Numerical study on the behavior of cables of cable-stayed ...€¦ · Numerical study on the...

Numerical study on the behavior of cables of cable-stayed bridges

P. G. Papadopoulos, J. Arethas & P. Lazaridis Department of Civil Engineering, Aristotle University of Thessaloniki, Greece

Abstract

A simple method is proposed for the nonlinear static analysis of cable-stayed bridges, with emphasis on the analysis of cables. The bridge is simulated by a plane truss model. A short computer program, with less than 200 Fortran instructions, is used, for the nonlinear static analysis, by incremental loading, of the above truss model. Geometric nonlinearities are taken into account by writing the equilibrium conditions with respect to deformed structure, within each step of the algorithm. The sag effect of cables is considered by assuming them to be the axial structural members with the equivalent elasticity modulus of Ernst. The above method is applied on a typical cable-stayed bridge with a central span of about 200m. The results are found to be in satisfactory approximation with other published data. Seismic inertia loads, parallel to the bridge axis, are applied to all the nodes of the truss. These loads combined with dead loads only, cause, in the back-stay, a state near slackening, whereas when they are combined with traffic in the central span they cause overstress of the exterior cable of the central span. The above two affected cables are analysed by another short computer program, with less than 200 Fortran instructions, for the 3D nonlinear static analysis of a discretized isolated cable, for gradual prescribed displacements of supports. Inclusion of geometric stiffness allows the study of a taut cable as a stable structure. Even a state of the cable near slackening can be studied. The 3D analysis allows the consideration of wind pressure perpendicular to the cables’ plane. Keywords: cable-stayed bridge, truss model, incremental loading, sag effect, Ernst equivalent elasticity modulus, pylon, deck, seismic inertia loads, geometric stiffness, slackening.

© 2005 WIT Press WIT Transactions on The Built Environment, Vol 81, www.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V 473

1 Introduction

The cable-stayed bridges, with small up to medium spans, up to 200m, and total lengths up to 400m, became popular because of their easy erection, economy and aesthetics. And a lot of them have been built all over the world the last 30 years (Virlogeux [1], Walther et al. [2]). In the preliminary design of a cable-stayed bridge, the required pretensions and sections of the cables can be pre-estimated by hand, by considering the steps of erection of the bridge, that is the suspension of successive parts of the deck from the inclined cables. Then, the bridge is analysed for the dead load (mainly self-weight of deck) and its initial shape is found, which usually slightly differs from the designed shape (Wang and Yang [3]). And the analysis for various cases of live loading (mainly due to traffic), in central and side spans, follows, which gives larger deformations and reactions. The structural analysis of cable-stayed bridges exhibits the following sources of geometric nonlinearities: Large displacements of beams (pylons and deck), which can lead to significant N – M (axial force-bending moment) interaction and possibly to buckling, as well as the sag effect of the cables. To take into account the above geometric nonlinearities, the equilibrium conditions must be written with respect to the deformed structure and the stiffness matrix must be updated, within each step of an incremental loading procedure. Whereas, the cables are assumed, within each loading step, as linear axial structural members, with a variable equivalent elasticity modulus obtained from Ernst formula (Ernst [4]), which is valid for a taut cable with a sag-to-span ratio f/L less than 1/12. However, for car (not railway) bridges, which are considered here, in which the live load is a small fraction of dead load, the above geometric nonlinearities of beams are usually not significant, thus remains as the only one significant geometric nonlinearity the sag effect of cables (Nazmy and Abdel-Ghaffar [5], Wang and Yang [3]).

Figure 1: Simulation of a pylon element and a deck element by a plane truss element.

A special problem in cable-stayed bridges is the possibility of slackening of stay-cables for some loading cases. In Wu et al. [6], a 2D step-by-step nonlinear dynamic analysis of a PC cable-stayed bridge was performed, for a real


474 Earthquake Resistant Engineering Structures V

earthquake parallel to bridge axis, and loosening of some internal cables in the central span was observed. Aim of present work is to develop a simple method for the nonlinear static analysis of cable-stayed bridges by use of a truss model, with emphasis to the analysis of cables. And to investigate how the seismic inertia loads affect the cables.

2 The proposed truss model

The usual finite elements, used for the discretization of a structure, have complicated stiffness matrices and exhibit difficulties, particularly in the nonlinear problems. A simple alternative is the truss models, which have very simple stiffness matrices, as the bars of a truss are the simplest possible finite elements. The truss models can take into account material nonlinearities, simply by the nonlinear uniaxial stress-strain laws of the bars (Papadopoulos and Karayannis [7]) and geometric nonlinearities by simply writing the equilibrium conditions with respect to the deformed truss and updating the simple stiffness matrix within each step of an incremental loading procedure (Papadopoulos and Xenidis [8]). The truss models have been proved reliable by comparison of their numerical results with other relevant published experimental and analytical data. In the 2D structural analysis, the local stiffness matrix of an isolated bar, with respect to a reference axes system OXY, can be written:

Kℓ = KE + KG =

+

=+

1001N

ccccccEAINccEA2yyx

yx2x

o3

t

o (1)

where KE elastic stiffness, KG geometric stiffness, E elasticity modulus. A cross-section area, ℓo undeformed length, c = {cx cy} direction cosines of bar axis, N axial force (tensile force is assumed positive) and ℓ is the present length of bar axis.

Figure 2: An inclined cable with sag and its equivalent axial member.



The global stiffness matrix of a truss can be written:

Kg = B diag (kℓi) Bt i = 1 … nb (2)

where B = (Bik) i = 1 … nn, k = 1 … nb is the linkage (Boolean) matrix, nn number of nodes, nb number of bars. Any element Bik is equal to +1 if i is left node of bar k,–1 if i is right node of bar k and 0 if there is no connection between bar k and node i. When we have to simulate an element of a pylon of a bridge by a plane truss element, as shown in fig. 1a, the following formulae can be used for the determination of bar sections:

( ) ( ) 3

23 A

31

2/bd8/9sincosA

→=ϑϑ

113

3 A2

bd89AcosA →=+ϑ (3)

33 2 2

9sin8 2

blA A Aϑ + = →

These formulae are valid for ϑ>21°. If ϑ<21°, negative values result for bar sections A1, which is inadmissible. So, for an element of a slender deck of a bridge, where usually ϑ<<21° (fig. 1b), the following simplified formulae can be used for the determination of bar sections: A1 = A3 = bd / 4 A2 = b ℓ / 2 (4) Sometimes, even in a pylon element, possibly ϑ<21°, so the above simplified formulae (4) are used in this case, too. A short, thus transparent, computer program, with less than 200 Fortran instructions, has been developed for the nonlinear static analysis of a plane truss model by an incremental loading technique. The bars of the truss, which has been declared as cables, have as initial equivalent elasticity modulus Eeq that of a cable without sag Eo. Within each step of incremental loading algorithm, for every cable, the following iteration is performed: By using the equivalent elasticity modulus Eeq of previous step, we find the present stress σ = Eeqε of the cable, which is substituted in the formula of Ernst for the equivalent elasticity modulus of a cable (Ernst, H. [4]), see fig. 2:

( )

3o

2o

eq

12

EL1

EE

σ

ρ+

= (5)

where ρ = 78.5 kN/m3 is the density of steel and L is the horizontal span of the cable.



By this new value of Eeq, we find a new σ = Eeqε, which is again substituted in eqn. (5) and so on. Usually, for a sufficiently tensioned cable, the convergence of this iteration is very fast. In 2-3 steps, an accuracy ∆σ < 0.001 kN/cm2 is attained. The above formula is valid for a taut cable with a sag-to-span ratio f/L less than 1/12.

Figure 3: Given data of the application.

3 Application

3.1 Given data

The above procedure is applied on a typical cable-stayed bridge with two pylons, a central span of about 200m and two end spans of about 100m each. The half of this symmetric bridge is shown in fig. 3. The height of pylon is 30m below deck and about 50m above deck. The stay-cables have a harp layout and there are 15 in each side of half central span and 15 in each side of end span, uniformly distributed. The pylons and deck are made of prestressed concrete with elasticity modulus E = 2,800 kN/cm2 and density γ = 24 kN/m3, whereas the cables are made of steel with a high yield strength σy = 160 kN/cm2, elasticity modulus E = 2×104 kN/cm2 and density γ = 78.5 kN/m3. The primary curve of the axial stress-strain σ – ε law of the cables is shown in the right lower part of fig. 3. Each pylon consists of two vertical legs with rectangular cross-section 3.0m×4.5m and two cross beams of the same section as shown in fig. 3. The deck is a slender plate of rectangular cross-section with thickness only 0.4m and width 13.0m as shown in fig. 3. The deck is connected to the pylon by an articulating joint. This typical cable-stayed bridge has also been analysed in Walther, R. et al [2], in the



chapter 4, Parametric study, pages 77-80. A very similar bridge has also been built in reality in 1993 in Evripos, Greece, designed by Professor Jorg Slaich (Virlogeux, M. [1]).

Figure 4: Discretization of the bridge.

Figure 5: Estimation of pretension of cables.

3.2 Discretization

A plane truss model, as shown in fig. 4, simulates the above cable-stayed bridge. Each stay-cable of the model represents five real cables. The bar sections of the deck, as well as those of the pylon, are determined by the simplified formulas (4), because, even in the pylon elements, the angle ϑ is less than 21° (see fig. 1). There are only 28 nodes, thus an algebraic system 56×56 have to be solved within each step of the incremental loading. It is also shown in fig. 4, with a larger scale, how the articulating joint, between deck and pylon, is described by the model.



Table 1: Estimation of pretensions and sections of cables.

Cable number 1 2 3 4 5 6

1 λ m 21.7 31.0 27.9 27.9 31.0 31.0 2 W/2 = wλ/2 kN 1354 1934 1741 1741 1934 1934 3 No = W/2sinα kN 2952 4217 3796 3796 4217 4217 4 A = No/σo cm2 73.8 105.4 94.9 94.9 105.4 105.4

3.3 Estimation of cables pretensions by hand

By considering the way of erection of the cable-stayed bridge, that is the suspension of successive parts of the deck from the inclined cables (fig. 5), we can easily estimate the pretension forces of stay-cables by a hand calculator, and then their required cross-section areas. This procedure is described in table 1. The numbering of cables is that shown in fig. 4. First, for each cable, the length λ of the corresponding part of the deck, suspended by it, is determined. Then, the self-weight W = wλ of this part of deck is found, where w = ρbd, ρ = 24 kN/m3 density of concrete, and b, d sides of rectangular deck section. The W is received by the pretension forces of two inclined cables according the formula No = W/2sinα, where α is the inclination angle of the cables (fig. 5). The required cross-section area of a cable is A = No/σo, where σo = 40 kN/cm2 is a usual value of pretension stress, half of permissible stress σP = 80 kN/cm2, which, in turn, is half of yield stress σy = 160 kN/cm2 (see right lower part of fig. 3). In order to declare, in the input of computer program, the pretensions of the cables, we find, for each one, the elongation ∆ℓo = Noℓ / EA, where ℓ its designed length and the undeformed length ℓo = ℓ−∆ℓo, which is included in the input of the program.

3.4 Loading cases

Nonlinear static analyses are preformed for the truss model of fig. 4 of the bridge under consideration of fig. 3, by the proposed program, for the following loading cases: a. Dead loads: These are mainly due to the self-weight of the deck, w/2 = ρbd/2 = 24 kN/m3×0.4m×13.0m/2 = 62.4 kN/m. The results are shown in fig. 6a. The deformations are small, even in the center of central span, less than 0.5m. Thus, the initial shape of the bridge only slightly deviates from the designed shape (see also Wang and Yang [3]). A remarkable agreement is observed between the pretension forces of cables of fig. 6a, found by computer analysis, and those of table 1, pre-estimated by hand. b. Live loads in central span: These are mainly due to traffic and are represented by a uniform distributed loading p/2 = 22 kN/m as recommended in Walther, R. et al [2], p.77. The results are shown in fig. 6b. The maximum deformation in the center of the central span does not exceed 1.0m. The tensile axial stresses of central span are large, up to σ = 5880/105 = 56 kN/cm2, as was expected and increased up to σ = 3556/74 = 48 kN/cm2 in the back-stay (exterior cable of side span).



c. Live loads in side span: The results are shown in fig. 6c. Small deformations are observed and the tensile axial stresses of cables are, as was expected, large in the side span.

Figure 6: Loading cases. (a) Dead load, (b) Live load in central span, (c) Live load in side span.



3.5 Comparison with other published data

Some representative results of the above loading cases analysed by the proposed truss model and computer program of incremental loading, as shown in fig. 7, are compared with corresponding results of the same bridge analysed in Walther et al [2], p.77-80.

Figure 7: Comparison with results in [2]. (a) Bending moments in pylon due to live loads in central span, (b) Axial forces in pylon due to dead loads, (c) Axial forces in deck due to dead loads, (d) Bending moments in deck due to live loads in central span, (e) Deformations of deck for live loads in central and side span. − − − results in [2], results in present work.



In fig. 7a, the bending moments of pylon, in case of live loads in central span, found in present work, are in a satisfactory approximation with those of Walther et al [2]. Whereas, in fig. 7b, the compressive axial forces of pylon, due to dead loads, are found by the proposed here method, about 30% larger than those in Walther et al [2]. This means that, in the truss model, more than 90% of vertical loads are received by the pylon and the rest by the abutment, which is reasonable for the static system of a cable-stayed bridge and it is with the side of safety as regards the proportioning of pylon sections. Satisfactory approximations are observed between the results of proposed here method and the corresponding ones of Walther et al [2], concerning the compressive axial forces of the deck due to dead loads (fig. 7c), and the deformations of the deck for live loads in central span, as well as in side span (fig. 7e). Whereas, in fig. 7d, where the variation of bending moments along the deck is represented, due to live loads in central span, some acceptable differences are observed between the results of present method and those of Walther et al [2], which are explained by the sparse spacing of cables in the model. Here attention must be paid to the fact that second order moments due to large axial forces and large deformations are included in the diagram of fig. 7d. This effect is shown by the jumps in the bending moments of deck in fig. 7d at the points where cables are connected and it is less pronounced in the bending moments of pylon (fig. 7a).

3.6 Earthquake parallel to bridge axis

We are going to analyse statically the above bridge for seismic inertia loads, to all the nodes of truss model, parallel to bridge axis, directed to side span, equal to mγx, where m mass, γx =0.6g and g gravity acceleration. This is an antisymmetric loading. In order to avoid solving the whole bridge, we first linearly analyse the half of it by posing the appropriate constraint on the deck at the symmetry axis, and by assuming temporarily that the cables can also receive compression (fig. 8a). However, a pure antisymmetric loading in the bridge is not possible, because the symmetric dead and even live loading always exist. And the superposition principle does not hold because of the nonlinearities. For this reason, we assume that the only one nonlinearity is the sag effect of cables and perform an iterative procedure. That is, we repeatedly linearly analyse the half bridge for symmetric and antisymmetric loadings, we superpose the results and change, each time, the elasticity moduli E of the cables by use of Ernst formula, until the corresponding E of cables, of symmetric and antisymmetric loading, approximate each other. By this approximate method, we first combine the above seismic inertia loads with dead loads only and find a state near slackening in the back-stay (exterior cable of side span). Then, by combining these inertia loads with live loads in the central span, we find an overstressing in the exterior cable of the central span.

3.7 Analysis of isolated cables

A second computer program has been developed for the 3D nonlinear static



analysis of a, discretized to successive small segments, isolated cable, for gradual prescribed displacements of supports. This program, too, is short, with less than 200 Fortran instructions. The prescribed displacements of supports are those found by the first program at the connection of the cable under consideration with a point on the pylon and another one on the deck. The inclusion of geometric stiffness allows studying a taut cable as a stable structure, because perpendicularly to its axis only the geometric stiffness exists. On the other hand, the 3D analysis allows including wind pressure perpendicular to cables plane, as well as seismic inertia loads in all directions.

Figure 8: (a) Seismic inertia forces parallel to bridge axis, (b) Loosening of back-stay, (c) Overstress of exterior cable in central span.

By using this computer program, we analysed the two mainly affected cables of the last loading case of seismic inertia loads. In fig. 8b is shown in detail the deformed configuration and the free body diagram of the back-stay (exterior cable of side span), which is near slackening. Whereas, in fig. 8c the same are shown in detail for the overstressed exterior cable of the central span.

4 Conclusions

1. The proposed truss model has been applied on a typical cable-stayed bridge of medium size. The satisfactory approximation of the results with other



relevant published data showed that the truss model is reliable and can be proved useful in the preliminary design of cable-stayed bridges. 2. Two computer programs have been developed. The first one for the nonlinear static analysis of a plane truss model of a cable-stayed bridge by incremental loading. And the second for the 3D nonlinear static analysis of a discretized isolated cable, for gradual prescribed displacements of supports. Both programs are very short, thus transparent. Each one has less than 200 Fortran instructions. 3. Seismic inertia loads, parallel to the axis of bridge, may cause to the back-stays (exterior cables of side span) a state near slackening. To avoid this, a longer central span is suggested, which acts as a ballast producing additional tension in the back-stays. 4. The above seismic inertia loads, when combined with traffic in the central span, may cause overstress to the exterior cables of central span. To avoid this, increase of sections of these cables is suggested. 5. In the program for nonlinear static analysis of an isolated cable, the inclusion of geometric stiffness allows studying a taut cable as a stable structure. Even a state near slackening can be studied. 6. The 3D analysis of an isolated cable allows including wind pressure perpendicular to cables plane, as well as seismic inertia loads in all directions. 7. As regards wind pressure, fewer cables, with larger sections, are obviously preferable than more cables with smaller sections, because, in the first case, the surface area, perpendicularly to wind pressure, is drastically reduced.

References

[1] Virlogeux, M., Recent evolution of cable-stayed bridges, Engineering Structures, 21, pp. 737-755, 1999.

[2] Walther, R., Houriet, B., Isler, W., Moia, P. & Klein, J.F., Cable-stayed bridges, 2nd edition, Th. Telford, 1999.

[3] Wang, P.-H. & Yang, C.-G., Parametric studies of cable-stayed bridges, Computers and Structures, 60(2), pp. 243-260, 1996.

[4] Ernst, H.J., Der E-Modul von Seilen unter Berücksichtigung des Durchhanges, Der Bauingenieur, 40(2), pp. 52-55, 1965.

[5] Nazmy, A. & Abdel-Ghaffar, A., Three-dimensional nonlinear static analysis of cable-stayed bridges, Computers and Structures, 34(2), pp. 257-271, 1990.

[6] Wu, Q., Takahashi, K. & Nakamura, S., The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge, Journal of Sound and Vibration, 268, pp. 71-84, 2003.

[7] Papadopoulos, P. G. & Karayannis, C., Seismic analysis of RC frames by network models, Computers and Structures, 28, pp. 481-494, 1988.

[8] Papadopoulos, P. G. & Xenidis, H., A truss model with structural instability for the confinement of concrete columns, European Earthquake Engineering, 2, pp. 57-88, 1999.



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