AN ISOGEOMETRIC SOLID-SHELL FORMULATION OF THE KOITER METHOD FOR BUCKLING AND INITIAL POST-BUCKLING ANALYSIS OF COMPOSITE SHELLS

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1 th Eropean Conference on Comptational Mechanics (ECCM ) 7th Eropean Conference on Comptational Flid Dynamics (ECFD 7) 5 Jne 28, Glasgow, UK AN ISOGEOMETRIC SOLID-SHELL FORMULATION OF THE KOITER METHOD FOR BUCKLING AND INITIAL POST-BUCKLING ANALYSIS OF COMPOSITE SHELLS Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica Università della Calabria, 873, Rende (Cosenza), Italy, giovanni.garcea@nical.it Key words: Bckling, isogeometric analysis, composite shells, Koiter method Abstract. Nmerical formlations of the Koiter theory allow the efficient prediction, throgh a redced order model, of the behavior of shell strctres when failre is dominated by bckling. In this work, we propose an isogeometric version of the method based on a solid-shell model. A NURBS-based interpolation is employed on the middle srface of the shell to accrately describe the geometry and the high continity typical of the displacement field in bckling problems and to directly link the CAD model to the strctral one. A linear interpolation is then adopted throgh the thickness together with a modified generalized constittive matrix, which allows s to easily eliminate thickness locking and model mlti-layered composites. Redced integration schemes, which take into accont the continity of the shape fnctions, are sed to avoid interpolation locking and make the integration faster. A Mixed Integration Point strategy makes it possible to transform the displacement model into a mixed (stress-displacement) one, reqired by the Koiter method to obtain accrate predictions, withot introdcing stress interpolation fnctions. The reslt is an efficient nmerical tool for bckling and initial post-bckling analysis of composite shells, characterized by a low nmber of DOFs and integration points and by a simple and qick constrction of the redced order model. INTRODUCTION The failre of composite shells often occrs becase of bckling phenomena which make them sensitive to material, geometrical and load imperfections []. Thosands of eqilibrim path evalations can be reqired in order to detect the worst imperfection case in terms of failre load. Frthermore, the stacking seqence has proven to strongly affect the bckling and post-bckling response of the shells and the design of an optimal layp can significantly increase the load-carrying capability. Conseqently, the need for an optimization process leads to a frther comptational brden and reqires more efficient tools of analysis and design. For these reasons, a great amont of research has focsed on developing redced order models (ROMs) based on the finite element (FE) implementation [2, 3] of the Koiter theory of elastic stability, capable of frnishing, with an acceptable

2 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea comptational cost, an accrate prediction [] of the limit load vale and the initial postcritical behavior for a very large nmber of imperfections. A solid-shell FE model [4] is particlarly convenient for the constrction of the ROM, since it allows s to avoid the se of finite rotations. Althogh a large nmber of locking free linear solid-shell elements are available, their behavior is not sfficiently accrate when modeling crved geometry and, as a conseqence, a fine discretization is reqired. An interesting alternative is given by the isogeometric analysis (IGA) [5] based on NURBS shape fnctions. IGA seems very attractive in particlar in bckling problems, where a highly continos soltion is often expected [, 7, ]. It has been shown in [3] that the Koiter method reqires a mixed formlation in order to avoid a locking phenomenon in the evalation of the forth-order coefficients of the redced system of eqations and to increase the range of validity of the ROM, which gets worse in displacement formlations when the slenderness of the strctre increases [3] and the pre-bckling path exhibits even small nonlinearities. This aspect can also be observed in path-following analyses [3, 4, 8] where displacement formlations lead to a slow convergence rate of the Newton scheme when slender strctres are analyzed. In Koiter analysis this phenomenon is mch more evident becase the eqilibrim path is directly extrapolated sing the ROM, and an eqilibrim error is not corrected by an iterative scheme, so affecting the accracy of the method. On the contrary mixed formlations avoid this drawback becase the stresses are directly extrapolated. Frthermore, the joint se of a Green-Lagrange strain measre and of a mixed Hellinger-Reissner variational formlation [2, 3], leads to a 3rd order polynomial dependence of the strain energy on the discrete DOFs with the conseqence of the zeroing of all the forth order strain energy variations. In this work, we propose an isogeometric nmerical formlation of the Koiter theory for the analysis of composite shells which exploits the advantages of a solid-shell model. A linear throgh-the-thickness interpolation is considered for geometry and displacements. The nonlinear model is based on a Total-Lagrangian formlation with the se of the Green-Lagrange strain measre. Inspired by the FE approach proposed by Sze [4], the Green-Lagrange strains are linearized along the thickness direction, allowing the definition of a modified generalized constittive matrix which effectively eliminates thickness locking and leads to accrate predictions for mlti-layered composites, withot introdcing additional throgh-the-thickness DOFs. The displacement field and the geometry are rewritten in terms of semi-sm and semi-difference of the top and bottom srface qantities. The model so obtained is described by middle srface coordinates only, allowing s to interpolate geometry and displacements sing bivariate NURBS of generic order and continity. Each control point is eqipped with six DOFs bt, in contrast to traditional shell models, only displacement DOFs are employed. Different patch-wise redced integration rles [9, ], previosly proposed for linear analysis, are investigated in stability problems with the aim of eliminating interpolation lockings and increasing the comptational efficiency when C and C 2 NURBS are adopted. To obtain the mixed description of the problem, reqired by the Koiter formlation, the Mixed Integration Point (MIP) strategy, recently proposed in [8] for finite element path-following analyses, is here extended to the proposed isogeometric Koiter analysis. 2

3 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea 2 KOITER IGA USING MIXED INTEGRATION POINTS 2. The nonlinear model and the nmerical integration We consider a slender hyperelastic strctre sbject to conservative loads p[] proportionally increasing with the amplifier factor. The eqilibrim is expressed by the virtal work eqation Φ[] δ ˆp δ =, U, δ T () where U is the field of configration variables, Φ[] denotes the strain energy, T is the tangent space of U at and a prime is sed to express the Frechèt derivative with respect to. We assme that U will be a linear manifold so that its tangent space T will be independent of. When a mixed format is adopted the configration variables collect both displacement and stress fields. The displacement based IGA formlation previosly presented allows s to express the strain energy of the element as a sm of element contribtions Φ[] e Φ e[d e ] ( ) Φ e [d e ] 2 εt C ɛ ε dω e (2) Ω e where Ω e is the element domain and a nmerical integration is sally adopted. The shell is modeled sing the isogeometric solid-shell model proposed in []. The d-dimensional target space of order p and reglarity r, labeled as Sr p, is exactly integrated by a nmber of ((p r)/2) d integration points per element, distribted over the patch, significantly lower than in standard Gass qadratre rles. Their positions and weights are not eqal for each element, bt are evalated, once and for all, in a pre-processing phase and depend on r, p and patch mesh [, 9]. They also open p new possibilities for patch-wise redced integration schemes. In fact p and r can be selected by the ser and are not reqired to be those for the exact integration of the problem space. If the integration space presents sprios modes, a certain nmber of qadratre points are added near the bondary elements in order to remove them and the approximation space is said to be over-integrated and labeled as S r p [, 9]. The nmber of integration points n can be different element-by-element and the strain energy can be evalated as Φ e [d e ] 2 n ε g [d e ] T C g ε g [d e ] w g (3) g= where sbscript g denotes qantities evalated at the integration point [ξ g, η g ], w g is the prodct of the corresponding weight and the determinant of the Jacobian matrix J evalated at the integration point and C g is C ɛ at the integration point. 2.2 The Koiter method sing mixed integration points The fndamental idea of the MIP strategy [8] is to relax the constittive eqations at the level of each integration point. This is made by rewriting the strain energy in a 3

4 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea psedo Hellinger-Reissner form on the element n ( Φ e [ e ] σ T g ε g [d e ] ) 2 σt g C g σ g w g (4) g= where the stresses at each integration point σ g are now independent variables being σ e =. σ n d e (5) The Koiter method is based on a third order Taylor expansion of Eq.(), in and the modal amplitdes α i [, 2]. Letting i T be a generic variation of the displacement field and denoting with a bold symbol the discrete FEM conterpart of the continm qantities, and referring to the solid-shell finite element model presented in [2], the redced model constrction for the perfect strctre consists of the following steps.. The fndamental path is evalated as f [] = + û, K û = f, K K[ ] (a) where K and f are obtained from the following energy eqivalence T K 2 := Φ 2 T f = p. and reqires the soltion of a linear system to evalate the initial path tangent û. A sbscript will denote, from now on, the point in which the qantities are evalated, i.e. Φ Φ [ ] and so on. 2. The bckling modes and loads are obtained from the linearized critical condition consisting of the eigenvale problem K[] v (K + K [û]) v = (b) where K is obtained from the following energy eqivalence T K 2 = Φ û The (m (m + ))/2 + qadratic corrective FE vectors w ij, ŵ are obtained by the soltion of the linear systems K b ŵ +ˆf + m c kˆf k = with c k = v T kˆf k= K b w ij + f ij + (c) m c kˆf k = with c k = v T k f ij k= 4

5 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea in which K b K + b K, ˆf k = K v k, b is a reference vale of the bifrcation clster, sally the first bckling load and f ij, f are defined as a fnction of modes v i and û by the energy eqivalences δw T f ij = Φ v j v j δw δw Tˆf = Φ û 2 δw 4. The constrction of the redced system of eqations r k [, α i ] µ k [] + ( k )α k 2 2 m i= + m α i α j α h B ijhk =, k = m i,j,h= α i C ik + 2 m α i α j A ijk i,j= (d) is carried ot by evalating the energy terms for i, j, h, k = m, being Φ b 2 = (Φ + b Φ û) 2, 2 as sm of element contribtions A ijk =Φ v i v j v k C ik = Φ bŵw ik B ijhk = Φ b(w ij w hk + w ih w jk + w ik w jh ) µ k [] = 2 2 Φ û 2 v k. The evalation of the eqilibrim path, to be repeated for each additional imperfection, is obtained by solving the modified redced system r k [, α i ] + µ k [, α i ] = where µ k represents the effect of the imperfection, and can be evalated as in []. 2.3 Strain energy variations sing mixed integration points In the following ig = {σ ig, d ie } will denote the vector representation on the integration point g of i. The first variation of (4) is n [ ] T [ ] { Φ σg sgσ sgσ ε g [d e ] C g σ g e = w d e s g with (7a) gd s gd B g [d e ] T σ g. g= and B g [d e ] = L g + Q g [d e ]. The second variation of (4) is Φ e 2 = n { } σ T g B g [d e ]d 2e + σ T 2gB g [d e ]d e + σ T g Q g [d e ]d 2e wg. (7b) g= Letting ε Q gk = dt eψ gk d 2e the kth component of vector Q g [d e ]d 2e we have σ T g Q g [d e ]d 2e k σ gk ε Q gk = dt eg[σ g ]d 2e with G[σ g ] = k σ gk Ψ gk. (7c) 5

6 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea In matrix form the second variation of (4), letting G g G e [σ g ], becomes Φ e 2 = n g= [ σg d e ] T [ C g B T g B g G g ] [ σ2g d 2e ] w g = g T gk g 2g. (7d) The second variation can also be written in vector form introdcing the incremental force vector so defined Φ e 2 = [ ] C T gs g[ 2g ] with s g σ g[ 2g ] 2g + B g d 2e B T (7e) g g σ 2g + G g d 2e The third variation of the strain energy is Φ e 2 3 = n { } σ T g Q g [d 3e ]d 2e + σ T 2gQ g [d 3e ]d e + σ T 3gQ g [d e ]d 2e wg (7f) g= that can also be written in vector form introdcing the secondary force vector as Φ e δ δ 2 δ 3 = [ ] T gs g[ 2g, 3g ] with s Q g[ 2g, 3g ] g [d 3e ]d 2e Q g [d 3e ] T σ 2g + G g [σ 3g ]d 2e g (7g) 3 NUMERICAL RESULTS 3. Composite crved panel nder compression The first test regards a crved panel nder compression whose geometry, loads, and bondary conditions are depicted in Fig.. The material properties can be fond in Table. Two different layps are considered: [] and [45, 45, ] s. The lamination significantly inflences the shape of the bckling modes as illstrated in Fig.2. This is confirmed by Tables 2, 3, 4 and 5 which show the convergence of the first 4 linearized bckling loads. The high continity together with the exact representation of the geometry leads to very good reslts with all the integration strategies. Again, however, the S 2 for C and S 4 and S 3 for C 2 represent the best choices in terms of accracy and efficiency. The stdy of the initial post-bckling behavior of the panel is carried ot considering the presence of a geometrical imperfection ẽ that is a combination of the first and the second bckling modes. In particlar, it is the difference between them scaled in order to obtain ẽ =.t. The Koiter soltion is evalated sing a ROM based on the first two bckling modes only, since the higher ones are far from the first two, and it is compared with reference paths. For both the layps, the initial post-bckling exhibits a limit load as shown in Fig.3 and Fig.4 for [] and in Fig.5 and for [45, 45, ] s. C - S, 2 C 2 - S 3 and C 2 -S 4 are the best performing strategies, providing a good estimate of the limit loads with a 8 8 mesh, which became practically exact sing a mesh.

7 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea l/ C.5 v= v= l/ y, v z, w A B v= v=, w= x, C= R 3 Figre : Composite crved panel: geometry and bondary conditions. E E 22 = E 33 ν 2 = ν 3 ν 23 G 2 = G 3 G Table : Composite crved panel: material properties. 4 elm. 8 elm. elm. S 2 S 3 S 4 ANS S2 S 3 S 4 ANS S2 S 3 S 4 ANS > 2 Table 2: Composite crved panel: first 4 normalized bckling loads for [] with C interpolation. 4 elm. 8 elm. elm. S 2 S 3 S 4 ANS S2 S 3 S 4 ANS S2 S 3 S 4 ANS > 2 Table 3: Composite crved panel: first 4 normalized bckling loads for [45, 45, ] s with C interpolation. 4 elm. 8 elm. elm. S 3 S 4 S S 3 S 4 S S 3 S 4 S Table 4: Composite crved panel: first 4 normalized bckling loads for [] s with C 2 interpolation. 7

8 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea 4 elm. 8 elm. elm. S 3 S 4 S S 3 S 4 S S 3 S 4 S Table 5: Composite crved panel: first 4 normalized bckling loads [45, 45, ] s with C 2 interpolation. (a) mode [] (c) mode 2 [] (b) mode [45/ 45/] s (d) mode 2 [45/ 45/] s Figre 2: Composite crved panel: first and second bckling mode for two layps.5 C - S 2 C -S 3 C -S 4 C -ANS 8 8 elements C - S 2 C -S 3 C -S 4 C -ANS elements Figre 3: Composite crved panel: eqilibrim path for [] and C interpolation 8

9 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea.5 C 2 - S 3 C 2 -S elements.5 C 2 - S 3 C 2 -S 4 elements C 2 -S C 2 -S Figre 4: Composite crved panel: eqilibrim path for [] and C 2 interpolation C - S 2 C - S 2.5 C -S 3 C -S 4 C -ANS 8 8 elements C -S 3 C -S 4 C -ANS elements Figre 5: Composite crved panel: eqilibrim path for [45/ 45/] and C interpolation.5 C 2 - S 3 C 2 -S elements.5 C 2 - S 3 C 2 -S 4 elements C 2 -S C 2 -S Figre : Composite crved panel: eqilibrim path for [45/ 45/] and C 2 interpolation 9

10 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea 2 elm. 24 elm. 48 elm. S 2 S3 S4 ANS S 2 S3 S4 ANS S 2 S3 S4 ANS >2 Table : Laminate composite cylinder: first 4 normalized bckling loads for C. 2 elm. 24 elm. 48 elm. S 3 S4 S S 3 S4 S S 3 S4 S Table 7: Laminate composite cylinder: first 4 normalized bckling loads for C Laminate composite cylinder sbjected to axial compression The cylinder considered in the following and labelled Z33 was manfactred and tested by DLR (German Aerospace Center). Geometry, material properties and bondary conditions are reported in [2]. The lowest bckling loads of the perfect strctre are reported in Table and Table 7 for the C and the C2 interpolation respectively. The corresponding bckling modes are depicted in Fig.7. De to problem symmetries they occr in coples. Also for this test, C -S 2, C2 -S 3 and C2 -S4 trn ot to be particlarly accrate and provide good reslts with a relatively coarse mesh in comparison with those sally employed in the FE literatre [2]. In evalating the initial post-bckling behavior a load imperfection is introdced by a concentrated force halfway p the cylinder axis. By inclding jst one mode in the ROM, a good prediction of the limit load is obtained as shown in Fig.8 and Fig.7. (a) mode and mode 2 (b) mode 3 and mode 4 Figre 7: Laminate composite cylinder: first 4 bckling modes.

11 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea C - S 2 C -S 3 C -S 4 C -ANS C C C - S 4 2 C 2 - S 3 C 2 -S 4 C 2 -S C C Figre 8: Laminate composite cylinder: eqilibrim path sing 24 9 mesh 4 CONCLUSIONS This work explored the se of IGA for the constrction of a redced model for composite shells ndergoing bckling. The reslts showed that the isogeometric model is able to frnish an excellent approximation of the bckling loads, the limit loads and the initial post-bckling behavior employing a very low nmber of DOFs and a redced nmber of integration points. In particlar the C - S 2 and C 2 - S 3 approaches proves to be convenient becase it reqires jst one integration point per element withot sprios modes. REFERENCES [] G. Garcea, F. S. Ligori, L. Leonetti, D. Magisano, A. Madeo, Accrate and efficient a posteriori accont of geometrical imperfections in Koiter finite element analysis, International Jornal for Nmerical Methods in Engineering 2 (9) (27) doi:.2/nme.555. [2] D. Magisano, L. Leonetti, G. Garcea, Koiter asymptotic analysis of mltilayered composite strctres sing mixed solid-shell finite elements, Composite Strctres 54 (2) doi:./j.compstrct [3] D. Magisano, L. Leonetti, G. Garcea, Advantages of the mixed format in geometrically nonlinear analysis of beams and shells sing solid finite elements, International Jornal for Nmerical Methods in Engineering 9 (9) (27) doi:.2/nme [4] K. Sze, W. Chan, T. Pian, An eight-node hybrid-stress solid-shell element for geometric non-linear analysis of elastic shells, International Jornal for Nmerical Methods in Engineering 55 (7) (22) doi:.2/nme.535. [5] Y. B. J. Astin Cottrell, Thomas J. R Hghes, Isogeometric Analysis: Toward Integration of CAD and FEA, 29. doi: [] G. Garcea, R. Gonçalves, A. Bilotta, D. Manta, R. Bebiano, L. Leonetti, D. Magisano, D. Camotim, Deformation modes of thin-walled members: A comparison be-

12 Leonardo Leonetti, Domenico Magisano, Francesco Ligori and Giovanni Garcea tween the method of generalized eigenvectors and generalized beam theory, Thin- Walled Strctres (2) doi:./j.tws [7] G. Garcea, L. Leonetti, D. Magisano, R. Gonçalves, D. Camotim, Deformation modes for the post-critical analysis of thin-walled compressed members by a Koiter semianalytic approach, International Jornal of Solids and Strctres - (27) doi:./j.ijsolstr [8] D. Magisano, L. Leonetti, G. Garcea, How to improve efficiency and robstness of the Newton method in geometrically non-linear strctral problem discretized via displacement-based finite elements, Compter Methods in Applied Mechanics and Engineering 33 (27) doi: [9] K. A. Johannessen, Optimal qadratre for nivariate and tensor prodct splines, Compter Methods in Applied Mechanics and Engineering 3 (27) 84 99, special Isse on Isogeometric Analysis: Progress and Challenges. doi: [] C. Adam, T. Hghes, S. Boabdallah, M. Zarrog, H. Maitornam, Selective and redced nmerical integrations for NURBS-based isogeometric analysis, Compter Methods in Applied Mechanics and Engineering 284 (25) doi:./j.cma [] Leonardo Leonetti and Francesco Ligori and Domenico Magisano and Giovanni Garcea, An efficient isogeometric solid-shell formlation for geometrically nonlinear analysis of elastic shells, Compter Methods in Applied Mechanics and Engineering 33 (28) doi: [2] Magisano, D. and Liang, K. and Garcea, G. and Leonetti, L. and Ress, M., An efficient mixed variational redced-order model formlation for nonlinear analyses of elastic shells, International Jornal for Nmerical Methods in Engineering 3 (4) (28) doi:.2/nme