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EXTEUSIAIL RITUAL P1RIESSUPE
November 1954
This Report is Cno of a Series
Issued in Cooperation with
AIR FCRCE-NAVY-CIVIL SULCOMMITTEE
on
AIRCRAFT DESIGN CRITERIA
Under the Supervision of the
,AIRCRAFT COMMITTEE
of the
MUNITIONS MARI)
No. 1844'
UNITED STATES DEPARTMENT OF AGRICULTURE
FOREST SERVICE
FOREST PRODUCTS LABORATORY
Madison 5,Wisconsin
In Cooperation with the University of Wisconsin
OTEi."3.0t1 FOREST PRODUCTS LABORATORY
LIBRARY
ANALYSIS OF LONG CYLINDERS OF SANDWICH CONSTRUCTION
1
UNDER UNIFORM EXTERNAL LATERAL PRESSURE -By
MILTON E. RAVILLE, Engineer
Forest Products Laboratory, Forest Service
U. S. Department of Agriculture
Summary
A theoretical solution is obtained for the stresses in long sandwich
cylinders subjected to externally applied uniform lateral pressure.
The analysis is extended to take into account failure due to elastic
instability, and an equation is derived for the determination of critical loads on long sandwich cylinders. The application of this analysis
to long curved panels (portions of a long cylinder) is discussed. The
sandwich construction is assumed to consist of isotropic membrane facings and an orthotropic core.
Introduction
In this report it is assumed that the sandwich cylinder is comprised of
thin, isotropic facings of a relatively stiff material separated by an
orthotropic core of a relatively weak material. The facings are assumed
to be so thin that bending and shear in the individual facings may be
neglected. It is further assumed that the only stress components present
in the core are the normal stress on surfaces parallel to the facings and
the transverse shear stress. Since elasticity theory is used in regard
to the core, the solution is not limited with respect to core thickness.
Results based on the assumption of membrane facings are somewhat limited
-This report is one of a series prepared by the Forest Products
Laboratory under U. S. Navy, Bureau of Aeronautics Order No. 01595.
Maintained at Madison, Wis., in cooperation with the University of
Wisconsin.
Report No. 1844
Agriculture-Madison
in applicability, but they should be sufficiently accurate for the
majority of sandwich panels. It is felt that the core assumptions
represent reasonably well the cores presently in use, especially those
of the honeycomb type.
The method of analysis is based primarily on the fact that the core assumptions permit the direct determination of the core displacement functions
insofar as their dependence on the radial distance is concerned. The usual
requirement of continuity of displacements at the interfaces is specified.
The stability analysis is based on the method used by Timoshenko2 in the
analysis of the stability of homogeneous isotropic cylinders. The stresses
in the sandwich cylinder are determined for loads less than the critical
load; and, in discussing buckling, only small deformations from this unifatally compressed form of equilibrium are considered.
Notation
r, e, z
radial, tangential, and longitudinal coordinates, respectively
a
radius to middle surface of outer facing
b
radius to middle surface of inner facing
t
thickness of facings
E
modulus of elasticity of facings
Poisson's ratio of facings
E
modulus of elasticity of core in direction normal to facings
G
re
modulus of rigidity of core in r-e plane
q
intensity of uniform external lateral loading
p
intensity of uniform external lateral pressure, equal to -q
Q
ar, Q',
r re
normal stress in the radial direction in outer facing, inner
facing, and core respectively
n't
T
arc
re
re
small normal stresses in the radial direction in outer facing,
inner facing, and core respectively
small shear stress in the plane of the middle surface of the
outer and inner facing, respectively
3S. Timoshenko. Theory of Elastic Stability, p. 446.
Report No.
1841
-2-
F!PE2f:”v FOREST PRODUCTS LABORATORY
small transverse shear stress in the core
rec
direct stress resultants in the tangential direction in the
plane of outer and inner facing, respectively
Ne , Net
small direct stress resultants in the tangential direction
in the plane of outer and inner facing, respectively
.49
U, U
radial displacements of outer facing, inner facing, and
core, respectively
Lie
u, u , u
small radial displacements of outer facing, inner facing,
and core, respectively
V V , V
small tangential displacements of outer facing, inner facing,
and core, respectively
Ee
tangential strains in outer and inner facing, respectively
C
,
o
C
e C ec
small tangential strains in outer facing, inner facing, and
core, respectively
n
number of waves in circumference of cylinder
a
one-half the central angle of curved panel
k
log
(1 + b/a)
1
Et log b/a
E a
c
natural logarithm
A, B, A, B, C, H, A, B, C, H
n n n n n n n n
arbitrary constants
Theoretical Analysis
The first step in the analysis is that of determining the stresses in the
sandwich cylinder for pressures up to the critical pressure. The cylinder
remains circular and is in a state of uniform compression. Throughout the
analysis, q represents a uniform, lateral load acting in the positive radial
direction. The pressure, p, is equal to -q.
Report No. 1844
-3-
Equilibrium of Core
As previously stated, it is assumed that the core can transmit only normal
stresses in the radial direction and transverse shear stresses. When the
cylinder is in a state of uniform compression, the only stress present in
the core is the normal stress in the radial direction, a. Considering
rc
the equilibrium of the differential element of the core shown in figure 1, the
summation of forces in the radial direction results in the following equation:
- a
rc +
rc r de dz + (a rc
da t
dr)(r + dr) de dz = 0
drr
This reduces to the following differential equation of equilibrium:
da
dr
rc
arc = 0
The solution of equation (1) is
a = E
rc
c r
in which E c A represents the constant of integration. The radial displacement
of the core, u e , is related to the radial stress, arc ' by the following equation:
du c
(3)
E c dr
a
rc
From equation (2) it follows that
duc A
.
dr
r
Integration gives
u
c
(4)
= A log r/a + B
The use of log r/a 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 2,
it is seen that, when the cylinder is uniformly stressed by the action of the
external load, q, the only stresses present in the facingsare the radial
stresses, a and a , which are exerted by the core, and the tensile forces
r
r
per unit length of facing, N e and Ne . These stresses are assumed to be
acting on the middle surfaces of the facings, an assumption which is justified since the facings are assumed to be thin. An equilibrium equation
Report No. 1844
-4-
can be obtained for each facing by'summing forces in the radial direction
on each element. The equilibrium equation which pertains to the outer
facing is
qa de dz - a a de dz - N9 de dz = 0
r
This reduces to
N = qa - a a
0
r
or, since a r
= (arc)
r = a
N = qa - a(a )
0
(5)
r = a
In a similar manner, the equilibrium equation of the inner facing is obtained as
N
1
e
=
b(a )
rc
(6)
r = b
The requirement is now made that the displacements of the core and facings
be equal at the respective interfaces. It is assumed that the core extends
to the middle surfaces of the facings. Thus
u = (u)
(7)
r = a
and
(8)
u' = (u )
c r = b
From Hooke's law,
N
e
=
Et(E )
Since e e = , and, in view of equation (7),
No = Et1
a
By using equation
Report No. 1844
(u0)
Ee
=
u
r = a
a
r = a
(4)
in conjunction with the above equation
-5-
then
(9)
Et(32)
a
N
Also, from Hooke's law,
N' = Et(e)
0
0
u
(8) and
Since c e = , the above equation in conjunction with equations
(4) leads to:
(10)
Ne = Et( TA; log b/a +
Since, from equation (2), (arc)
r = a
A
= E - ' equations
- and (a rc )
cb
Eca
r = b
(5) and (6) may be written as
Ne qa - E A
c
and.
1
(12)
N E A
c
e
equations (9) and (11) and the
(12), two equations are obtained
A and B may be determined. These
qa -
(13)
Et
EA = --B
a
c
and.
From equation (14)
(15)
Substituting the above value of B into equation (13) and multiplying
t,
through byE
—
Et'
Report No. 1844
-6-
qa2
mca
Et
Et
A =
[E
.cb
- log bial A
Et
from which
A =
Eca
Et
Et
(1 + b/a) - log b/a
or
A =
1
mss
Ec
+ b/a)
Et -log b/a
(16)
E a
Substitution of the above value of A into equation (15) gives
Et log b/a
B = 2a
.13
Et
E cb
(1 + b/a) -
Et log b/a
E a
( 17 )
When the value of A given by equation (16) is substituted into equations
(2), (11), and (12), respectively, the following expressions for the
stresses in the cylinder are obtained:
clafir)
w rc --\"/
N
N
where k =
e
qa
e
q k
a
(18)
- k)
(19)
(20)
1
Iva
+ b/a) Et log
E ca
Also, by using the values of A and,B given by equations (16) and (17),
equation (4) fox the core displacement, after some simplification,becomes
2
ue - Et
Report No. 1844
E
- k) + ---k log rid
E ta
c
-7-
(21)
Since u, the radial displacement of the outer facing, equals (u c )
u , the radial displacement of the inner facing, equals (uc)
r = a
and
r = b
u = 22." (1 - k)
(22)
qa2
(k b/a)
u =- Et
(23)
- Et2
and
The analysis thus far is not limited strictly to membrane facings. The
only restrictions on facing thicknesses are those imposed by the usual
thin-walled-cylinder theory plus the additional requirement that the radial
displacement of the middle surface of the facings be equal to the radial
displacement of the corresponding interfaces.
The next step in the theoretical analysis concerns the development of the
stability criteria. In this analysis, which follows, the facings are
assumed to be membrane-type cylindrical shells. As mentioned previously,
it is assumed that the stress situation in the deformed cylinder differs
only slightly from the stress situation which exists just before buckling.
A bar is placed over the appropriate symbol to denote the small stresses,
strains, and displacements which occur when the cylinder goes from the
initially uniformly stressed circular form to the slightly deformed configuration. These quantities are dependent upon 0 as well as r.
Equilibrium of the Core
Since the cylinder is how considered to be in a slightly deformed state,
the core is also slightly deformed. Figure 3 shows that, in addition to
the radial stress, 4k, which exists just before buckling, an additional
small radial stress, a rc
and a small transverse shear
stress, T
must
rc ,YrOcY
be considered. These stresses result when the core takes on small deformations from the initially uniformly compressed, circular shape, and they
are dependent upon 0 as well as r. The differential element shown in
figure 3 is considered to be in equilibrium under the stresses shown. Summation of forces in the radial direction yields the following equilibrium
equation:
(cik + d m ) r do dz +
T
Report No. 1844
rec
dr dz + (T
rec
1 ++
- rc
d(q111)
6°rc
dr dr + ar dr)(r + dr) de dz
+ rrec do) dr dz = 0
60
-8 -
If terms containing the products of more than three differentials are neglected, the above equation reduces to
6
66 -r dr de dz + --119-dr de dz = 0
rc dr de dz + –g7:9
6e
or, dividing through by r dr de dz,
a& rcrc
6r
r
1 67rec
–
- o
r 6(9
(24)
Summation of forces in the tangential direction yields the following equation of equilibrium:
-
rec
r de dz + ( T
+
rOc 6T6 rrec
dr)(r + dr) de dz + T rec dr de dz = 0
which reduces to
e rec 2T rec
r
(25)
o
The core displacements are related to the core stresses by the following
equations:
a re
E
65c
(26)
c
and
rec = G re
1 6u c
6v c
7/7'
(27)
r
where 11 e and 1 c are the small radial and tangential core displacements
from the uniformly compressed form of equilibrium. Equations (24 - 27)
are sufficient to enable the direct determination of the displacement
functions 11 and 1 insofar as their dependence on r is concerned. Thus,
e
c
it is assumed that a and 1 are expressible in the following form:
c
c
(r) cos ne
c = f1
(28)
= f (r) sin ne
2
(29)
e
Report No. 1844
-9-
This assumes that during buckling the circumference of the cylinder is
subdivided into n waves; and the lowest value of the critical pressure
will be obtained for n = 2, as in the case of homogeneous isotropic
cylinders. The functions, f (r) and f (r) are now determined.
2
l
,
From equations (26), (27), (28), and (29) it is seen that d
rc
and T
rec
may be expressed as follows:
T
(30)
= f (r) cos ne
3
rc
(31)
f (r) sin n8
rec
L.
The use of the expression for T
rec
results in the following equation:
2 f4(r)
d f (r)
1
dr
-
given by equation (31) in equation (25)
0
The solution of the above equation is
n
A
f (r) = —
2 Y where A
r
4
n
is an arbitrary constant.
Substitution of the above expression into equation (31) yields
T
n
reC
A
= - sin ne
r
(32)
2
When the expressions for d
re and T ree given
by equations (30) and (32)
are substituted in equation (24), the result is
d f
3
(0
dr
f (r)
3
- r
A
n
r
The solution of the above equation is
n
nA B
f3(r) - 9n +—
r
r-
Report No. 1844
- 10-
and, from equation (30),
nA'
n
arc -
+
B'
n
cos ne
(33)
Referring to equation (26), the following equation may be written:
B'
6U 1 ' nA'
n
n
c
- —
+
r
6r
E r
cos ne
611cd
Since, from equation (28), TT.-
d f (r)
1
1
nA 'B'
n
n)
dr
Ec
r
f l( r)
cos ne,
dr
r
Integrating
nA
1
f1 (r) = — (- --n + B' log r +
• r
Ec
Therefore
=
nA
1
c E
n
+ Bn log r + C n ) cos ne
(34)
Rewriting equation (27),
6;.
c
car
1 611
c
c = rec
r
r G
re
Substituting for T rec and
6v
6r
c
1
r
Report No. 1844
c
ac their values ,Wen by equations (32) and (34),
n2A'
nB' log r C' .
1 A'
1
n
nn
+ –n) sin ne
—2- sin'ne + — ( - r2 +
r
r
E
G
r
c
re
-11-
Using equation (29) for ire,
d f (r)
2
dr
f (r)
1
_ (
2
G
r
n2 A'
.. __.) n
r0
E
c r
nB I log r
n
2E
c
r
1 C'
E
c r
The solution of the above equation is
1
f (r) = Hnr 2
(2G
n2
r0 2%
An'I
) r
E
c
c'
(1 + log r)- n
Therefore
n2
vc =
A'
G - 2E
ci rn
[- (21
re
CI
nB'
E
c
(1 + log 0 - En + Hnr sin ne (35)
The functional dependence on r of the displacement's a
and ir as expressed
c
c
in equations (34) and (35) having been determined, it is convenient at this
point to redefine the constants A , B , C , and H. Equation (34) may be
n
n
n
written as follows:
2
(36)
ac = (Ana + Bn+ C na log r/a) cos ne
Then equation (35) must be replaced by
ir
1 E
n
a2
= - nA a + 7-- —
c - 2) B r - nC a (1 + log r/a) + Inrl sin ne
n
n
c
n
(2n G
JJ
re
[
(37)
The displacement equations are thus expressed in terms of the simpler nondimensional arbitrary constants A, B , C , and H. Equations (32) and.
n
n
n
(33) for the core stresses are expressed in terms of the new constants as
follows:
a2
E
c
TrOc = - — Bn– sin ne
r
n
(38)
and
Report No. 1844
-12-
a re
E c( - B
a2a
—) cos nO
C nr
nr
(39)
—F
The validity of the above equations is easily verified by the substitution
(1)f equations (38) and (39) into the equilibrium equations, equations (24)
and (25), and further by the substitution of the expressions given by equations (36 - 39) into the stress-displacement relations, equations (26) and
(27). It is of interest to note that the manner in which the core stresses
and displacements vary with respect to r is not so simple as is sometimes
assumed in problems of this nature.
Equilibrium of Facings
Immediately before buckling, the stress situation in the facings is as
, the
r
=
a
= b
stresses shown in figure 2 can be determined from equations (18), (19), and
(20). As stated previously, the stress situation in the facings when the
cylinder is in a slightly deformed state is assumed to differ only slightly
from the situation which existed just before buckling. In figure 4, which
illustrates the differential elements of the deformed facings, the symbols
illustrated in figure 2. Since a'r
_
= (arc)
and
an a = (a )
r
rc
t
•
and a are used to indicate this small stress variat e) fi e) Tre) T
a
r
re) 10
tion which has taken place. The assumption of membrane facings eliminates
the necessity of considering bending and transverse shear in the individual
facings. In formulating the applicable equilibrium equations of the facings, account must be taken of the rotation and stretching of the facing
elements during buckling. This effect was found to be negligible in regard
to the core. Due to the difference in rotation between the longitudinal
elements M and N of the outer facing and , the difference in rotation between the longitudinal elements M and N of the inner facing, the initial
l
-'
W 1 6 u
de and (1 + –--7-) de
b 6e
b 60
a 60 a 6e
4
for the outer and inner facings, respectively. — As a result of the tan-
central angle, de, becomes (1 +
1 6v
-
l 62a,
gential strain in the outer and inner facings, the areasof their differential elements become (1 + ge) a:de dz and (1 + ge) b de dz, respectively.
a, V, G , v, z e , and g o represent small displacements and strains which
result when the cylinder goes from the undeformed to the deformed state.
With reference to figure 4 it is seen that two equilibrium equations can
be written for each facing. Considering the outer facing first, the summation of forces in the direction normal to the differential element of
the outer facing yields
4
Timoshenko. Theory of Elastic Stability, p. 431.
Report No. 1844
-13-
a r )(1
q (1 + E e ) a de dz - (qk +
- [
qa (1 - k) +
iie
(
+ E e ) a de dz
1 + .1-...i _ 2,
a ae a
4
de
dz = 0
a
If terms containing products of small quantities, i.e., products of barred
quantities, are neglected and the above equation is divided by de dz, the
result may be written as
= - (3
Since
„1 6-G1 62a + qa (1 - k)Z - qa (1 - m
k- a 66)
a 661 ,
CI + 1 6V
Je =
.
76, the above equation further reduces to
+ qa (1 - k)( 2 + 1
N-
a a
22)
2
(4o)
a6
The second equilibrium equation of the outer facing is obtained by summing
forces in the direction tangent to the element. Thus,
6Re
---de
661
dz -
re
(1 +
' 19 ) a
de
dz = 0
Again by neglecting the term containing the product of the barred quantities
and dividing through by de dz, the above equation is reduced to
e
(4 1)
= Trea
The two equilibrium equations which apply to the inner facing are obtained
in a manner exactly analogous to that used in obtaining equations (40) and
(41). These equations are:
-I
N e=
as°
-I
fa' + ---)
6211'\
a rb + qak
b b 60 2
(42)
- - Treb
(43)
In addition to the four equilibrium equations (40 - 43), the following two
equations, based on Hooke's law and the strain-displacement relations, may
be written:
Report No.
1844
-14-
fl e =
Et
1
N
Et
e
1 -
Et 2 ( 11
2 ` ei
1 - u
61)
(44)
a a 6(9
(04.1 61.)
q' NEt
2
2 ` ei
b 68
1_ u '‘ b
(45)
The following relations enable the right-hand sides of equations (40 - 45)
to be expressed in terms of the constantsnA n
B Cn, and H which appear
in the equations for the core displacements, equations- (.36) and (37):
r
re
and
= (a )
Cr = (arc)
r a
r = b
or, on the basis of equation (39),
r
= E (- B
C
a2
E (- Bn b + C
+ C ) cos ne and a
n
n
r
C
a
n
TO
cos ne
(46)
T
re
= (T
rec
)
and
T =
rec
re
r = a
r =.b
or, on the basis of equation (38),
T
re
Ec
=
B
n
sin ne
a . (a )
c
r = a
and
T
and.
H
-
re
Ec
a2
Bsin ne
n n b
= (11c ) r
(47)
b
or, on the basis of equation (36),
a = (A la + B a) cos ne and
n
_1
a
u = (A a +Ba— +Calog
n
n b
n
Ws.)
cos ne
(48)
= (17 )
and
r = a
v =
)
r = b
or, on the basis of equation (37),
1 Ec n
V = - n A a + (
a-nC a+H a sin ne
n
'2n G - &Bn
n
n
re
[
Report No. 1844
-15-
and
Vt .
1 E cn
n A a + (--- –) B a
n
n
2
2n G
-
- n C a (1 + log b/a) + H b
n
n
r0
[
sin
ne
(49)
The relationships given by equations (46 - 49) express the assumed continuity conditions on the stresses and displacements at the interfaces.
The further assumption that the core extends to the middle surfaces of the
facings is implied.
On the basis of the above-listed continuity conditions, equations (40 - 45)
can be transformed, respectively, to equations (50 - 55), shown below.
fl
e
E
=
c
a (Bn - C n ) - (n2 - 1) qa (1 - k)(An + Bn ) cos
(50)
ne
{
ile
Ec a
7– = —717Bn sin ne
ie
(51)
l - Cn)
= E- E c a (Bn i
- (n2 - 1) qak
f: (An
+ B
n
+ C
(52)
log b/a)j cos ne
n
2
Ec b m a
,n — sin ne
=
n
b2
di e
)0
1 - [1
An
(n21\
Et
9
(53)
2
n2
( Ec
I
2G re
a
Et
2
{-((n - 1) A – +
N =
1- 11 2
n b
e
2
+ 1) B
n
2
- n C
n
+ n H
n
cos
(54)
a2
n2
- — + 1) B —
n b2
2
2G re
- [n2 + (n2 - 1) log b/a] C
n
a/b + n H
n
cos ne
(55)
The integration of equations (51) and (53) yields
Ec a
B cos ne
n
n2
Fre
(56)
and
E
_
-
Report No. 1844
2
a
c b B cos ne
n 2
2
b
n
(57)
- 16-
ne
The integration constants are zero, since Re and
•
on e, as shown by equations (50) and (52).
are dependent entirely
Four independent equations containing the constant A n , Bn , e n , and Hn , as
well as the load q, can be obtained from equations (50), (52), (54), (55),
(56), and (57). If the right-hand sides of equations (50) and (56) are
equated, the result is
E
c
a (B
- C ) - (n2 - 1) qa (1 - k)(A + B ) n
n
n
n
E, a
n
2
B
n
The above equation may be written as follows:
- (n 2 - 1)(q/E c )(1 - k) An
- [(n 2 - 1)(0 c )(1 - k) - ( n
2
,
; . ) ] Bn
J
o
(58)
If the right-hand sides of equations (54) and (56) are equated, the result
is
Et[_ ,
kn - - I) A
2
1 - 11
E
2G
rO 2
2,
n
- n u
n
+ n H
n
a
c
n
n
/ Ec
n2
+ k---- — + 1) B
2
n
which reduces to
(n 2 _ I) A 4.
/ n
- n
2
C
E c a (1 - 11 2 )
n2
,
2
- 2" ' j" 2
re
n Et
[
Ec
+ n H
n
n
B
n
=0
(59)
If the right-hand side of equations (52) and (57) are equated, the result
is
Ec a (Bnb
Cn ) - (n2 - 1) qak .f.-;" (An + Bn + Cn log b/a)
Ec a
n2
B
a
n -
b
which reduces to
Report No. 1844
- 17-
(n2 - 1)(q/E )k A + (n2 - 1) (0 )k(n2
b
n
c
4
13
- 1)
n2
n
(6o)
Cn = 0
(n2 - 1)(q/E c )k log b/a -
Finally, if the right-hand sides of equations (55) and (57) are equated,
the result is
(n2
Et
1•
a
1) A
n b
( Ec
2G
r0
-
n2
2
a2
,
n
+ 1) B --
n
a
- [n2 + (n2 - 1) log b/s] C n + n Hn = -
E
c2
a
Bn
a
which reduces to
, a Ec a (1 n2
— + 1) +
• n2 Et
re2
Ec
- (n2 - l) 5113. An + [( 2G
- [n2 + (n2 - 1) log b/al
2.
)
a
b
Bn
(61)
C n + n Hn = 0
Since Hn appears only in equations (59) and (61), it is eliminated immedi
ately by subtracting equation (59) from equation (61). The result is
2
2
E a (1 - u2)
a
a
c
- — + 1)(-- - 1)
- (n2 - 1)( – - 1) A +
2
n Et
b2
2
2G
•1.0
( 12;- +
1)] Bn - [n2- 1) + (n2 - 1) -161' log b/a]
= 0
(62)
Equations (58), (6o), and (62) comprise a system of three equations conn
Bn , and C which occur in the expressions for
taining the constants A,
the displacement functions
ti
c and l c . A buckled form of equilibrium is
possible only if equations (58), (60), and (62) yield non-zero solutions
for these constants; this requires that the determinant of the coefficients
of An ,Bn , and C n be equal to zero. The equation used for the determination of the critical load is obtained from this determinant. Specifically,
Report No. 1844
-18-
(n2 - 1) q/E
c
(1 - k)
(n2 - 1) q/E
n
( n2 - 1)(cliEdk
c
(1 - k)
1
2
- 1
2
n
(n 2 - 1) (q/E )k
(n2.- 1) (q/E)k log b/a
c b
n
2 - 1
2
_
Ec
n 2a 2
( 2Gre
1)(132 - 1)
n2 - 1)(121_, - 1)
= 0
a
n
n2 (1-!..
E c a (1 - µ 2 ) a
( + 1)
n 2 Et
1)
(n2 1) – log b/a
The following quadratic equation is obtained from the expansion of this
determinant:
n (1 - b/a)2
E cn 1 2- b2/a2
2
(q2 /E c 2) k (1 - k
(b/a)2
2G re 4- 2 )( b 2/a 2 )
f
/3/ +
+ E c' a (1 - p 2 ) ( 1 + b/a) log /
c
n Et
b/a
2 1
-k (1 + b/a) + n 2
n
L
-k (1
1
, [( Ec
n
[b/a + k (1 - b/a)]
2
n - 1
2G
E c a (1 - µ 2 %)
2
n Et
1 + b/a)
b/a
11 - n2
(1 b ia)2
_
bla
(b/a)e
Iva) ] log bia
bia
2 + 1)(
(1 - b/a)2
b/a
1
1 - 102/a2‘
- 0
2
(63)
Analysis of Results
The results of this report are contained in equations (18), (19), and (20),
the equations for determining the stresses in a long sandwich cylinder under
Report No.
1844
-19-
uniform external loading; and in equation (63), the equation for determining the critical load on a long sandwich cylinder. These equations
are repeated below:
a
= q — k
(1 8)
N qa (1 - k)
(19)
a
rc
e
N
e
= qak
(q2/E 2) k
c
(1 - k)
n2
(1 - b/a)
2
2Gr
r9
,
E c a (1 - 2) 1 + b/a
( b/a
n 2 Et
- k (1 + b/a)
E
2
c
n2
b/4
2
1\
+ q/Ec
r
Ll
- k (1 - b/a)]
b
- 1
[b/a + k (1 - b/ad (--c-- 2
2G
re
a (1 - 4
n
2
(1 +
Et
n
b/a
2
/a
2
t
(b/a)2
[b/a
b/a
b/a
1 _ b2/a2
+ 1)(
(1 - b/a)2
1
2
(20)
2
(1 - b/a)2
n2
1
n
_
log
2
n 1 - b /a
E,
a
(b/)2
n2
2
- 0
b2 / a2
)
(63)
b/a
where, in each of the above equations,
k =
(
+ b/a) 1
Et log b/a
E
c a
and log b/a signifies the natural logarithm of b/a.
The application of equations (18), (19), and (20) to specific examples is
E a
very simple. If the ratios, c and b/a, are known, the value of k can
Et
be determined. Substitution of this value of k into equation (18) yields
an expression for the core transverse normal stress in terms of the uniform lateral load, q, and the ratio of the radial distance to the middle
surface of the outer facing, a, to the variable radial distance, r. The
Report No. 1844
-20-
maximum normal stress in the core always occurs at the inside surface of
the core, that is, at r = b. Substitution of the k value into equations
(19) and (20) yields expressions for the forces per unit length of the
facings, N o and N o , in terms of the load, q, and the radial distance to
the middle surface of the outer facing, a. Examination of the expression
for k shows that, in general, Ne and Ne are approximately equal since k
is approximately equal to one-half in the practical range of dimensions
and elastic constants. If the value of the modulus of • elasticity of the
1
core approaches infinity, k becomes equal to 1--4-.7
3a ; then Ne =,q
a
and N 0 = q 1 + b/a. Thus, the statement that N o and N o are equal if Ec
is infinite and b/a As 1 is approximately correct and becomes less accurate
as the thickness of the cylinder increases. If E c —) 0, k --10; and the
external load is supported entirely by the outer facing. The solution
given by equations (18), (19), and (20) may be compared to Reissnerts
solution of the problem of the closed circular sandwich ring acted upon
by a uniform radial load.2 If desired, equations similar to equations
(18), (19), and (20) can be easily obtained for the case of internal
pressure, or for a combination of internal and external pressure, by the
methods used in this report.
Equation (63), for the determination of the critical load on long sandwich cylinders, is of practical interest only for n = 2. A value of n = 1
represents, physically, a rigid-body translation of the cylinder, and
values of n >2 result in larger values of the critical load than those
obtained if n = 2. If, in equation (63), the integer 2 is substituted for
n and the symbol p is used to represent a uniform lateral pressure, that
is, -q, the result is
(p2 /E k (1 - k) 4
2G rO
(b/a)2
E
c
,
a (1 - 2 ) 1 + b/a
Et
1
2G
E re
)]log
b2/a2,
2 )( 1 - b2/a2
piEc
I[(1 - b/a)2
b/a
[1 - k (1 -
- k (1 + b/a)] +
1) (
Ec +
(1 - b/a) 2(
b/a)1
logbb/ a
/
a.
b2/a2) + E c a (1 - 4 2 )
2
2
b /a 4
Et
(b/a)
13 ba
[b/a
+ k (1 - b/a.
(1 - b/a)2
1 + b/a1•
)
b/a
4
=0
b/a
(64)
5
-National Advisory Committee on Aeronautics Technical Note No. 1832, p. 44.
Report No. 1844
-21-
In the practical range of the values of the, dimensions and the physical
constants of the sandwich cylinder, equation (64) yields two widely separated positive roots for p, the lower of which represents the critical
pressure p . Consequently, p may be obtained with very good accuracy
cr
cr
if the term containing p 2 in equation
(64)
is neglected.
It is of interest to examine the results given by equation (64) in certain limiting cases. First, it is noted that the critical pressure becomes equal to zero if either E c or Gre is set equal to zero. This is to
be expected in view of the assumption of membrane facings. It is of
greater interest to examine equation (64) with E c and G taken to be
infinite. In this case, equation (64) becomes
a
(1 -
cr
Et
11 2 )
_
3
(1 - bia)2
2, 2
1 + b2/
re
(65)
The value of p obtained from equation (65) should be comparable to the
cr
value obtained from the formula for the critical pressure on a long homogeneous cylinder if the moment of inertia used is equal to the moment of
inertia of the spaced facings of the sandwich cylinder. The formulafor
the critical pressure on a long homogeneous thin cylinder i02.
3 E I
Pcr - 3 (1
r o- 2)
(66)
If the mean radius, r o , is taken equal to.the mean radius of the sandwich
cylinder, a + b , and I is made equal to the moment of inertia of the spaced
2
t ( a - b)
sandwich cylinder facings,
equation (66) may be written as
2
follows:
p
a (1 -
cr
Et
11
2 )
3 (1 - b/a), 2
(67)
(1 + b/a)3
Comparison of equations (65) and (67) shows that they would be equal if
h2
(1 + b/a) ),,,,, (1 + b/a )3
(1 + =–)
(1 + b 2 /a 2 )
for thin
2 were equal to
•
4
a
b
cylinders, i.e., for T Pd 1. As the b/a ratio decreases from a value of 1,
6
–S. Timoshenko. Theory of Elasticity, p. 216.
Report No. 1844
-22-
equation (67) becomes less accurate since it is derived on the basis of
its being a thin cylinder with the pressure applied on the middle surface
and thus fails to represent althicker cylinder with the pressure applied
on the outer surface.
Equation (63) may also be applied to problems concerning the stability of
long sandwich panels in the form of a portion of a cylinder hinged along
the edges e = 0 and e = 2 a (fig. 5). If, in equation (63), o/a is substituted for n and -p is substituted for q, the smaller value of p given
by equation (63) represents the critical pressure on a panel whose dimensions and properties are known. This solution presupposes the unsymmetrical
type of buckling indicated in figure 5, and therefore' it' may not yield the
lowest critical pressure for panels of small curvature, that is, relatively
flat panels. A relatively flat sandwich panel may buckle symmetrically with
no inflection points between the supports, as in the case of homogeneous
panels.?
Conclusions
It is felt that the solutions presented in this report are accurate for
sandwich cylinders with materials and dimensions such as to render the
basic assumptions applicable. The core assumptions are probably sufficiently representative of all practical sandwich construction. The range
of applicability of the solutions could be increased somewhat if the
effect of the thicknesses of the facings on the overall stiffness of the
cylinder were taken into account in the stability analysis. This would
entail the use of shell theory rather than membrane theory in regard to
the facings. Another extension of the problem which is of greater practical importance would be to obtain the corresponding solutions for
cylinders of finite length. A supplementary report containing the solutions for cylinders of finite length is planned for the near future.
Numerical computations and curves based on the solutions contained in
this report have been omitted since they will appear as limiting cases in
the supplementary report.
7S. Timoshenko. Theory of Elastic Stability, p. 230.
Report No. 1844
-23-