Stowell, Elbridge Z

English

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NATIONAL ADVISORY COMMITTEE
TICS t(FOR AERONA1
TECHNICALNOTE
No.
CRZTICAL SHEAR STRESS OF AN lXl?RQITELY
LONG PLATE IN THE PLASTIC REGION
By Elbridge Z. Stowell
Langley Aeronautical Lalxmatory
Langley Field, Va.
W!!!Iii=
Washington
August 1948
-. . . .. . .. . ----
-.—. —-. .-. . . .. ... . . . .
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TECHNICAL m!rE Nol 1681
cRmIcAL sHEARsTKFssoFAN13wtmmm
LONG PLATE IN THE 21ASTIC REGION
By Elhridge Z. StowelJ.
The plasticifq reduction factor in the formula for the lmckling
stress of a long plate under uniform shear has %een computed from the
properties of the stress-train cmve for the materisl. Some 13mited
tests on the shesr hckling of 24-S-0alumin~w plates in the
plastic region tend to confirm the value of the reduction factor
predicted %y the theory. Restiaht against rotation along the edges
of the plate hsa little m no effect uqon the mlue of the reduction
factor accord3ng to theoretical calculation.
INTRODUCTION
The problem of computing the critical sheer t3tre9sfor an inf~tely
long plate with edges elastically restratied against rotation was solved
for the elastic stress rsnge in reference 1. me theom of refereme 1
is no longer applicable, however, in cases where buckling occurs beyond
the elastic range; in such cases, the theory ~t ~e m~ified to mow for
curvature of the stress+train curve. The problem consists h the compk
tation of the plastici@ reduction factor q in terns of the properties
of the stress+train curve for the material.
The general equations for the Wckling of a thin plate under combined
compression and shear in the plane of the plate were derived in reference 2.
Upon the assumption that the plate remdns in the plastic state during
luckling, the differential equation of equilibrium of the plate was derived,
together with the correspondbg expressions for the vmiatio~ in ener~ at
lnlckling. These general expressions were then applied to the case of a
plate in which one of the compressive forces and the shear force were zero,
that is, to a plate under simple compression.
The same general expressions msy also be applied to the case of a
plate under pure shesr in the plane of the plate hy setting both core
pressive forces equal to zero. This application has leen made in the
present paper.
.-—.— .—..——— -——.—.— -— ——— —
..—— —. .—— ——...— _
213EsuL’Is
l’heoretdcalreml ts.– The plate is ~s~a to be infinitely long
with side edges elaaticaUy restrained against rotation. The applied
shear stress is assumed to be uniformly distributed and, as in
reference 2, no unloading of the plate is assumed to occur during
lnlckling. By virtue of these assumptions, the plate remains in the
plastic state during Wckling and the relations developed in reference 2
apply.
The procedure followed Is to employ the same deflection surface in
the energy expression for the plastic caae that was used in reference 1
for the elastic case. From the resulting expressions the approxhate
vslne of the critical sheer stress in the plastic region is computed.
The expression for the critical shear stress in such a case is
I’&?&
T= Ii
12(1 –l12)b’2
inwhich
T critical shear stress
k8 coefficient depend3qupon conditions of edge restraint
and shape of plate
t thickness of plate
b width of plate
E YO~tS moamus
v Yoissonts ratio
n coefficient which
plastic range
allows for reduction in shear stress in
The value of the wave length required to make the critical.shear stress
a minimum is found snd is sulmtituted in the expression for the critical
shear stress, which is then known in terms of the angle of the waves, the
_*@ of the e~stic res~~t, md the plasticitivcoefficients. .
Division of this expression for
spending apprmimate expression
the coefficient q to a closer
vidual critical stresses.
the critical shear stress by the corre-
in the elastic range gives the value of
apprm3mation than either of the indi- k
The coefficient q is given formally in the appendix m a function
of the angle of the waves ~, the magnitude of the ela8tic restraint e,
_______ .
and the plaetioity coefficients E~/E ad C3. 5a ree.ultbg quation L9
where
EB secant modulus of material
fl(6), f2(G) fmotion8 of elastio mstratit 6
C3 plastlci~ coefficient
‘#o wave angle in elastic -
The angle of the wa~es @ is that angle which wll.1 make the critical shear stress a
minimum. As soon as the magnitude of the elastio restraht is aeleotad, the mgle of
the waves, end therefore q, w be cmput’ed for any stress.
Computations of o were made for a plate of 2494 aluminum alloy in the oases where
the edges of the plate were simply supper-kl and c-a. The procedures umd are given
h detail in the appendLr. The cdmd.atione with both edge oondltlone resulted in
sulmtantiaJJy the mm curve, shuwn in figure 1. Curvesof q= E# and q= E&as
obtdned for ocmqrressim in reference 2 are IILcluded for ocmperison.
..— ..— _
NA.C.ATN No. 1681
Experimental check.— The critical shear stress of ei@t long
plates of 24S-0 aluminum alloy has been measured in the plastic
region by Gerard (reference 3) . An attempt was made to secure a
clamped cond.ition along the edges and buckling was detemined by the
difference in strain on opposite sides of the sheet as mea8Ured with
electrical gages. Gerard.*B data are represented by the test points
in figlre 2. The solid-line cwve of figure 2 gives the theoretical
values of q as c~uted by equation (1) fram the axial stres~
strain curve supplied by Gerard for 24S-0 aluminum a3My In reference 3.
This theoretical cmve confirms the trend of the expertient.alpoints in
a satisfactorymanner, although it lies in the upper part of the scatter
bend.
fi reference 3 Gerard suggested that the shear secant tiulus be
used as the effective modulus for a long plate under shear. Gererd.3s
assumption is, from equation (2) of reference 3,
where the G*s refer to the slope of a shear stress-strain curve
derived from the Well stress~train curve. The dasl+line curve in
figure 2, which was co~uted. frolnfigure 8 of reference s,.shows the
Vd.ue Of q that results from this suggestion. The agre~nt with
the solid–line curve, ~ch represents the values of ~ compute~ fram
the theory of the present paper, is seen to be fair.
The necessazy conditions for use of the shear secant modulus,
however, should %e exmined further. In the appenti, equation (1)
is shown to reduce to the form
q = Cknlstant E*
if C3 isconstantl For Gerardts curve for 24S-0 aluminum alloy the
requirement for a constant Cq is closely fulfilled. When the v~ue
of q is computed.from equation (1),
~ = 0.89
This relation is plottmd as the solid
it 1s found that
E8
r
line in fi~e 2. ReDults of
.
thecalculation are also given in a more familier form in figure 3.
Caution should be employed h generalizing the use of the shear
secant modulus to materials, such as 24S41!aluminum slloy, for which
C3 is not a constant in the plastic range. The upper curve in
I’w.ATN No. 1681
figure 1 represents E~/% for 24S-T aluminum alloy and therefore
describes the axial stress+train curve for that material. The center
curve, which represents q, cannot be obtained from the upper curve by
multiplya the ordinates hy a constant factor throughout the plastic
range; therefore ~ for this material is not the shear secant tiulus.
CONCLUSIONS
The theo~ of plastic bucklhg previously reported has been applied
to the determination of the plasticim reduction factor in the formula
for the lnzcklingstiess of a long plate under uniform shear from the
properties of the stres~train curve for the material. Some limited
data on the sheer buckling of 24S-0 alW~W plates ti the plastic
region tend to confirm the value of the reduction factor predicted by
the theo~. The theory indicates tliatrestraint againat rotation along
the edges of the plate has little or no effect on the value of the
reduction factor.
Langley Aeronautical Lalmratory
National AdvisoH Committee for Aeronautics
Wey Fiad, Va., My 24, 1948
-—— .— ..- —.--— ----—
—.. -— ——-
.- — .-
6
JiETlmDrx
ANAIXSIS
General theory.– Reference 2 shows that for a plate under shear
alone the work T done on the plate, expressed in a Cartesian (xt, y!)
system, is
(2)
,
where
T applied shear stress
w deflection of plate at (xY, y!)
1. half-zave length
The expression for the strain energy in the ylate VI, frcxn
reference 2, is
(3)
where
L Lx%C3’2+2E6
.
Et tmgent modulus of materisl
E, in addition, the plate has equsl elastic restrednts c against
1.
rotation along the side edges, the strain energy in these restraints V2
is, from reference 2,
—
——-—..__— ___ _______
NACA TN No. 1681
I& a more suita%le oblique coordinate system he deftied by
X=x f+yttmq
-1 1
-%’Y =~oB I
as in reference 1. (See fig. 4.)
Then T, VI, and V2 inthenew system are
-2p-c3)B
]( )
@ 52
[
a% a%
+ l+2tany-(l– Q] ~ p
(4)
(5)
(6)
(7)
.. —._ .—. ——.
._— _ — ———. . ———. -—. ——
8where
A comparison with the corresponding formulas in appendix B of reference
will show that the two grouys are identical if Poisson’s ratio is taken
1
aso.5Emaifc3=l; that is, if only the elastic range is considered.
As in reference 1, the assumed deflection-surface is
in which
stress a
the half’ve length k willhe adjusted to make the critical
minimum. Substituting this deflection surface into
expressio~ (6), (7), and (8) and perfoming the indicated integrations
gives
~_w21#b@h&
o 2A
V1 = W02
~~, {Y-12(v41)~kCos q T –2(1–c09%’
‘3) “w]’ ‘m
}
1 + 2 Binzp - (1- C3)cos~ =
+2 (9)
cos2#
#D’eAV2= W02
2b13 cos3@
.
u
.
-—— --.—— .-—
2A
NACA‘I!lVNo. 1681
.
where
‘=(*-*)’’+(:-$)’+*
N=(--++&-&+i
Setting w the equali~ T = VI + v~, which holds at buckling, and
solving for the critical shear stress T gives
T 1
[
r) [
42
1
+– ‘(1 - c~) Sin$d + L
‘sin’@ h CoS’@
r’ ‘::’7
M+$
or, replacing bl cos ~ ly b, and writing ~ = fl(~)
and ;. f2(c) yields
. .
(lo)
—.
.--. .—— -.—- — ————— ---- --. — -——
. —.
10 NACA ~ NO. 1681
h the elastic renge t3’9 this expression reduces to the ccwrespondingformula in reference 1.
The helf+ave length A and.thewave angle ~ will adilust Wemselves
to make r a minimum; that is,
&_.—_
(?ak2
which gives
fromwhich
[+21+2sin~-(l-
.1
Cj) Cosw] f*(6) ~
The angle ~ is found from the relation
which gives
(U)
(m
(13)
.
———. —.. _. -— ——. .
II
I
The wlue of ~ is thus fbd as mon as the restmaint Is chosen end the ‘value of
assumed. This value of ~ is the one to be used in equation (12) for computation
minimum critioal sheer dress Tti.
I
\ In the ektlc range, the corresponding mimlmum criticeJ sheer stress (7O)ti
Et/EB iS
of the
iB Obt.%bd.
~
,
by putting C3 = 1 end D’ = D = ‘~ In equation (12):
(14)
where the subscript O sigmlf lea that the quantiti is In the elastic range. Thus, fram
equation (13),
%(6)
Cos #o =
~~ + 2f,(6)
(15)
The value of q Is the ratio of the orftical 13hear slmem T* to the oritioal shear
0tres5 (To)- that would be obtalnad if the materiel were wholly elastic; thus
The velue of q ae a funotlon of atreas’wil.l now be cmmidered for the two extreme
oases of simply supported ami olemped. edges:
(a) For the case of simply supported edges,
Cos +0 = $
sin 290 = 0.943
%sin o = 0.333
3– C3
COB 2$$= o
*+3+C3
This yal.ueof v is ylotted against stress in figure 1 for 24SJT aluminum
Woy .
(b) For the case of clamped edges,
6 ==
fl(d = 5.14
f2(e) = 1.24
cos@o = 0.26
sin 290 = 0.966
sin~() =
Cos 2$$=
0.37
1.24(3 – C3)
*+ 1“24(3 + C3)
5?
—.. —
NACA m 190. 1681 13
This value of q is nearly identical nmnerically tith the value from
cese (a) and.is ‘representedby the same curve in figure 1.
Conditions under which GerardYH shear secant modulus applies .-
Equation (1) gives the general formula for q, which involves,
besides Es@, the following quantities:
C3
Functions only of restraint
coefficient C
Angle of waves on assumption of
elastici~ (by equation (15))
depends only on e
Angle of waves in plastic range
(by equation (13) ) deyends on e
and plastici~ coefficient C3
I’lastici@ coefficient
The magnitude of the restraint coefficient 6 is considered to be
pert of the data furnished tith the problem. Thus, fl(e) and fz(c)
are lmown constants; the angle #o is therefore also a known constant.
If, in addition to these quantities entering into equation (l), the
plastici~ coefficient C3 should also he a constant, equation (1)
would reduce to the form
Gerardts assumption is, frmn equation (2) of reference 3,
~=~
——---- —--—————- — -——- -——-—- —-———
—. —.—
——
14 NACA ~ NO. 1681
where we Gts refer to the slope of a shear stress~train curve derived
from the exiel stress-strain curve.
~ + * ~J and therefcmeThe plastici~ coefficient C3 is aefllledas
a
the condition for constancy of C3 is tha-t
Et = XE~
or
.,
where X is sane constant. The relation al . Keix, where K is a
constant, satisfies this reqtiement. For Gerardts curve for 24S-0 alundmm
elloy, X is apprmimately 0.28 over the plastic renge, and the requirement
for a constant C3 is closely fulfilled. WEn the value of q is computed “
fran equation (1) it is fouad that q = 0.89 ~. This relation is plotted
as the solid We h figures 2 snd 3.
.
.—. .—— .—— -. .- ,-
NACA TN No. 1681
lumERENcEs
15
1. Stowell, Elbridge Z.: Critical Shear S~ess for an ~initely Long
Flat Plate with Equal Elastic Restraints agatist Rotation along the
Parallel Edges. NACA KKR No. 3K12, 1943.
2. Stowell, Elbridge Z.: A Unified Theory of Plastic Bucklimg of
ColumnE and Plates. NACA TN I?o. 1556, 19M.
3.Gerard, George: Critical Shear Stress of Plates above the I&oportional
Limit. Jour. Appl. Mech., vol. 15, no. 1, Wch 1948, pp. 7–12.
.
#
..
. — ..—. ... —_ ———.
_ —— — .—-— —
————
16
.
NAC.A m No. 1681
1.0
.8
.6
?
.4
.2
\
Es
T
I
All edge conditions
\
0 10 20 30 40 50
Figure L- Values of ~ for a long plate of
24 S-T aluminum alloy under u“nif rm
1shear stress. (The curves for Es E and
A
Et/E for compressive stress u are c
included for comparison.)
——. . —
IA
.
17
Lo
.8
.6
T
.4
.2
‘\
Suggested by \.
Gerard *
(fig.8 of \I
reference 3 ) \\
Oi.
@ \\
~t
Computed from \ \
equation (1)
“\
o Gerard (reference 3 )
T
o 0 2 4 6 8
T, ksi
Figure 2.- Values of ~ for a long plate of
24 S-0 aluminum alloy under uniform
shear stress.
10
. ——. —. ——. -— ——
II
I
I
10
8
Skar
stress
6
k:
4
~.
----
-.
.-”
..-
<
.“
./”.
.
0
C@/
//
0
Derived shear stress-strain curve
~o~~r$r~ shear buckling curve
I
“o
Figure
,
2 4 6 8 10 12 14xlo-a
Shear strain, y
3,- Comparison of derived shear stress-strain curve
and theoretical buckling curve with test data of
Gerard.
NACA ~ NO. 1681
/
x
/
(’
/
/
0
/
I
0 I
I
1- Y’
Figure 4.- Oblique coordinate system.
NACA - I=@v ~e~. v~
—..-
..— ——--- — —————————-—— -—-—’—
——.—.


Critical shear stress of an infinitely long plate in the plastic region

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