PLAXIS 3D TUNNEL. Scientific Manual. version 2

Transcription

1 PLAXIS 3D UNNEL Scentfc Manual verson 2

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3 ABLE OF CONENS ABLE OF CONENS 1 Introducton Deformaton theory Basc equatons of contnuum deformaton Fnte element dscretsaton Implct ntegraton of dfferental plastcty models Global teratve procedure Groundwater flow theory Basc equatons of steady flow Fnte element dscretsaton Flow n nterface elements Consoldaton theory Basc equatons of consoldaton Fnte element dscretsaton Elastoplastc consoldaton Element formulatons Interpolaton functons and numercal ntegraton of lne elements node lne elements Numercal ntegraton of lne elements Interpolaton functons and numercal ntegraton of area elements node trangular elements node quadrlateral elements Structural elements Interpolaton functons and numercal ntegraton of volume elements node wedge elements Numercal ntegraton over volumes Dervatves of shape functons Calculaton of element stffness matrx References Appendx A Calculaton process Appendx B Symbols

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5 INRODUCION 1 INRODUCION In ths part of the manual some scentfc background s gven of the theores and numercal methods on whch the PLAXIS 3D unnel program s based. he manual contans chapters on deformaton theory, groundwater flow theory and consoldaton theory, as well as the correspondng fnte element formulatons and ntegraton rules for the varous types of elements used n the 3D unnel program. In the Appendx a global calculaton scheme s provded for a plastc deformaton analyss. hs part of the manual stll has the character of an early edton. Hence, t s not complete and extensons wll be consdered n the future. More nformaton on backgrounds of theory and numercal methods can be found n the lterature, as amongst others referred to n Chapter 6. For detaled nformaton on stresses, strans, consttutve modellng and the types of sol models used n the PLAXIS program, the reader s referred to the Materal Models Manual. 1-1

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7 DEFORMAION HEORY 2 DEFORMAION HEORY In ths chapter the basc equatons for the statc deformaton of a sol body are formulated wthn the framework of contnuum mechancs. A restrcton s made n the sense that deformatons are consdered to be small. hs enables a formulaton wth reference to the orgnal undeformed geometry. he contnuum descrpton s dscretsed accordng to the fnte element method. 2.1 BASIC EQUAIONS OF CONINUUM DEFORMAION he statc equlbrum of a contnuum can be formulated as: L σ + p = 0 (2.1) hs equaton relates the spatal dervatves of the sx stress components, assembled n vector σ, to the three components of the body forces, assembled n vector p. L s the transpose of a dfferental operator, defned as: L x y z = y x z z y x In addton to the equlbrum equaton, the knematc relaton can be formulated as: ε (2.2) = L u (2.3) hs equaton expresses the sx stran components, assembled n vector ε, as the spatal dervatves of the three dsplacement components, assembled n vector u, usng the prevously defned dfferental operator L. he lnk between Eq. (2.1) and (2.3) s formed by a consttutve relaton representng the materal behavour. Consttutve relatons,.e. relatons between rates of stress and stran, are extensvely dscussed n the Materal Models Manual. he general relaton s repeated here for completeness: σ& = M ε& (2.4) he combnaton of Eqs. (2.1), (2.3) and (2.4) would lead to a second-order partal dfferental equaton n the dsplacements u. 2-1

8 SCIENIFIC MANUAL However, nstead of a drect combnaton, the equlbrum equaton s reformulated n a weak form accordng to Galerkn's varaton prncple (see among others Zenkewcz, 1967): ( ) 0 δ u L σ + p dv = (2.5) In ths formulaton δu represents a knematcally admssble varaton of dsplacements. Applyng Green's theorem for partal ntegraton to the frst term n Eq. (2.5) leads to: δ ε σ dv = δ u p dv + δ u t ds (2.6) hs ntroduces a boundary ntegral n whch the boundary tracton appears. he three components of the boundary tracton are assembled n the vector t. Eq. (2.6) s referred to as the vrtual work equaton. he development of the stress state σ can be regarded as an ncremental process: σ = -1 σ + σ σ = & σ d t (2.7) In ths relaton σ represents the actual state of stress whch s unknown and σ -1 represents the prevous state of stress whch s known. he stress ncrement σ s the stress rate ntegrated over a small tme ncrement. If Eq. (2.6) s consdered for the actual state, the unknown stresses σ can be elmnated usng Eq. (2.7): -1 δ ε σ dv = δ u p dv + δ u t ds δ ε σ dv (2.8) It should be noted that all quanttes appearng n Eqs. (2.1) to (2.8) are functons of the poston n the three-dmensonal space. 2.2 FINIE ELEMEN DISCREISAION Accordng to the fnte element method a contnuum s dvded nto a number of (volume) elements. Each element conssts of a number of nodes. Each node has a number of degrees of freedom that correspond to dscrete values of the unknowns n the boundary value problem to be solved. In the present case of deformaton theory the degrees of freedom correspond to the dsplacement components. Wthn an element the dsplacement feld u s obtaned from the dscrete nodal values n a vector v usng nterpolaton functons assembled n matrx N : u = N v (2.9) 2-2 PLAXIS 3D UNNEL

9 DEFORMAION HEORY he nterpolaton functons n matrx N are often denoted as shape functons. Substtuton of Eq. (2.9) n the knematc relaton (2.3) gves: ε = L N v = B v (2.10) In ths relaton B s the stran nterpolaton matrx, whch contans the spatal dervatves of the nterpolaton functons. Eqs. (2.9) and (2.10) can be used n varatonal, ncremental and rate form as well. Eq. (2.8) can now be reformulated n dscretsed form as: -1 ( ) = ( ) p + ( ) t ( ) σ B δv σ dv N δv dv N δv ds B δv dv (2.11) he dscrete dsplacements can be placed outsde the ntegral: (2.12) t -1 δ v B σdv = δv N p dv + δv N t ds δv B σ dv Provded that Eq. (2.12) holds for any knematcally admssble dsplacement varaton δv, the equaton can be wrtten as: -1 B σ d V = N p d V + N t d S B σ d V (2.13) he above equaton s the elaborated equlbrum condton n dscretsed form. he frst term on the rght-hand sde together wth the second term represent the current external force vector and the last term represents the nternal reacton vector from the prevous step. A dfference between the external force vector and the nternal reacton vector should be balanced by a stress ncrement σ. he relaton between stress ncrements and stran ncrements s usually non-lnear. As a result, stran ncrements can generally not be calculated drectly, and global teratve procedures are requred to satsfy the equlbrum condton (2.13) for all materal ponts. Global teratve procedures are descrbed later n Secton 2.4, but the attenton s frst focussed on the (local) ntegraton of stresses. 2.3 IMPLICI INEGRAION OF DIFFERENIAL PLASICIY MODELS he stress ncrements σ are obtaned by ntegraton of the stress rates accordng to Eq. (2.7). For dfferental plastcty models the stress ncrements can generally be wrtten as: e p ( ) σ = D ε ε (2.14) 2-3

10 SCIENIFIC MANUAL In ths relaton D e represents the elastc materal matrx for the current stress ncrement. he stran ncrements ε are obtaned from the dsplacement ncrements v usng the stran nterpolaton matrx B, smlar to Eq. (2.10). For elastc materal behavour, the plastc stran ncrement ε p s zero. For plastc materal behavour, the plastc stran ncrement can be wrtten, accordng to Vermeer (1979), as: p g g ε = λ ( 1 ω) + ω σ σ -1 (2.15) In ths equaton λ s the ncrement of the plastc multpler and ω s a parameter ndcatng the type of tme ntegraton. For ω = 0 the ntegraton s called explct and for ω = 1 the ntegraton s called mplct. Vermeer (1979) has shown that the use of mplct ntegraton (ω = 1) has some major advantages, as t overcomes the requrement to update the stress to the yeld surface n the case of a transton from elastc to elastoplastc behavour. Moreover, t can be proven that mplct ntegraton, under certan condtons, leads to a symmetrc and postve dfferental matrx ε/ σ, whch has a postve nfluence on teratve procedures. Because of these major advantages, restrcton s made here to mplct ntegraton and no attenton s gven to other types of tme ntegraton. Hence, for ω = 1 Eq. (2.15) reduces to: ε p = g λ σ (2.16) Substtuton of Eq. (2.16) nto Eq. (2.14) and successvely nto Eq. (2.7) gves: σ = tr e g σ λ D σ wth: tr -1 e σ = σ + D ε (2.17) In ths relaton σ tr s an auxlary stress vector, referred to as the elastc stresses or tral stresses, whch s the new stress state when consderng purely lnear elastc materal behavour. he ncrement of the plastc multpler λ, as used n Eq. (2.17), can be solved from the condton that the new stress state has to satsfy the yeld condton: f ( σ ) = 0 (2.18) 2-4 PLAXIS 3D UNNEL

11 DEFORMAION HEORY For perfectly-plastc and lnear hardenng models the ncrement of the plastc multpler can be wrtten as: where: tr f ( σ ) λ = d + h (2.19) σ f e g d = D σ σ tr (2.20) he symbol h denotes the hardenng parameter, whch s zero for perfectly-plastc models and constant for lnear hardenng models. In the latter case the new stress state can be formulated as: σ = tr σ - tr ( σ ) f e g D d + h σ (2.21) he -brackets are referred to as McCauley brackets, whch have the followng conventon: x = 0 for: x 0 and: x = x for: x > GLOBAL IERAIVE PROCEDURE Substtuton of the relatonshp between ncrements of stress and ncrements of stran, σ = M ε, nto the equlbrum equaton (2.13) leads to: K = v f -1 f ex n - (2.22) In ths equaton K s a stffness matrx, v s the ncremental dsplacement vector, f ex s the external force vector and f n s the nternal reacton vector. he superscrpt refers to the step number. However, because the relaton between stress ncrements and stran ncrements s generally non-lnear, the stffness matrx cannot be formulated exactly beforehand. Hence, a global teratve procedure s requred to satsfy both the equlbrum condton and the consttutve relaton. he global teraton process can be wrtten as: K j δ v j = f - f ex j-1 n (2.23) 2-5

12 SCIENIFIC MANUAL he superscrpt j refers to the teraton number. δv s a vector contanng subncremental dsplacements, whch contrbute to the dsplacement ncrements of step : v = n j δv (2.24) j= 1 where n s the number of teratons wthn step. he stffness matrx K, as used n Eq. (2.23), represents the materal behavour n an approxmated manner. he more accurate the stffness matrx, the fewer teratons are requred to obtan equlbrum wthn a certan tolerance. In ts smplest form K represents a lnear-elastc response. In ths case the stffness matrx can be formulated as: e K = B D B dv (elastc stffness matrx) (2.25) where D e s the elastc materal matrx accordng to Hooke's law and B s the stran nterpolaton matrx. he use of an elastc stffness matrx gves a robust teratve procedure as long as the materal stffness does not ncrease, even when usng nonassocated plastcty models. Specal technques such as arc-length control (Rks, 1979), over-relaxaton and extrapolaton (Vermeer & Van Langen, 1989) can be used to mprove the teraton process. Moreover, the automatc step sze procedure, as ntroduced by Van Langen & Vermeer (1990), can be used to mprove the practcal applcablty. For materal models wth lnear behavour n the elastc doman, such as the standard Mohr-Coulomb model, the use of an elastc stffness matrx s partcularly favourable, as the stffness matrx needs only be formed and decomposed before the frst calculaton step. hs calculaton procedure s summarsed n Appendx A. 2-6 PLAXIS 3D UNNEL

13 GROUNDWAER FLOW HEORY 3 GROUNDWAER FLOW HEORY In ths chapter we wll revew the theory of groundwater flow as used n the PLAXIS 3D unnel program. In addton to a general descrpton of groundwater flow, attenton s focused on the fnte element formulaton. 3.1 BASIC EQUAIONS OF SEADY FLOW Flow n a porous medum can be descrbed by Darcy's law. Consderng flow n a vertcal x-y-plane the followng equatons apply: qx = - k x φ x qy = - k y φ y qz = - k z φ z (3.1) he equatons express that the specfc dscharge, q, follows from the permeablty, k, and the gradent of the groundwater head. he head, φ, s defned as follows: p φ = y - (3.2) γ w where y s the vertcal poston, p s the stress n the pore flud (negatve for pressure) and γ w s the unt weght of the pore flud. For steady flow the contnuty condton apples: q q x y qz + + = 0 (3.3) x y z Eq. (3.3) expresses that there s no net nflow or outflow n an elementary area, as llustrated n Fgure 3.1. q y q + dy y q z q x q x q + dx x q z q + dz z q y Fgure 3.1 Illustraton of contnuty condton 3-1

14 SCIENIFIC MANUAL 3.2 FINIE ELEMEN DISCREISAION he groundwater head n any poston wthn an element can be expressed n the values at the nodes of that element: (, ) e φ ξη = Nφ (3.4) where N s the vector wth nterpolaton functons and ξ and η are the local coordnates wthn the element. Accordng to Eq. (3.1) the specfc dscharge s based on the gradent of the groundwater head. hs gradent can be determned by means of the B-matrx, whch contans the spatal dervatves of the nterpolaton functons. In order to descrbe flow for saturated sol (underneath the phreatc lne) as well as non-saturated sol (above the phreatc lne), a reducton functon K r s ntroduced n Darcy's law (Desa, 1976; L & Desa, 1983; Bakker, 1989): r qx = - K k x φ x = r qy - K k y φ y = r qz - K k z φ z (3.5) he reducton functon has a value of 1 below the phreatc lne (compressve pore pressures) and has lower values above the phreatc lne (tensle pore pressures). In the transton zone above the phreatc lne, the functon value decreases to the resdual permeablty rato α. he default value for α n PLAXIS 3D UNNEL s 0.01, the default heght for the transton zone, β, s 0.3. In the transton zone the functon s descrbed usng a lnear relaton: ( 1 α r ) p K = 1 (3.6) β γ w hs above relatonshp s llustrated n Fgure 3.2. It s not based on physcal experence or experments, but t s a numercal artfact that s necessary to solve unconfned problems. K r dry sol β saturated sol tenson pressure α 0 p γ w Fgure 3.2 Adjustment of the permeablty between saturated and unsaturated zones (K r = rato of permeablty over saturated permeablty) 3-2 PLAXIS 3D UNNEL

15 GROUNDWAER FLOW HEORY he reducton functon K r has the value of 1 underneath the phreatc lne and a small value α above the phreatc lne, whereas around the phreatc lne (transton zone) there s a lnear transton from 1 to α. In order to avod eventual flow above the phreatc lne, the value of α should be taken small. he parameter β defnes the sze of the transton zone. In the numercal formulaton, the specfc dscharge, q, s wrtten as: where: r e K RBφ q= (3.7) q q q x = y q z and: kx 0 0 R = 0 k y k z (3.8) From the specfc dscharges n the ntegraton ponts, q, the nodal dscharges Q e can be ntegrated accordng to: e Q = - B q dv (3.9) n whch B s the transpose of the B-matrx. On the element level the followng equatons apply: e Q e e = K φ wth: e K = r K B R B dv (3.10) On a global level, contrbutons of all elements are added and boundary condtons (ether on the groundwater head or on the dscharge) are mposed. hs results n a set of n equatons wth n unknowns: Q = K φ (3.11) n whch K s the global flow matrx and Q contans the prescrbed dscharges that are gven by the boundary condtons. In the case that the phreatc lne s unknown (unconfned problems), a Pcard scheme s used to solve the system of equatons teratvely. he lnear set s solved n ncremental form and the teraton process can be formulated as: K j-1 δφ j = Q K j-1 φ j-1 φ j = φ j-1 + δφ j (3.12) n whch j s the teraton number and r s the unbalance vector. In each teraton ncrements of the groundwater head are calculated from the unbalance n the nodal dscharges and added to the actve head. From the new dstrbuton of the groundwater head the new specfc dscharges are calculated accordng to Eq. (3.7), whch can agan be ntegrated nto nodal dscharges. 3-3

16 SCIENIFIC MANUAL hs process s contnued untl the norm of the unbalance vector,.e. the error n the nodal dscharges, s smaller than the tolerated error. 3.3 FLOW IN INERFACE ELEMENS Interface elements are treated specally n groundwater calculatons. he elements can be on or off. When the elements are swtched off, there s a full couplng of the pore pressure degrees of freedom. When the nterface elements are swtched on, there s no flow from one sde of the nterface element to the other (mpermeable screen). 3-4 PLAXIS 3D UNNEL

17 CONSOLIDAION HEORY 4 CONSOLIDAION HEORY In ths chapter we wll revew the theory of consoldaton as used n PLAXIS. In addton to a general descrpton of Bot's theory for coupled consoldaton, attenton s focused on the fnte element formulaton. Moreover, a separate secton s devoted to the use of advanced sol models n a consoldaton analyss (elastoplastc consoldaton). 4.1 BASIC EQUAIONS OF CONSOLIDAION he governng equatons of consoldaton as used n PLAXIS follow Bot's theory (Bot, 1956). Darcy's law for flud flow and elastc behavour of the sol skeleton are also assumed. he formulaton s based on small stran theory. Accordng to erzagh's prncple, stresses are dvded nto effectve stresses and pore pressures: where: ( steady pexcess ) σ = σ + m p + (4.1) σ = (σ xx σ yy σ zz σ xy σ yz σ zx ) and: m = ( ) (4.2) σ s the vector wth total stresses, σ' contans the effectve stresses, p excess s the excess pore pressure and m s a vector contanng unty terms for normal stress components and zero terms for the shear stress components. he steady state soluton at the end of the consoldaton process s denoted as p steady. Wthn PLAXIS p steady s defned as: psteady = Mweght pnput (4.3) where p nput s the pore pressure generated n the nput program based phreatc lnes or on a groundwater flow calculaton. Note that wthn PLAXIS compressve stresses are consdered to be negatve; ths apples to effectve stresses as well as to pore pressures. In fact t would be more approprate to refer to p excess and p steady as pore stresses, rather than pressures. However, the term pore pressure s retaned, although t s postve for tenson. he consttutve equaton s wrtten n ncremental form. Denotng an effectve stress ncrement as σ& ' and a stran ncrement as ε&, the consttutve equaton s: where: & σ ' = M ε& (4.4) ε = (ε xx ε yy ε zz γ xy γ yz γ zx ) (4.5) and M represents the materal stffness matrx. 4-1

18 SCIENIFIC MANUAL 4.2 FINIE ELEMEN DISCREISAION o apply a fnte element approxmaton we use the standard notaton: u = N v p = N p n ε = B v (4.6) where v s the nodal dsplacement vector, p n s the excess pore pressure vector, u s the contnuous dsplacement vector wthn an element and p s the (excess) pore pressure. he matrx N contans the nterpolaton functons and B s the stran nterpolaton matrx. In general the nterpolaton functons for the dsplacements may be dfferent from the nterpolaton functons for the pore pressure. In PLAXIS, however, the same functons are used for dsplacements and pore pressures. Startng from the ncremental equlbrum equaton and applyng the above fnte element approxmaton we obtan: wth: B d σ d V = N d f d V + N d t d s + r B d σ d V = N d f d V + N d t d s + r (4.7) 0 0 r = f d V + d s d V (4.8) N N 0 t B σ 0 where f s a body force due to self-weght and t represents the surface tractons. In general the resdual force vector, r 0, wll be equal to zero, but solutons of prevous load steps may have been naccurate. By addng the resdual force vector the computatonal procedure becomes self-correctng. he term dv ndcates ntegraton over the volume of the body consdered and ds ndcates a surface ntegral. Dvdng the total stresses nto pore pressure and effectve stresses and ntroducng the consttutve relatonshp gves the nodal equlbrum equaton: K d v + L d p = d f (4.9) n n where K s the stffness matrx, L s the couplng matrx and df n s the ncremental load vector: K = B M B d V (4.10a) L = B m N d V (4.10b) n d f = N d f d V + N d t d s (4.10c) 4-2 PLAXIS 3D UNNEL

19 CONSOLIDAION HEORY o formulate the flow problem, the contnuty equaton s adopted n the followng form: ε n p R ( γ w y - psteady - p ) / γ w-m + = 0 t K w t where R s the permeablty matrx: kx 0 0 R = 0 k y k z (4.11) (4.12) n s the porosty, K w s the bulk modulus of the pore flud and γ w s the unt weght of the pore flud. hs contnuty equaton ncludes the sgn conventon that p steady and p are consdered postve for tenson. As the steady state soluton s defned by the equaton: R ( γ y - p ) / γ = 0 (4.13) w steady w the contnuty equaton takes the followng form: ε n p R p / γ w + m - = 0 (4.14) t K w t Applyng fnte element dscretsaton usng a Galerkn procedure and ncorporatng prescrbed boundary condtons we obtan: where: d v - H p + L - S n d t d p d t H = ( N ) R N / γ d V, n = q (4.15) w n S = N N d V (4.16) K w and q s a vector due to prescrbed outflow at the boundary. However wthn PLAXIS t s not possble to have boundares wth non-zero prescrbed outflow. he boundary s ether closed or open wth zero excess pore pressure. Hence q = 0. In realty the bulk modulus of water s very hgh and so the compressblty of water can be neglected n comparson to the compressblty of the sol skeleton. In PLAXIS the bulk modulus of the pore flud s taken automatcally accordng to (also see Reference Manual): K w n = 3( ν u - ν ) Kskeleton (4.17) (1-2 ν u )(1 + ν ) 4-3

20 SCIENIFIC MANUAL Where ν u has a default value of For draned materal and materal just swtched on, the bulk modulus of the pore flud s neglected. he equlbrum and contnuty equatons may be compressed nto a block matrx equaton: d v K L d t d f 0 0 n v L -S = d p d t n 0 H p + n q d t n (4.18) A smple step-by-step ntegraton procedure s used to solve ths equaton. Usng the symbol to denote fnte ncrements, the ntegraton gves: K L v * L - S p n = 0 0 v0 f n * 0 t H p + n0 t q n (4.19) where: * S = α t H + S * q = q + α q (4.20) n n0 n and v 0 and p n0 denote values at the begnnng of a tme step. he parameter α s the tme ntegraton coeffcent. In general the ntegraton coeffcent α can take values from 0 to 1. In PLAXIS the fully mplct scheme of ntegraton s used wth α = ELASOPLASIC CONSOLIDAION In general, when a non-lnear materal model s used, teratons are needed to arrve at the correct soluton. Due to plastcty or stress-dependent stffness behavour the equlbrum equatons are not necessarly satsfed usng the technque descrbed above. herefore the equlbrum equaton s nspected here. Instead of Eq. (4.9) the equlbrum equaton s wrtten n sub-ncremental form: K δ v + L δ p = r n n (4.21) where r n s the global resdual force vector. he total dsplacement ncrement v s the summaton of sub-ncrements δv from all teratons n the current step: r n = N f d V + N t d s - B σ d V (4.22) wth: f = f + f and: t = t 0 + t (4.23) PLAXIS 3D UNNEL

21 CONSOLIDAION HEORY In the frst teraton we consder σ = σ 0,.e. the stress at the begnnng of the step. Successve teratons are used on the current stresses that are computed from the approprate consttutve model. 4-5

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23 ELEMEN FORMULAIONS 5 ELEMEN FORMULAIONS In ths chapter the nterpolaton functons of the fnte elements used n the PLAXIS 3D unnel program are descrbed. Each element conssts of a number of nodes. Each node has a number of degrees of freedom that correspond to dscrete values of the unknowns n the boundary value problem to be solved. In the case of deformaton theory the degrees of freedom correspond to the dsplacement components. In addton to the nterpolaton functons t s descrbed whch type of numercal ntegraton over elements s used n PLAXIS. 5.1 INERPOLAION FUNCIONS AND NUMERICAL INEGRAION OF LINE ELEMENS Wthn an element the dsplacement feld u = (u x u y u z ) s obtaned from the dscrete nodal values n a vector v = (v 1 v 2... v n ) usng nterpolaton functons assembled n matrx N: u = N v (5.1) Hence, nterpolaton functons N are used to nterpolate values nsde an element based on known values n the nodes. Interpolaton functons are also denoted as shape functons. Let us frst consder a lne element. Lne elements are the bass for dstrbuted loads on vertcal planes n the 3D model. he extenson of ths theory to areas and volumes s gven n the subsequent sectons. When the local poston, ξ, of a pont (usually a stress pont or an ntegraton pont) s known, one can wrte for a dsplacement component u: n u ( ξ) = N ( ξν ) (5.2) = 1 where: v the nodal values, N (ξ ) the value of the shape functon of node at poston ξ, u(ξ ) the resultng value at poston ξ and n the number of nodes per element NODE LINE ELEMENS In Fgure 5.1, an example of a 3-node lne element s gven, whch s compatble wth the sde of a 6-node trangle, an 8-node quadrlateral or a 15-node volume element n the PLAXIS 3D unnel program, snce these elements also have three nodes on a sde. he shape functons N have the property that the functon value s equal to unty at node and zero at the other nodes. 5-1

24 SCIENIFIC MANUAL For 3-node lne elements, where nodes 1, 2 and 3 are located at ξ = 1, 0 and 1 respectvely, the shape functons are gven by: N 1 = ½ (1 ξ ) ξ (5.3) N 2 = (1+ξ ) (1 ξ ) N 3 = ½ (1+ξ ) ξ 3-node lne elements provde a second-order nterpolaton of dsplacements. Fgure 5.1 Shape functons for a 3-node lne element NUMERICAL INEGRAION OF LINE ELEMENS In order to obtan the ntegral over a certan lne, the ntegral s numercally estmated as: 1 k F ( ξ) d ξ F ( ξ ) w (5.4) ξ=-1 = 1 where F(ξ ) s the value of the functon F at poston ξ and w the weght factor for pont. A total of k samplng ponts s used. A method that s commonly used for numercal ntegraton s Gaussan ntegraton, where the postons ξ and weghts w are chosen n a specal way to obtan hgh accuracy. For Gaussan-ntegraton a polynomal functon of degree 2k-1 can be ntegrated exactly by usng k ponts. he poston and weght factors of the two types of ntegraton are gven n able 5.1. Note that the sum of the weght factors s equal to 2, whch s equal to the length of the lne n local coordnates. he type of ntegraton used for the 3-node lne elements n PLAXIS s shaded. 5-2 PLAXIS 3D UNNEL

25 able 5.1 Gaussan ntegraton ELEMEN FORMULAIONS ξ w max. polyn. degree 1 pont ponts ± (±1/ 3) ponts ± (± 0.6) ponts ± ± ponts ± ± (5/9) (8/9) INERPOLAION FUNCIONS AND NUMERICAL INEGRAION OF AREA ELEMENS Areas and surfaces n the PLAXIS 3D unnel program are ether formed by 6-node trangular elements or by 8-node quadrlateral elements. he nterpolaton functons and the type of ntegraton of these elements s descrbed n the followng subsectons NODE RIANGULAR ELEMENS he 6-node trangles are created n the 2D mesh generaton process and used n the vertcal planes of the 3D model to form the faces of the 15-node wedge elements for sol. he 6-node trangles are also the bass for z-loads on clusters n the 3D model. For trangular elements there are two local coordnates (ξ and η). In addton we use an auxlary coordnate ζ = 1 ξ η. 6-node trangular elements provde a second-order nterpolaton of dsplacements. he shape functons can be wrtten as (see the local node numberng as shown n Fgure 5.2) : N 1 = ζ (2ζ 1) (5.5) N 2 N 3 N 4 N 5 N 6 = ξ (2ξ 1) = η (2η 1) = 4 ζ ξ = 4 ξ η = 4 η ζ 5-3

26 SCIENIFIC MANUAL ζ=0.0 3 ξ=0.0 ξ=0.5 ξ=1.0 η=1.0 ζ= η=0.5 ζ= η=0.0 Fgure 5.2 Local numberng and postonng of nodes ( ) and ntegraton ponts (x) of a 6-node trangular element As for lne elements, one can formulate the numercal ntegraton over areas as: k F ( ξη, ) d ξ d η F ( ξ, η) w (5.6) = 1 he PLAXIS 3D unnel program uses Gaussan ntegraton wthn the area elements. For 6-node trangular elements the ntegraton s based on 3 sample ponts (Fgure 5.2). he poston and weght factors of the ntegraton ponts are gven n able 5.2. Note that the sum of the weght factors s equal to 1. able pont Gaussan ntegraton for 6-node trangular elements Pont ξ η w 1 1/6 2/3 1/3 2 1/6 1/6 1/3 3 2/3 1/6 1/ NODE QUADRILAERAL ELEMENS he 8-node quadrlateral elements are created n the 3D mesh extenson process and they are used at the faces of the 15-node wedge elements n the z-drecton. hese elements are the bass for dstrbuted loads on slces n the 3D model and for structural elements (plates and geogrds) and nterface elements. 8-node quadrlateral elements provde a second-order nterpolaton of dsplacements. Quadrlateral elements have two local coordnates (ξ and η). he shape functons of 8- node elements can be wrtten as (see the local node numberng as shown n Fgure 5.3): 5-4 PLAXIS 3D UNNEL

27 ELEMEN FORMULAIONS N 1 = (1 ξ ) (1 η) ( 1 ξ η) / 4 (5.7) N 2 = (1+ξ ) (1 η) ( 1+ξ η) / 4 N 3 = (1+ξ ) (1+η) ( 1+ξ+η) / 4 N 4 = (1 ξ ) (1+η) ( 1 ξ+η) / 4 N 5 = (1 ξ ) (1+ξ ) (1 η) / 2 N 6 = (1 ξ ) (1+ξ ) (1+η) / 2 N 7 = (1 η) (1+η) (1+ξ ) / 2 N 8 = (1 η) (1+η) (1 ξ ) / 2 For 8-node quadrlateral elements the numercal ntegraton s based on 4 (2x2) Gauss ponts (Fgure 5.3), equvalent to the ntegraton of lne elements, but n two drectons. he poston and weght factors of the ntegraton ponts are gven n able 5.3. he sum of the weght factors s equal to 4, whch s equal to the area of the quadrlateral n local coordnates. able pont Gaussan ntegraton for 8-node quadrlateral elements Pont ξ η w 1 1/3 3 1/ /3 3 1/ /3 3 +1/ /3 3 +1/ ξ=-1.0 ξ=0.0 ξ=1.0 η= η= η=-1.0 Fgure 5.3 Local numberng and postonng of nodes ( ) and ntegraton ponts (x) of an 8-node quadrlateral element 5-5

28 SCIENIFIC MANUAL SRUCURAL ELEMENS Structural area elements n the PLAXIS 3D unnel program,.e. plates and geogrds, and nterfaces are based on the 8-node quadrlateral elements as descrbed n the prevous secton. However, there are some dfferences. Geogrd elements are not dfferent from 8- node quadrlaterals, as descrbed before. Geogrd elements have eght nodes and three dsplacement degrees of freedom per node (u x, u y, u z ). hese elements are numercally ntegrated usng 4-pont Gaussan ntegraton (see able 5.3). Plate elements are slghtly dfferent from 8-node quadrlaterals n the sense that they have sx degrees of freedom per node nstead of three,.e. three translatonal d.o.f.s (u x, u y, u z ) and three rotatonal d.o.f.s (φ x, φ y, φ z ). hese elements are numercally ntegrated usng 2x4 pont Gaussan ntegraton. he poston of the pars of ntegraton ponts corresponds wth able 5.3, but these ponts are postoned ½d eq / 3 outsde (perpendcular to) the plane of the element (Fgure 5.4). ξ=1.0 3 η=1.0 ξ= η=0.0 ξ= ½d eq / ½d eq / η=-1.0 Fgure 5.4 Local numberng and postonng of nodes ( ) and ntegraton ponts (x) of an 8-node plate element ξ= ξ=0.0 ξ=1.0 6 η= η= η= Fgure 5.5 Local numberng and postonng of nodes ( ) and ntegraton ponts (x) of a 16-node nterface element 5-6 PLAXIS 3D UNNEL

29 ELEMEN FORMULAIONS Interface elements are dfferent from the 8-node quadrlaterals n the sense that they have pars of nodes nstead of sngle nodes. Moreover, nterface elements have a 3x3 pont Gaussan ntegraton nstead of 2x2. he poston and numberng of the nodes and ntegraton ponts s ndcated n Fgure 5.5 (see also able 5.4). he dstance between the two nodes of a node par s zero. Each node has three translatonal degrees of freedom (u x, u y, u z ). As a result, nterface elements allow for dfferental dsplacements between the node pars (slppng and gappng). For more nformaton see Van Langen (1991). able pont Gaussan ntegraton for 16-node nterface elements Pont ξ η w INERPOLAION FUNCIONS AND NUMERICAL INEGRAION OF VOLUME ELEMENS he sol volume n the PLAXIS 3D unnel program s modelled by means of 15-node wedge elements. he nterpolaton functons, ther dervatves and the numercal ntegraton of ths type of element are descrbed n the followng subsectons NODE WEDGE ELEMENS he 15-node wedge elements are created n the 3D mesh extenson procedure. hs type of element provdes a second-order nterpolaton of dsplacements. For wedge elements there are three local coordnates (ξ, η and ζ). he shape functons of these 15-node volume elements can be wrtten as (see the local node numberng as shown n the Fgure 5.6): N 1 = (1 ξ η) (1 ζ) (+2ξ+2η+ζ) / 2 (5.8) N 2 = ξ (1 ζ) (2 2ξ ζ) / 2 N 3 = η (1 ζ) (2 2η+ζ) / 2 N 4 = (1 ξ η) (1+ζ) (+2ξ+2η ζ) / 2 5-7

30 SCIENIFIC MANUAL N 5 = ξ (1+ζ) (2 2ξ+ζ) / 2 N 6 = η (1+ζ) (2 2η ζ) / 2 N 7 = (1 ξ η) ξ (1 ζ) * 2 N 8 = ξ η (1 ζ) * 2 N 9 = η (1 ξ η) (1 ζ) * 2 N 10 = (1 ξ η) (1 ζ) (1+ζ) N 11 = ξ (1 ζ) (1+ζ) N 12 = η (1 ζ) (1+ζ) N 13 = (1 ξ η) ξ (1+ζ) * 2 N 14 = ξ η (1+ζ) * 2 N 15 = η (1 ξ η) (1+ζ) * ζ 5 12 η ξ Fgure 5.6 Local numberng and postonng of nodes ( ) and ntegraton ponts (x) of a 15-node wedge element NUMERICAL INEGRAION OVER VOLUMES As for lnes and areas, one can formulate the numercal ntegraton over volumes as: k F ( ξηζ,, ) d ξd ηd ζ F ( ξ, η, ζ) w (5.9) = PLAXIS 3D UNNEL

31 ELEMEN FORMULAIONS he PLAXIS 3D unnel program uses Gaussan ntegraton wthn the wedge elements. For 15-node wedge elements the ntegraton s based on 6 sample ponts. he ntegraton s a mxture between the 3-pont ntegraton of a 6-node trangular element and the 4- pont ntegraton of an 8-node quadrlateral. he poston and weght factors of the ntegraton ponts are gven n able 5.5. See Fgure 5.6 for the local numberng of ntegraton ponts. Note that the sum of the weght factors s equal to 2. able pont Gaussan ntegraton for 15-node wedge element Pont ξ η ζ w 1 1/6 2/3 1/3 3 1/3 2 1/6 1/6 1/3 3 1/3 3 2/3 1/6 1/3 3 1/3 4 1/6 2/3 +1/3 3 1/3 5 1/6 1/6 +1/3 3 1/3 6 2/3 1/6 +1/3 3 1/ DERIVAIVES OF SHAPE FUNCIONS In order to calculate Cartesan stran components from dsplacements, such as formulated n Eq. (2.10), dervatves need to be taken wth respect to the global system of axes (x,y,z). ε = B v (5.10) where B N 0 0 x N 0 0 y N 0 0 z = N N 0 y x N N 0 z y N N 0 z x (5.11) 5-9

32 SCIENIFIC MANUAL Wthn the elements, dervatves are calculated wth respect to the local coordnate system (ξ,η,ζ). he relatonshp between local and global dervatves nvolves the Jacoban J: N x y z N N ξ ξ ξ ξ x x N x y z N N J = η η η η y = y N x y z N N ς ς η η z z Or nversely: N N x ξ N 1 N = J y η N N z ς (5.12) (5.13) he local dervatves N / ξ, etc., can easly be derved from the element shape functons, snce the shape functons are formulated n local coordnates. he components of the Jacoban are obtaned from the dfferences n nodal coordnates. he nverse Jacoban J -1 s obtaned by numercally nvertng J. he Cartesan stran components can now be calculated by summaton of all nodal contrbutons: ε xx ε (5.14) yy vx, ε zz = B v y, γ xy v z, γ yz γ zx where v are the dsplacement components n node PLAXIS 3D UNNEL

33 ELEMEN FORMULAIONS CALCULAION OF ELEMEN SIFFNESS MARIX he element stffness matrx, K e, s calculated by the ntegral (see also Eq. 2.25): K e e = B D B dv (5.15) he ntegral s estmated by numercal ntegraton as descrbed n Secton In fact, e the element stffness matrx s composed of submatrces K j where and j are the local nodes. he process of calculatng the element stffness matrx can be formulated as: K e e = B D B w (5.16) j j k k 5-11

34 SCIENIFIC MANUAL 5-12 PLAXIS 3D UNNEL

35 REFERENCES 6 REFERENCES [1] Aubry D., Ozanam O. (1988), Free-surface trackng through non-saturated models. Proc. 6th Internatonal Conference on Numercal Methods n Geomechancs, Innsbruck, pp [2] Bakker K.J. (1989), Analyss of groundwater flow through revetments. Proc. 3rd Internatonal Symposum on Numercal Models n Geomechancs. Nagara Falls, Canada. pp [3] Bathe K.J., Koshgoftaar M.R. (1979), Fnte element free surface seepage analyss wthout mesh teraton. Int. J. Num. An. Meth Geo, Vol. 3, pp [4] Bot M.A. (1956), General solutons of the equatons of elastcty and consoldaton for porous materal. Journal of Appled Mechancs, Vol. 23 No. 2 [5] Brnkgreve R.B.J. (1994), Geomateral Models and Numercal Analyss of Softenng. Dssertaton. Delft Unversty of echnology. [6] Desa C.S. (1976), Fnte element resdual schemes for unconfned flow. Int. J. Num. Meth. Eng., Vol. 10, pp [7] L G.C., Desa C.S. (1983), Stress and seepage analyss of earth dams. J. Geotechncal Eng., Vol. 109, No. 7, pp [8] Rks E. (1979), An ncremental approach to the soluton of snappng and bucklng problems. Int. J. Solds & Struct. Vol. 15, pp [9] Van Langen H. (1991), Numercal analyss of sol-structure nteracton. Dssertaton. Delft Unversty of echnology. [10] Vermeer P.A. (1979), A modfed ntal stran method for plastcty problems. In: Proc. 3rd Int. Conf. Num. Meth. Geomech. Balkema, Rotterdam, pp [11] Vermeer P.A., De Borst R. (1984), Non-assocated plastcty for sols, concrete and rock. Heron, Vol. 29, No. 3. [12] Vermeer P.A., Van Langen H. (1989), Sol collapse computatons wth fnte elements. Ingeneur-Archv 59, pp [13] Van Langen H., Vermeer P.A. (1990), Automatc step sze correcton for nonassocated plastcty problems. Int. J. Num. Meth. Eng. Vol. 29, pp [14] Zenkewcz O.C. (1967), he fnte element method n structural and contnuum mechancs. McGraw-Hll, London, UK. 6-1

36 SCIENIFIC MANUAL 6-2 PLAXIS 3D UNNEL

37 APPENDIX A - CALCULAION PROCESS APPENDIX A - CALCULAION PROCESS Fnte element calculaton process based on the elastc stffness matrx Read nput data Form stffness matrx K = B D B d V New step + 1 Form new load vector f = -1 + f ex Form reacton vector f n = σ d V ex e B - 1 c Calculate unbalance f = f - f ex n Reset dsplacement ncrement v = 0 New teraton j j + 1 Solve dsplacements δ v = K -1 f *) ex Update dsplacement ncrements Calculate stran ncrements = + δ v ε = B v ; δ ε = B δ v v j v j- 1 Calculate stresses: Elastc σ tr = σ - 1 D ε c + e Equlbrum σ eq = σ, j - 1 D δ ε σ, j c + e tr f ( σ ) c = tr σ - Consttutve d, j Form reacton vector f n = B d V σ c e g D σ Calculate unbalance f = f ex n Calculate error f e = f ex Accuracy check f e > e tolerated new teraton Update dsplacements v = v 1 + v Wrte output data (results) If not fnshed new step Fnsh A-1

38 SCIENIFIC MANUAL *) he soluton of the system of equatons s done usng a sparse teratve soluton procedure wth smart pre-condtonng. For normal elastoplastc deformaton calculatons the soluton s based on the Conjugate Gradent method (CG), whereas for consoldaton calculatons (resultng n an ndefnte matrx) the soluton s based on SYM-QMR 1. he precondtonng s based on the elastc materal stffness matrx wth dagonal scalng and usng a varable drop tolerance. 1 Freund R.W., Jarre F. (1996). A QMR-based nteror-pont algorthm for solvng lnear programs. Mathematcal Programmng Seres ~ B 76, pp A-2

39 APPENDIX B - SYMBOLS APPENDIX B - SYMBOLS B : Stran nterpolaton matrx D e : Elastc materal stffness matrx representng Hooke's law f : Yeld functon f : Load vector g : Plastc potental functon K : Stffness matrx L : Dfferental operator M : Materal stffness matrx N : Matrx wth shape functons p : Body forces vector t : me t : Boundary tractons u : Vector wth dsplacement components v : Vector wth nodal dsplacements V : Volume w : Weght factor γ : Volumetrc weght ε : Vector wth stran components λ : Plastc multpler ξ η ζ : Local coordnates σ : Vector wth stress components ω : Integraton constant (explct: ω=0; mplct: ω=1) B-1

40 SCIENIFIC MANUAL B-2