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Machine Direction Strength Theory of Corrugated Fiberboard

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Machine Direction Strength Theory of Corrugated Fiberboard Thomas J. Urbanik1 Machine Direction Strength Theory of Corrugated Fiberboard REFERENCE: Urbanik.T.J.,“Machine Direction Strength Theory of Corrugated Fiberboard,” Journal of Composites Technology & Research, JCTRER, Vol. 18, No. 2, April 1996. pp. 80-88. ABSTRACT: Linerboard elements between the corrugations of corru- gated fiberboard can be viewed as short wide columns when the fiber- board is loaded perpendicular to the axes of the corrugations. Column ends are elastically restrained by the corrugated medium. A theory of buckling of nonlinear corrugated fiberboard material, with compression perpendicular to the corrugation axes, was developed. This theory is consistent with a previous theory applied to fiberboard wish compres- sion parallel to the corrugations. The theory matched strength data of corrugated fiberboard using paper compression strength, extensional stiffness, and bending stiffness data as inputs. The theory was further improved by empirically correcting for interactions between material crush failure and structure buckling failure. The correction equation predicts an optimum form of the linerboard stress-strain curve from initial slope and maximum stress data and predicts an element slender- ness that varies with the mode of failure. KEYWORDS: plate structure, elastic stability, buckling, fiber- board, paper The fabrication of corrugated fiberboard yields a sandwich struc- ture in which a linerboard material is glued to a corrugated medium. French researchers [3] experimentally corroborated the importance The direction of rnachining (MD) coincides with the fiber align- of MD paper strength and advocated maximizing the geometric ment of the paper and is perpendicular to the principal axes of the mean of MD and CD paper strengths to maximize box strength. corrugations (Fig. 1). The direction parallel to the corrugation axes When optimizing paper properties to maximize top-to-bottom is called the cross-machine direction (CD). Standard corrugating box strength, the user should consider the end-use loading condi- geometries, A, B, C flute (Fig. 1), have traditionally been tions of the box and insure that MD fiberboard strength does not employed, although it is beeoming increasingly popular to produce suffer. The use of linerboard marketed as a “high strength” material optimum geometries with respect to paper properties. Corrugated and the greater options for customizing the corrugating geometry fiberboard boxes are normally stacked in the top-to-bottom orienta- make this issue particularly relevant. Also, in considering MD tion where fiberboard compression strength in the CD parallel to ECT strength, the interaction between linerboard and corrugated the corrugations relates to box strength. To facilitate the tilling medium becomes more critical. and dispensing of interior packages, boxes are sometimes stacked Buckling theory can be used to prove that linerboard strength side-to-side or end-to-end or handled as unit loads by clamp trucks in CD-loaded corrugated fiberboard is a function of corrugated where MD strength becomes equally important. medium stiffness and that its strength can be increased by, at most, Performance-based changes made to shipping regulations for 1.75 times due to rigid support from the corrugating medium. With corrugated boxes have motivated linerboard producers to maximize MD loading of fiberboard the linerboard becomes more unstable, both CD paperboard strength and the expected edgewise crush test but its strength can be increased up to 4 times due to rigid support (ECT) strength of the combined board. Researchers at the Forest from the corrugating medium. These predictions, between the Products Laboratory applied buckling theory to CD paperboard extreme conditions of zero and infinite medium rigidity, quantify [1] and to CD corrugated fiberboard [2] and predicted the impor- the importance of medium stiffness to the performance of liner- tance of both MD and CD stress-strain properties. Subsequently, board material with CD and MD loading of corrugated fiber- board. 1Research engineer. USDA Forest Service, Forest Products Laboratory, Madison, WI 53705-2398. The Forest Products Laboratory is maintained Objective and Scope in cooperation with the University of Wisconsin. This article was written and prepared by U.S. Government employees on official time, and it is During paper production the compression strength of paper sam- therefore in the public domain and not subject to copyright. ples measured off-line is commonly used es a quality control 1996 by the American Society for Testing and Materials 80 URBANIK ON CORRUGATED FIBERBOARD 81 criterion. Other paper srress-strain properties that contribute to along the linerboard-medium attachment points. Bending stiffness box strength might fortuitously correlate with paper strength. An moduli of each microplate were functions of respective plate curva- understanding of how linerboard material and corrugating medium tures in the x and y directions related to fiberboard strain in they material affect box strength (according to mechanistic principles) direction. Element stiffness coefficients Kij related the nodal rota- can provide the rationale to rank various paper properties by impor- tion along a y- direction plate edge to the external y- direction force tance and manage quality control. The objective of this paper is and varied nonlinearly with strain and bending moduli. Finite to broaden the buckling theory previously applied to CD-loaded element stiffness matricies for each linerboard microplate and corrugated fiberboard and thereby predict the strength of MD- medium microplate were added to construct a global structure loaded fiberboard. The global buckling of corrugated fiberboard matrix for the corrugated subsection. The lowest compressive strain pastels in a box is not analyzed in this report. For such an analysis, that yielded a 0-value determinant of the global stiffness matrix the theory developed in this paper can provide a prediction of corresponded to buckling. material strength for consideration when boxes are loaded in the The following general linear equation for the buckling perturba- MD of the corrugated fiberboard. tion of a thin plate with curvatures in the x and y directions caused In the treatment of the mechanics of honeycombs [4], cellular by general loads is obtained by substituting Eq 2 into Eq 1 walls were considered to buckle like columns with elastic ends when the honeycomb structure was compressed in a plane perpen- dicular to the cell axes. In the theory of corrugated fiberboard developed here, linerboard elements are considered to buckle by the same mechanism, but a nonlinear material characterization applicable to paper is added. An interaction between material crushing and buckling, found applicable to wood columns in [5] is used to broaden the failure mechanism and correct for apparent deviations from buckling theory. The solution technique applied to Eq 3 for CD buckling in Ref 2 Localized Buckling Theory was to reduce its form to a characteristic equation having four Background Terminology roots and to derive expressions for the Kij in terms of the roots. This paper implements the same technique to analyze MD buckling, but Several equations and terminology from previous publications first makes the general perturbation more specific. are useful in the development of equations used in this paper. In The stress-strain relation proposed by Johnson and Urbanik [6] the nonlinear theory for elastic plates [6], on which this paper is 2 and used here to characterize the MD edgewise compression of based, equation (2.37) in the form paper is (4) where a is stress, El is MD strain, and d l and d 2 are material constants (Fig. 2). Equation 4 is written for the MD of paper so that d, and d2 are appropriate to that direction. The same form of relation with different material constants holds for uniaxial where x and y are material Cartesian coordinates, Nij force resul- compression in any dkection. Constants d 1 and d 2 are used to tants, Mij moment resultants, and w is the transverse displacement, distinguish them fmm constants c1 and c2 used for the CD. gives the equation of transverse force equilibrium. When applied to a component of corrugated fiberboard, the x direction is in the MD of paper; the y direction is in the CD. The moment resultants Nonlinear Beam-Column Formulas used here are derived from the moment-curvature relations given A subsection of corrugated fiberboard (Fig. 3) can be treated by equation (2.34) of reference [6] as as a repeating sequence of linerboard microplates and core where h is plate thickness and H ij bending stiffness moduli. The CD edgewise crush analysis in Johnson and Urbanik [2] considered the elastic stability of a subsection of corrugated fiber- board characterized by a repeating sequence of microplates joined 2All equations with decimal notation represent expressions from the referenced sources. 82 JOURNAL OF COMPOSITES TECHNOLOGY & RESEARCH FIG. 4- Sign convention for positive edge moments and positive slopes along element edge. In the equations taken from Ref 6, Subscript 1 relates to the y direction and Subscript 2 relates to the x direction. The general solution to Eq 5 is given by FIG. 3— Structural section with flat plate elements. The x direction is aligned with the MD: the y direction is aligned with the CD. in which ß is given by microplates subjected to compression and bending. When a section of corrugated fiberboard is compressed in the plane perpendicular (9) to
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