MIETI-IDS F CALCULATING TE STRENGT AND MDULUS f!elasticity. F PLYWD IN CMIPPESSIN Revised September 1955 INFRMATIN REVIEWED AN REAFFIRMED 196 No. 1315 UNITED STATES DEPARTMENT F AGRICULTURE FREST SERVICE FREST PRDUCTS LABRATRY Madison 5, Wisconsin In Cooperation with the University, of Wisconsin
METDS F CALCULATING TE STRENGT AND MDULUS 1 F ELASTICITY F PLYWD IN CMPRESSIN- By J. A. LISKA, Engineer Forest Products Laboratory,? Forest Service U. S. Department of Agriculture Summary The experimental determination of the compressive strength properties of plywood, involving hundreds of possible variations in construction and combinations of species, would require a tremendous amount of time and money. Furthermore, such a procedure would mean that each proposed plywood construction would have to be fabricated before information on its compressive strength properties could be determined. To eliminate both of these difficulties, a theoretical analysis was developed that permits computation of the compressive strengths and moduli of elasticity of plywood when the applied stress is parallel or perpendicular to the grain directions of the face plies, and the grain directions of adjacent plies are at 9 to each other. This analysis was checked by 448 compressive tests on plywood specimens. Two species of wood, in 1 veneer thickness and 4 constructions, were evaluated in this test program. Comparisons of experimental and theoretically computed results show good agreement, and indicate that the formulas developed will give accurate indications of the compressive strength properties of plywood. Material Test specimens were fabricated from veneer cut from Douglas-fir and Sitka spruce. Although the exact region of growth is not known, the selected -his report originally was issued during World War II, in cooperation with the Army-Navy-Civil Committee on Aircraft Design Criteria of the Aeronautical Board, for use by the aircraft industry. This revision evaluates the original data to substantiate the theoretical formulas. 2Maintained at Madison, Wis., in cooperation with the University of Wisconsin. Rept. No. 1315 (revised) Agriculture-Madison
veneer bolts were cut into rotary-cut veneer at Tacoma, Wash., in accordance with commercial practice. ne 8-foot bolt was selected from each of 2 trees of each species. The veneer was of high quality and had few defects. Several thicknesses of veneer were cut from each bolt, but only one thickness, 3/16 inch, was used in this study. The veneer was clipped into consecutively numbered sheets 8 feet long and 4 feet wide, and then dried by conventional methods. Upon arrival at the Forest Products Laboratory, the veneer was conditioned to constant weight in a room maintained at 72 F. and 52 percent relative humidity. This conditioning resulted in a moisture content of about 1 percent in the veneer. After conditioning, the veneer was cut to provide closely matched material for a set of four matched test specimens. The specimens were originally fabricated for bending tests. After the bending tests were completed, an undamaged portion of each specimen was selected for compression tests. Specimens were bonded with casein glue (Forest Products Laboratory formula 413) with an average spread of adhesive of.32 pound per square foot of glue line. The individual specimens in each matched set of four contained the same number of plies, but the ply orientation varied as follows: ne specimen had the grain directions of all plies parallel to each other and parallel to the specimen length. Another specimen had similar orientation between plies, but the grain directions were perpendicular to the specimen length. In a third specimen, the grain directions of adjacent plies were at 9 to each other, with the grain direction of the outer plies parallel to the specimen length. The fourth specimen was similar in construction to the third, but the grain direction of the outer plies was perpendicular to the specimen length. Thus, two specimens might be considered to be of laminated construction, while the other two were of conventional plywood construction. The specimen orientation provided for tests on the laminated constructions both parallel and perpendicular to the grain direction and for tests on the plywood constructions both parallel and perpendicular to the grain direction of the face plies. The number and nominal size of specimens of each species and construction tested are given in table 1. The complete program involved 448 specimens. Method of Test All specimens were tested with load applied parallel to the specimen length through a spherically seated loading-head. The load was applied continuously throughout the test at a rate of movement of the movable crosshead of.3 inch per inch of specimen length per minute. The loaded ends of the specimens were smooth, parallel to each other, and at right angles to the specimen length. Deformation data was obtained with a Lamb's roller compressometer mounted over the central portion of the Rept. No. 1315-2-
specimen length. The gage length of the compressometer was 2 inches for the 3- and 4-inch specimens and 6 inches for the.8-inch specimens. Loaddeformation data were obtained throughout each test to the point of maximum load. Theoretical Analysis An expression for computing the effective modulus of elasticity of a plywood prism in compression developed by March-3- is: E y (11 xy ) hi xi i=n (E ) y hi xi (1) 1=1 X i where: E the effective modulus of elasticity in the direction parallel to the X-axis, as indicated in figure 1 (E.x )i = the modulus of elasticity of the ith ply in the X direction (E ) = the modulus of elasticity of the ith ply in the Y direction Y h = the overall thickness of the plywood prism h i = the thickness of the i th ply X i = 1 - (pxy )i (pyx )i which is approximately equal to unity (p ) = the Poisson's ratio of the i th ply denoting the ratio of the xy expansion in the Y direction to the contraction in the X direction due to a compressive stress in the X direction (p ) = the Poisson's ratio of the i th ply denoting the ratio of the yx expansion in the X direction to the contraction in the Y direction due to a compressive stress in the Y direction. 3 March,. W. Stress Strain Relations in Wood and Plywood Considered as rthotropic Materials. Forest Products Laboratory Report No. 153. Feb. 1944. Rept. No. 1315-3-
This expression neglects irregularities in the state of stress at the edges of the prism and assumes that the strain components are uniform across the thickness of the prism. The expression is a general one for plywood in which the grain directions of adjacent plies are parallel or perpendicular to each other and parallel or perpendicular to the edges of the prism. The plies may be of any species and of rotary-cut or quarter-sliced veneer, as long as proper consideration is given to these factors when substituting values for the elastic moduli and Poisson's ratios of the individual plies. It can be shown that the term following the minus sign in the general expression is small and may be neglected without significant error. Substituting unity for Xi, the expression may then be written: (2) This formula may be expressed in terms of the longitudinal, tangential, and radial grain directions of the individual plies. Conventional nomenclature for the effective modulus of elasticity of the plywood -- Ew when the load is applied parallel to the face grain, and E x when the load is applied perpendicular to the face grain -- can be substituted for E x. The expression for the effective modulus of elasticity may then be expressed as: i=n Ew, E x = Ei hi (3) i=1 where E i = either E Le the compressive modulus of elasticity of the individual veneers parallel to the grain; E T, the compressive modulus of elasticity of the individual veneers in the tangential direction; or E R, the modulus of elasticity of the individual veneers in the radial direction; depending on the orientation of the grain in the i uh ply and whether the veneer is rotary-cut or quarter-sliced. If all of the veneers in the plywood are of the same species and are rotary-cut, as was the case in this study, formula (3) may be written: Rept. No. 1315-4-
1, E E = + EThT) W X (4) where hii = total thickness of the longitudinal plies ht = total thickness of the tangential plies. Formula (4) may also be expressed in terms of the ratio of the 2 moduli of elasticity. E Lc Ew, E x T E T E Lc (5) When quarter-sliced veneer is used instead of rotary-cut veneer, the subscript Twill be replaced by R and appropriate properties substituted. The moduli of elasticity in known exactly for many wood ET may be made by assuming ELc the tangential and radial directions are not species. In these instances, an approximation Ep and ==- to be.5 and.1 respectively. E Lc The assumptions that the strain components are uniform across the thickness of the loaded prism and that irregularities in stress at the edges of the prism are neglected lead to the general assumption that the unit strain in the prism is equal to the unit strain in the individual plies. Since the unit strain is equal to the stress divided by the modulus of elasticity, the following expressions for stresses within the proportional limit are valid when the load is applied parallel to the face grain and perpendicular to the face grain, respectively. CW f cl E w ELc (6) and f f cx = E x ELe (7) Formulas (6) and (7) may be written: and E w f ew = ELc f cl E, f = f CX ELc cl (8) (9) Rept. No. 1315-5-
where fcw the average stress in a plywood prism when a load is applied parallel to the face grain. It is the average stress at the proportional limit or ultimate respectively, when fel, is the proportional limit strength or ultimate strengtr-- of the solid wood species in compression parallel to the grain. f the average stress in a plywood prism when a load is applied cx perpendicular to the face grain. It is the average stress at the proportional limit or ultimate respectively, when f is the proportional limit strength or ultimate strength of the solid wood species in compression parallel to the grain. When formulas (8) and (9) are used to compute stresses at ultimate, it is assumed that the stress-strain curves to maximum stress for the prism and the individual parallel plies are similar, or that the ratios of the secant moduli are the same as those of the elastic moduli. Since the deformation of the prism, including that at ultimate,will be influenced primarily by the behavior of the, parallel plies, this assumption appears reasonable and only small errors will be introduced through its use. When more than one species is used in the plywood, the species having the lowest ratio of f cl E must be used to compute the stress in the plywood prism. Lc Thus formulas (5), (8), and (9) provide methods of computing the stresses and moduli of elasticity in plywood prisms. Values computed by these formulas were compared with those obtained by the tests in this program to verify the theoretical relationships. Analysis of Data and Discussion of Results Comparisons of theoretical and experimentally determined properties of four constructions of Sitka spruce plywood are presented in table 2, and similar data for Douglas-fir plywood are given in table 3. The first line of data under the column "Test Properties" for each construction and log lists the strength and elastic properties obtained from compression tests on specimens with all the plies parallel to each other and parallel to the direction of load application. These data are similar to those obtained from compression parallel to the grain tests on solid wood. The values for ultimate crushing strength and fiber stress at the proportional limit may be substituted for f_, in formulas (8) and (9). The expression for "22. the modulus of elasticity obtained is E lie. Test data obtained on specimens with the grain direction of all plies parallel to each other and perpendicular to the direction of applied load are given for each construction and log. Rept. No. 1315-6-
Since the veneers used in this program were rotary-cut, the expression for the modulus of elasticity is E T. Thus the data needed to compute the strenth and modulus of elasticity of plywood by means of formulas (4), (5), (8), and (9) are available. The last 2 lines in each group of 4 are the compressive properties for specimens with the grain direction of adjacent plies at right angles and the face grain direction parallel or perpendicular, respectively, to the direction of load. The average moisture content and specific gravity values shown indicate good matching between the laminated veneer and plywood specimens. Therefore, no adjustments need to be made for either of these factors when computing the plywood properties. Using formulas (4) or (5) and (8) and (9), the average properties of each plywood construction and log were computed from the data from the laminated veneer tests. These values, as well as the ratios of computed to test values, are presented in tables 2 and 3. For a more rapid comparison of the accuracy of the theoretical formulas, the experimental and theoretical results are plotted in figure 2. Perfect agreement between theoretical and experimental results would mean that all points would lie along the 45 line in figure 2. This agreement was not obtained, but most of the test data were within 1 percent of the computed values. The envelope lines on figure 2 establish the 1 percent limits. From the data presented in figure 2 and tables 2 and 3, it is evident that the theoretical formulas give a good estimate of actual plywood properties. Thus the average strength properties of plywood of any construction can be computed from the properties of the species of wood used. Normally, the properties of plywood in compression will be computed from the compressive properties already obtained for solid wood. Such properties may be found in U. S. Department of Agriculture Technical Bulletin No. 479, "Strength and Related Properties of Woods Grown in the United States" or the ANC-18 Bulletin, "Design of Wood Aircraft Structures." Published values for solid wood should be adjusted for the specific gravity and moisture content of the plywood according to accepted procedures before computations of plywood strength properties are made. The above publications do not give a modulus of elasticity value for compression parallel to the grain. This value can be obtained by increasing the bending modulus of elasticity by 1 percent to correct for shear deformation. In formulas (4) and (5) the second term in the parentheses is small in comparison with the first term. Thus, approximate values can be obtained by considering only those plies with grain direction parallel to the direction of stress and omitting the second term in these formulas. Rept. No. 1315-7-
Conclusions 1. Comparisons of experimentally determined compressive strength values of plywood, with the grain direction of the face plies parallel or perpendicular to the direction of applied stress and the grain directions of adjacent plies at 9 to each other, show good agreement with compressive strength values computed with theoretically developed formulas. Therefore, these plywood properties can be determined theoretically from basic strength data already available for solid wood. This means a tremendous saving in cost and time over that which would be required if each proposed plywood construction had to be evaluated by test. 2. The effect on compressive properties of the plies with grain direction perpendicular to the direction of applied stress is small compared to the effect of the plies with grain direction parallel to the direction of applied stress. Therefore, a good approximation of the compressive properties can be obtained by considering only the plies with grain direction parallel to the applied stress direction when using the theoretical formulas. Rept. No. 1315-8- 1.-14
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2,4 6, 2, 5, :::1 Q: /6 it 4, ti LEGEND: DUGLAS- FIR S1TKA SPRUCE ULTIMATE CRUSING STRENGT n FIBER STRESS AT PRPRTINAL LIMIT LI MDULUS F ELASTICITY 1, 2, 3, 4, 5, 6, CMPUTED ULTIMATE CRUSING STRENGT R STRESS AT PRPRTINAL LIMIT 4 B /,2 1,6 2, 2,4 CMPUTED MDULUS F ELASTICITY (1, P.5.1.1 Figure 2. --Comparison of experimental and computed properties of plywood. z 14 16 791