Supplement to
ANALYSIS Or LONG CYLINIARS Of SANUWICI1
CONSTRUCTION UNDER UNIFCRM EXTERNAL
LATERAL PRESSURE
Facings of Moderate and Unequal Mid-messes
February 1955INFORMATION
REVIEkla
'I 60 •
LOAN COPY
Please return to:
Wood Engineering Research
Forest Products Laboratory
Madison, Wisconsin 53705
This Report is One of a Series
Issued in Cooperation with the
AIR FORCE-NAVY-CIVIL SUBCOMMITTEE
on
AIRCRAFT DESIGN CRITERIA
Under the Supervision of
AIRCRAFT COMMITTEE
of the
MUNITIONS BOARD
No. 1844-A
OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison 5,Wisconsin
UNITED STATES DEPARTMENT
In Cooperation with the University of Wisconsin
Supplement to
ANALYSIS OF LONG CYLINDERS OF SANDWICH CONSTRUCTION
UNDER UNIFORM EXTERNAL LATERAL PRESSUREFacings of Moderate and Unequal Thicknesses
By
MILTON E. RAVILLE, Engineer
2 Forest Service
Forest Products Laboratory, –
U. S. Department of Agriculture
Summary
A previous analysis of the problem of sandwich cylinders subjected to
uniform external lateral loading is extended in order that results may
be applied to sandwich cylinders having relatively thick facings of
unequal thickness. In the development of the stability criteria, the
effect of the stiffness of the individual facings on the stability of
the composite cylinder is taken into account. Solutions are obtained
from which the stresses and displacements in a stable sandwich cylinder
may be determined, and an expression for the determination of the load
at which a sandwich cylinder becomes elastically unstable is derived.
The sandwich cylinder is assumed to consist of isotropic shell facings
and an orthotropic core.
Introduction
This report is a supplement to a previous report that contains a
theoretical analysis of sandwich cylinders acted upon by uniform external lateral pressure. Formulas were derived for the stresses in
the cylinder, and an expression was obtained from which critical
-This report is one of a series prepared by the Forest Products
Laboratory under U. S. Navy, Bureau of Aeronautics Order No. 01595
2 and U. S. Air Force Order No. A.F. 18(600)-102.
—Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
Report No. 1844-A
Agriculture-Madison
pressures may be determined. In that report it was assumed that the
facings of the cylinder were thin enough to render membrane theory
applicable and that the facings were of equal thickness.
The purpose of this supplementary report is to present solutions for
the stresses and for the critical pressures that apply to sandwich
cylinders having moderately thick facings of unequal thickness. This
requires that the bending moment and the transverse shear in the
individual facings be considered in the development of the stability
criteria. This extension of the previous work is felt to be of importance in view of the fact that an analysis based on membrane facings may
prove inadequate as a design criterion in cases where relatively thick
facings are used. It is assumed throughout that buckling takes place
at stresses below the elastic limit of the sandwich materials.
The method of analysis used here follows closely the method used in
the original report. The same core assumption, namely, that only
transverse shear stress and normal stress on planes parallel to the
facings are present, is used. It is felt that this assumption applies
very well to all practical sandwich constructions because of the relatively low load-carrying capacity in the tangential direction, of the
core materials as compared to that of the facing materials. The facings
are assumed to be homogeneous and isotropic, and, as indicated previously, they are analyzed on the basis of shell theory rather than
membrane theory.
Notation
r, 9
a
b
polar coordinates
radius to middle surface of outer facing
radius to middle surface of inner facing
t othickness of outer facing
ti
t
E
thickness of inner facing
thickness of either facing when t o = ti
modulus of elasticity of facings
Poisson's ratio of facings
modulus of elasticity of core in direction normal to facings
modulus of rigidity of core in r - 9 plane
Report No. 1844-A
-2-
intensity of uniform external lateral loading
q
rc normal stress in the radial direction acting on the outer
a
ar,
a
r, r
, re
facing, on the inner facing, and in the core, respectively
11-r
,
ar, arc small normal stress in the radial direction acting on the
outer facing, on the inner facing, and in the core,
respectively
Tre,
small shear stresses acting on outer and inner facings,
respectively
Tr()
small transverse shear stress in the core
direct stress resultants in the tangential direction in
the plane of outer and inner facing, respectively
small direct stress resultants in the tangential direction
in the plane of outer and inner facings, respectively
small bending moments per unit length of outer and inner
facing, respectively
small resultant shear forces per unit length of outer and
inner facing, respectively
u $ ucradial displacements of outer facing, inner facing, and
core respectively
_t
small radial displacements of outer facing, inner facing,
and core, respectively
u,
I
, V
vc
small tangential displacements of outer facing, inner
facing, and core, respectively
unit tangential strains in outer and inner facing,
respectively
€0,
• e ,
e' e
e
ec
small unit tangential strains in outer facing, inner facing, and core, respectively
small changes in curvature of outer and inner facing,
respectively
n
number of waves in circumference of cylinder
a
one-half the central angle of curved panel
P
E c a (1 - 112)
Et o
Report No. 1844-A
-3-
l
c a
7
(1 - .L2)
Et o
5n
2
Ec
n
G re - 2
t 2
o
0
12a
t
Si
2
2
i
12b
2
Et o (1 - b2/a2)
2G re b (1 - p.2)
A, B, An , Bn , C n , Hnarbitrary constants
Stress Analysis
The sandwich cylinder is assumed to be long enough that the effect of
the constraints at the ends is negligible. Under a condition of uniform external loading, each cross section remains circular. The dimensions of a cross section of the cylinder and the positive directions of
the polar coordinates, r and e, are indicated in figure 1. In order to
avoid any confusion in regard to signs, the intensity of the external
load is assumed to act in the positive r direction; obviously, a negative value of q signifies compressive loading. In the analysis which
follows, the assumption is made that the core extends to the middle
surfaces of the facings and that the load q is applied to the middle
surface of the outer facing. This assumption amounts to neglecting the
half-thicknesses of the facings as compared to their radii.
Equilibrium of Core
As previously stated, the assumption is made that, in general, the core
transmits only normal stresses in the direction perpendicular to the
facings and transverse shear stresses. In the case of uniform external
loading, it is noted that the transverse shear stress is zero from considerations of symmetry, and the only stress present in the core is the
normal stress in the radial direction, Erc . Considering the equilibrium
Report No. 1844-A
-4-
of the differential element of the core shown in figure 2, the summation of forces in the radial direction results in the following
equation:
- arc r de + (a rc
dard)
+ dr de = 0
dr
The differential element is considered to be of unit length in the
longitudinal direction. The above equilibrium equation reduces to
a re
da rc
dr
(1)
=0
The solution of equation (1) may be written as
A
arc
Ec –
r
rc
(2)
where, for convenience, the constant of integration is represented by
Ec A. E c represents the modulus of elasticity of the core in the radial
direction, and A is an arbitrary constant. The radial displacement of
the core, uc , is related to the radial stress, arc , by the following
equation:
a rc
E
du c
c dr
(3)
From equation (2) it follows that
duc A
dr
Integration of the above equation yields
(4)
uc = A log r/a + B
The use of log EL2 instead of log r merely alters the arbitrary constant of integration.
Equilibrium of Facings
With reference to the differential elements of the facings shown in
figure 3, it is seen that, when the cylinder is acted upon by an external loading of uniform intensity, so the only forces in the facings
are the tensile forces per unit length N 9 and Ne. ar and al.. in figure
Report No. 1844-A -5-
3 represent the stresses exerted by the core upon the outer and inner
facings, respectively. An equilibrium equation can be obtained for
each facing by summing forces in the radial direction on each differential element, considered to be of unit length in the longitudinal
direction. The equilibrium equation which pertains to the outer facing
is
qa de - ara d6 - No d9 0
This equation reduces to
Ne = a (q - ar)
or, since /r = ((Ire)
r =a
ii9 = a [q - (circ)
[
r = ]
(5)
In a similar manner, the equilibrium equation of the inner facing is
found to be
Ne = b
(6)
(a )
rc
r=b
From the application of Hooke's law along with the assumption that the
tangential stress in the facings is uniformly distributed through their
thicknesses, the following relationships are obtained:
Ne = Et o E e
(7)
and.
Ne Et i E
o
(8)
e
If the right-hand sides of equations (5) and (7) and the right-hand
sides of equations (6) and (8) are equated, the following two equations
result:
a
E e = Et
L q
(cfrc ) r =
and
e l
b (a )
e Eti rc/
Report No. 1844-A
r=b
-6-
Since
,
u
and ee
ee = T.
a2
u Et
=)
c
cr(a
,
these two equations become
(9)
r=
and.
u =
b
2
,
ka
Et.ra
(10)
r=b
If continuity of displacements at the interfaces is assumed, then
u = (u )
c
r=a
and
u ' = (uc)
r=b
The above relationships in conjunction with equations (2) and (4) enable
equations (9) and (10) to be written as follows:
.B =
2 01
,
Eo
v,
Eto
(11)
and
A log b/a + B
Eb A
= Et i
(1 2)
Equations (11) and (12) may be solved for A and B with the following
results:
A = 523: k
Ec
and
2
B = Et
ga ((1 - k)
(14-)
where
k-
1
b to Eto
a t i
Ea
o
Report No. 1844-A
(15)
b/ a
-7-
The substitution of the value of A given by equation (13) into equation
(2) yields the following expression for the radial stress in the core:
a
Q rc
(i6)
When the above value of arc is substituted into equations (5) and (6),
the following two equations result:
Ne = qa (1 - k)
(17)
and
N qak
e
(18)
The substitution of the values of A and B given by equations (13) and
(14) into equation (4) yields the following expression for the core
displacement:
u
qa2
[1 - k (1
ET log r/a)]
c = EnE
[1
ta
(19)
Since u = (u )
and u = (u)
, then
c r=a
r=b
u =
Et2
(1 - k)
(20)
and
qab
- Et k
i
(21)
Equations (16) - (21) completely describe the stresses and displacements
in the sandwich cylinder subjected to uniform external, lateral loading.
In each of these equations the value of k is given by equation (15).
Stability Analysis
In discussing the stability of the sandwich cylinder under uniform
external lateral pressure, the equilibrium of a slightly deformed
element of the cylinder must be considered. It is assumed that the
stress situation that exists in this deformed element differs only
slightly from the stress situation that existed just before buckling.
The stresses before buckling are given by equations (16), (17), and
(18). Since the cross section of the deformed cylinder is no longer
circular, the small changes in the stresses and the displacements resulting from buckling will be functions of e as well as r. Following the
Report No. 1844-A
-8-
system of notation used in the original report, a bar is placed over the
appropriate symbol to denote the small stresses, strains, and displacements that occur when the cylinder goes from the initial uniformly
stressed circular form to the slightly deformed configuration. Again,
it is assumed that the core extends to the middle surfaces of the facings and that the load q is applied to the middle surface of the outer
facing. This assumption is now somewhat more restrictive than it was
in the case of axial symmetry, since it now means that the effect of
the interface shear on the bending of the facings and the displacements
due to the bending of the individual facings are neglected. However,
it is felt that, for cylinders with shell-type facings, the results
based on this assumption should be of sufficient accuracy.
Equilibrium of the Core
Since the cylinder is now considered to be slightly deformed, the core
is also slightly deformed: A differential element of - this core is shown
in figure 4. In addition to the radial stress, q r k, given by equation
(16), a small radial stress, 7
-rc) and a small transverse shear stress,
recY due to buckling must be taken into account. The differential
element is considered to be in equilibrium under the action of the stresses
shown. Since the small change in the geametry .of the core introduces only
small terms of higher order into the equilibrium equations, the differential element in figure 4 is shown in its undeformed state. The analysis
of the core is exactly the same as that given in-the original report,
and for this reason only the basic equations will be repeated here.
The equilibrium equations that apply to the core are obtained by summing
forces in the radial and tangential directions in figure 4. Theseequilibrium equations are:
1 .r rec = 0
rc
r
3cir C
-37 ( 22 )
r
and
67rec 2Trec n
(23)
r
The following stress-displacement relations are also applicable:
a
Ec
rc
auC
(24)
3T—
and
T
rec
Report
=
No.
G
re
1 Lu c
c
T 36-
1844-A
77
or
r
(25)
c]
-9-
It was shown in the original report that the small radial and tangential
displacements of the core can be completely determined insofar as their
dependence on r is concerned. Then, on the assumption that during buckling the circumference of the cylinder subdivides into n waves, the displacement functions are written as follows:
a
(26)
uc = [An + B n —
]
r + Cn log —
a cos ne
and
vc = [ - n An + 711- bn Bn a - n C n (1 + log -8 ) + H n
4
sin n9 (27)
where
E c
6n
= 2G
n
2
re
2
and
An , Bn , Cn , and Hn are arbitrary constants.
After the substitution of the expressions for Tic and vC given by equations
(26) and (27) into equations (24) and (25), the expressions for the small
core stresses become
a
rc
c
=—
r - B 2 + C n cos ne
n
(28)
and
1 Ec
(29)
n = - — — B 2 sin ne
ruc
n r n r
It may be easily verified that equations (28) and (29) satisfy the equilibrium equations, equations (22) and (23).
Equilibrium of Facings
Figure 5 shows the differential elements of the facings of the slightly
deformed cylinder. In addition to the forces that exist just before
buckling (given by equations (16), (17), and (18)), the small forces and
moments that arise during buckling are shown in the figure. In considering the equilibrium of the facing elements, account is taken of the
rotation and stretching of the facings which occurs during buckling. As
described in the original report, the initial central angle, de, becomes
Cl
a
1 a y. 1 gii)
+——- de
6e a 6e2
Report No. 1844-A
and
-10-
(
_r
01
1 6v 1 6-u
) de
+ — - —
b 6e b 6e2
for the outer and inner facings, respectively, and the, areas of the
differential elements of the outer and inner facings become (1 + Zo) a dB
and (1 + Ze) b de l respectively. Three equations of equilibrium can be
written for each facing; the differential elements are considered to be
of unit length in the longitudinal , direction.
Considering first the differential element of the outer facing, the summation of forces in the direction normal to the surface yields
q (1 + T) a dO - (qk + "dr )(1 + Te ) a d
/
1
+ Fi ]
e
-
If the relationship,
6V
271)
a-ge
+ —
d9-[qa(1k) + — de = 0
6e 6e2
6e
OV
ce =
'ET 51-P is used and small quantities of higher
order -- that is, products of barred quantities -- are neglected, the above
equation becomes
a
6 ce
isire
= -
a
+ qa (1 - k)
The summation of forces
equation:
in
+
TO
6271 )
(30)
a 602
the tangential direction-yields the following
a
- [a (1 - k) + Re]
+
- Fre (1
+
[q
a dO
+
de]
(1 - k) + Fre +
+
a
E -
2-J-1 ) de
a 692
0
If small quantities of higher order are again neglected, the above equation May be written as
Ogg,
de
7_
a
=
Tre
The third equilibrium equation of the outer facing is obtained by. equating
to zero the summation of moments about point 0, shown in figure 5. Thus
- me +(me
+ ao
dt4) + ;9 a de 0
6146
The above equation reduces to
OR
0+aQe=O
&?
Report No. 1844-A
(32)
-11-
Similarly the three equilibrium equations which pertain to the inner
facing are obtained by summing forces in the normal and tangential
directions and by summing moments about the point 0' using the differential
element of the inner facing shown in figure 5. These equilibrium equations
are:
I at 1)
....,
3%
4
11
N - - — - b o + qak (r
9 60
b
,
b 602
(33)
T
;
-1-
I
I
(34)
Cle . - b 7re
and
(35)
,10 Q0 = 0
If equation (32) is solved for
and the resulting value is substituted
into equations (30) and (31), the three equilibrium equations of the outer
facing are reduced to the following two equations:
Ne +
1 %
a6 302
a
(74)
+ q (1 - k)(71
.1521
60)
(36)
and
6% 1 6Re
-
(37)
a 1-re
In like manner, if equation (35) is solved for '47:9 and this value is substituted into equations (33) and (34), the three equilibrium equations
of the inner facing are reduced to the following two equations:
6
Nn + -
s"
-
a ( b 602
b
0
62—'
bcrr+q-k u + u
68
(38)
and
6R;
IC 75—
6N; 1
(39)
b :Fre
Fran the application of Hooke's law, the following twy expressions relating
the small tangential forcesper unit length, fi re and Ne, to the small tangential strains, Te and ee, are obtained:
Report No. 1844-A
-12-
Eto
Ne - 2
-
-E-0
(40)
and
,71
14
9
Et
r
_ (a)
- 42
Also, the following two equations relating the bending moments in the
facings, Re and Re, to the changes in curvature in the facings, 5-",;,(9
are applicable:2—
Et o 3
–
and –t
(42)
2 ) xe
12(1 Et 3
m
1
e_
(43)
12(1 - p-) xe
Since
and
_ 1 (E-1
-
b2 69 66?7
equations (40) - (43) may be written as follows:
47:
e
Eto
(1.1
1 6Ti.
(14-0
1 - 4 2 a a 4)e)
2Timoshenko, S., and Goodier, J. Theory of Elasticity. New York, 1951.
Timoshenko, S. Theory of Plates and Shells, First Edition. New York
1940.
Report No. 1844-A
-13-
Eti
N
1
1.1
(45)
2
Et o
Me-
12 (1
1
-
.L2)
v
2–
a e
ae;
–
(16
a2
(46)
and
Et i
Me
3
2
12 (1 - p )
b
2
/6:71
62u,
\ae
ae
(47)
2 )
The requirement is now made that there be continuity of displacements at
the interfaces; that is,
U = (Tic)
(4 8)
r = a
11 =
r
= (ve)
(49)
(ac)
b
(50)
r=a
and
(51)
r=b
The use of equations
written as follows:
(48) - (51)
Et o (u_7
▪ /6
- 2
ae )
° a (1 - p )
N e
Et i
b (1 -
(44) - (47)
to be
(52)
r = a
aiT
p 2 )
▪
c
Et o 3
Me —
enables equations
12a2 (1 -
(53)
6(9
r=b
A71
)
C\
37:r c _
69
602 r = a
(54)
and
Eti3
_A
(-7/770
(55)
12b2 (1 - p2 )
Report No. 1844-A
6f)
692 )
-14-
r=b
With the aid of equations (4 8 ) - (55) and the fact that
ere
=
(7rOc)
(56)
7.re = (Tree
r =a
(57)
r=b
(58)
ar = (arc)
r=a
and
Ur = (_arc)
(59)
r=b
the equilibrium equations of the facings, equations (36) - (39), may be
expressed entirely in terms of the core displacements and stresses. For
example, if the right hand sides of equations (52), (54), (56), (58), and
(48) are substituted for Re, Me , 7 re, Fr r1 and IT, respectively, in equilibrium equation (36), the result is
Et o61-To
a (1 -
)
= a 12 (1 - p2 ) a3 ae 3
6(9
= - a (5". rc )
3
Et03
a)r
C
+ q (1 - k) (a0 + ;Or
r =a
=a
(6o)
In a similar manner, equations (37) - (39) may be transformed, respectively,
to the following three equations:
Et o
Et o 3
a (1 =a
=a
12 (1 - p2)
(T ree )
(61)
r=a
Eti
Eti3
b (1 - p 2 )
a0 c)
r=b
(33-170
12 (1 - 4 2 ) b 3 6c) 3,377)
r =b
621.7
= b (Fre)
r = b
+ q k
b
+
c ae2
and
Report No. 1844-A
-15-
(62)
r=b
)ric
Eta b (1 -
) (6,9
a2.T7
Eti3
662 )
(6c
12 (1 112 ) b 3 602
60
r=b
r =.• b
=
b (7r6c)
(63)
r=b
If the expressions given by equations (26) - (29) for the core displacements and stresses are substituted into equations (60) - (63), four equations containing the parameters An , Bn „ C n, and Hn are obtained. These
four equations aret
[(n2 - 1) y (1 - k) - (n2 -
An 4. [(n2 - 1) y (1 - k) +n (1
n2 0 )
o
n2 00 )] Cn
+ (1 + n4 o o ) -p] Bn + [3 - n2 (1
(64)
+ [n (1 + n 2 0 0 )] H n = o
(n2 - 1) An + [ - 6 n (1 + 0 o ) -
(1 +
n2 00) + ---]
n2Bn
+ [n2 (1 + 0 o )] C n + [ - n (1 + 0 o )] Hn = 0
(65)
to
4. rtn2 - iN 8 ,k -2 + bp, EL-(1 + n2 0i)
- (n2 - 1 N1 -IA
[(n2 - 1) yk —
L ‘
n
I b f t i- b
t i‘
.h t
t
t
p -= 2
log
b/a
+ tt- 1 + n4 0 ) + p t o ] B + [(n2 - 1) yk —°ti
t ia
i
i n
t
- (n2 - 1) log b/s. - n2 (1 + n2 00] Cn + [n b/a (1 + n 2 00] Hn = o
(66)
and.
a
a
(n2 - 1) An + [ - 611 .T0- (1 + 0 i ) - 17. (1 + n2 0 i)- n2t1. ] Bn
+ [(n2 - 1) log b/a + n2 (1 + 00] C n + [- n b/a (1 + 00] H n = 0
(67)
where, in each of the above equations,
7
qa (1 - u2)
Et o
Report No. 1844-A
-16-
2
to
12a2
2
ti
(I)
=
2
12b
and
a (1 - 112)
E-t o
Each of the terms in equations (64) - (67) contains one of the parameters An) Bn) Cn0 and Hn that appear in the displacement equations of
the cylinder. A buckled form of equilibrium is possible only if equations (64) - (67) yield solutions for these parameters which are different from zero. This requires that the determinant of the coefficients
of these parameters must be equal to zero. This determinant may be
written as follows:
Report No. 1844-A
-17-
0'1
CO
nC).
0
..---.
H
.--..
0
+
.e.H
+
H
ia-
a)
0
,0
cci 0
.......,
0
rar
.e.
I
ai
•n.,..„
-e.
I
4:1)
.0 $4
O
N
0
i
01 ,-i
----.
H
.e•
4-) i
4-)+
.01 Ed
H
al 1
r-f
cial I
N
0
+
.-1
--N
O
N
all 0
cal%
+
EO
-e.
0
43.
N
O
(Cr'
0
+
H
.....
90)
Q)
•
1-)
I) 0
0 't
a) r.) ,--1 EU
rd 1-1 cd 0
-P y •-1
ry vd
5
a)
410
H
H
•-I
r '-1 2 5-I
4-)
...--..
N
cd 1 ra
1
..-..
H
e-
FA
8
?q0 •ri ;-I
0.
-P CO -I
1
H
H
H
I
v0
N
?...
0
Report No. 1844—A
0 I ..-1
I
-4-) -P
co
M
cu
r.
0
G.)
0
-r-i
cd
-I-, i:d a) -P
W H
rip
la) -I-) 0
O
0
•r-1 0 0 0 La
1-
ca
bp 4-, -,–r
-18-
Tj
;•
0 0
ra
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r1
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0
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r
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e
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14)
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s-1
54
a)
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• .,,
.
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The use of equations (16) through (21) is self-explanatory; if the dimensions and material properties of a given sandwich cylinder are known, the
stresses and displacements in terms of the external load, 2) may be easily
computed. In the case of sandwich cylinders having facings of equal thickness, equations (16) - (21) reduce to equations (18) - (23) of the original
report. It is of interest to compare equations (16) - (21) with results
obtained by Reissner. a If Reissner's equations are expressed in the notation
used here, the following equations for the cylinder stresses result:
1 +
N
e
=qa
N = qa
e
E t ,
kl - b/a)
Eca
E
2 + Et
Eca
- b/a)
1.
2 4. Et ( 1 _ b/a)
Ea
and
a rc (at middle surface of core) = q
Equations (16)
follows:
21
(1 b/a) [ 2 + Et
— (1 - b/a)
E
(18), for facings of equal thickness, may be written as
b
- --- log b/a
a
EE ca
Et
Ne = qa
1 + b/a
N e = qa
log b/a
ca
Et
1
1 +
/
Et
Et log b/a
- ---
and
re
arc
middle surface of core) = q
(
1 + bia) [1 4. b/a Et
--- log b/a]
Eca
2
1
is - (1 - b/a), it
Since the first term in the series expansion of log
may be noted that, for cylinders with b/a ratios close to 1, the equations
of Reissner yield results that are very nearly the same as those given by
equations (16), (17), and (18). Reissner's results are based on the assumption that a - b < < a and hence become increasingly less accurate as the
cylinder thickness is increased.
b/a
2Reissner, Eric, Small Bending and Stretching of Sandwich-type Shells.
National Advisory Committee on Aeronautics, Tech. Note 1832. 1949.
Report No. 1844-A
-19-
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a)
cp
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aS Ea 0 ........
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d
al
0
01,0
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a
CO
(0
(r)
tO
H 0 0
O a)
a)
rW
of
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r--
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P,
qEi 8I g
° 15-4.4
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41
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per -P
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a) H 0
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al
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la() 03
H
4
CPI
0
a)
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c)
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C4 .r.1
r--1 -4->
co •,-4
0
lE1
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11 0
01
I
I
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LI E
q -r-1 .0
-I-)
Ed Z e I EI--i
ED
-.-1
-.t 3
P4 0
0
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01 •H
I
a)
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I+)
.0
0
....,
4-)
H
-1-3 0 r0
H
O -I-) ----.. z
C-1
0 a)
.14
CI) d
a)
S.4 0 03
0
4-1
ON
0
0
14'N
.. ,.0
?..
H (1) ‘.0 . 1-.1
O 0 0
?-..
, ,0
r•r")
r•
EH -P -P
-I-'
rz4 0 r-I
CO H '----' +2
Report No. 1844-A
-20-
where
Et o (1 * - 2G
r0
b2
)
;5
2
b (1 - p )
Equation (70), with n = 2, yields values of y er within 3 percent of the
values obtained from equation (69) for usual sandwich constructions.
For cylinders having very thin facings (membrane facings) of equal thickness, 0 0 and O i are assumed to be zero, and equation (70) reduces to
(3.
y
cr
.12y
= - (n 2 - 1)
/
n 2 Et kl
a
2
1 +
2G re b (1 -
(71)
b)
2
)
Equation (71), for n = 2, becomes
3
(
_)2
(72)
7cr
2Et (1 +
2
G re b (1 - p )
The value of ger is then determined from the definition previously given,
q cr
-
Et
a (1 - p.2)
7cr
Equation (72) may be used as a good approximation to the expression obtained in the original report (equation 6!-) for the critical load on
cylinders with membrane facings.
It is of interest to examine the results given by equation (68) for certain
limiting cases other than that of membrane facings. Some of these results
are given in table 1. In each of these
limiting cases, 0 0 and O i are
neglected as compared to 1.
Equation (68) may be used for determining the approximate value of the
critical load on long sandwich panels in the form of a portion of a cylinder
hinged along the edges e 0 and e = 2a as shown in figure (7). If, in
Report No. 1844-A
-21-
th
equation (68),
is substituted for n, the smaller absolute value of
obtained from equation (68) represents the critical load on a panel
whose dimensions and properties are known. This solution applies to the
unsymmetrical type of buckling shown in figure (7). If, as in the case
of relatively flat panels, symmetrical buckling with no inflection point
between the supports occurs, equation (68) is not applicable.
a
Conclusions
The purpose of this report was to extend the previous work done in connection with sandwich cylinders subjected to uniform lateral loading by
taking into account the effect of the stiffnesses of the individual facings of the cylinder and also by making the results applicable to sandwich cylinders having facings of unequal thickness. The results indicate
that for the majority of sandwich cylinders the analysis based on membrane
facings is sufficiently accurate. However, for cylinders having relatively
thick facings, say in the neighborhood of one-fourth of the core thickness,
the facing stiffnesses have an appreciable effect on the critical load on
the composite cylinder. Since the magnitude of the effect of the facing
stiffnesses is dependent not only on the dimensions of the cylinder but
also on the mechanical properties of the core and facing materials, equation (68) of this report should be used for computing the critical load
if indications are that this effect may be important. In all cases of
sandwich cylinders having facings of unequal thickness, the equations of
this report should be used for computing stresses prior to buckling and
for the determination of critical loads.
Report No. 1844-A
-22-
Table 1.--Results obtained with equation 68 for certain
limiting cases
•
:
:
: t o : t ii :
:
E c : G
gcr
•
Eo
o : ti
Gre
Et
:
4
.
:
•
E
c
•
Gre
3
i
2
(1 - p ) b
3
Ca)
Eto3
to
:
:
o
:
:
4•(1 -
)
o5
11 2 )
a3
1.1 2
:
0
::
G
re
:
t
•
o •
4
:
:
Et o 3
ti •
(1 -
:
E
00
:
0
i
o
:
•
3
o
a (1 -
[1 2 )
A a tio
V , —2—tio
j_ b
2
t
C
b
o o a +
t4)
to
a
co
;
:
:
•
:
: co
:
:
:
:
: -
:
•
t
:
-
Report No. 1844 A
t
:
t 2 ( 1 4.
(1 b)2 +
Et
3
a (1 - 1.12)
a/
12ab
b2
1+ -.a'
14
a