Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 161-169, Warsaw 2007

ON THE STABILITY BEHAVIOUR OF ELASTIC BEAMS

UNDER INTERNAL PRESSURE

Lena Zentner

Valter Böhm

Hartmut Witte

Faculty of Mechanical Engineering, Technical University of Ilmenau, Germany

e-mail: Lena.Zentner@tu-ilmenau.de; Valter.Boehm@tu-ilmenau.de; Hartmut.Witte@tu-ilmenau.de

Scientific reception of the term ”stability” stresses steady adaptation to
its changing fields of application. Nevertheless, the determination of cri-
tical forces remains one of themain tasks of stability theories.We exem-
plify some classes of the stability loss in beams under internal pressure
for the static case. Additionally, we analyse in more detail the dynamic
stability of beams under internal pressure and demonstrate means to
keep an equilibrium.

Key words: stability, elastic beam, dynamic modelling

1. Static cases of loss of stability

Bestknownexamples for the loss of stabilityunder static loads are theEulerian
cases of stability (Leipholz, 1968). For undercritical loads, the equilibrium
is determinate, while at critical loads bifurcations in the solutions to state
equations occur. Solutions are nomore bijective, one load situation may lead
to more than one possible geometric configuration of the system. Figure 1
exemplifies this phenomenon in the case of a half-cycle shaped bending beam
subject to loading bya single forcewith constant direction (conservative force)
but with a 2D-free floating location of the site of application under load. Two
possible trajectories of this point and two realisations of the equilibrium are
illustrated. Solutions have been determined numerically, a current application
is the design of compliant grasping devices.

Another well-known problem of the loss of stability is buckling. As an
example, Fig.2 shows a valve coherent in thematerial; due to symmetry only
one half of it is drawn. Other examples for statical stability of fully compliant
mechanisms are presented in Huba et al. (2002).



162 L. Zentner et al.

Fig. 1. Equilibrium situations of a half-cycle shaped beam under external load. A
force constant in amount and direction traces the free end of the beam

Fig. 2. A valve using compliant mechanisms: pressure-displacement relation;
(a) principle of the valve, (b) configuration of the membrane below the lower

pressure threshold of the two-point switchmechanism (”open”), (c) configuration of
the membrane above the threshold (”closed”)

Thevalve serves to prevent region (4) inFig.2 frompressure overload. Due
to different diameters and thus different cross–sectional areas in regions (1)
to (3), different pressures occur. Depending on pressure differences along the
flow line from (1) to (3), elasticmembrane (5) is deflected towards plate (6). If
the pressuredifference between (1) and (3) exceeds a predefined threshold, the
valve closes. Calculation of the stress strain relations in themembranemay be



On the stability behaviour of elastic beam... 163

based on a quasi-static attempt, if conditions allow to neglect inertial effects
of the fluid.

High deflections provoke only small stresses and strains, functions and
shape constancy are guaranteed for a large number of cycles, providing short
andconstant switching timeswith constantdiameters of thevalve inall ”open”
states. Figures 2b and 2c compare the unswitched and switched states as far
as the geometry is concerned. Additionally, an example of this problem is
described in Huba et al. (2002).

2. Dynamic cases of loss of stability

Anumber of systems, especially non-conservative ones,may only insufficiently
be described by static approaches (Djanelidse, 1958). For example, if a stra-
ight bending beam is subject to a load like illustrated in Fig.3, calculations
based on static approaches do not identify unstable situations, which, never-
theless, may be found when using dynamic methods. Below the threshold of
the critical load, one stable equilibrium exists. The critical load induces mo-

Fig. 3. Deformation of compliant beam structures loaded by external forces and/or
bending moment applied to the free-moving end of the beam. Resistance is provided
by elasticity of the beam in combination with internal pressure and/or influence of

filaments

tion, whichmay lead to stable situations as well as to unstable ones.Methods
to quantitatively identify these situations and to derive related stability crite-
ria are mostly lacking in literature, thus development of methodology in this
field is an important task for the future. Our contribution to this duty will
be the dynamical analysis of a hollow beam under the influence of internal
pressure and of non-conservative forces. The technical application we aim at
is a pneumatic ”finger” for grasping, especially for manipulation. Due to the
underlying technology, wewould like to know the influence of internal pressure



164 L. Zentner et al.

or length-constant filaments in the wall of the structure on the stability beha-
viour of the structure. The effects of internal pressure on deformationmay be
represented by one force and a bendingmoment acting on the moving end of
the beam.
The deformation of the beam is expressed as a function of time and spacial

coordinate x (Fig.4). The external force is provoked by a point-like mass m
attached to the moving end of the beam, which performs small movements v
parallel to the y-axis. Thus, redistribution of the mass reduces the kinetic
energy balance to the analysis of states of the point-likemass, without neglec-
ting the elasticity of the beam. As a consequence, the reducedmodel provides
higher amplitudes and smaller critical loads than amore complexmodel based
oncontinuummechanics.Thus, the results of the calculations assureadditional
security, especially as far as critical loads are concerned.

Fig. 4. A dynamic case of loading of the beam by non-conservative forces and the
bending moment

The force acting on the beam generated by the point-like mass is

Fk =−mv̈l (2.1)

The bendingmoment of the beam M =EIzv
′′ may be expressed as

EIzv
′′ =F(vl−v)+(Fk −Fθlδ)(l−x)+Ml (2.2)

In further considerations a dynamic model of the material will be used.
Suppose that Fk, vl and θl are independent of x, then the solution of the

previous equation is

v(x) =Asinλx+Bcosλx+
(
θlδ−

Fk
λ2EIz

)
x+(vl−θlδl)−

Fkl+Ml
λ2EIz

(2.3)

with

λ2 =
F

EIz

The bendingmoment Ml is supposed to be proportional to the angle θl by a
factor k

Ml =−kθl (2.4)



On the stability behaviour of elastic beam... 165

In combination with boundary conditions v(0) = 0 and v′(0) = 0, these
equations yield

v(x) =
(
θlδ−

Fk
λ2EIz

)(
x−
sinλx

λ

)
+
(
vl−θlδl−

Fkl+Ml
λ2EIz

)
(1−cosλx) (2.5)

Taking into account the two additional boundary conditions v(l) = vl and
v′(l)= θl allows the derivation of the following equations

vl =
(
θlδ−

Fk
λ2EIz

)(
l−
sinλl

λ

)
+
(
vl−θlδl−

Fkl+Ml
λ2EIz

)
(1− cosλl)

(2.6)

θl =
(
θlδ−

Fk
λ2EIz

)
(1− sinλl)+

(
vl−θlδl−

Fkl+Ml
λ2EIz

)
λsinλl

Knowing conditions (2.4) for the moment and force Fk, we eliminate θl and
get the equation of motion for the point-like mass

v̈l+ω
2vl =0 (2.7)

with the natural frequency ω

ω2 =
λ3EIz

(
λ(δ−1)cosλl− k

EIz
sinλl− δλ

)

mBigl[
(
2 k
EIz
+λ2l

)
cosλl−λ

(
1− k

EIz
l
)
sinλl−2 k

EIz

] (2.8)

The sign of ω2 6=0 decides the solution to equation (2.7)

vl =

{
K1 sinωt+K2cosωt for ω

2 > 0

K3 sinhωt+K4coshωt for ω
2 < 0

(2.9)

In the case of ω2 > 0, the amplitude is constant. For ω2 < 0 the solution
tends to infinity. The sign of ω2 may change if the sign of the denominator or
of the numerator in (2.8) changes. These events occur if

λ(δ−1)cosλl−
k

EIz
sinλl− δλ=0 (2.10)

or if (
2
k

EIz
+λ2l

)
cosλl−λ

(
1−

k

EIz
l
)
sinλl−2

k

EIz
=0 (2.11)

Introduction of dimensionless parameters F̃ = Fl
2

EIz
and k̃= kl/(EIz) facilita-

tes the further procedure.We transform λ and k to dimensionless parameters

λ=

√
F̃

l
k=
k̃EIz
l



166 L. Zentner et al.

Using (2.10) and (2.11), we get relationships for k̃(F̃) (Fig.5)

k̃=−

√
F̃

sin
√
F̃
[cos

√
F̃(1− δ)+ δ]

(2.12)

k̃=
sin
√
F̃ − cos

√
F̃

√
F̃ sin

√
F̃ +2cos

√
F̃ −2

Fig. 5. Critical loads from equation (2.12)1 (dashed line) and (2.12)2 (solid line) in

the (k̃, F̃) plane

The asymptote of the graph of equation (2.12)2 coincides with the second
asymptote described by equation (2.12)1 and is determined from the formula

√
F̃ sin

√
F̃ +2cos

√
F̃ =2k̃

and the bendingmoment are indeterminate for F̃ =0. In Fig.5, the straight
line of (2.12)2 intersects the function derived in equation (2.12)1. Consecutive-
ly, an a priori exclusion of the critical loads is impossible. In the case of k̃=0,
a simple change of the force direction increases the critical force by a factor of
ten. Dashed line I in Fig.6 illustrates the strategy to avoid instability. Line II
shows the positive effect of an additional external moment on the avoidance
of the critical loads.
Graphic representations of k̃(F̃,δ) allow the derivation of strategies to

avoid the critical forces by guiding the processes on ”uncritical pathes”. One
of those solutions is given by the dashed line in Fig.7. The price to be pa-
id for this avoidance is control: preemptive (planning) or current (Zentner,
2003).
Introduction of a bending moment acting as an extended load not only

providesmore general solutions to stability problems, but also offers access to
new strategies for the avoidance of instable phases in the processes. Superpo-
sition of loads helps in ”shifting” the critical loads into uncritical zones of the



On the stability behaviour of elastic beam... 167

Fig. 6. Critical loads from equation (2.12)1 (dashed line) and (2.12)2 (solid line) in

plane (δ,F̃) for k̃=0 (line I) and k̃=15 (line II). Adaptive change of loads allows
for avoidance of critical loads

Fig. 7. Critical loads, determined by equations (2.12); dashed line: one possible
strategy for the guidance of the loading process avoiding critical situations

load condition, as illustrated in Fig.6. Physical realisation of such additional
loads enable one to use embedded filaments in the basis structure (Fig.3),
which provokemoments under an external force. Thus, themoment is realised
under dangerous load conditions, without the use of additional sensors.

Our results cover those of Djanelidse (1958) for a static case without an
external bending moment (k̃ = 0). For k̃ = 0 and δ = 0, equation (2.12)1

provides a formula cos
√
F̃ =0, where the eigenvalues equal the critical force

in the Euleian sense.



168 L. Zentner et al.

3. Conclusions

Ourattempts tomodel thebehaviourof compliantmechanisms, likepneumatic
manipulator ”fingers”, mechanically identified the needs to extend methods
for the determination of critical loads. For this purpose, we analysed beams
under a load by internal pressure in combination with external forces and/or
bending moments. In some loading situations of the pneumatic ”finger”, the
analysis of the derived equations allowed a multifold increase of the critical
loads. Guidance of the loading process was required to trace uncritical regions
of parameter combinations describing the equilibrium situation.

Of special impact on the increase of the critical loads is the controlled
overlay of bending moments onto the forces applied. In situations with given
external loads, this overlay allows the minimisation of structural weight.

References

1. DjanelidseG., 1958,Obustojchiwosti sterschnjapri dejstwii sledjatschej sily,
Trudy Leningradskogo Politechnicheskogo Instituta, 192

2. Huba A., Molnár L., Takács Á., 2002, Hohlraumsondemit wurmförmiger
Bewegung, ISOM 2002. Advanced Driving Systems, Ist. Internat. Symp. on
Mechatronics, Chemnitz, 466-470

3. HubaA.,Molnár L., 2002,Dynamicmodels of silicone rubbers based on the
synthesis method, Materials Science, Testing and Informatics, Trans. Tech.
Publications Uetikon/Zrich, 95-100

4. Leipholz H., 1968, Stabilitätstheorie, B.G. Teubner, Stuttgart

5. ZentnerL., 2003,Untersuchung undEntwicklung nachgiebiger Strukturen ba-
sierend auf innendruckbelasteten Röhren mit stoffschlüssigen Gelenken, Ilme-
nau ISLEVerlag, ISBN 3-932633-77-6

O stateczności podatnych belek poddanych obciążeniu ciśnieniem

wewnętrznym

Streszczenie

Naukowepodejście do kwestii stateczności cechuje niezmienna adaptacja tego ter-
minu do różnych dziedzin zastosowań. Niemniej, wyznaczanie obciążeń krytycznych



On the stability behaviour of elastic beam... 169

wciąż pozostaje jednym z głównych zadań teorii stateczności. W pracy zaprezento-
wano kilka przykładów różnych typów utraty stateczności w belkach poddanych ob-
ciążeniu ciśnieniem wewnętrznym w przypadku statycznym. Dodatkowo, szczegóło-
wo przeanalizowano zagadnienie stateczności dynamicznej takich układów i pokazano
sposoby utrzymywania ich w równowadze.

Manuscript received August 3, 2006; accepted for print August 18, 2006