JOURNAL OF THEORETICAL AND APPLIED MECHANICS 42, 2, pp. 335-348, Warsaw 2004 BUCKLING OF I-CORE SANDWICH PANELS1 Maciej Taczała Waldemar Banasiak Ship Structures and Mechanics Chair, Technical University of Szczecin e-mail: waldemar.banasiak@ps.pl Necessity of optimisationof shiphull structuralmass calls for application of innovative materials and structural components. One option is based on using structural components with internal structure. The considered sandwich panels are composed of two plates stiffened by vertical ribs (I- core) or ribs of different shape (V-core). Such panels are applied as the ship hull structural components, replacing the conventional stiffened pa- nels. They are subject to typical loadings acting in the ship hull; tension, compression and lateral loading.Analysis of stability of sandwich panels subject to compressive loading is presented in the paper. Stabilities of conventional and innovative ship panels were compared. Influence of the filling foamwas also investigated. Key words: I-core sandwich panels, buckling 1. Introduction Innovative structural components such as sandwich panels have been re- cently applied in shipbuilding.The sandwich panels have proven to havemany advantages over traditional plates; lowweight, modular prefabrication, decre- ase of labour demand etc. The panels are used in production of walls, decks, bulkheads, staircases anddeckhouses on the ships.One of the first examples of application, according to the Lloyd’s Register publication (2000) was a vehicle deck section made of the Sandwich Plate System (SPS), assembled into the RoPax vessel. Presently after a year in service, the ship has not experienced 1 A part of this contribution was presented on the Xth Symposium ”Stability of Struc- tures” in Zakopane, September 8-12, 2003. 336 M.Taczała, W.Banasiak any problems concerning the strength or degradation of the innovative struc- ture. Since the sandwich panels are quite a new application in shipbuilding, the knowledge concerning the behaviour of this type of structure is still insuf- ficient. They are even not referred to by the rules of classification societies. Analysis of sandwich panelswithout core andwith different cores has been a subject investigated and presented by many authors. The local buckling of sandwichpanelsmade up of hybrid laminated faces andflexible core is investi- gated byAiello andOmbres (1997). Stability analysis of sandwich panels with a flexible core is presented byFrostig (1998). The analysis uses high-order the- ory anddetermines the bifurcation loads and local and overall bucklingmodes of panels.Wrinkling analysis of sandwich panels containing holes is presented byHadi andMatthews (2001). Razi et al. (1999) used cylindrical holes tomo- del the sandwich panels with damages. They present an analytical method to determine the stress distribution in panels with arbitrarily located damage. Most sheet faces of sandwich panels aremodelled using two-dimensional plate and cores cosisting of three-dimensional solid elements. The face plates were modelled with a nine-node isoparametric elements based on the Mindlin pla- te theory of bending and vibration analysis presented by Lee and Fan (1996). Philippe et al. (1999) developed a newmodel of sandwich structure referred to as a tri-particle model. The tri-particle solution was compared with the exact Pagano solution and the results were obtained by using the Mindlin-Reissner plate theory. The core material is considered to be isotropic for cellular cores or orthotropic for honeycomb. Review of the analytic solutions for bending and buckling of flat rectangular orthotropic plates is presented by Bao et al. (1997). The experimental research of double skin composite elements under lateral and axial loadswas carried out byOduyemi andWright (1989),Wright et al. (1991). Laser-weldedpanels, knownas I-core panels, producedbyMeyer-Werft shi- pyard inPapenburgwere presented byKozak (2002). He described the tests of sandwichpanelsdevelopedunder theEuropeanUnionProject ”SANDWICH”. The purpose of experimental tests of I-core and V-core sandwich panels is to define the strength properties of such innovative structures. Naar et al. (2002) in their paper analysed the strength of various types of double bottom struc- tures. Among other types compared are conventional ship and steel sandwich structures. Behaviour of fibre-reinforced plastic deck of bridge structures is described byQiao et al. (2000). To simplify the analysis of the bridge deck, an equivalent orthotropic plate was used instead of an exact model of the actual deck geometry. In the present paper stability of the I-core panel under compressive loading using the finite elements method is analyzed. The considered I-core plate is a Buckling of I-core sandwich panels 337 laser-welded steel sandwich panel producedby theMeyer-Werft shipyard.The panel is composed of two thin face plates joined by ribs. 2. Theoretical background A mathematical model is considered for the structure composed of the plate and the filling foam – Fig.1. Fig. 1. Model of compressed I-core panel The following assumptionsaremade: (i)materials of theplate and foamare linear elastic and (ii) strain components are small. Assuming the compressive stresses to act in the direction of X1 axis of the global coordinate system, the principle of virtual work is ∫ Vp σ p ijδεij dV + ∫ Vf σ f ijδεij dV =Pp ∫ l δ∆dx+Ps ∫ l δ∆dx (2.1) where [σ p ij] is the stress tensor for the plate, [σ f ij] – stress tensor for the foam, [δεij] – variation of the strain tensor, Pp and Pf are compressive forces acting in the plate and foam, respectively, δ∆dx – variation of the change of elementary length. Expressions on the left-hand side of Eq. (2.1) are virtual works of stresses in the plate and foam, respectively, and the expressions on the right-hand side are virtual works of compressive loading. Ordering the components of the stress andstrain tensors in the formtypical for the finite element method, Eq. (2.1) can be rewritten as 338 M.Taczała, W.Banasiak ∫ Vp σ p iδεi dV + ∫ Vf σ f i δεi dV =Pp ∫ l δ∆dx+Pf ∫ l δ∆dx (2.2) The compressive forces are given by Pp = ∫ Ap σ p 1 dA Pf = ∫ Af σ f 1 dA (2.3) Material properties are defined by the constitutive matrices [C p ij], [C f ij] used for the definition of stress-strain relationships for the plate and foam σ p i =C p ijεj σ f i =C f ijεj (2.4) It is usually assumed in the analysis of the linearised buckling that the com- pressive stresses are constant over the cross-sectional area of a homogenous material. Since in the present approach we have a combination of two dif- ferent materials, the stresses are assumed to be proportional to the Young moduli of materials σ p 1 =Epε1 σ f 1 =Efε1 (2.5) with the strain being the same for bothmaterials. The change of the elemen- tary length is expressed in the linearised analysis as a function of derivatives of the out-of-plane displacements ∆dx= 1 2 (U22,X1 +U 2 3,X1 ) (2.6) and the variation of the elementary length change is δ∆dx=(U2,X1δU2,X1 +U3,X1δU3,X1) (2.7) where index (·),X1 denotes differentiation with respect to X1. Employing Eqs (2.3)-(2.7), Eq. (2.2) becomes ∫ Vp Cijεjδεi dV + ∫ Vf C f ijεjδεi dV = = ∫ l [ ∫ Ap Epε1(U2,X1δU2,X1 +U3,X1δU3,X1) dA ] dx+ (2.8) + ∫ l [ ∫ Af E f ε1(U2,X1δU2,X1 +U3,X1δU3,X1) dA ] dx Buckling of I-core sandwich panels 339 Displacements U2 and U3 given in the global local coordinate system are re- lated to the displacements {uj} in the local coordinate system via the trans- formation matrix [Tij] U2 =Tj2uj U3 =Tj3uj (2.9) Consequently, U2,X1 =Tj2uj,X1 δU2,X1 =Tk2δuk,X1 U3,X1 =Tj3uj,X1 δU3,X1 =Tk3δuk,X1 (2.10) Employing the chain rule for differentiation uj,X1 = ∂uj ∂X1 = ∂uj ∂xm ∂xm ∂X1 =Tm1uj,xm (2.11) Eq. (2.8) becomes ∫ Vp C p ijεjδεi dV + ∫ Vs C s ijεjδεi dV = = ε1 ∫ Vp Ep(Tj2Tm1uj,xmTk2Tn1δuk,xn+Tj3Tm1uj,xmTk3Tn1δuk,xn)dV +(2.12) +ε1 ∫ Vs Es(Tj2Tm1uj,xmTk2Tn1δuk,xn +Tj3Tm1uj,xmTk3Tn1δuk,xn) dV Two types of finite elements will be employed in the finite element modelling: plate elements for the plating and solid elements modelling the foam. Defor- mations of the plating are consistent with Kirchoff-Love plate theory, while three-dimensional stress and strain is assumed for the foam. Displacement field for the plate is approximated using shape functions of the four-noded rectangular element ui =N p ijdj (2.13) and for the foam eight-noded hexahedral element ui =N s ijdj (2.14) where {dj} is the nodal displacement vector and [N p ij], [N s ij] are matrices of the shape functions for plate and solid elements, respectively. Strains for plates and solids are obtained using thematrices of derivatives of the shape functions εj =B p jqdq εj =B s jqdq (2.15) 340 M.Taczała, W.Banasiak Employing the finite element formulations, Eqs. (2.13)-(2.15), Eq. (2.12) is rewritten ∫ Vp C p ijB p jpdpB p iqδdq dV + ∫ Vp C s ijB s jpdpB s iqδdq dV = = ε1 ∫ Vp E p(Tj2Tm1N p jp,xm dpTk2Tn1N p kq,xn δdq + +Tj3Tm1N p jp,xm dpTk3Tn1N p kq,xn δdq) dV + (2.16) +ε1 ∫ Vs Es(Tj2Tm1N s jp,xm dpTk2Tn1N s kq,xn δdq + +Tj3Tm1N s jp,xm dpTk3Tn1N s kq,xn δdq) dV Since the above equation is true for arbitrary variation of displacements, it follows [ ∫ Vp C p ijB p jpB p iq dV + ∫ Vs C s ijB p jpB p iq dV ] dp = = ε1 [ Ep(Tj2Tm1Tk2Tn1+Tj3Tm1Tk3Tn1) ∫ Vp N p jp,xm N p kq,xn dV + (2.17) +Es(Tj2Tm1Tk2Tn1+Tj3Tm1Tk3Tn1) ∫ Vs Nsjp,xmN s kq,xn dV ] dp which can be written in the form [Kpqp+K s qp− (K Gp qp +K Gs qp )]dp =0 (2.18) where [Kpqp], [K s qp] are stiffness matrices for the plate and solid, respectively, and KGpqp , K Gs qp are geometrical matrices. Equation (2.18) is a typical formu- lation of an eigenvalue problem for linearised buckling, with strain ε1 being the searched value instead of the stress as in the standard formulation. The problem in the present approach is solved by the subspace iteration method, originally developed by Bathe (1996). 3. Numerical examples 3.1. Reference example Cold-formed steel lipped channels investigated numerically byDubina and Goina (1997) were taken as reference cases. Three beamswith sections presen- ted in Fig.2 subject to axial compression were analysed using ANSYS. The Buckling of I-core sandwich panels 341 beams were modelled with four-noded shell elements. Pinned support was as- sumed at the beam ends. In the present analysis, the pinned support was realised by the diaphragms situated at the ends of the beams having thickness 20mm, considerably larger than the thickness of the profileswhich is less than 1.5mm. Model Bl Bf Bw t L [mm] [mm] [mm] [mm] [mm] L36P0280 12.6 37.1 97.2 1.48 279.9 L36P0815 12.7 37.0 97.4 1.48 814.6 L36P1315 12.4 36.9 97.1 1.47 1316.4 Fig. 2. Section of investigated profiles Buckling modes obtained for the present analysis are presented in Fig.3. Fig. 3. Bucklingmode of L36P0280 (a), L36P0815 (b) and L36P1315 (c) 342 M.Taczała, W.Banasiak The referencevalues andresults of thepresent analysis are given inTable 1. Comparing the results we note that the reference cases were analysed in a geometrically non-linear range. The results thus refer to the conjugation of the local and global buckling modes. In the present analysis it is the local buckling mode which was found to be the first buckling mode for models L36P0280 and L36P0815 while the global buckling mode is the first one for model L36P1315. Table 1.Comparison of results Critical force [kN] Model Reference Presented example formulation L36P0280 75 66.9 L36P0815 68 63.6 L36P1315 40 51.1 3.2. Comparison of conventional and I-core ship panels An essential idea of application of the innovative I-core panels in structu- ral ship design is to replace the conventional structures composed of plating, stiffeners and girders. Due to increased stiffness and strength of I-core panels under lateral loading, the stiffeners can be eliminated from the structural de- sign to simplify assembling of the structures – Fig.4. A method of selection of the scantlings of the I-core panel equivalent to the conventional ship panel was described by Pyszko (2002), who considered requirements concerning mi- nimal thickness, sectionmodulus and stability according to theRules ofPolish Register of Shipping. Fig. 4. Conventional ship panel and I-core panel Buckling of I-core sandwich panels 343 Comparative analysiswasperformed for the twopanels: conventional panel of size 2400×2400mmstiffenedwith three angles 100×50×8 and equivalent I-core panel of the same size andscantlings facing thickness 2mm, rib thickness 3mm, distance between facings 50mm. Structural design with application of the I-core panel is more advantageous as its mass is 235.9kg what should be comparedwith themass of the conventional panel – 451.2 kg. Critical stresses are also in favour of the innovative panel – 266.7MPa – against 207.4MPa for the conventional panel. The bucklingmodes of both panels is presented in Fig.5 and Fig.6 Fig. 5. Bucklingmode of conventional ship panel Fig. 6. Bucklingmode of equivalent I-core panel 3.3. Buckling of I-core and V-core panels The presented examples are I-core and V-core panels which were taken from the catalogue of the panel series. The I-core panels (Fig.7) of size 600× 600mmwere analysed assuming that the edges were clamped. The panels were compressed in the direction in accordance with the po- sition of the ribs. Buckling modes of the analysed I-core panels are shown in Fig.8. Another innovative structural design is a V-core panel, where the ribs are not situated vertically but at a certain angle with respect to the facings. The models of such apanel taken for investigation are presented inFig.9.A typical local buckling mode of the analysed V-core panels is shown in Fig.10. 344 M.Taczała, W.Banasiak Type t ts hs p σcr [mm] [mm] [mm] [mm] [MPa] 2/3/40 2 3 40 120 304 3/4/40 3 4 40 120 655 4/5/50 4 5 50 120 1122 5/6/50 5 6 50 120 1722 6/7/60 6 7 60 120 2429 7/8/70 7 8 70 120 3227 8/9/80 8 9 80 120 4112 9/9/90 9 9 90 120 4815 10/10/100 10 10 100 120 5696 Fig. 7. Scantlings of analysed I-core panels Fig. 8. Bucklingmode of panel 2/3/40 (a), 5/6/50 (b), 7/8/70 (c), 10/10/100 (d) Buckling of I-core sandwich panels 345 t ts hs σcr [mm] [mm] [mm] [MPa] 0.8 1 80 73 0.8 1 100 64 2 3 80 492 2 3 100 315 4 5 80 1800 4 5 100 1154 Fig. 9. Dimensions of analysed V-core panels Fig. 10. Typical local bucklingmode of V-core panel 3.4. Buckling of I-core panels filled with foam I-core panels are also offered in a variant in which the structure is filled with foam. Models of such structures were built using plate elements for mo- delling facings and ribs and eight-noded solid elements formodelling the foam. Two types of isotropic foamswere applied: foamwithYoungmodulus 20MPa denoted foam 1 and 100MPa denoted foam 2. Poisson’s ratio in both cases is equal to 0.3. Bucklingmode of I-core panel filledwith foam is shown inFig.11. The comparison of critical stresses of I-core panels with and without foam is given in Table 2. 346 M.Taczała, W.Banasiak Fig. 11. Bucklingmode of I-core panel with foam Table 2.Critical stresses of analysed I-core panels with foam Dimensions of the I-core panel Panel without Panel with Panel with foam foam 1 foam 2 t ts hs p σcr σcr σcr [mm] [mm] [mm] [mm] [MPa] [MPa] [MPa] 2 3 40 120 304 433 781 3 4 40 120 655 796 1063 4 5 50 120 1122 1305 1558 7 8 70 120 3227 3365 3668 Strengthening effect of the foam can be observed. Even the application of the foamwith small elasticmodulus significantly increases the buckling stress. The stabilizing effect of the foam with larger elastic modulus is large and increases the buckling stress for the panels with the thinnest elements almost twice. 4. Conclusions A method of investigation of linearised buckling for structures with ribs modelled by plate elements was presented. The method was implemented in the finite element code. Examples of the analysis of buckling of the I-core and V-core panels subject to compressive loading with a possibility to detect both the overall and local bucklingmodes were given. Advantageous aspects of the application of the I-core panels as compared to the conventional structural design in terms of their stability were indicated. The stabilizing effect of the foam filling the I-core panels was also proven. Buckling of I-core sandwich panels 347 References 1. AielloM.A.,OmbresL., 1997,Localbuckling loadsof sandwichpanelsmade with laminated faces,Composite Structures, 38, 191-201 2. 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Steel Research, 19, 97-110 Wyboczenie paneli typu I-core Streszczenie Optymalizacja masy kadłuba okrętowego wymaga zastosowania innowacyjnych materiałów i elementówkonstrukcyjnych.Możliwe jestwykorzystanie elementówkon- strukcyjnych ze strukturą wewnętrzną. Analizowane w pracy panele składają się z dwóch płyt usztywnionych żebrami pionowymi (I-core) lub żebrami innego kształtu (V-core). Panele takie stosowane są jako elementy konstrukcyjne kadłuba okrętowego zastępując konwencjonalne usztywnione panele. Poddane są one obciążeniu występu- jącemuwkadłubie statku: rozciąganiu, ściskaniu i obciążeniupoprzecznemu.Wpracy przedstawiono analizę stateczności sandwiczowych paneli typu I-core i V-core pod- danych ściskaniu. Porównano stateczność konwencjonalnych i innowacyjnych paneli okrętowych. Zbadano także wpływ piany wypełniającej panele typu I-core. Manuscript received December 30, 2003; accepted for print February 4, 2004