Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 1, pp. 15-24, Warsaw 2013 STRENGTH AND BUCKLING OF SANDWICH BEAMS WITH CORRUGATED CORE Krzysztof Magnucki Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland and Institute of Rail Vehicles “TABOR”, Poznań, Poland Paweł Jasion Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland e-mail: pawel.jasion@put.poznan.pl Marcin Krus, Paweł Kuligowski, Leszek Wittenbeck Institute of Rail Vehicles “TABOR”, Poznań, Poland The subject of the paper is a sandwichbeamwith a crosswise or lengthwise corrugated core. The beam ismade of an aluminiumalloy. The plane faces and the corrugated core are glued together. Geometrical properties and rigidities of the beams are described. The load cases investigated in the work are pure bending and axial compression. The relationship between the applied bending moment and the deflection of the beam under four-point bending is discussed. The analytical and numerical (FEM) calculations as well as experimental results are described and compared.Moreover, for the axial compression, the elastic global buckling problem of the analysed beams is presented. The critical loads for the beams with the crosswise and lengthwise corrugated core are determined. The comparison of the analytical and FEM results is shown. Key words: sandwich beam, corrugated core, strength, elastic buckling, deflection 1. Introduction The theoretical model of sandwich structures was formulated in the middle of XX century. Reissner (1948) presented deflection of a sandwich rectangular plate. Plantema (1966) andAllen (1969) described strength and stability problems of sandwich structures. Libove and Hubka (1951) analysed properties of corrugated-core sandwich panels. Noor et al. (1996) and Vinson (2001) discussed elastic behaviour of sandwich structures. Luo et al. (1992) presented results of analytical analysis of the bending stiffness of a corrugated board and compared them with expressions given by other authors. Cheng et al. (2006) used the finite element method to determine an expression for the equivalent stiffness of sandwich structures with various types of cores.Amoreaccurate expression for the stiffness of corrugated sheetswasderivedbyBriassoulis (1986).McKee et al. (1963) analysedbending stiffness of a sandwichplatewith a corrugated core for three-point and four-point bending tests. Gilchrist et al. (1999) applied the finite element method to analyse bendingand twisting of a corrugated board.Buannic et al. (2003) presented a homogenization method for a sandwich plate with a corrugated core and compared thismethod with results of finite element analysis. Analytical homogenization of a corrugated cardboard under torsion was described by Abbes and Guo (2010). An equivalent stiffness of a corrugated plate under bending and torsion treated as an orthotropic plate was discussed by Liew et al. (2007). Rubino et al. (2009) presented the comparison of dynamic behaviour of a clamped monolithic plate and a clamped sandwich plate with an Y-frame and corrugated core. Seong et al. (2010) showed bending results of sandwich plates with bi-directionally corrugated cores. An application of a sandwich corrugated plate as abridge structurewas describedbyJi et al. (2010). 16 K.Magnucki et al. Wittenbeck andMagnucki (2009) andMagnucki andWittenbeck (2010) determined rigidities of a double-layered cylindrical vessel in which the internal layer was corrugated. Stability of that structure was also investigated. Magnucki et al. (2011, 2012) described behaviour of sandwich beamswith corrugated cores. The authors presented the comparison of results of the numerical- analytical analysis and experimental investigation discussed by Wasilewicz and Jasion (2010). The subject of the present studyare sandwichbeamswith crosswise or lengthwise corrugated cores (Fig. 1).The simply supported sandwichbeamswith the length L carries two concentrated transverse forces F1 oracompressiveaxial force F0.Thefirst configurationof forces andsupports is known as four-point bending and gives a constant bendingmoment over the area between the forces. The distance between the forces F1 is marked as L0, and the distance from the support to the force F1 is marked as L1. Fig. 1. Sandwich beams with lengthwise (a) and crosswise (b) corrugated cores In the following analyses, the relationship between the bending moment and the deflection of the beam with longitudinal and transverse orientation of the corrugation is investigated. The following material and geometrical properties were assumed: E = 69000MPa, ν = 0.3, L=750mm, L1 =L0 =250mm. 2. Rigidities of the sandwich corrugated beams The geometry of the sandwich corrugated beam is shown in Fig. 2. The beam consists of two plane faces of the thickness tf and a corrugated core of the depth tc and the thickness t0. The total thickness of the beam equals h. The pitch of the corrugation equals a0. Fig. 2. The cross section of the sandwich corrugated beam Themiddle surface of the corrugated core is assumed as a sine curve and is described by the following expression fc(ξ)= 1 2 tc(1−x0)sin(2πξ) (2.1) Strength and buckling of sandwich beams... 17 where x0 and ξ are dimensionless variables defined as follows x0 = t0 tc ξ= x a0 The dimensionless middle surface length S0 of one pitch of the corrugated core can be written as S0 = 1 ∫ 0 √ 1+c20cos 2(2πξ) dξ (2.2) where c0 =πtc(1−x0)/a0. Expressions for the total area and themoment of inertia of the cross-section per unit length with respect to the y axis have the following form Ax =(2+k)tf Ix = 1 12 t3c[2x1(4x 2 1+6x1+3)+3x0(1−x 2 0)S1] (2.3) where x1 = tf tc k= x30 4x1[S2x 2 0+3(1−x 2 0)S3] S1 = 1 ∫ 0 sin2(2πξ) √ 1+ c20cos 2(2πξ) dξ S2 = 1 4 ∫ 0 dξ √ 1+ c20cos 2(2πξ) S3 = 1 4 ∫ 0 sin2(2πξ) √ 1+ c20cos 2(2πξ) dξ The expression for the total area and the moment of inertia of the cross-section per unit length with respect to the x axis are given by Ay = tc(2x1+x0S0) Iy = 1 12 t3c [ 2x1(4x 2 1+6x1+3)+ x30 S0 ] (2.4) Thus the flexural rigidity of the sandwich beamwith the corrugated core can be written as Dx =EIy Dy =EIx (2.5) where E is Young’s modulus. 3. Analytical and numerical analysis of bending of sandwich beams From the Euler-Bernoulli beam bending theory the relationship between the bending mo- ment Mg and the deflection function w(y) of the neutral axis, for the beam with longitudinal corrugation, is described by the following formula Dy d2w dy2 =− Mg a (3.1) where a denotes width of the beam. 18 K.Magnucki et al. Assuming cylindrical bending, the radius of curvature R can be determined based on displa- cements of three arbitrary points spaced at an equal distance from each other. Here the distance is marked as c (see Fig. 3). The relationship between the bending moment Mg and the radius of curvature R is in the form 1 R = Mg aDy (3.2) Fig. 3. Schema of the measuring procedure for beam deflection For a small deflection, there is: f ≪ c. Then the stiffness of the beam defined as the ra- tio Mg/f, can be written as Mg f = 2aDy c2 (3.3) Replacement of the variable y by x in (3.2) and (3.3) gives expressions for the maximum deflection and the ratio Mg/f for transverse corrugation. Thestiffness of sandwichbeamsunderpurebendingwasalsoanalysedusingthefiniteelement method. The necessary calculations were carried out by means of ABAQUS system. The glue connection between the outer faces and the core was modelled with the use of tie constraints. The thin shell elements S4Rwere used to elaborate the three-dimensionalmodel of the sandwich beam. The arrangement of loads and supports is illustrated in Fig. 1. Five cases investigated in the work, in which different combination of the face and corrugated core thicknesses are taken into account, are presented in Table 1. Table 1. Investigated cases No. t0 [mm] tf [mm] 1 0.3 1.0 2 0.3 1.5 3 0.3 2.0 4 0.4 1.0 5 0.5 1.0 Other parameters are: width a = 100mm, depth of the core tc = 9.5mm, pitch of the corrugation a0 = 14mm, distance between the reference points c = 50mm. The applied lo- ad F1 = 800N causes a constant bending moment in the central section of the beam equal to Mg =200Nm. Examples of deformation of FE models of sandwich beams with the corrugated core are shown in Fig. 4. The results of analytical and FEM calculations of the beams stiffness are given in Section5 and compared with the results of experimental tests. Strength and buckling of sandwich beams... 19 Fig. 4. Deformation of FEmodels of beams with lengthwise (a) and crosswise (b) corrugated core 4. Experimental bending tests of sandwich beams Experimental investigations of stiffness of sandwichbeamsunder four-pointbendingwere carried out on a special test stand, whichwasmounted on Zwick Z100 testingmachine. The view of the test stand is shown inFig. 5.Details of experimental investigationswere describedbyWasilewicz and Jasion (2010). Fig. 5. View of the test stand The four-point bending was realized by means of a special system of tensioned beams. The load F1 wasmeasured bymeans of a dynamometer.An inductive detector of displacementHBM WA/10mm was used to measure deflection of the beam. The distance between the inductive detectors was c=50mm.The scheme of themeasurement system is presented in Fig. 6. During the tests, the data fromthedynamometer anddetectorswere recorded.Thedependencebetween the bendingmoment and displacement was plotted. 5. Comparison of the results of strength investigation of sandwich beams The results from analytical, numerical and experimental analyses are listed in Tables 2 and 3. Thedependencybetween the thickness of particular layers of the sandwichbeamandthe stiffness of the beam is shown inFigs. 7 and 8 for crosswise corrugation, andFigs. 9 and 10 for lengthwise corrugation. 20 K.Magnucki et al. Fig. 6. Scheme of measuring detectors Table 2.Results of investigations for the crosswise corrugated core (Mg/f [Nm/mm]) No. t0 tf Analyt. FEM Error Exper. Error 1 0.3 1.0 331.22 334.16 0.88% 330.1 −0.34% 2 0.3 1.5 530.08 533.47 0.64% – – 3 0.3 2.0 763.42 767.16 0.49% – – 4 0.4 1.0 339.75 343.35 1.06% – – 5 0.5 1.0 348.17 352.61 1.28% – – Fig. 7. Comparison of analytical, numerical and experimental results for the crosswise corrugated core (thickness of the corrugated core t0 =0.3mm) Fig. 8. Comparison of analytical, numerical and experimental results for the crosswise corrugated core (thickness of the faces tf =1mm) Strength and buckling of sandwich beams... 21 Table 3.Results of investigations for the lengthwise corrugated core (Mg/f [Nm/mm]) No. t0 tf Analyt. FEM Error Exper. Error 1 0.3 1 305.21 307.65 0.8% 304.9 0.1% 2 0.3 1.5 504.05 506.46 0.48% – – 3 0.3 2 737.38 738.27 0.12% – – 4 0.4 1 305.22 308.50 1.08% – – 5 0.5 1 305.24 309.21 1.29% – – Fig. 9. Comparison of analytical, numerical and experimental results for the lengthwise corrugated core (thickness of the corrugated core t0 =0.3mm) Fig. 10. Comparison of analytical, numerical and experimental results for the lengthwise corrugated core (thickness of the faces tf =1mm) 6. Analytical and numerical FEM calculations of elastic global buckling of sandwich beams The simply supported beam is compressed by the axial force F0 (F1 =0, see Fig. 1). According to Euler’s formula, the critical load for the beam is defined in the following form FCR,E = π2EI L2 (6.1) According to that formula, examplary calculations of the critical load have been carried out for the following parameters: • geometry of the cross section: tf =9.5mm, t0 =0.3mm, tc =9.5mm, a0=14mm • length of the beam L=750mm • Young’s modulus E =69000MPa • moment of inertia of the cross section – for the sandwich beamwith the crosswise (CW) corrugated core, in accordance with Eq. (2.3)2 I(CW) =5971.6mm4 22 K.Magnucki et al. – for the sandwich beamwith yhe lengthwise (LW) corrugated core, in accordancewith Eq. (2.4)2 I(LW) =5529.0mm4 The obtained values are shown in Table 4 and compared with those given by the finite element method. The FE model of the beams was the same as that presented in Section 3. The first buckling modes for sandwich corrugated beams are shown in Fig. 11. Table 4.Buckling loads obtained analytically and with the use of the FEmethod CW-crosswise LW-lengthwise F (Anal) CR,E [N] 80688 74707 F (FEM) CR,E [N] 79281 73314 Fig. 11. The first buckling mode of beams with lengthwise (a) and crosswise (b) corrugated core 7. Conclusions In the paper, the strength and stability problems of a sandwich beam with a crosswise and lengthwise corrugated core have been studied. The geometry of the cross section as well as rigidities of the beams have been described. For both load cases, that is for pure bending and axial compression, the analytical and FEM calculations have been carried out. The comparison of the results obtained frombothmethods is shown inTables 2, 3 and 4.Thebiggest discrepancy in the case of pure bending is about 1.3%, and for axial compression – 1.9%. Moreover, for thepurebending load, experimental tests have beenperformed.Thedifferences between the stiffness of the beams obtained in the test and from the analytical model were less than 0.4%. It shouldbenoted that according toTables 2 and3, the stiffness of the beamwith the crosswise corrugated core is higher when compared to the beamwith the lengthwise corrugated core. The difference varies within the range 3-12% depending on the thicknesses of particular layers of the beam. Acknowledgements The investigation has been supported by theMinistry of Sciences andHigher Education in Poland – grant No. 10004706. References 1. Abbes B., Guo Y.Q., 2010, Analytic homogenization for torsion of orthotropic sandwich plates: Application to corrugated cardboard,Composite Structures, 92, 699-706 2. Allen H.G., 1969,Analysis and Design of Structural Sandwich Panels, PergamonPress: Oxford, London, Edinburgh, NewYork, Toronto, Sydney, Paris, Braunschweig Strength and buckling of sandwich beams... 23 3. Briassoulis D., 1986, Equivalent orthotropic properties of corrugated sheets, Computers and Structures, 23, 2, 129-128 4. Buannic N., Cartraud P., Quesnel T., 2003, Homogenization of corrugated core sandwich panels,Composite Structures, 59, 299-312 5. Cheng Q.H., Lee H.P., Lu C., 2006, A numerical analysis approach for evaluating elastic con- stants of sandwich structures with various cores,Composite Structures, 74, 2, 226-236 6. Gilchrist A.C., Suhling J.C., Urbanik T.J., 1999, Nonlinear finite elementmodeling of cor- rugated board,Mechanics of Cellulosic Materials, AMD 231/MD 85, 101-106 7. Ji H.S., Song W., Ma Z.J., 2010, Design, test and field application of a GFRP corrugated-core sandwich bridge,Engineering Structures, 32, 9, 2814-2824 8. Libove C., Hubka R.E., 1951, Elastic constants for corrugated core sandwich plates, J. Struct. Eng., ASCE, 122, 8, 958-966 9. Liew K.M., Peng L.X., Kitipornchai S., 2007, Nonlinear analysis of corrugated plates using a fsdt and amesh free method,Comput. Methods Appl. Mech. Eng., 196, 2358-2376 10. Luo S., Suhling J.C., Considine J.M., Laufenberg T.L., 1992, The bending stiffness of corrugated board,Mechanics of Cellulosic Materials, AMD 145/MD 36 11. Magnucki K., Wittenbeck L., 2010, Stability of elastic orthotropic circular cylindrical ves- sel, Proceedings of the ASME 2010 Pressure Vessels and Piping Division Conference, Bellevue, Washington, USA, PVP2010-25221, 1-7 12. Magnucki K., Krus M., Kuligowski P., Wittenbeck L., 2011, Strength of sandwich beams with corrugated core under pure bending, The 2011 World Congress on Advances in Structural Engineering and Mechanics (ASEM’11+), Volume of Abstracts, Seoul, Korea (CD, 321-330) 13. MagnuckiK.,Kuligowski P.,Wittenbeck L., 2012, Zginanie sprężystychbelek trójwarstwo- wych z rdzeniem falistym,Pojazdy Szynowe, 1, 1-4 14. McKee R.C., Gander J.W., Wachuta J.R., 1963, Flexural stiffness of corrugated board, Paperboard Pack, 48, 149-159 15. Noor A.K., Burton W.S., Bert C.W., 1996, Computational models for sandwich panels and shells,Applied Mechanics Reviews, ASME, 49, 3, 155-199 16. Plantema F.J., 1966, Sandwich Construction: the Bending and Buckling of Sandwich Beams, Plates and Shells, JohnWiley & Sons: NewYork, London, Sydney 17. Reissner E., 1948, Finite deflections of sandwich plates, Journal of the Aeronautical Science, 435-440 18. Rubino V., Deshpande V.S., FleckN.A., 2009,The dynamic response of clamped rectangular Y-frame and corrugated core sandwich plates,European Journal of Mechanics A/Solids, 28, 14-24 19. Seong D.Y., Jung C.G., Yang D.Y., Moon K.J., Ahn D.G., 2010, Quasi-isotropic bending responses ofmetallic sandwich plates with bi-directionally corrugated cores,Materials and Design, 31, 2804-2812 20. Vinson, J.R., 2001, Sandwich structures,Applied Mechanics Reviews, ASME, 54, 3, 201-214 21. Wasilewicz P., Jasion P., 2010, Strength analyses of sandwich beams, Poznan University of Technology, Poznan, Report in Polish, 21-361/2010 22. Wittenbeck L.,MagnuckiK., 2009,Elastic buckling of orthotropic cylindrical vessel,The 12th International Conference on Pressure Vessel Technology, ICPVT-12, Jeju, Korea, Abstract Book, pp16, (CD, 98-100) 24 K.Magnucki et al. Wytrzymałość i stateczność belek trójwarstwowych z rdzeniem falistym Streszczenie Przedmiotempracy sąbelki trójwarstwowezrdzeniemfalistym.Poszczególnewarstwybelekwykonane są ze stopu aluminium i połączone klejem.Wyróżniono dwa kierunki pofałdowania: wzdłużny i poprzecz- ny. Opisano geometrię przekroju poprzecznego i wyznaczono sztywności belek. W pracy rozpatrzono dwa przypadki obciążenia. Pierwszy z nich to czyste zginanie realizowane poprzez tak zwane zginanie czteropunktowe. Dla tego typu obciążenia zbadano zależność między przyłożonym momentem gnącym a ugięciem belki. Dla drugiego przypadku obciążenia, osiowego ściskania, omówiono problem globalnej stateczności sprężystej i wyznaczono obciążenia krytyczne. Ponadto zaprezentowano porównanie wyni- kówuzyskanych z zaproponowanegomodelu analitycznego zwynikami otrzymanymi zmetody elementów skończonych i z badań laboratoryjnych. Manuscript received November 29, 2011; accepted for print February 20, 2012