Monitoring and system identification of suspension bridges: An alternative approach

Transcription

1 Monitoring and syste identification of suspension bridges: An alternative approach Erdal Şafak Boğaziçi University, Kandilli Observatory and Earthquake Reseach Institute, Istanbul, Turkey Abstract This paper introduces a new approach for onitoring and syste identification of suspension bridges. The ethodology is based on the theoretical forulation of the dynaic response of a suspension bridge and its coponents. The vibration records are used to identify the boundary conditions and the unknown coefficients in the equations. The goal is to identify the forces in the ain coponents of the bridge, rather than its odal properties. By proper placeent of right-type of sensors, it is possible to identify the tie-varying forces on the ain suspension cables, hangers, towers, and the deck fro the records. 1 Introduction The standard approach for vibration onitoring and syste identification of suspension bridges is to install sensors on key coponents of the bridge (e.g., pylons, deck, and the cables), and analyse the records to identify the odal characteristics (e.g., odal frequencies, daping ratios, and ode shapes) of the vibrations. Calibrated analytical odels of the bridge are developed by atching the identified odal properties. Due to their large size, geoetric nonlinearities, non-proportional daping, and oving traffic loads, ost suspension bridges do not eet the requireents and assuptions of classical odal analysis. Also, since there are an infinite nuber of odes, it is not possible to identify all of the and to develop a single odel that atch the recorded data. This paper presents an alternative approach for onitoring and syste identification of suspension bridges. The objective is to identify the physical paraeters of the response, such as the axial forces in the ain suspension cables, axial forces in the hangers, forces and oents in the pylons, and the flexibility of the suspended deck.

2 2 2 Coponents and dynaic response of suspension bridges The ain coponents of a suspension bridge are the suspension cables, pylons, the deck, and the hangers that connect the deck to the suspension cables. Theoretically, the bridge is a single dynaic syste because of the interaction aong the coponents,. However, due to the differences in their flexibilities and the degrees of freedo, the response of soe of the coponents can be studied independently by separating the frequency bands (i.e., by band-pass filtering) in the records that are pertinent only to the vibrations of that coponent. In the following sections, we present the analytical expressions for the static and dynaic response of each ain coponent of the bridge. The analytical expressions are based on the assuption of first-order approxiation and linear behaviour. These expressions can be used to calculate the dynaic forces in the cables, the deck, hangers, and the pylons. Identified forces provide a good starting values for the developent of ore accurate nonlinear odels for the bridge, which can atch the recorded response uch better than those developed based on odal properties. 3 Cables The two ain suspension cables can be odelled as an elastic cable with unifor cross-section subjected to axial tension H, and a uniforly distributed vertical load p, as shown in Fig. 1. The vertical load p represents the su of the cable weight plus the weight of the deck and the hangers. When held at the sae elevation at both ends, such cables sag into a catenary, whose vertical profile can be expressed analytically in ters of hyperbolic cosine (cosh) functions. For shallow catenaries, where L/d > 8 (see Fig. 1), the axial tension in the cable is approxiately constant, and the vertical profile w(x) can be approxiated by a shallow parabola, as given below [1]: w(x) = p L2 2H x L x 2 L where: p : Unifor load (dead+live load) per unit length of the cable. H : Mean tension force on the cable. L, d: Cable span and idspan sag. d L / 8 (shallow catenary)

3 3 Fig. 1 Notation and the loads for the cables. The easure of sag is defined by the following paraeter: α 2 = 8d 2 L EA H L where L L e L 1+ 8 d 2 e L The axiu sag at idspan: d = p L2 8H ; (d / L)2 << 1.0 and α 2 0 as d 0. Assuing that: (1) the cable profile is a shallow catenary, (2) longitudinal (i.e., x direction in Fig.1) coponents of the otion are negligible, and (3) the second order ters can be neglected, the natural frequencies of the cable can be calculated fro the following equations [2,3]: In vertical-plane, syetric ode: = λ i 2L where λ i is found by solving the equation: H tan π λ i 2 = π λ i 2 4 α π λ i 2 2 Note that as α 2 0, λ i 1, and as α 2, λ i

4 4 In vertical-plane, antisyetric ode: = i L H Out-of vertical-plane ode: = i 2L H The first order dynaic coponent of the cable force, H d, is calculated as [2]: ( ) 2 πλ H d = H i B 8d i where B i is the aplitude of dynaic in-plane displaceent in the i'th ode. 3 Backstay cables Backstay cables are the cables spanning between the pylon tops and the ground anchorages at both ends of the bridge. Typically, these sections do not include hangers, as the decks in these parts are carried by concrete coluns. Fig. 2 shows the sketch of a typical backstay cable, and the notations and diensions. The distributed load, q=g, represents the weight of the cable, where is the cable ass per unit length and g is the gravitational acceleration. Again, we assue first-order linear behavior and unifor cross-section subjected to axial tension H b. Note that H b is different than H (i.e., the axial force on centre cables) because of the different angles of slope with respect to the pylon, and the friction between the cable and the saddle at the top of the pylon. Fig. 2 Sketch and the notation for a backstay cable.

5 5 By using the notation in Fig. 2, the following can be written for the static configuration of the backstay cable [1]: Static vertical profile: w(x) = The sag at id-point: s = g x 2H b cos(θ) (L x) + d L x g L 2 8H b cos(θ) Length of backstay cable with sag: L s = The elongation of the cable: The corresponding change in sag: ΔL = H b L AE L x=0 or H b = 1+ ( dw / dx) 2 16s2 1+ 3L 2 Δs = 3H b L2 16sAE g L2 8s cos(θ) 16s2 1+ 3L 2 dx L 1+ 8s2 3L 2 The natural frequencies of the in-plane and out-of-plane vibrations of a backstay cable can be estiated fro the following equations [2,3] In vertical-plane, syetric ode: where = λ i 2L * H b in Hz L * = L / cos(θ), H b is the axial force, and is the ass per unit length. λ i is found by solving the equation: where α * 2 is defined as α * 2 = gl2 H b 2 EA 8H b 2 + (gl) 2 tan π λ i 2 = π λ i 2 4 α π λ i 2 * 2 3 Note that as α * 2 0, λ i 1, and as α * 2, λ i 2.86 In vertical-plane, antisyetric ode: = i L * H b Out-of vertical-plane (i.e., swing) ode: = i 2L * H b

6 6 4 Deck 4.1 Vertical deck vibrations: A siple, first-order linear elastic odel for the deck is a bea on elastic foundation, as shown in Fig. 3. The springs in the elastic foundation represent the vertical flexibility of the suspension cables and the hangers. k(x) Fig. 3 The deck odeled as a bea on elastic foundation. Assuing that the spring stiffness k(x) is approxiately constant (k k(x) ), along the deck, the natural frequencies of the vertical vibrations are [5,6]: where: = 1 2π iπ L 4 EI + k E, I, : Modulus of elasticity, oent onertia, and ass per unit length of the deck. k(x) : Spring stiffness (i.e., ain cable and hanger stiffness) per unit length of the deck. x k, F k : Location and agnitude of live load on the deck. 4.2 Torsional deck vibrations: Assuing that hangers are always in tension, and using the notation in Fig. 4, the equations for torsional vibrations of a deck segent can be written as follows:

7 7 Fig. 4 Torsional vibrations of the deck.!! θ = c θ θ! 6k cos(θ)sin(θ) + f (t)!!w = c y!w 2k w + g where θ = θ(t) : Rotation with respect to the center-line of the deck w = w(t) : Vertical displaceent of the center-line of the deck k = Vertical stiffness of ain cables = Mass of the deck per unit length c θ,c y = Daping for rotation and vertical translation f (t) = External dynaic load (e.g., wind turbulence) on the deck For sall rotations (i.e., cos(θ) 1 and sin(θ) θ) :!! θ = c θ θ! 6k θ + f (t)

8 8 It should be noted that for a large initial torsional push (e.g., large initial θ(0) value), which can happen in sudden and strong wind stors, this approxiation ay grossly underestiate θ(t). For sall rotations, the local torsional natural frequency of the deck segent is: f θ = 1 2π 6k 5 Pylons Pylons of a suspension bridge are basically cantilever beas subjected to large axial copression forces. The i th natural frequencies of such a cantilever is given by the following equation [6]: = where λ 2 i 2πh 2 EI 1 P λ 2 1 P 2 b λ i λ 1 = 1.875, λ 2 = 4.694,!,λ i (2i 1) π 2 P b = π 2 EI / (4h 2 ) (Buckling load of the pylon) P = Axial copression load on the pylon h, EI = Height and the flexural rigidity of the pylon 6 Hangers For hangers with L h / D > 100, where L h is the free hanger length (after the sockets) and D is the hanger diaeter, the hangers can be assued to behave like a taut string. For a taut string, the relationship between the hanger force N h and the i th natural frequency,, are given by the following equations [2,4] = i 2L h 1000 N h, or N h = h h L h i 2

9 9 where: = i th odal frequency of hanger in Hz; i = 1,2,, N h = Hanger load in kn h = Hanger ass per unit length in kg/ L h = Hanger length in eters.. The equation indicates that the hanger has an infinite nuber of odes and they are all integer ultiples of the first ode, that is: f 1 = 1 2L h 1000 N h h and = i f 1 In other words, the difference between any two successive odal frequencies in the spectru should be equal to the first odal frequency (i.e., +1 - =f 1 ). For short hangers, where L h / D < 100, the end conditions and bending deforations influence the natural frequency. In such cases, the natural frequencies are calculated by considering the hanger as a thin bea subjected to axial tension. Natural frequencies of such beas for various end conditions can be found in [6]. 7 Conclusions The dynaic behavior of the ain coponents (i.e., the cables, pylons, the deck, and the hangers) of a suspension bridge can be approxiated fairly accurately by analytical odels. Theoretically, the recorded vibrations of a suspension bridge represent data fro a single dynaic syste because of the interaction aong the coponents. However, due to the differences in their flexibilities and the degrees of freedo, the dynaic characteristics of the ain coponents of the bridge can be identified independently by studying specific frequency bands (i.e., by bandpass filtering) in the records that are pertinent only to the vibrations of that coponent. This paper presents analytical expressions for the static and dynaic response of each ain coponent of the bridge. The equations are based on the assuption of first-order approxiation and linear behaviour. These expressions can be used to calculate the dynaic forces in the cables, the deck, hangers, and the pylons. They provide good starting values for the developent of ore accurate nonlinear odels for the bridge. Those odels would atch the recorded response uch better than those developed based on the calibration with odal properties.

10 10 References 1. Tioshenko, S.P. and Young, D.H. (1965). Theory of Structures, McGraw-Hill Book Co., New York. 2. Irvine, M. (1981). Cable Structures, The MIT Press Series in Structural Mechanics. 3. Abdel-Ghaffar, A.M. (1976) Dynaic Analysis of Suspension Bridge Structures, California Institute of Technology, EERL Report 76-01, Pasadena, California. 4. Gerardin, M. and Rixen, D. (1997). Mechanical Vibrations, John Wiley & Sons, New York. 5. Tioshenko, S.P., Young, D.H., and Weaver, JR,W. (1974). Vibration Probles in Engineering, John Wiley & Sons, New York. 6. Blevins, R.D. (1979). Forulas for Natural Frequency and Mode Shape, Von Nostrand Reinhold Co., New York.