Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling ﬁnite element analysis of plates S. G. P. Castro,E. L. Jansen 𝑎 𝑥𝑦 𝑏 𝐴1 𝐴2 𝑥𝑦 𝐴3 𝑥𝑦 𝐴4 𝑥𝑦 C1 𝑥𝑦 C2 𝑦 𝑥B2 𝑥𝑦 B1 𝑥𝑦 Anisotropy effect on post-buckling Highlights Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling ﬁnite element analysis of plates S. G. P. Castro,E. L. Jansen •Derivation of a complete formulation of Koiter’s method for multi-mode analysis •Achievement of a transparent correspondence between formulation and numerical implementation of Koiter’s method •Application to single-mode and multi-mode initial post-buckling analysis of isotropic and composite plates •Development of an enhanced Bogner-Fox-Schmit (BFS) ﬁnite element with enriched in-plane displacements to verify the formulation •Veriﬁcation by comparison with literature •Calculation and report of reference values for multi-modal expansion coeﬃcients and second-order ﬁelds Displacement-based formulation of Koiter’s method: application to multi-modal post-buckling ﬁnite element analysis of plates S. G. P. Castroa, E. L. Jansenb aDelft University of Technology, Group of Aerospace Structures and Computational Mechanics, Kluyverweg 1 2629 HS Delft, Netherlands bRotterdam University of Applied Sciences, School of Engineering and Applied Science, G.J. de Jonghweg 4-6, 3015 GG Rotterdam ARTICLE INFO Keywords : Koiter method Displacement-based Buckling Post-buckling Finite elements Classical formulation PlateABSTRACT Koiter’sasymptoticmethodenablesthecalculationanddeepunderstandingoftheinitialpost-buckling behaviourofthin-walledstructures. Forthesingle-modeasymptoticanalysis,Budiansky(1974)pre- sentedaclearandgeneralformulationforKoiter’smethod,basedontheexpansionofthetotalpoten- tialenergyfunction. TheformulationfromBudianskyishereinrevisitedandexpandedforthemulti- modal asymptotic analysis, of primordial importance in structures with clustered bifurcation modes. GiventheadmittedlydiﬃcultimplementationofKoiter’smethod,especiallyformulti-modalanalysis andduringtheevaluationofthethird–andfourth–ordertensorsinvolvedinKoiter’sanalysis;thepre- sented study proposes a formulation and notation with close correspondence with the implemented algorithms. Theimplementationisbasedonstate-of-the-artcollaborativetools: Python,NumPyand Cython. ThekinematicrelationsarespecializedusingvonKármánshellkinematics,andthedisplace- ment ﬁeld variables are approximated using an enhanced Bogner-Fox-Schmit (BFS) ﬁnite element, modiﬁedtoreachthird-orderinterpolationalsoforthein-planedisplacements,usingonly4nodesper element and 10 degrees-of-freedom per node, aiming an accurate representation of the second-order ﬁelds. The formulation and implementation are veriﬁed by comparing results for isotropic and com- posite plates against established literature. Finally, results for multi-modal displacement ﬁelds with up to 5 modes and corresponding post-buckling factors are reported for future reference. 1. Introduction The asymptotic theory originally proposed by Koiter [1] allows a rapid evaluation of the initial post-buckling behav- ior of structures. The method has been used within a semi- analytical context [2, 3] and has also been applied within a ﬁnite element framework [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. In recent years, the method has been applied in the context of variable stiﬀness of panel-type structures [17, 18, 14, 16, 19, 20] and in the analysis of imperfection sensitive shells [21, 22, 23]. In particular, the multi-modal formulation of Koiter’s approach provides a systematic and eﬃcient procedure to assess the nonlinear behavior of the structure in cases whereseveralbucklingmodesinteract,suchasinstructures highlyoptimizedforbuckling[24,25,26]andimperfection- sensitive shell structures [27, 28, 29, 30, 31, 32]. In such designs, small imperfections due to variations in manu- facturing parameters can induce diﬀerent bifurcation paths [33,34],whichcanbestudiedbyKoiter’sperturbationanal- ysis. Plates may also show clustered buckling modes, usually whenhighaspectratiosareinvolved[16]. Madeoetal. [16] evaluated variable stiﬀness plates where 4 modes were re- quiredinthemulti-modalexpansiontoobtainasatisfactory approximation of the post-buckling behavior. In these stud- ies, the authors used a bi-linear mixed 4-node plate element (MISS-4[35,36,37])withﬁrst-ordersheardeformationthe- ory. The aim of the present work is to derive a complete ORCID(s): 0000-0001-9711-0991 (S.G.P. Castro); 0000-0001-7511-2910 (E.L. Jansen)displacement-based formulation for the single- and multi- modal Koiter’s asymptotic analysis. The formulation and notationhereinproposedattemptstokeepaclosecorrespon- dence between the theory and the implemented algorithms, and thereby helpful in addressing the issues experienced in the past with the ﬁnite element implementation of Koiter’s asymptotic method, as highlighted by Casciaro [9]. The following discussion starts revisiting the double ex- pansion of the total potential functional using Koiter’s the- ory, as proposed by Budiansky [38] for a single-mode asymptotic analysis. The formulation is then expanded for the more general case of a multi-modal asymptotic analy- sis. General-purpose functional derivatives are presented andexpressionsforthesefunctionalderivativesarelaterob- tainedusingvonKármánnon-linearplatekinematics. These expressions can be used with any displacement-based an- alytical, semi-analytical or numerical approach; and in the present study it is proposed to approximate the displace- ment ﬁelds using the Bogner-Fox-Schmit (BFS) element [39], one of the most accurate rectangular ﬁnite elements for predicting the deﬂection of thin-walled shells, as stated by Zienkiewicz & Taylor [40, p. 153]. The conventional BFSachievesthird-orderinterpolationforthedeﬂectionus- ing only 4 nodes per element and 6 degrees-of-freedom per node. An enrichment is proposed to the conventional BFS element to achieve third-order interpolation also for the in- plane displacements, keeping only 4 nodes per element and increasing to 10 degrees-of-freedom per node. This enrich- ment of the in-plane displacements enable high accuracy while approximating the second-order displacement ﬁelds, required in Koiter’s method. The developed formulation andimplementationisveriﬁedagainstexistingliteraturefor SGP Castro, EL Jansen: Preprint submitted to Elsevier Page 1 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates single- and multi-modal asymptotic expansion of metallic and composite plates, under various loading and boundary conditions. Reference results for multi-modal analysis that arenotreadilyavailableintheliteraturearereported,includ- ing the second-order displacement ﬁelds. 2. Expansion of the Total Potential Energy Functional Koiter’s asymptotic theory was ﬁrst presented in 1945 [1]. Budiansky [38] presents a complete review of Koiter’s asymptotic method, proposing a general-purpose formula- tionbasedonthetotalpotentialenergyfunctionalofthesys- tem. The present work elaborates on this formulation and expands it to the complete multi-modal Koiter’s analysis. Given a total potential energy functional that has the dis- placementsuuuandtheascalarloadmultiplier asunknowns [uuu;]. At a pre-buckling static equilibrium:  [u0u0u0;0]= ¨[u0u0u0;0]uuu=0 (1) whereu0u0u0is a known solution corresponding to the pre- buckling load 0. The formulation herein presented is com- patiblewitheitheralinearoranonlinearpre-bucklingstate. In Eq. 1, the notation ¨uuuis used instead of  to conve- niently express the functional variation as a tensor product between the Fréchet derivative ¨and the variation of the vector containing all degrees-of-freedom uuu[38]. The ﬁrst functionalvariationbecomesaﬁrst-ordertensormultiplying a vector; the second variation becomes a second-order ten- sormultiplyingtwovectors,andsoforth,asdemonstratedin Eq. 2.  = ¨uuu 2 = ¨¨uuuuuu 3 = ¨¨¨uuuuuuuuu 4 = ivuuuuuuuuuuuu(2) Assume that there exists at least one point of equilibrium that intersects [uuu./;]at a bifurcation point [ucucuc;c], such thatucucuc=cu0u0u0. Thisbifurcationpointisevaluatedaccording toAppendixA.Hence,atthebifurcationpoint,thefollowing static equilibrium ought to exist:  [ucucuc;c]= ¨[ucucuc;c]uuu=0 (3) Assumingthat iscontinuousatthevicinityof ucucuc,aTay- lor expansion around the equilibrium point [ucucuc;c]can be applied on Eq. 3 to ﬁnd approximations for new state of equilibrium[uuu./;],Å > c. The error of this approx- imation becomes zero when limc. For this ﬁrst Taylor expansion, a displacement perturbation vvvis assumed such that: vvv=uuu./*ucucuc (4)withlimcvvv= 000becauseuuucrosses the bifurcation point. The notation .n/ c= .n/[ucucuc;c], is adopted to obtain the result of the ﬁrst Taylor expansion shown in Eq. 5, that approximatesthetotalpotentialenergyfunctionalat [uuu;c], with the new displacement state being uuu=ucucuc+vvv. ¨[uuu;c]uuu= ¨ cuuu+ ¨¨ cuuuvvv+1 2 ¨¨¨ cuuuvvv2 +1 6 iv cuuuvvv3+O.4/=0(5) ItisworthemphasizingthateachFrechétderivativesfrom the ﬁrst expansion of Eq. 5 is calculated at [ucucuc;c], and not [u0u0u0;c]; whereucucuc=cu0u0u0andu0u0u0is a linear or nonlinear pre-buckling state. Terms .n/ crepresentnth–order Frechét derivativesofthetotalpotentialenergyfunctionalaboutthe displacement degrees of freedom uuu. Equation 2 demon- strateshowtheseFrechétderivativesofthetotalpotentialen- ergy functional ultimately generate nth–order tensors. Vec- torsuuuandvvvare1st–order tensors, such that .n/ cvvv.n*1/uuu should be read as nth–order tensor products. Deﬁning the followingnotationforthederivativesof intermsof[38]: .n/ c= .n/[ucucuc;c] Ö .n/ c=d d .n/ c ÖÖ .n/ c=d2 d2 .n/ c(6) To approximate the total potential energy functional at [uuu;]each Frechét derivate of Eq. 5 undergoes a second Taylor expansion that assumes uuu./continuous for any load increment*c, resulting in: .n/[uuuc;]= .n/ c+Ö .n/ c.*c/+1 2ÖÖ .n/ c.*c/2+O.3/(7) TheexpressionsinEq. 7arereadilyappliedintoEq. 5to obtaintheapproximationforthetotalpotentialenergyfunc- tional at the new equilibrium, i.e. ¨[uuu;], which becomes: ¨[uuu;]uuu= ¨¨ c+Ö ¨¨ c.*c/+1 2ÖÖ ¨¨ c.*c/2+5 vvvuuu +1 2 ¨¨¨ c+Ö ¨¨¨ c.*c/+1 2ÖÖ ¨¨¨ c.*c/2+5 vvv2uuu +1 6 iv c+Ö iv c.*c/+1 2ÖÖ iv c.*c/2+5 vvv3uuu +5 (8) 3. Single-Mode Asymptotic Analysis Koiter[1]proposedtoexpress uuu*ucucucand*cusingthe following asymptotic expansion: uuu*ucucuc=vvv=uIuIuI+2uIIuIIuII+3uIIIuIIIuIII+5 *c=aIc+bIc2+5(9) SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 2 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates where:(1)is a scalar parameter. (2)uIuIuIis a ﬁrst-order ﬁeld,takendirectlyfromoneoralinearcombinationofmul- tiple linear buckling modes. Vector uIuIuIis customarily re- scaled dividing by the maximum normal displacement am- plitude and multiplying by the plate or shell thickness. (3) uIIuIIuIIis a second-order ﬁeld that provides a correction to the ﬁrst-order ﬁeld. (4)The third-order ﬁeld uIIIuIIIuIII, and higher, areassumedtohaveanegligiblecontribution. (5)aIandbI arerespectivelyﬁrst-andsecond-orderloadparameterstobe determined. Equation9isareduced-ordermodel(ROM)re- latingtheload anddisplacement uuuaroundtheequilibrium point[ucucuc;c]. Note that this ROM could have been built around any equilibrium point, a property that is explored in the Koiter-Newton approach [41, 42, 43, 44]. Koiter’s ex- pansiongivenbyEq. 9isdirectlyusedintoEq. 8torender: 21 2uIuIuI2 ¨¨¨ c+uIIuIIuII ¨¨ c+aIcuIuIuIÖ ¨¨ c uuu +31 6uIuIuI3 iv c+aIcuIIuIIuIIÖ ¨¨ c+1 2aIcuIuIuI2Ö ¨¨¨ c +1 2a2 I2 cuIuIuIÖÖ ¨¨ c+ :0uIIIuIIIuIII ¨¨+uIuIuIuIIuIIuII ¨¨¨ c+bIcuIuIuIÖ ¨¨ c1 uuu +5=0 (10) Note in Eq. 10 that only the terms up to 3are shown. Equation 10 must be satisﬁed regardless the values of and uuu,suchthateachtermmustvanishseparately. Thearbitrary valueuuu=uIuIuIcanbeused[38],with uIuIuIbeingtheﬁrsteigen- vector when I= 1, or a composition of eigenvectors ob- tainedatthebifurcationpoint [ucucuc;c]. Withuuu=uIuIuItheor- thogonality of the second-order ﬁeld leads toÖ ¨¨ cuIuIuIuIIuIIuII=0. Hence,Eq. 10canbeusedtoobtaintheequationsfor aIand bIin the single-mode expansion: aI=*1 2cuIuIuI3 ¨¨¨ c uIuIuI2Ö ¨¨ c bI=*1 6uIuIuI4 iv c+1 2aIcuIuIuI3Ö ¨¨¨ c +1 2a2 I2 cuIuIuI2ÖÖ ¨¨ c+uIuIuI2uIIuIIuII ¨¨¨ cF cuIuIuI2Ö ¨¨ c(11) Equation11isinagreementwithCasciaro[9]. NoteinEq. 11 thataIcan be calculated using only uIuIuI, whereasbIad- ditionally needs the second-order ﬁeld uIIuIIuII. Budiansky [38] suggests to compute uIIuIIuIIusing the terms for 2in Eq. 10: ¨¨ c uII uII uIIuuu+1 2uIuIuI2 ¨¨¨ cuuu+aIcuIuIuIÖ ¨¨ cuuu=0 which must hold for all arbitrary variations uuu, such that: ¨¨ c uII uII uII+1 2uIuIuI2 ¨¨¨ c+aIcuIuIuIÖ ¨¨ c=0 (12)For problems in solid mechanics ¨¨ cwill generally be an invertible positive-deﬁnite square matrix [38], such that the solution of Eq. 12 can be:  uII uII uII= ¨¨ c*1 *1 2 ¨¨¨ cuIuIuI2*aIcuIuIuIÖ ¨¨ c (13) Note in Eqs. 12 and 13 that vector  uII uII uIIis appropriately usedinsteadof uIIuIIuIIbecauseEq. 13allowsmultiplesolutions of uII uII uIIfor diﬀerent multipliers applied to uIuIuI. However, one must guarantee that any calculated  uII uII uIIis orthogonal to uIuIuI suchthatthesecond-orderdisplacementvector uIIuIIuIIbecomes a valid vector basis for the reduced-order model of Eq. 9. Gram-Schmidtorthogonalization[45]canbeusedtoobtain the orthogonal component of  uII uII uII, where ﬁrst the projection of uII uII uIIontouIuIuIis obtained as: projuIuIuI uII uII uII=uIuIuIê uII uII uII;uIuIuIë êuIuIuI;uIuIuIë Next,theorthogonal uIIuIIuIIcanbecalculatedsubtractingthe projection from  uII uII uII, using: uIIuIIuII= uII uII uII*uIuIuIê uII uII uII;uIuIuIë êuIuIuI;uIuIuIë(14) 4. Multi-Modal Asymptotic Analysis Thepossibilitytoperformpost-bucklinganalysisinstruc- tures with clustered buckling modes is one of the apparent advantages of Koiter’s method compared to other methods [9, 16]. The necessity of considering multiple modes in the asymptotic expansion has been demonstrated by many authors, e.g. when the vibration frequencies are closely spaced [46]; variable-angle tow plates with clustered buck- ling modes [16]; and imperfection sensitive structures [47, 38, 48, 28, 49, 29, 50, 22]. Thesingle-modeasymptoticexpansionofEq. 9canbegen- eralizedtoamulti-modalasymptoticexpansion,asshownin Eqs. 15 and 16 [51, 12]: uuu*ucucuc=vvv=iuiuiui+ijuijuijuij+5 (15) I.*I/=IaIjkjk+IbIjkljkl+5(16) where summation convention is applied for repeated in- dicesj;k;l= 1;2;5mis applied. Equation 16 is a reduced-order model consisting of a system of mequations obtained for I= 1;2;5;m, which has 1;2;5;mun- knowns. The value Icorrespond to the Ithlinear buck- ling eigenvalue uIuIuI, always re-scaled by dividing with the SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 3 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates maximumout-of-planedisplacementandmultiplyingbythe platethickness. Findingtherightnumberoflinearbuckling modesmin the multi-modal analysis is a common question [9],andanacceptedcriterionistoselectanumberofmodes that lies within 10%-20% departing from the ﬁrst critical load [9]. NotethatEq. 15consistsonareduced-ordermodeltocal- culate displacements uuubased on a pre-buckled state ucucucwith known linear buckling modes uiuiuiand known second-order displacement ﬁelds uijuijuij. As in the case of the single-mode expansion, for plates and shells it is customary to re-scale uijuijuijdividing by the maximum normal displacement ampli- tude ofuijuijuijand multiplying by the plate or shell thickness. The coeﬃcients ifori= 1;§;mare found for each load after solving the system of mequations given by Eq. 16. Solving Eq. 16 requires the calculation of all coeﬃcients aijkandbijkl. The expressions given by Eqs. 15 and 16 are applied to the expanded total potential energy functional of Eq. 8. Thetermsmultiplying jkandjklarecollected, analo- gouslytothetermsmultiplying 2and3forthesingle-mode expansion in Eq. 10. The collected terms for the multi- modal expansion are shown in Eq. 17, where the follow- ing orthogonality property of the linear buckling modes is used:êuiuiui;ujujujë= 0;Åij; leading to ¨¨ cuiuiuiujujuj= 0;Åij; Ö ¨¨ cuiuiuiujujuj=0;Åij; andÖÖ ¨¨ cuiuiuiujujuj=0;Åij. Moreover, col- lected terms in brackets that are multiplying any of the per- turbation parameters j;k;lultimately vanish, knowing that j;k;l0. jk aijk+aikjiuiuiuiÖ ¨¨ c+ ¨¨¨ cujujujukukuk+ ¨¨ cujkujkujk+ ¨¨ cukjukjukj uuu +jkl4 iÖ ¨¨ cuiuiuibijkl+biklj+bilkj +bijlk+bikjl+biljk+ iv cujujujukukukululul + ¨¨¨ cujujujukluklukl+ujujujulkulkulk+ukukukujlujlujl +ukukukuljuljulj+ulululujkujkujk+ulululukjukjukj +ÖÖ ¨¨ c2 iuiuiuiaiijaikl+aijiaikl+aiijailk+aijiailk +aiikaijl+aikiaijl+aiikailj+aikiailj +aiilaijk+ailiaijk+aiilaikj+ailiaikj +Ö ¨¨¨ ciaiijukukukululul+aijiukukukululul+aiikujujujululul+aikiujujujululul +aiilujujujukukuk+ailiujujujukukuk+aijkuiuiuiululul+aikjuiuiuiululul +aijluiuiuiukukuk+ailjuiuiuiukukuk+aikluiuiuiujujuj+ailkuiuiuiujujuj5 uuu +5=0 (17) For the expanded equilibrium to be stationary, each term in Eq. 17 must vanish separately, similarly to the single- mode expansion. Assuming uuu=uiuiuiin Eq. 17, the expres- sionsforaijkandbijklcanbeobtained,asrespectivelygivenin Eqs. 18 and 19. aijk=*1 2i ¨¨¨ cuiuiuiujujujukukuk Ö ¨¨ cuiuiuiuiuiui(18) bijkl=*1 6iÖ ¨¨ cuiuiuiuiuiui iv cuiuiuiujujujukukukululul +3 ¨¨¨ cuiuiuiujujujukluklukl+ulululujkujkujk +iÖ ¨¨¨ cuiuiuiaiijukukukululul+aijkulululuiuiui+aikluiuiuiujujuj +2 iÖÖ ¨¨ cuiuiuiuiuiuiaiijaikl+aijkaili++aiklaiij (19) Noteinthecalculationof bijklthatthesecond-orderﬁelds uijuijuijareneeded. Asinthesingle-modeexpansioncase,anon- orthogonal second-order ﬁeld  uij uij uijis calculated ﬁrst. Note that the terms in brackets multiplying jkin Eq. 17 are obtained for any ithmode. Therefore, the contribution for alli=1;§;mmodesareaddedandthefollowingequation for uij uij uijis obtained:  ujk ujk ujk= ¨¨ c*1H *1 2 ¨¨¨ cujujujukukuk*1 mi=mÉ i=1aijkiuiuiuiÖ ¨¨ cI (20) The orthogonal second-order ﬁeld vectors in the multi- modal asymptotic expansion can be obtained after succes- sive Gram-Schmidt orthogonalization [45] operations, used toremovethecomponentsof  ujk ujk ujkthatareparalleltoalllinear buckling modes used in the multi-modal expansion uiuiui, with i=1;2;§;m. This orthogonalization procedure is formu- lated in Eq. 21. ujkujkujk= ujk ujk ujk*i=mÉ i=1uiuiuiê ujk ujk ujk;uiuiuië êuiuiui;uiuiuië(21) 5. Functional Derivatives using Von Kármán Kinematics This section demonstrates how to calculate the func- tional derivatives leading to the nth–order tensors .n/ c,Ö .n/ c andÖÖ .n/ c, previously introduced during the single-mode and multi-modal asymptotic expansions. The discussion focus onvonKármánnon-linearkinematics,andcanbeeasilyex- tendedtoothernon-linearkinematicequations,suchasthose proposed by Sanders [49, 52] and Timoshenko & Gere [52, section 2.2.4]. 5.1. Strains VonKármánproposedakinematicrelationforplatesthat neglectvariousnon-lineartermsthatcomefromthefullnon- linear Green-Lagrange strain-displacement relations [52, section 2.2.2]. Von Kármán kinematics are also referred to in the literature as Donnell-type [53, 54] or Kirchhoﬀ- Lovenon-linearequations. Usingclassicalequivalentsingle SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 4 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates layer theory, the three-dimensional strains are expressed as .x;y;z/=""".x;y/+z.x;y/,suchthattheextensional """and rotationalstrains can be deﬁned in terms of the in-plane displacement ﬁeld variables u.x;y/;v.x;y/;w.x;y/as: """=h n l nj"xx "yy xyi n m nk=h n l nju;x+1 2w;2 x v;y+1 2w;2 y u;y+v;x+w;xw;yi n m nk =h n l njxx yy xyi n m nk=h n l nj*w;xx *w;yy *2w;xyi n m nk(22) Assuming the following approximation for the displace- ment ﬁeld: u;v;w=SSSuuuuuu SSSuuu=b f fdSSSu SSSv SSSwc g ge(23) whereSSSu;v;wareknownshapefunctions;eachcomponent u;v;wcan be written adopting summation convention for repeated indices as: u=Su aua v=Sv aua w=Sw aua(24) witha= 1;2;5;n. The strain variations can be repre- sented as: """="""¨uuu =¨uuu(25) where the¨(prime) symbol is used to denote a Frechét’s diﬀerentiation. Adopting index notation to represent the strains, such that "1="xx;"2="yy;"3= xy;1= xx;2=yy;3=xy; the ﬁrst and second variations of the extensional and rotational strains become: "i="¨ iaua i=¨ iaua ."i/="¨¨ iabuaub .i/=¨¨ iabuaub(26) Withthesedeﬁnitions,theﬁrstFrechét’sdiﬀerentiationof the strains can be represented as: """¨ a=h n l njSu a;x+w;xSw a;x Sv a;y+w;ySw a;y Su a;y+Sv a;x+w;xSw a;y+w;ySw a;xi n m nk ¨ a=h n l nj*Sw a;xx *Sw a;yy *2Sw a;xyi n m nk(27)The second diﬀerentiation: """¨¨ ab=h n l njSw a;xSw b;x Sw a;ySw b;y Sw a;ySw b;x+Sw a;xSw b;yi n m nk ¨¨ ab=000(28) Note in Eq. 28 that "¨¨ iabrepresents a symmetric second- ordertensor,whichisanimportantpropertytobeconsidered while implementing the method. For the diﬀerentiations with respect to , we must recall that all functional expansions were calculated about the bi- furcationpoint [ucucuc;c],suchthatthestrainsandstressesare thosecorrespondingtothedisplacement ucucuc=u0u0u0,with= c. Starting with """and, using the notation ). /_)=Ö . / and)2. /_)2=ÖÖ . /: Ö"""=h n l nju0;x+w0;2 x v0;y+w0;2 y u0;y+v0;x+2w0;xw0;yi n m nk Ö=h n l nj*w0;xx *w0;yy *2w0;xyi n m nk(29) Ifthepre-bucklingstate u0u0u0isevaluatedlinearlyforaplate with no bending-extension coupling and only in-plane pre- buckling loads, w0=0and the initial post-buckling analy- sis is greatly simpliﬁed. Nevertheless, the formulation pre- sented next is valid for the more general case of w00. The second diﬀerentiation with respect to gives: ÖÖ"""=h n l njw0;2 x w0;2 y 2w0;xw0;yi n m nk ÖÖ=000(30) For"""¨ aand¨ a, the ﬁrst diﬀerentiation with respect to  gives: Ö"""¨ a=h n l njw0;xSw a;x w0;ySw a;y w0;xSw a;y+w0;ySw a;xi n m nk Ö¨ a=000(31) For the second diﬀerentiations with respect to ,ÖÖ"""¨ a=000 andÖÖ¨ a=000. For"""¨¨ aband¨¨ aball derivatives with respect to are zero. SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 5 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates 5.2. Stresses Based on Eqs. 22 - 31 it is straightforward to compute the corresponding stresses. Using classical constitutive re- lations for laminated composites [55] and adopting the in- dex notation: N1=Nxx,N2=NyyandN3=Nxy; M1=Mxx,M2=MyyandM3=Mxy; the stress-strain relations can be written as: Ni=Aij"j+Bijj Mi=Bij"j+Dijj(32) whereAijrepresentstheplatemembranestiﬀness; Bijthe membrane-bending coupling stiﬀness; and Dijthe bending stiﬀness; all for i;j=1;2;3. The ﬁrst Frechét derivative of the stress terms are: N¨ ia=Aij"¨ ja+Bij¨ ja M¨ ia=Bij"¨ ja+Dij¨ ja(33) Recalling from Eq. 28 that ¨¨ jab= 0, the second Frechét derivatives are: N¨¨ iab=Aij"¨¨ jab M¨¨ iab=Bij"¨¨ jab(34) NotethatN¨¨ iab;M¨¨ iabaresymmetricsecond-ordertensors. The ﬁrst derivatives with respect to can be readily com- puted as: ÖNi=AijÖ"j+BijÖj ÖMi=BijÖ"j+DijÖj(35) ÖN¨ ia=AijÖ"¨ ja ÖM¨ ia=BijÖ"¨ ja(36) ÖN¨¨ iab=0 ÖM¨¨ iab=0(37) Finally, the second derivatives about are: ÖÖNi=AijÖÖ"j ÖÖMi=BijÖÖ"j ÖÖN¨ ia=0 ÖÖM¨ ia=0 ÖÖN¨¨ iab=0 ÖÖM¨¨ iab=0(38)5.3. Functional Derivatives Assumingageneralloadingvectorwithdistributedforces NNNattheplateboundaries  ,thetotal potentialenergycan be written as: =1 2Ê Ni"i+Miid *Ê NNNuuud. /(39) whered =dxdyandsummationconventionisadopted for terms with repeated index iwithi= 1;2;3. The stationary total potential energy at [ucucuc;c]is deﬁned as ¨ c= ¨[ucucuc;c], calculated using the ﬁrst variation of c= [ucucuc;c]: ¨ cuuu=4 1 2Ê 0 Ni"i+Ni"i+Mii+Mii1 d *Ê NNNuuud. /5 (40) The variation uuuis deﬁned as uuu=uauaua= ^5;ua;5`, suchthattheﬁrstFrechétderivativeofthetotalpotentialen- ergy becomes: ¨ cuauaua=4 1 2Ê 0 N¨ ia"i+Ni"¨ ia+M¨ iai+Mi¨ ia1 d *Ê NNNSuuu ax=lxd. /5 ua (41) ResumingwiththesecondFrechétderivative,nowreplac- inguuuwithuuu=ububub=^5;ub;5`: ¨¨ cuauauaububub=4 1 2Ê 0 N¨¨ iab"i+N¨ ia"¨ ib+N¨ ib"¨ ia+Ni"¨¨ iab +M¨¨ iabi+M¨ ia¨ ib+M¨ ib¨ ia+Mi >0 ¨¨ iab1 d 5 uaub =4 1 2Ê 0 N¨¨ iab"i+N¨ ia"¨ ib+N¨ ib"¨ ia+Ni"¨¨ iab +M¨¨ iabi+M¨ ia¨ ib+M¨ ib¨ ia1 d 5 uaub (42) Note that ¨¨ cin Eq. 42 represents a second-order tensor with all geometrically non-linear terms present. If a linear pre-bucklingstateisassumed, ¨¨ cbecomesthelinearconsti- tutivestiﬀnessmatrixofthesystem[9]. Continuingwiththe thirdFrechétderivative,nowusing uuu=ucucuc=^5;uc;5`: ¨¨¨ cuauauaubububucucuc=4 1 2Ê 0  *0 N¨¨¨ iabc"i+N¨¨ iab"¨ ic+N¨¨ iac"¨ ib SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 6 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates +N¨ ia"¨¨ ibc+N¨¨ ibc"¨ ia+N¨ ib"¨¨ iac+N¨ ic"¨¨ iab +Ni >0 "¨¨¨ iabc+ *0 M¨¨¨ iabci+M¨¨ iab¨ ic+M¨¨ iac¨ ib +M¨ ia >0 ¨¨ ibc+M¨¨ ibc¨ ia+M¨ ib >0 ¨¨ iac1 d 5 uaubuc =4 1 2Ê 0 N¨¨ iab"¨ ic+N¨¨ iac"¨ ib+N¨ ia"¨¨ ibc +N¨¨ ibc"¨ ia+N¨ ib"¨¨ iac+N¨ ic"¨¨ iab +M¨¨ iab¨ ic+M¨¨ iac¨ ib+M¨¨ ibc¨ ia1 d 5 uaubuc(43) Lastly, using uuu=ududud=^5;ud;5`, the fourth Frechét derivative gives: iv cuauauaubububucucucududud=4 1 2Ê 0  *0 N¨¨¨ iabd"¨ ic+N¨¨ iab"¨¨ icd+ *0 N¨¨¨ iacd"¨ ib +N¨¨ iac"¨¨ ibd+N¨¨ iad"¨¨ ibc+N¨ ia >0 "¨¨¨ ibcd+ *0 N¨¨¨ ibcd"¨ ia +N¨¨ ibc"¨¨ iad+N¨¨ ibd"¨¨ iac+N¨ ib >0 "¨¨¨ iacd+N¨¨ icd"¨¨ iab +N¨ ic >0 "¨¨¨ iabd+ *0 M¨¨¨ iabd¨ ic+M¨¨ iab >0 ¨¨ icd+ *0 M¨¨¨ iacd¨ ib +M¨¨ iac >0 ¨¨ ibd+ *0 M¨¨¨ ibcd¨ ia+M¨¨ ibc >0 ¨¨ iad1 d 5 uaubucud =4 1 2Ê 0 N¨¨ iab"¨¨ icd+N¨¨ iac"¨¨ ibd+N¨¨ iad"¨¨ ibc +N¨¨ ibc"¨¨ iad+N¨¨ ibd"¨¨ iac+N¨¨ icd"¨¨ iab1 d 5 uaubucud (44) It is now possible to compute the functional diﬀerentia- tionscorrespondingtothesecondexpansionofthetotalpo- tential energy functional, about . From Eq. 42, the ﬁrst diﬀerentiation about becomes: Ö ¨¨ cuauauaububub=4 1 2Ê 0  >0ÖN¨¨ iab"i+N¨¨ iabÖ"i+Ö N¨ ia"¨ ib +N¨ iaÖ"¨ ib+ÖN¨ ib"¨ ia+N¨ ibÖ"¨ ia+ÖNi"¨¨ iab +Ni >0Ö"¨¨ iab+ >0 ÖM¨¨ iabi+M¨¨ iabÖi+ÖM¨ ia¨ ib +M¨ ia 70 Ö¨ ib+ÖM¨ ib¨ ia+M¨ ib 0 Ö¨ ia1 d 5 uaub =4 1 2Ê 0 N¨¨ iabÖ"i+Ö N¨ ia"¨ ib+N¨ iaÖ"¨ ib +ÖN¨ ib"¨ ia+N¨ ibÖ"¨ ia+ÖNi"¨¨ iab+M¨¨ iabÖi +ÖM¨ ia¨ ib+ÖM¨ ib¨ ia1 d 5 uaub(45) The second diﬀerentiation of ¨¨ caboutbecomes:ÖÖ ¨¨ cuauauaububub=4 1 2Ê 0  >0ÖN¨¨ iabÖ"i+N¨¨ iabÖÖ"i+ 0 ÖÖN¨ ia"¨ ib +2ÖN¨ iaÖ"¨ ib+N¨ ia 70 ÖÖ"¨ ib+ 0 ÖÖN¨ ib"¨ ia+2ÖN¨ ibÖ"¨ ia +N¨ ib 70 ÖÖ"¨ ia+ÖÖNi"¨¨ iab+ÖNi >0Ö"¨¨ iab+ >0 ÖM¨¨ iabÖi +M¨¨ iab 70 ÖÖi+ 0 ÖÖM¨ ia¨ ib+ÖM¨ ia 70 Ö¨ ib + 0 ÖÖM¨ ib¨ ia+ÖM¨ ib 0 Ö¨ ia1 d 5 uaub =4 1 2Ê 0 N¨¨ iabÖÖ"i+2ÖN¨ iaÖ"¨ ib +2ÖN¨ ibÖ"¨ ia+ÖÖNi"¨¨ iab1 d 5 uaub(46) Theﬁrstdiﬀerentiationof ¨¨¨ ccanbecalculatedbasedon Eq. 43 as: Ö ¨¨¨ cuauauaubububucucuc=4 1 2Ê 0  >0ÖN¨¨ iab"¨ ic+N¨¨ iabÖ"¨ ic+ >0ÖN¨¨ iac"¨ ib +N¨¨ iacÖ"¨ ib+ÖN¨ ia"¨¨ ibc+N¨ ia 0 Ö"¨¨ ibc+ >0ÖN¨¨ ibc"¨ ia +N¨¨ ibcÖ"¨ ia+ÖN¨ ib"¨¨ iac+N¨ ib >0Ö"¨¨ iac+ÖN¨ ic"¨¨ iab +N¨ ic >0Ö"¨¨ iab+ >0 ÖM¨¨ iab¨ ic+M¨¨ iab 70 Ö¨ ic+ >0 ÖM¨¨ iac¨ ib +M¨¨ iac 70 Ö¨ ib+ >0ÖM¨¨ ibc¨ ia+M¨¨ ibc 0 Ö¨ ia1 d 5 uaubuc =4 1 2Ê 0 N¨¨ iabÖ"¨ ic+N¨¨ iacÖ"¨ ib+ÖN¨ ia"¨¨ ibc +N¨¨ ibcÖ"¨ ia+ÖN¨ ib"¨¨ iac+ÖN¨ ic"¨¨ iab1 d 5 uaubuc(47) The second diﬀerentiation of ¨¨¨ caboutcan be calcu- lated from Eq. 47: ÖÖ ¨¨¨ cuauauaubububucucuc=4 1 2Ê 0  >0ÖN¨¨ iabÖ"¨ ic+N¨¨ iab 70 ÖÖ"¨ ic+ >0ÖN¨¨ iacÖ"¨ ib +N¨¨ iac 70 ÖÖ"¨ ib+ 0 ÖÖN¨ ia"¨¨ ibc+ÖN¨ ia 0 Ö"¨¨ ibc+ >0ÖN¨¨ ibcÖ"¨ ia +N¨¨ ibc 70 ÖÖ"¨ ia+ 0 ÖÖN¨ ib"¨¨ iac+ÖN¨ ib >0Ö"¨¨ iac + 0 ÖÖN¨ ic"¨¨ iab+ÖN¨ ic >0Ö"¨¨ iab1 d 5 uaubuc=0(48) The ﬁrst and second diﬀerentiation of iv cwith respect to can be calculated based on Eq. 44: SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 7 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates Ö iv cuauauaubububucucucududud=4 1 2Ê 0  >0ÖN¨¨ iab"¨¨ icd+N¨¨ iab >0Ö"¨¨ icd+ >0ÖN¨¨ iac"¨¨ ibd +N¨¨ iac >0Ö"¨¨ ibd+ >0ÖN¨¨ iad"¨¨ ibc+N¨¨ iad 0 Ö"¨¨ ibc+ >0ÖN¨¨ ibc"¨¨ iad +N¨¨ ibc >0Ö"¨¨ iad+ >0ÖN¨¨ ibd"¨¨ iac+N¨¨ ibd >0Ö"¨¨ iac + >0ÖN¨¨ icd"¨¨ iab+N¨¨ icd >0Ö"¨¨ iab1 d 5 uaubucud=0(49) ÖÖ iv cuauauaubububucucucududud=0 (50) 6. Modiﬁed Bogner-Fox-Schmit Finite Element The Bogner-Fox-Schmit (BFS) ﬁnite element [39] is a classical C1 contiguous conﬁrming plate element obtained bytakingtensorproductsofcubicHermitesplines. TheBFS is one of the most accurate rectangular ﬁnite elements for thin-walledshells,asstatedbyZienkiewicz&Taylor[40,p. 153], and therefore has been chosen to implement the for- mulation herein developed. With only 4 nodes per element, thestandardBFSelementapproximatestheout-of-planedis- placementsusing 3rd–orderpolynomials,whichisstillarea- sonablelow-orderinterpolationforplatesandverysimpleto implement [56], in contrast with triangular elements which use higher order polynomials [56], such as the Argyris ele- ment [57]. A drawback of the standard BFS element when used in the context of Koiter’s method is the linear interpolation of the in-place displacements [39], resulting in a poor conver- gence when trying to represent the second-order displace- mentﬁeldsuijuijuijandalldependentquantitiessuchasthe bijkl factors of Eq. 19. AmodiﬁedBFSelementisproposedtoenablethird-order interpolationalsoforthein-planedisplacements. Additional 4 degrees-of-freedom per node are included, being the ﬁrst derivatives of the in-plane displacements: u;x,u;y,v;xand v;y. Hence, no nodes are added and the resulting element hasstill4nodeswith10degrees-of-freedompernode. This enhancedBFSelementisreferredtointhefollowingdiscus- sionasBFSC:Bogner-Fox-Schmit-Castro. Thein-plane u;v and out-of-plane wdisplacements are approximated using: u;v;w=³4 i=1SSSu;v;w iueueuei(51) whereueueueicontains the 10 degrees-of-freedom of the ith node. The matrices containing the shape functions SSSu;v;w i are deﬁned as: SSSu i=HiHx iHy i0 0 0 0 0 0 0 SSSv i=0 0 0HiHx iHy i0 0 0 0 SSSw i=0 0 0 0 0 0 HiHx iHy iHxy i(52)withHi;Hx i;Hy i;Hxy icalculated using natural coordi- nates [58, 59, 60]: Hi=1 16.+i/2.i*2/.+i/2.i*2/ Hx i=*lx 32i.+i/2.i*1/.+i/2.i*2/ Hy i=*ly 32.+i/2.i*2/i.+i/2.i*1/ Hxy i=lxly 64i.+i/2.i*1/i.+i/2.i*1/(53) wherelx;lyare respectively the ﬁnite element dimen- sions alongx;y. The values of i;igiven in Eq. 54 were adopted for each of the four nodes. Node ii 1 *1 *1 2 +1 *1 3 +1 +1 4 *1 +1(54) Inthepresentstudy,onlyrectangularelementswereused, such that the natural coordinates can be deﬁned simply as: = 2x_lx*1; and= 2y_ly*1. All derivatives of SSSu;v;w irequiredinthestrainequationscanthenbecalculated intermsofthenaturalcoordinatesusingEq. 55. Allintegra- tions over the ﬁnite element domains are performed numer- ically using standard Guass-quadrature and a mesh of 44 integration points per element, unless otherwise speciﬁed. ) )x=lx 2) ) ) )y=ly 2) )(55) 7. Implementation ThemainchallengetoimplementKoiter’sasymptoticap- proach is the calculation of the third- and fourth-order ten- sors ¨¨¨ cand iv c. A naive implementation can easily blow the memory of any modern computer. For example, a mesh with only 20 nodes, or 200 degrees of freedom using the BFSC element, would generate a complete iv ctensor with 2004=1:6109elementstobestored,requiring11.92GB of memory if double precision is used. Moreover, calculat- ingall1:6109termsiscomputationallytooexpensive. For slightly larger problems found in any engineering applica- tionsuchnaiveapproachbecomesjustimpracticable. Many drawbacks experienced in the past with Koiter’s asymptotic method are due to mistakes in implementation more than to intrinsic defects of the asymptotic approach itself, as stated by Casciaro [9]. The implementation herein proposed care- fullytakesintoaccountallnon-zerotermsderivedinSection 5.3 while calculating the higher-order tensors. The strategy adopted consists on evaluating the products involving high- order tensors ¨¨¨ cand iv con-the-ﬂy, already during the nu- merical integration of each element. For instance, iv cde- ﬁned in Eq. 44 is never fully calculated. Instead, the tensor SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 8 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates product iv cuauauaubububucucucudududisevaluatedandstoredduringtheﬁnite elementassembly,usingthefollowingrearrangementofEq. 44: iv cuauauaubububucucucududud=1 2kÉ e=1Ê e0 uaubN¨¨ iab"¨¨ icducud +uaucN¨¨ iac"¨¨ ibdubud+uaudN¨¨ iad"¨¨ ibcubuc +ubucN¨¨ ibc"¨¨ iaduaud+ubudN¨¨ ibd"¨¨ iacuauc +ucudN¨¨ icd"¨¨ iabuaub1 d e(56) where erepresents the domain of one ﬁnite element. Notethatonlytensorsuptosecond-orderarefullycalculated and stored, e.g. Niab;"ibd. Moreover, for the implementa- tion one should note that Niab=Niac=Niad=Nibc= Nibd=Nicdand"iab="iac="iad="ibc="ibd="icd, suchthat"iabiscalculatedwithEq. 28and NiabwithEq. 34. Theimplementationcarriedoutforthepresentstudyisbased onPython[61],NumPy[62]andCython[63],whereallten- sor products are eﬃciently evaluated using NumPy [62], as illustrated in the following pseudo-code: from numpy import einsum ... phi4 = 0 for point inintegration_points : weight = compute_weight ( point ) Niab = compute_Niab ( point ) eicd = compute_eicd ( point ) phi4 += 1/2* weight * einsum ( /quotesingle.Variab ,icd ,a,b,c,d /quotesingle.Var, Niab ,eicd ,ua ,ub ,uc ,ud) The authors could verify the beneﬁt of the proposed formulation and notation while implementing the method, achievingaone-to-onecorrespondencebetweenthemethod and implemented algorithms. 8. Results for single-mode Koiter’s asymptotic expansion 8.1. Isotropic plates Lanzo et al. [64] performed initial post-buckling analy- sisinisotropicplateswithvariousaspectratios,loadingand boundaryconditions,asillustratedinFigure1. Table1com- pares the convergence of c_refwith ref=2 b2Eh3 121*2 Table 2 shows the convergence of bI, often referred to in the literature dealing single-mode asymptotic expansion as the "b*factor". The results from Lanzo et al. [64] are com- pared against the BFSC implementation with progressively reﬁned meshes. The mesh reﬁnement is controlled by the numberofelementsalong y,namedny,calculatingthenum- ber of elements along xusingnx=a_bny. The numeri- calintegrationwasdoneusing4x4integrationpoints,which 𝑎 𝑥𝑦 𝑏 𝐴1 𝐴2 𝑥𝑦 𝐴3 𝑥𝑦 𝐴4 𝑥𝑦 C1 𝑥𝑦 C2 𝑦 𝑥B2 𝑥𝑦 B1 𝑥𝑦Figure 1: Mo dels fo r isotropic plates, adapted from Lanzo et al. [64] providedaconvergedandstableintegrationscheme,i.e. we veriﬁed that adding more integration points did not change theresultsandusinglessintegrationpoints(3x3)resultedin unstable behaviour for models B1, B2, C1 and C2. The ﬁrst buckling mode u1u1u1and corresponding second- order ﬁeldu11u11u11for each all models with a_b= 3are given in Figures 3 - 4. Those results correspond to a mesh of .nx=60/.ny=20/BFSCelements. Theplotsarecreated using a visualization mesh of 900300 points along xand y, respectively; calculating the consistent displacement ac- cordingtoEq. 52ateachplottingpoint. Notethecomplexity ofthesecond-orderdisplacement,whicharein-planeforthis case. The fast convergence for bIshown in Table 2 is a di- rectconsequenceofthefastconvergenceofthesecond-order ﬁelduIIuIIuII. The enrichment on uandvenabled by the BFSC elementisthemainreasonforthisfast-convergingbehavior. 8.2. Composite plates A composite plate presented by Phan & Reddy [65] and Bilotta et al. [66] will be used as a reference for veriﬁca- tionofthesingle-modeexpansionforcompositeplates. The following material properties are used: E2=8GPa;G12 E2=0:6; 12=0:25 Using the load and boundary conditions of model A1of Fig. 1, the plate geometry is deﬁned based on a=b=1m, andthicknessof h=0:1m. Thelaminatestackingsequence is0_90_90_0,withtheprincipaldirectionbeingtheplate’s x*axis. Table3showstheconvergenceofthecriticallinear bucklingeigenvaluefortheBFSCelement. Theresultsfrom Phan&Reddy[65]areexactsolutionsobtainedwithclassi- cal plate theory, whereas the results from [66] are based on SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 9 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates T able 1 Convergence of BFSC element fo r rst buckling mo de, isotropic case Mo del CaseLanzo et al. [64] BFSC convergence fo r c_ref nxnyc_refny=4ny=6ny=8ny=10ny=12ny=14ny=16ny=18ny=20 A1a_b=12525 4.00263 3.99358 3.99921 3.99983 3.99995 3.99998 3.99999 4.00000 4.00000 4.00000 a_b=24921 4.00323 3.99419 3.99929 3.99985 3.99996 3.99998 3.99999 4.00000 4.00000 4.00000 a_b=34915 4.00674 3.99439 3.99932 3.99985 3.99996 3.99998 3.99999 4.00000 4.00000 4.00000 A2a_b=13333 7.71346 7.66910 7.69296 7.69263 7.69203 7.69171 7.69154 7.69145 7.69139 7.69136 a_b=24921 7.01142 6.95834 6.97190 6.97210 6.97191 6.97178 6.97171 6.97167 6.97165 6.97163 a_b=3 - - 7.03814 7.05577 7.05595 7.05565 7.05547 7.05537 7.05531 7.05528 7.05526 A3a_b=1 - - 1.69586 1.69768 1.69806 1.69817 1.69821 1.69823 1.69824 1.69825 1.69825 a_b=23317 1.38808 1.38371 1.38566 1.38602 1.38612 1.38616 1.38618 1.38619 1.38619 1.38619 a_b=3 - - 1.33821 1.33894 1.33908 1.33912 1.33914 1.33914 1.33915 1.33915 1.33915 A4a_b=1 - - 6.73286 6.74461 6.74418 6.74373 6.74350 6.74337 6.74331 6.74327 6.74324 a_b=24921 4.85495 4.83580 4.84603 4.84698 4.84713 4.84715 4.84716 4.84716 4.84716 4.84715 a_b=3 - - 4.39779 4.40549 4.40627 4.40642 4.40645 4.40646 4.40646 4.40646 4.40646 B1a_b=13333 25.57659 24.19007 25.40526 25.50502 25.52282 25.52721 25.52846 25.52882 25.52889 25.52887 a_b=24923 23.94518 23.30597 23.83356 23.87513 23.88171 23.88292 23.88303 23.88291 23.88275 23.88261 a_b=34915 24.2542 23.51109 24.07096 24.10957 24.11329 24.11364 24.11343 24.11316 24.11292 24.11273 B2a_b=13333 48.15739 50.37432 48.28268 47.93517 47.83308 47.79429 47.77688 47.76807 47.76321 47.76033 a_b=24923 42.21176 42.88262 41.91507 41.73877 41.68699 41.66723 41.65827 41.65367 41.65109 41.64952 a_b=34915 41.72252 41.53774 40.70907 40.55857 40.51447 40.49763 40.48996 40.48601 40.48377 40.48241 C1a_b=12525 9.35185 8.71909 9.24793 9.31349 9.32538 9.32769 9.32784 9.32749 9.32708 9.32670 a_b=24523 6.56822 6.54374 6.55314 6.55058 6.54903 6.54813 6.54757 6.54720 6.54695 6.54678 a_b=34515 5.88460 5.95400 5.89677 5.87237 5.86074 5.85441 5.85060 5.84813 5.84645 5.84525 C2a_b=12525 14.78220 14.92525 14.74298 14.68146 14.66129 14.65341 14.64965 14.64755 14.64623 14.64535 a_b=24523 10.34334 10.74294 10.44685 10.35511 10.31516 10.29405 10.28156 10.27355 10.26811 10.26424 a_b=34515 9.74613 9.67930 9.56774 9.54651 9.54032 9.53787 9.53667 9.53599 9.53557 9.53530 T able 2 Convergence of BFSC element fo r bI (b* facto r), isotropic case Mo del CaseLanzo et al. [64] BFSC convergence fo r bI (b* facto r) nxnybI=1ny=4ny=6ny=8ny=10ny=12ny=14ny=16ny=18ny=20 A1a_b=12525 0.18244 0.18655 0.18373 0.18311 0.18289 0.18279 0.18273 0.18270 0.18267 0.18266 a_b=24921 0.21177 0.21517 0.21277 0.21222 0.21202 0.21193 0.21188 0.21185 0.21183 0.21182 a_b=34915 0.22167 0.22519 0.22302 0.22251 0.22232 0.22224 0.22219 0.22216 0.22214 0.22212 A2a_b=13333 0.19576 0.21278 0.26446 0.19664 0.21681 0.19589 0.20598 0.19571 0.20176 0.19565 a_b=24921 0.26541 0.27179 0.26607 0.26500 0.26467 0.26452 0.26447 0.26443 0.26441 0.26440 a_b=3 - - 0.33573 0.33077 0.32955 0.32911 0.32904 0.32897 0.32893 0.32891 0.32889 A3a_b=1 - - 0.00619 0.00529 0.00504 0.00493 0.00488 0.00485 0.00482 0.00481 0.00480 a_b=23317 0.00881 0.00951 0.00912 0.00900 0.00895 0.00892 0.00891 0.00890 0.00889 0.00889 a_b=3 - - 0.02572 0.02600 0.02507 0.02528 0.02497 0.02509 0.02494 0.02501 0.02492 A4a_b=1 - - 0.17106 0.17037 0.17023 0.17015 0.17011 0.17009 0.17007 0.17006 0.17005 a_b=24921 0.26083 0.31749 0.27225 0.26075 0.26085 0.26367 0.26339 0.26047 0.25998 0.26069 a_b=3 - - 0.18543 0.18392 0.18358 0.18346 0.18341 0.18337 0.18336 0.18334 0.18333 B1a_b=13333 0.21935 0.21531 0.33212 0.22298 0.24930 0.22145 0.23232 0.22108 0.22622 0.22094 a_b=24923 0.21960 0.24769 0.23083 0.23598 0.22395 0.23024 0.22380 0.22462 0.22473 0.22321 a_b=34915 0.21148 0.21735 0.26961 0.22659 0.21428 0.22178 0.21473 0.21194 0.21577 0.21416 B2a_b=13333 0.29241 0.39369 0.34365 0.39354 0.30060 0.31165 0.30953 0.30172 0.29923 0.29945 a_b=24923 0.28583 0.34901 0.33925 0.32215 0.28414 0.29305 0.28055 0.28671 0.28312 0.28326 a_b=34915 0.27654 0.33651 0.34539 0.31122 0.28495 0.28404 0.27896 0.27751 0.27998 0.27734 C1a_b=12525 0.11453 0.10765 0.11914 0.11926 0.11897 0.11877 0.11864 0.11855 0.11849 0.11845 a_b=24523 0.07171 0.07680 0.07527 0.07465 0.07438 0.07424 0.07417 0.07412 0.07409 0.07407 a_b=34515 0.07992 0.11259 0.09853 0.09254 0.08860 0.08632 0.08505 0.08429 0.08383 0.08353 C2a_b=12525 0.11686 0.15020 0.12852 0.12391 0.12229 0.12159 0.12124 0.12104 0.12092 0.12084 a_b=24523 0.13282 0.11883 0.17133 0.15872 0.15760 0.14789 0.14515 0.14331 0.14308 0.14044 a_b=34515 0.08651 0.10179 0.09192 0.08984 0.08913 0.08883 0.08868 0.08860 0.08855 0.08851 bi-quadratic ﬁnite elements and a mesh of 25x25 elements. Note the good agreement between the BFSC element and theliteratureforallanisotropicratios E1_E2,alreadywitha mesh of only 4x4 elements. Theconvergenceofthe bIcoeﬃcientisshowninTable4. Thesecond-orderdisplacementﬁeld u11u11u11wassolvedwithEq.13andorthogonalizedwithEq. 14. NoteinFigure6howthe u,v,ù u2+v2displacementcomponentsof u11u11u11changewith the anisotropic ratio E1_E2, clearly showing that in the ini- tial post-buckling the displacements are progressively more focused with the increase of E1_E2, and that these focused post-bucklingdisplacementsarealignedwiththe 0ýand90ý SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 10 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates A1 A2 A3 A4 B1 B2 C1 C2 Figure 2:uIuIuI fo r isotropic plates with a_b=3 ; each ro w co rresp onds to one mo del; columns from left to right a re linea r buckling mo des fo r: I=1;2;3;4 T able 3 Convergence of BFSC element fo r rst buckling mo de and mo del A1 witha_b=1 , anisotropic case E1_E2Phan & Reddy [65] Bilotta et al. [66] BFSC convergence fo r c_ref c_ref ny=4ny=6ny=8ny=10ny=12ny=14ny=16ny=18ny=20 1 - - 4.1596 4.1662 4.1670 4.1671 4.1671 4.1672 4.1671 4.1671 4.1671 3 5.7538 5.7561 5.7501 5.7534 5.7537 5.7538 5.7537 5.7537 5.7538 5.7540 5.7540 10 11.492 11.4976 11.4923 11.4922 11.4919 11.4919 11.4921 11.4919 11.4914 11.4916 11.4910 20 19.712 19.7148 19.7167 19.7135 19.7127 19.7127 19.7119 19.7124 19.7122 19.7144 19.7036 directions of the laminate stacking sequence. 9. Results for multi-mode Koiter’s asymptotic expansion 9.1. Isotropic plates Tiso[10]presentedmulti-modalresultsformodelA1us- ing a geometry of: a= 0:14m,b= 0:10mand thickness of1mm; with isotropic material properties E= 70GPa and= 0:3. With this conﬁguration, the ﬁrst two buck- ling loads are close and respectively equal to 2828:15N and2866:50N. Tiso’s deﬁnition of coeﬃcients bijklcan betranslatedtothenotationhereinpresentedasgiveninEq. 57, which has a diﬀerent index ordering when compared toEq. 19. Despitethediﬀerentindexordering,Eq. 19andEq. 57 are equivalent in cases where aijk=0. bTiso ijkl=*1 6 iv cuiuiuiujujujukukukululul+1 2 ¨¨¨ cukukukulululuijuijuij+1 2 ¨¨¨ culululuiuiuiujkujkujk iÖ ¨¨ culululululul (57) Table 5 show the veriﬁcation of the formulation herein presented against the results from Tiso [10] for a multi- modal expansion using 2 modes. Jansen&Rahman[67]usedthegeneralpurposeﬁniteele- mentcodeDIANA[68]toinvestigatethedynamicresponse of plates and shells using reduced-order models based on SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 11 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates A1 A2 A3 A4 B1 B2 C1 C2 Figure 3:u11u11u11 fo r isotropic plate with a_b=3 ; each ro w co rresp onds to one mo del; columns from left to right a re: u ,v ,ù u2+v2 T able 4 Convergence of BFSC element fo r bI (b* facto r), mo del A1 witha_b=1 , anisotropic case E1_E2Bilotta et al. [66] BFSC convergence fo r bI (b* facto r) bI=1 (b* facto r) ny=4ny=6ny=8ny=10ny=12ny=14ny=16ny=18ny=20 1 - 0.17403 0.17138 0.17078 0.17056 0.17046 0.17041 0.17038 0.17036 0.17034 3 0.19865 0.20384 0.20011 0.19933 0.19906 0.19894 0.19887 0.19883 0.19880 0.19878 10 0.17075 0.17792 0.17277 0.17172 0.17135 0.17117 0.17108 0.17103 0.17100 0.17098 20 0.12595 0.13393 0.12809 0.12689 0.12646 0.12627 0.12616 0.12609 0.12604 0.12607 Koiter’s theory. The plate therein investigated has a geom- etry ofa= 0:6m,b= 0:2m, total thickness of 1mm. The same plate was used to verify the formulation and im- plementation herein presented for the multi-modal asymp- totic expansion of a composite plate. Using the loading and boundary condition of model A1 (Figure 1, the anal- yses in DIANA are carried out using two element types: 1) Theeight-nodequadrilateralshellelementCQ40L,basedon ﬁrst-ordersheardeformationkinematics(FSDT),meshwith 16x48elements; 2)Allman-typetriangularelement[69,70], with kinematics based on the classical laminated plate the-ory (CLPT). The Allman-type element triangular mesh has 24x72x2 elements. Table 6 presents the linear buckling eigenvaluesandTable7presentstheresultsforthe bijklco- eﬃcients using 5 modes, calculated per Eq. 19 for a plate with isotropic properties E= 70GPaand= 0:3. The relative error is calculated about the results using DIANA’s Allman-typeelement. TheBFSCmeshhas16x48elements, andtheauthorsveriﬁedthatallmodelsareconverged. There isclearlyacloseagreementbetweenDIANA’simplementa- tionandthepresentformulation,withamaximumdiﬀerence of 9%, attributed to diﬀerences veriﬁed for the linear buck- SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 12 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates A1 A2 A3 A4 B1 B2 C1 C2 Figure 4:u22u22u22 fo r isotropic plate with a_b=3 ; each ro w co rresp onds to one mo del; columns from left to right a re: u ,v ,ù u2+v2 T able 5 Multi-mo dal expansion co e cients bTiso ijklfo r isotropic plate bTiso 1jklTiso [10] BFSC bTiso 2jklTiso [10] BFSC b1111 0.1353 0.1347 b2111 0 0 b1112 0 0 b2112 0.0663 0.0682 b1121 0 0 b2121 0.2796 0.2543 b1122 0.0663 0.0692 b2122 0 0 b1211 0 0 b2211 0.2747 0.2729 b1212 0.0675 0.0644 b2212 0 0 b1221 0.2747 0.2766 b2221 0 0 b1222 0 0 b2222 0.2221 0.2176 lingeigenvalues i,showninTable6. Formanycoeﬃcients, there is a sign switch because uiuiuiare symmetric eigenmodes about the plate mid-plane, providing interchangeably posi- tive and negative solution for some of the bijklcoeﬃcients. Figure 7 illustrates the second-order ﬁeld modes uijuijuijfor the isotropic plate used in the multi-modal asymptotic ex-T able 6 Eigenvalues i fo r isotropic plate used in multi-mo dal analysis, units in[N_m] EigenvalueDIANA DIANABFSCCQ40L Allman 1 6307 6337 6328 2 6844 6885 6866 3 7405 7432 7426 4 8105 8165 8127 5 9863 9955 9886 6 12040 12182 12066 7 14606 14822 14635 8 17542 17584 17575 pansion. 9.2. Composite plates Table 8 presents the linear buckling eigenvalues and Table 9 present results for a composite plate with layup of+45_*45_90_0 symand material properties E11= SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 13 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates T able 7 Multi-mo dal expansion co e cients bijkl fo r isotropic plate using 5 mo des bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC ErrCQ40L Allman CQ40L Allman CQ40L Allman CQ40L Allman CQ40L Allman b1111 0.21531 0.21756 0.21564 1% b2111 0 0 0 b3111 0 0 0 b4111 0.01072 0.01079 0.01084 0% b5111 0 0 0 b1112 0 0 0 b2112 0.11844 0.11830 0.11971 1% b3112 -0.11414 0.11583 0.10853 6% b4112 0 0 0 b5112 -0.01003 0.01004 -0.01012 1% b1113 0 0 0 b2113 -0.03165 0.03132 0.02935 6% b3113 0.15488 0.15732 0.16034 2% b4113 0 0 0 b5113 0.00353 0.00349 -0.00352 1% b1114 0.03791 0.03852 0.03868 0% b2114 0 0 0 b3114 0 0 0 b4114 0.09640 0.09693 0.09810 1% b5114 0 0 0 b1115 0 0 0 b2115 -0.03296 0.03256 -0.03278 1% b3115 0.04183 0.04190 -0.04218 1% b4115 0 0 0 b5115 0.07800 0.07834 0.07940 1% b1121 0 0 0 b2121 0.10201 0.10181 0.09619 6% b3121 -0.13375 0.13454 0.14553 8% b4121 0 0 0 b5121 -0.01008 0.01006 -0.01014 1% b1122 0.22357 0.22823 0.23092 1% b2122 0 0 0 b3122 0 0 0 b4122 -0.02560 -0.02558 -0.02360 8% b5122 0 0 0 b1123 -0.05975 0.06042 0.05661 6% b2123 0 0 0 b3123 0 0 0 b4123 0.02485 -0.02539 -0.02385 6% b5123 0 0 0 b1124 0 0 0 b2124 -0.09139 -0.09116 -0.09643 6% b3124 0.14851 -0.15510 -0.14599 6% b4124 0 0 0 b5124 -0.01931 0.01934 -0.01798 7% b1125 -0.06222 0.06282 -0.06322 1% b2125 0 0 0 b3125 0 0 0 b4125 -0.07448 0.07506 -0.07873 5% b5125 0 0 0 b1131 0 0 0 b2131 -0.03709 0.03637 0.03935 8% b3131 0.17117 0.17440 0.16386 6% b4131 0 0 0 b5131 0.00333 0.00331 -0.00333 1% b1132 -0.05975 0.06042 0.05661 6% b2132 0 0 0 b3132 0 0 0 b4132 0.03334 -0.03378 -0.03594 6% b5132 0 0 0 b1133 0.08108 0.08206 0.08364 2% b2133 0 0 0 b3133 0 0 0 b4133 0.00267 0.00238 0.00239 0% b5133 0 0 0 b1134 0 0 0 b2134 0.04119 -0.04193 -0.03948 6% b3134 0.02392 0.02267 0.02277 0% b4134 0 0 0 b5134 0.02974 0.02990 -0.03152 5% b1135 0.02190 0.02185 -0.02200 1% b2135 0 0 0 b3135 0 0 0 b4135 0.02999 0.03073 -0.02920 5% b5135 0 0 0 b1141 0.03791 0.03852 0.03868 0% b2141 0 0 0 b3141 0 0 0 b4141 0.08099 0.08163 0.07762 5% b5141 0 0 0 b1142 0 0 0 b2142 -0.04796 -0.04733 -0.04364 8% b3142 0.22521 -0.23119 -0.24585 6% b4142 0 0 0 b5142 -0.02136 0.02158 -0.02012 7% b1143 0 0 0 b2143 0.04655 -0.04697 -0.04411 6% b3143 0.01803 0.01631 0.01636 0% b4143 0 0 0 b5143 0.02118 0.02148 -0.02031 5% b1144 0.34088 0.34603 0.35002 1% b2144 0 0 0 b3144 0 0 0 b4144 0.04186 0.04112 0.04142 1% b5144 0 0 0 b1145 0 0 0 b2145 -0.13952 0.13888 -0.14561 5% b3145 0.20255 0.21029 -0.19972 5% b4145 0 0 0 b5145 0.03251 0.03196 0.03228 1% b1151 0 0 0 b2151 -0.03314 0.03261 -0.03286 1% b3151 0.03946 0.03965 -0.03989 1% b4151 0 0 0 b5151 0.06468 0.06506 0.06222 4% b1152 -0.06222 0.06282 -0.06322 1% b2152 0 0 0 b3152 0 0 0 b4152 -0.03748 0.03783 -0.03525 7% b5152 0 0 0 b1153 0.02190 0.02185 -0.02200 1% b2153 0 0 0 b3153 0 0 0 b4153 0.03716 0.03765 -0.03557 6% b5153 0 0 0 b1154 0 0 0 b2154 -0.06348 0.06273 -0.05827 7% b3154 0.35252 0.35857 -0.37768 5% b4154 0 0 0 b5154 0.02982 0.02916 0.02949 1% b1155 0.48408 0.49009 0.49626 1% b2155 0 0 0 b3155 0 0 0 b4155 0.05705 0.05601 0.05654 1% b5155 0 0 0 b1211 0 0 0 b2211 0.11844 0.11830 0.11971 1% b3211 -0.11414 0.11583 0.10853 6% b4211 0 0 0 b5211 -0.01003 0.01004 -0.01012 1% b1212 0.19257 0.19643 0.18555 6% b2212 0 0 0 b3212 0 0 0 b4212 -0.04879 -0.04927 -0.05214 6% b5212 0 0 0 b1213 -0.07002 0.07018 0.07591 8% b2213 0 0 0 b3213 0 0 0 b4213 0.02199 -0.02266 -0.02134 6% b5213 0 0 0 b1214 0 0 0 b2214 -0.04796 -0.04733 -0.04364 8% b3214 0.16784 -0.17373 -0.16313 6% b4214 0 0 0 b5214 -0.04244 0.04283 -0.04494 5% b1215 -0.06255 0.06291 -0.06338 1% b2215 0 0 0 b3215 0 0 0 b4215 -0.03388 0.03390 -0.03150 7% b5215 0 0 0 b1221 0.22357 0.22823 0.23092 1% b2221 0 0 0 b3221 0 0 0 b4221 -0.02560 -0.02558 -0.02360 8% b5221 0 0 0 b1222 0 0 0 b2222 0.26634 0.27587 0.27403 1% b3222 0.06669 -0.07876 -0.07909 0% b4222 0 0 0 b5222 -0.02111 0.02278 -0.02296 1% b1223 0 0 0 b2223 0.01850 -0.02129 -0.02139 0% b3223 0.26692 0.27435 0.27908 2% b4223 0 0 0 b5223 -0.01844 -0.01904 0.01800 5% b1224 -0.09054 -0.09132 -0.08418 8% b2224 0 0 0 b3224 0 0 0 b4224 0.14102 0.14590 0.14725 1% b5224 0 0 0 b1225 0 0 0 b2225 -0.06939 0.07388 -0.07438 1% b3225 -0.21859 -0.22834 0.21563 6% b4225 0 0 0 b5225 0.11457 0.11821 0.11945 1% b1231 -0.05975 0.06042 0.05661 6% b2231 0 0 0 b3231 0 0 0 b4231 0.03334 -0.03378 -0.03594 6% b5231 0 0 0 b1232 0 0 0 b2232 0.01850 -0.02129 -0.02139 0% b3232 0.29211 0.30076 0.28465 5% b4232 0 0 0 b5232 -0.02975 -0.03041 0.03205 5% b1233 0 0 0 b2233 0.07402 0.07417 0.07547 2% b3233 0.00700 -0.00521 -0.00520 0% b4233 0 0 0 b5233 -0.00283 0.00235 -0.00237 1% b1234 0.08787 -0.09062 -0.08509 6% b2234 0 0 0 b3234 0 0 0 b4234 0.01855 -0.01893 -0.01906 1% b5234 0 0 0 b1235 0 0 0 b2235 -0.06062 -0.06173 0.05831 6% b3235 -0.04564 0.04096 -0.04134 1% b4235 0 0 0 b5235 0.01383 -0.01407 -0.01422 1% b1241 0 0 0 b2241 -0.04796 -0.04733 -0.04364 8% b3241 0.22521 -0.23119 -0.24585 6% b4241 0 0 0 b5241 -0.02136 0.02158 -0.02012 7% b1242 -0.17252 -0.17588 -0.18602 6% b2242 0 0 0 b3242 0 0 0 b4242 0.09860 0.10277 0.09780 5% b5242 0 0 0 b1243 0.07775 -0.08090 -0.07615 6% b2243 0 0 0 b3243 0 0 0 b4243 0.01781 -0.01853 -0.01868 1% b5243 0 0 0 b1244 0 0 0 b2244 0.26417 0.26994 0.27235 1% b3244 0.12529 -0.12952 -0.13036 1% b4244 0 0 0 b5244 0.01864 -0.01991 0.01807 9% b1245 -0.11982 0.12102 -0.11240 7% b2245 0 0 0 b3245 0 0 0 b4245 0.10346 -0.10630 0.11089 4% b5245 0 0 0 b1251 -0.06222 0.06282 -0.06322 1% b2251 0 0 0 b3251 0 0 0 b4251 -0.03748 0.03783 -0.03525 7% b5251 0 0 0 b1252 0 0 0 b2252 -0.06939 0.07388 -0.07438 1% b3252 -0.35269 -0.36471 0.38396 5% b4252 0 0 0 b5252 0.07955 0.08253 0.07916 4% b1253 0 0 0 b2253 -0.06062 -0.06173 0.05831 6% b3253 -0.03351 0.02823 -0.02844 1% b4253 0 0 0 b5253 0.01144 -0.01186 -0.01201 1% b1254 -0.26338 0.26794 -0.28089 5% b2254 0 0 0 b3254 0 0 0 b4254 0.03271 -0.03489 0.03165 9% b5254 0 0 0 b1255 0 0 0 b2255 0.37666 0.38331 0.38701 1% b3255 0.16398 -0.16879 -0.17032 1% b4255 0 0 0 b5255 -0.05715 0.05705 -0.05765 1% b1311 0 0 0 b2311 -0.03165 0.03132 0.02935 6% b3311 0.15488 0.15732 0.16034 2% b4311 0 0 0 b5311 0.00353 0.00349 -0.00352 1% b1312 -0.07002 0.07018 0.07591 8% b2312 0 0 0 b3312 0 0 0 b4312 0.02199 -0.02266 -0.02134 6% b5312 0 0 0 b1313 0.08961 0.09097 0.08547 6% b2313 0 0 0 b3313 0 0 0 b4313 0.00354 0.00331 0.00333 1% b5313 0 0 0 b1314 0 0 0 b2314 0.06246 -0.06251 -0.06648 6% b3314 0.01803 0.01631 0.01636 0% b4314 0 0 0 b5314 0.01709 0.01753 -0.01667 5% b1315 0.02066 0.02068 -0.02081 1% b2315 0 0 0 b3315 0 0 0 b4315 0.05219 0.05239 -0.05522 5% b5315 0 0 0 b1321 -0.05975 0.06042 0.05661 6% b2321 0 0 0 b3321 0 0 0 b4321 0.02485 -0.02539 -0.02385 6% b5321 0 0 0 b1322 0 0 0 b2322 0.01850 -0.02129 -0.02139 0% b3322 0.26692 0.27435 0.27908 2% b4322 0 0 0 b5322 -0.01844 -0.01904 0.01800 5% b1323 0 0 0 b2323 0.08101 0.08131 0.07697 5% b3323 0.00700 -0.00521 -0.00520 0% b4323 0 0 0 b5323 -0.00385 0.00342 -0.00345 1% b1324 0.11790 -0.12059 -0.12824 6% b2324 0 0 0 b3324 0 0 0 b4324 0.01855 -0.01893 -0.01906 1% b5324 0 0 0 b1325 0 0 0 b2325 -0.09781 -0.09860 0.10382 5% b3325 -0.03351 0.02823 -0.02844 1% b4325 0 0 0 b5325 0.01383 -0.01407 -0.01422 1% b1331 0.08108 0.08206 0.08364 2% b2331 0 0 0 b3331 0 0 0 b4331 0.00267 0.00238 0.00239 0% b5331 0 0 0 b1332 0 0 0 b2332 0.07402 0.07417 0.07547 2% b3332 0.00700 -0.00521 -0.00520 0% b4332 0 0 0 b5332 -0.00283 0.00235 -0.00237 1% b1333 0 0 0 b2333 0.00194 -0.00141 -0.00141 0% b3333 0.12149 0.12257 0.12130 1% b4333 0 0 0 b5333 0.01391 0.01388 -0.01386 0% b1334 0.00944 0.00851 0.00854 0% b2334 0 0 0 b3334 0 0 0 b4334 0.06196 0.06203 0.06314 2% b5334 0 0 0 b1335 0 0 0 b2335 -0.00929 0.00763 -0.00769 1% b3335 0.16495 0.16646 -0.16606 0% b4335 0 0 0 b5335 0.04960 0.04936 0.05034 2% b1341 0 0 0 b2341 0.04655 -0.04697 -0.04411 6% b3341 0.01803 0.01631 0.01636 0% b4341 0 0 0 b5341 0.02118 0.02148 -0.02031 5% b1342 0.07775 -0.08090 -0.07615 6% b2342 0 0 0 b3342 0 0 0 b4342 0.01781 -0.01853 -0.01868 1% b5342 0 0 0 b1343 0.01252 0.01182 0.01188 0% b2343 0 0 0 b3343 0 0 0 b4343 0.06741 0.06758 0.06429 5% b5343 0 0 0 b1344 0 0 0 b2344 0.03475 -0.03502 -0.03525 1% b3344 0.41852 0.42450 0.43188 2% b4344 0 0 0 b5344 0.01553 0.01522 -0.01538 1% b1345 0.18455 0.18703 -0.19700 5% b2345 0 0 0 b3345 0 0 0 b4345 0.02995 0.02878 -0.02905 1% b5345 0 0 0 b1351 0.02190 0.02185 -0.02200 1% b2351 0 0 0 b3351 0 0 0 b4351 0.03716 0.03765 -0.03557 6% b5351 0 0 0 b1352 0 0 0 b2352 -0.06062 -0.06173 0.05831 6% b3352 -0.03351 0.02823 -0.02844 1% b4352 0 0 0 b5352 0.01144 -0.01186 -0.01201 1% b1353 0 0 0 b2353 -0.01266 0.01107 -0.01118 1% b3353 0.16495 0.16646 -0.16606 0% b4353 0 0 0 b5353 0.05255 0.05240 0.04992 5% b1354 0.10603 0.10969 -0.10418 5% b2354 0 0 0 b3354 0 0 0 b4354 0.02725 0.02667 -0.02694 1% b5354 0 0 0 b1355 0 0 0 b2355 0.04548 -0.04564 -0.04606 1% b3355 0.58796 0.59206 0.60319 2% b4355 0 0 0 b5355 0.02128 0.02076 -0.02102 1% b1411 0.03791 0.03852 0.03868 0% b2411 0 0 0 b3411 0 0 0 b4411 0.09640 0.09693 0.09810 1% b5411 0 0 0 b1412 0 0 0 b2412 -0.04796 -0.04733 -0.04364 8% b3412 0.16784 -0.17373 -0.16313 6% b4412 0 0 0 b5412 -0.04244 0.04283 -0.04494 5% b1413 0 0 0 b2413 0.06246 -0.06251 -0.06648 6% b3413 0.01803 0.01631 0.01636 0% b4413 0 0 0 b5413 0.01709 0.01753 -0.01667 5% b1414 0.28640 0.29139 0.27692 5% b2414 0 0 0 b3414 0 0 0 b4414 0.04186 0.04112 0.04142 1% b5414 0 0 0 b1415 0 0 0 b2415 -0.07022 0.06999 -0.06519 7% b3415 0.25104 0.25769 -0.24333 6% b4415 0 0 0 b5415 0.03251 0.03196 0.03228 1% b1421 0 0 0 b2421 -0.09139 -0.09116 -0.09643 6% b3421 0.14851 -0.15510 -0.14599 6% b4421 0 0 0 b5421 -0.01931 0.01934 -0.01798 7% b1422 -0.09054 -0.09132 -0.08418 8% b2422 0 0 0 b3422 0 0 0 b4422 0.14102 0.14590 0.14725 1% b5422 0 0 0 b1423 0.11790 -0.12059 -0.12824 6% b2423 0 0 0 b3423 0 0 0 b4423 0.01855 -0.01893 -0.01906 1% b5423 0 0 0 b1424 0 0 0 b2424 0.18470 0.19014 0.18089 5% b3424 0.12031 -0.12684 -0.12780 1% b4424 0 0 0 b5424 0.05896 -0.06065 0.06330 4% b1425 -0.13255 0.13504 -0.12575 7% b2425 0 0 0 b3425 0 0 0 b4425 0.03271 -0.03489 0.03165 9% b5425 0 0 0 b1431 0 0 0 b2431 0.04119 -0.04193 -0.03948 6% b3431 0.02392 0.02267 0.02277 0% b4431 0 0 0 b5431 0.02974 0.02990 -0.03152 5% b1432 0.08787 -0.09062 -0.08509 6% b2432 0 0 0 b3432 0 0 0 b4432 0.01855 -0.01893 -0.01906 1% b5432 0 0 0 b1433 0.00944 0.00851 0.00854 0% b2433 0 0 0 b3433 0 0 0 b4433 0.06196 0.06203 0.06314 2% b5433 0 0 0 b1434 0 0 0 b2434 0.03337 -0.03429 -0.03456 1% b3434 0.45534 0.46249 0.43973 5% b4434 0 0 0 b5434 0.01706 0.01642 -0.01658 1% b1435 0.13142 0.13441 -0.12692 6% b2435 0 0 0 b3435 0 0 0 b4435 0.02725 0.02667 -0.02694 1% b5435 0 0 0 b1441 0.34088 0.34603 0.35002 1% b2441 0 0 0 b3441 0 0 0 b4441 0.04186 0.04112 0.04142 1% b5441 0 0 0 b1442 0 0 0 b2442 0.26417 0.26994 0.27235 1% b3442 0.12529 -0.12952 -0.13036 1% b4442 0 0 0 b5442 0.01864 -0.01991 0.01807 9% b1443 0 0 0 b2443 0.03475 -0.03502 -0.03525 1% b3443 0.41852 0.42450 0.43188 2% b4443 0 0 0 b5443 0.01553 0.01522 -0.01538 1% b1444 0.14804 0.14680 0.14777 1% b2444 0 0 0 b3444 0 0 0 b4444 0.31532 0.31939 0.31842 0% b5444 0 0 0 b1445 0 0 0 b2445 0.06128 -0.06455 0.05853 9% b3445 0.18407 0.18253 -0.18424 1% b4445 0 0 0 b5445 0.16826 0.17003 0.17164 1% b1451 0 0 0 b2451 -0.06348 0.06273 -0.05827 7% b3451 0.35252 0.35857 -0.37768 5% b4451 0 0 0 b5451 0.02982 0.02916 0.02949 1% b1452 -0.26338 0.26794 -0.28089 5% b2452 0 0 0 b3452 0 0 0 b4452 0.03271 -0.03489 0.03165 9% b5452 0 0 0 b1453 0.10603 0.10969 -0.10418 5% b2453 0 0 0 b3453 0 0 0 b4453 0.02725 0.02667 -0.02694 1% b5453 0 0 0 b1454 0 0 0 b2454 0.19382 -0.19668 0.20510 4% b3454 0.20229 0.19694 -0.19871 1% b4454 0 0 0 b5454 0.10377 0.10517 0.10118 4% b1455 0.20173 0.19992 0.20174 1% b2455 0 0 0 b3455 0 0 0 b4455 0.29529 0.29799 0.30066 1% b5455 0 0 0 b1511 0 0 0 b2511 -0.03296 0.03256 -0.03278 1% b3511 0.04183 0.04190 -0.04218 1% b4511 0 0 0 b5511 0.07800 0.07834 0.07940 1% b1512 -0.06255 0.06291 -0.06338 1% b2512 0 0 0 b3512 0 0 0 b4512 -0.03388 0.03390 -0.03150 7% b5512 0 0 0 b1513 0.02066 0.02068 -0.02081 1% b2513 0 0 0 b3513 0 0 0 b4513 0.05219 0.05239 -0.05522 5% b5513 0 0 0 b1514 0 0 0 b2514 -0.07022 0.06999 -0.06519 7% b3514 0.25104 0.25769 -0.24333 6% b4514 0 0 0 b5514 0.03251 0.03196 0.03228 1% b1515 0.40139 0.40706 0.38885 4% b2515 0 0 0 b3515 0 0 0 b4515 0.05234 0.05111 0.05166 1% b5515 0 0 0 b1521 -0.06222 0.06282 -0.06322 1% b2521 0 0 0 b3521 0 0 0 b4521 -0.07448 0.07506 -0.07873 5% b5521 0 0 0 b1522 0 0 0 b2522 -0.06939 0.07388 -0.07438 1% b3522 -0.21859 -0.22834 0.21563 6% b4522 0 0 0 b5522 0.11457 0.11821 0.11945 1% b1523 0 0 0 b2523 -0.09781 -0.09860 0.10382 5% b3523 -0.03351 0.02823 -0.02844 1% b4523 0 0 0 b5523 0.01383 -0.01407 -0.01422 1% b1524 -0.13255 0.13504 -0.12575 7% b2524 0 0 0 b3524 0 0 0 b4524 0.03271 -0.03489 0.03165 9% b5524 0 0 0 b1525 0 0 0 b2525 0.26154 0.26762 0.25647 4% b3525 0.13559 -0.14228 -0.14394 1% b4525 0 0 0 b5525 -0.05715 0.05705 -0.05765 1% b1531 0.02190 0.02185 -0.02200 1% b2531 0 0 0 b3531 0 0 0 b4531 0.02999 0.03073 -0.02920 5% b5531 0 0 0 b1532 0 0 0 b2532 -0.06062 -0.06173 0.05831 6% b3532 -0.04564 0.04096 -0.04134 1% b4532 0 0 0 b5532 0.01383 -0.01407 -0.01422 1% b1533 0 0 0 b2533 -0.00929 0.00763 -0.00769 1% b3533 0.16495 0.16646 -0.16606 0% b4533 0 0 0 b5533 0.04960 0.04936 0.05034 2% b1534 0.13142 0.13441 -0.12692 6% b2534 0 0 0 b3534 0 0 0 b4534 0.02725 0.02667 -0.02694 1% b5534 0 0 0 b1535 0 0 0 b2535 0.03760 -0.03847 -0.03892 1% b3535 0.62291 0.62848 0.59811 5% b4535 0 0 0 b5535 0.02128 0.02076 -0.02102 1% b1541 0 0 0 b2541 -0.13952 0.13888 -0.14561 5% b3541 0.20255 0.21029 -0.19972 5% b4541 0 0 0 b5541 0.03251 0.03196 0.03228 1% b1542 -0.11982 0.12102 -0.11240 7% b2542 0 0 0 b3542 0 0 0 b4542 0.10346 -0.10630 0.11089 4% b5542 0 0 0 b1543 0.18455 0.18703 -0.19700 5% b2543 0 0 0 b3543 0 0 0 b4543 0.02995 0.02878 -0.02905 1% b5543 0 0 0 b1544 0 0 0 b2544 0.06128 -0.06455 0.05853 9% b3544 0.18407 0.18253 -0.18424 1% b4544 0 0 0 b5544 0.16826 0.17003 0.17164 1% b1545 0.18507 0.18245 0.18430 1% b2545 0 0 0 b3545 0 0 0 b4545 0.18211 0.18432 0.17724 4% b5545 0 0 0 b1551 0.48408 0.49009 0.49626 1% b2551 0 0 0 b3551 0 0 0 b4551 0.05705 0.05601 0.05654 1% b5551 0 0 0 b1552 0 0 0 b2552 0.37666 0.38331 0.38701 1% b3552 0.16398 -0.16879 -0.17032 1% b4552 0 0 0 b5552 -0.05715 0.05705 -0.05765 1% b1553 0 0 0 b2553 0.04548 -0.04564 -0.04606 1% b3553 0.58796 0.59206 0.60319 2% b4553 0 0 0 b5553 0.02128 0.02076 -0.02102 1% b1554 0.20173 0.19992 0.20174 1% b2554 0 0 0 b3554 0 0 0 b4554 0.29529 0.29799 0.30066 1% b5554 0 0 0 b1555 0 0 0 b2555 -0.18789 0.18501 -0.18679 1% b3555 0.25230 0.24895 -0.25181 1% b4555 0 0 0 b5555 0.35181 0.35393 0.35447 0% SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 14 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates A1 A2 A3 A4 B1 B2 C1 C2 Figure 5:u33u33u33 fo r isotropic plate with a_b=3 ; each ro w co rresp onds to one mo del; columns from left to right a re: u ,v ,ù u2+v2 127:62900GPa,E22= 11:30740GPa,G12=G23= G13= 6:00257GPaand12= 0:300235. Results using DIANA’s CQ40L and Allman-type elements are present to- gether with the BFSC results. The error of the BFSC im- plementation is again calculated about the results obtained with the Allman-type element. The same mesh, load and boundaryconditionsofthepreviousisotropiccaseareused. Again, a maximum error of 9% is veriﬁed for the bijklco- eﬃcients, attributed to the diﬀerences in ivalues between the Allman-type and BFSC elements. Figure 8 illustrates the second-order ﬁeld modes uijuijuijfor the composite plate used in the multi-modal asymptotic ex- pansion. 10. Conclusion A complete displacement-based formulation for the Koi- ter’s multi-modal asymptotic expansion was presented. The formulationandnotationadoptedenabledaclosecorrespon- dencebetweentheformulationandimplementedalgorithms, facilitating further developments on the ﬁeld of initial post-T able 8 Eigenvalues i fo r comp osite plate used in multi-mo dal analy- sis, units in[N_m] EigenvalueDIANA DIANABFSCCQ40L Allman 1 5257 5312 5303 2 5420 5483 5467 3 6011 6088 6060 4 6132 6185 6178 5 6877 6976 6928 6 7967 8100 8021 7 9258 9441 9317 8 10740 10993 10804 buckling analysis and reduced-order models based on Koi- ter’smethod. Thepresentworkusedstate-of-the-artcollab- orative tools to implement the developed methods: Python [61], NumPy [62] and Cython [63]. A modiﬁed Bogner-Fox-Schmit (BFS) ﬁnite element SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 15 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates T able 9 Multi-mo dal expansion co e cients bijkl fo r comp osite plate using 5 mo des bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC Err bijklDIANA DIANABFSC ErrCQ40L Allman CQ40L Allman CQ40L Allman CQ40L Allman CQ40L Allman b1111 0.18993 0.18806 0.18646 1% b2111 0 0 0 b3111 -0.01064 -0.01116 0.01117 0% b4111 0 0 0 b5111 0 0 0 b1112 0 0 0 b2112 0.10871 0.10467 0.10615 1% b3112 0 0 0 b4112 0.10064 0.09356 0.08835 6% b5112 -0.01046 -0.01025 0.01036 1% b1113 -0.03384 -0.03531 0.03531 0% b2113 0 0 0 b3113 0.09543 0.09339 0.09447 1% b4113 0 0 0 b5113 0 0 0 b1114 0 0 0 b2114 0.02851 0.02771 0.02604 6% b3114 0 0 0 b4114 0.13833 0.13851 0.14054 1% b5114 -0.00367 -0.00358 0.00359 0% b1115 0 0 0 b2115 -0.02944 -0.02877 0.02901 1% b3115 0 0 0 b4115 -0.03644 -0.03392 0.03409 1% b5115 0.08213 0.08050 0.08152 1% b1121 0 0 0 b2121 0.09250 0.08810 0.08345 5% b3121 0 0 0 b4121 0.11881 0.10997 0.11925 8% b5121 -0.01053 -0.01010 0.01013 0% b1122 0.19890 0.18587 0.18916 2% b2122 0 0 0 b3122 0.02464 0.02214 -0.02063 7% b4122 0 0 0 b5122 0 0 0 b1123 0 0 0 b2123 0.08446 0.08087 -0.08539 6% b3123 0 0 0 b4123 0.13451 0.13344 -0.12660 5% b5123 0.02016 0.01891 0.01763 7% b1124 0.05216 0.04922 0.04641 6% b2124 0 0 0 b3124 0.02454 0.02422 -0.02281 6% b4124 0 0 0 b5124 0 0 0 b1125 -0.05386 -0.05109 0.05169 1% b2125 0 0 0 b3125 0.07375 0.06968 0.07309 5% b4125 0 0 0 b5125 0 0 0 b1131 -0.03384 -0.03531 0.03531 0% b2131 0 0 0 b3131 0.07951 0.07781 0.07424 5% b4131 0 0 0 b5131 0 0 0 b1132 0 0 0 b2132 0.04284 0.03944 -0.03659 7% b3132 0 0 0 b4132 0.20449 0.19737 -0.20872 6% b5132 0.02261 0.02198 0.02072 6% b1133 0.30355 0.29547 0.29851 1% b2133 0 0 0 b3133 -0.04021 -0.03494 0.03563 2% b4133 0 0 0 b5133 0 0 0 b1134 0 0 0 b2134 0.04266 0.04314 -0.04045 6% b3134 0 0 0 b4134 -0.01420 -0.01106 0.01132 2% b5134 0.02246 0.02297 0.02172 5% b1135 0 0 0 b2135 0.12820 0.12414 0.12960 4% b3135 0 0 0 b4135 0.18158 0.18143 0.17310 5% b5135 -0.03298 -0.02875 0.02942 2% b1141 0 0 0 b2141 0.03365 0.03258 0.03515 8% b3141 0 0 0 b4141 0.15212 0.15288 0.14439 6% b5141 -0.00338 -0.00331 0.00337 2% b1142 0.05216 0.04922 0.04641 6% b2142 0 0 0 b3142 0.03332 0.03282 -0.03470 6% b4142 0 0 0 b5142 0 0 0 b1143 0 0 0 b2143 0.03810 0.03953 -0.03732 6% b3143 0 0 0 b4143 -0.02012 -0.01827 0.01900 4% b5143 0.03179 0.03211 0.03355 4% b1144 0.07169 0.07286 0.07382 1% b2144 0 0 0 b3144 -0.00231 -0.00184 0.00188 2% b4144 0 0 0 b5144 0 0 0 b1145 -0.01888 -0.01784 0.01790 0% b2145 0 0 0 b3145 0.02959 0.03017 0.02878 5% b4145 0 0 0 b5145 0 0 0 b1151 0 0 0 b2151 -0.02964 -0.02832 0.02838 0% b3151 0 0 0 b4151 -0.03360 -0.03137 0.03199 2% b5151 0.06765 0.06635 0.06365 4% b1152 -0.05386 -0.05109 0.05169 1% b2152 0 0 0 b3152 0.03659 0.03461 0.03273 5% b4152 0 0 0 b5152 0 0 0 b1153 0 0 0 b2153 0.05671 0.05304 0.04936 7% b3153 0 0 0 b4153 0.31573 0.30416 0.31875 5% b5153 -0.02991 -0.02495 0.02547 2% b1154 -0.01888 -0.01784 0.01790 0% b2154 0 0 0 b3154 0.03635 0.03617 0.03431 5% b4154 0 0 0 b5154 0 0 0 b1155 0.42279 0.40110 0.40683 1% b2155 0 0 0 b3155 -0.05338 -0.04528 0.04646 3% b4155 0 0 0 b5155 0 0 0 b1211 0 0 0 b2211 0.10871 0.10467 0.10615 1% b3211 0 0 0 b4211 0.10064 0.09356 0.08835 6% b5211 -0.01046 -0.01025 0.01036 1% b1212 0.16924 0.15646 0.14871 5% b2212 0 0 0 b3212 0.04859 0.04539 -0.04815 6% b4212 0 0 0 b5212 0 0 0 b1213 0 0 0 b2213 0.04284 0.03944 -0.03659 7% b3213 0 0 0 b4213 0.15060 0.14565 -0.13722 6% b5213 0.04557 0.04424 0.04627 5% b1214 0.06157 0.05785 0.06264 8% b2214 0 0 0 b3214 0.02192 0.02219 -0.02104 5% b4214 0 0 0 b5214 0 0 0 b1215 -0.05423 -0.05030 0.05057 1% b2215 0 0 0 b3215 0.03262 0.02977 0.02784 6% b4215 0 0 0 b5215 0 0 0 b1221 0.19890 0.18587 0.18916 2% b2221 0 0 0 b3221 0.02464 0.02214 -0.02063 7% b4221 0 0 0 b5221 0 0 0 b1222 0 0 0 b2222 0.25341 0.24575 0.24462 0% b3222 0 0 0 b4222 -0.06973 -0.08089 -0.07988 1% b5222 -0.02405 -0.02558 0.02557 0% b1223 0.07839 0.07004 -0.06520 7% b2223 0 0 0 b3223 0.14389 0.13969 0.14099 1% b4223 0 0 0 b5223 0 0 0 b1224 0 0 0 b2224 -0.01975 -0.02396 -0.02355 2% b3224 0 0 0 b4224 0.23989 0.22601 0.23077 2% b5224 0.01987 0.01990 -0.01893 5% b1225 0 0 0 b2225 -0.06766 -0.07177 0.07160 0% b3225 0 0 0 b4225 0.19737 0.18845 -0.17989 5% b5225 0.12450 0.12107 0.12234 1% b1231 0 0 0 b2231 0.04284 0.03944 -0.03659 7% b3231 0 0 0 b4231 0.20449 0.19737 -0.20872 6% b5231 0.02261 0.02198 0.02072 6% b1232 0.15454 0.14361 -0.15216 6% b2232 0 0 0 b3232 0.10056 0.09946 0.09525 4% b4232 0 0 0 b5232 0 0 0 b1233 0 0 0 b2233 0.25015 0.24887 0.25000 0% b3233 0 0 0 b4233 -0.11461 -0.10795 -0.10874 1% b5233 0.02115 0.02293 -0.02096 9% b1234 0.06971 0.07019 -0.06650 5% b2234 0 0 0 b3234 -0.01857 -0.01900 -0.01911 1% b4234 0 0 0 b5234 0 0 0 b1235 0.10376 0.09420 0.08797 7% b2235 0 0 0 b3235 0.10470 0.10162 -0.10542 4% b4235 0 -0.00001 0 b5235 0 0 0 b1241 0.05216 0.04922 0.04641 6% b2241 0 0 0 b3241 0.03332 0.03282 -0.03470 6% b4241 0 0 0 b5241 0 0 0 b1242 0 0 0 b2242 -0.01975 -0.02396 -0.02355 2% b3242 0 0 0 b4242 0.26106 0.24556 0.23420 5% b5242 0.03241 0.03214 -0.03365 5% b1243 0.07805 0.07661 -0.07208 6% b2243 0 0 0 b3243 -0.01867 -0.01795 -0.01808 1% b4243 0 0 0 b5243 0 0 0 b1244 0 0 0 b2244 0.06795 0.06695 0.06802 2% b3244 0 0 0 b4244 -0.00373 0.00075 0.00030 b5244 -0.00238 -0.00116 0.00129 11% b1245 0 0 0 b2245 0.05590 0.05582 -0.05302 5% b3245 0 0 0 b4245 -0.03551 -0.02317 0.02485 7% b5245 -0.01483 -0.01425 -0.01444 1% b1251 -0.05386 -0.05109 0.05169 1% b2251 0 0 0 b3251 0.03659 0.03461 0.03273 5% b4251 0 0 0 b5251 0 0 0 b1252 0 0 0 b2252 -0.06766 -0.07177 0.07160 0% b3252 0 0 0 b4252 0.32196 0.30442 -0.31968 5% b5252 0.08671 0.08617 0.08325 3% b1253 0.23458 0.22045 0.23094 5% b2253 0 0 0 b3253 0.03422 0.03612 -0.03311 8% b4253 0 -0.00001 0 b5253 0 0 0 b1254 0 0 0 b2254 0.05590 0.05582 -0.05302 5% b3254 0 0 0 b4254 -0.02367 -0.01102 0.01226 11% b5254 -0.01274 -0.01274 -0.01283 1% b1255 0 0 0 b2255 0.35028 0.33969 0.34263 1% b3255 0 0 0 b4255 -0.14729 -0.13501 -0.13719 2% b5255 -0.05897 -0.05460 0.05557 2% b1311 -0.03384 -0.03531 0.03531 0% b2311 0 0 0 b3311 0.09543 0.09339 0.09447 1% b4311 0 0 0 b5311 0 0 0 b1312 0 0 0 b2312 0.04284 0.03944 -0.03659 7% b3312 0 0 0 b4312 0.15060 0.14565 -0.13722 6% b5312 0.04557 0.04424 0.04627 5% b1313 0.25292 0.24617 0.23460 5% b2313 0 0 0 b3313 -0.04021 -0.03494 0.03563 2% b4313 0 0 0 b5313 0 0 0 b1314 0 0 0 b2314 0.05792 0.05846 -0.06152 5% b3314 0 0 0 b4314 -0.01420 -0.01106 0.01132 2% b5314 0.01828 0.01915 0.01822 5% b1315 0 0 0 b2315 0.06360 0.06167 0.05803 6% b3315 0 0 0 b4315 0.22313 0.21757 0.20638 5% b5315 -0.03298 -0.02875 0.02942 2% b1321 0 0 0 b2321 0.08446 0.08087 -0.08539 6% b3321 0 0 0 b4321 0.13451 0.13344 -0.12660 5% b5321 0.02016 0.01891 0.01763 7% b1322 0.07839 0.07004 -0.06520 7% b2322 0 0 0 b3322 0.14389 0.13969 0.14099 1% b4322 0 0 0 b5322 0 0 0 b1323 0 0.00001 0 b2323 0.17482 0.17720 0.16891 5% b3323 0 0 0 b4323 -0.11396 -0.11429 -0.11493 1% b5323 0.06470 0.06452 -0.06675 3% b1324 0.10598 0.10382 -0.10963 6% b2324 0 0 0 b3324 -0.01867 -0.01795 -0.01808 1% b4324 0 0 0 b5324 0 0 0 b1325 0.11638 0.10951 0.10342 6% b2325 0 0 0 b3325 0.03422 0.03612 -0.03311 8% b4325 0 0 0 b5325 0 0 0 b1331 0.30355 0.29547 0.29851 1% b2331 0 0 0 b3331 -0.04021 -0.03494 0.03563 2% b4331 0 0 0 b5331 0 0 0 b1332 0 0 0 b2332 0.25015 0.24887 0.25000 0% b3332 0 0 0 b4332 -0.11461 -0.10795 -0.10874 1% b5332 0.02115 0.02293 -0.02096 9% b1333 -0.12792 -0.11056 0.11258 2% b2333 0 0 0 b3333 0.31619 0.30635 0.30539 0% b4333 0 0.00003 0 b5333 0 0 0 b1334 0 0 0 b2334 -0.03246 -0.03198 -0.03205 0% b3334 0 0 0 b4334 0.37384 0.36459 0.37020 2% b5334 -0.01611 -0.01421 0.01451 2% b1335 0 0.00001 0 b2335 0.05950 0.06434 -0.05871 9% b3335 0 0 0 b4335 -0.15999 -0.13457 0.13791 2% b5335 0.17925 0.17269 0.17412 1% b1341 0 0 0 b2341 0.03810 0.03953 -0.03732 6% b3341 0 0 0 b4341 -0.02012 -0.01827 0.01900 4% b5341 0.03179 0.03211 0.03355 4% b1342 0.07805 0.07661 -0.07208 6% b2342 0 0 0 b3342 -0.01867 -0.01795 -0.01808 1% b4342 0 0 0 b5342 0 0 0 b1343 0 0 0 b2343 -0.03228 -0.03385 -0.03388 0% b3343 0 0 0 b4343 0.40462 0.39410 0.37601 5% b5343 -0.01705 -0.01366 0.01403 3% b1344 -0.00736 -0.00582 0.00594 2% b2344 0 0 0 b3344 0.06091 0.06062 0.06154 2% b4344 0 0 0 b5344 0 0 0 b1345 0.11564 0.11445 0.10841 5% b2345 0 0 0 b3345 -0.02607 -0.02237 0.02292 2% b4345 0 0 0 b5345 0 0 0 b1351 0 0 0 b2351 0.05671 0.05304 0.04936 7% b3351 0 0 0 b4351 0.31573 0.30416 0.31875 5% b5351 -0.02991 -0.02495 0.02547 2% b1352 0.23458 0.22045 0.23094 5% b2352 0 0 0 b3352 0.03422 0.03612 -0.03311 8% b4352 0 -0.00001 0 b5352 0 0 0 b1353 0 0.00001 0 b2353 0.18202 0.18104 -0.18693 3% b3353 0 0 0 b4353 -0.16932 -0.12937 0.13328 3% b5353 0.11015 0.10635 0.10249 4% b1354 0.09411 0.09544 0.09092 5% b2354 0 0 0 b3354 -0.02607 -0.02237 0.02292 2% b4354 0 0 0 b5354 0 0 0 b1355 -0.16978 -0.14324 0.14682 2% b2355 0 0 0 b3355 0.29010 0.27197 0.27501 1% b4355 0 0.00001 0 b5355 0 0 0 b1411 0 0 0 b2411 0.02851 0.02771 0.02604 6% b3411 0 0 0 b4411 0.13833 0.13851 0.14054 1% b5411 -0.00367 -0.00358 0.00359 0% b1412 0.06157 0.05785 0.06264 8% b2412 0 0 0 b3412 0.02192 0.02219 -0.02104 5% b4412 0 0 0 b5412 0 0 0 b1413 0 0 0 b2413 0.05792 0.05846 -0.06152 5% b3413 0 0 0 b4413 -0.01420 -0.01106 0.01132 2% b5413 0.01828 0.01915 0.01822 5% b1414 0.07884 0.08042 0.07584 6% b2414 0 0 0 b3414 -0.00328 -0.00304 0.00316 4% b4414 0 0 0 b5414 0 0 0 b1415 -0.01741 -0.01650 0.01680 2% b2415 0 0 0 b3415 0.05144 0.05057 0.05299 5% b4415 0 0 0 b5415 0 0 0 b1421 0.05216 0.04922 0.04641 6% b2421 0 0 0 b3421 0.02454 0.02422 -0.02281 6% b4421 0 0 0 b5421 0 0 0 b1422 0 0 0 b2422 -0.01975 -0.02396 -0.02355 2% b3422 0 0 0 b4422 0.23989 0.22601 0.23077 2% b5422 0.01987 0.01990 -0.01893 5% b1423 0.10598 0.10382 -0.10963 6% b2423 0 0 0 b3423 -0.01867 -0.01795 -0.01808 1% b4423 0 0 0 b5423 0 0 0 b1424 0 0 0 b2424 0.07394 0.07274 0.06903 5% b3424 0 0 0 b4424 -0.00373 0.00075 0.00030 b5424 -0.00357 -0.00245 0.00262 7% b1425 0 0 0 b2425 0.09119 0.09017 -0.09423 4% b3425 0 0 0 b4425 -0.02367 -0.01102 0.01226 11% b5425 -0.01483 -0.01425 -0.01444 1% b1431 0 0 0 b2431 0.04266 0.04314 -0.04045 6% b3431 0 0 0 b4431 -0.01420 -0.01106 0.01132 2% b5431 0.02246 0.02297 0.02172 5% b1432 0.06971 0.07019 -0.06650 5% b2432 0 0 0 b3432 -0.01857 -0.01900 -0.01911 1% b4432 0 0 0 b5432 0 0 0 b1433 0 0 0 b2433 -0.03246 -0.03198 -0.03205 0% b3433 0 0 0 b4433 0.37384 0.36459 0.37020 2% b5433 -0.01611 -0.01421 0.01451 2% b1434 -0.01043 -0.00961 0.00998 4% b2434 0 0 0 b3434 0.06593 0.06552 0.06251 5% b4434 0 0 0 b5434 0 0 0 b1435 0.16363 0.16000 0.16743 5% b2435 0 0 0 b3435 -0.02759 -0.02151 0.02216 3% b4435 0 0 0 b5435 0 0 0 b1441 0.07169 0.07286 0.07382 1% b2441 0 0 0 b3441 -0.00231 -0.00184 0.00188 2% b4441 0 0 0 b5441 0 0 0 b1442 0 0 0 b2442 0.06795 0.06695 0.06802 2% b3442 0 0 0 b4442 -0.00373 0.00075 0.00030 b5442 -0.00238 -0.00116 0.00129 11% b1443 -0.00736 -0.00582 0.00594 2% b2443 0 0 0 b3443 0.06091 0.06062 0.06154 2% b4443 0 0 0 b5443 0 0 0 b1444 0 0 0 b2444 -0.00106 0.00022 0.00009 b3444 0 0 0 b4444 0.10759 0.11005 0.10873 1% b5444 -0.01463 -0.01509 0.01503 0% b1445 0 0 0 b2445 -0.00671 -0.00326 0.00361 11% b3445 0 0 0 b4445 -0.14533 -0.14292 0.14283 0% b5445 0.05128 0.04950 0.05049 2% b1451 -0.01888 -0.01784 0.01790 0% b2451 0 0 0 b3451 0.03635 0.03617 0.03431 5% b4451 0 0 0 b5451 0 0 0 b1452 0 0 0 b2452 0.05590 0.05582 -0.05302 5% b3452 0 0 0 b4452 -0.02367 -0.01102 0.01226 11% b5452 -0.01274 -0.01274 -0.01283 1% b1453 0.09411 0.09544 0.09092 5% b2453 0 0 0 b3453 -0.02607 -0.02237 0.02292 2% b4453 0 0 0 b5453 0 0 0 b1454 0 0 0 b2454 -0.01006 -0.00686 0.00733 7% b3454 0 0 0 b4454 -0.14533 -0.14292 0.14283 0% b5454 0.05417 0.05240 0.05007 4% b1455 0 0 0 b2455 -0.04172 -0.03999 -0.04044 1% b3455 0 0 0 b4455 0.50937 0.46891 0.47971 2% b5455 -0.02072 -0.01554 0.01611 4% b1511 0 0 0 b2511 -0.02944 -0.02877 0.02901 1% b3511 0 0 0 b4511 -0.03644 -0.03392 0.03409 1% b5511 0.08213 0.08050 0.08152 1% b1512 -0.05423 -0.05030 0.05057 1% b2512 0 0 0 b3512 0.03262 0.02977 0.02784 6% b4512 0 0 0 b5512 0 0 0 b1513 0 0 0 b2513 0.06360 0.06167 0.05803 6% b3513 0 0 0 b4513 0.22313 0.21757 0.20638 5% b5513 -0.03298 -0.02875 0.02942 2% b1514 -0.01741 -0.01650 0.01680 2% b2514 0 0 0 b3514 0.05144 0.05057 0.05299 5% b4514 0 0 0 b5514 0 0 0 b1515 0.34824 0.33057 0.31766 4% b2515 0 0 0 b3515 -0.04840 -0.03930 0.04023 2% b4515 0 0 0 b5515 0 0 0 b1521 -0.05386 -0.05109 0.05169 1% b2521 0 0 0 b3521 0.07375 0.06968 0.07309 5% b4521 0 0 0 b5521 0 0 0 b1522 0 0 0 b2522 -0.06766 -0.07177 0.07160 0% b3522 0 0 0 b4522 0.19737 0.18845 -0.17989 5% b5522 0.12450 0.12107 0.12234 1% b1523 0.11638 0.10951 0.10342 6% b2523 0 0 0 b3523 0.03422 0.03612 -0.03311 8% b4523 0 0 0 b5523 0 0 0 b1524 0 0 0 b2524 0.09119 0.09017 -0.09423 4% b3524 0 0 0 b4524 -0.02367 -0.01102 0.01226 11% b5524 -0.01483 -0.01425 -0.01444 1% b1525 0 0 0 b2525 0.24395 0.24177 0.23317 4% b3525 0 0 0 b4525 -0.12652 -0.12069 -0.12194 1% b5525 -0.05897 -0.05460 0.05557 2% b1531 0 0 0 b2531 0.12820 0.12414 0.12960 4% b3531 0 0 0 b4531 0.18158 0.18143 0.17310 5% b5531 -0.03298 -0.02875 0.02942 2% b1532 0.10376 0.09420 0.08797 7% b2532 0 0 0 b3532 0.10470 0.10162 -0.10542 4% b4532 0 -0.00001 0 b5532 0 0 0 b1533 0 0.00001 0 b2533 0.05950 0.06434 -0.05871 9% b3533 0 0 0 b4533 -0.15999 -0.13457 0.13791 2% b5533 0.17925 0.17269 0.17412 1% b1534 0.16363 0.16000 0.16743 5% b2534 0 0 0 b3534 -0.02759 -0.02151 0.02216 3% b4534 0 0 0 b5534 0 0 0 b1535 -0.15396 -0.12432 0.12713 2% b2535 0 0 0 b3535 0.17826 0.16750 0.16188 3% b4535 0 0.00001 0 b5535 0 0 0 b1541 -0.01888 -0.01784 0.01790 0% b2541 0 0 0 b3541 0.02959 0.03017 0.02878 5% b4541 0 0 0 b5541 0 0 0 b1542 0 0 0 b2542 0.05590 0.05582 -0.05302 5% b3542 0 0 0 b4542 -0.03551 -0.02317 0.02485 7% b5542 -0.01483 -0.01425 -0.01444 1% b1543 0.11564 0.11445 0.10841 5% b2543 0 0 0 b3543 -0.02607 -0.02237 0.02292 2% b4543 0 0 0 b5543 0 0 0 b1544 0 0 0 b2544 -0.00671 -0.00326 0.00361 11% b3544 0 0 0 b4544 -0.14533 -0.14292 0.14283 0% b5544 0.05128 0.04950 0.05049 2% b1545 0 0 0 b2545 -0.03584 -0.03575 -0.03594 1% b3545 0 0 0 b4545 0.53806 0.49633 0.47575 4% b5545 -0.02072 -0.01554 0.01611 4% b1551 0.42279 0.40110 0.40683 1% b2551 0 0 0 b3551 -0.05338 -0.04528 0.04646 3% b4551 0 0 0 b5551 0 0 0 b1552 0 0 0 b2552 0.35028 0.33969 0.34263 1% b3552 0 0 0 b4552 -0.14729 -0.13501 -0.13719 2% b5552 -0.05897 -0.05460 0.05557 2% b1553 -0.16978 -0.14324 0.14682 2% b2553 0 0 0 b3553 0.29010 0.27197 0.27501 1% b4553 0 0.00001 0 b5553 0 0 0 b1554 0 0 0 b2554 -0.04172 -0.03999 -0.04044 1% b3554 0 0 0 b4554 0.50937 0.46891 0.47971 2% b5554 -0.02072 -0.01554 0.01611 4% b1555 0 0 0 b2555 -0.16590 -0.15320 0.15563 2% b3555 0 0 0 b4555 -0.20584 -0.14718 0.15310 4% b5555 0.36636 0.34222 0.34399 1% SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 16 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates Figure 6:u11u11u11 fo r anisotropic plate A1 witha_b=1 ; ro ws from top to b ottom a re: E1_E2=1;3;10;20 ; columns from left to right a re:u ,v ,ù u2+v2 was proposed and referred to as Bogner-Fox-Schimt-Castro (BFSC), which has enriched in-place displacements when compared to the conventional BFS element to enable an accurate representation of the second-order displacement ﬁelds. This element has only 4 nodes per element and 10 degrees-of-freedom per node, which are: u,u;x,u;y,v,v;x, v;y,w,w;x,w;yandw;xy. The results herein presented showed that the third-order interpolation of wprovided led to an ultra-fast convergence of the linear buckling eigenval- ues, whereas the third-order interpolation of uandvled to a ultra-fast convergence of the second-order displacement ﬁeldsandconsequentlyofthe bijklfactorsobtainedinKoi- ter’s analysis. Futurestudiesshouldinvestigatethesamestrategyherein adopted to enrich the in-plane interpolation of the BFS ele- menttocreatemodiﬁedversionsoftheTUBA3family[57]. Otherapproachesforenrichingthemembranedisplacement ﬁeld could also be investigated, such as the interpolation covers suggested by Jun et al. [71] for triangular shell el- ements. Another alternative is to use ﬁfth-order Hermite polynomials enhancing the BFS element to also interpolate the rotational strains w;xxandw;yy, keepingC1continuity even with distorted meshes, thus keeping the high conver- gencerateshereinobservedfornon-rectangularmeshes[72]. In the present study, Von Kármán nonlinear kinematics were used to derive the nth–order tensors obtained from the totalpotentialenergyofthesystem. Theeﬀectofusingmore completenonlinearkinematicssuchasSanders[49,52]and Timoshenko & Gere [52] should also be investigated, espe- cially for shells with single and double curvature. The displacement-based notation herein presented sim- pliﬁes formulating and implementing the compatibility be-u11 u12 u13 u14 u15 u22 u23 u24 u25 u33 u34 u35 u44 u45 u55 Figure 7:uijuijuij fo r isotropic plate used in multi-mo dal asymp- totic expansion analysis; columns from left to right a re: u ,v ,ù u2+v2 tween diﬀerent semi-analytical domains [73, 74, 75, 76], even for Koiter’s asymptotic analysis. Such possibilities would allow one to apply semi-analytical methods for stiﬀ- ened panels and shells using eﬃcient displacement repre- sentations such as those provided by Legendre hierarchical polynomials [77, 78, 79, 80]. Finally, the eﬀect of initial imperfections should be in- cluded in the presented formulation to enable asymptotic perturbation analyses, especially important in imperfection sensitive shells. CRediT authorship contribution statement S.G.P.Castro: Conceptualization,methodology,formal analysis, investigation, validation, software, data curation, writing - original draft preparation, writing - editing. E. L. Jansen:Validation, writing - review. SGP Castro, EL Jansen: Preprint submitted to Elsevier P age 17 of 21 Displacement-based fo rmulation of K oiter's metho d: application to multi-mo dal p ost-buckling nite element analysis of plates u11 u12 u13 u14 u15 u22 u23 u24 u25 u33 u34 u35 u44 u45 u55 Figure 8:uijuijuij fo r comp osite plate used in multi-mo dal asymp- totic expansion analysis; columns from left to right a re: u ,v ,ù u2+v2 A. Linear Buckling Analysis In Eq. 5 it is known from the equilibrium condition that ¨[ucucuc;c]=0. Dividing the remaining terms by ñvvvñ: ¨[uuu;c]uuu= ¨¨[ucucuc;c]vvv ñvvvñuuu+1 2 ¨¨¨[ucucuc;c]vvv21 ñvvvñuuu +1 6 iv[ucucuc;c]vvv31 ñvvvñuuu+O.4/=0 (58) Itbecomesconvenienttodeﬁneanormalizedeigenvector uiuiuiati, such that êuiuiui;uiuiuië=1: uiuiui=vvv ñvvvñ(59) Thus, one can change Eq. 58 using the deﬁnition of uiuiuito obtain: ¨[uuu;i]uuu= ¨¨[ucucuc;i]uiuiuiuuu+1 2 ¨¨¨[ucucuc;i]uiuiuivvvuuu +1 6 iv[ucucuc;i]uiuiuivvv2uuu+O.4/=0 (60) FromEq. 60,thefollowingrelationisobtainedwhen  i, i.e.vvv000: ¨¨[ucucuc;i]uiuiuiuuu= ¨¨[iu0u0u0;i]uiuiuiuuu=0 (61) Equation61representstheNeutralEquilibriumCriterion [29], which can be used to compute ianduiuiui, i.e. the lin- ear buckling eigenvalues and eigenmodes, based on a pre- buckling state [u0u0u0;0]. For a system with Ndegrees-of- freedomthereare Npairsi;uiuiui,sothatoneshouldselectthe modeuiuiuithat corresponds to the lowest positive i, named cin the previous discussions for single-mode asymptotic expansion. In most commonly used eigenvalue solvers, the eigenvaluescanbecomputedandsortedinascendantorder, such that only a few ﬁrst pairs of eigenvalues and eigenvec- tors need to be calculated.