876
ICULTUR: R0014
COMPRESSIVE BUCKLING CURVES FOR
SANDWICH CYLINDERS HAVING
CRUCTROPIC FACINGS
July 1960
No. 1876
This Report Is One of a Series
Issued in Cooperation with the
ANC-23 PANEL ON COMPOSITE CONSTRUCTION
FOR FLIGHT VEHICLES
of the Departments of the
AIR FORCE, NAVY, AM) COMMERCE
UNITED STATES DEPARTMENT OF AGRICULTURE'
FOREST PRODUCTS LABORATORY
FOREST SERVICE
MADISON 5 WISCONSIN
In Cooperation with the University of Wisconsin
COMPRESSIVE BUCKLING CURVES FOR SANDWICH
1
CYLINDERS HAVING ORTHOTROPIC FACINGS—
By
CHARLES B. NORRIS, Engineer
and
JOHN J. ZAHN, Engineer
2
Forest Products Laboratory, — Forest Service
U. S. Department of Agriculture
Abstract
Design curves are presented for determining the axial buckling stress
of sandwich cylinders having orthotropic facings and orthotropic cores.
They apply particularly to sandwich having glass-fabric laminate
facings and honeycomb cores. Sandwich constructions with isotropic
cores are included.
Introduction
The derivation of the formulas for the buckling stress of axially loaded
circular cylinders of sandwich construction is•given in Forest Products
Laboratory Report No. 1830. 3 The formulas apply to sandwich
-This progress report is one of a series (ANC-23, Item 58-1) prepared
and distributed by the Forest Products Laboratory under U. S. Navy,
Bureau of Aeronautics Order No. NAer 01974 and U.S. Air Force
Contract No. DO 33(616) 58-1. Results reported here are preliminary and may be revised as additional data become available.
2
Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
3
—March, H. W., and Kuenzi, Edward W. Buckling of Cylinders of
Sandwich Construction in Axial Compression. Forest Products
Labdratory Report No. 1830. Revised December 1957.
Report No. 1876
constructions having similar orthotropic facings of unequal thicknesses
and orthotropic cores. The natural axes of the orthotropic materials
are parallel to or perpendicular to the axis of the cylinder. These
formulas are simplified to apply to sandwich constructions having
isotropic facings and orthotropic cores. Design curves are given that
apply to isotropic cores and to orthotropic cores for which the moduli
of rigidity associated with radial and axial directions and radial and
tangential directions are related by the ratio 2. 5 or 0. 4. These ratios
apply approximately to most honeycomb-core materials.
In the present report the formulas of Report No. 1830 .3— are put in
terms of three groups of the elastic properties of the orthotropic 4
facings. It is shown in Forest Products Laboratory Report No. 1867—
that two of these groups have about the same values for all of the glassfabric laminates currently in use and that three values of the third
group will suffice for the same laminates. Design curves are
presented for sandwich cylinders having glass-fabric laminates of
these types and honeycomb or isotropic cores.
Mathematical Development
The three groups of elastic properties of the facings are:
Ex
y
g a (ryx
X 1-txy
=-
V Ex Ey
(3)
3
The nomenclature of Report 1830 — is used so that E x and Eyare the
moduli of elasticity of the facings in the axial and tangential directions,
±Norris, Charles B.
Compressive Buckling Curves for Simply
Supported Sandwich Panels with Glass-Fabric Laminate Facings and
Honeycomb Cores. Forest Products Laboratory Report No. 1867.
December 1958.
Report No. 1876
-2-
!i
is the modulus of rigidity associated with these directions, o- yx is
xY
the Poisson's ratio of the contraction in the x direction to extension in
the y direction due to a tensile stress in the y direction, o- xy is similarly
defined, and X = 1- oyxo-xy•
It is required to express equation (79) of Report No. 1830
K -
64
9 y2
"Y
( 3
2
+
(4)
Y41)
3z2
3
3ZN11
in terms of cy, (3, and .1 and to minimize it with respect to y and z. To
do this,
Y2, y 3 , and y 4 , according to equations (80), (81), (82),
and (86) of Report No. 18303, are expressed in these terms. Substituting
equations (1), (2), and (3) into equations (80), (81), and (82) of Report
3
No. 1830- and letting
6 -
1-13 2
Y
the following expressions are obtained:
YI -
1
[ a
2 z4+ 1 + 4096a2
24
8a z
2
(az + 96)
24
8a z
2
+ 81
24
34a z
2
(81az + 98) az 2 + 1
(a z 2 + 6) az .' + 1
24
8
2
4096a
16
V3
_
4096a
Report No. 1876
1+
16a z
2
2
(az + 6) a z
+
1
24
8a z
2
2
' + 6) az + 1
-3-
(5)
3
The expression for y 4 given by equation (86) of Report No. 1830 —
d , and T
3
given by equations (57) and (83) of Report No. 1830— are used. These
expressions are:
remains unchanged if the proper expressions for d l,
2
2
= z E (3z
x
d 2 = z2Ex
)
a
a
(6)
d = E
x
3
1
T
3
1
2
+
)
[ (3 az 2 + 20) oz 2 + 3
cr
and from equation (86) of Report No. 18303
T
V
4
X
f2)
32th2(f 1
I+
I (d d
1
d
2
)
T Ex2
rid i Sx
1 +
Ez
2
1/47
lidsSy
+
(8)
1 2" (d i d 3 d 2 2 ) S S
x y
+
Ex
in which
=
f 1f2
f
(c+
2
f +f
2
1
1f
Report No. 1876
-4-
2 Z 2,
3
3
f l
12
12
2 eflf2
Sx
E
x
3Xrhp.' zx
f1 + £2
2 cf f
l 2
S=
y
f 1 + f 23Xrhi'
yz•
Where f, f , and c are the thicknesses of the two facings and the core,
1
2
h is the thickness of the sandwich, r the radius of the cylinder, p t zx
and p.' yz the moduli of rigidity of the core associated with the radial
and axial and the radial and tangential directions.
Equation (4) can now be written as:
2
K =
1
1 +
21
riM 3
(f+ f )
1 + MeS x + M 5ri SxI
2
1
2
in which:
M
M
M
M
9y2
64
-62
2
3z
2
3
=
3
x
2 2
I
f
+ — M 2
2
32
1
--y
z`
[(3ycy z
=
1
4
S
[7 ( 3 a z
Report No. 1876
2
+ 9 + 213y -
2
2
g ) az + 3y
,+ y) + 0 (3 + ya z 2)
-5-
1
(-2z
+ 0)
(9)
M = M
5
0
31+ 0
Z2
S
zx
0
=
S
r-- yz
y
x
3
Equation (9) is exactly the same as equation (93) of Report No. 1830 —
but applies to sandwich having orthotropic facings as well as isotropic
facings because of the more general expressions given for M 1 , M2,
M 3 , M4 , and M 5 . This equation is to be minimized with respect to
both y and z. It is conynient to proceed in a manner similar to that
used in. Report No. 1830—. If the core has moduli of rigidity so great
that the shear deformations in it may be neglected, S x becomes zero
and equation (9)/becomes
M1
K0 = — +
2 (I + If)
(10)
M
2
3\0.1 (f + f )
1
2
This equation can readily by minimized with respect to Lo
K =
K0
'1
Thus,
n-
Where L
o
1 2
(f
and
0
1
minimized with respect to z
M M
)h
2
12
Then equation (9) may be written
N- M
1Q1
L o 71,
21
3L0 Q 1 (1 + Q2)
M2 M TIS
3x
{2
1 + KI T'S + M 1 S 2
5
x
x
Report No. 1876
X
+Q M
2 2
-6-
Where N =
KQ1
If
Q=—
2 I
and
Values of N minimized with respect to and z are given by the curve
sheets presented.
Design Curves
The critical buckling stress in the facings is given by:
f '1
o
E h
xr
(12)
in which h is the thickness of the sandwich, r is the mean
radius of the cylinder, and E is the modulus of elasticity of the
x
facings in the direction of the natural axis directed parallel to the
axis of the cylinder. Values , of Q a re given by equation (119) of
1
3
Report No. 1830- or
z
Q
12X(1 -
l
3
3c (f 1 - f2 )4
[1 h
3
Or, for facings of equal thickness,
3X,
Q
1
c % 2
c
+1
Where c is the thickness of the core, f 1 , and f are the thicknesses
2
of the facings, and X is one minus the product of the two Poisson's
ratios of the facings (about 0.99 for the glass-fabric laminates
considered in this , report4).
Report No. 1876
-7-
c
Values of N are plotted in figures 1 to 9 against values of Sx and -for particular values of a, 0, y, and 0. These particular values
were chosen because they apply to sandwich panels having glass— There are three values of a,
fabric facings and honeycomb cores. 4
1. 5, 1. 0, and 0. 67; one value of 0, 0. 6; one value of y, 0. 2; and
three values of 0, O. 4, 0. 1, and 2. 5.
Values of the parameter S are given by:
x
2cf f
l 2
x fl f2
S -
Ex
3 krhp.' zx
where p' zx is the modulus of rigidity of the core associated with the
radial and axial directions of the cylinder.
Values of the parameter 0 are given by:
=
zx
yz
where p.' is the modulus of rigidity of the core associated with the
yz
radial and tangential directions of the cylinder.
The values of L are associated with the values of a according to
o
the following tabulation:
a
1.
50
1. 00
0. 67
Lo
0. 530
0.795
1. 193
In calculating the values of N, sandwich having facings of equal
thickness was assumed; however, the curves can be used for sandwich having facings of unequal thickness with very little error.
The abscissas of the curves in figures 1 to 9 are plotted to two
scales, a large scale to the right and a small scale to the left. The
two scales meet at an abscissa of unity and the large one overlaps
Report No. 1876
-8-
the small one so that some of the data from zero to unity are plotted
twice, once to the large scale and once to the small scale. This was
done so that more accurate values of N may be obtained in the range
of S from zero to unity.
Report No. 1876 -9-
1.5-19