buckling and integrity analysis of a cable...

1

BUCKLING AND INTEGRITY ANALYSIS OF A CABLE STAYED TOWER Diego Orlando 1, Paulo Batista Gonçalves 2, Giuseppe Rega 3, Stefano Lenci 4

1 Department of Civil Engineering, Catholic University, PUC-Rio, 22451-900, Rio de Janeiro, RJ, Brazil, [email protected]

2 Department of Civil Engineering, Catholic University, PUC-Rio, 22451-900, Rio de Janeiro, RJ, Brazil, [email protected] 3 Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza Università di Roma, 00197, Roma, Italy, [email protected]

4 Dipartimento di Architettura, Costruzioni e Strutture, Università Politecnica delle Marche, 60131, Ancona, Italy, [email protected]

Abstract: Cable stayed structures are widely used to build towers, cover wide spans and in off-shore structures, among others. The prevention of buckling of these structures is a major concern in engineering design. In this paper the stability analysis of a simplified two-degree-of-freedom model of a guyed tower is studied. The results show that the system displays a strong modal coupling, resulting in several unstable post-buckling solutions, thus leading to high imperfection sensitivity. For applied loads lower than the theoretical buckling load, the system is supposedly in a safe position, according to the classical theory of elastic stability. However this is not completely true. The system may buckle at load levels much lower than the critical value due to the simultaneous effects of imperfections and dynamic disturbances. For systems liable to unstable post-buckling behavior, the safe pre-buckling well is delimited by the saddles associated with the unstable post-buckling paths. The heteroclinic or homoclinic orbits emerging from these saddles define the safe region. The geometry and size of the safe region is here analyzed using the mathematical methods of classical mechanics, in particular Lagrangian or Hamiltonian mechanics. This is a first step in the evaluation of the integrity of the dynamical system. Keywords: Cable stayed tower, post-buckling behavior, safe conservative basin.

1. INTRODUCTION

Cable stayed masts are used in several engineering areas including off-shore, mechanical, telecommunications and aero-space engineering. The efficiency of these structures to support axial loads is due to the stayed cables and their behavior is characterized by large displacements associated with high load bearing ratios. As cable stayed structures show large displacements, high non-linearities are associated to their static and dynamic behavior. Therefore, the knowledge of their non-linear behavior is of great interest to engineers and scientist. The analysis of cable stayed structures has been object of several investigations in the last decades. Among the most important studies we can mention the by Neves [1], Xu et al. [2], Wahba et al. [3], Madugula et al. [4], Kahla [5], Chan et al. [6] and Yan-Li et al. [7]. In the present paper the stability and integrity of simplified 2DOF model of a guyed tower is studied. The

static stability analysis of the model was performed by Thompson and co-workers [8-9]. They showed that this model display a complex post-buckling behavior with a strong modal coupling leading to several unstable post-buckling paths. In such cases, the load-carrying capacity of the structure in governed by the unstable branches of the post-buckling response. Also, in these structures the imperfections may substantially decrease the load capacity of the structure and the choice of a safe load level for design becomes usually a complex and difficult task for the engineer. The nonlinear analysis of the imperfect structure may help the engineer in the design process. However, most of the studies in this area relies on the local stability analysis of an equilibrium configuration and no additional information is given on the safety of a given equilibrium state. The aim of the present work is to show that a global stability analysis using the mathematical methods of classical mechanics, in particular Lagrangian or Hamiltonian mechanics can help the engineer in the understanding of this problem and of the degree of safety of the safe pre-buckling configuration through the analysis of the safe conservative basin [10].

The understanding of the global behavior of structures subjected to unstable post-buckling behavior is particularly important in structural systems where modal interaction of different buckling modes with equal or nearly equal buckling loads may lead to new unstable and sometimes unexpected equilibrium paths. This has been observed in several structural systems such as bars, plates and shells [11-16].

2. PROBLEM FORMULATION

Figure 1 illustrates a simplified model of a cable stayed tower. It is an inverted spatial pendulum composed of a

slender, rigid (but massless) bar of length l , pinned at the

base and with a tip-mass m . The lateral displacements are

restricted by three linear springs, initially inclined at °45 .

The first spring of stiffness 1

k , is located in the zy × plane,

while the others, 2

k and 3

k , are located symmetrically

about the y axis, with their positions defined by the angle

β . The two degrees of freedom are 1

u and 2

u , where

11sinθ=u and

22sinθ=u .

Proceedings of the 9th Brazilian Conference on Dynamics Control and their Applications Serra Negra, SP - ISSN 2178-3667 593

Free Vibration Analysis of a Cable Stayed Tower Diego Orlando, Paulo Batista Gonçalves, Giuseppe Rega, Stefano Lenci.

2

z

l

m

yx

−−−−ββββ

ββββl

k3

k1

k2

θ 1 θ 2

Fig. 1. Model of a cable stayed tower

The potential energy the system is given by

( )

(

(

)11(

1)(cos)sin(

22

1

1)(cos)(sin

22

1

2222

1

2

2

2

1

22

2

2

1

2

2

2

1

3

22

2

2

1

2

2

2

1

2

2

21

uuPl

uuuul

lk

uuuul

lk

ullkV

−−−−

−−+−+−−

−+

−−+−+−

−+

−−=

ββ

ββ (1)

where mgP = .

Following Thompson and Gaspar [8], it is assumed that the stiffness of the second and third springs are equal, ie,

Kkk υ==32

where υ is a positive constant, the stiffness of

the first spring is given by Kk )21(1

υ−= and °=120β .

Thus, we can rewrite (1) as:

( )

)11(

322

3222

1

2222

1

2

2

2

1

2

2

21

2

2

212

2

21

uuPl

uull

uullk

ullkV

−−−−

++−+

+−−+

−−=

(2)

The kinetic energy is written as

( )

−+

+++=

12

12

2

2

1

2

22112

2

22

1

2

uu

uuluulululmT

(3)

The Hamiltonian function is thus given by

( ) ( )

)11(

322

3222

1

2222

1

1

2

1),(

2

2

2

1

2

2

21

2

2

212

2

212

2

2

1

2

2211

2

2

22

1

2

uuPl

uull

uullk

ullkuu

uuluul

ululmVTuuHii

−−−−

++−−

+−−+

−−+

−+

+

++=+=

(4)

Here i

u are the generalized coordinates, i

u , the

generalized velocities. The equations of motion of the system are obtained,

using Hamilton´s principle of least action, as the extremals

of the functional =Φ2

1

t

t

Ldt . Then the evolution of i

u with

time is subjected to the Euler-Lagrange equations of motion:

2,10)()()(

==∂

∂+

∂

∂−

∂

∂i

u

V

u

T

u

T

dt

d

iii

(5)

3. BEHAVIOR OF A CABLE STAYED TOWER

In the following the static and dynamic behavior of the conservative 2 dof model is discussed.

3.1. Stability Analysis and Safe Pre-Buckling Region

The behavior of the model for different values of the

angle β was studied by Thompson and co-workers [8-9].

They found that the value of β has a significant influence

on the stability of the model. Here we restrict our attention

to the case with °= 120β , which correspond to a usual

configuration in practical applications. Considering

Kkk υ==32

, Kk )21(1

υ−= and 3/1=υ , the model of

cable stayed tower displays two coincident buckling loads,

4/21

KlPcrPcrPcr === , and orthogonal buckling modes

1u {1,0} and

2u {0,1}. The non-linear post-buckling

behavior in this case can be described by the following set of two coupled non-linear equations:

01

32

322

32

322

3

2

2

2

2

1

1

2

21

2

21

2

21

2

21

=−−

−

++

++−

−

+−

+−−

uu

u

uu

uu

uu

uu

λ

(6)


3

( )

01

223

2224

32

322

32

322

3

2

2

2

2

1

2

2

2

2

21

2

21

2

21

2

21

=−−

−−

−−+

++

++−

−

+−

+−−

−

uu

u

u

u

uu

uu

uu

uu

λ

(7)

where PcrP /=λ .

Figure 2 shows the fundamental path ( 0.021

== uu ),

which is stable up to the static critical load ( 0.1=cr

λ ) and

the three possible post-buckling paths: two coupled unstable solutions (6-7), and one uncoupled unstable solution with

0.01

=u . The interaction of the buckling modes leads to

increased imperfection sensitivity. Three different projections of the equilibrium paths are shown in Figure 3.

Fig. 2. Equilibrium paths of the system

(a) Plane λ×

1u

(b) Plane λ×

2u

(c) Plane

21uu ×

Fig. 3. Projections of the equilibrium paths onto three different planes

Figure 4, where the potential energy surface, expression

(2), and its projection onto the 21

uu × plane is shown for

7.0=λ and 25.1=λ , illustrates the variation of the

potential energy with the static load levels. Figure 4(a) show

that, for values of λ between zero and the critical load,

there is one local minimum, corresponding to the stable pre-buckling solution, surrounded by three saddles at the same energy level. The three saddles are associated with the three unstable equilibrium paths. So, due to the modal coupling, for static load levels lower than the critical load the system may loose stability by escaping from the pre-buckling well if any external perturbation exceeds the safe region delimited by the three saddles. This safe region decreases as the static load increases and becomes zero at the critical

load. Figure 4(b) shows the potential energy for 25.1=λ , a

value higher than the critical load ( 0.1=cr

λ ). In this case,

after the critical load, both eigenvalues change sign simultaneously and the local minimum associated with the trivial solution is transformed into a local maximum. Three saddles associated with the uncoupled and coupled post-buckling solutions are also observed in Figure 4(b). The potential energy landscape gives the engineer a global picture of the problem and shows how the unstable solutions influence the safety of the system. As the load increases and the stable region decreases, the range of allowable disturbances decreases accordingly. So, near the bifurcation point even very small perturbation may lead to escape from



4

the pre-buckling well. Since all structures work in a dynamical environment, a safety factor must be used in design. Having in mind that structural systems are usually lightly damped, the response of the real structure will only depart lightly from the conservative case.

(a) 7.0=λ

(b) 25.1=λ

Fig. 4. Curves of equal potential energy

Much of the global behavior and dynamics of a

structural system can be understood from the topologic structure of its potential and total energy functions. However, in a high-dimensional conservative dynamical system, it is difficult to make direct analysis of the geometrical structure of the phase space and the high-dimensional basin boundary. Safe basins are objects of the same dimension of the phase space, four-dimensional in our case. Therefore they cannot be fully visualized here. Useful insight can however be derived from observation of two- and three-dimensional cross-sections and from the geometry of the stable and unstable manifolds emerging from the saddles. In the present case three independent homoclinic orbits connect the three saddles shown in Figure 5 for

7.0=λ . They separate the initial conditions that lead to

bounded solutions surrounding the pre-buckling configuration from the unbounded escape solutions. The knowledge of these frontiers helps the designer to separate the phase space into safe and unsafe domains.

Fig. 5. Curves of equal energy for 7.0=λ . PS: Saddles. PMi: Stable

position corresponding to a local minimun.

So, the three homoclinic orbits lie on a hypersurface that

bounds the initial conditions leading to bounded solutions around the trivial pre-buckling solution, that is, the interior of this region is filled with a continuous family of stable trajectories. The equation of this surface can be obtained by the conservation of the total energy principle, equating the sum of expressions (2) and (3) to the value of the total energy at one of the saddles, that is

saddleiiiCuVuuT =+ )(),( (8)

Two three-dimensional and one two-dimensional sections of this four-dimensional region are shown in Figure 6 and Figure 7, respectively. One projection of the stable and unstable manifolds of the three saddles that lie on the hyper-surface defined by expression (8) and define the safe region is shown in Figure 8. As one can observe, the safe region is bounded by three homoclinic orbits. This 4D region is defined as the conservative safe basin of the pre-buckling configuration. This safe hyper-volume decreases swiftly in all planes as the static load increases and vanishes at the critical point. An analysis of this region gives information on the maximum allowable displacements and velocities. They also can help in vibration control by giving the upper bound of allowable disturbances.

(a) Plane dtduuu /

121××


5

(b) Plane

121// udtdudtdu ××

Fig. 6. Two three-dimensional sections of safe pre-buckling region for

7.0=λ .

Fig. 7. One two-dimensional section of safe pre-buckling region for

7.0=λ , on the plane 21

uu × .

Fig. 8. One two-dimensional projection of the stable and unstable

manifold of the three saddles for 7.0=λ , on the plane 21

uu × .

4. CONCLUSION

This paper analyses the buckling behavior of a simplified 2DOF model of a cable stayed tower with emphasis in the safety of the static pre-buckling configuration whose stability must be preserved for a safe design. Due to inherent symmetries of the model, it displays two coincident buckling loads. This leads to a strong modal coupling and the existence of several unstable post-buckling paths emerging from the bifurcation point. These unstable branches limit the safe region surrounding the equilibrium configuration. This region decreases as the load increases and becomes zero at the critical load. So, as the applied load increases the magnitude of the allowable perturbations decreases. The safe region is defined by the saddles associated with the unstable post-buckling paths. The paper shows that the tools of analytical dynamics can give the engineer a satisfactory understanding of the safety and integrity of the model, essential for a safe design of the structure.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the Brazilian research agencies CAPES, CNPq and FAPERJ.

REFERENCES

[1] F.A. Neves, “Vibrações de Estruturas Aporticadas Espaciais”. Doctorate Thesis, COPPE, University Federal of the Rio de Janeiro, Rio de Janeiro. 168 p, 1990.

[2] Y.L. Xu, J.M. Ko, and Z. Yu, “Modal Analysis of Tower-Cable System of Tsing Ma Long Suspension Bridge”. Engineering Structures, Vol. 19, No. 10, pp. 857-867, 1997.

[3] Y.M.F. Wahba, M.K.S. Madugula, and G.R. Monforton, “Evaluation of Non-Linear Analysis of Guyed Antenna Towers”. Computers & Structures, No. 68, pp. 207-212, 1998.

[4] M.K.S. Madugula, Y. M. F. Wahba, and G.R. Monforton, “Dynamic Response of Guyed Masts”. Engineering Structures, Vol. 20, No. 12, pp. 1097-1101, 1998.

[5] N.B. Kahla, “Nonlinear Dynamic Response of a Guyed Tower to a Sudden Guy Rupture”. Engineering Structures, Vol. 19, No. 11, pp. 879-890, 1997.

[6] S.L. Chan, G.P. Shu, and Z.T. Lü, “Stability Analysis and Parametric Study of Pre-Stressed Stayed Columns”. Engineering Structures, No. 24, pp. 115-124, 2002.

[7] H. Yan-Li, M. Xing, and W. Zhao-Min, “Nonlinear Discrete Analysis Method for Random Vibration of Guyed Masts Under Wind Load”. Journal of Wind



6

Engineering and Industrial Aerodynamics, No. 91, pp. 513-525, 2003.

[8] J.M.T. Thompson, and Z. Gaspar, “A Buckling Model for the Set of Umbilic Catastrophes”, Math. Proc. Camb. Phil. Soc., Vol. 82, pp. 497, 1977.

[9] J.M.T. Thompson, and G.W. Hunt, “Elastic Instability Phenomena”, John Wiley and Sons, London, 1984.

[10] P.B. Gonçalves, F.M.A. Silva, and Z. J. G. N. Del Prado, “Global Stability Analysis of Parametrically Excited Cylindrical Shells through the Evolution of Basin Boundaries”, Nonlinear Dynamics, Vol. 50, No. 1-2, pp. 121-145, 2007.

[11] V. Tvergaard, “Imperfection-Sensitivity of a Wide Integrally Stiffened Panel Under Compression”, Int. J. Solids Structures, Vol. 9, pp. 177-192, 1973.

[12] G. Kiymaz, “FE Based Mode Interaction Analysis of Thin-Walled Steel Box Columns Under Axial Compression”, Thin-Walled Structures, Vol. 43, No. 7, pp. 1051-1070, 2005.

[13] J.G. Teng, and T. Hong, “Postbuckling Analysis of Elastic Shells of Revolution Considering Mode Switching and Interaction”, International Journal of Solids and Structures, Vol. 43, No. 3-4, pp. 551-568, 2006.

[14] D.D. Quin, J.P. Wilber, C.B., Clemons, G.W. Young, and A. Buldum, “Buckling Instabilities in Coupled Nano-Layers”, International Journal of Non-Linear Mechanics, Vol. 42, No. 4, pp. 681-689, 2007.

[15] Z. Kolakowski, “Some Aspects of Dynamic Interactive Buckling of Composite Columns”, Thin-Walled Structures, In Press, Corrected Proof, 2007.

[16] P.B. Dinis, D. Camotim, and N. Silvestre, “FEM-Based Analysis of the Local-Plate/Distortional Mode Interaction in Cold-Formed Steel Lipped Channel Columns”, Computers & Structures, Vol. 85, No. 19-20, pp. 1461-1474, 2007.


buckling and integrity analysis of a cable...

Documents