In previous research, a thin-plate theory was derived for analyzing corrugated fiberboard under edgewise compression and subjected to localized buckling. In this note, buckling formulas for thin paperboard plates characterized by an approximate in-plane shear modulus of elasticity are further generalized to allow for arbitrary levels of shear modulus. The results have applications in the design of paper products made of composite plates

Urbanik, Thomas J.

English

Wood and Fiber Science (E-Journal)

EFFECT OF IN-PLANE SHEAR MODULUS OF ELASTICITY ON BUCKLING STRENGTH OF PAPERBOARD PLATES1 Thomas J. Urbanik Research General Engineer USDA Forest Service Forest Products Laboratory One Gifford Pinchot Drive Madison, WI 53705-2398 (Received August 199 1) ABSTRACT In previous research, a thin-plate theory was derived for analyzing corrugated fiberboard under edgewise compression and subjected to localized buckling. In this note, buckling formulas for thin paperboard plates characterized by an approximate in-plane shear modulus of elasticity are further generalized to allow for arbitrary levels of shear modulus. The results have applications in the design of paper products made of composite plates. Keywords: Plates, elastic stability, buckling, shear modulus, paper. INTRODUCTION analysis of Johnson and Urbanik (1 989), G is Previous research derived the theory appro- made a function of other stress-strain con- priate to analyzing corrugated fiberboard un- stants according der edgewise compression and subjected to lo- calized buckling. The thin-plate theory of E,\/Y,IU, = 2 ( 1 +  \/;) (1 Johnson and Urbanik (1987) considered the elastic buckling of long rectangular plate ele- ments of paper under longitudinal compres- sive loading. The thin plate theory was later incorporated in a theory of plate structures ap- plicable to modeling corrugated fiberboard short-column structures (Johnson and Urba- nik 1989). Basic stress-strain inputs to the analysis are the combination of properties c,, c, = E,, v,/v,, and v = fi for each plate element. Johnson and Urbanik (1987) assumed that the plate material is homogeneous and that the initial in-plane shear modulus of elasticity G could be approximated by a relationship due to St. Venant. As a result, instead of G being an independent input to the structure buckling The approximation made sense in that paper shear modulus is difficult to determine. In analyzing steel structures, the procedure for conceptually defining the elastic rigidities of a plate-stiffener combination with those of an equivalent constant thickness plate-suc- cessfully applied by Huffington (1 956)-re- duces to determining four independent rigid- ities of the equivalent plate. Likewise, the theory can have further applications to cor- rugated containers, spaceboard (Setterholm 1985), and other nonhomogeneous fiber prod- ucts if effective stress-strain properties char- acterizing the elastic constants ofthe panel ma- terial can be determined. The objective of this note is to extend the results of Johnson and Urbanik (I 987) and set forth the equations so that in applying the the- ' This article was written and prepared by U.S. Gov- ory to other than thin paper materials the effect ernment employees on official time, and it is therefore in the public domain and not subject to copyright. The Forest shear On structure Products Laboratory is maintained in cooperation with strength can be considered. Because this note the University of Wisconsin. supplements earlier work, the complete deri- Wood and Fibpr Science, 24(4), 1992, pp. 38 1-384 382 WOOD AND RBER SCIENCE, OCTOBER 1992, V. 24(4) TABLE I .  Forms of equations expanded to include in-plane shear. Equatlon numbcr" Generalized equation form New equation number a2S(E2 + I)* Initial 2 = 12(1 - u2) j = tan + 2X'C)1'2tanh(~2 + 2X2d)"2/fi] + a " From Johnson and Urbanik (1987). vations and definitions of terms leading to the -(2 + c)v results are not repeated. Unless otherwise in- = { dicated, equation numbers used within the text 2 - of this note refer to the equations of Johnson [(2 + c)u/2l2 - 1 and Urbanik (1987); equation numbers des- 1 - v2 ignated with a prime (') refer to their more generalized form. THEORY The new dimensionless shear constant is then The first equation from the set of Eq. (2.8) . (Johnson and Urbanik 1987) is retained in the introduced as form c^ = (2 + c)v 2Hl2 + H3, 2 (4) = v2(2 + c )  HI I (2) and instead of using Eq. (1.4), a more general Equation (2.1 1) is then derived as follows: The expression equal to Eq. (3.3) of Johnson and more generalized form for p becomes Urbanik (1984) is substituted for c: Urbanik-IN-PLANE SHEAR MODULUS OF ELASTICITY 383 TABLE 2. Stress-strain properties for four materials and dimensionless buckling stress when S = 0.3. EI El G Difference Material (GPa) (GPa) Y ,  v 2  (GPa) P Simple t (%) Steel 207 207 0.303 0.303 78.7 0.99 0.75 0 1 0.75 Paper 6.87 3.20 0.403 0.135 1.69 0.8 1 0.69 5.6 1 0.73 Plate 1 131 165 0.280 0.22 1 47.2 0.85 0.70 4.3 1 0.73 Plate 2 55.8 1 1  1 0.437 0.2 19 34.3 1.1 0.77 -2.5 1 0.75 The result is the expression given by Eq. (2.10') in Table 1. Other equations requiring further generalizations are also listed there. Letting 2 = 1 is equivalent to assuming the St. Venant approximation, and doing so reduces the gen- eralized equations to their forms given by Johnson and Urbanik (1987). FIXED-PLATE ALGORITHM In determining the buckling strain of a fixed- edge plate, the algorithm given by Johnson and Urbanik (1987) sometimes fails to converge when t # 1. A new algorithm, motivated by Fig. 5 of Johnson and Urbanik (1 987), makes use of the fact that in the vicinity of point P,, which defines the minimum i at the optimum X, the form of the curve Y,Y,  can be deter- mined from the interpolating polynomial ap- proximation to Eq. (2.13') given by A relative minimum occurs when dddx = 0 at Therefore, the estimate of an optimum x is updated from the values of b, and b2 deter- mined by fitting i(x) to each three most recent pairs of i-x estimates. Note from Eq. (2.11') (Table 1) that low val- ues of G input to Eq. (2.10') may cause the expression within brackets to become nega- tive. This complicates the plate structure buck- ling analysis of Johnson and Urbanik (1989). If the lowest singularity among the plates is used to upwardly bound the search for a root at a fixed A, the search may seek out complex roots. The form of the complex region appears in Fig. 5 of Johnson and Urbanik (1 987) as the region below and to the right of line R,-R,. The program for analyzing a plate system was therefore augmented by an additional subpro- gram that constrains the search for an opti- mum h [Eq. (4.3)] of Johnson and Urbanik (1 989) to the real region. RESULTS AND CONCLUSIONS Experimental data on steel from Huffington (1 956) and on paper from Suhling et al. (1 989) (summarized in Table 2) can be used to test the severity of the St. Venant approximation. The paper data are for a commercial 205 g/m2 linerboard. The plate data are effective stress- strain properties determined by assuming that the effective plate thickness equals the original thickness and equating experimental rigidities to the values predicted by bending moduli H I , ,  H,,, HI,, and H3,. Plate 1 is a steel plate with parallel ribs machined on one face by remov- ing 16.2% of the material to a depth of 25% of the plate thickness. Plate 2 has its second face equally machined. For comparison, data on mild steel are also given. Figure 1 shows how plate buckling stress would vary with plate stiffness according to the generalized theory of Johnson and Urbanik 384 WOOD AND F'IBER SCIENCE, OCTOBER 1992, V. 24(4) -- Fixed .---- Slmple support 0 0.2 0.4 0.6 0.8 1.0 Stiffness S FIG. I. Plots of L? as a function of S for three levels of (:, two support conditions, and u = 0.25. (1987). More specific comparisons are made in Table 2. Dimensionless buckling stress of a simple support plate is predicted for the case S = 0.3 using the ? value predicted from data for comparison with E = 1. For the limited data, using an approximate shear modulus seems to be sufficient. When the shear mod- ulus is known, the generalized equations from Table 1 can be used to improve the buckling strength predictions. NOMENCLATURE b,, b,, 6, Regression constants C Linear buckling constant c Shear constant c,, c, Constants in uniaxial stress-strain curve a = c, tanh(c,t/c,) t Dimensionless shear constant E,, E, Young's modulus in l -direction and loaded 2-direction f Function of i G Shear modulus of elastic- ity H I , ,  H,,, H,,, H,, Plate bending stiffness moduli P,, R,,  R,, Y,, Y, Points on graph S Dimensionless plate stiffness iu Dimensionless root of buckling equation b Dimensionless root of buckling equation p' Partial derivative of b Normalized buckling strain h Buckling wave periodic- ity 8 Dimensionless buckling stress x Dimensionless periodic- ity of buckling wave u Geometric mean of Pois- son's ratio u , ,  v, Poisson's ratios in l-di- rection and loaded 2-di- rection REFERENCES H U ~ N G T O N ,  N. J . ,  JR. 1956. Theoretical determination of rigidity properties of orthogonally stiffened plates. J. Appl. Mech. March: 15-20. JOHNSON, M. W., JR., AND T. J. URBANIK. 1984. A non- linear theory for elastic plates with applications to char- acterizing paper properties. J. Appl. Mech. 5 l(3): 145- 152. , AND- . 1987. Buckling of axially loaded, long rectangular paperboard plates. Wood Fiber Sci. 19(2):135-146. , AND - . 1989. Analysis of the localized buckling in composite plate structures with application to determining the strength of corrugated fiberboard. ASTM J. Composites Technol. & Res. 1 l(4): 121-127. SETTERHOLM, V. C. 1985. FPL spaceboard-A new structural sandwich concept. Tappi J. 66(6):40-42. SUHLING, J. C., D. E. GUNDERSON, M. W. JOHNSON, AND R .  E. ROWLANDS. 1989. Nonlinear elastic constitutive relations for cellulosic materials. I n  Proceedings, 3rd joint ASCE/ASME Mechanics Conf., July 1-15 1989. 

Effect of in-Plane Shear Modulus of Elasticity on Buckling Strength of Paperboard Plates

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Effect of in-Plane Shear Modulus of Elasticity on Buckling Strength of Paperboard Plates

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