PLAXIS Version 8 Scientific Manual

Transcription

1 PLAXIS Verson 8 Scentfc Manual

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3 ABLE OF COES 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 for lne elements Interpolaton functons for trangular elements umercal ntegraton of lne elements umercal ntegraton of trangular elements Dervatves of shape functons Calculaton of element stffness matrx heory of senstvty analyss & parameter paraton Senstvty analyss Defnton of threshold value heory of parameter varaton Bounds on the system response References Appendx A - Calculaton process Appendx B - Symbols

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5 IRODUCIO 1 IRODUCIO In ths part of the manual some scentfc background s gven of the theores and numercal methods on whch the PLAXIS 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 PLAXIS. In the Appendx a global calculaton scheme s provded for a plastc deformaton analyss. In addton to the specfc nformaton gven n ths part of the manual, 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 DEFORMAIO HEORY 2 DEFORMAIO 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 EQUAIOS OF COIUUM DEFORMAIO he statc equlbrum of a contnuum can be formulated as: L σ + p = (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 y z x In addton to the equlbrum equaton, the knematc relaton can be formulated as: (2.2) ε = 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) 2-1

8 SCIEIFIC MAUAL he combnaton of Eqs. (2.1), (2.3) and (2.4) would lead to a second-order partal dfferental equaton n the dsplacements u. 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): ( L σ + p) dv = δ u (2.5) In ths formulaton δ u represents a knematcally admssble varaton of dsplacements. Applyng Green's theorem for partal ntegraton to the frst term n Eq. (2.5) leads to: ε σ dv δ u p dv + δ = 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 = σ -1 + Δσ Δ 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 Δσ 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): δ ε Δσ dv = δ u p dv + δ u t ds δ ε σ -1 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 FIIE ELEME DISCREISAIO 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 : 2-2 PLAXIS Verson 8

9 u = DEFORMAIO HEORY v (2.9) he nterpolaton functons n matrx are often denoted as shape functons. Substtuton of Eq. (2.9) n the knematc relaton (2.3) gves: ε = L v = B v (2.1) In ths relaton B s the stran nterpolaton matrx, whch contans the spatal dervatves of the nterpolaton functons. Eqs. (2.9) and (2.1) can be used n varatonal, ncremental and rate form as well. Eq. (2.8) can now be reformulated n dscretsed form as: ( B δν ) Δσ dv = ( δν ) p dv + ( δν ) t ds ( B δν ) he dscrete dsplacements can be placed outsde the ntegral: δ ν B Δσ dv = δ ν p dv + δ ν t ds δ ν B σ -1 σ dv -1 dv (2.11) (2.12) Provded that Eq. (2.12) holds for any knematcally admssble dsplacement varaton δv, the equaton can be wrtten as: B Δσ d V = p d V + t d S B σ -1 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 Δσ. 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 IEGRAIO OF DIFFEREIAL PLASICIY MODELS he stress ncrements Δσ 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 ( Δε Δ ε ) Δ σ = D (2.14) 2-3

10 SCIEIFIC MAUAL In ths relaton D e represents the elastc materal matrx for the current stress ncrement. he stran ncrements Δε are obtaned from the dsplacement ncrements Δv usng the stran nterpolaton matrx B, smlar to Eq. (2.1). For elastc materal behavour, the plastc stran ncrement Δε p s zero. For plastc materal behavour, the plastc stran ncrement can be wrtten, accordng to Vermeer (1979), as: p Δ ε = Δλ -1 g g 1 + ω (2.15) σ σ ( ω) In ths equaton Δλ s the ncrement of the plastc multpler and ω s a parameter ndcatng the type of tme ntegraton. For ω = 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: Δ ε p g = Δλ (2.16) σ Substtuton of Eq. (2.16) nto Eq. (2.14) and successvely nto Eq. (2.7) gves: σ tr = σ Δλ D e g σ D e wth: σ tr = σ -1 + Δε (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 Δλ, 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 ) = (2.18) For perfectly-plastc and lnear hardenng models the ncrement of the plastc multpler can be wrtten as: 2-4 PLAXIS Verson 8

11 where: tr f ( σ ) Δ λ = d + h DEFORMAIO HEORY (2.19) σ f d = σ tr e g D σ (2.2) 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: he conventon: tr σ = σ - tr ( σ ) f e g D d + h σ (2.21) -brackets are referred to as McCauley brackets, whch have the followng x = for: x and: x = x for: x > 2.4 GLOBAL IERAIVE PROCEDURE Substtuton of the relatonshp between ncrements of stress and ncrements of stran, Δσ = M Δε, nto the equlbrum equaton (2.13) leads to: K Δ v = f - f ex -1 n (2.22) In ths equaton K s a stffness matrx, Δv 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 j v = f - f ex j-1 n (2.23) he superscrpt j refers to the teraton number. δv s a vector contanng subncremental dsplacements, whch contrbute to the dsplacement ncrements of step : 2-5

12 SCIEIFIC MAUAL Δ = δ v v n j= 1 j (2.24) 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: K = B e 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 (199), 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 Verson 8

13 GROUDWAER FLOW HEORY 3 GROUDWAER FLOW HEORY In ths chapter we wll revew the theory of groundwater flow as used n PLAXIS. In addton to a general descrpton of groundwater flow, attenton s focused on the fnte element formulaton. 3.1 BASIC EQUAIOS 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: φ q x = - k x x φ q y = - k y y (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: φ = p y - γ w (3.2) 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 + = (3.3) x y Eq. (3.3) expresses that there s no net nflow or outflow n an elementary area, as llustrated n Fgure 3.1. Fgure 3.1 Illustraton of contnuty condton 3-1

14 SCIEIFIC MAUAL 3.2 FIIE ELEME DISCREISAIO he groundwater head n any poston wthn an element can be expressed n the values at the nodes of that element: φ (ξ,η) = φ e (3.4) where 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): q x = - K r k x φ x q y = - K r k y φ y (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 mnmum of 1-4. In the transton zone the functon s descrbed usng a log-lnear relaton: or r K K h / h = 1 4 k 1 k r 4h log( K ) = (3.6) h where h s the pressure head and h k s the pressure head where the reducton functon has reached the mnmum of 1-4. In PLAXIS h k has a default value of.7 m (ndependent of the chosen length unt). In the numercal formulaton, the specfc dscharge, q, s wrtten as: r e q = K RBφ (3.7) r where: q = q q x y and: R = k x k y (3.8) 3-2 PLAXIS Verson 8

15 GROUDWAER FLOW HEORY 1,1,1,1 Kr (log),1 1,2 1,8,6,4,2,1 h/hk (a) 1,8,6 Kr,4,2 1,2 1,8,6,4,2 h/hk Fgure 3.2 Adjustment of the permeablty between saturated (a) and unsaturated (b) zones (K r = rato of permeablty over saturated permeablty) (b) From the specfc dscharges n the ntegraton ponts, q, the nodal dscharges Q e can be ntegrated accordng to: Q e = - B q dv (3.9) 3-3

16 SCIEIFIC MAUAL n whch B s the transpose of the B-matrx. On the element level the followng equatons apply: Q e e e e r = K φ wth: K K B R B dv = (3.1) 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 = Kφ (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 j j 1 j 1 δ φ = Q K φ j j 1 φ = φ + 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. 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 I IERFACE ELEMES 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 Verson 8

17 COSOLIDAIO HEORY 4 COSOLIDAIO 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 EQUAIOS OF COSOLIDAIO 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: ( ) σ = σ + m p steady + p excess (4.1) where: σ = ( σ σ σ σ σ ) xx yy zz xy 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: p = Mweght (4.3) steady p nput where p nput s the pore pressure generated n the nput program based phreatc lnes or on a groundwater flow calculaton. ote 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: & σ ' = M & ε (4.4) where: & ε = ( & ε & ε & ε & γ & γ & ) xx yy zz xy yz γ zx and M represents the materal stffness matrx. (4.5) 4-1

18 SCIEIFIC MAUAL 4.2 FIIE ELEME DISCREISAIO o apply a fnte element approxmaton we use the standard notaton: u = v p = 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 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 dσ d V = d f d V + d t d S + r (4.7) r = f d V + t d S - B σ d V (4.8) where f s a body force due to self-weght and t represents the surface tractons. In general the resdual force vector r 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 n = d f n (4.9) 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 (4.1a) L = B m d V (4.1b) d f n = d f d V + d t d S (4.1c) o formulate the flow problem, the contnuty equaton s adopted n the followng form: 4-2 PLAXIS Verson 8

19 COSOLIDAIO HEORY ε n p R ( γ w y - psteady - p ) / γ w - m + = (4.11) t K w t where R s the permeablty matrx: k x R= k y (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 = (4.13) w steady the contnuty equaton takes the followng form: w ε n p R p / γ w + m - = (4.14) t K t w Applyng fnte element dscretsaton usng a Galerkn procedure and ncorporatng prescrbed boundary condtons we obtan: where: d v d p p n - H + L - S = q (4.15) n d t d t H = ( ) R / n γ w d V, S = d V K w (4.16) and q s a vector due to prescrbed outflow at the boundary. However wthn PLAXIS Verson 8 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 =. 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 3 ( ν u -ν ) = K skeleton n ( 1-2ν )( 1+ ν ) u (4.17) 4-3

20 SCIEIFIC MAUAL Where u ν has a default value of.495. he value can be modfed n the nput program on the bass of Skempton's B-parameter. 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: t d p d t d v d -S L L K n = q t d f d + p v H n n n (4.18) A smple step-by-step ntegraton procedure s used to solve ths equaton. Usng the symbol Δ to denote fnte ncrements, the ntegraton gves: Δ Δ p v S - L L K n * = Δ Δ Δ q t f + p v H t n * n n (4.19) where: H + S t = S * Δ α q + q = q n n n * Δ α (4.2) and v and p n 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 to 1. In PLAXIS the fully mplct scheme of ntegraton s used wth α = ELASOPLASIC COSOLIDAIO 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: = r p v + L K n n δ δ (4.21) where r n s the global resdual force vector. he total dsplacement ncrement Δv s the summaton of sub-ncrements δv from all teratons n the current step: d V B - S d t d V + f r n σ = (4.22) 4-4 PLAXIS Verson 8

21 COSOLIDAIO HEORY wth: f = f + Δ f and: t = t + t Δ (4.23) In the frst teraton we consder σ = σ,.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 ELEME FORMULAIOS 5 ELEME FORMULAIOS In ths chapter the nterpolaton functons of the fnte elements used n PLAXIS 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, whereas n the case of groundwater flow the degrees-of-freedom are the groundwater heads. For consoldaton problems degreesof-freedom are both dsplacement components and (excess) pore pressures. In addton to the nterpolaton functons t s descrbed whch type of numercal ntegraton over elements s used n PLAXIS. 5.1 IERPOLAIO FUCIOS FOR LIE ELEMES Wthn an element the dsplacement feld u = (u x u y ) s obtaned from the dscrete nodal values n a vector v = (v 1 v 2... v n ) usng nterpolaton functons assembled n matrx : u = v (5.1) Hence, nterpolaton functons 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 geotextle elements, plate elements and dstrbuted loads. 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: where: n ( ) = ( ξ ) v u ξ (5.2) v =1 the nodal values, (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. In the graph, an example of a 3-node lne element s gven, whch s compatble wth the 6-node trangle elements n PLAXIS, snce 6-node trangles have three nodes at a sde. he shape functons have the property that the functon s equal to one at node and zero at the other nodes. For 3-node lne elements, where the nodes 1, 2 and 3 are located at ξ = -1, and 1 respectvely, the shape functons are gven by: 5-1

24 SCIEIFIC MAUAL 1 = -½ (1-ξ ) ξ (5.3) 2 = (1+ξ ) (1-ξ ) 3 = ½ (1+ξ ) ξ Fgure 5.1 Shape functons for 3-node lne element Fgure 5.2 Shape functons for 5-node lne element When usng 15-noded trangles, there are fve nodes at a sde. For 5-node lne elements, where nodes 1 to 5 are at ξ = -1, -½,, ½ and 1 respectvely, we have: 1 = - (1-ξ ) (1-2ξ ) ξ (-1-2ξ ) /6 (5.4) 2 = 4 (1-ξ ) (1-2ξ ) ξ (-1-ξ ) /3 3 = (1-ξ ) (1-2ξ ) (-1-2ξ )(-1-ξ ) 4 = 4 (1-ξ ) ξ (-1-2ξ ) (-1-ξ ) /3 5 = (1-2ξ ) ξ (-1-2ξ ) (-1-ξ ) /6 5-2 PLAXIS Verson 8

25 ELEME FORMULAIOS 5.2 IERPOLAIO FUCIOS FOR RIAGULAR ELEMES For trangular elements there are two local coordnates (ξ and η). In addton we use an auxlary coordnate ζ = 1-ξ-η. For 15-node trangles the shape functons can be wrtten as (see the local node numberng as shown n Fgure 5.3) : 1 = ζ (4ζ-1) (4ζ-2) (4ζ-3) /6 (5.5) 2 = ξ (4ξ-1) (4ξ-2) (4ξ-3) /6 3 = η (4η-1) (4η-2) (4η-3) / = 4 ζ ξ (4ζ-1) (4ξ-1) = 4 ξ η (4ξ-1) (4η-1) = 4 η ζ (4η-1) (4ζ-1) 7 = ξ ζ (4ζ-1) (4ζ-2) *8/3 8 = ζ ξ (4ξ-1) (4ξ-2) *8/3 9 = η ξ (4ξ-1) (4ξ-2) *8/3 1 = ξ η (4η-1) (4η-2) *8/3 11 = ζ η (4η-1) (4η-2) *8/3 12 = η ζ (4ζ-1) (4ζ-2) *8/ = 32 η ξ ζ (4ζ-1) = 32 η ξ ζ (4ξ-1) = 32 η ξ ζ (4η-1) Fgure 5.3 Local numberng and postonng of nodes 5-3

26 SCIEIFIC MAUAL Smlarly, for 6-node elements the shape functons are: 1 = ζ (2ζ-1) (5.6) = ξ (2ξ-1) = η (2η-1) = 4 ζ ξ = 4 ξ η = 4 η ζ 5.3 UMERICAL IEGRAIO OF LIE ELEMES In order to obtan the ntegral over a certan lne or area, the ntegral s numercally estmated as: 1 ξ=-1 k F( ξ )dξ F( ξ ) w = 1 (5.7) 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. wo methods are frequently used n PLAXIS, frstly ewton-cotes ntegraton, where the ponts ξ are chosen at the poston of the nodes, and secondly Gauss ntegraton where fewer ponts at specal locatons can be used to obtan hgh accuracy. he poston and weght factors of the two types of ntegraton are gven n able 5.1 and 5.2 respectvely. ote that the sum of the weght factors s equal to 2. Usng ewton-cotes ntegraton one can exactly ntegrate polynomnal functons one order below the number of ponts used. For Gauss-ntegraton a polynomal functon of degree 2k-1 can be ntegrated exactly by usng k ponts. For nterface elements and geotextle elements PLAXIS uses ewton-cotes ntegraton, whereas for beam elements and the ntegraton of boundary loads Gaussan ntegraton s used. able 5.1 ewton-cotes ntegraton ξ 2 nodes ± nodes ± 1, 1/3, 4/3 4 nodes ± 1, ± 1/3 1/4, ¾ 5 nodes ± 1, ± 1/2, 7/45, 32/45, 12/45 w 5-4 PLAXIS Verson 8

27 ELEME FORMULAIOS able 5.2 Gauss ntegraton ξ 1 pont ponts ± (±1/ 3) 1 3 ponts ± (±.6)... 4 ponts ± ± ponts ± ± w (5/9) (8/9) UMERICAL IEGRAIO OF RIAGULAR ELEMES As for lne elements, one can formulate the numercal ntegraton over trangular elements as: F( ξ, η ) dξ dη F( ξ, η ) w = 1 k (5.8) PLAXIS uses Gaussan ntegraton wthn the trangular elements. For 6-node elements the ntegraton s based on 3 sample ponts, whereas for 15-node elements 12 sample ponts are used. he poston and weght factors of the ntegraton ponts are gven n able 5.3 and 5.4. ote that, n contrast to the lne elements, the sum of the weght factors s equal to 1. able pont ntegraton for 6-node elements Pont ξ η ζ w 1, 2 & 3 1/6 1/6 2/3 1/3 able pont ntegraton for 15-node elements Pont ξ η ζ w 1,2 &

28 SCIEIFIC MAUAL 5.5 DERIVAIVES OF SHAPE FUCIOS In order to calculate Cartesan stran components from dsplacements, such as formulated n Eq. (2.1), dervatves need to be taken wth respect to the global system of axes (x,y,z). B v = ε (5.9) where = x z y z x y z y x B (5.1) 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: = = z y x J z y x z y x z y x z y x ς ς ς η η η ξ ξ ξ ς η ξ (5.11) Or nversely: = ς η ξ J z y x 1 (5.12) 5-6 PLAXIS Verson 8

29 ELEME FORMULAIOS he local dervatves /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 yy ε zz = γ xy γ yz γ zx v B v v x, y, z, where v are the dsplacement components n node. (5.13) For a plane stran analyss, stran components n z-drecton are zero by defnton,.e. ε zz = γ yz = γ zx =. For an axsymmetrc analyss, the followng condtons apply: ε zz = u x / r and γ yz = γ zx = (r = radus) 5.6 CALCULAIO OF ELEME SIFFESS MARIX he element stffness matrx, K e, s calculated by the ntegral (see also Eq. 2.25): K e = B D B dv e (5.14) he ntegral s estmated by numercal ntegraton as descrbed n secton 5.3. In fact, the e element stffness matrx s composed of submatrces K where and j are the local nodes. he process of calculatng the element stffness matrx can be formulated as: e j K = B D B w (5.15) k e j k j 5-7

30 SCIEIFIC MAUAL 5-8 PLAXIS Verson 8

31 HEORY OF SESIIVIY AALYSIS & PARAMEER PARIAIO 6 HEORY OF SESIIVIY AALYSIS & PARAMEER PARIAIO hs chapter presents some of the theoretcal backgrounds of the senstvty analyss and parameter varaton module. he chapter does not gve a full theoretcal descrpton of the methods of nterval analyss. For a more detaled descrpton you are referred to the lterature e.g. [17] to [23]. 6.1 SESIIVIY AALYSIS A method for quantfyng senstvty n the sense dscussed n ths Secton and n the Secton Senstvty analyss and Parameter varaton of the Reference Manual s the senstvty rato η SR [15]. he rato s defned as the percentage change n output dvded by the percentage change n nput for a specfc nput varable, as shown n Eq. 6.1: η SR f(x L,R ) f(x) 1% f(x) = (6.1) x L,R x 1% x f(x) s the reference value of the output varable usng reference values of the nput varables and f(x L,R ) s the value of the output varable after changng the value of one nput varable, whereas x and x L,R n the denomnator are the respectve nput varables. For the senstvty rato, an nput varable, x L,R, s vared ndvdually across the entre range requrng 2+1 calculatons, beng the number of vared parameters consdered. An extenson to the senstvty rato and a more robust method of evaluatng mportant sources of uncertanty s the senstvty score, η SS, whch s the senstvty rato η SR weghted by a normalzed measure of the varablty n an nput varable, as gven by Eq. 6.2: (max xr mn xr ) η SS =η SR (6.2) x By normalsng the measure of varablty (.e. the range dvded by the reference value), ths method effectvely weghts the ratos n a manner that s ndependent of the unts of the nput varable. Performng a senstvty analyss as descrbed above, the senstvty score of each varable, η SS,, on respectve results A,B,,Z, (e.g. dsplacements, forces, factor of safety, etc.) at each constructon step (calculaton phase) can be quantfed as shown n able 6.1 (Senstvty matrx). he total senstvty score of each varable, Ση SS,, results from summaton of all senstvty scores for each respectve result at each constructon step. 6-1

32 SCIEIFIC MAUAL able 6.1 Senstvty matrx Respectve results A B Z Σ α Input varables % x 1 η SS,A1 η SS,B1 η SS,Z1 Ση SS,1 α(x 1 ) x 2 η SS,A2 η SS,B2 η SS,Z2 Ση SS,2 α(x 2 ) M M M M M M M x η SS,A η SS,B η SS,Z Ση SS, α(x ) It should be noted, that the results of the senstvty analyss appeared to be strongly dependent on the respectve results used and thus results relevant for the problem nvestgated have to be chosen based on sound engneerng judgement. Fnally, the total relatve senstvty α(x ) for each nput varable s then gven by [16] as Ση SS, α ( x ) = (6.3) Ση = 1 SS, Fgure 6.1 shows the total relatve senstvty of each parameter α(x ) n dagram form n order to llustrate the major varables. otal relatve senstvty hreshold value x 1 x 2 x 3... x α(x ) Fgure 6.1 otal relatve senstvty n dagram form 6-2 PLAXIS Verson 8

33 HEORY OF SESIIVIY AALYSIS & PARAMEER PARIAIO DEFIIIO OF HRESHOLD VALUE he beneft of such an analyss s twofold: Frstly, the results are the bass for a decson-makng n order to reduce the computatonal effort nvolved when utlzng a parameter varaton,.e. at ths end a decson has to be made (defnton of a threshold value), whch varables (parameters) should be used n further calculatons and whch one can be treated as determnstc values as ther nfluence on the result s not sgnfcant (Fgure 6.1). Secondly, senstvty analyss can be appled for example to desgn further nvestgaton programs to receve addtonal nformaton about parameters wth hgh senstvty n order to reduce the uncertanty of the system response,.e. the result may act as a bass for the desgn of an nvestgaton program (laboratory and/or n stu tests). 6.2 HEORY OF PARAMEER VARIAIO he parameter varaton used n the PLAXIS parameter varaton module refer to classcal set theory where uncertanty s represented n terms of closed ntervals (bounds) assumng that the true value of the relevant unknown quantty s captured (X [x mn,x max ]). In general, an nterval s defned as a par of elements of some (at least partally) ordered sets [24]. An nterval s dentfed wth the set of elements lyng between the nterval endponts (ncludng the endponts) and usng the set of real numbers as the underlyng ordered set (real ntervals). Hence, all ntervals are closed sets. hus, a (proper) real nterval x s a subset of the set of real numbers R such that: [ x, x ] = { x' R x x' x } x = (6.4) mn max mn where x mn x max and x mn = sup(x), x max = nf(x) are endponts of the nterval x. In general, x denotes any element of the nterval x. If the true values of the parameters of nterest are bounded by ntervals, ths wll always ensure a relable estmate (worst/bestcase analyss) based on the nformaton avalable. For the parameter varaton, the nput parameters x are treated as nterval numbers (x,mn /x,max ) whose ranges contan the uncertantes n those parameters. he resultng computatons, carred out entrely n nterval form, would then lterally carry the uncertantes assocated wth the data through the analyss. Lkewse, the fnal outcome n nterval form would contan all possble solutons due to the varatons n nput. max x * x * X Fgure 6.2 otal relatve senstvty n dagram form 6-3

34 SCIEIFIC MAUAL BOUDS O HE SYSEM RESPOSE Let X be a non-empty set contanng all the possble values of a parameter x and y = f(x), f : X -> Y be a real-valued functon of x. he nterval of the system response through f, can be calculated by means of a functon used n set theory. In fact, f x belongs to set A, then the range of y s f ( A) { f ( x) : x A}. = (6.5) Here, the set A s called the focal element. he basc step s the calculaton of the system response through functon f whch represents here a numercal model. In general, ths nvolves two global optmsaton problems whch can be solved by applyng twce the technques of global optmsaton (e.g. [25],[26]) to fnd the lower and upper bound, y mn and y max, respectvely, of the system response: where [ y ] f ( A) = y (6.6) mn, max y mn = mn f ( x) (6.7) x A y max = max f ( x) (6.8) x A In the absence of any further nformaton a so-called calculaton matrx, can be constructed by assumng ndependence between parameters x [11]. A s the Cartesan product of fnte ntervals x 1 x (calculaton matrx), therefore t s a - dmensonal box (nterval vector) whose 2 vertces are ndcated as v k, k = 1,,2, beng the number of parameters consdered. he lower and upper bounds y mn and y max of the system response can be obtaned as follows: { f ( ) : k 1,... } y = ν 2 (6.9) mn mn k = k { f ( ) : k 1,... } y = ν 2 (6.1) max max k = k If f(a) has no extreme value n the nteror of A, except at the vertces, Equatons (6.9) and (6.1) are correct n whch case the methods of nterval analyss are applcable, e.g. the Vertex method [27]. If, on the other hand, f(a) has one or more extreme values n the nteror of A, then Equatons (6.9) and (6.1) can be taken as approxmatons to the true global mnmum and maxmum value. 6-4 PLAXIS Verson 8

35 REFERECES 7 REFERECES [1] Aubry D., Ozanam O. (1988), Free-surface trackng through non-saturated models. Proc. 6th Internatonal Conference on umercal Methods n Geomechancs, Innsbruck, pp [2] Bakker K.J. (1989), Analyss of groundwater flow through revetments. Proc. 3rd Internatonal Symposum on umercal Models n Geomechancs. agara Falls, Canada. pp [3] Bathe K.J., Koshgoftaar M.R. (1979), Fnte element free surface seepage analyss wthout mesh teraton. Int. J. um. 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 o. 2 [5] Brnkgreve R.B.J. (1994), Geomateral Models and umercal Analyss of Softenng. Dssertaton. Delft Unversty of echnology. [6] Desa C.S. (1976), Fnte element resdual schemes for unconfned flow. Int. J. um. Meth. Eng., Vol. 1, pp [7] L G.C., Desa C.S. (1983), Stress and seepage analyss of earth dams. J. Geotechncal Eng., Vol. 19, o. 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), umercal analyss of sol-structure nteracton. Dssertaton. Delft Unversty of echnology. [1] Vermeer P.A. (1979), A modfed ntal stran method for plastcty problems. In: Proc. 3rd Int. Conf. um. Meth. Geomech. Balkema, Rotterdam, pp [11] Vermeer P.A., De Borst R. (1984), on-assocated plastcty for sols, concrete and rock. Heron, Vol. 29, o. 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. (199), Automatc step sze correcton for nonassocated plastcty problems. Int. J. um. Meth. Eng. Vol. 29, pp [14] Zenkewcz O.C. (1967), he fnte element method n structural and contnuum mechancs. McGraw-Hll, London, UK. [15] U.S. EPA: RIM (1999). RIM, otal Rsk Integrated Methodology. RIM FAE echncal Support Document Volume I: Descrpton of Module. EPA/43/D- 99/2A, Offce of Ar Qualty Plannng and Standards. [16] Peschl, G. M. (24). Relablty Analyses n Geotechncs wth the Random Set Fnte Element Method. Insttute for Sol Mechancs and Foundaton Engneerng, Graz Unversty of echnology, Dssertaton. [17] Moore, R. E. (1966). Interval Analyss. Prentce Hall: Englewood Clffs. 7-1

36 SCIEIFIC MAUAL [18] Moore, R. E. (1979). Methods and Applcatons of Interval Analyss. SIAM Studes n Appled Mathematcs: Phladelpha. [19] Alefeld, G.; Herzberger, J. (1983). Introducton to Interval Computatons. Academc Press, ew York. [2] Goos, G.; Hartmans, J. (eds.) (1985). Interval Mathematcs Sprnger Verlag, Berln. [21] eumaer, A. (199). Interval Methods for Systems of Equatons. Cambrdge Unversty Press, Cambrdge. [22] Kearfott, R. B.; Krenovch, V. (eds.) (1996). Applcatons of Interval Computatons. Kluwer Academc Publshers, Dordrecht. [23] Jauln, L.; Keffer, M.; Ddrt, O.; Walter, E. (21). Appled Interval Analyss. Sprnger, London. [24] Kulpa, Z. (1997). Dagrammatc representaton of nterval space n provng theorems about nterval relatons. Relable Computng, Vol. 3, o. 3, [25] Ratschek, H.; Rokne, J. (1988). ew Computer Methods for Global Optmzaton. Ells Horwood Lmted, Chchester. [26] uy, H. (1998). Convex Analyss and Global Optmzaton. Kluwer Academc Publshers, Dordrecht. [27] Dong, W.; Shah, H. C. (1987). Vertex method for computng functons of fuzzy varables. Fuzzy Sets & Systems, Vol. 24, PLAXIS Verson 8

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

38 SCIEIFIC MAUAL A-2

39 APPEDIX B - SYMBOLS APPEDIX 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 : Permeablty K r : Permeablty reducton functon K : Stffness matrx K : Flow matrx L : Dfferental operator L : Couplng matrx M : Materal stffness matrx : Matrx wth shape functons p : Pore pressure (negatve for pressure) p : Body forces vector q : Specfc dscharge Q : Vector wth nodal dscharges r : Unbalance vector R : Permeablty matrx 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 φ : Groundwater head ω : Integraton constant (explct: ω=; mplct: ω=1) B-1

40 SCIEIFIC MAUAL B-2