m m V m We can now defne several tensors from equaton (.9): Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

13 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 8 of (m) M IacJ = H J δ ac ρ H I dv m (.3) V m (m) K IacJ = H I,b E abcd H J,d dv m (.3) V m (m) FIa B = fa B H I dv m (.3) V m (m) FIa S = fa S H I ds m (.33) S m (m) F ǫ mn Ia = E abcd ǫ cd H I,b dv m (.3) V m (m) F σ mn Ia = σab H I,b dv m (.35) V m where (m) K IacJ s the element stffness tensor, (m) FIa B s the tensor of element body forces, (m) FIa S s the tensor of element surface forces, (m) F ǫ mn Ia tensor of element ntal stress effects. Now equaton (.9) becomes: s the tensor of element ntal stran effects, (m) F σ mn Ia (m) M IacJ ū Jc + (m) K IacJ ū Jc = (m) FIa B + (m) FIa S + (m) m m m (m) By summng 7 all the relevant tensors, a well known equaton s obtaned: s the (m) F ǫ mn Ia (m) F σ mn Ia (.36) m 7 Summaton of the element volume ntegrals expresses the drect addton of the element tensors to obtan global, system tensors. Ths method of drect addton s usually referred to as the drect stffness method. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

14 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 8 of M AacB ū Bc +K AacB ū Bc = F Aa (.37) A,B =,,...,# of nodes a,c =,...,# of dmensons (, or 3) where: M AacB = m (m) M IacJ ; K AacB = m (m) K IacJ (.38) are the system mass and stffness tensors, respectvely, ū Bc s the tensor of unknown nodal acceleratons, and ū Bc s the tensor of unknown generalzed nodal dsplacements, whle the load tensor s gven as: F Aa = m (m) F B Ia + m (m) F S Ia + m (m) F ǫ mn Ia (m) F σ mn Ia (.39) m After assemblng the system of equatons n (.38) t s relatvely easy to solve for the unknown dsplacements ū c ether for statc or fully dynamc case. It s also very mportant to note that n all prevous equatons, omssons of nertal force term (all terms wth ρ) wll yeld statc equlbrum equatons. Descrpton of solutons procedures for statc lnear and nonlnear problems are descrbed n some detal n chapter 8. In addton to that, soluton procedures for dynamc, lnear and nonlnear problems are descrbed n some detal n chapter 9. A note on the fnal form of the tensors used s n order. In order to use readly avalable system of equaton solvers equaton (.38) wll be rewrtten n the followng from: M PQ ū P +K PQ ū P = F Q P,Q =,,...,(#ofdofs)n (.) where M PQ s system mass matrx, K PQ s system stffness matrx and F Q s the loadng vector. Matrx form of equaton.38, presented as equaton. s obtaned flattenng the system mass tensor M AacB, system stffness tensor K AacB, unknown acceleraton tensor ū Bc, unknown dsplacement tensor ū Bc and the system loadng tensor F Aa. Flattenng from the fourth order mass/stffness tensors to two Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

15 Jeremć et al..3. FINITE EEMENT DISCRETIZATION 83 of dmensonal mass/stffness matrx s done by smply performng approprate (re ) numberng of nodal DOFs n each dmenson. Smlar approach s used for unknown acceleratons/dsplacements and for loadngs. Statc Analyss: Internal and External oads. Internal and external loadng tensors can be defned as: (f Ia ) nt = (m) (m) K IacJ ū Jc = m V m σ ab H I,b dv m (.) (f Ia ) ext = m (m) F B Ia + m (m) F S Ia + m (m) F ǫ mn Ia (m) F σ mn Ia (.) m where (f Ia ) nt s the nternal force tensor and (f Ia ) ext s the external force tensor. Equlbrum s obtaned when resdual: r Ia (ū Jc,λ) = (f Ia (ū Jc )) nt λ(f Ia ) ext (.3) s equal to zero, r(u,λ) =. The same equaton n flattened form yelds: r(u,λ) = f nt (u) λf ext = (.) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

16 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 8 of. Isoparametrc Sold Fnte Elements.. 8 Node Brck r r 8 5 r Fgure.3: 8 node brck element Table.: Values of r, r, and r 3 at each of the eght nodes Node r r r Shape functon of the nodes whch ndcates the node number: N (e) = 8 (+r (r ) )(+r (r ) )(+r 3 (r 3 ) ) (.5) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

17 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 85 of.. Node Brck r r r Fgure.: node brck element Table.: Values of r, r, and r 3 at each of the 9 th to th nodes Node r r r Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

18 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 86 of Shape functon of the 8 corner nodes ( to 8) whch ndcates the node number: N (e) = 8 (+r (r ) )(+r (r ) )(+r 3 (r 3 ) )(r (r ) +r (r ) +r 3 (r 3 ) ) (.6) Shape functon of the node numbers 9,, 3, and 5 whch ndcates the node number: N (e) = ( r )(+r (r ) )(+r 3 (r 3 ) ) (.7) Shape functon of the node numbers,,, and 6 whch ndcates the node number: N (e) = ( r )(+r (r ) )(+r 3 (r 3 ) ) (.8) Shape functon of the node numbers 7, 8, 9, and whch ndcates the node number: N (e) = ( r 3)(+r (r ) )(+r (r ) ) (.9) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

19 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 87 of..3 7 Node Brck r r r Fgure.5: 7 node brck element Table.3: Values of r, r, and r 3 at each of the th to 7 th nodes Node r r r Shape functon of the 8 corner nodes ( to 8) whch ndcates the node number: N (e) = 8 (+r (r ) )(+r (r ) )(+r 3 (r 3 ) )(r (r ) )(r (r ) )(r 3 (r 3 ) ) (.5) Shape functon of the node numbers 9,, 3, and 5 whch ndcates the node number: N (e) = ( r )(+r (r ) )(+r 3 (r 3 ) )(r (r ) )(r 3 (r 3 ) ) (.5) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

20 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 88 of Shape functon of the node numbers,,, and 6 whch ndcates the node number: N (e) = (+r (r ) )( r )(+r 3 (r 3 ) )(r (r ) )(r 3 (r 3 ) ) (.5) Shape functon of the node numbers 7, 8, 9, and whch ndcates the node number: N (e) = (+r (r ) )(+r (r ) )( r 3)(r (r ) )(r (r ) ) (.53) Shape functon of the node number : N (e) = ( r )( r )( r 3) (.5) Shape functon of the node numbers and whch ndcates the node number: N (e) = ( r )(+r (r ) )( r 3)(r (r ) ) (.55) Shape functon of the node numbers 3 and 5 whch ndcates the node number: N (e) = (+r (r ) )( r )( r 3)(r (r ) ) (.56) Shape functon of the node numbers 6 and 7 whch ndcates the node number: N (e) = ( r )( r )(+r 3 (r 3 ) )(r 3 (r 3 ) ) (.57).. Isoparametrc 8 Node Fnte Element The basc procedure n the soparametrc 8 fnte element formulaton s to express the element coordnates and element dsplacements n the form of nterpolatons usng the local three dmensonal 9 coordnate system of the element. Consderng the general 3D element, the coordnate nterpolatons, usng ndcal notaton are: 8 name soparametrc comes from the fact that both dsplacements and coordnates are defned n terms of nodal values. Superparametrc and subparametrc fnte elements exsts also. 9 n the case of element presented here, that s soparametrc 8 node fnte element. Ensten s summaton rule s mpled unless stated dfferently, all lower case ndces (,j,p,q,m,n,o,r,s,t,... ) can have values of,, 3, and values for captal letter ndces wll be specfed where need be. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

21 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 89 of x = H A (r k ) x A (.58) where A =,,...,n and n s the total number of nodes assocated wth that specfc element, x A s the -th coordnate of node A, =,,3, k =,,3 and H A are the nterpolaton functons defned n local coordnate system of the element, wth varables r, r and r 3 varyng from to +. The nterpolaton functons H A for the soparametrc 8 node are the so called serendpty nterpolaton functons manly because they were derved by nspecton. For the fnte element wth nodes numbered as n Fgure (.6) they are gven n the followng set of formulae: x x 3 r 3 3 r r 7 x Fgure.6: Isoparametrc 8 node brck element n global and local coordnate systems H = sp() (+r ) ( r ) ( r3 ) H 8 = sp(8) ( r ) (+r ) ( r3 ) H 6 = sp(6) (+r ) ( r ) ( r3 ) H = sp() ( r ) ( r ) ( r3 ) H = sp() (+r ) ( r ) (+r3 ) H = sp() ( r ) ( r ) (+r3 ) for more detals see Bathe (98) H 9 = sp(9) ( r ) ( r ) ( r3 ) H 7 = sp(7) (+r ) (+r ) ( r3 ) H 5 = sp(5) ( r ) ( r ) ( r 3 ) H 3 = sp(3) ( r ) (+r ) ( r 3 ) H = sp() ( r ) ( r ) (+r 3 ) H 9 = sp(9) ( r ) (+r ) (+r 3 ) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

22 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 9 of H 8 = (+r ) ( r ) ( r 3 ) 8 H 7 = ( r ) ( r ) ( r 3 ) 8 H 6 = ( r ) (+r ) ( r 3 ) 8 H 5 = (+r ) (+r ) ( r 3 ) 8 H = (+r ) ( r ) (+r 3 ) 8 H 3 = ( r ) ( r ) (+r 3 ) 8 H = ( r ) (+r ) (+r 3 ) 8 H = (+r ) (+r ) (+r 3 ) 8 + H 5 H 6 H + H H 5 H 9 + H 3 H H 8 + H 3 H 6 H 7 + H H H + H H H 9 + H H 8 H 9 + H H 7 H 9 where r, r and r 3 are the axes of natural, local, curvlnear coordnate system and sp(nod num) s booleanfunctonthatreturns+fnodenumber(nod num)spresentandfnodenumber(nod num) s not present. To be able to evaluate varous mportant element tensors, we need to calculate the stran dsplacementtransformatontensor 3. Theelementstransareobtanedntermsofdervatvesofelement dsplacements wth respect to the local coordnate system. Because the element dsplacements are defned n the local coordnate system, we need to relate global x, x and x 3 dervatves to the r, r and r 3 dervatves. In order to obtan dervatves wth respect to global coordnate system,.e. x a we need to use chan rule for dfferentaton n the followng form: = r a = J x k x k r ak a whle the nverse relaton s: r a (.59) r k = x a r k x a = J ak x a (.6) where J ak s the Jacoban operator relatng local coordnate dervatves to the global coordnate dervatves:.e. (m) K IacJ, (m) FIa, B (m) FIa, S (m) F ǫ mn Ia, (m) F σ mn Ia, that are defned n chapter (.3). 3 from the equaton ˆǫ ab = ((H I,b ū Ia)+(H I,a ū Ib )) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

23 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 9 of J ak = x a r k = x r x r x 3 r x r x r x 3 r x r 3 x r 3 x 3 r 3 (.6) Theexstenceofequaton(.59)requresthatthenverseofJ ak exstsandthatnverseexstsprovded that there s a one to one correspondence between the local and the global coordnates of element. It should be ponted out that except for the very smple cases, volume and surface element tensor 5 ntegrals are evaluated by means of numercal ntegraton 6 Numercal ntegraton rules s qute a broad subject and wll not be covered here Isoparametrc 8-7 Node Fnte Element r r Fgure.7: 8-7 varable node brck element unque. 5 as defned n chapter (.3) by equatons (.3), (.3), (.33), (.3) and (.35). 6 Gauss egendre, Newton Coates, obatto are among the most used ntegraton rules. 7 nce explanaton wth examples s gven n Bathe (98) r Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

24 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 9 of H = (+r )(+r )(+r 3 ) 8 H = ( r )(+r )(+r 3 ) 8 H 3 = ( r )( r )(+r 3 ) 8 H = (+r )( r )(+r 3 ) 8 H 5 = (+r )(+r )( r 3 ) 8 H 6 = ( r )(+r )( r 3 ) 8 H 7 = ( r )( r )( r 3 ) 8 H 8 = (+r )( r )( r 3 ) 8 H 9 +H +H 7 H 9 +H +H 8 H +H +H 9 H +H +H H 3 +H 6 +H 7 H 3 +H +H 8 H +H 5 +H 9 H 5 +H 6 +H H +H 5 +H 6 H +H 3 +H 6 H 3 +H +H 6 H +H 5 +H 6 H +H 5 +H 7 H +H 3 +H 7 H 3 +H +H 7 H +H 5 +H 7 H 8 H 8 H 8 H 8 H 8 H 8 H 8 H 8 H 9 = ( r )(+r )(+r 3 ) H +H 6 H = ( r )( r )(+r 3 ) H 3 +H 6 H = ( r )( r )(+r 3 ) H +H 6 H = ( r )(+r )(+r 3 ) H 5 +H 6 H 3 = ( r )(+r )( r 3 ) H +H 7 H = ( r )( r )( r 3 ) H 3 +H 7 H 5 = ( r )( r )( r 3 ) H +H 7 H 6 = ( r )(+r )( r 3 ) H 5 +H 7 H 7 = ( r 3)(+r )(+r ) H +H 5 H 8 = ( r 3)( r )(+r ) H +H 3 H 9 = ( r 3)( r )( r ) H 3 +H H = ( r 3)(+r )( r ) H +H 5 H = ( r )( r )( r 3) H H H H H H H H H H H H Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

25 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 93 of H = ( r )(+r )( r 3)r H 3 = ( r )( r )( r 3)r H = ( r )( r )( r 3)r H 5 = (+r )( r )( r 3)r H 6 = ( r )( r )(+r 3 )r 3 H 7 = ( r )( r )( r 3 )r 3..6 Surface oads for Sold Brcks In order to apply surface load on brck elements, equvalent nodal forces have to be appled nstead of the surface load. The equvalent force of the -th node F s gven by the followng equaton wth shape functon H and load dstrbuton functon f. F = S fh ds (.6) When we assume the load dstrbuton s unform, t turns nto F = f H ds S (.63) Furthermore, when the magntude of the load per unt area s, and the sze of the element s, equvalent nodal forces are gven as shown n Fgure.8 for 8 node brck element, brck element, and 7 nodes brck element. Fgure.8 shows cases of normal loads on vertcal upper surface (wth nodes:,, 3, for 8 node brck;,, 3,, 9,,, and for node brck; and,, 3,, 9,,, and 6 for the 7 node brck). Nodal loads from unform surface loads for 7 node brck are obtaned as: Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

26 Jeremć et al... ISOPARAMETRIC SOID FINITE EEMENTS 9 of F 3 F F 3 F F F 3 F F F F 9 F F 6 F 9 F F F F F F F F a) b) c) Fgure.8: Nodal loads for brck elements: (a) F = F = F 3 = F = +/; (b) F = F = F 3 = F =, F 9 = F = F = F == + 3 ; (c) F = F = F 3 = F = + 36, F 9 = F = F = F == + 9,F 36 = 9. for nodes,, 3, and, N (e) = 8 (+r (r ) )(+r (r ) )(+r 3 (r 3 ) )(r (r ) )(r (r ) )(r 3 (r 3 ) ) + H ds = = 8 (+r 3(r 3 ) )(r 3 (r 3 ) ) + + = 8 (+r 3(r 3 ) )(r 3 (r 3 ) )((r ) ) ((r ) ) ( 3 ) (+r (r ) )(+r (r ) )(r (r ) )(r (r ) )(r (r ) )dr dr = 8 (+r 3(r 3 ) )(r 3 (r 3 ) )((r ) ) ((r ) ) (.6) for nodes 9,, and, N (e) = ( r )(+r (r ) )(+r 3 (r 3 ) )(r (r ) )(r 3 (r 3 ) ) + H ds = = (+r 3(r 3 ) )(r 3 (r 3 ) ) + = 8 (+r 3(r 3 ) )(r 3 (r 3 ) )((r ) ) ( 3 )( 3 ) ( r )(+r (r ) )(r (r ) )dr dr = 9 (+r 3(r 3 ) )(r 3 (r 3 ) )((r ) ) (.65) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

27 Jeremć et al..5. TWO NODE, 3D TRUSS FINITE EEMENT 95 of for nodes 6 N (e) = ( r )( r )(+r 3(r 3 ) )(r 3 (r 3 ) ) + H ds = = (+r 3(r 3 ) )(r 3 (r 3 ) ) + + = 8 (+r 3(r 3 ) )(r 3 (r 3 ) )( 3 )( 3 ) ( r )( r )dr dr = 8 9 (+r 3(r 3 ) )(r 3 (r 3 ) ) (.66).5 Two Node, 3D Truss Fnte Element Bathe and Wlson (976); Bathe (98).6 3D Beam-Column Fnte Element, Degrees of Freedom Bathe and Wlson (976); Bathe (98); Przemeneck (985) Stffness Matrx: Equaton.67 Mass Matrx: Equaton.68 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

28 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5 [K] = EA EA EI z 3 6EIy 6EI z EIz 3 EI y 3 6EIy EIy 3 6EIy GJ x 6EI z GJx EI y EI y 6EI y 6EI z EI z 6EIz EA EA EIz 3 6EIz EIy 3 GJx 6EIy 6EI z EI z 6EI y EI y EI z 6EIz 3 EI y 3 GJ x 6EI y 6EI y EI z 6EIz EI y EI z (.67) Jeremć et al..6. 3D BEAM-COUMN FINITE EEMENT, of

29 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5 [M] = ρa Iz l 5Al + Iz 9 Al 7 6Iz 3l 5Al + Iz Al Iy l 5Al Iy 9 Al 7 6Iy 3l 5Al Iy Al J x 3A l Iy l Al 5 + Iy 5A 3l + Iy Al l + Iz l Al 5 + Iz 3l 5A Iz J x 6A l Iy 3A Al l 6 3 Iz 3A 9 7 6Iz 3l 5Al Iz 3 Al Iz l 5Al Iz Al 9 7 6Iy 3l 5Al + Iy 3 Al Iy l 5Al + Iy Al J x J 6A x 3A 3l Iy Al l Iy l 3A + Iy l Al 5 + Iy 5A 3l + Iz Al l Iz 3A l Iz l Al 5 + Iz 5A (.68) Jeremć et al..6. 3D BEAM-COUMN FINITE EEMENT, of

30 Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, of.7 3D Beam-Column Fnte Element, 9 Degrees of Freedom Przemeneck (985) Condensaton Formulaton: Equatons.69 to.75 Rearranged dof Stffness Matrx: Equaton.76 K rr part of stffness matrx: Equaton.77 K rc part of stffness matrx: Equaton.78 K cr part of stffness matrx: Equaton.79 K cc part of stffness matrx: Equaton.8 Stffness Matrx: Equaton.8 T Matrx: Equaton.8 Rearranged Mass Matrx: Equaton.83 Mass Matrx: Equaton.8 k rr k rc k cr k cc d r d c = r r r c (.69) ] [ ([ k rr [ [k condensed ] = { r condensed = [T] = K rc ][ [ k rr ] [ r r } K cc ] [ K rc ][ K rc ][ { K cr ]) K cc ] [ K cc ] { } { d r = K cr ] r c } [ r r } K rc ][ K cc ] { r c } (.7) (.7) (.7) I [ ] [ ] (.73) K cc K cr [K condensed ] = [T] T [K][T] (.7) K condensed should gve same results usng ether method. [M condensed ] = [T] T [M][T] (.75) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

31 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5 [K rearranged ] = EA EA EI z EIz 3 3 6EI z EI y 3 EIy 3 6EIy 6EIy EA EA EI z EIz 6EIz 6EIz 3 3 EIy EI y 6EI y 6EI y 3 3 6EI y 6EIy 6EI z 6EIz 6EI z GJ x GJx EI y GJx 6EI y 6EIy 6EI z 6EIz EI z GJ x EI y EI y EI z EI z EI y EI z (.76) Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, of

32 Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, 9... of [K rr ] = EA EA EI z EIz 3 3 6EI z EI y 3 EIy 3 6EIy EA EA EI z EIz 3 EIy 3 6EIz 3 EI y 3 GJ x 6EI y 6EI y 6EIy 6EI z 6EIz EI y EI z (.77) [K rc ] = [K cr ] = [K cc ] = 6EI z 6EIy 6EIz GJx 6EI y EI y EI z GJx 6EI y 6EIy 6EI z 6EIz GJ x EI y EI z EI y EI z (.78) (.79) (.8) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

33 Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, 9... of [K condensed ] = EA EA 3EI z 3EIz 3 3 3EI z 3EIy 3EIy 3EIy 3 3 EA EA 3EI z 3EIz 3 3EIy 3 3EIz 3 3EI y 3 3EI y 3EI y 3EIy 3EI z 3EIz 3EI y 3EI z (.8) [T] = (.8) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

34 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5 [M rearranged ] = ρa Iz 9 5Al 7 6Iz 3l 5Al + Iz l Al + Iz Al Iy 9 5Al 7 6Iy 3l 5Al Iy Al l Iy Al Iz 3 5Al Iz l 5Al Iz 3l Al Iz Al 9 7 6Iy 3 5Al Iy l 5Al + Iy Al 3l + Iy Al J x J 3A x 6A 3l Iy l Al + Iy l Al 5 + Iy 5A l Iy 3A 3l + Iz Al l Iz l Al 5 + Iz 5A l Iz 3A J x J 6A x 3A l Iy Al 3l + Iy Al l Iy l 3A 5 + Iy 5A l + Iz 3l Al Iz Al l Iz l 3A 5 + Iz 5A (.83) Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, 9... of

35 Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5 [M condensed ] = ρa Iz 39 5A 8 6Iz 5A 8 + Iz 5A Iy 39 5A 8 6Iy 5A 8 Iy 5A Iz 5A Iz 5A 3 35 Iz 5A Iy 7 5A Iy 3 5A 35 + Iy 5A J xa 8 Iy 5A Iy 5A 5 + Iy 5A 8 + Iz 5A 3 35 Iz 5A 5 + Iz 5A (.8) Jeremć et al..7. 3D BEAM-COUMN FINITE EEMENT, of

36 Jeremć et al..8. SHEAR BEAM FINITE EEMENT of.8 Shear Beam Fnte Element.9 Quadrlateral Shell Fnte Element wth 6DOFs per Node Based on works by Bergan and Felppa (985); Alvn et al. (99); Felppa and Mltello (99); Felppa and Alexander (99); Mltello and Felppa (99). Stffness matrx for ths element s obtaned by averagng two quad shells made up of two ANDES trangular shells (wth alternatng orentaton of dagonals, Stošć (98-3)). Two Node, 3D, Gap-Frctonal, Dry Contact Fnte Element Haraldsson and Wrggers (); Wrggers (); Sheng et al. (7a) In most sol-structure systems, there are nterfaces between dfferent materals whch are n contact. These contact areas can get detached durng the loadng so there wll be gaps n the model. The focus of ths secton s on the contact of concrete and sol. In fact transferrng the loads to structure s defned based on the contact areas whch wll affect eh response of the sol-structure systems (Kkuch and Oden (988)). The man objectve at the contact areas s to defne normal and shear forces, and dsplacements at the common surfaces. The dea of contact element has become from elements so called jont elements whch was ntally developed for modelng of rock jonts. Typcally normal and tangental stffness were used to model the pressure and frcton at the nterface (Sheng et al. (7b); Haraldsson and Wrggers (); Desa and Srwardane (98)). The study of two dmensonal and axsymmetrc benchmark examples have been done by Olukoko et al. (993) for lnear elastc and sotropc contact problems. Study was done consderng Coulomb s law for frctonal behavor at the nterface. In many cases the nteracton of sol and structure s nvolved wth frctonal sldng of the contact surfaces, separaton, and reclosure of the surfaces. These cases depend on the loadng procedure and frctonal parameters. Sheng et al. (7b) dscussed how frctonal contact s mportant for structural foundatons under loadng, ple foundatons, sol anchors, and retanng walls. Khoe and Nkbakht (6) used the extended fnte element method (X-FEM) for modelng of frctonal contact problems. In order to be able to smulate dscontnutes n model durng the analyss wthout consderng the boundary condtons of doman, specal functons should be ncluded wthn standard fnte element framework. A fnte element method for analyss of frctonal contact problems was developed by Chandrasekaran et al. (987). The nonlnear problem s solved by mposng geometrc constrants on equlbrum con- Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

37 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... 5 of fguraton whch compatblty condtons are volated. Dfferent condtons happenng at contact area, such as stckng and slppng, are determned from the relatve magntudes of the normal and tangental nodal forces. Developng contact fnte element for analyss of connectons between lamnated composte plates has been done by Barbero et al. (995). Such element s compatble wth a three-dmensonal plate element based on constant shear theory. The contact element formulaton s based on a smple regularzaton of the unlateral contact wth a Coulomb frcton problem. Two-dmensonal frctonal polynomal to segment contact elements are developed by Haraldsson and Wrggers () based on non-assocated frctonal law and elastc-plastc tangental slp decomposton. Several benchmarks are presented by Konter (5) n order to verfy the the results of the fnte element analyses performed on D and 3D modelngs. In all proposed benchmarks the results were approxmated pretty well wth a D or an axsymmetrc solutons. In addton, 3D analyses were performed and the results were compared wth the D solutons. Hjaj et al. () presented an algorthm for solvng frctonal contact problems where sldng rule s non-assocated. The algorthm s a combnaton of the classcal unlateral contact law and an ansotropc frcton model wth a non-assocated slp rule. It has been observed that slp rule has a strong nfluence on the frctonal behavor. Formulaton The smplest formulaton for contact s a dscretzaton whch establshes constrant equatons and contact nterface consttutve equatons on a purely nodal bass. Such a formulaton s called node-to-node contact. For ths dscretzaton the frctonal contact formulaton s developed below. Wrggers (). The varables adopted to formulate the model are shown n Fgure.9: the force(f) and dsplacement vectors (u). Each vector s composed of three terms: the frst one acts along the longtudnal drecton whereas the other two components le on the orthogonal plane. The total relatve dsplacement s the summaton of the elastc and plastc dsplacements (.86). F = [p ; t] T ; u = [v; g s ] T (.85) [ ] T [ ] T u el = v el ; gs el ; u pl = v pl ; gs pl ; u = u el +u pl (.86) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

38 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... 6 of Fgure.9: Forces and relatve dsplacements of the element. Elastc behavor The elastc behavor s defned by the relaton: K N (p) df = E du el ; E = K T (.87) K T The normal dsplacement-normal force relatonshp, vald for loadng and unloadng condtons, s the same that has been ntroduced by Gens et al. (988, 99) and gven by: p = C N v el (v max v el ) (.88) v el s the normal relatve elastc dsplacement, v max represents the maxmum closure value and C N s a constant. Ths relaton s vald only whenever the dfference (v max v el ) s greater than zero, otherwse p s set equal to zero. The dervatve of (.88) gves the normal stffness K N (.89), that depends on both the maxmum closure v max and the current value of the elastc normal dsplacement v el. The tangental stffness K T s assumed to be constant. K N = C Nv max (v max v el ) ; K T = const (.89) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

39 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... 7 of An alternatve, smplfed normal force deformaton relatonshp s gven as f n (u x,u y,u z ) = k nu x +.k n u x,f u x <,f u x < (.9) Force dsplacement equaton f n (u x ) = k nu x +.k n u x s a parabola that has a non-zero tangent at u x =, as ths helps n solvng the problem. The value of stffness.k n s chosen as stffness at / of mlmeter (.m)... Ths formulaton s desgned to provde Smooth transton from separaton to contact response (zero tangent at ntal contact). Penalty stffness that grows contnuous and unboundedly wth penetraton dstance. Smple shear response Plastc model The plastc model s defned n terms of yeld surface and plastc potental represented n Fgure.. Snce the proposed model reproduces an elasto-perfectly plastc behavor of the nterface, no hardenng rule has been ntroduced. The expresson for the mplemented yeld surface corresponds to a partcular case of the Mohr-Coulomb crtera (.9), n whch the only parameter s the frctonal one (r 3 = tanφ) and the coheson s neglected. On the contrary, the expresson adopted for the plastc potental s a von Mses crtera (.9), n whch k s a postve constant. The drecton of the normal to the plastc potental G s s not orthogonal to the yeld functon (unless the frcton angle φ = ), so the flow rule s non-assocated and the plastc normal relatve dsplacement s always zero. Ths means that the normal elastc dsplacements (v el ) are equal to the total normal dsplacements (v). f s t r 3 p = (.9) G s t k = (.9) Implementaton The formulaton used for the mplementaton s the Implct General Backward Euler algorthm n stress space (GBE). GBE algorthm solves the problem n two steps: Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

40 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... 8 of Fgure.: Yeld surface f s, plastc potental G s and ncremental plastc dsplacement δu p. the elastc problem s solved - wth the prescrbed ntal condtons - to obtan a tral elastc state; f the tral state s outsde the yeld locus, the plastc problem s solved - at a fxed current confguraton - assumng the tral state as ntal condtons. In ths smple model the evoluton equatons, the constrants and the ntal condtons are respectvely: F n+ = E u el n+ = E[ u n+ u pl u pl n+ ] n+ = λ (.93) n+ Gsn+ F n+ λ n+ ; f s (F n+ ) ; λ n+ f s (F n+ ) = (.9) F = F n ; u = u n (.95) Durng the fst step (elastc predctor) the plastc flow s frozen ( λ = ), the ncremental plastc dsplacements are set to zero and the soluton s called elastc tral state (.96): F tr n+ = F n + F tr n+ = F n +E u n+ (.96) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

41 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... 9 of If the yeld functon fs tr (F tr n+ ), the soluton s correct and the force vector s updated; on the contrary, the tral soluton has to be corrected durng the second step. In fact, the volaton of the consstency condton s restored by solvng the system of equatons durng the second step, and t s called return mappng (.97): F n+ = F tr n+ λ n+ E G sn+ F ; f tr s n+ = t tr n+ r3 p tr n+ (.97) In ths case, the yeld functon and the plastc potental have a partcular expresson that the process of return mappng has an nterestng geometrcal nterpretaton. The state F n+ represents the closest pont projecton of the tral state F tr n+ to the yeld surface f sn+ n the plane to whch the tangental force belongs. Snce the shape of the yeld functon and the plastc potental s crcular n ths plane, the return mappng procedure s called radal return mappng. The algorthm s lsted below. g sn+ = g sn + g sn+ ; g sn+ = g el s n+ +g pl s n+ (.98) v n+ = v n + v n+ ; δ n+ = v max +v n+ (.99) f δ n+ > K N = CKNvmax K (v max v n+ ) T = const else K N = K T =. (.) t tr n+ = t n +K T g sn+ ; p tr n+ = K N v n+ +p n (.) check f f tr s n+ = t tr n+ r 3 pn+ tr (.) f f tr s n+ > t n+ = t tr n+ λk T n n+ n n+ = n tr λ = K T ( t tr p n+ = p tr n+ n+ = ttr n+ t tr n+ n+ r 3 p n+ ) update g pl s n+ = λn n+ ; g el s n+ = g sn+ g pl s n+ g el s n+ = g el s n + g el s n+ ; g pl s n+ = g pl s n + g pl s n+ (.3) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

42 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, DRY... of Geometrc descrpton z Global coordnate system contact plane y z l x l J J J J x I Denton of local axes y l I Nodes n contact I Movement n normal drecton I Movement n tangent drecton Fgure.: Descrpton of contact geometry and dsplacement responses Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

43 Jeremć et al... TWO NODE, 3D, GAP-FRICTIONA, COU... of. Two Node, 3D, Gap-Frctonal, Coupled (Saturated) Contact Fnte Element (Segura and Carol,, 8, ; Zh Fu and Hong u, ). Sesmc Isolator Fnte Elements Base solaton system are used to change dynamc characterstcs of sesmc motons that excte structure and also to dsspate sesmc energy before t exctes structure. Therefor there are two man types of devces: Base Isolators (Kelly, 99a,b; Toopch-Nezhad et al., 8; Huang et al., ; Vasslou et al., 3) are usually made of low dampng (energy dsspaton) elastomers and are prmarly meant to change (reduce) frequences of nput motons. They are not desgned nor modeled as energy dsspators. Base Dsspators Kelly and Hodder (98); Fad and Constantnou (); Kumar et al. () are developed to dsspate sesmc energy before t exctes the structure. There two man types of such dsspators: Elastomers made of hgh dsspaton rubber, and Frctonal pendulum dsspators Both solators and dsspators are usually developed to work n two horzontal dmensons, whle motons n vertcal drectonare not solated or dsspated. Ths can create potental problems, and need to be carefully modeled. Modelng of base solaton and dsspaton system s done usng two node fnte elements of relatvely short length. Base Isolaton Systems are modeled usng lnear or nonlnear elastc elements. Stffness s provded from ether tests on a full szed base solators, or from materal characterzaton of rubber (and steel plates f used n a sandwch solator constructon). Dependng on rubber used, a number of models can be used to develop stffness of the devce Ogden (98); Smo and Mehe (99); Smo and Pster (98). Partcularly mportant s to properly account for vertcal stffness as vertcal motons can be amplfed dependng on characterstcs of sesmc motons, structure and stffness of the solators Hjkata et al. Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5

44 Jeremć et al..3. FUY COUPED, POROUS SOID PO... of (); Arak et al. (9). It s also mportant to note that assumpton of small deformaton s used n most cases. In other words, stablty of the solator, for example overturnng or rollng s not modeled. It s assumed that elastc stffness wll not suddenly change f solator becomes unstable (rolls or overturns). Base Dsspator Systems are modeled usng nelastc (nonlnear) two node elements. There are three basc types of dsspator models used: Hgh dampng rubber dsspators Rubber dsspators wth lead core Frctonal pendulum (double or trple) dsspators Each one s calbrated usng tests done on a full dsspator. It s mportant to be able to take nto account nfluence of (an ncrease n) temperature on resultng behavor. Energy dsspaton results n heatng of devces, and an ncrease n temperature nfluences materal propertes of dsspators... Two Node, 3D, Rubber Isolator Fnte Element Kelly (99a,b) Behavor of rubber (Ogden, 98; Smo and Mehe, 99; Smo and Pster, 98).. Two Node, 3D, Frctonal Pendulum Fnte Element.3 Fully Coupled, Porous Sold Pore Flud Fnte Elements.3. u-p-u Formulaton Background For a sngle-phase materal encountered n structural mechancs, the response under ultmate/load load can be predcted usng smple calculatons, at least for statc problems and for rgd plastc materal. For sol mechancs, smple, lmt-load calculatons cannot be fully justfed under statc stuaton. However, for problems of sol dynamcs, the use of smplfed methods s almost never admssble as too much modelng uncertanty s ntroduced n results of such calculatons. The relatonshp between effectve stress, total stress and pore pressure s(assume tensle components of stress as postve, hence a compressve pressure, p as negatve) (Zenkewcz et al., 999a) σ j = σ j αδ j p (.) Draft Book, UCD/BN FOR EXCUSIVE USE BY UC DAVIS STUDENTS Aprl, 7, 9:5