LONGITUDINALLY STIFFENED JUIN-WALLED
'PLYWOOD CYLINDERS IN AXIAL
COMPRESSION
April 1948
LOAN CO 'Y
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L-goratory
Forest Produc
705
Madison, Wis onsin
INFORMAi
AND RLAt
1956
This Report is One of a Series
Issued In Cooperation with the
ARMY-NAVY-CIVIL COMMITTEE
on
AIRCRAFT DESIGN CRITERIA
Under the Supervision of the
AERONAUTICAL 130ARID
No. 1562
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
LONGITUDINALLY STIFFENEDTHIN-W AT,TAID PLYWOOD
CYLINDERS IN AXIAL COMPRESSION-
By EDWARD W. KUENZI, Engineer
and
C. B. NORRIS, Engineer
S ummary
Results are presented on tests of 1,050 specimens of, longitudinally
stiffened thin-walled plywood cylinders in axial, compression. The specimens
were cylindrical shells of thin plywood (approximately 0.05 inch thick) 18
inches in diameter with inner longitudinal stiffenerscf veneer, plywood, or
solid wood. The stiffener thicknesses varied from 1/48 inch to 1 inch, the
spacings between stiffeners from 1/4 inch to 6 inches, and the number of stiffeners from 9 to 28.
The test data indicate that the average shell buckles at the critical
stress of a complete unstiffened cylinder, provided the spacing between its
stiffeners is not too small and the stiffeners are sufficiently rigid. If the
stiffeners are more closely spaced and sufficiently rigid, the buckling stress
of the shell is greater than that of ttn unstiffened cylinder. No theory is
presented regarding specimens with closely spaced, rigid stiffeners.
Most of the failures occurred, when the shell buckled. A few specimens
having heavy stiffeners sustained loads subsequent to buckling of the shell:
and finally failed because of general instability. No theory is presented
regarding the load at which general instability may occur.
When the efficiency (ratio of load to weight) of stiffened shells was
compared to that of unstiffened shells, it was found that an unstiffened shell
having the same weight and curvature but greater thickness than the stiffened
shell would theoretically buckle at the same, or greater, loads as the stiffened shell. This was particularly true if the stiffeners were heavy, in
which case an unstiffened shell of the saMe weight was theoretically capable
of supporting two to three times the initial buckling load of a stiffened
shell.
-This progress report is one of a series prepared and distributed by the
Forest Products Laboratory under U, S. Navy, Bureau of Aeronautics
No, NBA,PO-11Aer 00619; Amendment No. 1 and U, S. Air Force No.,
USAY-P0-(33-038)48-41E. Results here reported are pteliMinary , and may
be revised as additional data become available,
Report go. 1562
-1-
Introduction
At the time this work was begun at the Forest Products Laboratory (July
1943), it was thought that more information on the behavior of longitudinally
stiffened shells in axial compression was needed for the design of aircraft
structures, Previous work at the Forest Products Laboratory on unstiffened
shells showed that plywood is particularly useful in the experimental determination of the buckling stresses of shell structures, since it is smooth and
can be made thicker than metal shells of the same size and still exhibit
buckling. Accordingly, plywood was used in the extensive series of specimens
included in the work covered by this report.
A theoretical analysis applicable to a stiffened structure of this type
should include the behavior of plates supported at the edges by stiffeners
that are not rigid but that 'can-bend and twist, but as an analysis of this
sort would be exceedingly comgicated, g it was not included in the work covered
by this report. Description of Specimens
The 'specimens Were
circular shells of plywciod'18 inches in diameter
All stiffeners were placed
with-their length parallel to the AXiS of the cylinder : and were equally spaced
around the circumference of the shell. 'The'stiffeners'were of three types,
veneer,- plywood, and solid woad.' Tables 1 (cols. 25to17) and 2 (cols, 14 to
16) give the sty* and'number4COL 3) Ofetiffeners-for each specimen.
having 'stiffeners glued to'their'inner'surfacet.
The plywood'was made *f yellow,--poplar Oz i yeilOW birch veneers of aircraft grade, rotary cut at the Forest Ptahota HLAbOratoty. All of the plywood
was made by bonding the veneers with film glue set in a standard hot-plate
veneer press. The two ends: Of the flat sheet that were to be later joined to
form the circular shell were scarfed. The stiffeners of veneer and those of
send Woed4Orethen placed' with their grain directions parallel to the axis
Oftheehell and were: glued to the plywood in ahot-plate press. The sheet
Was finally bent'tO Cylindrical form, and the scarf joint was glued with a
thermosetting synthetic-resin glue
Th&lAyWebd'etIffener6 were made of three plies, of whieh:thefacmplies
were 1/3a inch thick with' their grain in the circumferential direction, and
the core ply was 1/16 inch thick with its grain in the axial direction. They
were bag molded to the curvature of the shell and finally glued to the shell
by internal bag molding in a female mold.
1Timoshenko, S. "Theory of Elastic Stability," p. 337, 1936.
Chwalla, E. Ingenieur-Archiv, vol. 5. p. 54, 1934.
Report No. 1562
-2-
The ends of specimens having stiffeners 1/8 inch thick or less were
cut square and true in a turning lathe. Those of specimens having thicker
stiffeners were cut by placing the specimen on a mandrel and rotating it into
a fine-toothed, circular, cross-cut saw.
A stiffened plywood shell in which the face grain of the plywood is in
the circumferential direction and the grain of the stiffeners in an axial
direction will develop internal stresses if its moisture content is increased.
Because the stiffeners do not expand so much in the axial direction of the
shell as does the plywood shell, the shell is subjected to compression and the
stiffeners to tension. If the change in moisture content is great enough, the
compression in the shell will cause it to buckle between the stiffeners.
Since a thin stiffener is stretched more easily than a heavy one, buckles in
the plywood shell between stiffeners will not be so likely to appear in a
shell having light stiffeners. Also, if the stiffeners are made of the same
plywood construction as that of the plywood shell, they will expand by the
same amount as the shell and no stresses due to the expansion will occur.
Since the stiffeners were bonded to the shell by means of heat and
pressure, the shell and the stiffeners were quite dry immediately after manu.
facture. An attempt was made to condition all specimens to a constant weight
in a controlled atmosphere of 64 percent relative humidity at a temperature
of 75° F. Buckles in the plywood shell then appeared between the stiffeners
that were 3/8 inch, 1/2 inch, and 1 inch thick. The plywood'of specimens
having lighter stiffeners did not buckle. In order that smooth specimens could
be tested, those that did buckle under conditioning were placed in a dry kiln
until they became smooth, when they were tested immediately:
Coupons of flat plywood were made of material cut from portions of the
log adjacent to the portions used in making the corresponding cylindrical
shell or corresponding plywood stiffener.
Coupons of,stiffeners thicker than 1/8 inch were cut from the ends of
the stiffeners priQr to gluing the stiffeners to the plywood shell.
Coupons orstiffeners of veneer were made of veneers cut from portions
of the log adjacent to the portions used in making the stiffeners. These were
prepared for testing in compression by gluing together six 1- by 4-inch pieces.
The pieces were not glued over their entire surface but were only glued for a
distance of 1/2 inch from each end. The resulting compression specimen was
made up of six veneers having their ends glued together. Thus end failures
during test were prevented.
Each of the coupons received the same conditioning before test as that
of the corresponding stiffened cylinder.
Report No. 1562
-3-
Methods of Testing
All of the stiffened plywood cylinders, except those having 19 or 28
stiffeners 1/2 by 1 inch in cross section, were tested in a hydraulic testing
machine. Those excepted, being heavier specimens, were tested in a screwoperated testing machine of 1,000,000-pound capacity.
The ends of each'cylinder were fitted with external plywood rings,
lightly clamped around the specimen. The purpose of these rings was to'hold
the shell to its circular shape.
With the exception of the heavy specimens having 19 or 28 stiffeners
1/2 by 1 inch in cross section, the cylinders, with end rings attached, were
placed on aflat machined steel plate that rested on a spherical bearing on
the bottom head of the testing machine. The top of the shell was allowed to
bear on the machined surface of the upper head of the testing machine. A
small initial,load was applied. The bottom plate was then adjusted by
tapping on its corners with a mallet. If the bearing was uniform, the
initial load was sustained. If the bearing was not uniform, the load dropped.
Then load was applied again and the bottom plate tapped. This procedure was
continued until no drop in initial load occurred. Screw jacks in the form of
bolts were then placed under the corners of the lower bearing plate to prevent its tilting during the test. The load was applied until failure occurred.
The heavy specimens were tested in the same manner except that no
spherical bearing was used. The ends of these specimens were brought to bear
uniformly by using paper shims.
Methods used to test the coupons are described in detail in the appendix
of this report.
Computation of Results
In order to compare the buckling stress of a stiffened shell to the
theoretical buckling stress of an unstiffened shell the buckling constant k
was computed for each specimen.
The buckling stress of a long plywood cylinder under axial compression
is given by the formula4§.
p =1:ET
r
where EL = modulus of elasticity of the veneers of the plywood, measured
parallel to the grain.
h = thickness of plywood.
r
radius of curvature of cylinder.
3
-This is formula (1) of Forest Products Laboratory Report No. 1322.
Report No. 3.562
-4-
If pp and p s denote the stress in the plywood shell and the stress in
the stiffeners, respectively, and if it is assumed that deformations in the
plywood shell and stiffeners are equal until buckling occurs,
P- -
PEp
(1) (col. 20, table 1)
(col. 19, table 2)
v EsAs + EpAp
PE
and.
s
Ps - E s A s fi EpAp
(2) (col. 21, table 1)
(col. 20, table 2)
where P = load on the specimen.
Es = compressive modulus of elasticity of the stiffener in the
axial direction.
E = compressive modulus of elasticity of the plywood in the axial
direction.
A . cross-sectional area of plywood shell.
A . sum of cross-sectional areas of the stiffeners.
s
Values of E s and Ep were determined from compression tests of coupons.
Then a value of k for any specimen can be found by the formula
k=
(3)
ELh
(col, 22, table 1)
(col. 21, table 2)
Values of EL were determined from bending tests of coupons, and
values of p from equation (1).
1
If the specimen is so short that T z< 1.5, the buckling constant will
be higher than for a long specimen. In this inequality
1 =
length of specimen.
E 1/4
z = ()
, where
El
E1 bending, modulus of elasticity of the plywood in the axial
direction.
E2 = bending modulus of elasticity of the plywood in the circumferential direction.
The effect of length on the buckling stress of unstiffened cylinders is discussed in Forest Products Laboratory Report No. 1514 (revised), The curve of
that report ehowing -- for values of
Report No. 1562
/1
k--z)
-5-
is repeated as figure 1 in this
report. In order to:compare : values of ,buckling constants for short specimens,
kx was computed by dividing the value of k as determined by formula 3 by the
ratio of
for the corresponding value of — z given by figure
The only
ICX)
short specimens were several of those having Sitka spruce stiffeners. Data
for these are given in table 1.
In order to obtain a comparison with unstiffened shells of the same
weight, the buckling stresses and buckling loads of equivalent unstiffened
cylinders were computed by the following formulas:
A)
P
pp equiv. = 0.115 EL(Ass
2
(4)
277x
(AS
P equiv. = 0.115 EL (col. 25, table 1)
(col. 23, table 2)
p)2
(col. 26, table 1)
(5)
2wr 2(col, 24, table 2)
These formulas are obtained from formula: (1) of Forest Products
Laboratory Report No. 1322. They assume that the stiffeners are replaced by
material added to the original shell so that the cross-sectional area is the
same as that of the total area of the stiffened specimen. Thus the equivalent
A, + A,
thickness is '
Y. The mean radius of curvature is assumed to be the same
2 77 r
as that of the stiffened shell. The modulus of elasticity EL is also assumed
to be that of the stiffened shell. The constant k = 0.115 is that obtained
from Forest Products Laboratory Report No. 1322 for all plywood having values
E1
of greater than 0.20 but less than 0.80.
E1 + E 2
Presentation of Results
The results of the 'tests are presented in tabular and graphical form.
Table 1 contains the data obtained from tests of cylinders of yellowpoplar plywood stiffened with Sitka spruce stiffeners. Table 2 contains the
data from tests of yellow birch plywood cylinders stiffened with yellow birch
veneer, plywood, or solid-wood stiffeners, The values presented are averages
of observed and computed values of groups of specimens. In addition to
information relating to dimensions of the specimens, buckling loads, and the
like, the tables contain the computed shell and stiffener stresses, the
buckling constant k, the computed stresses of an equivalent unstiffened
cylinder, ratios of the buckling load of an equivalent cylinder to that of
the stiffened specimen, and compressive proportional-limit stresses of the
shell and stiffener material. The computed values were obtained by using the
formulas previously presented. They were computed for each specimen and then
averaged, thus the tabulated values will not necessarily be obtained exactly
if average observed values are used in the computations.
Report No. 1562
-6-
2
Graphs showing the buckling constant plotted against the parameter
s , where s is the space between the stiffeners, are presented in figures
hrz
2, 3, 4, 5, 6, 7, and 8. Figure 2 shows the constant km for yellow-poplar
plywood cylinders stiffened with Sitka spruce stiffeners. figures 3 to 8
show the constant k for yellow birch plywood cylinders stiffened with
1/48-inch veneers 1/32-inch veneers, 1/24-inch veneers 1/16-inch veneers,
1/8-inch plywood, and 1/4- by 3/8-inch, 1/2, by 1/2-inch, and 1/2- by 1-inch
birch stiffeners, respectively.
Figure 9 shows frequency-distribution curves for the buckling constants
of the stiffened yellow birch plywood cylinders in compression. The number of
occurrences of values of k are plotted versus the values of k for each different size of stiffener,'and for all stiffeners.
Discussion of Results
Behavior of Specimens Under Test
The plywood of all of the specimens buckled between the stiffeners,
This buckling occurred suddenly and involved most of the shell,
If the stiffeners were less than 1/8 inch thick the stiffener also
buckled and additional compressing in the testing machine increased the depth
of the buckles until failure appeared in the plywood or stiffeners or both.
The load after initial buckling continued to decrease as further compression
occurred.
The specimens having heavier stiffeners,exhitited buckles between the
stiffeners; but it was not always apparent whether the buckles grew and involved the stiffeners, because the failure was of an explosive kind that .
shattered the entire specimen. Compression failures were found in several of
the stiffeners. Some of the specimens with heavystiffeners carried loads
greater than the load at which initial buckling occurred. The performance
after initial buckling, see figures 10 and 11, mas , an increase in the depth .
of the initial buckles and also the formatis* Of larger buckles involving
several stiffeners (figs. 12 and 13). Failure occurred by a tearing of the
plywood at the stiffener or by a breaking of several stiffeners.
Discussion of Data
The cylindrical shell between the stiffeners will tend to buckle in a
manner similar to that of a complete cylinder as described in Forest Products
Laboratory Reort No. 1322-A, but the buckling will be restrained by the
presence of the stiffeners. If the spacing between the stiffeners is sufficiently great to allow the formation of complete natural buckles, the critical
stress will be equal to that of a complete cylinder, as is given by Report No.
1322. For more closely spaced stiffeners, an increase in critical stress
might be expected.
Report No; 1562
-7-
The following discussion of the expected increase is based on the
curves shown in figure 14, which is a reproduction of figures 10 to 13, inclusive, of Report No. 1322-A. The ordinates of these curves are proportional
to the critical stresses, and the abscissas are proportionate amplitudes of
the buckles. It is assumed in Report No. 1322-A that the plywood shells are
not perfect cylinders, but were originally buckled, and that the amplitudes
of the original buckles are proportional toe ° . A family of curves is drawn
in each graph, each curve of which represents a certain size of buckle and is
designated by a particular value of
A hr
= ft7r
e
b2'
in which b is the width of the buckle, h is the thickness of the plywood, and
T is the radius of the cylinder, The critical stress associated with each
value of n is the minimum value of the stress indicated by the corresponding
curve. Pour different families of curves are shown, one for each of four
values off°.
It is evident from these four families of curves that the critical
stress of complete cylinders of the plywood for which the curves were drawn,
is given by 211- 0.095, since this is the minimum value of the envelopes of
ELh
the families of curves; that is, the critical stress of a complete cylinder is
independent of the original amplitude, ' 0 1, of the buckles. The value of 71
associated with the critical stress is 0.35 in all of these families, since
this value yields a minimum, and the width of the natural buckle is, therefore, given by 0.35 = 477 21 E .
b2
Now, if the space between the stiffeners, s, is greater than the value
of b given by this equation, there should be room for natural buckles to form
between the stiffeners, and the critical stress will be that of an unstiffened
cylinder. However, if the space between the stiffeners is less than b, the
size of the buckles will be limited,n will have a value greater than 0.35,
and the critical stress will be given by the minimum value of 2E- determined
EL h
on the curve associated with this greater value of 71. An examination of the
four families of curves shown in figure 14 indicates that a considerable increase of critical stress may be expected if the original amplitude of the
buckles, 6 0' is small but practically no increase if e o is large.
This discussion indicates that the effect of stiffener spacing can best
2
be shown by plotting k = 2E- against the parameterfir ,
plots indicated
hr.
ELh
E i/4
(.2)
;
that this parameter should be divided by the aspect ratio z El
2
therefore, the parameter -11-- was employed in figures 2 to 8, inclusive.
hrz
Report 1662
These figures indicate that, with the exception of the cylinders stiffened with the 1/8-inch plywood stiffeners, very little increase in the
critical stress was obtained by the use of stiffeners except for very close
spacings. It seems probable that the differential swelling between the wood
stiffeners and the plywood cylinders caused original waves of sufficient
amplitude to yield this result. The plywood stiffeners were of the same construction as the plywood of the cylinders, and, therefore, differential
swelling did not occur, It is possible, also, that the lighter stiffeners
did not limit the buckle size as sharply as the heavy stiffeners.
The ordinates of figures 2 to 8, inclusive, are values of k = Pr
ELh
corrected by means of figure 1 to the proper values for long cylinders. Thus
2
for large values of s
the values of k should agree with values for unhrz
stiffened cylinders. The agreement obtained is shown by comparing the position
of the plotted points with the short horizontal lines drawn in the figures.
These lines are drawn at positions indicating the values for unstiffened
cylinders as obtained from Report No, 1322. The agreement is shown more
clearly by the frequency curves of figure 9. The value of k for unstiffened
cylinders is 0.115.
The variation in the experimental results can be expected to be large
because of the presence of small local imperfections in the cylinders tested.
This matter is discussed in Reports Nos. 1322, 1322-A, and 1322-B.
A comparison was made between the critical loads obtained from the tests
of the stiffened cylinders and critical loads computed for unstiffened
cylinders of the same weights as the stiffened ones. The critical loads of the
unstiffened cylinders were computed according to equation (5). In this com
putation a value of k of 0.115 was employed as a , good average value for the
types of plywood that would normally be used.
column of tables 1 and
2 gives the ratios of the critical loads of the unstiffened cylinders to those
of the stiffened cylinders. Average values of these ratios, taken from table
2, for cylinders stiffened with stiffeners of different depths are as follows:
Stiffener
depth in.
1/4.8
1/32
1/24
1/16
1/8
3/8
1/2
1
Average values
of ratios
0.93
1.04
1.12
1.24
1.55
1.69
2.30
2.21
These average values indicate that higher critical loads can, in
general, be obtained by the use of unstiffened cylinders than by the use. of
stiffened cylinders. The plywood of which the stiffened cylinders were made
was of 1:2:1 construction ('core twice as thick as one of the faces) with face
grain circumferential. The slight advantage over unstiffened cylinders of
the cylinders stiffened with stiffeners 1/48 inch deep, is probably similar
to the advantage over cylinders of 1:2:1 construction of cylinders made of
plywood for which the stiffness in the axial direction is equal to that in the
circumferential direction.
Report No. 1562 -9-
Conclusions.
-
The, buckling stress of the curved:sheet:between the stiffeners of a,
stiffened 044-!walied plywood cylinder may be taken to be the same as that
of the same cylinder , without stiffeners. If the stiffeners are very closely
spaced,the value of the stress determined by the method described in this
report will be _conservative.
An unstiffened cylinder of the same weight and curvature and, therefore,,,:of_a greater thickness than, the stiffened cylinder, will in ;general,
buckle at a load equal to or greater than that applied to the stiffened
cylinder,
APPENDIX
Testing of Minor Coupons
Bending Specimens
Bending specimens 1 inch wide were tested on a span length 48 or. mOre.
times the thickness of the specimen for face-grain directions parailelte the
span and 24'or more times the'thibkness'for fade-grain directions perpendicular to the span, so that the necessity of correcting the deflection for
transverse'shearintdeformations was eliminated. The ends of the specimens
rested on supports rounded to about 1/16-inch radius. The ;pad was applied:.
through a 1/4-inch-diameter bar placed at the center of the span. Specimens
having their face-grain directions parallel to the span were tested on an
apparatus that measured the load by means of a platform scale reading to
1/100 pound. The rate of loading and driving force was obtained from a
testing machine on which the scale was mounted. Specimens having their
face-grain direction perpendicular to the span were tested on an apparatus
that used water as a load. The load increments could be measured to 10 grams.
_
Load-deflection curves were plotted as the test , was run. Immediately
after testing, 4 sample was Out from the specimen for moisture-content determination. Properties computed from test data were the modulus of elasticity
and moisture content.
Compression Specimens
The plywood -cempreseten Specimens were 1 inch wide. and 4 inches long
with the face-grain direction oriented the same with , respect to the direction
of the applied load AS;it was to thecorresponding part of the cylindrical
shell. Because Of - the thinness of the specimen, an appara tu s consisting of
thin spring fingers was used to give lateral support, The deformation was
Report
156k
measured over a 2-inch gage length with a Marten's mirror apparatus. Loaddeformationciata were recorded as the test was made. Properties computed from
the test data were the modulus of elasticity, the compressive stress at
proportional-limit load, the compressive strength, and the moisture content.
The stiffener compression specimens were 4 inches long. The veneer
stiffener specimens were made as described in "Description of Specimens."
They were tested in the same apparatus as were the plywood coupons. Coupons
of the heavy stiffeners were tested without the lateral-support apparatus.
Report No. 1562
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Figure 11.--Initial buckling of a plywood cylinder having twentyeight 1- by 1/2-inch stiffeners.
Z M 77257 F
Figure 12.--Failure of specimen shown in figure 10. Note the
appearance of large buckles.
z /4
77258 F
Figure 13.--Failure of specimen shown in figure 11. Note the
appearance of large buckles.
2 14
77259 F