Seide, Paul

English

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NATIONALADVISORY COMMITTEE
FOR AERONAUTICS
TECHNICAL NOTE 2760
DERIVATION OF STABILITY CRITERIONS FOR BOX BEAMS
WITH LONGITUDINALLY STIFFENED COVERS
CONNECTED BY POSTS
By Paul Seide
i
Langley Aeronautical Laboratory
Langley Field, Va.
Washington
August 1952
https://ntrs.nasa.gov/search.jsp?R=19930083535 2020-06-17T20:18:45+00:00Z
IN
.
NATIONAL ADVISORY CWTTEE FOR AERONAUTICS
TECHNICAL NOTE 2760
l
DERIVATION OF STABILITY CRITERIONS FOR BOX BEAMS
WITH LONGITUDINALLY STIFFENED COVERS
CONNECTED BY POSTS
By Paul Seide
SUMMARY
An investigation has been made of the elastic stability of an
idealized box beam with longitudinally stiffened covers connected by
posts and subjected to end moments and axial loads. Stability crite-
. rions which give the axial stiffness of the posts required to achieve
desired stress values in the box-beam covers are derived.
#
INTRODUCTION
In a previous investigation of the use of posts as internal
supporting elements in box beams (ref. 1), the results indicated that
supports in the form of posts had possibilities for decreasing struc-
tural weight. The compression and tension covers of the idealized box
beam considered in that investigation were unstiffened. Since unstiff-
ened sheet is not the most efficient compression resistant element, the
structural efficiency may be considerably improved when the sheet is
stiffened by longitudinal stiffeners which, in turn, are supported by
posts. An idealization of such a box beam is investigated in the -.
present paper.
SYMBOLS
am
. c
.
D
general coefficient in Fourier series for w
integer defining location
l~c~(M-1)
plate flexural stiffness,
of post in X-direction,
Et3
=7=7
2d
E
EI
F
g,h,i,j,m,1n)r>s)q>cz
k
L
1
M
N
Nx
Px
P
R
s
t
u
‘s
w
NACATIJ 2760
integer defining location of post-in Y-direction, .
l~d~(N-1)
Young’s modulus of elasticity of plate material
l
—
flexural stiffiess of longitudinal stiffeners
axial stiffness of posts, force per unit deflection
integers, also used a-ssubscripts
plate-load coefficient, 12Nx/fi2D
stiffener-load coefficient, ZPx/YC2D
longitudinal distance between posts
transverse distance between posts
number of bays
number of bays
plate load per
stiffener load
in X-direction
in Y-direction
unit width
integer defining number of buckles in Y-direction,
l<p<(N-1)
integer defining number of buckles in X-direction,
l~q~(M-1)
plate-flexural-stiffness ratio,
‘T/DCr
post=axial-stiffhess parameter, FZ2/#DC
plate thickness
potential energy of box beani
potential energy of longitudinal stiffeners
plate deflection in Z-direction
—
—
., —
.
NACA TN 27%0
X,Y,Z
.
x,y
~
$
7
P
Subscripts:
c
T
coordinate axes
distances along
aspect ratio of
posts, L/z
(see fig. 1)
coordinate axes
flexural-stiff’ness
Poisson’s ratio of
plate between adjacent lines and rows of
compression cover
tension cover
ratio, EI/ZD
plate material
.-
Those parameters of the present paper that,have the same symbols as
in reference 1 have been redefined in terms of the transverse distance
between rows of posts rather than the cover width because more natural
parameters are thereby obtained.
.——
.
DESCRIPTION OF STRUCTURE
t
The structure of the present paper (fig. 1) differs from that of
reference 1 in that longitudinal stiffeners of equal area and flexural
stiffness are attached to the tensioriand compression covers and coincide
with each row of posts. The stiffeners attached to the tension cover may
differ from those attached to the compression cover. Each compression-
cover stiffener is subjected to the same compressive load and each
—
tension-cover stiffener, to the same tensile load. The stiffeners are
considered to be concentrated at the middle surface of the cover to which
they are attached and to have no torsional stiffness. As in reference 1,
the covers consist of two simply supported flat rectangular plates of
equal length and width which are connected in the interior by identical
axially elastic posts in rectangular array. The plates may have unequal
thicknesses and material properties amd are subjected to uniform edge —
loadings, which may be ofdifferent intensities for the two plates. The
connection between the plates and the posts is one of simple support.
Both the longitudinal and transverse spacing of the posts are uniform;
the two spacings, however, may differ. (See fig. 1.)
The principal results of the investigation of this structure consist
of elastic stability criterions which give the post axial stiffness
.
required to achieve desired.stress values in the covers of a given box
beam at buckling.
.
k NACATN 2760
DERIVATION OF STABILITY CRITERIONS
The method of derivation of the stability criterions of the present
paper is similar to that of reference 1. Fourier series for the deflec-
tions of the compression and tension covers are substituted into the
potential-energy expression for the structure; the potential-energy
expression is then minimized with respect to the unknown coefficients
in the deflection functions. A series of linear homogeneous equations
is obtained from which stability criterions”forthe structure are deter-
mined. The algebraic manipulations that yield these criterions are
somewhat more complex than those of reference 1 and are given in detail.
In oi’derto obtain the potential energy of the present stz’ucture,
it--isnecessary only to add the potential ener~ of the stiffeners to
the potential energy of the structure considered in reference 1. The
potential energy of the compression-cover stiffeners is given by
and of the tension-cover stiffeners, by
r
{
N-1 (EI)T ~m
UsT=r ~
d=l ‘o
With WC and WT
reference 1, these
given by equaiions
equations become
Yx [Y?-m ‘2 2
-j(z
)
2
m2u(EI)c - pUSC . —- s~n @
% n=l% N (3)4MLd.1m.1 M2L2
(1) and (2), respectively, of
and
e—
—
—
.
.
a--
NACA TN 2760
The potential energy of the structure is then
and (4) of the present paper and equation (6)
given by the expression
r
5
the sum of equations (3)
of reference 1 and is
1
(5)
The buckling load is determined by the condition that the potential
energy be a minimum; that is,
au NJ o
—= —=
Z3aijC %jT
(i=l,2, . ..w)
(j=l,2, . ..rn)
.
.
or
6 NACA TN 2760
— .-
.-
.
(6)
and
[(i2+J2~~+i+2B2qaiJT+ .—
9
(7)
(i=l,2, . .. co)
(J=1,2, . ..W)
Equations (6) and (7) my be separated into sets containing only
particular coefficients ~. If j is of the form
J = CIN (c1 =1, 2, , , . m) .
.
NACATN 2760 7
&
that is, the covers buckle with longitudinal nodes through the stiffeners
Y
nfidsin
-.
‘fid is shown in reference 2and posts, the summation sin —
d=l N T
to be equal to zero. Then, particular equations for j = CIN of equa-
tions (6) and (7) become
—
[(i2 + cfMw)2 - 1i?M2P2~ (aiclN)c = 0 (8)
and
—
.
[
(
1
22 2)2+ i%2~2~ (a~~lN)T = 0i2+c1M~ (9)
(i=l,2, ...,00)
(c1 =1,2, . ..m )
The stability criterions for this case are
or
(lo)
(11)
or both, in which case ~ = -~.
If i is of the form
i=cp “
and
(C2 =1,2, . ..m )
(c1 =l,2,0. .m )
.8 NACATIIJ2760
that is, the covers buckle with transverse nodes through the posts, the
particular equations for i = c#4 of equations (6) and (7) reduce to
and
(%”u),+
These equations are similar in form to
the stability criterions for this case
B2ksc+
7~-—
C22
.
.-
—
(12)
.-
1
.
—
(13)
(C2 =1,2, ..*”W )
(J#cIN; C1= 1,2,...03)
equations (A4-)of reference 3 and
can be written immediately as
=0 (14)
(C2 =1, 2, . , . co) .
.
2N NACA TN 2760 9
.
and
.
-—
(C2 =1,2, ..0CU )
In the cases considered thus far - buckling of the structure with
longitudinal or transverse nodes through the posts - the posts were not
compressed. The remaining equations of equations (6) and (7) determine
the criterions for buckling of the box beam with compression of the
posts. ‘Theseequations can also be broken up into sets; each set of
. equations corresponds to a particular mode of buckling.
The values of i and j that have not been considered can be
4 expressed as —
i =2sM+q
or
i =2(s+l)M-q
(s=o, 1,2, . ..m)
(q=l,2, . ..1)l)
and
j=2rN+p
or
j=2(r+l)N-p —
(r=0,1,2,0. .~)
(P=1,2, . .. 1)1)
.
When these values of i and j are substituted into the remaining equa-
M-1
.
tions of equations (6) and (7) and the summations ~ sinm~ sin #
C=l
10
N-1
and ~ sin
d=l
01 reference
equations of
NACA TN 2760
nfidsin ~ are evaluated by means of the table on page 8
..
T
2, the equations separate into eight groups. The remaining
.
equations (6) separate into :_ —
—
.
[
(2s
1
+ p%~q = o(a2sM+q,2(h+l)N-p)C
-i- +
[
2(r + 1)
22
1}Ef-N (2s
(l&)
---
.
)C12 2)+E %P (a2sM+q,2(r+l)N-p)c
(2S+ir[’s+h-] m[‘@’ ~ (a2sM+q,2hN+P), -
1P2sM+q, 2(h+l)N-p)c - p%~q = o (16b)
NACA TN 2760 11
1(a2(s+l)M-q,2(h+l)N-p)C ‘P3sKl?q=o
({F(s+,)-i2+k(r+l)-12~f-E(’+1)-“
2
1)
[# ~p2 (a2(s+l)M-q,2(r+l)N-p)C - 2(s + 1, -
2
1{[ 1
2
}[
‘2(s +1)-; 7c-qcP’
~ (a2(S+l)M-q,’hN+P)c -;
=
1(a2(s+l)M-q,’(h+l)N-p)c
(16c)
+ ~3SKpq = O (l&l)
(s=0,1,2, .o. m)
(r=0,1,2, . ..m)
(p=l,2, . ..1)l)
(q=l,2, . ..1)l)
.
.
—
12 NACATN 2760 —.
where -—. ..— .
1“””~q = ~ ~. (a2@+q,2~+p)C - (a2gM+q,2(h+l)N-p)C- —“==
(a2(g+l)M-q,2hN+P)C+
(a@$f+@(h+l)N-P)T +
and the remaining equations
.
(a2(g+l)M-q,2(h+l)N-p)c- (a2&W+%2hN+P)T+
.—
(a2(g+l)M,2~+p
1
)T - (a2(g+l)M-q,2(h+l)N-P)T
of equations (7) separate into --
9
1[(2. + ;y[% + ;y,, +k’s,~2~ (’2sM+q,2klN+P)T-
1(a2sM+q,2(h+l)N-p)T -P3:~q=o (lya) ‘.-
.
.
NACATN 2760 13
1(a2sM+q,2(h+l)N-p)~ +pj:~q=o
-.
(1P)
({p(s+,)-f12+kr+;yp2~
[ 22
1)
2(s +1)-2 ) [(M %P ~ (a2(s+l)M-q,2rN+p ~ +2s+ l)-
q2
1 {[ 1
2
2(s + 1)
- ; 7~ + k@
}[
~ (a2(s+l)M-q,2hN+p)T -E
=
1
3s
(a2(s+l)M-q,2(h+l) N-p)T + p ~ ‘l?q = 0 (17C)
14 NACATN 2760
q21)~ %P2 (a2(s+l)M-q,2(r+l)N-p)T-~(s+1)-]2{~,s+ l)-
1 }[: ~2(s+l)M-q,2hN+p)T -q 27T + %TP2 ~K
(a’(s+l)M-q,2(h+l)N-pT ‘P2i KPq=0 .)]
(17d)
(S=o, l, ’,. ..m)
(r=0,1,2, . ..~)
(p=l,2, . ..1)l)
(q=l,2, . ..1)l)
where Kpq is as previously defined. Each group of equations (16)
and (17) corresponding to particular values of. p and q “is inde-
pendent of the others inasmuch as only the deflection-function coef-
ficients corresponding to these particular values ofip and q appear
in each group.
Consider equations (16a) and (16b) correspotiing to some particular
values of p and q. Divide each equation (16a) by the coefficient~f
(% )sM+q,2rN+p ~
and each equation (16b) by the coefficient of
( ) This process yieldsa2sM+q,2(r+l)N-p~“
.
—
-—
—
NACATN 2760
1(%M+q,2(h+l)N-&+ p3s
i~+:f+@+:yBj2- (%+:)2,%%’=0
and
~2sM+q,2(r+l)N-p)c -
—
(18a) _
—
3O@.
(%sM+q,2(h+l)N-p)c -
] *+#f+~(r+l)c~2p~
%’”0
.
- (2”+N%
(18b)
(s=o, 1,2, . .. &l)
(r=0,1,2, . ..m)
to a given value of s and
of all equations (18a)
Add all of the equations (18b) corresponding
subtract the resulting equation from the sum
corresponding to a given value of s. The first terms of equations (18a)
and (18b) then combine to yield
cm
E
r=o
which is the same
m
.x
h=O
[( )(a2sM+q,2rN+p ~ - 1a2sM+q,2(r+l)N-P)~
as
[
(a2sM+qj2hN+p 1)C-(a2sM+q,2(h+l)N-p)C
.—
--
The resulting equation can then be written as
“[‘+(’”+:)’ 2~)2s +,; 7c -
1
L.
Nmi
function
( )]a2sM+q,2(h+l)N-p ~
Idivide each equation
coefficients and add
I
I
a ,
I 11’
=() (19)
(s=o, 1,2, . ..~)
(19) by the coefficient of the first summation of deflection-
all of the resulting equations. Thus,
,
1 I
,
I
,, I 1
i-
CFl
, ,
m.
Zq( )(a2sM+q, am+p ~ - )]a%M+q, 2(h+l)N-p ~ +E=O h=O
Similarly,
,
(20)
equationa (16c) and (l&i) for particular values of p and q yield
~ ~ lj~(s+l)M,m+p)c- (a2(s+mL%2(h+l)N-.p)J -
(21)
equations (17a) and (1~) for particular values of p and q yield
[
~~ (a2sM+q,m+,)T-(a2M+%2(h+l)N-,, -)]
and equatimm (17c) and (17d) for particular
L
values of p
1
~
) -(a2(s+l)M-q,2(4+l)N-p)Tl+a2(s+l)M-q,2TtJ+p ~
z
S=l
-1
(22)
W q yield
P
02
.
I . .
Fina14, add equations (21) and (22) and subtract the resulting equation I&cm the sum of
!5
equations (20) and (23). The first terms of equations (20) to (23) combine to give ~q. The
~
stability criterion for buckling of the structure with q buckles in the X-direction (longi- H
tudinal) and, p buckles in the Y-direction (transverse) can then be obtained as z
8
Since repetition of the methcxl. of solutiou for the
and q would give the same stability criterion as
can take on the values
(24)
equations corresponding to other values of p
equation (24), p and q in this equation
p=l,2, . .. l-l
q = 1,2,...1-1
G
20 NAW TN 2760
—
Equations (10), (11), (14), (1>), and (24) constitute the complete k.
set of stability criterions. IF
7~ = 7T = kSC
‘%,=”
the stability criterions reduce to those of reference 1, after allowances
have been made for differences in the parameters used,
CONCLUDING REMARKS
Criterions for the elastic stability under end mom&t-and akial load
of an idealized box beam with longitudinally stiffened covers cotiected
by posts have beer-derived. These stability criterions give the axial
stiffness of the posts required to achieve desired stress values in the
box-beam covers.
The parameters used in the present paper are defined in terms of the
transverse spacing of the rows of posts rather than the cover width
because more natuml parameters are thereby obtained.
Langley Aeronautical Laboratory
National Advisory Conmittee for Aeronautics
1.
‘2,
3.
Langley Fieid, Vs., June 3, 1952
REFERENCES
Seide, Paul, and Barrett, Paul F.: The
.
.—
.
Stability of the Com~ression
Cover of Box Beams Sti?fened by Posts. ‘~ACA R~_p.1047, 19~1. —
(SupersedesNACA TN 2153,) .—
Budiansky, Bernard, Seide, Paul, and Weinberger, Robert A.: The
Buckling of a Column on Equally Spaced Reflectional and Rotational
Springs. NACA TN 1519, 1948.
Seide, Paul, and Stein,
Supported Plates With
Manuel: Compressive Buckling o&Simply
Longitudinal Stiffeners, NACA.TN 1825, 1949. .
.NACA TN 2760 21
z
‘k N ‘XC/ ‘c /
PXT
x
Figure l.- Idealization of
covers
.
a box beam with longitudinally stiffened
connected by posts.
-.
—
--
—
.
NACA-Langley -8-19.52-1000


Derivation of stability criterions for box beams with longitudinally stiffened covers connected by posts

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