Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading
The out-of-plane stability of tapered cross-sectioned thin curved beams under uniformly distributed radial loading is investigated by using the Finite Element Method. Solutions referred to as Bolotin’s approach are investigated for the dynamic stability analysis and the first unstable regions are...
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| Zitieren: | Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading / M. Sabuncu, H. Osturk, S. Cimen // Прикладная механика. — 2010. — Т. 46, № 11. — С. 133-144. — Бібліогр.: 27 назв. — англ. |
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Sabuncu, M. Osturk, H. Cimen, S. 2016-02-26T19:02:59Z 2016-02-26T19:02:59Z 2010 Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading / M. Sabuncu, H. Osturk, S. Cimen // Прикладная механика. — 2010. — Т. 46, № 11. — С. 133-144. — Бібліогр.: 27 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/95458 The out-of-plane stability of tapered cross-sectioned thin curved beams under uniformly distributed radial loading is investigated by using the Finite Element Method. Solutions referred to as Bolotin’s approach are investigated for the dynamic stability analysis and the first unstable regions are examined. Out-of-plane vibrations and out-plane buckling analyses are also considered. In addition, the results obtained in this study are compared with the results of other investigators in existing literature for the fundamental frequency and critical lateral buckling load. The effects of subtended angle, variations of cross-section and dynamic load parameter on the stability regions are shown in graphics. За допомогою методу скінчених елементів проаналізовано бокову стійкість викривленої звуженої у кінці тонкої балки при однорідно розподіленому радіальному навантаженні.Розв‘язки у вигляді наближення Болотіна досліджені в рамках аналізу динамічної стійкості. Також вивчені бокові коливання і бокове випучення балки. Отримані результати щодо основної частоти та критичного навантаження при боковому випученні порівняні з іншими опублікованими результатами. На рисунках показано вплив утвореного дугою кута зміни поперечного перерізу та параметру динамічного навантаження на область стійкості. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading Анализ боковой динамической устойчивости искривленных балок при однородно распределенном радиальном нагружении Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| spellingShingle |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading Sabuncu, M. Osturk, H. Cimen, S. |
| title_short |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| title_full |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| title_fullStr |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| title_full_unstemmed |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| title_sort |
out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading |
| author |
Sabuncu, M. Osturk, H. Cimen, S. |
| author_facet |
Sabuncu, M. Osturk, H. Cimen, S. |
| publishDate |
2010 |
| language |
English |
| container_title |
Прикладная механика |
| publisher |
Інститут механіки ім. С.П. Тимошенка НАН України |
| format |
Article |
| title_alt |
Анализ боковой динамической устойчивости искривленных балок при однородно распределенном радиальном нагружении |
| description |
The out-of-plane stability of tapered cross-sectioned thin curved beams under
uniformly distributed radial loading is investigated by using the Finite Element Method.
Solutions referred to as Bolotin’s approach are investigated for the dynamic stability analysis and the first unstable regions are examined. Out-of-plane vibrations and out-plane buckling analyses are also considered. In addition, the results obtained in this study are compared with the results of other investigators in existing literature for the fundamental frequency and critical lateral buckling load. The effects of subtended angle, variations of cross-section and dynamic load parameter on the stability regions are shown in graphics.
За допомогою методу скінчених елементів проаналізовано бокову стійкість викривленої звуженої у кінці тонкої балки при однорідно розподіленому радіальному навантаженні.Розв‘язки у вигляді наближення Болотіна досліджені в рамках аналізу динамічної стійкості. Також вивчені бокові коливання і бокове випучення балки. Отримані результати щодо основної частоти та критичного навантаження при боковому випученні порівняні з іншими опублікованими результатами. На рисунках показано вплив утвореного дугою кута зміни поперечного перерізу та параметру
динамічного навантаження на область стійкості.
|
| issn |
0032-8243 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/95458 |
| citation_txt |
Out-of-plane dynamic stability analysis of curved beams subjected to uniformly distributed radial loading / M. Sabuncu, H. Osturk, S. Cimen // Прикладная механика. — 2010. — Т. 46, № 11. — С. 133-144. — Бібліогр.: 27 назв. — англ. |
| work_keys_str_mv |
AT sabuncum outofplanedynamicstabilityanalysisofcurvedbeamssubjectedtouniformlydistributedradialloading AT osturkh outofplanedynamicstabilityanalysisofcurvedbeamssubjectedtouniformlydistributedradialloading AT cimens outofplanedynamicstabilityanalysisofcurvedbeamssubjectedtouniformlydistributedradialloading AT sabuncum analizbokovoidinamičeskoiustoičivostiiskrivlennyhbalokpriodnorodnoraspredelennomradialʹnomnagruženii AT osturkh analizbokovoidinamičeskoiustoičivostiiskrivlennyhbalokpriodnorodnoraspredelennomradialʹnomnagruženii AT cimens analizbokovoidinamičeskoiustoičivostiiskrivlennyhbalokpriodnorodnoraspredelennomradialʹnomnagruženii |
| first_indexed |
2025-11-25T21:40:34Z |
| last_indexed |
2025-11-25T21:40:34Z |
| _version_ |
1850559934491525120 |
| fulltext |
2010 П РИКЛАДНАЯ МЕХАНИКА Том 46, № 11
ISSN0032–8243. Прикл. механика, 2010, 46, № 11 133
M . S a b u n c u 1 , H . O z t u r k 2 , S . C i m e n 3
OUT-OF-PLANE DYNAMIC STABILITY ANALYSIS OF CURVED BEAMS
SUBJECTED TO UNIFORMLY DISTRIBUTED RADIAL LOADING
Dokuz Eylul University, Faculty of Engineering, Department of Mechanical Engineering,
35100 Bornova, Izmır, Turkey; e-mail: 1 mustafa.sabuncu@deu.edu.tr;
2 hasan.ozturk@deu.edu.tr; 3 serefcimen@gmail.com
Abstract The out-of-plane stability of tapered cross-sectioned thin curved beams under
uniformly distributed radial loading is investigated by using the Finite Element Method.
Solutions referred to as Bolotin’s approach are investigated for the dynamic stability analy-
sis and the first unstable regions are examined. Out-of-plane vibrations and out-plane buck-
ling analyses are also considered. In addition, the results obtained in this study are compared
with the results of other investigators in existing literature for the fundamental frequency and
critical lateral buckling load. The effects of subtended angle, variations of cross-section and
dynamic load parameter on the stability regions are shown in graphics
Key words: dynamic stability, finite elements, buckling, curved beam, vibration.
§1. Introduction.
Curved beams are used in high technology applications especially in turbine blades,
bridges and outer space industry and this has made elastic instability a problem of great im-
portance. The problems, which are examined in this branch of elasticity, are related to those
in the theory of vibrations and the stability of elastic systems. The dynamic stability of me-
chanical systems according to Bolotin’s definition [4], represents a specific stability of mo-
tion. When Bolotin’s approach [4] examined, three stages, static stability (buckling analy-
sis), vibration analysis and dynamic stability analysis, are seen to be included in the equa-
tion of dynamic stability. Therefore, these three stages are studied in this paper. Some of the
research studies on the dynamic and static stabilities of structures that have been carried out
by other investigators can be summarized as follows:
Timoshenko and Gere [21] studied the buckling analysis of hinged-hinged Bernoulli-
Euler curved beams by using the analytical method. Papangelis & Trahair [15] developed a
flexural-torsional buckling theory for circular arches of doubly symmetric cross section.
Non linear expressions for the axial and shear strains were derived, and these were substi-
tuted into the second variation of the total potential to obtain the buckling equation. Banan
et al [1] discussed the Finite Element analysis for the buckling analysis of curved beams on
elastic foundations for both in-plane and out-plane. Bazant and Cedolin [2] examined the
buckling analysis of curved beams by employing analytical and energy methods. Yoo et al
[27] performed buckling analysis of curved beams by using the finite element method.
Tüfekci and Arpacı [22] studied exact analytical solutions for in-plane static problems of
planar curved beams with variable curvatures and variable cross-sections which are derived
by using the initial value method. The governing equations include the axial extension and
shear deformation effects.
Ojalvo & Newman [13], presented the frequencies of a cantilever ring segment. Sabir
and Ashwel [17] discussed the natural frequency analysis of circular aches deformed in
plane. The finite elements developed by using different types of shape functions are em-
ployed in their analysis. Petyt and Fleischer [16] analyzed free vibrations of a curved beam
mailto:hasan.ozturk@deu.edu.tr
mailto:serefcimen@gmail.com
134
for various boundary conditions. Belek [3] studied the vibration characteristics of a symmet-
rical cross section curved packet. The effect of disc curvature on the frequencies of vibration
was investigated and was shown that the vibration characteristics of a multi-bladed packet
can be predicted from the in-plane and out-of-plane inference diagrams of a symmetric
cross section two bladed curved packet. Sabuncu [18] investigated the vibration analysis of
thin curved beams. He used several types of shape functions to develop different curved
beam finite elements and pointed out the effect of displacement functions on the natural
frequencies by comparing the results. Silva and Urgueira [19] studied the dynamic stiffness
matrices for the out-of-plane vibration of curved beams using the Lagrange’s equations and
the dynamic equilibrium equations, respectively, and then solved for the natural frequencies.
Kang et al [7] discussed computation of the eigenvalues of the equations of motion govern-
ing the free in plane vibration including extensibility of the arch axis and the coupled out-of-
plane twist-bending vibrations of circular arches using Differential Quadrature methods
(DQM). Kawakami et al [8] derived the characteristic equation by applying the discrete
Green functions and using the numerical integration to obtain the eigenvalues for both the
in-plane and out-of-plane free vibrations of the non-uniform curved beams, where the for-
mulation is much complicated than that of the classical approaches. Yıldırım [26] performed
the in-plane and out-plane free vibration analysis with the program developed using the
transfer matrix method of a double-side symmetric and elastic curved beam. Huang et al [6]
investigated a systematic method for analyzing the out-of-plane dynamic behaviours of non-
circular curved beams the governing equations of which take into account the effects of
shear deformation, rotary inertia and viscous damping. Wang and Sang [23] derived an ana-
lytic method of examining the out-of-plane vibration of continuous curved beams on peri-
odical supports. Lee and Chao [10, 11] examined out-plane vibration of curved beams with
a non-uniform cross-section for constant angle and two physical parameters are introduced
to simplify the analysis, and the explicit relations between the torsional displacement, its
derivative and the flexural displacement were derived. Wu and Chiang [24] investigated the
natural frequencies and mode shapes for the radial bending vibrations of uniform circular
arches by means of curved beam elements. In this study, the standard techniques were used
to determine the natural frequencies and mode shapes for the curved beam with various
boundary conditions and subtended angles. An improved shear deformable thin-walled
curved beam theory to overcome the drawback of currently available beam theories was
newly proposed for the spatially coupled free vibration and elastic analysis by Kim and Kim
[9]. The governing differential equations for the free in-plane vibration of uniform and non-
uniform curved beams with variable curvatures, including the effects of the axis extensibil-
ity, shear deformation and the rotary inertia, were derived using the extended-Hamilton
principle by Yang et al [25].
Thomas and Abbas [20] examined dynamic stability of an axial periodic loaded Ti-
moshenko beam using the finite element method and Bolotin’s approach [4]. In this study
the effect of shear deformation and rotary inertia on the static buckling loads and the regions
of dynamic instability were studied. Nair et al [12] investigated the dynamic stability of a
curved rail under a constant moving load using a linear theory. Fukuchi and Tanaka [5]
studied a characteristic analysis on quasi-periodic and chaotic behavior of a circular arch
under follower forces with small disturbances. In this study, the stability region chart of the
disturbed equilibrium in an excitation field was calculated numerically. Ozturk, et al [14]
have investigated in-plane stability analysis of non-uniform cross-sectioned thin curved
beams under uniformly distributed dynamic loads by using the Finite Element Method. In
this study, two different finite element models, representing variations of cross-section, were
developed by using simple strain functions in the analysis.
This paper presents the out-of-plane stability analysis of tapered cross-sectioned thin
curved beams under uniformly distributed radial loading by using the Finite Element
Method. Solutions referred to as Bolotin’s approach are investigated for the dynamic stabil-
ity analysis and the first unstable regions are examined. Since natural frequency and buck-
ling load effect the determination of stability regions, the out-of-plane vibration and lateral
buckling analyses are also studied. In addition the results obtained from this study are com-
pared with the results of other investigators in existing literature for the fundamental fre-
quency and critical lateral buckling load. The effects of subtended angle, variations of cross-
section and dynamic load parameter on the stability regions are shown in graphics.
135
§2. Models of Curved Beams.
The model shown in Fig. 1is used for the
Finite Element analysis of curved beams. In
this study, C1, C2, C3, C4 and C5, represent
different type cross-sectioned curved beams.
The explanation of these cross-sections is as
follows:
C1: Uniform ( , ;R t R tt t b b= = Fig. 2a );
C2: Unsymmetric tapered with constant
width ( , ;R t R tt t b b≠ = Fig. 2b.),
C3: Double unsymmetric tapered ( ,R tt t≠
;R tb b≠ Fig. 2c.),
C4: Symmetric tapered with constant width ( , ;R t R tt t b b≠ = Fig. 2d.),
C5: Double symmetric tapered ( , R t R tt t b b≠ ≠ Fig 2e.).
The boundary conditions and applied loading on a curved beam are shown in Fig. 3.
The periodic uniformly distributed dynamic load is 0( )P t P= + costP tω+ where ω is the
disturbing frequency, the static and time dependent components of the load can be represented as
a fraction of the fundamental static buckling load crP ,
hence ( )P t becomes,
( ) ( cos ).cr dP t P tα β ω= + (2.1)
The displacement functions of the curved beam
were defined as [3];
1 2 3 4 6cos sin ;u a R a R a R a R a Rφ φ φ φ= + + − − (2.2)
1 2 5 6cos sin ;a a a aθ φ φ φ= + − − (2.3)
Fig. 1 Coordinate system and displacements of the
curved beam.
Fig. 2. Cross-sections of curved beams (a) uniform (C1, tR= tt, bR= bt ); unsymmetric tapered with constant width
( C2, tR ≠ tt, bR = bt ); (c) double unsymmetric tapered ( C3, tR ≠ tt, bR ≠ bt.); (d) symmetric tapered with constant
width( C4, tR ≠ tt, bR = bt ); (e) double symmetric tapered ( C5, tR ≠tt, bR ≠ bt ).
Fig. 3. Curved beam under uniformly distrib-
uted radial loading.
136
.y Rφ = (2.4)
Generalize coordinate vector of an element is obtained by Using Eqs. (2) and (3) for FEM
{ } [ ]1 1 1 ,T
o i i i i i iq u j u jθ θ + + += (2.5)
where and i i iu jθ are torsional displacement, displacement in X direction and slope of u
deflection in the ith node, respectively
.uj
y
∂
= −
∂
(2.6)
If the terms in Eq. (A.13) in the Appendix are substituted in the energy equations below, for
a finite element, Elastic Stiffness Matrix, Geometric Stiffness Matrix and Mass Matrix are
obtained, respectively.
The strain energy of the curved beam element is [3],
2'2'' '
0
1 ,
2
shl
o z
uV EI u GJ dy
R R
θ
θ
= + + −
∫ (2.7)
where
'
y
θ
θ
∂
=
∂
,
' uu
y
∂
=
∂
and
2''
2 ,uu
y
∂
=
∂
{ } [ ] [ ][ ] { }11
2
T T
o o o o o oV q C kk C q− −
= → [ ] [ ][ ] 1 ,T
o o o ok C kk C− −= (2.8)
where 0k is the elastic stiffness matrix of a single element ( [ ]okk given in Appendix).
The kinetic energy of the curved beam element is [3],
( ) ( )22
0
1 ;
2
shl
o pT A u I dyρ ρ θ = + ∫ && (2.9)
{ } [ ] [ ][ ] { }11
2
T T
o o o o o oT q C mm C q− −
= & & [ ] [ ][ ] 1 ,T
o o o om C mm C− −= (2.10)
where 0m is the mass matrix of a single element ( [ ]omm given in Appendix). goV denotes
the external work done by a uniformly distributed radial loading ( )P t and given by the
equation [15],
2'
2 2' ' '
2 2
0
1 ( ) ;
2
shl
go x z
uV P t R u r r dy
R
θ θ
= + + −
∫ (2.11)
2 2; ,x z
x z
I Ir r
A A
= = (2.12)
where xr and zr are the in-plane and out plane radii of gyration.
Eqs. (2.2) and (2.3) are substituted in to Eq. (2.11) and replacing the coefficients
1 6a a… the
{ } [ ] [ ] { }11 ( )
2
T T
go o o go o oV q P t C kk C q− − = → [ ] [ ] 1 ,T
go o go ok C kk C− − = (2.13)
where gok is the geometric matrix of a single element ( [ ]gokk given in Appendix).
137
Mass, elastic stiffness, geometric matrices of each beam element are used to form global
mass, elastic stiffness and geometric matrices. The dynamic response of a beam for a con-
servative system can be formulated by means of Lagrange’s equation of motion in which the
external forces expressed in terms of time-depended potentials, and then performing the
required operations the entire system leads to the governing equation of motion.
[ ]{ } [ ] { }
..
( ) 0,e gM q K P t K q + − = (2.14)
when ( )P t is substituted in equation Eq. (2.14). It becomes [20],
[ ]{ } [ ] { }
..
cos 0.e cr gs d cr gM q K P K P t K qα β ω + − − = (2.15)
Where, gsK and gtK matrices have shown influence of static and time dependent compo-
nents of the load respectively. Eq. (2.15) represents a system of second order differential
equation with periodic coefficients of the Mathieu – Hill type. From the theory of linear
equations with periodic coefficients, the boundaries between stable and unstable solutions of
Eq. (2.15) are formed by periodic solutions of period T and 2T where 2 / .T p w= It was
shown by Bolotin [4] that solutions with period 2T are the ones of greatest practical impor-
tance and that as a first approximation the boundaries of the principal regions of the dy-
namic instability can be determined from the equation:
21 0.
2 4e cr gs d cr gtK P K P K M qω
α β
− ± − =
(2.16)
The two matrices gsK and gtK will be identical if the static and time dependent component
of the loads are applied in the same manner gs gt gK K K ≡ ≡ then Eq. (2.16) becomes
[ ] [ ] { }
21 0 :
2 4e d cr gK P K M qω
α β
− ± − =
(2.17)
(i) Free vibration with 0, 0da b= = and / 2p w= the natural frequency
[ ] [ ] { }2 0;eK p M q − = (2.18)
(ii) Static stability with 1, 0da b= = and 0w =
[ ] { } 0;e cr gK P K q − = (2.19)
(iii) Dynamic stability when all terms are present
[ ] [ ] { }
21 0.
2 4e d cr gK P K M qω
α β
− ± − =
(2.20)
§3. Results and discussion.
The first four natural frequencies obtained with the present element are compared with
the analytical results of Ojalvo and Newman [13] and given in Table 1. Throughout this
investigation, 12 elements are used in modelling the beams. In addition, the critical lateral
buckling load of a uniform cross-section beam for various subtended angles are compared
with the results of Timoshenko and Gere [21] and given in Table 2.
As seen from Table 1 and Table 2, the agreement between the results obtained by using
the present finite element model and exact solution is good.
138
Table 1 Table 2
Table 1. Comparison of the first four natural frequencies of a cantilever curved beam obtained by using the pre-
sent finite element method and the results of Ojalvo and Newman [13]. ( t = 0.0127 m, b = 0.0127 m, R = 0.0254 m,
ρ = 2770 kg/m3, E = 6.89e10 N/m2, θ = 270˚).
Table 2. Comparison of the critical lateral buckling load obtained for various subtended angles of a fixed-
fixed curved beam with the results of Timoshenko and Gere [21] ( t = 0.001587 m, b = 0.02753 m, R =0.3556 m).
Fig. 4 shows that the effect of
variation of subtended angle of an
arch on the fundamental frequency
for various cross-sections. It can be
noticed from the figure that when the
subtended angle of an arch increases,
the fundamental frequency decreases
for all the cross-sections as expected.
It can also be noticed that the fre-
quency parameters of C2, C3 and C1,
C4 cross-sectioned curved beams are
fairly close. Between 30° and 60°
subtended angles, the fundamental
frequency of C5 cross-sectioned
curved beams are more apart than the
fundamental frequencies of other type
cross-sectioned curved beams.
When the subtended angle of an
arch increases, the fundamental fre-
quency of curved beams having the
same length but five different cross-
sections come closer. This phenome-
non can be explained as follows:
when the subtended angle of an arch
increases, the length of curved beams
also increases, consequently beams
become very flexible. The length
variation effect on the flexibility is
more dominant than the effect of
variation of cross-section.
As shown in Fig. 5, if the varia-
tion of the cross-section diminishes
and approaches the uniform cross-
section, the fundamental frequencies
of C4 and C5 cross-sectioned curved
beams increase and approach the fre-
quency parameter of C1 cross-
sectioned curved beam. On the other
hand, the fundamental frequencies of
C2 and C3 cross-sectioned curved
Natural Frequency (Hz)
Mode
Present Ref. [13]
1 9,7 8,48
2 23,19 22,26
3 70,48 72,56
4 176,46 171,5
Critical Lateral Buckling Load( Pcr)(kN/m)
Subtended
Angle Present Ref. [21]
30˚ 1815,79 1814,8
45˚ 777,73 775,40
60˚ 418,70 413,01
90˚ 167,71 164,1
Fig. 4. The effect of variation of subtended angle of an arch on
the fundamental frequency for various cross-sections. bR/bt =
=tR/tt=1( C1), bR/bt = 1and tR/tt = 0.5(O C2 and * C4), bR/bt =
= tR/tt =0.5(Δ C3 and ◊ C5).
Fig. 5. The variation of the free vibration frequency with various
ratio of cross-section. θ = 60˚, bR=bt, tR=tt (n C1), bR = bt and tR ≠ tt
(O C2 and * C4), bR/bt = tR/tt = 0.5 (Δ C3 and ◊ C5).
Fig. 6. The variation of the critical lateral buckling load with
various cross-section and curve radius. bR/bt = tR/tt=1( C1), bR/bt = 1
and tR/tt = 0.5(O C2 and * C4), bR/bt = tR/tt = 0.5 (Δ C3 and ◊C5).
139
beams decrease and approach the fundamen-
tal frequency of the C1 cross-sectioned
curved beam.
Effect of subtended angle of curvature
on the critical lateral buckling load is shown
in Figure 6. When the subtended angle of an
arch increases, as a result the curved beam
becomes more flexible. Thus, as shown in
the figure, the critical lateral buckling load
decreases. It can be noticed from the figure
that critical lateral buckling load values of
beams having C1, C2 and C4 type cross-
sections are close to each other. There is a
similar phenomenon between C3 and C5
type cross-sectioned curved beams. Critical
buckling loads of single tapered curved beams (C2 and C4) are higher than the double ta-
pered curved beams (C3 and C5), respectively, as expected. It can also be noticed that even
though the thickness of C4 tapers twice as much as C2 and the thickness and width of C5
tapers twice as much as C3 type beams. It seems that symmetric tapered beams are more
stable than expected.
Fig. 7 shows the effect of thickness variation of a curved beam on the critical lateral
buckling load for various cross-sections. It is seen that when the variation of cross-section
diminishes and approaches the uniform cross-section, the curved beam becomes stiffer; as a
result, the critical buckling load increases and takes the value of the uniform cross-sectioned
curved beam. From this figure, it can be said that static stability values of curved beams
having C5, C3, C4 and C2 type cross-sections increase, respectively. This increase de-
creases as the cross-section variation diminishes.
From Fig. 8, it can be noticed that the first dynamic instability region widens because of
the decrease in the subtended angle of the arch. C3 cross-sectioned curved beam is less sta-
ble compared to other cross-sections. In addition, when the dynamic load parameter in-
creases, the unstable region widens.
Fig. 8. The effect of dynamic load parameter on the first dynamic stability region of curved beams for various
cross-sections and two different subtended angles. α = 0 bR = bt, tR = tt ( C1), bR = bt and tR ≠ tt , (O C2 and * C4),
bR/bt = tR/tt = 0.5 (Δ C3 and ◊ C5), …… β = 90˚, ––––– β = 30˚.
Fig. 7. The variation of the critical lateral buckling load
with various ratio of cross-section. θ = 60˚, bR = bt, tR=
= tt(n C1), bR = bt and tR ≠ tt (O C2 and * C4), bR/bt =
=tR/tt = =0.5 (Δ C3 and ◊ C5).
140
Fig. 9. The effect of dynamic load parameter on the first dynamic stability region of curved beams for various
cross-sections and two different subtended angles. α = 0.2 bR = bt, tR = tt ( C1), bR=bt and tR ≠ tt, (O C2 and * C4),
bR/bt = tR/tt = 0.5 (Δ C3 and ◊ C5), …… β = 90˚, ––––– β = 30 .̊
Comparison of Fig. 8 and Fig 9 shows that, if the static load parameter increases, the
initial ratio of the disturbing frequency to the fundamental frequency moves towards origin.
It can be seen from the figures that the curved beam under periodic loading becomes unsta-
ble at a small disturbing frequency and small dynamic load parameter.
Fig. 10. The effect of dynamic load parameter on the first dynamic stability region of curved beams for various
cross-sections and two different subtended angles. α = 0, β = 30˚, –––– bR/bt = tR/tt = 1( C1), ––––– bR/bt = 1 and
tR/tt = 0.5 (O C2 and * C4), bR/bt = tR/tt = = 0.5 (Δ C3 and ◊ C5), ……. bR/bt = tR/tt = 1( C1), bR/bt = 1 and tR/tt = 0.8
(O C2 and * C4), bR/bt = tR/tt = 0.8 (Δ C3 and ◊ C5).
Figs. 10 and 11 show that when b1/b2 and t1/t2 ratios approach unity for 0 andα =
0, 2,α = the first unstable region approaches the region of C1 cross-sectioned curved beam.
For both figures, the order of the unstable regions of all cross-sections does not change from
the stability point of view.
141
Fig. 11. The effect of dynamic load parameter on the first dynamic stability region of curved beams for various
cross-sections and two different subtended angles. α = 0.2, β = 30˚, –––– bR/bt= tR/tt =1( C1), ––––– bR/bt = 1
and tR/tt = 0.5 (O C2 and * C4), bR/bt = tR/tt = 0.5 (Δ C3 and ◊ C5) , ……. bR/bt = tR/tt=1( C1), bR/bt = 1 and tR/tt = 0.8
(O C2 and * C4), bR/bt = tR/tt = 0.8 (Δ C3 and ◊ C5).
§4. Conclusions.
This paper presents the dynamic and static stability of non-uniform cross-sectioned
curved beams. The finite element method has been employed in the analysis. The effects of
subtended angle, variations of cross-section and dynamic load parameter on the out-plane
stability regions are examined. When the subtended angle of an arch increases the static sta-
bility (buckling) decreases and dynamic instability region widens but the initial ratio of the
disturbing frequency to the fundamental frequency moves up from the origin. If the static
load parameter is equal to 0.2, initial disturbing frequency moves towards origin. However,
the dynamic load parameter is bounded between the values of zero and 1.6. Consequently,
by changing the static and dynamic load parameters and the ratios of dimensions of thick-
ness and width at the tip cross-section to the ones at the root cross-section, the out-plane
dynamic stability of curved beam may be conserved.
Nomenclature:
A : Area of curved beam cross section; 1b : Width at the root of the curved beam;
2b : Width at the top of the curved beam; E : Modulus of elasticity of curved beam;
f : Natural frequency; G : Shear modulus of curved beam;
pI : Polar moment of area of cross-section; xI : Second moment of area of cross section
about axis X; zI : Second moment of area of cross section about axis Z;
J : St. Venant torsion constant of a curved beam; shl : Length of beam element;
R : Radius of curved beam; 1t : Thickness at the root of the curved beam;
2t : Thickness at the top of the curved beam; [ ]K : Global elastic stiffness matrix;
goK : Global geometric matrix; [ ]M : Global mass (inertia) matrix;
α : Static component of load; β : Subtended angle; dβ : Dynamic component of load;
ρ : Density of curved beam.
142
Р Е З ЮМ Е . За допомогою методу скінчених елементів проаналізовано бокову стійкість ви-
кривленої звуженої у кінці тонкої балки при однорідно розподіленому радіальному навантаженні.
Розв‘язки у вигляді наближення Болотіна досліджені в рамках аналізу динамічної стійкості. Також
вивчені бокові коливання і бокове випучення балки. Отримані результати щодо основної частоти та
критичного навантаження при боковому випученні порівняні з іншими опублікованими результата-
ми. На рисунках показано вплив утвореного дугою кута зміни поперечного перерізу та параметру
динамічного навантаження на область стійкості.
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APPENDIX
[ ]
1 2 1 3
5 4 5 1
1 4 2 1 3
1 2 1 3
1 1 1 1
0 / 0 /
0 / 0 /
0 / 0 /
;
1 / 1/ ( ) 0 1/ 1/ ( ) 0
1 / 0 /
1 / 2 / ( ) 1/ 2 / ( )
o o o o
o o o o
o o o o
o
o o o o
o o o o
B R B B R B
B R B B R B
B R B B R B
C
R R
B R B B R B
B R B B R B
β β β β
β β β β
−
−
− − −
− − −
=
− −
− −
− − − −
(A.1)
2 2cos sin ;oD φ β φ= − − (A.2)
( )1 cos 1 / ;o oB Dφ= − (A.3)
2 sin / ;o oB Dφ= (A.4)
( )3 sin /o oB Dφ φ= − (A.5)
( )4 1 cos sin / ;o oB Dφ φ φ= − − (A.6)
5 sin / ;o oB Dφ= (A.7)
{ } [ ] { }. ;o oq C a= (A.8)
{ } { }1 2 3 4 5 6 ;a a a a a a a= (A.9)
[ ] [ ]1 cos sin 0 ;P R R R R Rφ φ φ φ= − − (A.10)
[ ] [ ]2 cos sin 0 0 1 ;P φ φ φ= − − (A11)
[ ] [ ]3 sin cos 0 1 0 1 ;P φ φ= − − − (A12)
[ ] [ ][ ] { }1
1 ;o ou P C q−= [ ] [ ][ ] { }1
2 ;o oP C qθ −= [ ] [ ][ ] { }1
3 ;o oj P C q−= (A.13)
144
[ ]
'' '' '' ''
1 1 1 2 2 1 2 2
0
1 1 1 ;
shl T T T T
o zkk EI P P P P P P P P dyR R R
= + + +
∫ (A.14)
[ ] 1 1 2 2
0
;
shl T T
omm A P P P P dyρ
= +
∫ (A.15)
' ' ' '
2
1 1 2 2
' ' ' ' ' ' ' '0 2
2 2 2 1 1 2 1 1
( ) .
1 1 1
sh
T T
xl
go T T T T
z
P P r P P
kk P t R dy
r P P P P P P P PR R R
+ +
= − − +
∫ (A.16)
From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 14.05.2009 Утверждена в печать 08.06.2010
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