TORSIONAL CUCICILING Of LONGITUDINALLY/
STIFFENED, THIN-WALLED, !PLYWOOD
CYLINDERS
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INFORMATION REVIEWED
AND REAFFIRMED
1962
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 [CARD
No. 1563
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison, Wisconsin
In Cooperation with the University of Wisconsin
TORSIONAL BUCKLING OF LONGITUDINALLY STIFFENED,
THIN-WALLED, PLYWOOD CYLINDERS1
By EDWARD W. KUENZI, Engineer
and
C. B. NORRIS, Engineer
Summary
The results of torsion tests performed at the Forest Products Laboratory upon stiffened, thin-walled, plywood cylinders showed that the buckling
stress of the portion of the shell between the stiffeners is about 85 percent
of the buckling stress of an unstiffened cylinder of the same curvature and
thickness. The cylinders tested were about 0,04 inch thick, were curved to
a 9-inch radius, and had from 11 to 28 longitudinal stiffeners from 1/32 to
1/2 inch thick glued to the inner surface. The theoretical buckling torque
of an unstiffened cylinder of the same weight and curvature as a stiffened
cylinder was found to be greater than that for the stiffened cylinder.
Introduction
At the time this work was begun (July 1943), it was thought that more
information on the behavior of longitudinally stiffened shells 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 ex p erimental determination of the buckling stress 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
series of specimens included in the work covered by this report.
A theoretical analysis applicable to a stiffened structure 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 is
exceedingly complicated, it is not included in the work covered by this
report.
-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-NAer 00565 and Army Air Force No. AAF-P0-(33-038)46-1189. Results
here reported are preliminary and may be revised as additional data
become available.
21Timoshenko, S., "Theory of Elastic Stability," p. 337, 1936.
Chwalla, E., Ingenieur-Axchiv, vol. 5, p. 54, 1934.
Report No. 1563
-1-
Description of Specimens
The specimens consisted of cylinders of yellow birch plywood having
longitudinal yellow birch stiffeners glued to their inner surfaces. The
shell and the stiffener sizes are given in table 1. The plywood was of four
plies of 1/100-inch yellow birch veneer, in which the grain direction of
face plies was circumferential and the grain direction of core plies was
axial; it was thus equivalent to a three-ply, 1:2:1 construction. From
eleven to twenty-eight longitudinal stiffeners of different widths and of
thicknesses from 1/32 to 1/2 inch were glued to the inner surface of each
cylinder. The method of manufacture employed was identigal to that
described in Forest Products Laboratory Report No. 1562.2 This work included
tests on 320 plywood cylinders. Coupons of the plywood and the stiffeners
were also prepared and tested as described in that report.
Methods of, Tetting
The ends of the specimens were fitted with maple plugs 3 inches thick.
The plugs were of such diameter that they fitted inside the ring of stiffeners, and the spaces between the stiffeners were filled with shims the same
thickness as the stiffeners. The circumference of the plug was covered with
a thin, rubberized, glass cloth to increase the friction between the plug and
specimen. A slight taper on the pltg insured a tight fit. After the plugs
were placed, steel straps were clamped around the outside of the specimen.
A couple was applied through cross arms attached to each end plug as shown
in figure 1. The couple was slowly increased until failure occurred.
Computation of Results
The stresses in the plywood shell were computed by the formula
2R3T
T . 4
IT(R3 - R24) +
4
nb
(R2 - R14)
R1 4. R2
(1)
where T = torque
R3 = outside radius of curvature of shell
R2 = inside radius of curvature of shell
R1 = inside radius of curvature of stiffener
n = number of stiffeners
b = width of stiffeners
3
—
Forest Products Laboratory Report No. 1562, "Longitudinally Stiffened Thinwalled Plywood Cylinders in Axial Compression." In press.
Report No. 1563
-2-
This formula is derived as follows:
Figure 2 shows the notation in which x, r, and 0 are cylindrical
coordinates.
From figure 2
y=
If p. is the modulus of rigidity, then
=H
1.4
12_
where T is the shear stress.
The energy of a differential volume dV is
23
r
dEi = 1/ 2 - dr d clx
12
Then the total internal energy is
2nb
1 au R3t
R 1+R2 R2
r
3dr de dx + 1/2/1,s 2
r3dr de dx
i/a4c —2 Jf jjr
0 0 R2
0 0
111
where IL and
are the moduli of rigidity of the cylindrical shell and
stiffener, respectively.
Integrating,
1/4
2
4
R
c
4
4
RI )]
) +
2
2R1+ R
(R
2
The external energy is
e
e
= 1/2 TO
where T is the torque.
Report No. 1563 -3-
Equating the external energy to the internal energy
E
or
(k2
1/2
4
--
nbils
4.
R + R2 ( -2 - R1 )1 •
1
4
-2 )
= 1/4 7 [ Try- c k..3
4) R3
Wing the relationship Tc
and solving for
=Ac
T
T
C
2i1cTR3
= 7Tµ
4)
4
R3 - R2
nb As (B24
R1+ R2
p1
4)
Now if it is assumed that /lc
then
2TR
T = 3
4n
4
4 b
- R2 ) + (n2 - n14 )
R
1 R2
w(R3
which is equation (1). This equation is approximate because of the assumption that the shear strain in the cylinder and in the stiffeners is directly
proportional to the coordinate r and independent of coordinate 4), which is
not strictly true,
Previous work on the buckling of unstiffened thin-walled plywood
cylinders in torsion is presented in Forest Products Laboratory Report
No. 1529,1 The buckling stress can be calculated by the formula
7-
cr
=kE
Lr
-Forest Products Laboratory Report No. 1529, "Buckling of Thin-walled'
Plywood Cylinders in Torsion." 1945.
Report No, 1563
-4-
where EL is the modulus of elasticity in the direction of the grain
of the veneer of the plywood, k is a coefficient depending on the size and
construction of the cylinder, h is the thickness of the cylinder wall, and
r is the mean radius of the cylindrical shell of plywood. Values of E L were
computed, from data obtained from tests of coupons of the plywood.
For this report, values of k were taken from the curves of figure 3.
These curves were computed by the method of Forest Products Laboratory
Report No. 1529, for a three-ply plywood having the face plies half as thick
as the core and the grain direction of the face plies circumferential.
In order to compare a stiffened cylinder with an unstiffened one of
the same weight and curvature but greater thickness, the buckling torque of
such an equivalent cylinder was computed by the formula
rh 2
eqUiv. = auk
eEL
L e
where he h + 1.112-c-1 and k was determined for a J
e
2yr
12
he r
Presentation of Results
The results of tests are presented in tabular and graphical form.
Table 1 gives the sizes of the specimens, the buckling torques, and
the stresses in the plywood. It also contains the moduli of elasticity, as
determined from the test data of coupons, and the computed buckling stresses.
The ratios of T
• /T are tabulated beside values of
equiv.
Tequiv.
The graph of figure 4 shows the experimental buckling stresses in the
plywood plotted against theoretical values for an unstiffened cylinder of
the same dimensions and plywood construction.
Discussion of Results
All the specimens failed by buckling in the space between the stiffeners. The thinner stiffeners allowed the buckles to grow across the
stiffener. 1•iost of the specimens failed immediately at the time of buckling.
No additional load greater than that required to cause buckling was necessary
to promote failure. The failures were usually explosive'in nature, causing
the plywood to be broken into small pieces and separating the stiffeners from
the cylinder if they were too rigid to buckle.
Report No. 1563 -5-
A comparison of the experimental with the theoretical buckling
stresses, as shOwn in figure 4, indicates that the experimental buckling
stresses are about 15 percent lower than the computed values. No satisfactory explanation has been found for this discrepancy. The scatter of
the points is typical of data pertaining to the buckling of cylinders.
The buckling torques of stiffened cylinders and those of utstiffened
cylinders of the same weight may be compared by referring to the column
headed equiv.
T• iT in-table 1. The equivalent cylinder is stronger for all
the specimens, except one, and the ratios u'eauiv. range from 0.984 to 2.539.
Conclusions
The torsional buckling stress of the curved shell between the
stiffeners of a stiffened plywood cylinder is about 85 percent of that of
an unstiffenod cylinder of the same dimensions and plywood construction.
The theoretical buckling torque of an unstiffened, cylinder of the
same weight and curvature as a stiffened cylinder is greater than or equal
to that for the stiffened cylinder.
Report No. 1563
-6-
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Figure 1.--Sketch of testing device.
Figure 2.--Notation for stiffened cylinders in
torsion.
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Figure 4.--Experimental vs. computed buckling stresses
of stiffened plywood cylinders in torsion.
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