Nonlinear aerostatic stability analysis of Jiang Yin suspension bridge

Engineering Structures 24 (2002) 773–781 www.elsevier.com/locate/engstruct

Nonlinear aerostatic stability analysis of Jiang Yin suspension bridge Jin Cheng a,*, Jian-Jing Jiang a, Ru-Cheng Xiao b, Hai-Fan Xiang b a b

Department of Civil Engineering, Tsinghua University, Beijing, 100084, China Department of Bridge Engineering, Tongji University, Shanghai, 200092, China

Received 3 May 2001; received in revised form 13 December 2001; accepted 21 December 2001

Abstract A nonlinear aerostatic stability analysis of the Jiang Yin suspension bridge over the Yangtse River in China is carried out in this paper. We propose a new nonlinear method to analyze aerostatic stability of suspension bridges, based on both the three components of wind loads and geometric nonlinearity. A computer program NASAB, based on the proposed method, has been developed. The accuracy and efficiency of the computer program is examined by numerical examples. The effects of some important parameters on the aerostatic stability of the Jiang Yin bridge are studied. The results show that considering the effect of wind angle of attack on the slope of the curve of pitch moment coefficient is significant for the aerostatic stability analysis of the bridge. The displacement response under the displacement-dependent wind loads exhibits strong nonlinearity. The aerostatic instability of the bridge can exhibit asymmetric flexural–torsional instability in space. Wind angle of incidence and wind loads of cable have major effect on the aerostatic stability of the bridge.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Jiang Yin bridge; Aerostatic stability; Three components of wind loads; Geometric nonlinearity; Limit point instability

1. Introduction In recent decades, suspension bridges have been one of the most popular types of bridges. Longer and longer spans are being planned, such as the Messina bridge in Italy (3300 m) and the Gibraltar bridges between Spain and Morocco (3550 m) [14]. In China, five large straitcrossing projects have been planned. These crossing projects would consist of a series of large suspension bridges with span lengths of 2000–3000 m [10]. New problems are arising as spans become longer and bridge girders more flexible. One of the most important of these is aerostatic stability. Aerostatic instability appears when the deformed shape of the structure produces an increase in the value of the three components of displacement-dependent wind loads distributed in the structure. In the past, it was usually believed that the flutter onset wind velocity is generally much lower than the critical wind velocity under static wind load for suspension bridges. So many bridge engineers pay little

* Corresponding author.

attention to potential aerostatic instability of suspension bridges. Few studies have been done on the aerostatic stability of suspension bridges. However, past work suggests that aerostatic instability phenomenon is likely to take place. Hirai et al. [1] found that torsional divergence of suspension bridges could occur under the action of static wind loads in the wind tunnel test of the full bridge model. In addition, recently, this phenomenon was also been observed in the wind tunnel laboratory of Tongji University [9]. Therefore, it is necessary to investigate the aerostatic stability of suspension bridges. Previous aerostatic stability analyses of long-span suspension bridges generally used the linear method. This method is based on the assumptions of a linearized derivative of pitching moment and of a linear structural stiffness matrix [3,4]. However, it does not take into consideration the nonlinear effects arising from bridge structure and the three components of wind load. Therefore, the critical wind velocity causing aerostatic instability cannot be accurately calculated, the mode of instability as well as the coupling effect cannot be considered, and the wind velocity–deformation path of the bridge from applied wind velocity to divergence cannot be traced.

0141-0296/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 1 - 0 2 9 6 ( 0 2 ) 0 0 0 0 6 - 8

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Today, nonlinear methods of analysis are more promising. The rapid development of electronic computers and nonlinear finite element methods (NFEM) has made it possible to consider the nonlinear effects arising from bridge structure and the three components of wind load, and to analyze the aerostatic response by NFEM. Based on the bifurcation point instability concept, a nonlinear method that combines eigenvalue analysis and updated bound algorithms has been developed [2]. In that work, wind-induced nonlinear lateral-torsional buckling of cable-stayed bridges was analyzed. However, a suspension bridge or cable-stayed bridge is no longer a perfect structural system for two reasons: (1) Its elements, such as the girder and towers, are subject to both axial forces and bending moments. (2) Before the wind loads are applied, the bridge has sustained heavy dead loads and built-in construction loads so that initial deformations and stresses exist in every member. Therefore, the concept of bifurcation point instability based on the eigenvalue analysis will be invalid for suspension or cablestayed bridges [11]. Theoretically, aerostatic stability analysis of such bridges should be a limit point instability problem. In this paper, based on the concept of limit point instability, a nonlinear finite element method (NFEM) is presented to calculate directly the critical wind velocity for the aerostatic instability of suspension bridges, in which the three components of wind loads as well as geometric nonlinearity are involved. A computer program NASAB, based on the nonlinear method, has been developed. The accuracy and efficiency of the computer program is tested with numerical examples. The aerostatic stability of the Jiang Yin suspension bridge with a center span length of 1385 m is investigated by NFEM. The results give a critical wind velocity of NFEM higher than given by the linear method. The reasons for this difference are explained. Finally, the effects of some important parameters on the aerostatic stability of the bridge are studied. 2. Method of nonlinear analysis The three components of wind load are drag force, lift force and pitch moment. Consider a section of bridge deck in a smooth flow, as shown in Fig. 1. Assuming that under the effect of the mean wind velocity V with the angle of incidence a0, the torsional displacement of the deck is q. Then the effective wind angle of attack is a ⫽ a0 ⫹ q. The components of wind forces per unit span acting on the deformed deck can be written in wind axes as 1 Drag force: Fy ⫽ rV2Cy(a)D 2 1 Lift force: Fz ⫽ rV2Cz(a)B 2

(1a) (1b)

Fig. 1. Motion of the bridge deck and three components of wind load in different axes.

1 Pitch moment: M ⫽ rV2CM(a)B2 2

(1c)

Where Cy(a), Cz(a) and CM(a) are the coefficients of drag force, lift force, and pitch moment in local bridge axes, respectively; r is the air density; B is the deck width; D is the vertical projected area. The wind forces in (1) are the function of the torsional displacement of structure. They vary as the girder displaces. Therefore, the three components of wind load are displacement dependent. The equilibrium equation of structural system under wind load can be expressed as: [K(u)]·{u} ⫽ P(Fy(a),Fz(a),M(a))

(2)

where [K(u)] is the structural stiffness matrix including elastic stiffness matrix and geometrical stiffness matrix; {u} is the nodal displacement vector; P(Fy(a), Fz(a),M(a)) is the total wind load which includes drag force Fy(a),lift force Fz(a) and pitch moment M(a). To solve Eq. (2), an incremental-two-iterative method is used in this study. The procedure of calculating critical velocity by this method can be summarized as follows: 1. Assume an initial wind velocity V0; 2. Calculate wind load of the structure under V0; 3. Solve the global equilibrium Eq. (2) to get the displacement {u} by Newton–Raphon method; 4. Get the torsional angle of element from the displacement {u} by averaging the torsional displacement between left node and right node; 5. Recalculate wind load of the structure under V0; 6. Check if the Euclidean norm of static aerodynamic coefficients is less than the prescribed tolerance. The Euclidean norm is written as:

冘 Na

冦

j⫽1

冘 Na

j⫽1

[Ck(aj⫺1)]2

1/2

冧

[Ck(aj)⫺Ck(aj⫺1)]2

ⱕek(k ⫽ y,z,M)

(3)

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4. Description of Jiang Yin bridge

Fig. 2. Horizontal cantilever with a vertical point load at the free end.

where ek is the prescribed tolerance; Na is number of nodes subjected to the displacement-dependent wind loads. If satisfied, then add wind velocity according to scheduled change in wind velocity length. Otherwise repeat steps (3)–(6) until Eq. (3) is satisfied or the maximum number of iterations is reached. 7. If the iterations do not converge under certain wind velocity, then get back previous wind velocity and recalculate by shortening change length of wind velocity until the difference between two successive wind velocity is less than prescribed tolerance.

3. Computer implementation Based on the formulation described in Section 2, a computer program NASAB has been developed to perform the nonlinear aerostatic stability analysis of suspension bridges. The accuracy of the computer program has been verified in Cheng [9] through many examples although only two of them are reported here. The first example is, as shown in Fig. 2, taken from Mattiasson [6], In this case, a cantilever beam with a transversally acting point load at the free end is used. The results obtained by the computer program NASAB are compared to those of [6] in Table 1. The second example, as shown in Fig. 3, is taken from Bathe and Bolourchi [7]; In this case, a 45° circular bend is subjected to a cantilever-type loading. The results obtained by the computer program are compared to those of [7,8] in Table 2.

The Jiang Yin bridge, stretching from Jing Jiang city to Jiang Yin city in JiangSu Province of China, has a main span of 1385 m between Jing Jiang tower in the north and Jiang Yin tower in the south (see Fig. 4). The two towers differ in the height of the Jing Jiang tower is 184 m, and the height of the Jing Yin tower is 187 m. As shown in Fig. 4, the length of the Jing Jiang side span is 336 m, and the length of the Jing Yin side span is 309 m. This difference introduces some asymmetry with respect to the midspan of the bridge. The deck cross section (Fig. 5) is an aerodynamically shaped closed box steel girder 36.9 m wide and 3.0 m high. The distance between the two cables is 32.5 m; the spacing between the two hangers is 16.0 m; section material and geometrical features of the main members are indicated in Table 3. Complete structural data is given by Xiang et al. [5]. A three-dimensional finite element model has been established for the Jiang Yin bridge [9]. Three-dimensional beam elements were used to model the two bridge towers. The cables and suspenders were modeled by three-dimensional truss element accounting for geometric nonlinearity due to cable sag. The bridge deck is represented by a single beam and the cross-section properties of the bridge deck are assigned to the beam as equivalent properties. The connections between bridge components and the supports of the bridge were properly modeled. The static aerodynamic coefficients for the bridge studied are shown in Fig. 6, and were incorporated in computer program by using polynomial function representation. The three components of the displacement-dependent wind loads were only considered for the bridge deck while for the towers and cables only the initial drag force was considered.

5. Displacement response to varying wind velocity To investigate the displacement response of the Jiang Yin bridge to varying wind velocity, three scenarios are used. Type I: The initial wind velocity is 40 m/s and increases at 10 m/s up to a velocity of 110 m/s. Type

Table 1 Deflections at the free for the cantilever beam in Fig. 2 K⫽

2.0 4.0 6.0 8.0 10.0

P·L EI

U/L

q

V/L

NASAB

Mattiasson [6]

NASAB

Mattiasson [6]

NASAB

Mattiasson [6]

0.160528 0.328758 0.434363 0.504567 0.554703

0.16064 0.32894 0.43459 0.50483 0.55500

0.493639 0.670506 0.745480 0.786258 0.812252

0.49346 0.66996 0.74457 0.78498 0.81061

0.781839 1.12141 1.28389 1.37461 1.43045

0.78175 1.12124 1.28370 1.37443 1.43029

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Fig. 3.

Deformed configurations of a 45° circular bend.

Table 2 45° Bend displacements at the tipa Results

ADINA-1 ADINA-2 Spilers[8] NASAB a

Load P ⫽ 0 X Y

Z

Load P ⫽ 300 X Y

Z

Load P ⫽ 600 X Y

Z

29.3 29.3 29.3 29.3

0.0 0.0 0.0 0.0

22.5 22.2 22.9 22.1

39.5 40.4 43.8 40.48

15.9 15.7 19.1 15.48

53.4 53.6 59.2 53.62

70.7 70.7 70.7 70.7

59.2 58.5 60.0 58.5

47.2 46.8 46.2 46.88

ADINA-1 is the solution of Bathe and Bolourchi [7]; ADINA-2 is Bathe’s other solution presented in Table 1 of Spillers [8].

Fig. 4.

Elevation of the Jiang Yin bridge (unit: m).

II: The initial wind velocity is 70 m/s and increases at 20 m/s to a final velocity of 110 m/s. Type III: The initial wind velocity is zero, and suddenly increases to 110 m/s. The displacements at midpoint of the center span in the three types are shown in Table 4.

As can be seen in Table 4, displacements at the midpoint of the center span (V ⫽ 110 m/s) are almost the same in three types of applying wind velocity. This coincides with the fact that there is only one displacement response for any given wind velocity.

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Fig. 5.

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Cross-section of the deck of the Jiang Yin bridge (unit: m).

Table 3 Section geometrical and material feature of the main membera Substructures

Jd (m4)

I2(m4)

I3(m4)

m (t/m)

E (MPa)

g

Steel box girder Cable Hanger

4.82 – –

93.32 – –

1.844 – –

18.0 3.97 0.05

210,000.0 200,000.0 140,000.0

0.3 – –

a m: mass per unit length; E: modulus of elasticity; Jd: St. Venant constant; I2: out-of-plane moments of inertia; I3: in-plane moments of inertia; g: Poisson ratio.

6. Aerostatic stability In this paper, two different methods were adopted to analyze the aerostatic stability of this suspension bridge under displacement-dependent wind loads. Method I is the proposed nonlinear method in which three components of displacement-dependent wind loads as well as geometric nonlinearity are considered. Method II is the linear method [3,4]. The computed values of the critical velocity using the two methods are shown in Table 5. The torsional displacement behavior at midpoint of center span for the proposed method is shown in Fig. 7. The lateral displacement at the midpoint of the center span for the proposed method is shown in Fig. 8. The vertical displacement at the midpoint of the center span Fig. 6. Static aerodynamic coefficients as function of angle of attack.

Table 4 Displacements at the midpoint of the center span in three types of applying wind velocity Types

I II III

Table 5 Comparison of critical velocity in different methodsa Different methods

Displacements at midpoint of center span (V ⫽ 110 m/s) Vertical displacement (m)

Lateral displacement (m)

Torsional angle (°)

5.1555 5.1484 5.1284

14.758 14.761 14.771

2.8457 2.8402 2.8222

Critical wind velocity (m/s)

Linear method

Nonlinear method (proposed method)

Lateral– torsional buckling

Torsional divergence

101.4

97.2

113

a Two modes of aerostatic instability of suspension bridges: lateral– torsional buckling and torsional divergence.

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Fig. 7. The torsional displacement behavior at the midpoint of the center span.

the linear method is lower than that of the Jiang Yin bridge based on the nonlinear method. To explain this phenomenon, the following steps will be performed: (1) On the supposition that other parameters of the Jiang Yin bridge are not changed except the static aerodynamic coefficients of the bridge, the static aerodynamic coefficients of Hu Men bridge are adopted. The Hu Men bridge over the Pearl River (Zhu Jiang) is a long-span suspension bridge of 888 m central span length constructed in China. A detailed description of the bridge is given by Xiang et al. [12]. (2) Based on the static aerodynamic coefficients of Hu Men bridge, the critical wind velocity of the Jiang Yin bridge is calculated by both linear and nonlinear methods, respectively. The results are indicated in Table 6. The pitch moment coefficients of the Jiang Yin bridge and Hu Men bridge are shown in Fig. 11. The critical wind velocity of the linearized torsional divergence under pitching moment of long-span suspension bridges can be expressed as [4]: Utd ⫽ KtdftB

(4)

in which:

冪 2 m冉b冊 C⬘ p3

Ktd ⫽

r

2

1 M0

m⫽

m B ,b⫽ p·r·b2 2

冪m

r 1 ⫽ b b Fig. 8. The lateral displacement behavior at the midpoint of the center span.

for the proposed method is shown in Fig. 9. The instability configuration of a suspension bridge for the proposed method is shown in Fig. 10. From Table 5 and Figs. 7–10, it can be seen that the critical wind velocity of the Jiang Yin bridge based on

Fig. 9. The vertical displacement behavior at the midpoint of the center span.

Im

where m is mass per unit length; Im is mass moment of inertia about the centroidal axis per unit length; ft is the first symmetric torsion frequency;C⬘M0 is the slope of the curve of pitch moment coefficient when wind angle of attack is zero degree; B is the width of stiffened girder. As can be seen in Table 6, based on the static aerodynamic coefficients of the Jiang Yin bridge, the critical wind velocity obtained from the nonlinear aerostatic stability analysis is higher than that obtained from the linear aerostatic stability analysis of the Jiang Yin bridge. This result differs from the case of the Hu Men bridge. Based on the static aerodynamic coefficients of the Hu Men bridge, the critical wind velocity obtained from the nonlinear aerostatic stability analysis of the Jiang Yin bridge is lower than that obtained from the linear aerostatic stability analysis of the Jiang Yin bridge. The reasons for this phenomenon are mainly: (1) In the linear method, critical wind velocity obtained from Eq. (4) is in inverse proportion to the slope of the curve of pitch moment coefficient when the wind angle of attack is 0°. In other words, the effect of wind angle of attack on the slope of the curve of pitch moment coefficient is not taken into consideration. (2) The slope of the curve of pitch moment coefficient of the Hu Men bridge does not change with increasing wind angle of

J. Cheng et al. / Engineering Structures 24 (2002) 773–781

Fig. 10.

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The instability configuration of the Jiang Yin bridge.

Table 6 Comparison of critical velocity of Jiang Yin Bridge in different methods and different static aerodynamic coefficientsa Different method

Linear method (Eq. (4))

Nonlinear method

Critical wind velocity (I) Critical wind velocity (II)

97 106.43

113 93

a I: static aerodynamic coefficients of the Jiang Yin bridge; II: static aerodynamic coefficients of the Hu Men bridge.

This phenomenon is mainly related to the nonlinearity of three components of wind loads. 앫 From the instability configuration of the bridge, the aerostatic instability of the bridge exhibits asymmetric flexural–torsional instability in space. This phenomenon is probably related to the asymmetry of the structure.

7. Parametric study A variety of parameters such as wind angle of attack, wind load on the cable and cable sag can affect the aerostatic stability of the Jiang Yin bridge. In the following section, the effects of these parameters on the aerostatic stability of the bridge are investigated. 7.1. Effect of wind angle of incidence The aerostatic stability of the Jiang Yin bridge is computed by selecting different wind angles of incidence, i.e. 0°, 3° and 5°. The results are summarized in Table 7. Fig. 12 gives the torsional displacement behavior at the midpoint of the center span under different wind angles of incidence. As can be seen in Table 7 and Fig. 12, the angle of incidence has a major effect on both the critical wind velocity and the torsional behavior of the deck. The angle of incidence of a positive 5° reduces the critical wind velocity from 113 to 102 m/s.

Fig. 11. Pitch moment coefficients of the Jiang Yin bridge and the Hu Men Bridge.

attack. However, the slope of the curve of pitch moment coefficient of the Jiang Yin bridge decreases with increasing wind angle of attack. (3) The effect of wind angle of attack on the slope of the curve of pitch moment coefficient is taken into consideration in the nonlinear method. 앫 The displacement response under the displacementdependent wind loads exhibits strong nonlinearity.

7.2. Effect of wind load of cable and hanger To investigate the effect of wind load of cable and hanger on aerostatic stability of the Jiang Yin bridge, Table 7 Critical Wind Velocity under different angle of incidence Wind angle of attack (°)

0°

3°

5°

Critical wind velocity (m/s)

113

110

102

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Table 9 Effect of sagging of cable on critical wind velocity

Fig. 12. The torsional displacement behavior at the midpoint of the center span under different wind angles of incidence.

Table 8 Critical wind velocities in different cases Different cases

Case I

Case II

Case III Case IV

Critical wind velocity (m/s)

116

113

116

Different cases

Consideration of cable sag

No consideration of cable sag

Critical wind velocity (m/s)

113

113

critical wind velocity and torsional behavior of the deck are the same under Case I and Case III. However, the critical wind velocities under Case II and Case IV are lower than those of Case I and Case III. The common feature of Case II and Case IV is the consideration of the wind load of the cable. Therefore, when the aerostatic stability of the bridge is computed, the wind load on the hanger can be ignored but the wind load on the cable cannot be ignored. Otherwise, the design will not produce a safe bridge. 7.3. Effect of cable sag

113

four cases were selected as follows: (1) apply the wind load only on the stiffened girder; (2) apply the wind load on the stiffened girder and cable, simultaneously; (3) apply the wind load on the stiffened girder and hanger, simultaneously; (4) apply the wind load on the stiffened girder, hanger and cable, simultaneously. The results are summarized in Table 8. Fig. 13 gives the torsional displacement behavior at the midpoint of the center span under different cases. As can be seen in Table 8 and Fig. 13, both critical wind velocity and torsional behavior of the deck are the same under Case II and Case IV. Both

Fig. 13. The torsional displacement behavior at the midpoint of the center span under different cases.

This section investigates the effect of cable sag on the aerostatic stability of the bridge. An approach to account for the sagging of cable element is to consider an equivalent straight chord member with an equivalent modulus of elasticity, which was first suggested by Ernst [13]. The results are summarized in Table 9. Fig. 14 gives the torsional displacement behavior at the midpoint of the center span under different cases. As can be seen in the Table 9 and Fig. 14, the sagging of cable does not affect either the critical wind velocity or the torsional behavior of the deck. Therefore, when the aerostatic stability of the bridge is computed, the sagging of the cable can be ignored.

Fig. 14. The torsional displacement behavior at the midpoint of the center span under different conditions of sagging of cable.

J. Cheng et al. / Engineering Structures 24 (2002) 773–781

8. Conclusions Based on nonlinear aerostatic stability analysis of the Jiang Yin suspension bridge, it can be concluded that: 1. The effect of wind angle of attack on the slope of the curve of pitch moment coefficient should be consider for the aerostatic stability analysis of the bridge. 2. The displacement response under the displacementdependent wind loads exhibits strong nonlinearity. 3. From the instability configuration of the Jiang Yin bridge, the aerostatic instability of the bridge can exhibit asymmetric flexural–torsional instability in space. 4. The wind angle of incidence and wind loads of the cable have a major effect on the aerostatic stability of the bridge. 5. The effects of sagging of the cable and wind loads of the hanger can be ignored when the aerostatic stability of the bridge is computed. 6. Based on the concept of limit point instability, the proposed method of using an incremental-two-iterative procedures for analyzing the nonlinear aerostatic stability of suspension bridges is accurate, practical, and computationally efficient.

Acknowledgements The writers would like to thank to the National Nature Science Foundation of China for their financial support. The valuable comments of the anonymous reviewers of the paper are also acknowledged.

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