Principles of Minimum Cost Refining for Optimum Linerboard Strength

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1 Principles of Minimm Cost Refining for Optimm Linerboard Strength Thomas J. Urbanik and Jong Myong Won ABSTRACT The mechanical properties of paper at a single basis weight and a single targeted refining freeness level have traditionally been sed to compare papers. Understanding the economics of corrgated fiberboard reqires a more global characterization of the variation of mechanical properties and refining energy consmption with freeness. The cost of refining energy to increase paper performance is of the same magnitde as the cost of adding fiber to obtain an eqal increase in performance; the costs of energy and fiber vary dramatically with geography and market conditions. We provide formlas that can be programmed into a spreadsheet to graphically examine market condition and performance scenarios. We characterize a recycled linerboard and determine the market conditions that favor increased refining compared to increased fiber. KEYWORDS Cost optimization, Economics, Linerboard strength, Recycling. INTRODUCTION An important step inpapermaking is themechanical refining of plp fibers (conventionally via disc refiners) to generate optimm mechanical properties of the paper. Researchers have improved the strength properties of paper made from recycled fiber by blending the fiber with virgin fiber and chemical additives (1), sing chemical treatment, and modifying the paper machine. Refining is the most practical Urbanik is Research Engineer: USDA Forest Service. Forest Prodcts Laboratory, Madison, Wisconsin trbanik@fs.fed.s Won is Professor: Dept. of Paper Science & Engineering, College of Forest Sciences, Kangwon National University, Chnch , Korea. wjm@mail.kangwon.ac.kr process and has been adopted for mill prodction. However, the more the initial plp is refined, the more difficlt it is to restore the initial strength ofrecycled plp throgh refining (2). Tensile strength has been the property most investigated (2,3). Data are limited on how refining affects paper compressive strength. stiffness properties, or caliper. The Forest Prodcts Laboratory has researched the feasibility of sing small-diameter trees, waste wood, and nconventional wood species for papermaking sing an experimental refiner and formed handsheet specimens. The laboratory relationship between refiningvariablesandpaper performance provides insight into a process that is ndobtedly more complicated on an indstrial scale. Mechanical refining is an energy-intensive process at both the laboratory and indstrial scale. As we shall demonstrate, the cost ofrefining energy to increase paper performance is of the same magnitde as the cost of adding fiber to obtain an eqal increase in performance. In a stdy of important variables affecting corrgated fiberboard. Koning and Haskell (4) explored the effects and interactions of seven typical papermaking factors (wood species, plp yield, type of refiner, refining consistency, amont of refining, wet-press pressre, and srface) on varios strength properties of 205-g/m 2 linerboard handsheets. Among the properties investigated, compressive strength, modls ofelasticity, and thickness, which are known to relate to the compressive strength of corrgated fiberboard boxes, were consistently and significantly affected by the freeness level achieved by refining. In general. the athors fond that the lower of two freeness levels investigated reslted in better linerboards by virte of increasing overall strength. Greater linerboard strength can ths be obtained at a lower freeness by more refining, bt at a higher energy cost. Greater linerboard strength can also be obtained at a higher basis weight with a proportionally higher fiber cost. When the linerboard is considered as a component in corrgated fiberboard and the compressive strength of a box is important, the most economical linerboard in terms offiber Progress in Paper Recycling/Vol. 15, No. 4. Agst 2006 Page 13

2 and refining energy inpts is one with the minimm total cost of fiber and energy that yields the reqired box compressive strength. The generalization that lower freeness is better is not as evident. Linerboard in corrgated fiberboard specified by edgewise crsh strength and recycled corrgated medim are prodced throgh the se of higher process refining levels compared to that sed for conventional containerboards. Refining energy is among the most costly process variables in containerboard prodction. The benefits of process refining in combination with the basis weight of the components is important to nderstanding the economics of corrgated fiberboard. OBJECTIVE We compare the cost and performance of two kinds of linerboard: (a) expensive-to-process bt lightweight linerhoard made at a low hasis weight from highly refined plp and (b) cheaper-to-process bt heavier linerhoard made from less refined plp and more fiber. The empirical variations of refining energy consmption with freeness and variation in mechanical properties with freeness are sed to examine theoretically how the performance of the linerboard facing in Corrgated fiberboard relates to the total cost of energy and fiber. Reslts qantify the optimm refining level that yields fiberhoard with adeqate performance at the lowest cost of energy pls fiber. Understanding the effect of refining on paper properties at the laboratory scale is a necessary first step in optimizing indstrial prodction. Prodction costs emanating from paper machine speed, schedling efficiency, inventory and distribtion, etc., are ltimately important bt are not examined in this stdy. Refining to a low freeness level redces the drainage rate on the paper machine and might be constrained by a practical cost in actal prodction. Increasing a paper s hasis weight increases its drying time and shipping costs per nit area of paper. The final economics are a fnction of the effects and interactions among refining, basis weight, and prodction variables. The reslts of this stdy provide a methodology for qantifying the economic feasibility of new materials and for establishing bondaries in a more broadly designed indstrial experiment. MATERIALS AND TESTS A single commercial linerboard (94% kraft plp and 6% recycled cormgated containers) was recycled into plp and refined at three levels of consistency: 5%, 8%, and 12%. Linerboard material was first plped in a 50 liter Voith (Heidenheim, Germany) laboratory plper at 10% consistency for 30 mintes and with nominal hot water. The plp slrry was thickened and refrigerated for later se. Atmospheric refining was later carried ot on a 305-mm Sprot-Waldron(Mncy, Pennsylvania) single-disc refiner with C2976 plates. Refining at each level of consistency combined with each of three levels of refiner plate gap (102, 178, and 254 mm) was done at p to three freeness levels. The processed plps at 5% and 8% consistency were refined with one to three passes throgh the refiner. The processed plp at 12% consistency was refined with one or two passes throgh the refiner. Canadian standard freeness (CSF) measred according to TAPPI Test Method T 227 (5) and cmlative specific energy consmption were determined for the plp after each pass throgh the refiner. Handsheets at three levels of basis weight (BW) (nominally 126, 185, and 205 g/m 2 ) were made from each of the 24 refined plps. For comparison, the freeness of a control material was determined from nrefined plp, and control handsheets were made from nrefined plp at the three BW levels. Freeness and specific energy consmption levels determined for the plps and actal average BW level for the respective handsheets are given in Table 1. Compression load-straincrves and bending stiffness data that cold be sed to compte the compressive strength and bckling strength of a facing microplate in corrgated fiberboard were determined from handsheet specimens. Compression load-straincrves p to failre were prodced with the apparats described by Gnderson (6). Bending stiffness in a Taber instrment was determined following TAPPI T 489 (7). The srface-to-srface thickness (caliper) was determined by TAPPI T411 (8). Prior to testing, materials were preconditioned in a dry environment below 30% RH and then conditioned at 50% relative hmidity (RH). All tests were condcted in a conditioned environment at 23 C and 50% RH in accordance with ASTM Standard D (9). RESULTS AND DISCUSSION Energy Model Or data on freeness were fond to vary linearly with specific energy consmption and with interactions among consistency, plate refiner gap, and specific energy consmption according to the statistical eqation where F is freeness (ml CSF), SEC specific energy consmption (W h/kg), (1) Page 14 Progress in Paper Recycling / Vol. 15, NO. 4, Agst 2006

3 C G consistency (%), and plate refiner gap (µm). 4% of the actal determination of 704 ml CSF for the control material (Table 1). Statistically fitted coefficients obtained via a least sqares regression were determined as α = 732, β 0 = , b 1 = , β, = 7.37 x 10-5 The standard errors associated with α, β 0, β 1, and β 2 are 0.6%, 4.0%, 4.2%, and 5.8%, respectively, as a percent magnitde of the fitted coefficients. Expanding Eq. (1) with additional coefficients applied to C, G, CG, and G SEC was not fond to improve the fit to data significantly. The fit of Eq. (1) to or data (Table 1) is shown in Figre 1 in terms of the inverse expression where Eqation (2) with Eq. (3) as inpt expresses the behavior that for each combination of fixed C and G, the respective SEC-F plot passes throgh a common abscissa a and has a slope 1/β depending on C and G (Fig. 1). The fitted level of a predicts the plp freeness prior to refining and is within (3) Althogh Eq. (2) provides a way to accrately interpolate among the data of this stdy, it does not predict an increasing energy consmption approaching infinity as a zero freeness level is soght. Eqation (2) is sfficient for nderstanding the performance of or experimental refiner within the scope of this stdy. For commercial refiners and other inpt variables that affect refining energy consmption, more mechanistic eqations' wold need to be considered. Mechanical Properly Model Data on the variation of compression load P with compression strain ε were redced to average parameter vales c 1 and c 2 in a fit of the eqation p to the ltimate strength P at failre. The initial slope of Eq. (4) is given by c 2. Parameter c 1 is an asymptotic level of P if Eq. (4) is extrapolated to an infinite E. The fitted vales of c 1 and c 2 along with the determinations of P, bending stiffness D, and caliper t are given in Table 1. Strength and c stiffness are given in terms of a nit width of specimen. (4) Variation of each mechanical property Q with BW and F (Table 1) was statistically analyzed with the empirical formla (5) Fitted coefficients for each case of Q represented by c 1, c 2, P a, D, and t are given in Table 2 with normalizing inpts c in terms of the lowest nominal basis weight, BW 0 = 126 and α = 732 ml CSF. Eqation (5) is a statistical linear combination of variables BW, F, and interaction BW F with optimm exponential powers applied. The nmber of coefficients for each case of Q in Table 2 is for the most accrate fit of Eq. (5) that is statistically significant based on the statistical F-test at the Figre 1. Variation of total specific energy consmption (SEC) with freeness for linerboard plp refined at nine combinations of consistency at 5%, 8%, or 12% and refining plate gap at 102, 178, or 254 µm. The legend shown is for the three consistency levels and a 254 µm plate gap. Points on the graph are shaded white (102 µm), gray (178 µm), or black (254 µm) to indicate other levels of plate gap. The linear fits to data pass throgh a common abscissa a and have slopes 1/β that depend on consistency and plate gap according to Eq. (3). r 2 = Note that Eq. (2) shold not be extrapolated below the freeness levels of this stdy. For a more mechanistic relation between SEC and F inclding lower freeness levels, the eqation given by is noteworthy. This example of a more general eqation approximates Eq. (2) at or experimental levels of F and also approaches infinity as F approaches zero. Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006 Page 15

4 Table 1. Mechanical properties of handsheets made at three nominal BW levels from plps refined at varios levels of consistency and refiner plate gap (G), measred freeness level (F ), and total specific energy consmption (SEC) reslting from one to three refiner passes a 5% consistency 8% consistency G F SEC BW c 1 c 2 P D t c G F SEC BW c 1 c 2 P D t c µm ml W h/kg g/m 2 kn/m kn/m kn/m mnm µm µm ml W h/kg g/m 2 kn/m kn/m kn/m mnm µm 12% consistency Control G F SEC BW c 1 c 2 P D t c F BW c 1 c 2 P D t c µm ml W h/kg g/m 2 kn/m kn/m kn/m mnm µm ml g/m 2 kn/m kn/m kn/m mnm µm a BW is basis weight; c 1 and c 2 are parameters; P is ltimate strength; D, bending stiffness; t c, srface-to-srface thickness. Page 16 Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006

5 Table 2. Evalations of coefficients obtained from least sqares regression tit of Eq. (5) to mechanical property data in Table 1 sing BW 0 = 126 and α = 732 ml CSF. Mechanical property Eq. (5) c 1 c 2 P D t c coefficient (kn/m) (kn/m) (kn/m) (mn m) (µm) b g 95% confidence level. Mechanical properties c1, c2, and tc were fond to depend on b0, b1, b3, and b1; P depends on b0,b1,b2,b3,andb1;andddependson;b0,b1,b3,bg,andb1. Data on each mechanical property acqired at the actal BW level and respective predictions at the nominal BW levels are shown in Figres 2 to 6. For each fit of Q, it was frther verified that expanding Eq. (5) to inclde coefficients related to C and G did not reslt in a significant improvement. Figre 3. Variation of compressive load-strain constant c 2 with freeness ratio F/α among data in Table 1. See Figre 2 for explanation of points and legend. Cost Determination For prposes of this stdy, we consider the recycled linerboard cost as the sm of the costs attribtable Figre 4. Variation of ltimate compressive strength P with freeness ratio F/α See Figre 2 for explanation of points and legend. Figre 2. Variation of compressive load-strain constant c 1 with freeness ratio F/α among data in Table 1. Points represent respective mechanical property determined from handsheet at actal basis weight (BW). Lines are predicted property via Eq. (5) based on nominal BW. Legend refers to nominal BW (g/m 2 )andcontrol material(table 1). exclsively to fiber and the specific energy consmption of refining. The best example of a relevant commercial raw material representing the fiber wold probably be the recycled plp that enters the head box on the paper machine. The ltimate prchase price of linerboard by the corrgating converter will inclde other costs of marketing and prodction beyond the scope of this investigation. Costs of recycled fiber and energy vary dramatically with geography and market conditions. For instance, one measre of nprocessed fiber cost wold be the price of old Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006 Page 17

6 C f is the cost of fiber in linerboard at a specific basis weight and is compted from the conventional price C F reported per ton. (6) C is the cost of refining energy in linerboard at a specific c basis weight and is compted from the conventional energy rate C E. (7) Figre 5. Variation of bending stiffness D with freeness ratio F/α See Figre 2 for explanation of points and legend. C t is the combined cost to consider. For example, with fiber at $100/t and energy at $0.09/kW h, the cost components of a 200-g/m 2 linerboard made from plp refined at 1,000 W h/ kg wold he %20/10 3 m 2 + $18/10 3 m 2 = $38/10 3 m 2. (8) Performance Criteria The compressive strength of a corrgated box is related to the edgewise crsh strength of the combined board, which in trn depends on the localized compressive strength of the linerboard facings. If a facing component in cormgated fiberboard is adeqately spported against failre by local bckling, its load-carrying capacity approaches P of the linerboard material. P provides an pper bond on strength. A lower hond is the bckling strength B of a facing microplate nder simple spport conditions and can be determined from the theory by Johnson and Urbanik (12). Figre 6. Variation of srface-to-srface thickness t with c freeness ratio F/α See Figre 2 for explanation of points and legend. corrgated containers. Prices reported in the North American Factbook (10) vaned throghot the year 2000 from $42.25 per metric ton (t) in New York to $161.67/t in the San Francisco/Los Angeles area. Electrical energy rates dring the winter of2001 were reported by the Edison Electric Institte (11) to range from 4.4 cents/kw h for a 1000-kW demand in Missori to 13.7 cents/kw h for a 500-kW demand in New York. In adding these costs, it is helpfl to express them in terms of a nit area of linerboard: For conciseness, we provide eqations sfficient for prodcing the plots in Figres 7 to 9. The complete derivations are provided by Johnson and Urbanik (12, 13). We also limit or analysis to or isotropic handsheet data at standard test conditions as inpts, althogh the Johnson and Urbanik (12) theory is appropriate to anisotropic properties at arbitrary relative hmidity if sch data are available. Bckling strength is given by ^ where ε is a dimensionless bckling strain compted iteratively from (9) (10) Page 18 Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006

7 Figre 7. Variation of microplate strength with SEC of refining and dependence on flte size for linerhoard at two BW levels. Lower plot (points A-E), 185-g/m 2 linerboard; pper plot (point F), 204-g/m 2 linerboard. Solid lines, predicted P in B- or C-flte corrgated strctre; dashed lines, predicted B in A-flte strctre. Figre 9. Variation of microplate strength with combined cost components of fiber at C F = $140/t and energy at C E = 5 cents/kw h and dependence on flte se for linerboard at two BW levels. Lower plot (points C, J, G), 185-g/m 2 linerboard; pper plot (points H, I), 210-g/m 2 linerboard. See Figre 7 for explanation of line patterns. Dimensionless microplate stiffness (S) in terms of mechanical properties of the paper is inptted from (12) For simplicity, the second form of S in Eq. (12) is appropriate to handsheet data only. Figre 8. Variation of microplate strength with combined cost components of fiber at C F = $100/t and energy at C E = 9 cents/kw h and dependence on flte se for linerboard at two BW levels. Lower plots (points C, G), 185-g/m 2 linerboard; pper plot (point H), 210-g/m 2 linerboard. See Figre 7 for explanation of the line patterns. ^ An initial estimate of ε is given by (11) ^ Inpt Sand iteration of a involves the material Poisson s ratios v 1, v 2, their geometric mean v, thickness t appropriate to homogeneos material, and microplate width l. When v is nknown, the average v = considered in the calclations of Johnson and Urbanik (13) is sefl to apply. Using the corrgated flte pitch 1 as inpt l, the pper bond on strength at P is apt to apply to corrgated fiberboard having a narrow flte pitch, as in B-flte, or high stiffness facing relative to the linerboard thickness. The lower bond on strength at B is apt to apply to a wider flte pitch, as in A-flte, or low stiffness facing. The linerboard performance of interest is the strength of a facing microplate in a corrgated fiberboard strctre. The strength depends frther on the spport conditions provided by the corrgated medim bt lies somewhere between P and B. P is to be calclated directly from Eq. (5) sing 1 Typical vales given by Urbanik (14) are 6.35, 7.21, and 8.47 mm for B-, C-, and A-flte, respectively. Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006 Page 19

8 the parameters of mechanical property P from Table 2 as inpts. B is to be calclated from Eq. (9) as an iterative fnction of Eqs. (10-12) sing the parameters of mechanical properties c 1 and D from Table 2 as inpts and vales of v and l. Relation of Performance to Energy Consmption As Figre 1 shows, the specific energy consmption of refining varied with the refining freeness level for or specific linerboard plp. As Figres 2 to 6 show, mechanical properties varied with basis weight and freeness for handsheets made from the refined plp. If the facing components of corrgated fiberboard have properties represented by handsheets, performance wold be expected to vary with consmed energy, as shown in Figre 7. The predicted performance of linerboard facings at two BW levels is plotted in Figre 7. In addition. two ranges of flte size are analyzed. The conditions C = 8% and G = 178 µm are fixed. If a linerboard facing is made to 185 g/m 2 BW, its maximm strength from plp refined at varios levels wold follow the path shown by points A, B, C, and D (Fig. 7) corresponding to calclated freeness levels of 732, 648, 485, and 300 ml CSF, respectively. Increasing levels ofrefining redce plp freeness, increase energy consmption (Fig. 1), and increase strength (Fig. 4). Path A B C D (Fig. 7) qantifies maximm strength in terms of P for a 185-g/m 2 linerboard. Path A B C D (Fig. 7) is the predicted linerboard performance in a B-flte or C-flte corrgated fiberboard. If the same linerboard is fabricated into an A-flte strctre, strength is predicted to follow the path A B E (Fig. 7). Path B E reslts from a condition where B (Eq. (9)) is less than P. The actal intersection at point B depends on the vale of l = 8.47 mm assmed to apply. A crve predicting the performance of a 204-g/m 2 linerboard is also shown in Figre 7. Point F for a 204-g/m 2 linerboard with a calclated freeness level of 611 ml CSF is at the same strength level as point C for a 185-g/m 2 linerboard. In each case, P = 4.82 kn/m. However, point F reslts from plp refined to a freeness level consming 440 W h/kg, whereas point C reslts from plp refined to a freeness level consming 900 W h/kg. Strength at point F, which represents more fiber and less consmed energy, is the same strength as that at point C, which represents less fiber and more consmed energy. The better choice for corrgated fiberboard depends on the relative costs of fiber and energy. Optimization The qantification of point C as presented in Figre 7 is shown again in Figre 8 in terms of the cost C t, assming a fiber cost C F of $100/t and an energy cost C E of 9 cents/ kw h. Point C shows that the 185-g/m 2 linerboard wold have a strength P of 4.82 kn/m at a cost C t of $33.49/ 10 3 m 2 if sed in a B- or C-flte corrgated fiberboard. If the same linerboard is sed in an A-flte corrgated fiberboard, the expected strength is redced 9.5% to a B of 4.37 kn/m (point G, Fig. 8). The analysis of a 210-g/m 2 linerboard is also plotted in Figre 8. Point H is at the same strength level as point C bt at a lower cost C t of $26.99/10 3 m 2. For a reqired strength of 4.82 kn/m and for or inclded assmptions, the 210-g/m 2 linerboard is 19% less expensive than the 185-g/m 2 linerboard. The plp represented by point H is refined to a level of 663 ml CSF compared to 485 ml CSF for the plp at point C. In this example, it is less expensive to add fiber than to increase refining. The 210-g/m 2 linerboard (Fig. 8) has another inherent advantage. The Performance predicted by point H is the same for B-, C-, and A-flte strctres. By contrast, the 185 g/m 2 linerboard becomes weaker in the A-flte strctre, which cold generate additional costs. Additional analysis at or assmed conditions wold show that the 210-g/m 2 linerboard is more economical than the 185-g/m 2 linerboard at any reqired strength level. The analytical reslts cold change if market conditions change. Figre 9 predicts the performance at the same two BW levels assming a higher fiber cost C F of $140/t and a lower energy cost C E of 4.41 cents/kw h. For these market conditions and for a reqired strength of 4.82 kn/m, linerboards at both BW levels cost $33.25/10 3 m 2. Points C and H merge in Figre 9. However, if a lower strength, sch as 4.37 kn/m, were reqired in a B- or C-flte strctre, it wold cost $29.92/10 3 m 2 (point J, Fig. 9) to se 185-g/m 2 linerboard compared with $31.72/10 3 m 2 (point I, Fig. 9) to se 210-g/m 2 linerboard. The plp represented by point J (Fig: 9) was refined to a level of 597 ml CSF compared to 706 ml CSF for the plp at point I. In this example, when energy costs are low, it is less expensive to expend energy and increase refining than to add fiber. In contrast with the scenario in Figre 8, the 185-g/m 2 linerboard is 5.7% less expensive than the 210-g/m 2 linerboard. This cost advantage is limited to the flte size considered. If the linerboard is reqired to retain the same strength level in an A-flte strctre, it mst be refined to 485 ml CSF at a combined cost of $33.25/10 3 m 2 (point G, Figre 9). CONCLUSlONS The mechanical properties ofpaper at a single basis weight and a single targeted refining freeness level have traditionally been sed to evalate the feasibility of sing Page 20 Progress in Paper Recycling / Vol. 15, No. 4, Agst 2006

9 small-diameter trees, waste wood, and nconventional wood species for papermaking. While the conclsions drawn may have been valid, we see that a more global characterization of the variation of mechanical properties and refining energy consmption with freeness provides a more rational economic jstification of refining levels. At recent price levels, the cost of refining energy to increase paper performance is of the same magnitde as the cost of adding fiber to obtain an eqal increase in performance. The costs of energy and fiber vary dramatically with geography and market conditions. We provide scenarios for optimizing the performance of the linerboard component in corrgated fiberboard based on real world data, showing how the lowest cost linerboard depends on market conditions. For the single recycled plp investigated, an economic jstification for increased refining occrs only with a combination of relatively inexpensive energy and expensive fiber. Otherwise. it is more economical to se low refined plp and heavier hasis weight papers. LITERATURE CITED 1. Zhang, M., M. A. Hbbe, R. A. Venditti, and J. A. Heitmann Can recycled kraft fibers benefit from chemical addition before they are first dried? Appita J. 55(3): McKee, R.C Effect of replping on sheet properties and fiber characteristics. Paper Trade J. 155(21): Lmiainen, J Do recycled fibers need refining? Pap. P 74(4): Koning, J.W. Jr. and J.H. Haskell Papermaking factors that inflence the strength of linerboard weight handsheets. Res. Pap. FPL-323. U.S. Dept. Agricltre. Forest Service, Forest Prodcts Laboratory, Madison. WI. 5. TAPPI Official test method T 227, Freeness of plp (Canadian standard method). TAPPI, Atlanta, GA. 6. Gnderson, D Edgewise compression of paperboard: a new concept of lateral spport. Appita 37(2): TAPPI Official test method T 489, Bending resistance (stiffness) of paper and paperboard (Taber-type tester in basic configration). TAPPI. Atlanta, GA. 8. TAPPI Official test method T 411, Thickness (caliper) of paper, paperboard, and combined board. TAPPI. Atlanta. GA. 9. ASTM Standard D , Standard practice for conditioning paper and paper prodcts for testing. American Society for Testing and Materials, West Conshohocken, PA. 10. Plp & Paper North American Factbook. Paperloop Pblications. North Hollywood. CA, p Edison Electric Institte Typical bills and average rates report. Winter 2001, Washington, DC. 12. Johnson, M. W. Jr. and T. J. Urbanik Bckling of axially loaded. long rectanglar plates. Wood Fiber Sci. 19(2): Johnson, M. W., Jr. and T. J. Urbanik A nonlinear theory for elastic plates with application to characterizing paper properties. J. Appl. Mech. 51(3): Urbanik, T.J Effect ofcorrgated flte shape on fibreboard edgewise crsh strength and bending stiffness. JPPS 27(10): Manscript received for review March 3, Revised manscript received and accepted Agst 24, Progress in Paper Recycling / Vol. 15, No. 4. Agst 2006 Page 21