On the application of the energy method to stability problems


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On the application of the energy method to stability problems
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Physical Description:
41 p. : ill ; 27 cm.
Marguerre, Karl
United States -- National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:


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Aerodynamics   ( lcsh )
Aeroelasticity   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


In the two examples of the Euler strut and the slightly curved beam under transverse load it was shown that the difference between the stability problems and the problems of linearized elasticity theory rests upon the fact that in the stability problems the expression for the energy of deformation contains terms of higher than the second order in the displacements. This idea makes it possible to establish the connection between the energy method in the special form most used for stability investigations and the principle of virtual displacements in its general elasticity - theoretical version; besides, it permits the investigation of elastic behavior beyond the critical deflection.
Includes bibliographic references (p. 37).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Karl Marguerre.
General Note:
"Report date October 1947."
General Note:
"Translation of "Über die anwendung der energetischen methode auf stabilitätsprobleme" Jahrb. 1938 DVL, pp. 252-262."."

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0V j: 4 : ,2




By Karl Marguerre

Since stability problems have come into the field of vision of engi-
neers, energy methods have proved to be one of the most powerful aids in
mastering them. For finding the especially interesting critical leads
special procedures have evolved that depart somewhat from those customary
in the usual elasticity theory. A clarification of the connections seemed
desirable, especially with regard to the postcritical region, for tho
treatment of which these special methods are not suited as they are.

The present investigation2 discusses this question complex (made
important by shell construction in aircraft) especially in the classical
example of the Euler strut, because in this case -- since the basic fea-
tures are not hidden by difficulties of a mathematical nature the prob-
lem is especially clear.

The present treatment differs from that appearing in the Z.f.a.M.M.
(1938) under the title "Uber die Behandlung von Stabilitatsproblemen mit
Hilfe der energetischen Methode" in that, in order to work out the basic
ideas still more clearly, it dispenses with the investigation of behavior
at "large" deflections and of the elastic foundation; in its place the
present version gives an elaboration of the 6th section and (in its 7th
and 8th secs.) a new example that shows the applicability of the general
criterion to a stability problem that differs from that of Euler in
many respects.

"Uber die Anwendung der energetischen Methoie auf Stabilitats-
.probleme." Jahrb. 1938 D'L, pp. 252-262.
In the paper investigations were continued at the instigation of
Professor Trefftz (during his activity at the Deutschen Versuchsanstalt
fur Luftfahrt). For a large part of the work (especially in secs. 4 and
6) the author is very grateful to his colleague, R. Kappus, for his close
See the next to last paragraph of-the Introduction.

NACA TM No. 1138


In the two examples of the Euler strut and the slightly curved beam
under transverse load it was shown that the difference between the sta-
bility problems and the problems of linearized elasticity theory rests
upon the fact that in the stability problems the expression for the energy
of deformation contains terms of higher than the second order in the dis-
placements. This idea makes it possible to establish the connection be-
tween the energy method in the special form most used. for stability in-
vestigations and the principle of virtual displacements in its general
elasticity theoretical version; besides, it permits the investigation
of elastic behavior beyond the critical deflection.


Kirchoff's uniqueness law states: An elastic body can assume one
and. only one equilibrium configuration under a given (sufficiently small)
external loading. In the formulation from the energy point of view: the
potential II of the inner and outer forces has one and only one extremal

II = 0 (1.1)

and the extremal is a minimum.1

The 'niauerese law holds without restrictions in the realm of line-
arized theory of elasticity, that is, as long as the stresses a, T can
be expressed linearly in terms of the strains 7ik and the strains line-
arly in terms of the displacements u, v, w. Then the function II is of,
at most, the second degree in the displacements, and geometrical considera-
tions Ehow directly that a "parabola" of the second degree (positive def-
inite quadratic form) can have one ani only one minimum (or in mechanical
terms, equilibrium position). The situation changes, however, when struc-
tures are considered tho behavior of which can no longer be expressed with

lFor the derivation of the principle of virtual displacements (equa-
tion (1.1)) for elastic equilibrium, see, for example, reference 1, pp.
70 ff. A careful foundation of the general theory of the behavior at the
stability limit appears in reference 2, p. 160. (For further literature,
see reference 1, pp. 277 ff., or reference 2). The investigation in
section Stable and Unstable Equilibrium in particular make use of the
Trefftz point of view.

NACA TM No. 1138

sufficient accuracy by the linearied strain-displacement equations. Such
are, particular, bodies for which one dimension is small compared to the
others, structures in the form of shells, plates, or bars. For example,
a rod can, without exceeding the proportional limit, undergo bending de-
flections several times greater than its thickness, and under these cir-
cumstances the quadratic part of the (transverse) displacements in the
strain displacement equations is no longer small compared to the linear
part. Then the energy of deformation of the potential II becomes of
higher than the second degree in the displacements, and a parabola of
higher order can naturally have several extremals (equilibrium positions).

The problem of the theory of stability is usually considered to be
the determination of that external load under which several neighboring
equilibrium configurations are possible. The reason for limiting in-
vestigation to this "critical point" lies in the fact that the differential
equations describing the elastic behavior in the postcritical region are,
in general, no longer linear and an analytical treatment would therefore
be difficult; while at the critical point itself the problem can still be

This purely practical viewpoint has, however, led to a certain (as
is shown, from unfounded standpoint) systematic separation of the sta-
bility problem from the other problems cf the theory of elasticity, which
finds its mathematical expression in a formulation of the principle of
virtual displacements somewhat different from the usual one has also
for convenience led to the formulation of a special principle. (For ex-
ample; see reference 3.)

The principle of virtual displacements states that during a virtual
(that is, geometrically possible) displacement from an equilibrium posi-
tion, the energy of deformation taken up by the elastic body is equal to
the work done by the external forces. For the use of this principle in
the theory of elasticity it is convenient to express this fact in the
following way: An equilibrium state is distinguished by the fact that
for every virtual displacement from that state the potential of the inner
and outer forces

II = A, + V

Knowledge of the postcritical region was until now of secondary
practical interest, because buckled structural elements were considered
unpermissible. It has been only in recent years that in the shell con-
struction of aircraft critical loads have been permitted to be exceeded
by large amounts unhesitatingly.

NACA TM No. 1138

has a stationary value:

l= 8(Ai + V) =Q0 (1.2)

Therein the potential Ai of the inner forces is given by the energy
of deformation (inner work), and the potential V of the external forces
by the negative product of the external forces considered constant and
the displacements of their points of application. In the region of appli-
.cability of the proportionality law numerically V = -2Aa, where Aa
is the work done by the external forces as they increase from zero to
their final values in passing through only equilibrium states. The prin-
ciple (1.2) can therefore be written conveniently also in the form. (For
example, see reference 3.)

8(Ai- Aa) = 0 (1.3)

As against this there is often used as a "minimal principle" in stability
theory the condition

6(Ai -Aa) = 0 (1.4)

The author intends to show in the present paper (in the classical
example of the strut) that also stability investigations are best handled
in connection with the single main equation (1.2), wherein terms must be
retained of higher order in the deformations-logically only in the ex-
pression for the energy of deformation. This procedure is essential
from the practical standpoint, if the relationships are tc be investigated.
in the postoritical region, and desirable from the systematic standpoint,
because it becomes clear in this manner that no additional principles are
required. In particular, this consideration will clear up the apparent
contradiction between equations (1.3) and (1.4).

The calculation itself is carried out in the following manner:
First, the expression for the energy of deformation Ai is set up, then
the differential equations for the two components of displacement are
derived from the condition

1n = 0

NACA TM No. 1138 5

and the question of bhe stability of the equilibrium position is answered
by the restricted condition

II = minimum (1.5)

Then the same problem is treated with the help of a Ritz procedure; in
this way the result of the stability consideration is brought out in an
especially elegant manner. After a thorough discussion of the usual sta-
bility theory, it is shDwn in conclusion how the same considerations can
serve for the treatment of the snap-action problem of a slightly curved


If the customary assumptions of the beam theory are retained that
for small deflections of the beam the wcrk of stretching and work of
bending are independent of each other and that the part of the work re-
sulting from the shear forces is mall compared with the other two parts -
then the energy of deformation can easily be given.

As a result of the assumption of small displacements without at first
saying anything about the sizes of the displacements u and v (fig. 1)
relative to each other the strains u_ and wv(or wx2) may be neglected
in comparison with unity; that is, in a development of both quantities in
powers of ux and wx only the lowest power need be retained. If, there-
fore, the square of a line element of the beam centerline before deforma-
tion is


and after deformation

S/ ",2 2] 2
S+ uY, + vx idx (2.1)

(the subscripts on u an.i w indicate derivatives with respect to x),
then the strain of the beam centerline (reference 1, p. 57) is

S=/ + ux 1 (2.2)

NACA TM No. 1138

From Hookers law the corresponding stress is

= E ux 2 5/
G^E(^+2 )


and therefore the energy of stretching is

A = -
2 -o

+ 2
+ ---_ J dx
2 -

The incremental strain ex due to bending is, according to the
assumption that normal are preserved,

: x = -ZWxx


the bending energy is therefore given by


. E

dx dy dz a

vEf wdx


If El = u(0) u(Z) = -u(I) is the distance of approach of the ends,
P = pF = xF the compressive force, then (pF)(El) = V is the po-
tential of the external forces; the total potential IT (measured from
the stress-free state as the zero position just as the displacements
u, w, e !) is therefore

II = -


+ W-2 dx + -
.2 ) 2

I w2 dx plt I


The potential per unit length after division by EF can be written

SFn i2
SEF3 21

S2 2 2
us + )dx + f / X2 aX R e
2 / 2Z xx E



NACA.TM 9p.-1.-138

S.From equation (2.8) and. the condition (i5)-'

II = minimum (2.9)

there is obtained all information about the behavior of the strut at and
beyond the stability limit.


Consider a rod the left end of which is (x = 0) is supported and
the right end gives (x.= 1) is freely movable in a horizontal direc-
tion under a centrally placed compressive force P. (See fig. 1.) .As
given (that is, as the independent variable of the problem) take either
the horizontal displacement | or the compressive "
of the right-hand end stress

*u(M) = c1 -

According to the principle of virtual displacements, the displacements
are to be varied under a constant load in a manner compatible with the geo-
metrical conditions. If at a boundary point the displacement (in the pres-
ent problem, for example, e) is prescribed, the point is held fixed
during the variation, so that the work of the (unknown!) external forces
.doee not remain in the calculation; if on the other hand, the force is
given, then the (not fixed geometrically) end point is varied, and the
work of the external load (in the present problem P~u) enters into the

If in equation (2.8) the displacement u is varied (that is, if the
elements of the rod are.given a virtual displacement in the axial direc-
tion, while the boundary point or correspondingly the load is held fixed)
there follows from the rules of the calculus of variations

/ 2 o

together with the boundary conditions (3.1)

u(0) = 0 u(0) = 0
I wx
u(M) = -z I ux + -- + 0


8 NACA !'A-96. 1138

by varying v there is obtained' imindependently-f whether. or p is
considered as given):

Ux + ux + T wxx+ 0

with the boundary conditions (3.2)

w(O) vw() = xx(O) = wxx() = 0

'The exact integration of the system (3.'1), '(3.2) of nonlinear (!)
a imultaneous'equations offers no difficulty here. "
From (3.1) there follows'

Ux + constant e fo
PX 2 (3.3)
u = u(s) oX x

and with the use of (3.3), (3.2) becomes

x + W + XX 0 (3.4)

The solution of this linear equation with constant coefficients is

w = f sinkx +.g cos kx + fix + g

where k is the positive root of the quadratic equation

2 0
i 2

For the determination of the six constants of integration f, g,j fl, gi,
u(0), eo there are the six boundary conditions (3.1) and (3.2):

NACA TM No. 1138

g + g6 = 0

kg = 0

f sin k1 + g cos k1 + f1l + g, = 0

fk2 sin ki + gk2 cos kZ = 0

u(o) = 0

- El = u(0) co Lx --dx
o 2

-Co + = 0

It is found that

fI = gS = g = u(0) = 0

and either

f = 0, that is, w = 0

u = C

u = -EX

0 E
U/ X

f 0, w = f sin kx, k = -

E2 2
o ~12 -

Ir f 27rx
u = CX + g sin --

There are evidently two kinds of equilibrium positions: the straight
(f = 0) and the bent (f -0). The straight position is specified uniquely
by either p or e, the bent by for the amplitude f of the deflec-
tion can be determined uniquely from the left one of equations (3.5)

Of 2
,-= C E -= (3.8)
4il2 0




10 NAGA TM -No. 1138

not, hovweer, by p. From the right one of equations (3.5) there follows
rather that for f / 0 'a completely determined "critical" value

p = Ee p* (3.9)

cannot be exceeded. Therefore p is unsuited for an independent
parameter (the situation is different in the case of the corresponding
plate problem) (reference 4, p. 124). From equation (3.8) it can be seen
that f assumes real values only for e'> e*.

In figures 2 and 3 the quantities f and .p are.plotted against e.
The solid lines are for the solution (3.7), the dotted lines for equation

The result (3.9) that the load for the buckled strut-does not in-
crease beyond p* even when the .critical end shortening has been con-
siderably exceeded is a result of the limitation to "small" deflections.
For the present problem this limitation is not important because here it
was only a question of seeing that as a result of the appearance of higher
powers of w in equation (2.8) the elastic rod can assume several equi-
librium positions especially that the existence of a real multiplicity
is bound up with the exceeding of a certain "critical" strain e = e*.


In the "crd:nary" theory of elasticity it is necessary to consider
only the condition

IT = 0, that is, _TI = extremum (4.0)

'Reference 5, pp. 70 ff. Also the theory of the so-called exact
differential equation of strut buckling
EJ/p + Pw = 0
(reference 1, p. 280) shows that at large deflections (because of the in-
creased demand on bending energy) there is a very small increase in load.
Froa the energy -tandpoint, to be sure, this "exact" equation is not im-
portant, for if wj2 is taken as not small compared to -1 in the bond-
ing term (i, at is, the curvature l/o is used in place of wvx) it is.
necessary to proceed in a corresponding manner with the stretching term
(equation(2.2)) unless it is assumed that an incompressible strut exists
from the start, as is done in the theory of the Euler elastica.

NACA TM No. 1138 11

in the determination of equilibrium states, for the supplementary con-
dition 52II > 0 (mechanically: the stability of the equilibrium posi-
tion) is assured there because of the linearization (reference 1, pp. 71-
72); in the present problem the minimal condition must be set up explicitly,
since only through

o1I = 0, 52II > 0, that is, II = minimum. (4.1)

can the stable equilibrium positions be distinguished from other possi-
ble (the unstable) positions. The concept of stability is made precise
here by the following convention.1

1. An equilibrium state is called stable if for every neighboring
state the potential energy has a larger value."

2. An equilibrium state is called labile (unstable) if there is at
least one neighboring state for which the potential energy is smaller.

3. A stability limit (that is, a neutral equilibrium state) is
spoken of when there is at least one neighboring equilibrium state the
potential energy of which is equal to but none having potential energy
less than that of the given equilibrium state.

Return to equations (2.8) or.(2.9):

1 21 p
I= i + 1 dx + -- / dx c = Iir.iLm.
21 \ / 2 E

and perform e. variation; that is, replace u by u + Bu and w by
w + 5w; there results, after arranging in powers of Su, 6w and
stopping after terms of the second order,

The following definition was given, in substance by E. Trefftz in-
cidentally to his DVL lecture.

&ACA TM No. 1138

A A 1 2^
Ali = II (u + Su, w + w) (u,w) i + -II + .

0 2z+8xd

+ ux.


;!-) vx wx + i2 b wxx_ ]dx

dx + w 8u 6 w w dx
1z vt 1' a


+ ( )2

+ d.; J )2

The condition that the terms of first order shall vanish leads to equa-
tions (3.1) and (3.2); the question of stability is answered by the terms
of second order. By inserting for *u, w..the values obtained from
mII = 0 there results for (3.6) (straight position)



+u d (4.21)

for (3.7) (bent position)

+ f cos-. &wj
*}S2 -,/ X

+ 2()2

2 1 2
+ --- cos 8vx 8ux dx +- 5u dx
3 o I o


First the stability limit will be determined. According to the defi-
nition given above there must be at the stability limit a state for which
the second variation vanishes but none for which it becomes negative:
The value zero is therefore the smallest value that 8FII can assume at
the limiting pointt' If, therefore, S2IT has certain continuity prop-
6rties (the existence of which is obvious on mechanical grounds), then


(52 1A )


(82 )
\ )2

( *2



i, 5u

[(U + w!__2X
7 2

NACA TM No. 1138

the "characteristic" value 2fI = 0 is at the same time an analytical
minimum, compared with neighboring values, and the associated ("charac-
teristic")- displacement system Su, Sw is determined from the condition

-/( ITI= (4.3)
\ /

The straight position (see equation (4.21)) thus reaches the sta-
tility limit when bu, Sw satisfy the two differential equations

(6u) = 0

(SW) + (CV)x = 0

with the end conditions

Bw(0) = Bw(1( = 8w(O) ) = wf(2) = Su(O) = 0

and (4.5)
5u(l) = 0 Bux(l) = 0

The solution of this eigenvalue problem reads in both cases

Su = O, 5w = 5f sin (4.6)

The amplitude &f p 0 remains undetermined and.from the second of equa-
tions (4.-4) there is obtained for the critical value of

C i2i2
crit = --

that is exactly the expression crit = E* by which the branch point of
the equilibrium was characterized; stability limit and branch point coin-

The investigation proves to be somewhat more difficult for the bent
position. The two minimal conditions read

NACA I No. 1138

-- x u+ o- -8 = 0


25 3 + C(- os -1 cos -.6u 0

for the boundary condition there is retained

5w(O) = 5w(Z) = 8wxx(O) = 8wxx() = Bu(0) = 0

and, depending on whether or not the right-hand boundary point is pre-
scribed or not,

Bu(l) = 0 8U + cos 6) 0 (4.7)

It is recognized immediately that for f = 0, hence at the beginning of
buckling, the bar is in neutral equilibrium, for all conditions are satis-
fied by the solution (4.6). This result is trivial. It is not so directly
obvious, however, that also for f 0. there is a variation that makes
S271 a minimum; the homogeneous system of equations with homogeneous
boundary conditions permits of a non-vanishing solution also for f f 0.
In fact if the variation is so-performed that the second boundary con-
dition of (4.7') is satisfied there follows from the first of equations

bux =- cos 8w (4.8)

If this is put in the second of equations (4.7) the latter reduces to

i .W + C 12xx= (+ z 2 x = 0

and this equation (together with its boundary conditions) is satisfied 1y
89 = 5f sin -. For Bu there is obtained from (4.8)

NACA TM No. 1138

Iftff i 2Ixx
=u x + -1 sin---
212 21 I

in which 5f again represents the (not determinable by a system of homoge-
neous equations) arbitrary factor.

The question of the stability of the equilibrium positions below and
above the limit can now be answered.

1. Since the straight position (see equation (4.21)) is stable for
very small e, from considerations of continuity it is so for E < e*.

2. For e > 6* the straight position represents an unstable equi-
librium state, for a variation 5w, namely,

bw = 6f sin --

can be given for which 52 1 <0.

3. For e > c* the bent rod is against the variation

bw = 5f sin -(.
1 (4.9)
x f^f ( I 2nx>
Bu = x + sin -T-
21s 21 T

in neutral equilibrium.1 Since the variation (4.9) (and only this) makes
2e11 a minimum (of value zero) every other variation gives it a positive
value. If in some way the special variation (4.9) is prevented, then the
bent equilibrium position is stable. This holds especially in the impor-
tant case where not the force but the displacement of the end point is
prescribed; for the displacement system (.4.9) is in fact excluded by the
boundary condition Bu(I) = 0.

At this point a result discussed later in section Connection between-
the Ordinary Investigation of Stability and the Procedure Presented Here,
should be emphasized. The behavior of the rod beyond the stability limit
is different depending upon whether the load or the displacement is

1This result is naturally connected with the assumption that the
strut adheres strictly to the law of deformation established by the ex-
pression (2.7).

NACA TM No. 1138

considered as the prescribed quantity. More noteworthily, however, the
behavior at the stability limit is not affected by this difference. For,
although there are the two different boundary conditions (4.5)

bu(I) = 0
5ux(l) = 0

they both (together with the differential equation buxx = 0) lead to
the same result

Bu = 0

That is, it makes no difference whether a motion of the end points in the x-
direction is "permitted" or not: during the buckling they do not move.
Thus the result is arrived at that the two mechanically entirely different
problems: buckling under constant load and buckling under constant end
shortening, lead to exactly the same critical state bu, 5w, ecrit, Parit.


The results of sections The Differential Equations for the Displace-
ments u, w and Stable and Unstable Equilibrium can be illustrated very
elegantly if the variation problem is turned into an ordinary minimum
problem by the use of the Ritz method. In the case of the Ritz method
to be sure nothing certain can be said about the question of stability,
since from the start only quite definite displacements ar8 corn.Eiered and
thlerefor- r-.n. gn-el c:n b. c:oncli.- i .:'t the sln cf the second varia-
tion; nevertheless, with a Judiciously chosen deflection system the
question can be answered with great probability or the answer made very
plausible. In the present, especially simple case the earlier results
are found again exactly.

As a set of displacements satisfying all boundary conditions are
chosen,the solutions of the differential equations (3.2) and (3.1)

'This is a peculiarity of the problem. In general, the critical load
depends upon the boundary condj-tiorn whether are prescribed forces or
displacements, the system is supported, or guided, or built in, etc.; for
the "minimum" variation, from which the critical load follows, differs
according to the boundary conditions prescribed by the data of the problem.

NACA TM No. 1138

=z t2 g2f2
w= f sin -, u 2 + = + 2

Then the minimal condition

S 1I



2 1

2 2

dx + 2-

(Ce e*) +

wxx2 dx E

- -e = minimum

furnishes an equation for the "free value" f as a function of E

af 212 \ -1 2 0


The relationship between load and end displacement is obtained by means
of the stress-strain equation (2.3) from the second of equations (5.1);
it becomes

2 2
p n f


There are again the two possible solutions

f = O, p = Ee


Shence 2f p2
f V o, hence 2- = E*, p = E*

The question of stability is answered (with the above-mentioned
limitation) by the second variation. As hitherto, two cases are distin-

1. If the force is prescribed, e therefore left open, there are
two displacements to be varied, and the sign of the expression





NACA TM No. 1138


2 ---f8 + (8E)2
2f+ ((07

must be investigated..
with the coefficients

This expression is

2 2
-- E* -e
f2 212


a quadratic form in 8f,

+ 3 -

Since > 0, this expression is positive definite (that is,

never negative) as long as the discriminant

A -
f 2

( 2:&>2
85 2
af Iae


-e+ sr


is greater than zero. This is certainly the case below the critical point
(e < e*); here the system is therefore stable. On the other hand, A < 0
for f = 0 and e > e*; that is, the straight position is unstable above
the critical point. Finally, the bent position is neutral, because by
(5.6) the discriminant vanishes for f 0. This result agrees with that
of the previous section and is an expression of the fact that a buckled
strut in the elastic range can be bent arbitrarily farther without in-
-'creasing the load.

2. If the end displacement e is prescribed, only the quantity f
need be varied and it is found that:

S2 ( = 2 2 3 ) + f)322
2 2 2 -*. + ,



NACA W No. 1138

From this relation it follows fimmeilatealy thatt.

for e < e* 52A > 0

for' c'> e* and f j-0 62 > 0
for e > e* and f = 0 8t < 0 '

that is, the straight position is stable for s < c*, unstable for
e > e*, the bent position, as soon as it is mechanically possible (hence
for E > e*), alimys stable. Figure 4 shows the energy relationships in
this second case (prescribed displacement c = ae* of the right-hand end
of the rod). Plotted.as 'i

Ai 1 4 ( a-
(a-l)+ ---
<* 2 2

.as a function of f or = with or a= as a parameter.

(The. pbtentfal of the external forces is not included because it is not
affected by the minimal condition with respect to f.) It is seen that
the straight position (f = 0) is an equilibrium position under all
circumstances, for all curves start with a horizontal tangent. For
a < 1 associated with f = 0 is a minimum, for a > 1 a maximum; the
curves for a > 1 have further to the right also a minimum, whereby the
bent position f / 0 is characterized as a (stable) equilibrium position.
This figure shows especially well the "type" change of the curves in the
transition from the sub-critical to the supercritical region: the coin-
ciding 'of maximum and minim-an for a = 1. It is also clear here that,
although at the moment of transition the displacements are small, the
behaviQr of the body at the stability limit is nevertheless determined
b. y the "possibility" of greater deflections, expressed mathematically,
by-the existence of the f-terms of higher order in the expression for
* S '

*NACA TM N6. 1138



The ordinary stabiy theory is limited*.to an investigation of the
critical point. It was seen that the critical point is characterized by
two energy conditions'. The condition"

-. : -* ''-* *: :" 5&TI = 0 -

-forer advariatfoh Su, v, Sw (6.1)

characterizes it as an equilibrium position in general; the condition

52nI = 0

for a characteristic variation 5u, 5v, Sw (6.2)

as the critical one. Or, the critical point is distinguished by the fact
that.there a variation of the state of deformation can be made for which
the potential II remains unchanged to terms of the second order. Now
in .practical buckling problems it is usually a question of the transition
fripm.a-very simple (often independent of the coordinates) initial state
of stress .to .a comparatively very complicated one. It is therefore cuditm-
ary to .specify the initial .state-of stress directly without recourse to the
definition in terms of energy (6.1) and to proceed with the variation im-
mediately...in regard to the determination of the second state. There then
....remas as the-single important condition the statement (6.1), which can
beexpressed. in the form of a method-of-procedure as follows, for example:
Consider a system of infinitesimal distortions superimposed upon the
critical state of deformation, collect the parts of the potential energy
II quadratic in the added displacements and set the sum equal to'zero.
Thai such a procedure is at all possible rests upon the fact that as a
result .of the "large" initial stresses two types of quadratic term arise:
an'(alwayse positive) part that represents the work done by the stresses
caused by the added displacements, and a second part that comes from th6
work done by the stresses already present upon the quadratic part of the
added displacements. (See reference 2.)1

The fact that this statement-of-procedure concerning the vanishing
of the quadratic members is nothing more than the extended principle of
virtual displacements ("extended" in the sense of the statement about
S21I) does not come out clearly in the applications mostly for three
IIn equation (4.21), for example, the last two terms represent the
first type and the first term, the second type.


NACA TM No. 1138

1. Since a confusion with the very simple initial state is in general
not to be feared, it is possible to dispense with the designation bu,
Bv, Bw and write more briefly u, v, w for the added displacements.
This manner of writing does not express the fact that the added displace-
ments are to be not only small in the sense of the general hypotheses of
the theory of elasticity but also infinitesimal in the sense of the calcu-
lus of variations.

2. In close connection with the above, in considering the energy it
is customary to start not with the total potential II but directly with
the energy changes (appearing as the result of u, v, w) and to designate
these changes by1 A, V instead of by 5A, 8V; the (extended) principle
of virtual displacements becomes in this manner of writing

A + V_= O, or even A A = Aa (6.3)

since the potential difference V also represents the work of the
external forces on the infinitesimal2 displacements u, v, w. Equations
(6.3) can be put into words as follows: For the virtual displacement u,
v, w, through which the original equilibriuiA configuration goes over
into the neighboring ("buckled") configuration at the critical point.the
internal energy .Ai taken up by the system is'equal to the work done by
the external forces Aa taking into account the terms linear and quadratic
in, u, v, w. By this formulation the two conditions (6.1) and (6.2) are
combined into one; a procedure in which there is the danger of losing
sight of the difference between the (holding for any equilibrium position)
principle (6.1) and the (characterizing the critical position) extension

3. As the proper equation for the determination of the critical
system of virtual displacements u. v, w there follows (see sec. Stable
and Unstable Equilibrium) from (5.2) and the added requirement

52 1 > 0 for all other .u, 5v, 5w
the condition
85 21I) = 0 (6.4)

To distinguish 'them from those used earlier,' the quantities usually
written- Ai, A, VA are designated by Af, Aa, V.

2The second form of the law (6.3) theriTore does not represent the
special energy law Ai = Aa, by which is expressed the fact that for
conservative systems.the external work introduced by the transition from
the initial to the (not neighboring!) final state is stored up as elastic
energy in the body.

NACA TM No. 1138

If it is agreed to consider in A A,._ V on.y the (alone essential for
the critical behavior) quadratic terms, the condition.(6.4) is written
in the form

S(Ai + v) = o
(A -- A) = 0

This form, which is only a natural consequence of the original agree--
ment to write u, v, v instead of 3u, Bv, Fw, makes it quite clear to
what extent the simplified meaner of writing can lead to conceptual errors.
For the statement (6.4'), aside from the deceptive formal agreement, has
nothing to do with the principle of virtual displacement (1.2) or (1.3):
The principle (1.2), in content the same as the energy law (see the
Introduction), answers the question of the equilibrium positions under
given loads (or edge displacements), and equation (1.3) is a special form
of the same principle possible only in the realm of linearized elasticity
theory besides being very inexpedient1; equation (6.4'), on the other hand,
in content the same as the minimal condition (6.4) concerning the behavior
of the quadratic terms at the stability limit, gives the second equilibrium
position possible at the branch point and the sought-for value of the load
at which the equilibrium begins to be many-valued.

The difficulties so far discussed were difficulties in interpretation
arising from the symbolism of writing. There is another, more factual
circumstance that makes the question complex especially difficult to see
through. It was seen in section,Stable and Unstable Equilibrium that for
the rod t13re were two independent equations (4.4), with the likewise
independent boundary conditions (4.5), for the two added displacements
bu, 5w (which here would have been written u, w). From them it was
concluded that u vanished identically. This result and correspondingly
u = 0, v = 0 in the case of plates makes possible, when (as is tacitly
done in the stability theory of a bar) it is presupposed as known, a treat-
ment of the problems of bar and plate stability deviating from the general
methods of stability theory depicted above. Since, however, bars and plates
are the most well-known problems, being analytically the most tractable,
frequently ideas that were developed there are erroneously brought into

ISo, for instance, for the compressed strut below the critical point
twice the external work can be written in three forms: E12, pC, p2/E -
which is to be varied (with respect to ef)? The second form is meant,
but as a result of writing 2Aa in the place of -V that is no longer
uniquely discernible.

NACA TM No. 1138

more general stability problems. It is therefore necessary to examine
-more tho-ou.4ly the various special interpretations that can be given
to the occurrences at the stability limit in the case of bars and plates.

First of all outline the method by which it is necessary to proceed
according to the directions formulated at the beginning of this section.
If it is aseiuad a virtual displacement u, w at the critical point,
then, as can easily be seen1 the strain of a fiber to terms of the second
order is given by

x = x + --- z xx

Therefore the terms

= x zwxxT

of second order are: in the work done by the added

/ (ux -,xx) dx dy dz

in the work done by the already present (critical) compressive stress p
Ip --_ dx dy dz
.'~ p

(The external force
the second order.)
EF f (f
Ai =-a-(\ /

does work Pu(I); this makes no contribution of
After integration over y and z there results

^ i2 /Wxxc2 -

and the conditions (6.3) and (6.)1) become

Aa = 0

dx + 12 I w,2 dx

- wx dx = minimum = 0

The term ux /2 goes out in the expansion of. the radical (see
*1 xx
equation (2.2)), and the expansion of the curvature v

would give terms of the third order.




Vy2 dx)

NACA TM No. 1138

The expression coincides perfectly with the earlier expression (4-.2);
therefore the same differential equations and boundary conditions and es-
pecially the result- us 0 ,aro obtained, entirely independently of whether
a motion in the x-directipn of the right-hand end point during the buckling
is permitted or prevented. This double result (that u = 0, and that the
buckling is independent of the condition u(3) = 0 or ux(l) = 0) makes
possible the two following "customary" interpretations of the buckling

The first procedure consists in considering instead of the "natural"
problem, buckling under fixed load, the problem of buckling under fixed
end point and at the same time (what seems almost a natural consequence
of this stipulation) assuming from the start the vanishing of u also in
the interior. Hence there is superimposed upon the straight position
w = 0 a purely transverse displacement as a variation, keeping in mind
the presence of the still unknown longitudinal compressive stress -Eeo.
According to equation (2.2), as a consequence of the change in length
connected with the transverse displacement, the following stretching
energy is released

Ai = (--Eo) Jdx = w2 dx

at the same time a bending energy

A = T v 2 dx
1 2 2

must be added. This interplay between the two types of energy (and hence
the -two equilibrium positions) takes place when the values of Ai and A
are numerically exactly equal; that is, when

SAi =-T fw J X dX w dz ) 0 (6.6)
0 0

From the additional condition &A_ -= 0 there'follows as above the sine
equation.for w. This procedure thus leads to the correct end result
without, however, permitting. a guarantee of really having found the mini-
mal buckling load. For the "restraint" assumption u s 0 limits the
number of possible variations', and that it leads to the correct buckling
load for the rod (and plate) requires" at 'least a supplementary verifi-

NACA TM No. 1138

More important because in a still more special way a peculiarity
of the rol anid plate is another manner of thinking, which is almost
universally made the basis of derivation of the buckling equations. With
reference to the natural buckling process, the boundaries are considered as
movable; however and this is the characteristic mark of this method -
they cannot be allowed virtual displacement u (or u, v which, as has
been observed, would subsequently become zero) but are given a displace-
ment that is of a higher order of smallness (compared with w).

In the case of the beam it is customary to start this procedure with
the assumption that no additional stretching energy is taken up during
bending; that is, that the bent beam has the same length as the straight
one; it follows therefrom that as a result of the bending tihe ends must
approach each other by an amount

u- = ,wx2 dx (6.7)

(which in fact is of the second order in w!), so that the external forces
do the work poFu = / w2 dx. Now by formulating the quality

of inner and outer work (wherein by inner work is to be understood only
the bending energy)
EJ p oE 0
Ai = Aa or -- 2 X. dx 2 dx = 0
2 Jw 2 ,:,

and assuming as above the minimal property of this expression, thiE pro-
cedure leads to equation (6.5') naturally likewise without the u-term.1

In the assumption (6.7) there is anr. inconsistency: It cannot be
assumed a priori that a strut that changes its length elastically below
the buckling limit suddenly ceaees to do so beyond it. (In reality it
changes its length by quantities of higher orier.) It is more logical
to consider a perfectly incompresoable rigid against extension but
elastic in bending strut, for which the two hitherto independent dis-
placements u and w are related from the beginning by the (geometrical)
ux + 0 (6.7')
(Continued on p. 26)

NACA TM No. 1138

Since the procedure of equating the stretching energy to the external
work cannot be used in the case of the plate, a special auxiliary idea
has been used there in order to preserve the conceptually so similar idea,
that the boundaries are to move. (See reference 6.)

Without connecting the displacements u, v, w with each other nu-
merically proceed, in this method, from the assumption that u, v are
of a higher order of smallness than vw; that is, consider u, v not
as really independent virtual displacements but as connected with the
transverse displacement w by the order-of-magnitude condition 0

ux s Wx2 (etc.)

According to equation (2.2) there is obtained for the work done by the
critical stresses ax, ay, T on the displacements u, v, w to terms of
the "second order"

i = cxs fu x + dy + c oys vy

+ T lr

uy + vx + vwwy ix 'dy

+ dx dy


the bending energy is, as always, given by

Ai ) w)2 2(1l- )(wxx w v dx dy
12(1 2i) J)
(s = thickness)


(Continued from p. 25)
Such a strut permits no deformation at all below the critical load; above
it takes on only bending energy, which is furnished by the external work

lpCF WX2dx
Since up to the critical load no elastic deformation at all has taken place,
the two laws

Ai + V = 8(Ai + V) = 0 and

S(Ai + V) = &5 (A2 + V)

are here in content completely identical.


NACA TM No. 1138

SBoth parts together must be equal to the work Ag of the external force
in the sense of equation (6.3). The external work can now (and this is
the essence of Reissner's idea) be expressed generally in a very simple
manner, if it is remenbemedthatthe straight position is an equilibrium
position and that therefore in every virtual displacement A = Ai. On
taking the special displacement u* = u, v* = v, w* = 0 (with the bound-
ary displacements uj* = UR V = vR), A i *; therefore

A i = A f oxu + yVy + T (7 + V d id (6.8x)

and now on collecting terms in (6.8), the u- and v-terms cancel out;
there results the well-known Bryan plate equation (reference 1, p. 293),
exactly in the form obtained also under the assumption of purely trans-
verse displacements and Immovable boundaries.

The advantage of this method is that it offers the possibility of
formulating exactly the related presentation of a solution of the buckling
proceap by a boundary displacement. Its disadvantage is a double one:
The emphasizing of the boundary displacements gives the impression that
the participation of the external vork is universally important in a
* buckling process, which, as has been seen, is not so. But besides this it
is important for the entire consideration, just as for that of Bryan, that
u, v are of the second order with respect to w, which must be known
somehow beforehand;' therefore the interpretation of the external work
axux as a contribution of the second order is not transferable to
more general buckling problems. (See reference 6J

To summarize briefly the result of this section: In considering
the critical point it is-customary to dispense with the correct method
of writing the virtual displacements Bu, 8v, 8w in favor of the more
convenient u, v, w; thereby the connection between the customary
stability criterion and the principle of virtual displacements is con--
cealed. To be added is that the stability problem of the rod and of the
plate permits a special treatment which rests upon the fact that at the
critical point the tangential displacements u, v and the normal

1The very obvious conclusion, that can just be seen from the form
(2.2) of the strain that ux and Wx2 must be of the same order, is
not tenable; for an equation of the type (2.2.) holds, for instance, also
for the longitudinal, fibers of a cylinder, and yet here ux and wx
can become comparable because the tangential displacement v, which is
of the same order as w, is linearly coupled with u through the shear
and the transverse contraction.

2NACAW No. 1138

displacement w are of different orders ^of'magnitude'. Since, however,
the rod. or plate problem, 's the analytically simplest, is at the same
time the. best known, the need. easily arises of transferring methods of
thinking successful in these problems to more complicated problems, which,
as was to be shown,, is not possible.


In section Stable and Unstable Equilibrium, it was shown that for
the Euler strut the instability point (defined by 5(82II) = 0) coincided
with the branch point of the equilibrium. Branching problems are, however,
not the only kind of stability problem; a second'class, which is Just as
suited to the -energy definition of stability as are the brar.htirg
problems, comprises the so-called Dirchschlag problems.

In the Durchschlag problem the critical load is designated as that
load under which an (infinitesimal) displacement of the point of applica-
tion of the load is possible without an increase in the load, for which -
as in the branching problem there are therefore two (infinitesimally
close) equilibrium positions. Above the critical point an increasing
displacement is in general accompanied by a decreasing load the state
is unstable, the system snapsp" into1 or falls into a stable configuration.
*Prerequisite for such a phenomenon is a nonlinear relationship between
force and displacement even in the stable region.

The simplest Durchschlag problem is that of a slightly curved
beam under a transverse load. (See reference 7.)

If the ends of the initially curved beam are prevented from dis-
placing (fig. 5), then connected with the deflection caused by the
transver3e force Q is a shortening of the axis of the arc, as the
result of which a horizontal force R is made to act. Because the
effect of this (very large) compressive force upon the equilibrium of
forces in an element of the beam cannot be neglected, there arise
phenomena related to the buckling process in the Euler strut, insta-
bility phenomena.

Without carrying out all the details of the calculation (presented
completely elsewhere, reference 8) the principal method of solution for
this stability problem will be briefly outlined.

%In a manner similar to that in which a strut compressed beyond
the Euler limit at the least disturbance snaps or falls into the bent


NACA TM No. 1138

By a process that follows very closely that carried.
Energy of Deformation, is obtained, with the notation of
now taken positive downward), for the potential energy

2 40

S wx2 W )

out in section
figure 5 (w

2 1 2 x f
+ i Wxx dx- Qf




( ux


Wxx) ^ WUtex

+ i 2 w'xxX

I dx Q~f

there are obtained the two equilibrium conditions expressed in terms of
the displacements:

S +

W2 xx = 0

or, integrated once,



- WXw )= constant = h


and, with the use of equation (7.2)

1i xxxx + hwxx = hWxx


also the boundary conditions
u(O) = u(M) = w(O) = w() = wx(0) = wx(l) = 0

Wxxx(/2 + 0) wxxx(l/2 0) = Q/EJ


+ EF



NACA TM No. 1138

which together with the continuity requirements for

u, ux W, w, WV

at x = 1/2 give the 12' conditions that are necessary for the evaluation
of the 2 x 6 constants of integration in the 2 regions x 3 1/2.

The physical significance of the constant h can be recognized di-
rectly from (7.2): On the left-hand side is the stretching of the middle
line of the arch; therefore, to a factor 1/EF h is equal to the hori-
zontal force H, and (7.2) expresses the equilibrium condition that E
does not vary with x.

Just as in the case of the Euler strut the constant of integration
h = H/EF of the first equation enters into the second as the coefficient
of the unknown v; that is, the system (7.2) and (7.3.) is nonlinear.
(See equation (3.4) or (3.2)). Nevertheless, just as before the exact
solution can be given in this simple case without difficulty in terms of
the at first unknown

2 "-12h H EJx2
& H*- -
i2*2 H* 12

for example, for W = fo sin -:

2 3 ai-
a x Q sin a-- arx
Ssin -+ (7.5)
of0 _]- 22aEJ cos -

(x < 1/2)

and from (7.2) by another integration taking into account the boundary
conditions u(O) = u(l) = 0 there is obtained subsequently a transcen-
dental equation for the dependence of a upon Q and fo of the form:

S(x f W) dx (7.6)
EF I0 2

NACA.TM No. U138

The determination of the critical load can be carried out in two.
basically different ways.

The first method (see C. B. Biezeno,reference 7, pp. 21
from the condition 8Q/8f = 0, wherein the relation between
is established by equations (7.6) and (7.5) for x = 1/2

w(l/2) = f = fo


taIm sr
tan --
2 2

ff) proceeds
Q and f


A second method proceeds by way of the energy criterion (4.3).
For B2II is obtained from equation (7.1) after writing for brevity'

2W e + x + x

- Wx _- 1= -W (x,a)

the expression
52I = EF u ()+ 2 (W x 5uxWx -w (x, a)(&vx)
fL( 2 -
Th atiua dslceetsytm 5u 5 b hch0I

The particular displacement system Bu, 8w by which 52Il
equal to zero is obtained from

(521TI) = 0

that is, from the two homogeneous differential equations

B u + W Y x

Swx I=0




is Just made


- wx) Bux + i2Bxxx + W(a,x)Bwx = 0


Therein v is at first according to (7.5) a function of the tcw
parameters a and Q. Q for example, is considered as eliminated with
-the help of equations (7.6) and (7.51).

- _

NACA TM No. 1138

with the homogeneous. boundary conditions .

.. u() = u() = (O) = ) = (Z) = 8w(0) =8wv(l) = 0

The desired critical a-value is the lowest eigenvalue of the equations


Because of the great mathematical difficulties that equations (7.9)
present, the second method outlined is not suitable for an exact treat-
ment of the .problem but is well suited and therefore that procedure
will be considered here to an approximate treatment by the method of
Ritz or Galerkin.

This procedure can be started at either of two points: either,
make a Ritz approximation for 8w in (7.9), determine the correspond-
ing Su from the first of equations (7.9), and following Galerkin from
the condition 8(52II) = 0 obtain a (transcendental) equation for the
determination of a; or very much.more simply, if also necessarily
with a corresponding loss in accuracy introduce at the start
a Ritz approximation for w itself in place of (7.5) into the expression

It is well to use the second method but only indicate (reference 8)
the course of the calculation. If again

W = fo sin

is chosen and as a Ritz expression

xM 2irx
w = f, sin + fa sin -- (8.1)
2 2

then all boundary 'conditions are fulfilled, except for the one dis-
continuity requirement (7,4), the violation of which is however, un-
important. Further,by satisfying exactly equation (7.2) (obtained by
variation with respect to u) and calculating the horizontal force
E from (T.6), the integral in (7.1) can be evaluated and IT is
obtained as a function of the amplitudes fo, f1, f2 or the dimension-
less parameters

NACA TM No. 1138

7o = f
0 -

A2 =

(i = radius of gyration)


in the simple form

-t 4i4
n 1 EFi

",2 2+ 1622 +

Qj 2
EJ 2

1 413
q -
?o C


b4 Q
nafo H*



The equilibrium equations read

( 2AI + (?o 1) 2 ) qo = 0
O E 2,72


_N 22
= 321 2i hs02 ", = 0

They are in the two unknowns AN and N2 and of the third degree;
nevertheless a complete discussion is possible without numerical calcu-
lation, because the second equation may be written as a product

A2 ^ O 22 -- 8) = 0

Therefore the cases can be .distinguished

A 2=0

2X1 + (N- 7) ( 2- )=q o

and 7%a 0, that is,

2A2 = ?7o, .2/ 8
8No -- 67 = q o


- q 0A1M

NACA TM Wo. 1138

the discussion of which no longer requires any labor. The first system
provides a symmetrical deformation, the second a superposition of a sym-
metrical and an antisymmetrical deformation. The corresponding horizontal
compressive forces become

SH* NO / 2




The critical load q

F II (X 3) 2

As long as

crit is found from the condition

+2 (5 ,? 8 ).+ f (2) = 0
()Yk ~ ,. jy

and. the discriminant

()2fi )2
7,2 X 2 '.

are greater than zero, 52II as a positive-definite quadratic form in
5xi and 8A2 cannot be zero for any combination of these two variables.
Vanishing of the discriminant characterizes the pair of values Ax, 'M
for which there is exactly one combination AI, 052 for which 8211
becomes zero but none for which it is less than zero. The condition

22 2
on 9 2


gives, therefore, the stability limit and together with the two equations
(8.3) determines the three unknowns xi, 72, and qcrit. In this case
(8.7) reads


2' 2
( uN
- K^2


) = 0

NACA TM No. 1138

4 [8 -(Now

- 22 + ( -

+ 22 16X22a 0 )2 = 0

which condition becomes

A2 = 0


418- (Q 0

2- L2+

(^o- )2 .

and with


- 48 'o\- .2/2-) = 0

There are therefore two sets of values for A;i, ?a,

qcrit = 2 +

and qcrit

" 4\3/
3 /


72 = 0, 71 = No A/ 16,

qcrit = 2 Po 02 16

by which a critical state of the elastic system is characterized.

The physical significance of the relations (8.8), (8.9) and espe-
cially the (in this case unstable) behavior above the critical load vill
not be pursued in detail (see reference 8). (See fig. 6.) There the
load Q is plotted against the deflection f, of the point of application,

2 ./

X2 = 0, 7%i

7k)o .Xo2 4


2 ,2 = X0 o-


with the initial amplitude fo as a parameter. For fo/i = 70 2 Q.
increases monotonically with fl; instability is -not possible. In the

under the critical load given by (8.8) there enters definite instability,
increasing deflection f, without increase of load' -.Accordingly the
beam snaps under constant load until it finds a stable configura-
tioni at E7'. Since Az' > No the beam is now convex downward; it also
can be seen clearly that the system now-must be stable with respect to
an increase in load; a further deflection results in a longitudinal
pull. (See equation (8.4').)


AO -"22 (8.9')

the critical load is given by (8.9). Before the external load can
assume the value (8.8), the longitudinal compression according to (8.5')
reaches the value 4H*, that is, the second Euler load, under which the
strut assumes the S-shape configuration 'g J'0. It snaps again into
a stable position A?', this time, however, passing through an un-
symmetrical deformation. At the critical load there appears a branch-
ing of the elastic equilibrium; figure 6 shows the two branches of the
Q f, curve, both of which however -.and this is the noteworthy differ-
ence from the Euler problem are unstable.

For further details see the publication referred to. Here it was
just a question of.presenting the chain of ideas that led to the deter-
mination of the critical loads (8.8) and (8.9), in order to show the
application of the general stability criterion (4,3) to a stability
problem of an entirely different kind.

That is, at first vibrates about Mz' as a stable equilibrium

NACA TM No. 1138


1. Handbuch der Physik. Bd. 6 (Berlin), pp. 70 ff., 1928.:

2. Trefftz, E.: Zur Theorie der Stabilitit. Z.f.a.M.M. -Bd.-13, 1933,
p. 160.

3. Poschl, Z. B. Th.: Uber die Minimalprinzipe der Elastizitatstheorie.
Bau Ing. Bd. 17, 1936, p. 160.

4. Marguerre, Karl: The Apparent Width of the Plate in Compression.
NACA TM No. 833., 1937. -

5. Marguerre, Karl: Uber die Behandlung von Stabilitateproblemen mit
Hilfe der energetischen Methode. Zf.a.M.M. Bd. 38, 1938,
pp. 70 ff.

6. Reissner, H.: Z.f.a.M.M. Bd. 5, 1925, p. 475.'

7.-Biezeno, C. B.: Das Durchschlagen'eines schwach gekrummten Stabes.
Z.f.a.M.M. Bd. 18, 1938, p. 21.

Brazier, L. G.: The Flexure of Thin Cylindrical Shells and Other
"Thin" Sections. R. & M. No. 1081, British A.R.C., 1927.

Heck, 0. S.: The Stability of Orthotropic Elliptic Cylinders in Pure
Bending. NACA TM No. 834, 1937.

Weinel, E.: Ober Biegung und Stabilitat eines doppelt gekrumnten
Plattenstreifeus. Z.f.a.M.M. Bd. 17, Dec. 1937, pp. 366-369.

8. Marguerre, Karl: Die Durchschlagskraft eines schwach gebkr'smten
Balkens. Sitzunieberichte der Berliner Mathemiatischen Gesellschaft.
Bd. 37, June 1938, pp. 22-40

NACA TM No. 1138


Equation 3.7, last ,pat -

Trans, note: ?It appevrx that this equation, .ehuld ,be

.. f2 2ax ;.f2n2
*u 6* in -X*- .
x 8 1. 41 .

Equations (7.5) and (7.5')
Q} Q3
Trane'- note: It appears .that. the term shouXd be .,

Equation (8.8)
r1 /ot It -a"
Trae.,*note-., It- appears that 2 + 0 4I.... : hod'be
NO -3-

2+ o
,- o ,

Page 24

Trans. note:

It appears that Ai xx dx should be

OV EJ fr 2,


Page 36

Trans. note: It appears that (8.4') should be (8.5').

NACA TM No. 1138

1.o -- ~M )=CI

F.igure 1.- StrL.t under compression.

-j /

0 2 3
0 1 2 3

Fig-ure 2.- Amplitude f

against e.

I 4

0 1 2

FiLcure 3.- Load P against c.

NACA TM No. 1138

2.5 2'.

0 0 .5 1 1.5 2
= f/2i

Figure 4.- Variation of energy of deformation
with the amplitude f z 2it. Edge
compression = a* as a parameter.

Moment of inertia J, Section F.
Radius of gyration i, Elasticity modulus E

Figure 5.- Slightly curved beam under
transverse load Q-.


10-1- 7 -

/ .
8 --->/-'=6-- .--.-- --__-

0 6-" -.\L


0 2 3 4 6 8 9
-1 0 \ / 1

Figure 6.- Variation of the load 0 with the
displacement fl bf the point of
application, parameter = initial amplitude


II 11 11 0 10 6 II1 2
.3 1262 08106 308 2

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