1131UCIAING CIF RAG, MN, PLYWOOD
CYLINDERS IN AXIAL COMPRESSION
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March 1€&6
INFORMAIMN REV
AND REAFFIRML
1962
No. 1322
LOAN COPY
Please return to:
Wood Engineering Research
Forest Products Laboratory
Madison, Wisconsin 53705
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UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST PRODUCTS LABORATORY
FOREST SERVICE
MADISON
5,
WISCONSIN
In Cooperation with the University of Wisconsin
BUCKLING OF LONG, THIN, PLYWOOD CYLINDERS
IN AXIAL COMPRESSION!
By
H. W. MARCH, Head Mathematician
C. B. NORRIS, 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. Original 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
(1)
L7
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, E a , 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
E 1
against the ratio
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
E T =
113, 200 pounds per square inch
11 LT =
123,800 pounds per square inch
The values of Poisson's ratios, aLT and cr TL were taken to be 0.539
and 0.019; respectively. These are theva
ues_
—17
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 E T to E L is taken into account in the calculation of the argument
El
. A few calculations made for p li T increased by 50 percent
1+E (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 E T 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.
On 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 OA, 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
is small or
of types of plywood for which the ratio (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-
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