The elasto-plastic stability of plates


Material Information

The elasto-plastic stability of plates
Series Title:
Physical Description:
30 p. : ill ; 27 cm.
Ilʹi︠u︡shin, A. A ( Alekseĭ Antonovich ), 1911-
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


Subjects / Keywords:
Aerodynamics   ( lcsh )
Elastic analysis (Engineering)   ( lcsh )
Axial loads   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


The stress-strain relations developed by the author in previous work for use beyond the elastic range have been simplified and applied to thin plates. Different expressions are given for these relations in the elastic, the elasto-plastic, and the plastic zones which may arise in a thin plate. The stability of plates compressed beyond the elastic range is studied and examples are given of exact and approximate solutions.
Includes bibliographic references (p. 27).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by A.A. Ilyushin.
General Note:
"Report date December 1947."
General Note:
"Translation of "Uprugo-plasticheskaya ustoichivost plasteen" From Prikladnaya Matematika i Mekhanika X, 1946."

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University of Florida
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By A. A. Ilyushin

In this article are developed the results of my work (reference 1)
"The Stability of Plates and Shells beyond the Elastic Limit." A
significant improvement is found in the derivation of the relations
between the stress factors and the strains resulting from the
instability of plates and shells. In a strict analysis the problem
reduces to the solution of two simultaneous nonlinear partial differ-
ential equations of the fourth order in the deflection and stress
function, and in the approximate analysis to a single linear equa-
tion of the Bryan type. Solutions are given for the special cases
of a rectangular plate buckling into a cylindrical form, and of an
arbitrarily shaped plate under uniform compression. These solutions
indicate that the accuracy obtained by the approximate method is



On a moveable Darboux triheirci, relative to which we shall
study the element of the shell, we choose the xy plane to be
tangent to the middle surface, and the x and y directions
along orthogonal curves (fig. 1).

The state of stress of the element is determined by the
tensor of the stress S. Its components Z, Z,,, y, are
smr.ll compared to X, Yy, and X that is, each layer of the
shell element parallel to the middle surface is in a state of
plane stress. The intensity of stress in this layer rill be

""Uprugo-plasticheskaya Ustoichivost Plasteen." Prikladnaya
Mat'ematika i 1Mekhanika X, 196, pp 623-633.

NACA TM No 1188

Sy xy + 2 (1.1)

The state of strain of the element is determined by the
components of the tensor of the strains exx, eyy. and exy'
since the shears exz, eyz are small, but the relation to the
strain ezz may be found from the condition of constant volume
of the element,

exx + eyy + ezz = 0 (1.2)

The intensity of strain in this layer of the material is given
by the formula

ei = -- e2 + ey2 + e ey + exy2 (1.3)

In agreement with the laws for the elasticity and plasticity
of materials the stresses and strains are connected by the relations

1 (.i 1 Ci i
S X Yy = ex Sy 2 y 2 ei yy Xy 3e y

Here ai = ai(ei) is determined for each material as a function
of ei. The properties of this function are as follows. Within
the elastic limit, that is, for ai a' where a' is a physical
constant, Hooke's Law ai = Eei always holds. Beyond the elastic
limit ai = @(ei) is a certain curve (fic. 2). If at a certain
instant of time there occur infinitely small variations from the
state of strain, that is, the quantities ex receive
increments 5ex ., then the increments of stress below the
elastic limit are given by the formulae (1.4) by-setting i- = Eel.
Beyond the elastic limit the increments of stress for 8ei > 0 are
given by formulae (1.4) in accordance with the curve ai = 5(ei),
but for bei < 0 in-accordance with the law of unloading ai = Eei.

NACA TM No. 1188

and with
there is
close to

problem of the stability of shells (plates) is stated as
Given, a shell under given system of applied forces
the states of stress and. strain known. Required, the
value of the external forces "or which, at the same time,
equilibrium with other possible states of strain infinitely
the original state.

Let the change in the first and second quadratic form of the
middle surface of the shell relative to the given equilibrium
position be characterized by the parameters E1 2, 253, and
, X2, T where el', are length ratios and 2- is the

shear in the middle surface in the x,y plane, and ., T

are changes in curvature and twist. According to the Kirehoff
hypothesis the increments in length and shear at distance z
from the middle surface will be

6exx = E6 z 1

so y = E 2
y 2

,e = ,< 2sr
-- ..

We seek the stress increments corresponding to the strains (1.5).
For this it is necessary to take the variations of rel-tiono (l.h),
The variation of the intensity of strain may be found by making use
of (1.5), but artean.rd we write for the voriation of the work of the
internal forces ci ri, in tei-rm of the stroes coir-onentse,

cri 5'i = "*'- + Y y 50 Y 59+


We introduce the nondimensional quantities

X* = -

y h h
Y i a1


where h is the shell thiclnmcs. Then in agreement Trit, (I.1)

X '. + y -,2 y + 3Xy2 = 1

From (1.6) and (1.5) we have

S= e = -



NACA TM No 1188


E = XXE + Y*E + 2X *E
xl y 2+ Y2 y 3

*= X + fyX2

X = X*X + Y *X +
x 1 2

For the variation of formulae (1.4) we note that



do i/ 1

in which by the properties of the curve cr = 4(ei),
We denote by zo = Zo* the coordinate of the
the intensity of strain is unchanged (Bei = 0) d.uing
It is clear that

Zo -_
The variations of formulae (1.4) have the form

x Cr



S *X (z* z *) +
x 0

- d.0 0
ai da
ei dei O
layer for which


( eX z

Sy = -1I

S *y (Z* z*)



Xy* -:(z* z *) +

All quantities entering into the
except the strains and the curvatures
state of stress of the shell (whose s"



are Imown,
ability is

(E3 T*z

side of these equations
since the original
sou'1it) is supposed

+ y*T*

S2X *

+ e

5X = -
y 1

NACA TM No. 1188

a E daI
ai doi
given. The quantit2eQ E, E' = -, E"' = are shown in figure ?
Sei : de. i
as tangents of angles, Young's modulus being constant but E' and
E" depending on the state of stress.
Before instability the shell may find itself wholly beyond the
elastic limit, or it may.have elastic regions, elasto-plastic regions,
and purely-plastic regions. If the state of stress is ':onentless,
then the region of elasto-plastic strains, that is, the iogion where
part of the shell thickness is elastic, part plastic, is absent. In
this paper we confine ourselves to the detailed stability investigation
of compressed plates in which the state of stress is alir2 .; momentless
before instability. Hence we shall suppose that in the shells
considered below the region of elasto-plastic strain is missing,
before buckling (this ossu.mtion is not essential).

After instability the region of the shell where the stress was
originally elastic vill be, generally speaking, elosticilly deformed,
since the strain variations are assumed ilrfiiteslmal. Ths region of
purely plastic strain will be, generally speaking, resolved after
buckling into two one remaining pu-rerl plastic, the other elasto-
plastic. Figure 3 shows a section norna? to a hell i7trih the three
designated regions (the plastic region after instability : is shaded).

Let the surface z = represent the boundar.j between the
regions, one of which is elastic after instability, the other plastic.
For its determination we shall suppose that in the Jclasio-na-astic
zone, the plastic zone adjoins the shell surface = + s and the
elastic zone which originates as a result of unloadlnr joinss the
surface = -

In the region of elastic strain and in the zone of unlcading
(z zo) formulae (1.11) take the form

8Sx = E(E1 X1) Sy = E(c2 -2z) 5X = E(c ) (1.12)

In the region of plastic strain and in the Zone of 1.oading (z E o)
of the elasto-plastic region, these formulae may be presented in the

1From (1.12) and (1.13) it is seen that the variations BS .
on the boundary zo are continuous in the case where the original state
of stress corresponds to the beginning of flow, and equally so when, as
a result of variation, the state of stress changes in proportion to the
original state 'reference 2).

,A\CA TM No 1188

We proc
moments arise

5Sx = (E -E") SxX(z zo) + E'(E1- X1)

5Sy = (Et E") S *X(z zo) + E(E -2 ) (1

6X = (E E") X*X(z z),+ Tz)

:eed to the derivation of expressions for-forces and
ing in the shell during instability. For their deter-

mination we have

BT1 h
I h

M1 2-


In the region of pure
agreement with (1.13,

E1 1
S (T, 895

E'h (1 2

and for the moments

and. for the moments

-- M 1 .5M )-

-- {SM2 1 m^
3 4 ( M l 1 '6 2

X daz 5T2
x 2



Xz dz SM2 -

Yy dz

SYz daz


BS = h


5H h
y a


SX dz
y d

SX z dz

sly plastic strains we obtain for the forces, in

= l xlS *y

= 22 %iSY*e

l- + X'Sx *

= X, + X'S *X

1 S =

3 YX
.3 )

4l. 2
- 85 = -T + %$X *X
3D'3 .



IIACA TI No. 1188


D' = _


E' E"

In the region of purely elastic strains, formulae (1.1h) and
(1.15) hold, crny E' = E" = E, X' = 0.

Thus in the two regions, the forces are linear functions only
of e1, 2 and 2 ., the middle surface shear, and the moments
are linear functions only of the changes in curvature.

In the region of elasto-plastic strains, the stresses X .
have different expressions for z > zo and. for z zo. Hence,
the integrals in the expressions for the ''orces and moimenis must
be split into two parts. For example,

851 h 3S z dz = S z dz +
h' -- tIs h

SS.; z dz
i -0

in which for the region zo ? z g we take 9"' rccor'llng to
(1.12) and for the region z > z according to (1.13). As
a result of these calculations we obtain for the forces

S 1= = E + E') '+ (E *- (E E')X
6TT2) +2,.

+ E-- E" S (1 z )2

- 1 5T\ =[E + Et + (E -.E')z +
-o 2

z1 2 (E E')X *
2 2

E' E" 2 *
+ 2 S (1 Zo)

2 E (E E') 3 -
h SS =~E + El + (E E)zo +

(E E')T

E' -E_ Xy ( 1- o9 :
2 X 1



and for the moments

rACA TM No 1188

1 M 3M2 = E + E' +

+ -IE (( E"1 *) (2 + *) S)X*
2 .

2 = E + E' + (E E')o X* -* (E

- E')( 1 z42)

+ (1 zo*)2(2 + zo*)S yX

- E + E' + (E E')zo (E E')(1 z*23
3 E0

+E-- (1 zo*)2(2 + zo*)(yX*


The dependency between forces and strains is nonlinear, since
z enters into the formula and from (1.10) it depends on the strains.
From this fact proceed all the difficulties of solution of problems
in shell stability beyond the elastic limit.

Further, it is essential that the ordinate zo* de-ending on
both the changes in curvature X1, X2, T and on the strains el,
62', e3 be expressed only in the changes in curvature and the
forces T1;, 6T2, 5S. Multiplying the first equation of (1.17)
by ix*, the second by Y *, the third by 3X and adding, ye

Sx*ST1 + S *8T2 + 3Xy*8S


(E E')zo3 X (E-E')(i z *2

12 H =

(l o*)2 + 4z o 4

NACA TM No. 1188 9

By introduction of the notation 5 for the ratio of the thick-
ness hp of the plastic layer to the thickness of the shell

hp 1 zo
h 2


and solving equation (1.19) for w, we get

E EE"(1 +) (1 )(1 +
E E" l



(p =

x Sx*'T1 + Sy -ST2 + 3Xy*5S

1 -


E E"



Formula (1.17), (1.18) are appreciably simplified (otherwise
conserving the principal complications) if we consider onl- the
beginning of flow, that Is, we suppose that the shell material
before instability exceeds the elastic limit very slightly. In
this case

E' =E X' = =E

Therefore, in the notation of
have the form for the forces

S-- 2 BT )
Eh 12 2)

+= h
3S 3 2+ x 2
3 3 2 y

(1.20) the corresponding formulae

Xh 2
2 (1.23)

+h C! (1.23)
i22 -Y52

NACA TM No. 1188

for -:the. moments : -;

3D(1 -12

3D (M2

,;- -l


- i

-- + S 2(3 2t)X

S,,Sy* (3 .X

3 = I+ XX 3 + 2 )x
3 y

where D is the usual stiffness for Poisson's ratio equal to 1/2.


-Denoting the bending. of. the plate during instability by,. w(x,y)
and the displacements of points in the middle surface projected in
the x,y directions by u(x,y), .v(x,y), respectively, we have
expressions for the changes in curvature X, X, T, and the
strains E1, E2, E3:

2 2w
2 6Y2

T7 -
ox oy

3 2 y


1 x
c -^.

E1 ox

The forces applied in the-middle surface
may be written in the following form:

Tli = haiXx*

T2 = YciYy

before instability

S =hcr.X
Y i J

and their projection on the Z-axis- after instability in the form

T1X1 + T2X2 + 2ST = haiX


aVI\ ~


NACA TM No. 1188

Therefore, the condition of equilibrium of all forces applied to an
element and projected on the z-azis, gives

295M1 26H
-2 + 2- +

2 + haix', = 0


The condition of equilibrium of the middle surface forces after
instability will be

-+ --=0
5x by

2 + S
- + -- = 0
oy ox


Finally, the compatibility condition for the strains has the


E1 2
y2 x2
oy2 oix2



The combination of differential equations (2.2), (2.3), and (2.4)
is necessary and sufficient for the solution of the problem of
stability, if the corresponding boundary conditions are set up.
Indeed, according to (1.14), or to (1.21) and (1.20), the strains
.l, 2', 3 may be expressed in terms of the forces F'T1, 58T, 8S
and the curvatures (bending w), following which the moments
5M1, BM2, BH are functions of these same four arguments. Thus
the problem reduces itself to four differential equations with four
unknown functions, of which (2.2) is of the Bryan type, and (2.3),
(2.4) are of the type of equations in plane problems.

In the region of purely plastic strain of the plate, (that is,
such that the whole thickness, plastic before instability, remains
plastic after instability), the system of differential equations
is resolved into two. For simplicity we consider only the case
of the beginning of flow. Substitution of the values of 5M1, 5M2,
SH from (1.15) into (2.2) gives a differential equation for w
of the Bryan type:

4 hcri
V --X=

S'2-6 XX
4x x

+ 2--X *
x cy

+ Y- .;i XX


NACA M' No. 1188

where, in agreement with (l.9) and (-2.l),

" ~~

y ^'^r '

Y *_
* -^


The two boundary conditions on w agree with the usual.
boundary-conditions for the Bryan equation. '. ,

Solving equations (1.14) for the strains, w? get

- (.1
T \




S1 )

+ x

+ Y
(1 X)Eh


+ 3xy~

S *5T + S 2T + 3 Y*S
\~ -" -

2 = 3S
3 Eh

l- S -6T + S *T + 31 S
(1 x)Eh x 1 y 2 /

Equations (2.3) are satisfied'if the stress function:

ST1' 3 8T' .S ..
2Eh y2 Eh E x2 Eh F,.
,h y2 Eh 82 Eh .Ex;

F' is introduced:


following which, analogous to (2.6): we denote

t = sx
Sx *2

7 t2


3 ----
- Y a **o .r-

we obtain the compatibility condition for :strain in i-h 4i /*rm

S 3
S 3 *
y -~ I

> (a.7)

VhF x.





"'; '

TIACA TM No. 1188

In order to write the boundary conditions for this equation
it is necessary to compute the variations of the normal force jTv,,
and of the tangential force 5SV on a certain curvilineer contour
in the middle surface of the plate.

If the outward normal V and the tangent a to the contour
constitute a coordinate system such that by rotation the positive
direction of V coincides with that of y and the positive
direction of a coincides with that of x and if the angle
between the normal and the x-axis is denoted by a (fig. 4),
then our quantities have the known expressions

5T + ST ST, 5T
2 2
STV = 1 + + STcos 2a + 8S sin 2a

> (2.11)
2S, = sin 2a 5S cos 2a

The purely plastic region of the plate may be bounded by a
contour, part of which coincides with the boundary of the plate,
the part adjoins the elasto-plastic region. For the formulation
of the stability problem in the first part, the boundary conditions
have the form

s5T = SS = 0 (2.12)

and in the second nart STv, FS must be continuous.

It is easy to show that during instability the entire "late
may not remain in the pure'-y -plastic state; that is, an elasto-
pl.Estic region may come into belng. Indeed. going back we shall
have the uniform boundary conditions (2.12) on all external edges
of the plate. But the differential equations (2.3) and (2.4) for
conditions (2.7) will be also linear and homogeneous and so will
have the unique solution

5T1 =- ST2 = S = 0

It follows from (2.7) that E = e2 = 3 = 0, from which on
the basis of (1.9) and (1.10), z, = 0. But z = zo is the

NACA TM No, 1188

boundary between the elastic and plastic zones through the thickness
of the plate and the condition zo = 0 specifies that the middle
surface is thisboundary. It follows that a given region of a plate
is not purely plastic, but elasto-plastic, which contradicts the

During instability of a plate beyond the elastic limit it will
either go completely over to the elasto-plastic state or there will
remain purely plastic regions in it, which are not diffused through-
out the plate.

In the region of elasto-plastic strains, equation (2.2) on the
basis of expressions (1.24) may be presented in the form

ha.i 2 2
x = x* + 2 2(3 2)x (2.13)
D 4 x2 2x y + 2 7y

in which, as in equations (2.5), (2.10), the operator in parenthesis
acts like a multiplier on the quantity to its right.

The condition of compatibility of strain (2.4) on the basis of
(1.23) has the form

VF = S S 3x (2.14)
2^o y y

where the stress function F is determined by formulae (2.8). The
value of the ratio of the thickness of the plastic layer to
the plate thickness, enters into equations (2.13) and (2.14),
therefore they show compatibility; this quantity t is expressed
by formula (1.21) in which the function p is, if use is made of
the notation (2.9)

S 2 t (2.15)
h 1 \ X

Equations (2.13), (2.14) agree with the corresponding equations
(2.5) and (2.10) at the boundary of the purely plastic and the elasto-
plastic regions. Indeed, at this boundary, besides continuity in
the values of the forces 5Ty, 8Sy, the moments BMy, 5HV
(where 8HR V is the rotational moment according to the boundary

NIACA TM No. 1188

conditions of Kirchoff), the bending w and the slope of the tangent
plane, there must also hold the condition

hp= h



From (1.21) for this condition we have Q = -X and

following which the remarked coincidence of the equations

t = Xh,
to easily

The boundary conditions for equations (2.13), (2.14) on the
elasto-plastic pnrt of the contour, coinciding with the plate contour,
yield the usual requirement STV = 8Sv = 0 and two conditions relating
to the bending w.

Condition (2.15) or

t = 1 1 >h


represents in itself the equation of the boundary between the purely
plastic and the elasto-plastic regions.

The possibility of purely plastic regions arising at the same
with-the elasto-plastic regions follows from the fact that the
value of t in agreement with (1.21) and (2.15) ma.y take on values
not lying in the interval 1 C ;O. Certain exoi.wiles are given
below of exact solutions of the stability of plates and, in
particular, the problem of the compressed plate freely supported
along two sides; the edges of the plate near the free supports,
after instability, remain in the purely plastic sta.te.



The integration of the system of differential equations (2.13)
and (2.14) in the elasto-plastic region, and of (2.5) and (2.10) in
the plastic region with an undetermined boundary between them given

INCA TM No. 1188

by (2.16), is fraught with significant mathematical difficulties.
As was shown in 1, the stability problem simplifile when the
variations of the forces in the middle surface are zero every-
where. In that case the relative thickness of the plastic
layer is a known function of the coordinates, since from (1.22)
p = 0 and consequently

= -.- (3.1)

If the state of stress of the plate before instability is
uniform, the value of t will be constant, since in (1.22)
dai will be the same for the whole plate.

We call those solutions of stability problems anproximate,
for which the variations 5T1, 1T2, 5S of the forces are
identically zero. Thus, the equations (2.3) of equilibrium and
the boundary conditions (2.12) are satisfied, but, except in
special cases, the compatibility condition (2.4) is not satisfied.
The simplicity of such a solution arises from the fact that in
equation (2.13) the value of t is known and given by formula
(3.1), as a result of which this equation becomes linear with
constant or variable coefficients. It closely resembles the
equation for the elastic stability of an anisotropic plate.

The exact solution of the system (2.13), (2.14) are undoubtedly
of interest in their own right, but for us they have significance
because they can be made use of to estimate the degree of exactness
of approximate solutions.

We discuss a certain class of exact solutions of stability
problems for uniformly compressed i ratess of arbitrary shape and the
solution for a rectangular plate in the case when buckling into a
cylindrical shape is possible.

a. Stability of a Uniformly Compressed

Plate of Arbitrary Shape (Fig. 4)

In this case the state of stress of the plate before instability
is uniform and given by the formulae

Xx = Yy = ai, X = 0


NACA TM No. 1188

where Ci is the compressive stress Falong the edge and is also the
uniform stress intensity at any point in the plate. The resulting
stresses according to (1.7) and (1.4) will be

x* = Y =-1 X = 0 Sx* =
x Y 2

For the values of X and t we have the expressions from (2.6)
and (2.9)

,.:= -v t = 1- (3.4)

Equation (2.14) takes the form

V t, = o (3.5)

Neglecting the hnracnic function, we obtain a class of exact solutions

t -

as a result of which the value of .'p in (2.15) is expressed in terms
of and from (1.2) we find

= -- = 1 = const. (3.7)

The fundamental differential equation of stability (2.13) is now
linear with constant coefficients and hns the simple form

1 + 2(3 2j vw + V2w = 0 (?.8)

Its solution has been much studied for different shapes of
plates and for different boundary conditions, although in connection
with the elastic stability of comLpressed plates.

NACA TM No. 1188

The value of t (3.7) is little different from the approximation
(3.1), and characterizes the degree of deviation of the exact solution
from.the approximate,

In the general case we have from (3.5)

t X8 x + (3.9)

where ri is an arbitrary harmonic function. For continuous circular
plates, for example, P is a constant. According to (2.15) and (1.21)
we now have an expression for t in terms of X

=- 1 3\ + (3.10)
3% \ / 16X

following which equation (2.13), having in the given case the form

-- (3 2 X + X = 0 (3.11)
4 D

has only one unknown function X. By use of relations (3.4) it may
be integrated once

i \ 2 ha2
1 r(3 2) V + --- = r (3.12)
D 2

where 2 is a new harmonic function, also a constant for continuous
circular plates, insofar as w and V2w must be finite in the middle.
Equation (3.12), in view of (3.10), may be solved for V2w, after
which the problem reduces to the integration of only one linear partial
differential equation of the second order (for circular plates)

V w = ,(w,P1rr2)

The stress function F is now determined, in accordance with (3.9)
and (3.4), from the Poisson differential equation

NACA TM No. 1188

2 h =- (2x+ rl) (3.13)

As we see, the problem of the stability of circular plates may
be solved in comparatively simple fashion through to the end. The
details of a similar calculation will be clarified below for the
example of a rectangular plate compressed in one direction.

b. Stability of a Rectangular Plate Under the

Condition of Plane Strain (Fig. 5)

Such a case occurs if, when a rectangular plate o' length 7 is
compressed in the x-direction, the width b in the y--irection cannot
change as a result of walls along the boundaries :- = 0 and y = b.
The plane x = 0 shown in figure 5, where C = 2c and L = .2; will
evidently be a plane of esymetry of strains.

We assume the buckling to result in a cylindri1caL. shape. In such
a case,according to the conditions of the problem: we have for the
stresses before instability

9 i '3
x =--- Y =----! 1 3S =0
/3 3 2 Y

After buckling, w = w(x), E = e = 0.

From equations (1.24) we have

S3 = 0 1T = 8 oT

Since, in accordance with the equations of equilibrium )T1 = const.,
and ST1 = 0 from the condition at the edge x = ?, then we have

the case 5T1 = .To = 5S = 0. In consequence, the ap-nroximate
solution, as was noted at the beginning of 3 here becomes exact.

20 '".",. TM No. 1188

The thickness ratio 5 for the plastic layer is a constant
and is determined by formula (3.1). The stability equation (2.13)
takes the form

d4" hp d= 0 (3.15)
ax4 D -.x (3 2t) dax2

If the relative Karm`an modulus, expressed by
K = de l )L (3.16)
dai (1 + 1-X)2
\. dei

is introduced, then we get from (3.1)

S, i.(l. k) (3.17)
(2 ,/k)2 2 :

following which we may simplify the expression for the parameter in
equation (3.15)

2 hp hp (3.18)
D [- 2(3 2)J Dk

Since k = 1 up to the elastic limit, and k = 0 in a small
area where there is flow of the material, and since the character-
istic value of the parameter 7 must be the same in elastic and in
plastic problems, then it follows from (3.18) that the critical
stress corresponding to the small area of flow, is zero,

It is interesting to note that the Karman problem may be
considered as a limiting case of the stability of a rectangular
plate compressed in one direction, of small width b, for which
the parameter 7 will have the expression

2 4hp

NACA TM No. 1188

and consequently the critical stress is zero at the small area of
flow. As seen from the preceding end following examples of exact
solutions, the total loss of load-carrying ability of a plate,
predicted in the Karman problem, does not occur, generally speaking.
This circumstance has already been noted (reference 1).

c. The Stability of a Rectangular Plate Compressed

In One Direction (Fig. 5)

We shall suppose that the rectangular plate, sufficiently long
in the y-direction and compressed only in the -c-direction, buckles
into a cylindrical shape. In this case

7 = -r' Y = Xj = 0

S: ( .19)

= -1 Sy X* = -1 YV Z*
X y 0 -Y 0

By the conditions of the problem, all section o- the plJte
y = const. remain pl-.,ne after buckling and so we h-ve

C = 0 =o = const. (3.20)

on the basis of which from (1.24), SS = 0. Besides this, ST = 0
from the boundary condition at the edges x = + and consequently
it follows from (2.3) that sT1 = 0 everywhere.2

Since there are no forces in the y-direction, we must use the

/2 T dx = 0 (3.21)
L 2

From the second equation of the system (2.14) wo have

d2F T2 h 2 (.22)
dx2 Eh 2 4 1

NACA TM No. 1188

since X =.-Xl. It- :isa not difficult to convince one s self that
(3.22) is the integral of equation (2.14). The function c, by
which is found the value of from- (1.21), here has the form

p8-T2 -- 2 + X2 2 (3.23)
(1 Eh(1 X)h 4( x)

The bending moment in any section is

= D 2(3 2) X. (3.24)

and so the boundary condition on the edges x = + I is X, = 0.
It is clear from (3.23) that X1 cannot be zero in the elasto-
plastic region since e2 / 0 (this follows from the constancy of
sign of 2 X1, positive along the entire plate, necessitating
C2 0 to satisfy condition (3.22)). Thus the elasto-plastic
region does not go up to the edges of the plate and stops at the
section x = + c. The region adjoining this to the edge will be
purely plastic. Indeed, since t2X is positive, then e2 is also
positive. It follows from (2.7) that in the purely plastic and in
the purely elastic regions the force 5T2 has the same sign as
E2, that is, is a tensile force. But if, to the plate, compressed
beyond the elastic limit in the x-direction, there is applied a
tensile force in the y-direction, then the plate remains in the
plastic state. One may convince one's self of this by formally
calculating the value of 5e- according to (1.8), which at the
edges is equal to el, but the strain E, according to (2.7)
is negative, and so the value of bei will be positive, that is,
plastic strains before buckling remain plastic after buckling.

From (1.21) and (3.23) we now have

hX 1
-1 = p(Q) = 4 + 8 3%t2 (3.25)
42 p(n)

NACA TM No. 1188 23

From this we find the lower limit to the value of t ( > 0)

S> \(3.26)

The fundamental differential equation of stability (2.13) takes the

d2 3- ha,
dT 1K- 0 2 ---+f 1 = 0 (3.27)
d:t _

By introduction of the notation

Q() = 4 9X + 6~)3 = = (3.28)
2 2

we write equation (3.27) in the form

-+ = 0 (3.29)
d42 p P

where n is the basic parameter determining the critical stress

2 i
= (3.30)

The integral of equation (3.29) may be obtained by quadratures.
Through introduction of the notation

PE() = l + 12 2 3 (3.31)
(4 8i 3Xse)s

we obtain as a result

j T (a)2 = i2C r. ,/2 = c -J ,- p. (3.32)

NACA TM No. 1188

In the purely plastic-region we. have for- the force-' 8T2 and the
moment 5M1, in agreement with the results of 2 and (3.19):

XT )
Eh 4- 3x

'8M = -


The fundamental differential equation takes the form
S2 4 -
-- -+ X- =
aS2 4 31 1


The solution, satisfying the condition
is written in the form,

x 1 = 0

at the end

1 = C3E2 sin -
<,A 3X

in which as a result of symmetry we consider only deflections in the
right half of the plate (x 0).

For determination of the five undetermined constants namely,
the three integration constants C1, C2; C3, the boundary
coordinate a and the critical number i. we may, besides
equation (3.21), write four more conditions: Conditions of symmetry


conditions at the boundary region

S=a.* 5 =l


two continuity conditions, of moment and shear force, which in
accordance with (3.25) and (3.35) take the form

C3 sin = cos
3 45 -3. h(4 3%) \/4 37 \/4 3x

hP (1)
hP2 (1)

( M (3.38)

S= 1,


S= 0 -- = 0

TACA TM ITo. 1188

The constant e2 is not necessary and does not enter the
conditions insofar as they are independent of and -.

By making use of the prescribed conditions and introducinG
a new unimown t, the relative thickness of the plastic layer
at x = 0, we get for the values of p and 1 a (the relative
length of the purely plastic part) the following formul-e

XM 4 %, L
S-= I a (3.39)
.,.2(1 x)

where L and M are the integrals

I,'R() 1 + ,2 o) 1 .)2 &=
,R(I) ,/R( o) (() ,(1) ./(., ) ( P,)

in which the value of r is determined br the rel.?t.on

cot2 2 4 L3 2(4 3 r) r() -(1) (3.l1)
/2(1 0) .

As was already established., the value of 1 is positive,
therefore the integral L must be positive: and ',or this it is
required that 1 2 + X20 > 0, that is,

(< 0(3.42)

By considering the estimate (3.26), which is also reasonable for
Io, we see that this quantity is contained within nrrrrw limits and
close to the approximate value (3.1). It follows frcm this that the
critical stress will differ only slightly from the approximate value.

NACATM' No. 1188

d. Approximate Solution-of the Problem for a'Plate

in a Uniform State of Stress Before Buckling

In this case the stress components

Xx Y and- X and the

stress intensity ai are constant everywhere; the quantity X
will also be constant, and hence t by (3.1).

The x and. y axes in a given case may be so chosen that
the X stress is zero (principal axes of stress). The fundamental
stability equation (2.13) takes the form
stability equation (2.13) takes the form

3 -- k) XI*2 + 2 1F


+ ( k)Y *2

-3(l k)XYy* 1-
-x cy ox287

Hx* 4

y+ y*
v y -.2

in which the generalized Karman.modulus is introduced in accordance
with formulae (3.16) and (3.17), since the relation

W2(3 2t) = 1 k


The coefficients in equation (3,43) are all positive, since
the largest value of each of the quantities Xx*, Y is
-- and 1 > k > 0.
S Hence, the problem may be solved as a linear differential
equation of the Bryan type with constant coefficients, and in
difficulty is little different from the corresponding elastic

Translated by E. Z. Stowell
National Advisory Committee
for Aeronautics



NACA TM No. 1188 27


1 Ilyushin, A. A : Ustoichivost Plast inik i Obolochek za Prerelom
Uprugosti. Prikladnaya Mantra,?tika 1 Mlekhanika, H. S. 8, No. 5,
1944, pp. 337-360. (Also available as ITACA TIM Ho. 1116.)

2. Ilyushin, A. A.: K Teoria Malikh Uprugo-plasticheskikh Deformatsii.
Priklacdnaya Mateniatika i IMekhanika, X, 1946, p. 347.

NACA TM No. 1188

Figure 1.





Figure 2.

pl ast c

Figure 3.

30 NACA TM No. 1188


4_ 1-=0

Figure 4.

Figure 5.




o3 ol
0 4

2 lo d r r

10 K
e ii' *

4 (0

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0 H



0 q0

6H O

S n E
4 <* -

-2 T m lq g H

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H ^ K( '

.0 a
4 (W 0 i 0

-P P.- O u 4-
d g -P +P + co

U4S-1 4r- l W l
00 0aP -
P4 43 4-) M
4 x l d 4 p R
H 'd *d =P 4- PA
0C 0 0 4 '
,o 0 -H 0 A

O mPr *.c j
0 >!ar-I $4 r E-1 1-4

o 0e Cdo )s
W4 Mo a 0

$4em e 4 4
o 0 i a b-i4 o

1 a w p-r d
*r 0 *E = 0C

In pD, o 4e ) Ww 40
o D 0 1Q 4

b- 4 W4) 0 0 (--
0b- P+ o P

E- H r -4 6 i

4 0+0

) a r- O 1 0 0B o
91 u H f

a ,+ $4
.- +A'd H 4 )
4 e B -
op a o o

S 0 4 4-)+


Sma 0
P 10

r4 4 a
> In r-1 r 1D

4- 0e r-4 o

w3 p.
rt fr-l W q Q d
0P 04-;P Cd

p m *0 r3

044- P. e

A, 0 4,2
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3 1262 08106 659 8

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