DETERMINING PAPERBOARD STRENGTH--BIAXIAL TENSION, COMPRESSION, AND SHEAR

D. E. Gunderson R. E. Rowlands Research General Engineer Professor of USDA Forest Service Engineering Mechanics Forest Products Laboratory1 University of Wisconsin Madison, WI 53705 Madison, WI 53706

ABSTRACT

Paper used i n corrugated con ta iners and o t h e r s t r u c t u r a l a p p l i c a t i o n s is commonly sub jec ted t o b i a x i a l loading, inc lud ing shea r s t r e s s . S t reng th da ta under these cond i t ions a r e needed, b u t methods have no t been a v a i l a b l e t o t e a t paperboard under con t ro l l ed b i a x i a l compressive stress o r shea r s t r e s s p l u s compression. This paper desc r ibes techniques f o r making t h e s e measurements and c i t e s r e s u l t s . A cruciform specimen is used f o r compres­s i v e da ta and shea r stress is in t roduced through "off- axis" o r i e n t a t i o n . Off- axis da ta from u n i a x i a l and b i a x i a l tests a r e combined and i n t e r p o l a t e d by p l o t t i n g on normal s t r e s s- s h e a r stress coord ina tes . S t reng th da ta f o r a common paperboard have been obtained under completely genera l b i a x i a l i t y .

INTRODUCTION

It is customary and convenient t o s p e c i f y t h e s t r e n g t h of paper and paperboard i n terms of ten­s i l e and compressive s t r e n g t h i n t h e machine d i r e c t i o n and i n t h e c r o s s machine d i r e c t i o n . Data genera l ly come from l a b o r a t o r y t e s t s in which s t r i p s of paper o r i en ted i n t h e machine d i r e c t i o n (MD) o r cross machine d i r e c t i o n (CD) a r e u n i a x i a l l y s t r e s s e d t o f a i l u r e . When paper and board c a r r y load i n r e a l a p p l i c a t i o n s , however, t h e stresses which develop a r e r a r e l y u n i a x i a l . Consequently, it is appropr ia te t o a sk "What is t h e s t r e n g t h of paper when s t r e s s is app l i ed s imultaneously in more than one d i r e c t i o n ?" A t y p i c a l u n i a x i a l compression t e s t is represented i n Fig . 1A. I n Fig . 1B a t r ansverse compressive s t r e s s has been added t o t h e long i tud ina l stress o f Fig . 1A t o c r e a t e a b i a x i a l stress condi t ion. The a d d i t i o n of shea r stress (Fig. 1C) f u r t h e r complicates t h e problem. A l l t h r e e cases represen t " r e a l world" condi t ions , bu t t h e s t r e n g t h of paper under cond i t ions B and C has not been measured p rev ious ly (11).2 Yet t h e i n f o r mation is e s s e n t i a l f o r design (4,20). Our o b j e c t i v e is, t h e r e f o r e , t o acqu i re t h e miss ing experimental da ta and p r e d i c t b i a x i a l s t r e n g t h .

This a r t i c l e d e s c r i b e s determinat ion of t h e s t r e n g t h o f paper under b i a x i a l compression--and under combinations of b i a x i a l normal s t r e s s and shear s t r e s s . I n a subsequent s tudy , t h e au thors used t h e d a t a developed h e r e i n t o p r e d i c t mathe­m a t i c a l l y t h e b i a x i a l s t r e n g t h of paperboard (15). The test m a t e r i a l is a machine-run, p ine k r a f t paperboard: b a s i s weight 205 g/m2, d e n s i t y 670 kg/m3.

1Maintained i n Madison, Wis., i n cooperat ion w i t h t h e Univers i ty of Wisconsin.

2Underlined numbers i n pa ren theses r e f e r t o L i t e r a t u r e Ci ted a t end o f t h i s r e p o r t .

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F ig . 1. S t a t e s of stress.

BACKGROUND

Only r e c e n t l y has t h e b i a x i a l s t r e n g t h of paperboard been recognized as a key parameter i n determining corrugated- container performance. Set terholm e t a1. pioneered t h i s a r e a i n 1967-68 by measuring t h e b a s i c shea r c h a r a c t e r i s t i c s and t h e b i a x i a l t e n s i l e s t r e n g t h of paperboard (1,2). Pe te r son and Fox (3-5) demonstrated t h e b i a x i a l na tu re of t h e stresses i n t h e l ine rboard faces o f a corrugated pane l and developed a theory f o r c a l c u l a t i n g both t h e magnitude and d i r e c t i o n of t h e p r i n c i p a l stresses. deRuvo e t a1. (6) p ressur ized paper c y l i n d e r s t o measure t h e b i a x i a l t e n s i l e s t r e n g t h o f envelope and sack papers and r e l a t e d t h e i r r e s u l t s t o t h e Tsai-Wu theory (7-9 ) . Uesaka e t a l . (1 0 ) o r thogona l ly s t r e s s e d rec tangula r specimens t o examine t h e e l a s t i c behavior of paper under b i a x i a l t ens ion . Brezinski and Bardacker used a cruciform specimen for the same purpose (18,19). Fellers e t a1. (11) most recently extended available knowl­edge of b i a x i a l s t r e n g t h i n t o t h e t e n s i l e -compressive regimes and p red ic ted performance under b i a x i a l compression p l u s in- plane shear us ing the Tsai-Wu theory. To quote F e l l e r s , ". . . it remains t o be shown whether t h e mathematical express ion of t h e [ f a i l u r e ] envelope shape i n this [ b i a x i a l compressive] quadrant p r e d i c t s phys ica l r e a l i t y ." Nei the r d a t a nor a method f o r o b t a i n i n g t h e d a t a has been a v a i l a b l e i n t h e l i t e r a t u r e . T h e same problem has e x i s t e d f o r in- plane shea r stress p l u s compressive stress. Our c o n t r i b u t i o n f i l l s this gap.

METHODAND MATERIAL

B i a x i a l Normal S t r e s s

Both f l a t and c y l i n d r i c a l specimens were used t o c r e a t e b i a x i a l stress i n the MD and CD d i r e c t i o n . Cyl inders w e r e i n t e r n a l l y p ressur ized and a x i a l l y loaded t o provide a f u l l range of t ens ion- tens ion b i a x i a l stress r a t i o s i n t h e manner desc r ibed by deRuvo e t a l . (6) . Load was app l ied us ing t h e appara tus of r e f . 1 modified t o provide means f o r p r e s s u r i z i n g t h e c y l i n d e r s and continuous measure­ment of a x i a l fo rce . S t r e s s e s were c a l c u l a t e d a s i n r e f . 6 excep t t h a t t h e l o n g i t u d i n a l c r o s s s e c t i o n w a s ad jus ted t o inc lude t h e m a t e r i a l a t the seam overlap. Addi t iona l ly , ma te r ia l th ickness was introduced so t h a t stress i s expressed i n terms of

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force per unit of cross sectional area. Cylinders were used for tension-tension data exclusively. Failures were marked by explosive rupture of the cylinder. Fractures typically occurred in the direction perpendicular to the largest ratio of stress to strength.

Biaxial states of stress involving compression were developed using a cruciform specimen and a recently developed method of lateral support (3). The concept is illustrated in Fig. 2 where the cruciform specimen is shown supported on an array of slender, vertical rods. The rods are free to deflect in the x and y directions but form a rigid planar surface in the z direction. The specimen is held in contact with the rod tops by the pressure difference created when the enclosure is evacuated. Open areas on the planar surface must be masked. (Ref. 13 provides more details of the restraint mechanism and performance verification.) Tensile and/or compressive forces PX and PL are applied to opposite legs of the cruciform through loading tabs bonded to the face of the specimen at the ends of the legs (Fig. 3). The slots in the legs define the area over which the forces PX and PL act, to provide a determinate state of biaxial stress in the center of the specimen.' The longitudinal axis can be aligned with either the MD or CD of the specimen material. Load is applied by two inde­pendent load frames (Fig. 4). The longitudinal frame applies only compressive load. The trans­verse frame is capable of applying tensile or com­pressive load. Loads are recorded with commercial strain-gage load cells and displayed on an x-y recorder. Failure is marked by an abrupt drop in load and a classic shear-slip crease in the central portion of the specimen.

In both cylinder and cruciform tests, the ratio of longitudinal to transverse stress was approxi­mately constant as loads increased. Time to failure varied from 15 to 30 seconds depending on specimen type and load ratio. The effect of variations in load sequence on biaxial strength warrants investi­gation since it is likely that real structural products will be subjected to nonproportional loading.

Biaxial Normal Stress with Shear

Creating simple biaxial stress in a laterally sup­ported specimen involves considerable mechanical complexity. Adding yet another system of loading to create shear stress (Fig. 1C) seems, initially, to be so difficult as to be impractical. Fortu­nately, shear stress occurs naturally at any orien­tation other than the principal stress axes. Con­sequently, biaxial stress applied at an orientation other than the MD and CD results in biaxial normal stress and shear stress in the MD and CD. Thus, shear stress is produced in the MD and CD by orienting the machine direction of the paperboard at an angle to the longitudinal axis of the

3Transverse stress is calculated on the basis of 40 mm width. Longitudinal stress is based on the 25 mm minimum width of the longitudinal direc­tion. Slots in the transverse tabs cause the specimen to be necked in the longitudinal direction so that the highest level of biaxial stress is created in the central portion of the specimen.

specimen. Techniques for calculating the "off-axis" normal stress and shear stress are wel1 known. The relationship between stresses at various orienta­tions is often represented by a graphic construction known as Mohr's Circle (21) plotted on normal stress, shear stress coordinates. It enables the state of stress to be evaluated in any orientation if the stresses in the longitudinal and transverse directions are known.

Figure 5 illustrates an off-axis cruciform specimen. In Fig. 5a the MD of the test material is shown oriented at an angle to the longitudinal axis of the specimen. The specimen is subjected to longitudinal compression and transverse tension. The longitudinal and transverse stresses are plotted as on the normal stress coordinates

in Fig. 5b. For this particular case, the resultant normal stresses in the MD and CD are both compres­sive and include shear stress components of opposite sense, Fig. 5c. Uniaxial loading of an off-axis specimen can also be used to produce biaxial normal plus shear stress over a limited range.

Off-axis orientation can also be applied to pressurized cylinders. Fig. 6. shows the MD of the test material oriented at an angle to the longi­tudinal axis of the pressurized cylinder. For a pressurized capped cylinder, the transverse or "hoop" tensile stress is twice the longitudinal

tensile stress as illustrated by Mohr's Circle

of Fig. 6b. Other ratios of biaxial tension can be produced by adding an axial load A to the ends of the cylinder. Mohr's Circle shows that in this case (Fig. 6c) both the MD and CD normal stresses are tensile and a shear stress is developed in the MD and CD by virtue of the off-axis orientation. The limited lateral support of the cylinder restricts its use to tensile normal stresses.

A third method of obtaining biaxial normal stress with shear is that described by Setterholm (1), Fig. 7. The material axes are aligned with the cylinder axes. In-plane shear is developed by applying torques T of opposite direction at the ends of the cylinder. Various levels of longitudinal and transverse stress can be achieved by applying inter­nal pressure and axial tension A. Compressive nor­mal stresses must again be avoided because of the limited lateral support.

Graphic Interpolation

Figures 5 and 6 show how applying longitudinal and transverse stresses in the off-axis specimen pro­duces biaxial normal stress plus shear in the machine and cross machine directions. However, because the normal and shear stresses are not inde­pendent under a particular off-axis angle a method is needed to interpolate results at desired shear stress levels.

This can be achieved using the graphic approach of Fig. 8 in which several Mohr's circles of a common orientation are plotted on the same coordi­nates. Each of these circles represents a test con­ducted at a particular off-axis angle Methods of Figs. 5 and 6 enable numerous such circles to be obtained at the same Circles having subscripts 1, 2, and 3 are from off-axis cruciform tests and involve compressive stresses at some orientation. Subscripts 4, 5, and 6 represent cylinders subjected

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Fig. 2. Vacuum restraint concept in biaxial configuration. Rods are 3 mm by 3 mm by 115 mm. Space between rods is 0.7 mm.

ML835319 Fig. 3. Detail of cruciform specimen design.

to biaxial tensile stresses. The points designated obtained in coordinates for that by hand-MDn and CDn (n = 1 to 6) are the stresses in the MD Bitting a curve to the values of CDn and MDn. For

and CD at failure and involve different levels of a specified level of positive and negative shear shear stress Because all tests of Fig. 8 were stress, this strength envelope yields two sets of performed at the same off-axis orientation a paired values of normal stress for MD and CD. continuous failure envelope (dashed line) can be These acts can then be plotted in normal stress

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Fig. 4. Load frames and cruciform specimen with load blocks attached.

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Fig. 5. Off-axis cruciform specimen. a. Forces applied to the specimen. b. Determining stresses in MD and CD using Mohr’s circle. c. Resulting state of stress in MD and CD coordinates.

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Fig. 6. Off- axis c y l i n d e r specimen. a . P ressur ized c y l i n d e r . b. Determining stresses i n M D and CD using Mohr’s c i r c l e . c. Resu l t ing s t a t e of stress i n MD and CD coord ina tes .

coordinates as conceptually illustrated

in Fig. 9. Paired values of at a

fixed level of are taken from plot for 22° in Fig. 9a, and are plotted in Fig. 9d with a tri­angular symbol. Paired values of at the same level of are taken from the plot for 45° at Fig. 9b and are plotted in Fig. 9d with a circle symbol. In Fig. 9c the process is repeated for

= 67°. Results are plotted in Fig. 9d with a square symbol. In this manner i t is possible to construct the desired normal stress

plot of biaxial strength at a specified level of shear stress (Fig. 9d). The process can be repeated at different levels of

Material

A single material was used throughout. It is a pilot-run, machine-made 100% Lake States softwood, unbleached, kraft paperboard, basis weight 205 g/m2, density 670 kg/m3, modulus of elasticity 7.24 MPa (MD) and 3.37 MPa (CD)--also identified as material "A" in (13). All experiments were conducted at 23° C, 50% RH in accordance with Tappi Standard T 402 OS-70. Lateral restraint pressure for com­pression tests was 51 kPa (13).

RESULTS

Zero Shear Tests

Biaxial strength data obtained from on-axis (zero shear) tests are plotted in Fig. 10. The horizontal axis represents normal stress in the machine direc­tion; the vertical axis, normal stress in the cross machine direction. Tensile stress is positive; com­pressive negative. Except for the uniaxial data (square symbol), each data point represents a single specimen failure. Plotted uniaxial data represent the average of six specimens each. Cruciform speci­mens were used to obtain biaxial data in the three compression quadrants (II, III, and IV); cylinders were used to obtain biaxial data in the pure ten­sile quadrant (I). Jewett's method (14) and the vacuum restraint method (13) were used, respec­tively, for uniaxial tensile and compressive results.

Results of tests conducted with the MD and CD axes of the test material oriented at = 30°, 45°, and 60° are presented in Figs. 11-13. They are plotted in coordinates as in Fig. 8. The Mohr’s

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ML835322

Fig. 7. Torsion cylinder. a. Mechanical forces and internal pressure applied to the cylinder. b. Resulting state of stress in MD and CD coordinates. c. Mohr's circle representation of stresses.

ML83 5323

Fig. 8. Mapping the failure envelope using Mohr's circles derived from tests conducted at constant 8. Subscripts 1-3 identify cruciform tests; 4-6 are cylinder tests. Abscissa is normal stress Ordinate is in-plane shear stress

Circles of Fig. 8 are deleted in Figs. 11, 12, and 13 for clarity. Each data point in Figs. 11-13 represents a particular combination of normal and shear stress which causes failure. Circles repre­sent cylinder specimens, crosses represent cruciform specimens, and squares represent uniaxial compres­sion tests. Similar plots were developed for specimens tested at = 10° increments from 10° to

80° and at = 15° and 75° for a total of 11 plots. Superimposed failure envelopes at 10° intervals are shown in Fig. 14.

Normal Stress Limits at Fixed Shear Stress

From the 11 plots at the 11 discrete values of 47 sets of paired MD/CD failure stress values

were obtained at three levels of shear stress. These data include both tensile and compressive states of stress. To complement "off-axis" data scaled from the plots, 26 cylindrical torsion tests were conducted at corresponding shear stress levels. The 47 data points from the curves plus the 26 points from the cylindrical torsion tests and the 68 zero shear points shown in Fig. 10 provide the experimental data base for a subsequent analytical-experimental correlation (15).

DISCUSSION

Under biaxial loading at = 0, the CD tensile strength exceeds its uniaxial tensile strength (Fig. 10). This is compatible with previously observed behavior of other paper materials (2 , 6 , 11). Our particular material does not exhibit corres­ponding increase in MD strength. Compression-tension data in quadrants II and IV are consistent with Fellers zero-shear data (11). There are no other quadrant III data with which comparisons can be made.

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Fig. 9. Replotting from constant failure envelopes indicates

on coordinates to constant shear stress envelopes on normal stress coordinates Dashed curve strength at zero shear.

The cruciform specimen must be used with cau­tion because it can suffer from stress concentra­tions at the reentrant corners, transverse restraint, and end restraint of the longitudinal and transverse legs (11,16). The latter of these con­cerns applies only to off-axis tests. We examined the specimen photoelastically and conducted aux­iliary tests to address these concerns.

Transmission photoelastic examination4 showed that a large area of uniform biaxial stress exists in the central portion of the specimen and that the region of transition from uniaxial stress in the legs to biaxial stress in the central region is small. Our findings are similar to those of an earlier comprehensive photoelastic study by Monch and Galster (17).

Significance of stress concentration at re­entrant corners was evaluated by an auxiliary test. Necked tensile specimens of the test linerboard were shaped in two ways: By conventional die cutting and by repeated notching of a wide rectangular strip. Notches were sharp end, razor slit at spacings of 4 mm. Twenty specimens of each type (10 MD and 10 CD) were loaded to tensile failure in an Instron machine. We found no significant difference in the tensile strengths of the two sets of specimens--suggesting that common paperboard is remarkably insensitive to local stress concentrations.

4Photoelastic sheet PS-IE, Photolastic Div., P.O. Box 27777, Raleigh, NC 27611.

A second auxiliary test was performed to evaluate the effect of end restraint on the longi­tudinal and transverse legs. Sixteen specimens taken at 45° off-axis orientation were tested to failure in uniaxial compression on the vacuum restraint apparatus. Half of the specimens had flexible rubber end restraints; half rigid. Those with flexible end restraint averaged 6.8% stronger than those with rigid ends. The same test conducted on 12 0.5-mm-thick specimens of Sitka spruce (having a parallel grain to cross grain compression stiff­ness ratio of 40:1) showed a 19% greater strength for the flexible end specimens. All of these aux­iliary compression specimens had the same length and width. The results suggest that, while concern about end restraint is valid, the effect is small for paperboard having a low ratio (2:1) of anisot­ropy. We did not u s e flexible attachments at the load blocks in the experiments, but recommend them for more highly anisotropic materials. Cruciform performance is of particular concern when dealing with brittle or highly orthotropic materials--and when working in the tensile-tensile regime. The potential for problems is minimized in our study because: (a) the cruciform is not loaded in biaxial tension; (b) the shaped configuration places the zone of highest biaxial stress in the center of the specimen--well removed from the intersection of the longitude and transverse legs; and (c) paperboard is not a structure of highly directional laminated plies, but a network of fibers oriented in all directions with only a modest directional prefer­ence. The cruciform specimen has been successfully

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ML83 5325

Fig. 10. Biaxial strength at zero shear stress plotted on normal stress coordinates Machine-run pine kraft; basis weight 205 g/m2, density 670 kg/m3. Off-axis data are the zero shear intercepts of plots at constant

used with a variety of materials. We believe its use is appropriate here.

Confidence in the present work is further sub­stantiated by the following observations:

1. Zero shear cruciform and cylinder data in Fig. 10 are consistent with well-documented uniaxial results (2).

2. Zero shear intercepts of the 11 plots of off-axis data at constant agree well with each other, Fig. 14.

3. Zero shear intercepts referenced above agree well with the on-axis biaxial cruciform and on-axis cylinder data, Fig. 10.

4. Cylindrical torsion data agree very well with all other data at the three shear stress levels tested: = 6.9, 10.3, and 15.9 MPa (15).

In summary, we observe consistent results from seven different types of test: uniaxial tension with necked specimens, uniaxial compression with rectangular specimens, on-axis cruciforms, off-axis cruciforms, on-axis cylinders, off-axis cylinders, and torqued cylinders.

SUMMARY AND CONCLUSIONS

In structural applications, paper routinely encounters biaxial normal stress plus shear stress. its strength under these conditions, however, had previously been determined only in part. This study experimentally evaluates paperboard strength under conditions of general biaxial normal stress and shear stress. Compression-compression data and tensile and compressive data covering all four quadrants have been obtained at several levels of shear stress. On- and off-axis cruciform, cylinder, and uniaxial specimens were used. To the authors’ knowledge, this is the first use of the off-axis cruciform. Strength values were interpolated at specific shear levels by a novel graphic method. This research represents significant progress toward understanding the performance of paper­board structures, developing methods for efficient design, and optimizing use of wood resources. The time and expense of the experimental approach, how­ever, make it impractical for general use. Conse­quently, our next objective was to mathematically predict biaxial strength from limited uniaxial data (15).

The method developed here may also be suitable for testing other thin materials.

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Fig. 11. Failure envelope at a constant off-aria angle of 30° plotted on normal stress shear stress coordinates. Machine-run pine kraft; basis weight 205 g/m2, density 670 kg/m3.

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Fig. 12. Failure envelope at a constant off-axis angle of 45° plotted on normal stress shear stress coordinates. Hachine-run pine kraft; basis weight 205 g/m2, density 670 kg/m3.

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Fig. 13. Failure envelope at a constant off-axis angle of 60° plotted on normal stress shear stress coordinates. Machine-run pine kraft; basis weight 205 g/m2, density 670 kg/m3.

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Fig. 14. Failure envelope at constant off-axis angles from = 10° to = 80° plotted on plane stress shear stress coordinates. Machine-run pine kraft; basis weight 205 g/m2, density 670 kg/m3.

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NOMENCLATURE

Normal stress in the plane of the sheet Normal stress in longitudinal direction

Nomal stress in transverse direction

Shear stress in the plane of the sheet Specimen orientation, measured from longi­

tudinal axis of specimen to machine direc­tion of test material

Machine direction of test material Cross machine direction of test material Load applied to cruciform specimen in

transverse direction Load applied to cruciform specimen in

longitudinal direction Indicates longitudinal axis of specimen Indicates transverse axis of specimen Axial tensile load applied to cylinder specimen Torque applied to cylinder specimen As a subscript denotes the nth test in a series

of tests

ACKNOWLEDGMENT

This research vas jointly funded by the Forest Products Laboratory and the National Science Foundation (Grant No. MEA-8120393 monitored by Dr. C. J. Astill).

LITERATURE CITED

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Peterson, W. S., Tappi 63(10): 75-79 (1980).

Peterson, W. S., Tappi 63(11): 115-120 (1980).

Peterson, W. S., and Fox, T. S., Paperboard Packag. 65(11):68-76 (1980).

deRuvo, A., Carlsson, L., and Fellers, C., Tappi 63(5): 133-136 (1980).

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Fellers, C., Paper Properties in Compression. In: "Handbook of Physical and Mechanical Testing of Paper and Paperboard," (Richard Hark, ed.), Marcel Dekker, New York, 1983.

Gunderson, D. E., Edgewise Compression of Paperboard: A New Concept of Lateral Support, APPITA, [Submitted for publication, 1982].

Jewett, D. M., An Electrical Strain Gage for the Tensile Testing of Paper. USDA For. Serv. Res. Note FPL-03, Forest Products Laboratory, Madison, Wis, 1963.

Rowlands, R. E., Gunderson, D. E., Suhling, J. A., and Johnson, H. W., Hill-Type Predictions of the Biaxial Strength of Paper­board, [in preparation, University of Wisconsin-Madison].

Pagano, N. J., and Halpin, J. C., J. Compos. Mater. 2(1): 18-31 (1968).

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Brezinski, J. P., and Hardacker, K. W., Tappi 65(8): 114-115 (1982).

Hardacker, K. W., J. Phys. (E. Sci. Instr.) 14(5): 593 (1981).

Johnson, H. W., Jr., and Urbanik, T. J., A Nonlinear Theory for Elastic Plates with Application to Characterizing Paper Properties. J. of Appl. Mech. [In press].

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Gunderson, D. E.; Rowlands, R . E. De te rmin ing p a p e r b o a r d s t r e n g t h- - b i a x i a l t e n s i o n , compress ion , and s h e a r . In: P r o c e e d i n g s , 1983 I n t e r n a t i o n a l Pape r P h y s i c s Confe rence ; 1983 September 18-22; H a r w i c h p o r t , MA; A t l a n t a , GA: TAPPI P r e s s ; 1983: 253-263.