IB-UCINLINC OF SAStrif• - cytiNvikts
pi FINITE LENGTH UNDER
kX111MNAIL:
VAL PRESSURE
May 1955.
Wilt
EAFFatVi
1960
This Report is One of a Series
Cooperation with the
Issuod in
ANC-23 PANFIL ON SANDWICH CVNSTIPUCTICN
of the impartments of the
felPer, NAVY, ANID/COMAIEPCE
No. I844
UNITED Si'ATES DEPARTMg.NT'' OF AGRICULTURE
,
FOR-EST SERVICE
FOREST PRODUCTS
LABORATORY
‘<,
Nitictiso, 5, Wiscorv-siii
In Cooperation_with the University of Wisconsin,
BUCKLING OF SANDWICH CYLINDERS OF FINITE LENGTH
UNDER UNIFORM EXTERNAL LATERAL PRESSURE*
By
MILTON E. RAVILLE, Engineer
Forest Products Laboratory,** Forest Service
U. S. Department of Agriculture
Summary
A theoretical analysis is made of the problem of the buckling of
circular cylinders of sandwich construction acted upon by uniform ex. ternal lateral pressure. The solution obtained is based on the assumption that the sandwich cylinder is comprised of isotropic, membrane
facings and an orthotropic core. The mathematical solution of the
problem, which is in the form of a characteristic determinant of sixth
order, is applicable to sandwich cylinders of any length and of any core
thickness. Numerical results are obtained for various values of the
parameters that enter the problem, and curves are included which illustrate how the critical load varies as the values of these parameters are
varied.
*This report is one of a series prepared by the Forest Products
Laboratory under U. S. Navy, Bureau of Aeronautics Order No. 01593.
**Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
Report No. 1844-B
Agriculture-Madison
2.
The purpose of this thesis is to obtain the solution for the
critical load on circular sandwich cylinders of finite length subjected
to uniform external lateral pressure. The critical load is defined as
the intensity of pressure at which the cylinder buckles due to lack of
stiffness. This particular problem was chosen because it represents a
fundamental problem in sandwich analysis that has not previously been
solved. The need for such an analysis has arisen in connection with the
design of certain component parts of aircraft and guided missiles. It
is believed that the method of analysis used here is somewhat more rigorous than methods that have been used to obtain solutions to other
problems in sandwich analysis. For this reason, the general method of
approach may be of use in obtaining solutions to other new problems as
well as in obtaining better solutions to some problems that have already
been solved. The solution contained in this thesis represents an extension of a previous solution of the problem of the buckling of long sandwich cylinders (2).
In this thesis it is assumed that the sandwich cylinder is comprised of very thin, isotropic facings of a relatively stiff material
separated by an orthotropic core. The analysis of the facings is based
on membrane theory.* The core is considered to have such a low loadcarrying capacity in the tangential and longitudinal directions as compared to the facings that the normal stress in the core in these directions and the shear stress in the core on planes perpendicular to the
facings and in these directions may be neglected. This core assumption
*An analysis taking into account the stiffnesses of the individual facings may be made along the lines illustrated in Ref. 3.
has been widely used in sandwich analysis and is known to represent actual sandwich construction very well. The fact that no further simpli
fying . assumptions in regard to the core are needed, other than the assumption that it behaves as an elastic continuum, is emphasized. It is
felt that some analyses of sandwich construction are not sufficiently
accurate in certain ranges of physical properties and dimensions because
of additional simplifying assumptions made in regard to the core. The
action of the core and facings is related by the assumption that their
displacements are equal at the interfaces between the core and facings.
These interfaces are assumed to be at the middle surfaces of the facings.
The method used for establishing the stability criterion is similar in concept to that used by Timoshenko in the analysis of the buckling of homogeneous cylinders of finite length subjected to uniform external lateral pressure.* The assumption is made that, forpressures
less than the critical, the circular cylinder remains circular and the
only stresses present are a uniform circumferential stress in the facings and a uniform radial stress in the core. In discussing the buckling of the sandwich cylinder, only small deflections from this uniformly compressed form of equilibrium are considered; thus, the stresses induced in the cylinder as it goes from the circular to the slightly deformed configuration may be considered small as compared to the prebuckling stresses
*See Ref.
4, Art. 83.
Notation
r )) z
radial, tangential, and longitudinal coordinates,
respectively
a
radius to middle surface of outer facing
b
t
radius to middle surface of inner facing
thickness of each facing
length of cylinder
E
modulus of elasticity of facings
Poisson's ratio of facings
G
modulus of rigidity of facings
E c
modulus of elasticity of core in direction normal to
facings
G re
modulus of rigidity of core in re plane
Gr
modulus of rigidity of core in rz plane
intensity of uniform external lateral loading
q
1
k
1
a r
T
re
, T
,
7rec
n
rz
V , W
C
C
C
r
a
Et log
a
a
Ec
small normal stress in core in radial direction
u,
e
b
eczc
rz 7ez c
c
small transverse shear stresses in core
small radial, tangential, and longitudinal core displacements, respectively
small radial, tangential, and longitudinal core
normal strains, respectively
small radial, tangential, and longitudinal core
shearing strains, respectively
number of waves in circumference of buckled
cylinder
n2
2G - 2
E c
gne
re
Ec
rz
small normal strains in tangential direction in
outer and inner facings, respectively
8
E,
small normal strains in longitudinal direction i n
outer and inner facings, respectively
Up V, V
small radial, tangential, and longitudinal displacements of outer facing
Z Z
Irt
z
V'
small radial, tangential, and longitudinal displacements of inner facing
N
normal forces and shear force per unit length of
outer facing
Oz
normal forces, and shear force per unit length of
inner facing
Oz
E
c
a (1 -
Et
a
a
Et
natural logarithm
log
D , L , R An' Bn' Cn' n
n
n
arbitrary constants
6.
Mathematical Analysis
In the analysis of the sandwich cylinder, cylindrical coordinates
r, 8, and z are used; the dimensions of the cylinder and the positive
41•M•P
directions of the coordinates are indicated in figure 1. The radii to
the middle surfaces of the outer and inner facings are denoted by a and
h, respectively, and the thickness of each facing is denoted by t. The
origin of the coordinate system is placed at the middle cross-section of
the cylinder whose length is denoted by 2. Since q is considered to be
positive when it acts in the positive r-direction, buckling occurs at a
negative value of q.
As mentioned previously, the cylinder at the instant before buckling is assumed to be in a state of uniform compression with the same
stress condition existing in each cross-section. This assumption amounts
to saying that the ends of the cylinder are not supported until after
the cylinder has been initially compressed by a uniform load just less
than the critical load. The expressions for the pre-buckling stresses
are known,* and they are shown in figure 2. The radial stress in the
a
core is equal to q k and the circumferential force per unit length in
401•n••n•••••••nnn
the outer and inner facing is qa (1
and qak, respectively, where q
is the intensity of uniform external lateral loading in the positive r
direction and
1
k=
1
*See Ref. 2.
b
Et log aEa
In discussing the equilibrium of the cylinder after it has buckled into
a slightly deformed shape, it is assumed that the existing stress condition differs only slightly from the stress condition shown in figure 2.
The analysis of the equilibrium of a differential element of the core of
the deformed cylinder and of differential elements of the facings of the
deformed cylinder is considered next.
E2211112II2E2LLL2221-2
A differential element of the deformed core is shown in figure 3.
The stresses, a , T , and T , resulting from buckling are assumed to
r re
rz
a
be small as compared to the pre-buckling stress, q k. As indicated
previously, the assumption is made that the core and facing materials
are such that the stresses, a a and
Z
011..1•11.•
Tez'
in the core may be neglected.
.0.1111.01111•••
The effect of the changes in geometry in the core was found to be negligible; therefore, for simplification, the equilibrium equations are writ-
ten on the basis of the original geometry of the element. The summation
of forces in the radial direction results in the following equilibrium,
equation:
d qa k
r
(-q-9'-k-a)rdedz+(q a k+a + dr
T
(r + dr) dO dz +
de dr dz
69
6a
dr +
3r
r
dr)
rz (r + dr) dr de dz = 0
3r
If terms containing the product of more than three differentials are neglected, the above equation may be written as
aa r ar
+
)rr
)T
re
r 60
rz = 0
6z
(1)
The summation of forces in the tangential direction yields
)T
T
re
r dB dz + (T
re
627
dr) (r + dr) de dz + Tre dr de dz 0
or, neglecting differentials of higher order,
6T
re
6r
2T
re 0
(2)
r =
Finally, the summation of forces in the longitudinal direction yields.
rz
r de dz + (T +
rz
r z dr) (r + dr) de dz 0
or
67.
rz
r
rz = 0
r.
(3)
If it is assumed that during buckling the generators of the cylinder deflect into a half-wave of a sine curve and the circumference
subdivides into n waves, the core displacements take the following form:
u = f (r) cos ne cosgzc
1
v
c
f (r) sin ne cos /tz
2
w =f (r) cos ne sin
c
2
where u , v , and w represent the small displacements of the core in
c
c
c
the r, 19, and z directions, respectively. At the ends of the cylinder
2
u
c are both zero, which represents the conditions of simplyu and
c
2
6z
9
supported edges. The displacements u v, and w are related to the
c
c
c
core strains by the following equations*
duc
e
u
6v
c 1 c ;
r
°c r
= - •
r
6r
c
=
6u
1 _
c
7 e =
r 60
1
=
7ez r
c
Ni
c
6W
E =
C
6v .v614
6u
c
7
rz
6r.r
6z
dr
6v
6E)
(5)
c
6z
Also, since a9 and az are assumed to be zero,
c
.Ee =E
C
c r
or
in addition to
T
r0
G
7
r0 rec
6u
6v
,c
= Gre ( 21.7. w.2. 4. 177
v
(7)
and.
rz
=G
6u
6i4
f C
CN
rzrz c = Grz '6z 4' 6r
(8)
If the expressions for the displacements given by equations (4) are substituted into equations (6), (7), and (8) and the resulting expressions
for the stresses are required to satisfy the equilibrium equations(1),
*See Ref. 5, p. 305.
1 0.
(2), and (3), the functions of r that appear in equations (4) may be
determined explicitly, thus enabling equations (4) to be written as
follows:
r
a2
uc = [An+ Bna + Cna log a,
A
c
ne
n
,2
)
"
D n r] cos ne cos T
-
(9)
r.
n Ba n Ca n(1 + log -57)
n
cos "
a2
Dr
- 1) + X D n a
we = [X An a log -aI: + X B n r + C n r (log E.
a
(1 r 2
‘2a 22
where
6
r
z
7 log a)
+ Rna] cos ne sin /tz
ita
8 n0
5
2G
z
and
n2
2
E c
=
re
rz
A
h,
B
n,
C
and R n are arbitrary constants.
L
D
n' n' n'
If equations (9), (10), and (11) are substituted into equations (6), (7),
and (8), the expressions for the core stresses become
a = Ec
,2
A 0,
r
4. 0
"n
a n ) cos ne cos
(12)
11.
T=
re
A 2
n
c n r2
D
T =-E
rz
c
na
r
(13)
sin ne cos
cos ne sin
2
It may be easily verified that equations (12), (13), and (14) satisfy
the equilibrium equations, equations (1), (2), and (3). Furthermore,
because of the manner in which they were derived, it may be stated that
the right-hand sides of equations (9), (10), and (11) are unique functions insofar as their dependence upon r is concerned.
E22111111222L.5111EalaIl
Differential elements of the facings of the deformed cylinder
are shown in figure 4 • The quantities N N' N
e' 0 ,
N'
N
z' z' ez'
0..11•10
11n11.0.
and N'
ez
represent normal and shear forces per unit length of the facing upon
which they act. The core is assumed to extend to the middle surfaces
of the facings, and the stresses (a ) r r=a' (a r r=b' (Tre)r=a' (T rz ) r=a'
(
) r=b ,and (T )
are the . stresses exerted by the core on the facre'
rz r=b
ings as a result of buckling. All of the above quantities are assumed
to be small in comparison to the pre-buckling stresses shown in figure
4 as functions of the intensity of loading; a. The bending moments and
transverse shear forces in the individual facings are neglected, in accordance with membrane theory. In formulating the equilibrium equations
that apply to the facings, the effects of the rotation and stretching of
the facings must be taken into account. Because of the stretching of
the facings, the areas of the differential elements.of the outer and
12.
inner facings become (1 + e ) (1 + e ) a de dz and (1 + e') (1 + e')
z
b dO dz, respectively. Since the middle surface strains are small quantitles, their products may be neglected, and the differential areas of
the outer and inner facings are expressed as (1 + e + e ) a de dz and
e
z
(1 + e° + e') b de dz, respectively. The difference in the amount of
e
rotation of the sides AD andCD with respect to the z-axis of the
inder causes the central angle of the outer element to change from de
A2
to (1 + a ae a 2
de Similarly, the central angle of the inner
•
6e
2
6Ar'
element changes from de to (1 + -1z --77) de. Also, the difference in the amount of rotation of the sides AB and CD with respect to
A2„
the r-axis is
+ 11) ; the corresponding difference in rotation of
606z
6z,
the sides A' B? and C'D' of the inner element is
2vi
606z 6z
. The
foregoing expressions for the differences in rotation are as given by
Timoshenko* and verified by Osgood and Joseph.** Since N z and
are,
small quantities, the relative rotation of the sides upon which they act
may
be neglected; the inclusion of this effect leads only to small terms
of higher order.
The necessary. changes in the geometry of the facings having been
established, it is now possible to write three equilibrium equations
for each facing. Considering first the differential element of the
outer facing, the summation of forces in the direction perpendicular to
the surface of the element results in the following equation:
*See Ref. 4, Art, 79.
**See Ref. 6.
1
Nz (1
ee )ade
N,
uz
(Nz
dN
Oz de)
6e
6N
TE.z-
14.e)ade
-N
0
ez) dz
(r
)
1
rz r=a
1+ e ) dz
z
e
ee z
)
qa (1 .. k) ( 321/
) de d
dz = °
603z 3z
a de dz
If small quantities of higher order are again neglected, the above equation of equilibrium reduces to
6Nz 4. 1Nez
6e
'3
(r rz ) r=a
q (1 - k)
62v
3u\
dedz dz
(17)
The three equilibrium equations pertaining to the inner facing are obtained in a similar manner and may be written as follows:
N2
N
6N'
1 e
5-"T
+ q k (u'
(a)
r r=b
b
6N'
ez
az
30
u)
2
rer.b
(18)
(19)
and
6N'
z
6z
311'
2
Oz. 41. )
a k ( 6 vi
611)
rz
r=b
q 1;
6eaz 6z
b 3e
(20)
On the basis of Hooke's law, the following stress-strain relations in
the facings are applicable:
Et (€
No = 1.412
, 0
,,ez)
I.,
(21)
15.
N
Et (€ .4. nee)
Et
N=
ez
(22)
2(1+0 ez
ez
N(e,
Et
iter)
1-p. 2 e r z
e
=
Et
(23)
(24)
+
ket
z
(25)
and
N
Gt
ez
uz
Et
2(1+11) vz
(26)
where E, G, and a represent the modulus of elasticity, the modulus of
rigidity, and the Poisson's ratio of the facings, respectively. Since
the strain-displacement relations given by equations (5) are applicable
to the facings if r is replaced by a for the outer facing, r is replaced
by b for the inner facing, and the corresponding displacements of the
facings are substituted for the core displacements, equations (21)
through (26) may be written as follows:
N .
N
z
Et (u
&
2
a
a9
1-11
6z
+ a 6v 6w.‘
Et (
2 ‘ P4 a a de +
1-11,
(27)
(28)
N =
CZ
Et & 1w
2(1+4) 6z
a 6e
Et u t
f
1‘
v'
1* To. "aTi–
l-4
E t
N =
Z
1-4
u'
2
(29)
6w,
(30)
P' 57)
3w 1
6v'
b •b 6e
( 31 )
6z
and
N
_Et
6v' 1 6t4')
2(1+4) ` 6z
1-; 69
'
(
Oz
( 32 )
If continuity of displacements at the cylinder interfaces is assumed,
remembering that the core is considered to extend to the middle surfaces
of the facings, then
u
=
( uc)r=a
u' = (u)
r=b
v = (v)
v' = (v
)
r=a
c
w = (w
w y = (w
r=a
r=b
)
c
r=b
The above relations enable equations (27) through (32) to be expressed
as follows:
N
=
Et ( Ile , 1 6.7 2 a
a 66)
1.0.0
1-4
c
61,7p,
az r=a
( 33)
17,
Et ( u c)v c awc
2
a
a0
dz r=a
N=
TT
)v
Et
( c
ez 2(1+0 3z
=
a &30
Et (2_:. + 1 vc
i";
2 b
' N
(3)4)
=
z
( 1-p. 2
+
b
(35)
r=a
p, wc)
6z r=b
(36)
c +c)
b9
.6z r=b
(37)
3-v
and
N , .
Et ( c
ez
2(1.411) 6z
6T,/ c)
(38)
b 0 r=b
Also, if account is taken of the continuity of displacements at the
interfaces, the equilibrium equations of the facings, equations (15)
through (20), may be written as follows:
2u
N = - a (ar
r=a + q (1 - k) (u +
60
1
M
6z
(39)
r=a
N
e4. _ ez (T
a0
z
2
z
2IN
ez
a &)
re) r=a
(T)
rz r=a
(40
ci ( 1
2
v
3u
k)
.)03z
r=a
a)
18.
62u
N' b(a )
+
8
r r.b + q ba k (u0
69 2
6N'
6N'
113 6996 : z
(2)
r.b
(Tre)r=b
(14
3)
and
6N'
31q.'
z 1 ez
32,
b
32v
(T rz ) r=b "I" c111 k ( 603:
311
c
)r=-1
If the expressions for the facing stresses given by equatiam3(33) through
(38) are substituted into the above equations and then a further substitution is made of the expressions for the core displacements and stresses
given by equations (9) through (14), six equations containing the parameters A , B , C , D , L , and R that appear in the equations for the
h
n
n
n
n
n
displacements of the core, equations (4), are Obtained. These equations
may be written as follows:
19.
0
ro I oz;
N
stV
H
Cu
SI
11 -.1*
1
+
0
4
:I.
+
'17
-4'
0
1:::)
0
P1
4
+
0
=tg
r.---,
.14
_41
N
H
......-
ti
--,
H
N
o
1.....,
CM
+
CC1
I
(3)
(00
I
bf)
0
71
H
H
.....,
I N
dra I ai
L......t
CV IN
ro 1 cd
+
Ed
1.. . . 1
SI
0
0
ra i
0
H
N
+
0
A
Ed
tO
(\IP I Ncd
e4
O
on I cd
140
0 Ins
o
H
.---,
H.
I
(VP I Nai
+
al
+
Nei co
H
L..,
N41 No
,a la)
' C\I
HV.......-
-----,
to
0
H
do I cd
H I N
H
.........,
0
o
,---,
co.
+
to0
4-
ea i 03
co. 'No
o
co
I
rO
0
ro 03
H
I
r-4
o
-1:1:e03
N
,......,
CO
+
NrolNal
+
+
4-
N.
A cki
ai+
gi
,.........
1
0
ra
+
N
4:1
r-i
I
1
+
ri
+
0
+
1-4
N
I
N
4
re:0 I ni
N
4
1
Ackj
+
20.
ajr0 (\ja,
r4
o
Nro I Ncd
ro I (d
,•4
1
H
0
N
,c) 1 cti
o
ajr0 ( °jai
ci
";'
H
N
H
o
1
---,
H
1
0‘1
s:1
.........
C
P I d
N
$:1
I
,......
+
-1. I
ro
ad
N.11 al
11
H
+
+
0
Mici
0
COCIr---,
P.-1
cO
....---,
,........,
43 I d,....1
1
1
t/0
chi1
0
H
H
(0 0
......,
N lal
,--,
d
ro 03
I
CV
N
5:1
PI
+
+
+
A
NO I e.4
,--....
N
,--1
....._...
+
.4
.....,
+
21.
0
trN
ra cd
N
r4
o
,0 I cd
ta0
0
teN I rrN
ro I (LI
al
ICu
HI
4.r) I CO
bO
./....,
0
r-i
H
,0 I cd
+
0,-,
H
c)
0
9
1
ro 1 od
N
,4 I at
+
N IN
ro I 03
........
..----,
P I c0
ta0
0
H
airo rcti
N
.4
+
rri 1 ct
b0
0
H
Cu
i.
1
,-0 I cd
g
ro I (13
N
o
0
P.....4
+
rri0
Cl
Cu
i9 (NJ
1
cd
la0
0
i
tra I
a
; I....1H
.......
N
+
Nro I Ncd
i.
o
+
H
.......
+
0
0
P I cd
1.....9
to 03
tts
H
ra i cd
o i ai
,.....,
+
r.....1
Nro lalco
cal aje4
(r)
0
LIO
o
tsi
ti
to
,
co
H
0
r-4
00
;
;
aigl I al..4
lxic
NoN
.....,
....,
al
+
+
+
a
0
H
+
1
N
a
H +3
r4
.4,......
cd
C.)
,---.
N
a
I
H 4-)
....... ril
Cd
H
ca.
a)
a)
a)
;.4
04
rd
a)
e ir-1
riCH
a)
II
id
vi
ti
cd
22.
E
c
n2
6 .
nø2G2
re
b = Ec
z G
rz
x
Equations (45) through (50) are satisfied if the constants A n , Bn, Cn , Dn,
L n , and R are all equal to zero. This represents the uniformly comn
pressed circular form of equilibrium of the cylinder. A buckled form of
equilibrium is possible only if equations (45) through (50) yield nonzero solutions for the constants. This requires that the determinant of
the coefficients of these constants be equal to zero. This determinant
is shown on the following page.
:
ircti
4"
-.iN,T-,0
,,,
v
,,
•
•
-A
N .
.
'4, 1--,,,
-,
:
-
5-
:
5-
i
.
N.<
:
c V,
.
..
.
Si
+
-.
:
•
fi N
+
N .
A lm
•
%
N.
.
0
:
-0
fic,
,-
- :
•
Y,-T----.
-
24.
The lowest, real, negative value of a for which this determinant equals
zero is proportional to the critical load on the sandwich cylinder; this
value of a will be referred to hereafter as a . After this value is
cr
determined, the value of the critical load is obtained from the definition of a given previously, that is,
q
a
=
cr
a (1-p.2)
Et
Numerical Computations
\
Since the determinant in equation (51) contains the eigenvalue,
in four of the six rows, the expansion of this determinant would result
in a fourth order algebraic equation in a. Only the lowest of the four
roots of this equation, the root that corresponds to the type of buckling
illustrated in figure 5, is considered. Actually, for very short cylinders the root that corresponds to a face wrinkling type of buckling may
become the lowest; a satisfactory analysis of this type of buckling requires that the stiffnesses of the individual facings be taken into account. Since this analysis is based on the assumption of membrane facings, no attempt is made to determine the critical loads for this type of
buckling.
A literal expansion of the determinant in equation (51) was found
to be impractical. It was decided that numerical solutions obtained by
the use of machine methods would be the most practical way to obtain design data.* Originally, the sixth order determinant as expressed in
*The facilities of the Numerical Analysis Laboratory of the University of
Wisconsin were utilized for this purpose.
25.
equation (51) was used as a basis for numerical solution. However, due to
inherent difficulties involving the subtraction of large numbers of almost
equal magnitude, it was found that sufficient accuracy could not be obtained from this approach. The sixth order determinant was then reduced
in literal form to the fourth order determinant shown on the following
page. A procedure was devised from which the desired eigenvalue can be
determined from this determinant by trial and error. For a given set of
parameters containing the dimensions and physical constants of the cylinder, the assumed value of the eigenvalue is varied until the value of
the determinant becomes equal to zero. In this procedure each of the
twenty-four terms in the expansion of the fourth order determinant is computed independently and their sum is accumulated subsequently. This eliminates the previous difficulties in connection with loss of accuracy.
The numerical results obtained are shown in Tables 1 through
7.
The values are believed to be correct to three significant figures. More
significant figures can be obtained by simply increasing the number of
trials used. Curves based on the values given in the tables are shown in
figures 6, 7, and
8.
27.
Discussion of Results
The curves shown in figures 6, 7, and 8 are sufficient to illustrate how the value of the critical load varies when the parameters
representing the dimensions and physical constants of the sandwich cylE
E
inder are varied. The ratios, -2— = 4 and
= 10, were chosen since
Grz
they represent average values of these quantities for honeycomb cores,
the type most commonly used in sandwich construction. Computations were
also made with the values of these ratios interchanged since either situation is possible depending upon the orientation of the core in fabrication. The numerical results shown in Tables 1 through 4 and the curves
shown in figures 6 and 7 are based on the value of p . 1. This value
represents a cylinder that has a weak core; that is, a core having relatively low values of E G, and G • The numerical results shown in
c r0
rz
Tables 5 through 7 and the curves shown in figure 8 are based on the
MIP.116.06
Opop.m.66.M.
value of = 1 000. This value of
represents a cylinder having a
E
stiff core. Since p depends not only upon the ratio, FE, but also upon
13
MMIMM.
the ratio, a, the practical range of values of E is quite wide. Most
constructions have dimensions and physical properties such that they lie
in the range between p . 1 and p . 1 000. All of the computations are.
based on the value of .3 for the Poisson's ratio of the facings. Results were obtained for only two different ratios of the radius of the
inner facing to the radius of the outer facing; that is, .-=.95 and
a
•
a = .98. The procedure for machine computation was set up. so that any or
all of the values of these parameters may be changed without any
difficulty.
28.
The results indicate that ) for cylinders having relatively weak
cores, it is advantageous to have the higher value of the modulus of rigidity of the core in the tangential direction rather than in the longitudinal direction. The effect of the interchange of these values is evident from a comparison of the curves shown in figures 6 and 7. This effect is negligible in the case of cylinders having strong cores as evidenced by the fact that the values given in Tables 5 and 6 are essentially
the same. The fact that in all cases the overall stiffness properties of
the core become relatively more important as the length of the cylinder
decreases may also be noted. As the length of the cylinders is increased,
the values obtained for the critical loads approach the values previously
obtained on the assumption that the cylinder length is infinite. This is
to be expected because, if X is allowed to approach zero, the characteristic determinant for this problem reduces to the determinant obtained on
the basis of the assumption of infinite length.* As the cylinder decreases in length, the value of the critical load approaches the value
11N
G
re (1 - a and then falls off rapidly to zero. In any particular case,
if E is set equal to infinity, •the critical load becomes equal to
when the length becomes zero. This limiting value for the
critical load is characteristic of stability analyses of sandwich construction in general, if E c is assumed to be infinite. Examination of
011.1n11.
the curves shown in figures 6, 7, and 8 shows that the curves representing the cylinder having a weak core reach a maximum in the neighborhood
*See Ref. 2.
29.
f –= 1 while those representing the cylinder having a strong core cona
tinue to rise until the cylinder becomes very short. Indications are
that analyses based on the assumption of E c = co are sufficiently accurate
in most ranges, but for relatively short cylinders having weak cores such
an assumption may result in serious error.
Conclusions
The stability analysis of sandwich cylinders subjected to uniform
external lateral pressure presented here is believed to be sufficiently
accurate for use in design. The assumption of membrane facings limits
the range of applicability of the results somewhat since the effect of
the stiffnesses of the individual facings may be appreciable in some sandwich cylinders, particularly short cylinders having relatively thick facings. However, for cylinders of usual sandwich construction and with a
length equal to or greater than the radius, this theory is believed to
be adequate. The analysis of the problem with the stiffnesses of the
facings taken into account may be performed without a great amount of
additional work, and such an analysis is planned for the near future.
Additional curves giving the values of the critical loads for a greater
number of values of the parameters entering the problem are desirable
for design purposes. Such curves may be easily prepared in the future if
the time and expense involved in their preparatian seem justifiable. The
Forest Products Laboratory is at present planning a series of tests for
the purpose of correlating the results with the theoretical results Obtained here.
30.
Bibliography'
(1)
Nash, W. A. (2)
Raville, E. "Analysis of Long Cylinders of Sandwich Construction under Uniform External Lateral Pressure" (in:
Forest Products Laboratory Report No. 18)44, Nov.
1954).
(3)
Raville, M. E. "Supplement to the Analysis of Long Cylinders of
Sandwich Construction under Uniform External Lateral
Pressure" (in: Forest Products Laboratory Report
No. 18)44-A. Feb. 1955).
(4)
Timoshenko, S.
(5)
Timoshenko, S., and Goodier, ' J.
N. Y. 1951.
(6)
Osgood, W. R., and. Joseph, J. A.
"On the General Theory of Thin
Shells" (in: Journal of Applied Mechanics, Vol.
17, No. )4. Dec. 1950).
"Bibliography on Shells and Shell-like Structures"
(in: TMB Report 863, U. S. Navy Department, David
Taylor Model Basin, Washington. 195)4).
"Theory of Elastic Stability" 1st ed., N. 1936.
"Theory of Elasticity" 2nd ed.
Table 1. Values of Critical Pressure Expressed in Terms of d.
ct.
q a (1412)
Et
'a
c
Gro
E,
=10
rz
x
: n
x
0.00273
2.
0.25 .40
.5o
.50
.6o
.8o
1.00
.80
1.00
1.20
1.40
1.60
1.40
1.6o
1.80, 1.80
2.00
2.25
2.25
2.62
CI
If
ft
3
ft
•
ff
ft
• 4
ft
ft
ft
ft
5ft
•
•
6
ft
ft
7ft
•
•
.00292
.00373
.00496
.00,64
.00588
.00673
.00826
.00768
.00808
.00869
.00956
.0107
.0095o
.00993
.0105
.0104
.0107
.0112
.0110
.0115
2.62
3.15
3.15
3.15
3.15
3.15
3.15
3.15
3.50
3.50
3.5o
•
4.00
4.00
•
•
•
4.5o
4.5o
4.50
4.5o
4.50
5.00
6.00
10.00
•
•
•
8
ft
9
:
:
:
•
:
:
: o, 0114
.0119
.0117
:
10
11
12
:
:
:
13
14
13
14
15
:
:
:
:
:
15
• • 16
16
17
18
19
:
:
20
20
:
21
23
.0116
.0115
.0115
.0115
.0115
.0115
.0115
.0115
.0114
:
:.0114
.0113
:
.0112
.0112
:
.0112
:
.0112
•
.0110
:
.0106
:
.00930
:
Table 2. Values of Critical Pressure Expressed in Terms of d.
2
a g cr a (1"1,t
Et
E
12 = .98
a
x
0
0.0523
.25
.40
.25
.40
.60
.80
.40
.8o
1.20
1.20
1.60
1.60
2.25
G
= 10
rz
n
: 2
: If
If
:
: ft
: 3
: H
: n
: el
*. 4
il
:
: It
.. 5
: ff
: 6
: fl
x
: 0.000522
.000523
:
.
.000677
.
.00144
:
.00121
:
.00129
:
.00160
.00236
:
.00188
:
.00213
:
.00299
:
.002e4
:
.
.0051
:
.00339
:
.00439
:
:
: 2.25
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
3.15
2.25
3.15
3.15
3.15
3.15
3.50
3.50
3.50
4.00
4.5o
4.5o
4.5o
6.00
n
: 7
. ft
: 8
fl
:
: 9
: 10
: 11
: 12
: 13
: 14
: 15
: 16
: 17
: 18
: 23
ci
: 0.00403
.00552
:
.00401
:
.00488
:
.00461
:
.00450
:
.00447
:
.00458
:
.00458
:
.00457
:
.00462
:
.00471
:
.00470
:
.00469
:
.00460
:
Table 3. Values of Critical Pressure Expressed in Terms of U.
q
a (1-?)
d = cr
Et
Ir a
p. 1
E
= 10
orb
.95
b"
a
:
0
0.25
.
.
.50
:
Ec
rz
n
s 2
: It
.4o
If
: 0.00190
:
.00209
:
.00290
:
:
:
: 1.20
: 1.20
: 1.20
:
:
.00413
.00327
.00343
.00368
.00438
.00426
:
: 1.40
: 1.40
: 1.40
: 1.4o
: 1.40
: 1.60
.•
:
:
:
: 12
: 16
ft
.4o
.50
.60
.8o
.80
1.00
: 3
: ft
: II
: I,
: 4
•. "
1.20
1
.00
: ft
: 5
n
X
•.
•
:
:
:
:.
:
:
:.
.00472
.00543
.00462
:
..
:
:
.
:
5
6
7
8
9
10
11
: 1.60
: 17
: 1.60
: 1.80
: 18
: 18
-ri
•.
:
:
..
:
:
0.00492
:
.0o483
.00483
.00493
.00490
.00490
.00488
.00488
.0048,
:
:
x0485
.00477
:
.00485
Table 4. Values of Critical Pressure Expressed in Term ofd.
q C r a (14)
Et
=1
Ec = 10
Gr0
EC
Gr z
timmunemedmagsuplimforif
n
•
x
2 • 0.000432
0
: 1.20
0.25
:
.00°586
: 1.60
.11
.4o
.00135
:
: 1.60
.40
: 3ft : .000948
: 1.60
.00126
:
.60
:
1.60
ft
:
.00201
1.80
.60
.00128
4
.60
:
: 1.60
If
.80
:
.00146
•
2.00
ft
1.00
.00179
• 2.25
1.00
.00163
: .2.25
5
1,20
.00184
:
: 3.15
If
n
:
6
ft
: 7
: 8
: 9
10
11
:13
: 16
: 17
25
:
:
:
:
:
:
:
:
0.00177
.00212
.00198
.00194
.00193
.00193
.00193
.00198
.00197
.00197
.00196
Table 5. Values of Critical Pressure Expressed in Terms of d
qC
x
1 2)
a (Et
p l000
E
a
G
.
=
/JO
E,
.95
G
: n
«
5 o
0.25
.40
.50
.60
.75
.60
.75
1.00
1.20
1.40
1.60
1.80
1.80
2.00
2.25
2.62
3.15
2.62
3.15
14.00
: 2
: II
: If
: If
: "
: If
: 3
: If
: II
: ff
: It
: ft
: ft
: 4
: 91
: If
: If
: II
: 5
: "
: If
: 0.00394
.00420
:
:.
•
:
:
:
:
:
:
:
:
:
:
:
:
:
:
:
rz
X
X
:
:
10
:
.00510
:
.00640
.00852
.0137
.0117
.0126
.0151
.0182
.0224
.0279
.0347
.0315
.0352
.04o6
.0505
.0690
.054o
.0667
.0953
:
:
:
:
:
:
:
:
:
:
.
:
:
:
:
:
:
:
:
4.50
4.50
5.00
6.00
6.00
7.00
7.00
8.00
9.00
8.00
9.00
10.00
10.00
11.00
11.00
12.00
12.00
13.00
13.00
16.00
:
6
Ii
:
:
:
7I
I,
8
ft
: o.118
.117
:
.139
:
.194
:
.193
:
.257
:
.2.).8
:
:
If
::
:
:
9If
::
If
::
:
: 10
II
6•
11
:
12
:
: 13
If
:
If
if
.332
.427
.335
.419
0524
.517
.634
.627
.755
.748
.888
'0881
1.42
Table
Va:ueo of Critical Pressure Expressed in Term of a..
(1 -112)
=
p = 1000
.98
x
X
2
0
0.105
.25
.40
tt
It
ft
ft
.50
.50
.60
0.000612
.000620
:
1.20
1.40
1.60
2.00
2.00
ft
tt
0
•
it
.000771
.00153
.00272
It
ft
.75
1.00
1.00
2.62
3•15
3.15
.00189
.00209
.00261
.00422
.00386
.00444
.00526
.00637
.009,3
•00798
.0114
.0156
.0140
:
:
:
:
:
:
:
:
4.00
5.00
5.00
6.00
6.00
7.00
7.00
8.00
10.00
8.00
10.00
: 10.00
12.00
12.00
: 12.00
: 15.00
15.00
n
:ci
0.0197
.0292
ft
7
ft
8
ft
9ft
ft
: 10
:
:
:
:
:
:
:
.0265
.0361
.0349
.0452
.0448
•0562
.0876
.0565
•0843
it
: 12
13
15
16
•
.0836
:
.120
:
:
:
:
.118
.118
.181
.181
Table 7. Values of Critical Pressure Expressed in Terms ofa.
qc
x=
a (1-42)
Et
p = 1000
E
= 10
re
a
Eo =
= .95
Grz
••
n
o
0.25
.40
.50
.60
.75
.60
.75
8
1.00
: 2
: It
: ft
:
••
:
:
:
:
:
:
..
ft
If
tf
It
st
It
1.21,h
:
1.140
1.60
••
1.8o
1.8o
:
: 4
2.00
:
2,25
2662
3.15
2.62
:
If
ft
It
It
: ft
It
:
It
:
: 5
.00510
:
:
.0151
••
.0182
60224
:
3
:
•.
:
:
:
`•
••
:
•
0..00394
.00420
.0064o
.00852
.0137
.o116
.0126
.0279
.047
.0315
00352
A05
.004
.0689
.0538
S
•
•
•
3.15
4.00
4.5o
•
4.5o
•
•
5.00
6.00
6.00
•
7.00
7.00
8.00
9.00
8.00
9.00
10.00
10.00
•• 11.00
: 11.00
12.00
12.00
o•
:
It
:
:
•
•
•
If
:
.0951
.118
.117
.138
.194
.193
.257
.258
:
:
.332
*428
6
:
..
:
7
.•
ft
ft
ft
II
ft
tf
If
10
ft
ft
12
..
0,066,
.334
.10.9
:
•
:
•
:
•
.526
.517
.635
.626
.757
.747
Figure 1.--Sandwich cylinder.
Figure 2.--Differential elements of core and facings before buckling.
Figure 3.--Differential element of deformed core.
Figure 5.--Cross-section of buckled cylinder.
(for n = 4)