Pride, Richard A

Stowell, Elbridge Z

English

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i1
I
SECTION dOPY
TECHNICAL NOTE 1971
PLASTIC BUCKLING OF EXTRUDED COMPOSITE
SECTIONS IN COMPRESSION
By Elbridge Z. Stowell and Richard A. Pride
Langley Aeronautical Laboratory
Langley Air Force Base, Va.
NATIONAL ADVISORY COMMITTEE
FOR AERONAUTICS
.
1
, <
Washington
October 1949
9
https://ntrs.nasa.gov/search.jsp?R=19930082638 2020-06-17T22:14:09+00:00Z
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NATIONAL KDVIEXIRYCOMMITTEE FCR AERONAUTICS
TECHNICAL NOTE 1971
PIASTIC BUCKLING OF EXTRmED coMPoszr!13
,
SECTIONS IN COMPRESSION
.ByEllmidge Z. StowelL and Richard A. Eride
The theory of plastic buckling, cmiginslly derived for individual
plates, has been extended to apply to cumbtiatlmns of plates such as H-
end Z-sections. The theory has been applied specifically to extruded
H-sections of 75s-T6 aluminumell.oy;the calculated buckling stresses
agree satisfactorilywith the stresses obtained experimentally.
INTRODUCI’ION
The local buckling of aircraft section columns (H-, Z-, and chennel-
sections) has been studied extensively at the Nationel Advisory Committee
for Aeronautics. Experimental data representing the results of several
years of work in this field are summarized in reference 1; the collec-
tion of the facts about locel buckling in this reference may be considered
fairly ccmplete. In the field of theoretical studies, the situation is
not so fortunate. The calculation of the locsl buckling stress in the
elastic range has been described in reference 2; but so fer no rational
method has appeared in the literature for the corresponding calculation
in the plastic range. With the advent of the theory of plastic bucKLing s
described in reference 3, end the subsequent experimental confirmation
of this theory (reference 4), the basis has been laid for a method of
computation of local buckling stresses in the plastic range. The purpose
of the present work is to present such a xethod end to show that the
results obtained will predi,ctsatisfactorily the buckling stresses of
flanged section COl~S in the plastic range.
,
“
SYMBOLS
b
t
width, inches
thickness, inches
2D
NACA TN 1971
Young’s modulus of elasticity, psi
secent modulus, psi
tangent modulus,
half wave length
helf wave length
of elasticity
psi
of buckles
of buckles computed on basis
()Et3plate stiffness in bending ~
elastic buckling strain
buckling stress, psi
number of outstanding flanges
plasticity coefficient
($+fz)
ratio of actuel buckling stress to buckling stress
()
‘pl
computed on basis of elasticity —
’01
P Poif3son’sratio
k nondimensional coefficient deyendent upon plate
proportions end edge conditions
Subscripts:
w web
F - llen~e
pl
01
cr
in the plastic range
in the elastic renge
critical
.
—
.
.
.
—
—
—
.
.
..
NACA TN 1971 3
METHOD OF COMPUTATION
The theory of plastic buckl.lngdescribed in reference 3 applies
only to individual plates. A ccmposite aircraft section, such as an
H- or Z-section, on the other bend, is made up of en assembly of plates
Joined together, sndthe theory of reference 3 must be suitably modified
to take into account the interaction among the plates.
For the present study, a hypothetical section which consists of a
web plate with an unspecified num%er of flSnges Joined to it is considered.
Evidently three conditions must be met by the theory for buckling of this
composite section, whether in the elastic or in the plastic range:
(1) The critical stress of the section must be the seinewhether
computed on the basis of a web plate restrained by flanges or on the
basis of flange plates restrained by the web
(2) The half wave length of buckle must be the same in both web
end flsnge plates (condition of continuity at joint)
(3) The sum of the stiffnesses of the ind3vidueL pla%es must
vanish at the buckling stress
If the materiel were elastic, these three conditions would be
satisfied simultaneously by use of the manent-distribution method of
reference 2. This fictitious solution is assumed to be knuwn sndmay
therefore be used as auxiliery information in the solution of the
plastic problem.
h order to solve the problem of the buckling of a section such as
the H- or Z-section in the plastic rsnge, the first condition is met by
equating the expressions in reference 3 for the plastic %uckling stress
of a supported plate and of a flenge. The wave length of the buckles is
retained in these expressions and is left indeterminate. Both sides of
the equation may be divided by the corresponding expressions which would
be obtained in the elastic region without violating the equality. The
second con&Ltion is now met by’requiring the actti half wave length on
both sides of the equation to be the seine;the half wave length may then
be solved for in terms of the Mmension ratios, the restraints, the
plasticity coefficient, end the fictitious wave length which the structure
would assmne if the material were elastic. The third condition
establishes a certain necesssry relation between the restraints on the
web and on the flenges. With the half wave length known, the crlticel
stress mey then be ccmputed by a rapid trisl-and-error process.
4 NACA TN 1971 —-
RXSULTS AND DISCUSSION
The generel relations frm which the critical
any composite section -comprisinga web with flange
given in the appendix.
Table I shows certain typicel lasic relatims
c.crmpressfve stress
may be found ere
independent of
of
—
materiel, which hold for E-sections of specified thickn&s and width
ratios. The thickness ratios used were frcm O.~ to 2.0. The width_.
ratios used were from 0.4 to 1.0. Expressions for .(AlbJ2 are
presented for use in calculation of the plasticity factor n. This
factor is en over-all @asticity coefficient defined as”the”ratiloof the
true critical stress to the critical stress calculated on the basis of
perfect-elasticity. Since the expressions for q and (/)?dlF2 contain
the peremeter Cl, which is a plasticity coefficient obtained flromthe
stress-strain curve for the material, a trial-and-error method must be
used to obtain the critical stress frm these relations.
.
In order to illustrate the method, calculations have been made for
extruded H-sections of 75s-T6eluminum alJoy of vsrious dimension ratios.
The calculationsmade by the present method are presented in figure l(a).
The H-sections selected fcm e celcflations were ass~ea t-obe of’
(T
tF
constent-thickness — =1.0
%(
but of different width ratios
(
b
)
— =o.5endo.’8 .
%
Stresses are plotted against the critical elastic
buckling strain. The stress-strain curve used to ccmpute the buckling
curves are showm on the seinescale for comparison. The experimental
points were taken f!rcxnreference 1 end have been ”slightlyadJusted to a
single mean stress-strain curve agreeing with that shgwn in the figure. .
The width ratios of the actual specimens differed but in general lay .-
between the vslues assigned t-othe calculated curves which plot within
a band showd by-the hatched area alout 1/2 ksi in width. Most of the
experimental points ere.seen to lie within 1 ksi of the bend; at–the
extreme end of the curve, however, the experimental points lie 2 to & ksl
below the band.
In order to show the effect ofldimensi”onratios better, the upper
segment of figure l(a), with additiond. ccmputed “curves,is plotted to
a larger scale in figure l(b). The hatched region is the same as that
()
%
.—
shown in the previous figure. For constant thickiies8- t;=l , a range
—
.
b,
NACA TN 19’71 5
of tidth ratios &cm O.k to 1.0 is found to create a band only 0.6 ksi in
width. Decreasing the value of tW/tF raises the critical stresses,
since the strength of the flange plate is thus increased at the expense
of the web plate. Conversely, increasing the value of tW/tF lowers the
stresses.
CONCLUDING REMARBS
A method has been presented for determining the buckling stresses
in the plastic range for flanged-section coluqms. Calculations by this
method fa extruded H-sections sre compared with experimental results
end the agreement is found to be satisfactory.
Langley Aeronautical Laboratory
Nationsl Advismy Committee for Aeronautics
Langley Air Force Base, Vs., August 25, 1949
.
.
6 NACA TN 1971
DETAILS OF ANALYSIS
A composite section formed of a web plate end q outstanding
flanges (n = 2 for Z- or channel-sections,n = 4 for H-sections) is
considered.. The calculation of the critical stress in either the
elastic or the plastic region must satisfy the following conditions:
(1) The critical stress must be the seinewhether ccqnzt-edon the
basis of the wel or a flange plate
(2) The half wave length of the buc~es must be the seineIn all”
plates because of continuity atthb Joints
(3) The stability criterion must be satisfied
Formulas me given in reference 3 for the critical atrese
()‘N w
of a web plate
These formulas
flange,
and () of a flange plate ina@F
are, if subscripts W and F sre
the plastic region. _____ ._
used to denote web and
where
t
E@
D=—
.9
-.
—
width of plate
.-
.
thickness of plate
.-
-.
l
.
*

&-
.
-
. ..-
—.
———
—
--w
-<
.=
—
.
BNACA TN 1971
E
L
()fl Ew =
f2~w) =
f3f F) ‘
()
fk EF =
4
cl=;+
elastic restrtint
hslf wave length of buckles in the plastic region
0.0237cW2 + 0.297CW + O.w
o.oo46kw2 + 0.09476W+ 0.500
o.o~k W2 + o.189k W + 1.000
0.0046u; + 0.0947EW+ 0.500
0.oon&F2 + o.0506GF
0.0530er2+ 0.58>F + 1.647
0.OlgaFz + o.1742eF+ 0.500
0.0530eF2+ 0.X%F + 1.647
(Thefunctions of ~ ere close approximations, since they were found
from an energy method.)
If the materisl were elastic
Es =Et=E
Cl=l
8and
NACA TN 1971
.
L
(w= ;
: + 0.1184GF+
J —
where A Is the half wave length of the
I&cm the first condition to l).
()%1 w =
O.0107e,2 \b2tJF
buckles that would
satisfied
.
and
IYbm these relaticms it folluws that
.
(3)
(4)
%e obtained.
P
.
(5)
When the equality is wesented in this nondimensioti form, the
effect of taking Poisscmls ratio equal to 1/2 is minimized. In the
notation of reference 3, either side of equation (5) maybe recognized
as the over-all.plasticity coefficient q.
—
.
NACA TN 1971 9
Substitution of equations (1) to (k) in equation (5) yields
From the second condition, which requires continuity at the joints,
‘w=’”=’
and
so that equation (6) my be written
..
ga ‘.(:)2+ (Jf3(’”) + ‘m”)
=—
E (q+ ($f3(%) + ‘4(’”)
10
Solution
where
[
A=l+
[
B=l+
.- NACA TN 1971 ““
.
of equation (8) for h yields a relation of the form
.
(9)
—
#
.
The third condition expresses the stability criterian, in the notation
of reference 2:
or, in the restraint notatim of this paper
—
(lo)
2
(11)
NACA TN 1971
The restraint
on the flange
n
e ~ on the web is thus knuwn in terms of the restraint ~F
snd vice versa; the restraints, as well as A, ere found
&cm the soiution obtained on the basis of a-perfectly elastic material.
The sole remaining unknown quentity on the right-hand side of
equation (9) Is the plasticity coefficient Cl, which is a function of
stress. Relations of the type shown in table I, which hold for the
dimsnsion ratios shown, inde~endently of the materisl, are therefore
obtained. ,
Upon the assumption of a critical stress”as a first approximation,
h is determined ficm equation (9); either of the expressions (8) for
q, whichever is more convenient, mey now he computed snd the criticsl
stress found, to see if it agrees with the assumed first approximation.
If not, the trisl-snd-error procedure is followed in the usual wey.
Convergence is especially rapid if the stress given by the stress-strain
curve itself at the appro~iate strain is taken as the first apwoximation.
X2 —- NACA ‘TN1971- -
l
REFERENCES
..
.
1. Heimerl, George J.: Determination of”Plate C!amp?essiveStrengths.
NICATN 1480, 1947. ,
2. LTE@uist, Eugene ~, Stowkll, Ellri~e. -2.”,ant Schuette~ Evan H.: —
Principles of Moment Distribution Applied to Stability of Structures
Ccnn.osedof Bars or PIates.- NACA Rep. 809, lgh~.
30 Stowell, Elbridge Z.: A Unified @oqy of P1.asti&Buckling of
.—
Columns qnd Plates. NACA TN 1556, 1948.
.
4. Wide, RicharLA., and Heimerl, George J.: Plastic Buckling of Simply
Supported Compressed Plates. NACA TN 1817, 1949.
D
,
.-
.
.
I.-
14 . ... . NACA TN 1971
85
I
80
r
70
Stress,
ksi > ,
65
0
60
55
Stress-strain curve,
.
0
D
bw
o Experiment
I I I
.005 .007 .009 .Olt .013 .015
Caicuiated elastic buckiiig strain, eCr= kwm2tw2
i2(l-~2)bw2
(a) ~=l.o; ~=0.5 and 0.8.
Figure i.– Correlationof buckling curves and experimental resuits
for extruded H-section coiumns of 75S-T aiuminum alloy of
constant thickness Wlch fail by local instability.
.
,
15
82
81
80
Stress,
ksi
79
78
77
0.5
I.0
I.0
‘1.
1.8
I.0
.
2.0 .6
‘ fill .012 .013
kwm2tw2
Calculated elastic buckling strain, cCr=
12(1-~2)bw2
(b) Enlargement of part of figure I(a) for purpose of
showing effect of
Figure
additional dimension ratios.
l.- Concluded.
NkCA-Langley - 10-10-49-. a~s


Plastic buckling of extruded composite sections in compression

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