EFFECT OF AXIAL STIFFENERS ON THE
IBUCICLING PROPERTIES OF THIN CURVED
PLYWOOD PLATES iN AXIAL COMPRESSION
March 1948
Vjtri
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 MARV
No. 11507
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison 5, Wisconsin
In Cooperation with the University of Wisconsin
Ekil tCT
OF AXIAL STIFFENERS ON THE BUCKLING
PROPERTIES OF THIN CURVED PLYWOOD PLATES IN
AXIAL COMPRESSION1
By
FRED WERREN, Engineer
and
C. B. NORRIS, Engineer
Summary
This report presents the results of tests and analysis of data on 78 curved
plywood plates tested in axial compression. Each plate was unstiffened or had
a single stiffener parallel to the applied stress.
The results of the tests show that the addition of an axial stiffener to a
curved plywood plate in compression does not increase the buckling stress of
the plywood. The effect of the stiffener is to carry additional load and,if
stiff enough, to break up the buckle pattern. The theoretical buckling stress
can be determined by the same methods used for a cylinder of finite length.2
Introduction
In many of the structural uses of plywood, particularly in aircraft, it is
possible to provide additional strength and rigidity to a plywood shell by
the addition of stiffeners. The behavior of a specific section of a structure can be approximated by testing a prototype. If a sufficiently rigid
stiffener is added to a plywood plate, the plywood can be made to buckle on
either side of the stiffener instead of across the stiffener.
-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)46-1189E. Results here
-reported are preliminary and may be revised as additional data become
available.
?A report of the effect of circumferential stiffeners on curved plywood plates
in axial compression is now under preparation at the Forest Products Laboratory by T. B. Heebink.
Forest Products Laboratory Report No. 1514. "Effect of Length on the Buckling
Stress of Thin-walled Plywood Cylinders in Axial Compression," by E. W.
Kuenzi. (Revised 1948).
Rept. No. 1567
-1-
It was the purpose of the experiments cOn44PtecVfOithiereport to determine
the effect of an axial stiffener on the buckling characteristics of a curved
plywood p4te in axial compression. (Sitilarteate have been made on plywood
cylinders._) A comparison of the observed buckling stresses of the plates
and theoretical buckling stresses of comparable cylinders is presented in this
report. Variables in the curved plywood plates include (1) three constructions,
(2) two veneer thicknesses, (3) length to width ratios from 1.0 to 3.7, (4)
length to radius ratios from 1.2 to 1.8, (5) width to radius ratios from 0.25
to 1.0, and (6) grain direction of-face plies either parallel or perpendicular
to the direction of the applied streaé.
Description of Material
The veneers used in making the plywoodplates for these tests were rotary-cut
yellow birch and yellow-poplar of aircraft grade. Each plate was made from
veneers of a single species into three-, four-, or five-ply constructions,
The $rP411,4ireotion ofeaCh ply was perpendicular to the grainof the adjacent
13140S4 ' *; Cept in the fourplr,COnstruction in which thegrain-directiOna,pfthe two center plies were parallel to_ each. other. The plates were fOriqopcv
by bag-molding to the desired radius of curvature, using a phenolic-resin film
glue to bond adjacent plies.
Axial stiffeners were cut from a single Sitka spruce plank,. quarter-sawn into
strips 5/16 inch by 1 inch in cross section.: Test coupons-4 inches long were
cut from the ends of each strip. The strips were then ripped to-the proper size
of stiffener and glued to the concave face of the curved plate with cold-setting
resorcinol glue.
Static-bending and compression coupons were cut from the outside edge of each
panel while trimming to finished size.
All plates were made as shown in the sketches of figure.... The two curved,OP4
pieces were notched to a width and depth Sufficient to receive the axial
stiffener. After the stiffener had been glued to the panel, the panel was
nail-glued with a cold-setting resorcinol glue to the end pieces and conditioned
in an atmosphere at a temperature of 75' F. end a rel-atiye humidity of 64 percent. Prior to testing, the two curved ends of the plate were carefully
machined to insure flatness of the plywood and tb intlire equal length of plywood and! aial stiffener. The side rails were placed on the plate at the time
of test; thus providing simply supported edges:
Methods of Test
Compression tests of the plywood plates were made in a.100,000-pound hydraulic
testing machine. The specimen was centered on a 4e4YY,PP1 plate supported.
On'a 5.-inch knife edge (fig, 2), Thus the lower_ bearing.
in effect a
Forest Products Laboratory Report No. 1562. "Longitudinally Stiffened Thinwalled Plywood Cylinders in Axial Compression," by E. W. Kuenzi. 1947.
Rept. No. 1567
-2-
spherical head with negligible friction in a direction perpendicular to the
knife edge. It was thought that this type of head would not retard the formation of buckles when the buckling load was reached. The top of the specimen
rested squarely against the upper head of the testing machine.
A small initial load was put on the plywood plate and the plate carefully
centered and adjusted to obtain uniform bearing on the ends. The load was
then increased-to about one-fourth or one-half of the estimated buckling load,
and the lateral deflection measured by a dial reading to 0.001 inch. If the
dial reading changed appreciably, the load was reduced and the plate recentered
and readjusted. The load was then increased and lateral deflections again
observed. Corrections were made as necessary until the applied load produced
negligible deflection (of the order of 0 to 0.002 inch). Thus bending due to
eccentricity of loading was reduced to a minimum amount.
Two metalectric strain gages were attached at the center of the panel, the
inner gage being glued to the stiffener and the outer gage glued directly opposite on the convex face of the plate. Strain and lateral deflection readings
were taken at regular increments of load.
In all tests the plywood buckling was rapid and was accompanied by a drop in
load. Movement of the testing machine head was stopped when the plywood
buckled and the buckle pattern observed. If the stiffener had not buckled,
loading was resumed until the stiffener buckled or failed. In several instances, the load required to buckle or cause failure of the stiffener was
greater than the load required to buckle the plywood. The highest load supported by the specimen was recorded as the maximum load.
The static bending and compression coupons of plywood were tested in accordance
with the A.S.T.M. tentative methods.5 The properties of the stiffeners were
obtained from 1- by 4-inch coupons using the same apparatus and techniques
applied to the plywood compression coupons.
Computation of Results
The theoretical buckling stress of the cylinder comparable to each curved
plate was computed by the formula P = kEL
where:
p = buckling stress, pounds per square inch.
k = a constant, obtained as indicated below.
EL = modulus of elasticity,parallerto the grain of the wood in the
veneers of the plywood, pounds per square inch.
2 "Tentative Methods of Testing Plywood, Veneer, and Other Wood and Wood...base
Materials." A.S.T.M, Designation D805-4ST.
Rept. No. 1567
-3-
h = thickness of plywood, inches..
r = mean radius of curvature, inches.
The observed buckling stress of the plywood plate was calculated by dividing
the load oarrie4 . /py the plywood at, buCkling by the loaded. area. The load carried
by an axial stiffener .when the plywood:, buckled yap Calculated from the size,
modulus of, elaitio#7, and .average. strain of the stiffener. The total load
supported by the: Stiffened plate is the sum of , the load supported by the plywood and that supported by the stiffener.
The following
F
notation applies in addition to that above:
El = modulus of.elasticity in bending of the plywood in the axial
direction of the curved plate, pounds per square inch.
E 2 = modulus of elasticity in bending of the plywood in the circumferential direction of the curved plate, pounds per
square inch.
Eb = modulus of elasticity in compression of the plywood in the
axial direction of the curved plate, pounds per square inch.
E a = mbdulus of elasticity in compression of the plywood in the air,cumfeeential direction of the curved plate, pounds per
(square inch.
ET = modulus of elasticity tangential to the annual growth rings
of the wood in the veneers of the plywood, pounds per
square inch,
E L can be calculated from the relations
E1+ E 2 = Ea
EL =
+ E
b,
= EL + ET = EL (1 + C)
E l + E 2 + Ea +
2(1 + C)
Eb
ET
The ratio — or (C) is taken as 0.050 for yellow birch and
EL
0.045 for yellow-
poplar, based on recent values obtained at the Forest Products Laboratory.
Theoretical values for the buckling constant, k m. , can be obtained by enterE,
ing the curve of figure 3 with the proper • J-ratio. The value of k/km
El + E2
,
4
is plotted against the parameter E2 ) 1/ in figure4. The value of k can
vhr t 1
then be determined from the above relationships. ,These curves are a reproduction of those used in a previous report.2
Rept. No. 1567
-4-
Presentation and Analysis of Data
Table 1 presents the data used in determining the buckling stresses of the
plywood plates. The plates are arranged in 13 groups of six plates each, with
the plates of each group being of similar dimensions and plywood construction.
The first plate listed in a group was tested without a stiffener, the second
with a light stiffener, and the others with increasingly heavier stiffeners.
Examination of the observed buckling stresses of the plates within a group
showed no tendency of an increase in critical stress due to the addition of
the stiffeners. Variation between the buckling stresses within a group of
plates can be attributed largely to (1) initial imperfections in the plates,
(2) dimensional and physical differences between the minor specimens and the
plates, and (3) inevitable experimental errors.
A curved plate may be considered as a segment of a cylinder, and it would seem
likely that a relationship might exist between their critical stresses. In
figure 5 the observed buckling stress of each plate is plotted against the
theoretical buckling stress, corrected for length, of a comparable cylinder.
This figure shows that the observed values for the plates are only slightly
greater, on the average, than those computed for comparable cylinders. The
average observed stresses for the plates of groups "d" and "m" are about 35
percent greater than those for comparable cylinders.
If the length and/or width of a plate is sufficiently reduced to prevent the
formation of a full-size natural buckle, it would seem that the critical
stress would be increased. The effect of a decrease in length upon the critical stress for a complete cylinder is illustrated in figure 4. The stiffeners of a stiffened cylinder or of a stiffened curved plate and the restraints
at the edges of a curved plate impose a maximum limitation on the circumferential size of the buckle formed and therefore may be expected to increase
the critical stress. The width-to-radius ratios of the plates tested for
this report varied from 0.25 to 1.0; thus they were segments of comparable
cylinders varying from 0.04 to 0.16 of the complete cylinders. In spite of
this fact, as has been previously noted, the average observed critical values
for the plates were found to be only slightly higher than those computed for
the cylinders, indicating little increase due to the width limitation. This
failure in obtaining a consistent increase in critical stress over that of a
comparable cylinder may be understood from the theory developed in Forest
Products Laboratory Report No. 1322-A,2 upon which the following discussion
is based.
The buckling phenomenon in a cylinder subjected to axial compression is
greatly influenced by the presence of initial imperfections in the shape or
material of the cylinder and upon the circumferential width of the buckle.
Such initial imperfections are likewise found in curved plates, which may be
considered as sections of cylinders. The width of the buckle might be limited
by sufficiently reducing the width of the curved plate. Figure 6, which is
a reproduction of figures 10 to 13, inclusive, of Report No. 1322-A, shows
arch, g. W., "Buckling of Long, Thin Plywoo4 Cylinders in Axial Compression
(Mathematical Treatment),' 4_943•
Rept. No. 1567
-5.
the relation between the compressive stress, p, and e, Which is proportional
to the amplitude of the buckle. A family of curves for varying values of n
is drawn for four values of initial buckle amplitude, f • The value of nis
o
equal to 477 4
1 , where h is the plywood thickness, r is the radius of curvature,
13:4
and b is the width of the buckle. The critical stress associated with each
value of is the minimum value of the stress indicated by the corresponding
curve.
L
It will be noted from the curves that the minimum value of pr/E h, regardless
of the value of
is 0.095 at a value of -n of 0.350. This indicates that
the width of the natural buckle of a cylinder of plywood, of the construction
for which the curves were drawn, can be obtained from the relationship n = 0.350
= 417- hr .
If the full width ,of this natural buckle is not permitted to form,
b2
either by the use of heavy axial stiffeners or reduced panel width, the critical stress may be associated with some higher value of n and may, therefore,
be increased. For low values of
it may be noted that there is a sharp
increase in critical stress with an increase in n, but that for large values
of t , such as 0.5, the critical stress remains about the same for different
values of n.
eo,
eo,
o
Since the addition of axial stiffeners to the panels tested for this report
did not greatly increase the buckling stress, it, is evident that the values
of e were large. This is to be expected, particularly for thin cylindrical
forms, because
is the ratio of the amplitude of the original buckle to the
thickness. Although the above discussion is based on curves for three-ply
(1:2:1) plywood with axial face grain, a similar argument can be made for
other constructions.
o
eo
It would appear, from the above, that the buckling stress could be increased
by the addition of closely spaced heavy stiffeners, and tests on such Stiffened plywood cylinders have been made.± These, tests show that an increase
in the buckling stress Of the plywood is obtained from such stiffening if ule
stiffeners are very closely spaced.
Previous work on the buckling of unstiffened curved p1 mod plates in compassion" indicated a slight increase of the critical stress when the width Vat:,
decreased to about one-half radian. These plates were of smaller curvature:
than those tested for this report, and it. is therefore probable that the
14itial deflections .,
were smaller.
ea,
–Forest Products Laboratory Report No. 1508, "Buckling of Thin, Curved,
Plywood Plates in Axial Compression," by E. W. Kuenzi, 1944.
Rept. No. 1567 -6-
Conclusions
The following conclusions may be drawn from the results of the tests on thin
curved plywood plates simply supported at the edges and loaded in axial compreasion:
(1) An axial stiffener does not increase the buckling stress of the plywood.
The stiffener carries additional load and, if stiff enough, breaks up the
buckle pattern.
(2) The buckling stress is approximately equal to or slightly greater than
that of a thin-walled plywood cylinder, and may be calculated as such.
If local imperfections could be reduced and the curved plywood plates
perfectly, increases in critical stress might be attained by the
addition of a stiffener that would reduce the width of the buckle. The plates
tested were carefully made, however, and are thought to be as perfect as those
made commercially; yet no appreciable: increases were obtained.
(3)
made ztore
Rept. No. 1567
-7-
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2,660
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.500 1 .3/
3, 190
15.86
.121
.031
23.7
27.36
407 2,245 t 1,066
934
2,560 1,289
1.365
15.50
.11
27.40
424 2,552 2 2.07: 1,035 :::g
13.7
.126
.32
1.305
1, 502.
2610
,
1,272
53
23.4
15.89
.2299
2,176 1,126 11,062 2,301
2,640
1,360
27.39
.169 2 .32
2 ,640
1,2E2
-5
55
2).7
15.115
-2.11
2,740
-2
27.39
1,646
2,520
173
1,260
1.772
.217
.32
4:3 2,3302 1025 1,040 2.266
15 .51 27.35
.120
-6
2,255 1,10: 947 2,236
1,250
23.4
1.1212
2,540
2,620
1,260
2,1
.379
.32
2,256 1,187 g 1,031 2,256
15.45 2747
.U9
1,240
1,646
.3.7
.502
.32
1,415
2,670
-3
3.130
.051
12,96 277
'215
446 1. 150 1.279 2,411
4-la
Asia
30.5
2.209
1,560 1,560
475
31.1
14.97 21.011
2,171
44
944 2 1,237 2,276
-119
629
1,50
.126
1,710
1,710
.32
935
1210 1,120 2 1.129 2,275
.123
14.97 21.41
.166
1,520
31-1
-32
1.520
97
58
605
.3
946 1,130 r 1,279 2,411
14.97 27.36
-a'
.117
32. 5
.32
690
1,0155
1,r751
1,730
•253
75
1,630
1447 27.42
.121
2,
1110 1,120 : 1.169 2,270
1.2.32
31.1
2,120
121
1,010
-3e
.315
.32
1.950
975
14.95 27.37
.112
2.171
.29
-2.
30.5
2,276
1,674
610
949 1,237
.502
2,650
139
967
.31
1,570
14.93
lraantaraattal
.031
.120
2,123 1,074 1,006 2.206
30.1
1..4:
1,510
245
99/
111.97
31.6
2,251 1,047
77.g1
.120
-re
910
1,616
.1U
.32
1;620
936 2,169
904
.120
312,
-12
1,074 1,006 2,206
30.9
1,760
1./.1k7
1.720
.32
977
.250
77
950
:77.4
.122
1,022 1,002 2,171
31.5
1,540
.32
605
.507
-32
919
1.,
14.97 27.41
31.1
.119
119:
.32
1,690
911
60 2,166 1,022 1,002 2,176
295
-32
•379
1.932
31.1
14.96 27.37
.120
-2.
3362 2.251 1,047
936 2,159
2,220
1,727
.32
920
965
.900
1.930
Elreastereatlal
0.021
15.0
15.2
2
:11 :
22-1
-2
10.2G11-1,
2.01E1
6
.021
g::
15.0
35-0
15,0
Uloelma.ael aot recorded for • 1 524010 0f !, 7 660 3
F
15.59 e 15.01
15.57 11.56
15.59 2 92.99
15.56 I 16.97
15.56 18.99
15.61 111,99
1, and 4 6421.2.
.076
.072
.074
.064
.00,
2E1
256
269
261:
216.1672-2606.11 2676106
676 . 529 1.305
gin310 1.252
1,656
650. 129 1.391
402 1, 256
631
1,166
650
Z..
1.230
1,611
1,6,1
1,5E1
1,621
1,651
.126
.159
.;
.31
.32
.31
.31
.32
1,10D
1,225
1,140
1,100
1,550
1.110
2.659
1.730
1,225
1,340
1,950
2,190
162
279
715
7T.
762
844
604
905
1,030
910
1,075
1.165
1,200
Figure 2.--Axial-compression test of a curved plywood panel having
axial stiffener.
76776 F