Buckling Analysis of Corrugated Core Sandwich Panels

Transcription

1 Buckling Anali of Corrugated Core Sandwich Panel Toma Nordtrand SCA Reearch, Bo 3054, Sundvall, SWEEN Leif A. Carlon epartment of Mechanical Engineering, Florida Atlantic Univerit, Boca Raton FL 3343, U.S.A. ABSTRACT In an effort to improve tructural deign of corrugated board package under compreion load, buckling anali of implupported corrugated board panel, which contitute the main loadbearing component of a compreion loaded bo, ha been performed. Thi paper focue on prediction of effective (homogenied propertie of the corrugated core and the critical buckling load of a implupported board panel. An improved buckling load prediction ha been obtained b incorporation of the additional moment produced b tranvere hear deformation in the governing differential equation for equilibrium. The buckling load prediction are compared to previou analtical formulation, finite element anali and eperiment. INTROUCTION Corrugated board i a andwich emploed in a large variet of package. The dominating ue, however, i in boe ued in tranportation and torage. In order to predict the trength of boe under topload compreion, the buckling and collape of impl upported corrugated board panel, which are the main loadbearing component of a regular bo mut be well undertood. The corrugated core tpicall ehibit low tranvere hear tiffne perpendicular to the corrugation [] which ma reduce the buckling load and load bearing capabilit. Claical elatic buckling theor of orthotropic andwich plate [] ha been modified herein b incorporating of the additional moment produced b the membrane tree in the plate due to tranvere hear train in the governing equilibrium equation. Such an improvement wa firt included in the buckling theor of helical pring b Harin [3] and later applied to hear deformable plate b Bert and Chang [4] although their anali contain ome inconitencie that are corrected herein. Analtical prediction are compared to claical andwich theor, finite element calculation and eperiment. ue to the limited pace in thi paper, we will onl preent a brief outline of anali. For more detail, a more etenive paper i under preparation b the author [5]. ANALYSIS Figure how a corrugated board panel of thickne H and the Carteian coordinate tem., and 3 denote the curvilinear coordinate tem of the corrugated core o that the and ae are parallel and coincide with the core pipe. The principal ae of elatic mmetr of the face heet are aligned with the Carteian coordinate tem. For anali of the buckling load of a panel, it i necear to determine the etenional, bending, and H c t c p t f 3 z b,m,c Figure. Schematic of corrugated board

2 tranvere hear tiffnee, A ij, ij, and S 44 and S 55, where i,j =, and 6. Anali of the tiffnee ha been preented in Ref. [,7,]. The edge of the panel parallel to the ai are compreed uniforml b a line load of intenit, p, per unit length, ee Fig.. z b p a Figure. Schematic of a impl upported panel in edgewie compreion. The plate i impl upported at the edge, i.e. no rotational retraint and outofplane diplacement at the edge are allowed, but the are free to move inplane. Furthermore, the deflection and lope are aumed to be mall compared to the thickne of the plate. Tranvere hear deformation of the plate i accommodated b conidering the common aumption that croection remaining traight but not necearil normal to the midplane of the plate during bending []. Figure 3 how a croectional view of a deformed plate element in the zplane. d β N M w Q Q Q d M M d d z N N d w β β d w d Figure 3 Croection of a differential plate element The tranvere hear train at the left and right croection, z and z ( z d are indicated in Fig. 3. Becaue the inplane force are much larger than the tranvere load, the deformation of the plate require conideration if equilibrium i to be fulfilled in zdirection. After algebraic manipulation the end reult i obtained a,

3 q Q Q z β β N N N β β = 0 ( where q i an applied lateral preure in the zdirection. The main difference between the preent theor and the claical theor of Wiedmann [] lie in the derivation of moment equilibrium around the and ae of the differential element in Fig.3. Becaue the tranvere hear train are large the moment produced b the inplane force can not be neglected. Thu moment equilibrium around the ai in Fig. 3 i a follow M Q M M d d M d M d M d d Q z N z d dd N d dd z d = 0 z ( If both ide in eq. ( i divided b dd and letting d 0 and d 0, eq. ( reduce to M M Q N = 0 z z (3a Similarl moment equilibrium around the ai ield M M Q N = 0 (3b Thu moment equilibrium around the and ae along with the lateral force equilibrium of the plate form a tem of three imultaneou differential equation in term of the outofplane diplacement w and the tranvere hear train, at the centre point. z For the ubequent anali, the following dimenionle ratio of a andwich plate ued b Plantema [] are convenient, ζ =, ψ =, η =, S k z z =, p Sk = (4 p where, p, i the load intenit (load/unit with, and k i given b, k = b nπ p (5 Other parameter ued in the anali are the plate apect ratio a/b and contant A and B that determine the buckling mode defined b the integer m and n, i.e. the panel buckle into m and n half ine wave in the and direction. na =, mb mπ A =, a nπ B = b (6 3

4 The following anali i a corrected verion of the anali of Bert and Chang [4]. According to Navier procedure [], a olution of the preceding problem can be obtained b auming that the outofplane diplacement, w, and tranvere hear train and can be repreented b double trigonometric erie. In the preent anali a one z term olution i ued for each diplacement. Subtitution in the differential equation lead to the following tem of equation hown in matri form, k η ζ ζ η A A ζ A ( k ψ k B B B ( η ψ ( η ψ ζ z ψ z W 0 Γz = 0 (7 Γ Solution of eq. (7 different than the trivial one, W = Γ = Γ = 0 are onl poible when the determinant of the matri vanihe. Thi criterion lead to a econd order equation in k z 0 Pk Qk R = 0 (8 where ψ P = ζ z (9a ( ψ ζ η ( ζ η ( η ψ Q = P (9b R = ζ ( ζ η ψ O PO P ζ η (9c ζ and O = ψ ζ η η ζ (9d The olution of eq. (8 i thu Q = ± P Q 4P k R P (0 where onl the poitive value of k i valid. Uing the epreion (9, eq. (0 become 4

5 Q P = ψ ζ η ψ ζ z ( ζ η ( η ψ z (a R P = ψ ζ η η ( ζ η ζ ζ η (b ζ ψ ψ ζ ζ z z z It ma be hown [5] when the tranvere hear tiffnee become large that the buckling coefficient k, determined b eq. (0, i reduced to the buckling coefficient for orthotropic plate [] k = ς η ( ς ACKNOWLEGEMENT Financial upport provided to the firt author SCA Reearch i gratefull appreciated. Participation of the econd author acknowledge upport from the epartment of Mechanical Engineering of Florida Atlantic Univerit. REFERENCES. T. M. Nordtrand, H. G. Allen and L. A. Carlon, Tranvere Shear Stiffnee of Structural Core Sandwich, Compoite Structure, No. 7, 994, pp F.J. Plantema, Sandwich Contruction, John Wile & Son, Inc., New York, N.Y., J. A. Harin, On Buckling and the Lateral Rigidit of Helical Compreion Spring, Proceeding of the Konjlike Nederland Akademie Wettenchappen, Vol. 45, 94, pp , , Amterdam, Holland 4. C.W. Bert and S. Chang, ShearFleible Orthotropic Plate Loaded InPlane, Journal of Engineering Mechanic iviion, Proceeding of the American Societ of Civil Engineer, Vol. 98, No. EM6, pp , ecember, L. Carlon, C. Feller and P. Jonon, Bending Stiffne of Corrugated Board with Special Reference to Unmmetrical and MultiWall Contruction, (in German, a Papier, Vol. 39, No. 4, pp , 985 5

6 6 T. Nordtrand, Parametrical Stud of the Potbuckling Strength of Structural Core Sandwich Panel, Compoite Structure, Vol. 30, pp. 4445, C. Libove and R. E. Hubka, "Elatic Contant for Corrugated Core Sandwich Plate", NACA, TN 89, G. A. Baum,. C. Brennan and C. C. Habeger, "Orthotropic Elatic Contant of Paper", Tappi, Vol. 64, pp. 970, R. Paetow and L. Göttching, Poion Ratio of Paper, a Papier, Vol. 6, pp. 937, H. G. Allen and R. A. Shenoi, Fleural Fatigue Tet on Sandwich Structure, Second International Conference on Sandwich Contruction, March 9, Univerit of Florida, Gaineville, 99. T. Nordtrand and L.A. Carlon, Evaluation of Tranvere Shear Stiffne of Structural Core Sandwich Plate, Compoite Structure, Vol. 34, 997. S. Timohenko and S. WoinowkKrieger, Theor of Plate and Shell, McGrawHill, ANSYS Uer Manual, 989, Swanon Anali Stem Inc., Vol. 4. S. Ahmad, B. M. Iron and O. C. Zienkiewicz, Anali of Thick and Thin Shell Structure b Curved Finite Element, International Journal of Numerical Method in Engineering, No., 970, pp T. Nordtrand and L. A. Carlon, On the Anali of Buckling of Sandwich Panel, under preparation. 6