TORSIONAL BUCKLING OF LONGITUDINALLY 

STIFFENED, THIN-WALLED, PLYWOOD 

CYLINDERS 

January 1948 

INFORMATION REVIEWED 

AND REAFFIRMED 

1962 

This Report is One of a Series 

Issued In Cooperation with the 

ARMY-NAVY-CIVIL COMMIlTEE 

on 

AIRCRAFT DESIGN CRITERIA 

Under the Supervision of the 

AERONAUTICAL BOARD 

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 CYLINDERS 

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 experimental 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,2 it is not included in the work covered by this 

report, 

1This 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-NAer00565 and Army Air Force No. AAF-PO-(33-038)46-1189. Results 

here reported are preliminary and may be revised as additional data 

become available. 

2Timoshenko, S., "Theory of Elastic Stability," p. 337, 1936. 

Chwalla, E., Ingenieur-Archiv. 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 war 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 identical to that 

described in Forest Products Laboratory Report No. 1562. 3 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 Testing 

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 plug 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. 

The 

where T = 

R3 = 

R2 = 

R1 = 

n = 

b = 

Computation of Results 

stresses in the plywood shell were computed by the formula 

torque 

outside radius of curvature of shell 

inside radius of curvature of shell 

inside radius of curvature of stiffener 

number of stiffeners 

width of stiffeners 

3

Forest Products Laboratory Report No. 1662, "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 are cylindrical 

coordinates. 

From figure 2 

If μ is the modulus of rigidity, then 

where 7 is the shear stress. 

The energy of a differential volume dV is 

Then the total internal energy is 

where μc and μs are the moduli of rigidity of the cylindrical shell and 

stiffener, respectively. 

Integrating, 

The external energy is 

where T is the torque, 

Report No. 1563 -3-
Equating the external energy to the internal energy 

or 

Using the relationship 

and solving for 

Now if it is assumed that 

then 

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 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. 1529.4 The buckling stress can be calculated by the formula 

4 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 EL 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 

where was determined for a 

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 Tequiv./T are tabulated beside values of 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 dinensions 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. Most 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 unstiffened 

cylinders of the same weight may be compared by referring to the column 

headed Tequiv./T in table 1. The equivalent cylinder is stronger for all 

the specimens, except one, and the ratios 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 unstiffened 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-


Figure 1.--Sketch of testing device. Z M 75891 F 
Figure 2.--Notation for stiffened cylinders in 

tors ion. 

Z M 75892 F 

Figure 4.--Experimental vs. computed buckling stresses 

of stiffened plywood cylinders in torsion. 

Z M 75894 F