1131UCIAING CIF RAG, MN, PLYWD CYLINDERS IN AXIAL CMPRESSIN I nieftnatiett IkavictuciLn4-114cattirmyd March 1 &6 INFRMAIMN REV AND REAFFIRML 1962 No. 1322 LAN CPY Please return to: Wood Engineering Research Forest Products Laboratory Madison, Wisconsin 53705 r giii011111111111111ifinitin HiiiM1111 UNITED STATES DEPARTMENT F AGRICULTURE FREST PRDUCTS LABRATRY FREST SERVICE MADISN 5, WISCNSIN In Cooperation with the University of Wisconsin
BUCKLING F LNG, THIN, PLYWD CYLINDERS IN AXIAL CMPRESSIN! By H. W. MARCH, Head Mathematician C. B. NRRIS, Principal Engineer and E. W. KUENZI, Junior Engineer Forest Products Laboratory, Forest Service U. S. Department of Agriculture Introduction This preliminary report presents a curve for the buckling stress of long, thin, plywood cylinders plotted as a function of a parameter which is determined by the construction and elastic constants of the plywood. The curve expresses the results of an approximate mathematical analysis. The results of a rather extensive series of tests are compared with those predicted by the theoretical treatment. The buckling stresses determined by test show considerable scatter. This was to be expected from the results of tests that are to be found in the literature on the buckling of thin metal cylinders and also from theoretical considerations. It was pointed out by Coxl and by von Karman, Dunn, and Tsien± that the behavior of a cylinder buckling under axial compression is analogous to that of a column with a nonlinear elastic support. The buckling load of such This is one of a series of progress reports prepared by the Forest Products Laboratory relating to the use of wood in aircraft issued in cooperation with the Army-Navy-Civil Committee on Aircraft Design Criteria. riginal report published in 1943. 2 Maintained at Madison, Wis., in cooperation with the University of Wisconsin. 3 Cox, H. L. Jour. Roy. Aero. Soc. 44:230, 1940. 4 von Karman, Th.,, Dunn, L. G., and Tsien, H. S. Jour. Aero. Sciences 7:276, 1940. Also Tsien, H. S. Jour. Aero. Sciences 9:373, 1942. Report No. 1322-1- Agriculture-Madison
a column is highly sensitive to initial irregularities of shape or material properties. This sensitivity is greater in the case of a cylinder because of the additional dimension in which irregularities and their effects can be manifested. Theoretical Curve and Formula The buckling stress p of a long cylinder will be represented by the formula p.ke L7 (1) where k is a constant depending upon the construction of the plywood from which the cylinder is made, E L is the Young's modulus parallel to the grain of the wood In the veneei's of the plywood, all considered to be of the same species, h is the thickness of the cylinder, and r is its radius. The constants E 1, E2, Ea, Eb, 'LT, (TL,' and LT as defined in... Forest Products Laboratory Reports Nos. 1312 and 1316 all enter in the determination of the factor k of equation (1) for cylinders of rotary-cut plywood. It was found that a smooth curve could be drawn which would represent quite well the values of k plotted against the ratio E 1 The values of... k for various types of (E ie2) Douglas-fir plywood are given in table 1. The calculations were based upon the following tentative values of the constants of Douglasfir at 10 percent moisture content: E L = 1,960,000 pounds per square inch ET = 113, 200 pounds per square inch 11 LT = 123,800 pounds per square inch The values of Poisson's ratios, a- LT and cr TL were taken to be 0.539 and 0.019; respectively. These are theva 17 ues_ 5 for spruce. The results would not be materially affected by considerable changes in Jenkin, C. F. Report on Materials of Construction Used in Aircraft Aeronautics Research Committee. (London 1920). Report No. 1322-2-
these ratios. The values of k in table 1 are represented by the points denoted by crosses (x) in figure 1. The curve of this figure is a smoothed curve drawn through these points. In formula (1) the constant k appears as a factor multiplying EL. The use of the curve of figure 1 for cylinders made of plywood of other species than Douglas-fir implies that a moderate change in the ratio of the modulus of rigidity fl.lt to Young's modulus E L does not greatly affect the values of k and that the same statement is true for changes in the relevant Poisson's ratios. The effect of a change in the ratio of ET to E L is taken into account in the calculation of the argument El li 1 + E. A few calculations made for p T increased by 50 percent (E showed an increase in k of 5 percent or less. A survey of the steps in which the Poisson's ratios enter into the calculations indicates that the influence of moderate. changes in their values would be small. For the same reasons it appears that the curve of figure 1 can be used for cylinders made of plywood with quarter-sliced veneers. The difference in the values of ER and ET would be taken into account in the calculation of El and Ez while the usual differences between PLR and i LLT do not appear to be great enough to be serious. When a cylindei-77 made of plywood with veneers of different species, it appears that EL in formula (1) should be the modulus of the predominant spee-fg. Results of Tests with Grain of Face Plies in the Axial or Circumferential Direction The results of tests of birch and poplar plywood cylinders under axial compression are shown in figure le Each point represents the mean of the buckling stresses of several cylinders of the same construction and dimensions. In figure 2 each point represents the buckling stress of a single cylinder. A detailed description of the tests will appear in a separate report. For cylinders. 10.5 inches in diameter, the thicknesses ranged from. 0.03 inch to 0.07 inch while for those 18 inches in diameter the thickness ranged from 0.06 inch to 0.11 inch. The length of the cylinders varied from twice the diameter to four times the diameter for the cylinders 10.5 inches in diameter and was equal to about twice the diameter for the larger cylinders. From tests on isotropic cylinders made by other observers and from a few exploratory tests made on plywood cylinders to determine the influence of length on buckling stress, it appears that the buckling stress Report No. 1322-3-
of a cylinder whose length is equal to or greater than twice its diameter does not differ greatly from that of a very long cylinder. n this basis, the buckling stresses of the cylinders tested are compared with those predicted by formula (1) for infinitely long cylinders of the same construction. Further tests are in progress to determine the effect of length on buckling stress. Although the cylinders were well made and were free from obvious imperfections of shape or material, the observed buckling stresses show considerable scatter. There is ample reason from the theoretical standpoint for expecting this behavior because the buckling stress is extremely sensitive to minute initial imperfections. A similar scatter of experimental buckling stresses in the case of isotropic cylinders has been noted by previous observers. The curve of figures 1 and 2 was constructed using a certain minimum value of k for each type of plywood. With the aid of the theory to be presented in a later report, it can be seen that a buckling stress can be attained that is greater than the stress associated with this, minimum value of k. Also in the case of a considerable imperfection, failure may be initiated by large bending stresses in the vicinity of the imperfection. The failing compressive stress may under these circumstances be less than that given by the curve of figures 1 and 2. These reasons for scatter exist in addition to the unavoidable lack of complete agreement of the elastic constants of the plywood of the cylinder and those of the minor test specimens used for the determination of these constants. Reduction of Bucklin. Stress in Vicinity o Stress at Proportional Limit in Compression It is evident that the buckling stress may be expected to be reduced if the axial compressive stress approaches the stress of the plywood at proportional limit in compression. In figure 3 the ratio of the observed buckling stress to the stress at proportional limit is plotted for each group of cylinders against the ratio of the computed buckling stress from the curve of figure 1, to the stress at proportional limit. For each group of cylinders represented by a point in figure 3 the stress at proportional limit in compression was determined from minor test specimens. The general tendency of the plotted points to fall farther below the line A, whose slope is 1, as the point (1, 1) on this line is approached reflects the influence of the approach of the axial compressive stress to the stress at proportional limit in compression. Report No. 1322-4-
Results of Tests with Grain of Face Plies at an An le of 45 to the Axial Direction The mathematical analysis assumes that the grain of the face plies of the cylinders is either vertical or horizontal, that is, parallel or perpendicular to the axial direction. The results of tests, shown in figures 1, 2, and 3 are for such cylinders. Tests on a number of cylinders having the grain of the face plies at an angle of 45 * to the axial direction indicate that for these cylinders the buckling stress is intermediate between the buckling stresses of two cylinders of plywood of the same construction, one having the grain of the face plies vertical, the other having this grain horizontal. The results of such tests are shown in figure 4 which is similar to figure 3 where the grain of face plies was either horizontal or vertical. The computed stress was taken to be the average of the computed stresses of two cylinders of the same construction, one having the grain of the face plies horizontal, the other having this grain vertical. Conclusion In conclusion, it should be emphasized that the mathematical analysis leading to the construction of the curve of figure 1 involved the use of approximate methods. However, it appears to disclose the influence of important factors such as the construction of the plywood on the buckling stress of cylinders. Because of the steepness of the theoretical curve at the extreme right and left hand portions, a great deal of variability of test results is to be expected in these regions. Consequently, it appears advisable to avoid, when possible, the use of types of plywood for which the ratio is small or (E 1 +E2) nearly equal to unity. Suitable limiting values of this ratio would appear to be its values as shown in table 1 for three-ply construction with the face plies one-half as thick as the core and with these plies horizontal and vertical, respectively. Report No. 1322-5-.3-13
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SUBJECT LISTS F PUBLICATINS ISSUED BY THE FREST PRDUCTS LABRATRY The following are obtainable free on request from the Director, Forest Products Laboratory, Madison 5, Wisconsin: Box and Crate Construction and Packaging Data Chemistry of Wood and Derived Products Fungus Defects in Forest Products and Decay in Trees Glue, Glued Products, and Veneer Growth, Structure, and Identification of Wood List of 'publications on Mechanical Properties and Structural Uses of Wood and Wood Products Partial list of publications for Architects, Builders, Engineers, and Retail Lumbermen Fire Protection Logging, Milling, and Utilization of Timber Products Pulp and Paper Seasoning of Wood Structural Sandwich, Plastic Laminates, and Wood-Base Aircraft Components Wood Finishing Wood Preservation Partial list of publications for Furniture Manufacturers, Woodworkers and Teachers of Woodshop Practice Note: Since Forest Products Laboratory publications are so varied in subject no single list is issued. Instead a list is made up for each Laboratory division. Twice a year, December 31 and June 30, a list is made up showing new reports for the previous six months. This is the only item sent regularly to the Laboratory's mailing list. Anyone who has asked for and received the proper subject lists and who has had his name placed on the mailing list can keep up to date on Forest Products Laboratory publications. Each subject list carries descriptions of all other subject lists.