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Effect of Paperboard Stress-Strain Characteristics on Strength Of United States Department of Agriculture Effect of Paperboard Forest Service Stress-Strain Characteristics Forest Products Laboratory on Strength of Singlewall Research Paper FPL 401 Corrugated Fiberboard: A Theoretical Approach Abstract The stress-strain relationship for paperboard loaded in edgewise compression relates to the strength of singlewall corrugated containers. This relationship can be approximated from the paperboard characteristics of stress measured at the maximum load and the initial modulus of elasticity. Based on typical characteristics for both linerboard and corrugating medium material a design matrix is constructed for a factorial analysis. Using a computer, various stress-strain relationships are paired together like they might be on the cor- rugator, and the theoretical effects of the stress-strain characteristics are investigated. Computer drawn design curves show how these linerboard and medium characteristics affect combined board edgewise com- pressive strength and box top-to-bottom compressive strength. The interaction between the stress-strain characteristics and paperboard thickness is used to suggest new criteria for evaluating paperboard. United States Department of Agriculture Effect of Paperboard Forest Service Stress-Strain Characteristics Forest Products Laboratory1 on Strength of Singlewall Research Paper Corrugated Fiberboard: FPL 401 July 1981 A Theoretical Approach By THOMAS J. URBANIK, Engineer Introduction Koning (4) expanded on McKee, et al.'s formula to ex- amine the effects of linerboard characteristics on con- There are increasing demands on corrugated con- tainer strength. Though good correlation was obtained tainers to perform successfully in the service en- between theoretical and experimental values (5), ex- vironments. As the container’s environment is better tending the solution process which relied on an ap- defined, it becomes more rational to evaluate the con- proach by Moody (8) is inefficient for investigating the tainer based on end use performance. Design criteria large number of possible combinations of paperboard. that relate the corrugated components to the container’s performance in say, the warehouse environ- Johnson, Urbanik, and Denniston (3) devised an effi- ment, are likewise a more rational evaluation of paper- cient computer model that predicts the edgewise com- board. This report was written to characterize paper- pressive strength of singlewall from the compression board and then relate it to the container’s important stress-strain relationships for both facing and medium performance requirement of short term top-to-bottom material in the cross-machine direction (CMD) and the compressive strength. It is limited to balanced structural configuration of the board (fig. 1). They used singlewall under ideal test conditions. it to study the effect of combining different thicknesses of linerboards and corrugating mediums on the Measuring container compressive strength is probably edgewise compressive strength of A-flute corrugated. the most accurate evaluation of how well it will perform The results show how to balance facing and cor- in the warehouse. Full-scale container testing is not rugating medium thicknesses to minimize total fiber always practical and thus engineering models have weight and maximize edgewise compressive strength been devised that relate the components’ behavior to based on one characteristic pair of paperboard stress- box strength. The model developed by McKee, Gander, strain curves. This previous research thus evaluated and Wachuta (7)2 relates certain box and corrugated paperboard based on thickness but treated just one set board parameters, such as edgewise compressive of CMD stress-strain properties. strength, bending stiffness, and box perimeter, to con- tainer strength. Based on the alternate approaches to The purpose of this report is to extend the work by combining various paperboards and maximizing box Johnson, et al., and quantify how corrugated fiberboard strength, this model does not adequately explain the changes strength due to differences in linerboard and paperboards’ contribution to box compressive strength. corrugating medium stress-strain curves, and the in- teractions with thickness. To do this it was necessary 1 Maintained at Madison, Wis., in cooperation with the Univer- to numerically characterize stress-strain curves, thus sity of Wisconsin. making it possible to efficiently analyze different 2 Italicized numbers in parentheses refer to literature cited at curves. Then using experimental results to provide end of report. typical curves, numerical values were calculated using and compressive strength of paper (1, 6, 9, 11). It seem- ed reasonable to numerically quantify stress-strain curves with the familiar characteristics of the initial modulus of elasticity (MOE) and su . To obtain the relations between su and MOE and the stress-strain curve, this report characterizes the curve with a special form of the stress-strain model reported in (3) having 2 instead of 3 parameters and then relates su and MOE to these parameters. The model is s(e) = C1 tanh (C2e /C1) (1) Figure 1.–Cross sectional model of singlewall corrugated fiberboard. (M 149 722) where s is the predicted stress at a strain e; C1 is a horizontal asymptote that s approaches for very large e; a 2-parameter stress-strain model, and the resulting 4 C2 is the initial slope of the curve. Since MOE is the values used in a 2 factorial experiment to determine derivative of s relative to e evaluated at zero, the effects of these different paperboards on A-flute corrugated strength. Results are presented with com- 2 s´ (e) = C2/cosh (C2e/C1) (2-a) puterized plots, and these plots facilitate strength calculations for arbitrary theoretical stress-strain and at e = 0 curves. An additional benefit from the plots is that they provide theoretical information about the buckling and s´(0) = C2 = MOE (2-b) compression behavior of corrugated fiberboard and ex- plain how paperboard stress-strain characteristics in- Thus having a characteristic MOE automatically gives teract with thickness to initiate failure in either the fac- C2. To automatically determine C1 from the ing or medium component, thus suggesting new criteria characteristic parameters the correlation f between C1 for evaluating paperboard. This report also examines and su, where the effects on box top-to-bottom compressive strength as compared with those related to edgewise com- (3) pressive strength. was determined for a specific set of data and used. Us- The computer program developed in (3) is used here to ing su and MOE as experimental variables depicting enable over 10,000 computations to be made. Member each component, and solving eqs. (2-b, 3), parameters dimensions for the figure 1 structure are calculated C1, C2, and su for each component were thus entered to from the paperboard thicknesses and flute profile. the computer program to characterize the stress-strain Details are given in the appendix. behavior and analyze for buckling or compression failure in the fiberboard structure. Characterizing the Compression Stress-Strain Curve Experimental Check of Characterization With a method to numerically characterize stress- Stress-Strain Model strain curves in terms of su and MOE, it was necessary The computer program in (3) predicts edgewise com- to determine some reasonable values for use in the fac- pressive strength from certain fiberboard properties. torial design and check the accuracy of the Having the linerboard thickness, corrugating medium characterization. To do this, 13 linerboards and 9 cor- thickness, flutes per foot, flute height, and take-up- rugating mediums, with replicates, were tested for factor determines the widths of the paperboard edgewise compression in the cross-machine direction. elements between flute tips in figure 1. Then the buck- The paperboards were constrained against buckling in ling mode of failure of these elements is analyzed from a support fixture described by Jackson (2). Nominal their widths and their CMD stress-strain properties basis weights ranged from 26 through 90 pounds per which are entered to the program by parameters to a 1,000 square feet. Experimentally determined weights stress-strain model characterizing each component. To and effective thicknesses measured, using the Forest check for a compression mode of failure in an element, Products Laboratory (FPL) stylus apparatus (10), are the ultimate stress (su) is also entered for each compo- given in table 1. A linear regression of the table 1 data nent. Thus for input values to the 24 factorial experi- gives the basis weight W and thickness t relationship, ment it was necessary to numerically quantify the assuming equal slopes, as paperboard stress-strain curve so both buckling and 2 compression modes could be investigated; in essence, (lb11,000 ft ) = 2.224 + 3,562 (in.) (4-a) provide experimental variables (factors) that give the 2 continuous stress-strain curve shape and also yield su . Wcm (lb/1,000 ft ) = - 4.410 + 3,562 tcm (in.) (4-b) Numerous investigators have researched the effects of for the linerboard and corrugating medium respectively various papermaking variables on the elastic modulus (fig. 2). 2 All materials were preconditioned at less than 35 percent routine NREG (12). These best estimates of C1 are com- relative humidity (RH) and then conditioned at 73°F pared with the C1 values based on C1 = f(su) and are and 50 percent RH prior to testing at this same con- given in table 1. A regression analysis to determine the dition. The load was applied at a constant rate of relationship between C1 and su is as follows: deformation and recorded with a digital oscilloscope. C = 1523. + 1.028•10–7 s 3(lb/in.2) The compression machine speed produced a strain rate 1 u (5-a) in the paperboard of 0.01504 in./in./min. The experi- C = 44.91 + 1.120 s (lb/in.2) mental load deformation curves were converted into 1 u (5-b) stress-strain relationships. Stress is defined as the ap- Eq. (5-a) is used for linerboard and (5-b) for corrugating plied load per original cross sectional area based on ef- medium.
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