Advanced Finite Element Model of Tsing Ma Bridge for Structural Health Monitoring

ADVANCED FINITE ELEMENT MODEL OF TSING MA

BRIDGE FOR STRUCTURAL HEALTH MONITORING

Y. F. DUAN*,y,z, Y. L. XU*,x and Q. G. FEI*

*Department of Civil and Structural Engineering

The Hong Kong Polytechnic UniversityHung Hom, Kowloon, Hong Kong, China

yDepartment of Civil Engineering

Zhejiang University, Hangzhou, 310058, [email protected]

[email protected]@polyu.edu.hk

K. Y. WONG and K. W. Y. CHAN

Bridges and Structures Division

Highways Department, Hong Kong

Y. Q. NI and C. L. NG

Department of Civil and Structural Engineering

The Hong Kong Polytechnic University

Hung Hom, Kowloon, Hong Kong

Received 8 April 2008

Accepted 12 June 2010

The Tsing Ma Bridge is a cable suspension bridge carrying both highway and railway. A bridge

health monitoring system called wind and structural health monitoring system (WASHMS) has

been installed in the Tsing Ma Bridge and operated since 1997 to monitor the structural

performance and its associated loads and environments. However, there exists a possibility thatthe worst structural conditions may not be directly monitored due to the limited number of

sensors and the complexity of structure and loading conditions. Therefore, it is an essential task

to establish structural performance relationships between the critical locations/components of

the bridge and those instrumented by the WASHMS. Meanwhile, to develop and validatepractical and e®ective structural damage detection techniques and safety evaluation strategies,

the conventional modeling for cable-supported bridges by approximating the bridge deck as

continuous beams or grids is not applicable for simulation of real damage scenarios. To ful¯lthese tasks, a detailed full three-dimensional (3D) ¯nite element model of the Tsing Ma Bridge is

currently established for direct computation of the stress/strain states for all important bridge

components. This paper presents the details of establishing this full 3D ¯nite element model and

its calibration. The major structural components are modeled in detail and the connections andboundary conditions are modeled properly, which results in about half million elements for the

complete bridge model. The calibration of vibration modes and stresses/strains due to passing

‡Corresponding author.

International Journal of Structural Stability and DynamicsVol. 11, No. 2 (2011) 313�344

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455411004117

313

http://dx.doi.org/10.1142/S0219455411004117

trains is carried out, and a good agreement is found between the computed and measured

results.

Keywords: Structural health monitoring; ¯nite element model; cable suspension bridge; modal

analysis; ambient vibration test; model calibration.

1. Introduction

The Tsing Ma Bridge is a key component of the transportation network system in

Hong Kong. It is a cable suspension bridge carrying a dual three-lane highway on the

upper level of the bridge deck and two railway tracks and two carriageways on the

lower level within the bridge deck. The strength and integrity of the bridge during

the serviceability stage will, however, decrease due to the degradation induced by

tra±c, wind, temperature, stress corrosion, and environmental deterioration. To

protect such an immense capital investment and to assure user comfort and bridge

safety during the serviceability stage, a wind and structural health monitoring sys-

tem (WASHMS)1 has been installed and operated in the Tsing Ma Bridge since 1997

to monitor the integrity, durability, and reliability of the bridge. The WASHMS in

the Tsing Ma Bridge is composed of 276 sensors in seven types, namely, anem-

ometers, accelerometers, temperature sensors, strain gauges, global positioning sys-

tems, displacement transducers, and level sensing stations. However, the number of

sensors is always limited for such a large structure and locations of structural defects

or degradation may not be at the same positions as the sensors. A possibility exists

that the worst structural condition may not be directly monitored. Therefore, the

development of a structural performance relationship model for relating the struc-

tural performance conditions of the Tsing Ma Bridge to the measurement results at

limited locations from the current WASHMS through numerical modeling, statistical

analysis, and criticality and vulnerability analyses becomes an imperative task.

Based on the calibrated structural performance relationship model, an e®ective

bridge rating system can be developed as a rational basis for rating risk of major

bridge structural components and for selecting types and frequencies of inspection

and maintenance.

To establish such a rating system, comprehensive researches have been being

carried out by the authors. In order to develop the structural performance

relationship model, a detailed full three-dimensional (3D) ¯nite element model for

performance evaluation at stress/strain level for all important bridge components is

needed. Only based on such a ¯ne model, the criticality analysis can be carried out to

identify of the critical locations and components and to quantify the corresponding

levels of criticalities under various loading conditions including dead load, live load,

temperature load, wind load, seismic load, and even accidental load. Using such a

model, various damage scenarios can be simulated for developing and validating

practical and e®ective damage detection techniques. A bridge rating system can be

¯nally established for structural health monitoring, safety evaluation, and decision

making for inspection and maintenance of the Tsing Ma Bridge.

314 Y. F. Duan et al.

Previous researches lay solid foundation for this study. During the construction

stages of the Tsing Ma Bridge, ambient vibration measurements had been carried out

on the free standing towers and tower-cable system before erection of deck units,2 on

the tower-cable-deck system in erection stage,3 and on the bridge after the com-

pletion of deck welding connections.4 Finite element models were developed for

analytical study and comparison with the measured results, in which simpli¯ed

spinal beams are used to simulate the complicated bridge-deck truss system. Such

models were reasonable and suitable for the investigation of global structural

dynamic characteristics, but inappropriate for the present structural health moni-

toring, particularly for the monitoring of local components. For the purpose of health

monitoring, a hybrid 3D ¯nite element was developed by the Highways Department

of Hong Kong.1 Through modal analysis using the ¯nite element model and exper-

imental modal identi¯cation based on health monitoring measurement data, modal

frequencies and mode shapes were identi¯ed within 0�3.8 Hz frequency band.

The fruitful results were used for verifying and calibrating dynamic models for the

Tsing Ma Bridge, and for better understanding dynamic characteristics of the bridge.

However, this hybrid 3D model was still not ¯ne enough for criticality analysis

requiring results at strain/stress level to be directly compared with the measured

results. For example, the orthotropic decks (steel deck-plates supported by U-shape

troughs) were modeled by plate elements with equivalent depths so that the

measured results from strain gauges at the surfaces of deck plates or U-shape troughs

had no counterparts in computation results. A detailed full 3D ¯nite element model

for performance evaluation at stress/strain level is still needed.

The studies on long span cable supported bridge using ¯nite element and ¯eld

measurement technologies can be classi¯ed into four categories: (1) to determine the

dynamic characteristics of the bridges5�8; (2) to study particular advanced dynamic

issues, e.g., dynamic response of suspension bridge to typhoon and trains,9,10 and

bu®eting response of long span cable-supported bridges under skew winds11�14; (3) to

establish baseline model for future damage detection and safety evaluation by model

updating15,16; (4) to study particular issues on structural health monitoring, for

example damage detection17�19 and fatigue evaluation.20�22 However, none of the

reported ¯nite element models meets the requirement that stress/strain should be

directly computed for most structural components in the global model. With the

development of system design methodologies, sensing technologies, damage detection

algorithms, and safety evaluation methods,23,24 the structural health monitoring

system is gradually becoming technically available to provide information for eval-

uating structural integrity, durability, and reliability throughout the bridge life cycle

and to help to prioritize bridge inspection and maintenance.

This study aims to establish a ¯ne 3D ¯nite element model for structural per-

formance evaluation at stress/strain level, based on which a bridge rating system can

be developed for structural health monitoring, safety evaluation, and decision

making for inspection and maintenance of the Tsing Ma Bridge.

Advanced Finite Element Model of Tsing Ma Bridge 315

This paper presents the details of establishing a full 3D ¯nite element model, and

carrying out calibration of vibration modes and stresses/strains. To ful¯ll the

requirement of structural performance analyses at the level of stress and strain, all

structural components, such as bridge deck, towers, main cables, suspenders, saddles,

piers and anchorages, are modeled in detail. The connections and boundary con-

ditions including main cable-saddle connections, main cable-suspender connections,

rocker bearings, and sliding bearing are modeled properly. These modeling e®orts

assure that stresses/strains in major structural components can be directly computed

and validated. As a result, about half-million elements are used in the complete

bridge model.

2. Main Features of the Tsing Ma Bridge and Computer Programs

The Tsing Ma Bridge, stretching fromMaWan Island to Tsing Yi Island (Fig. 1), has

a main span of 1377m between MaWan tower in the West and Tsing Yi tower in the

east, Ma Wan approach span of 455m from Ma Wan anchorage to Ma Wan Tower,

and Tsing Yi approach span of 300m fromTsing Yi tower to Tsing Yi anchorage. Two

parallel main cables 36m apart in the north and south are accommodated by two pairs

of saddles at the top of Tsing Yi tower and MaWan tower, with their lower ends ¯xed

at Ma Wan anchorage and Tsing Yi anchorage. Both anchorages are gravity struc-

tures resting on the underlying rock.25 On the Ma Wan side, the main cables are also

held on the saddles on Pier M2, at a horizontal distance of 355.5m from Ma Wan

Tower. The bridge deck in Ma Wan approach span are supported by 19 pairs of

suspender units hung from the main cables, Pier M2 and another Pier (M1) at a

horizontal distance of 76.5m away from Pier M2. In the main span, 76 pairs of

suspender units from the main cables support the bridge deck to make a minimum

clearance of 62m. On the Tsing Yi side, the deck is supported by three piers (T3, T2,

and T1, with intervals of 72m) rather than by suspender units. This arrangement

introduces asymmetry with respect to the mid span of the bridge.

Modeling and simulation work is executed by MSC/PATRAN as model builder

and MSC/NASTRAN as ¯nite element solver. Among many advantages of MSC/

PATRAN as model builder, the unique one is its function of integrating multiple

1377m355.5m76.5m23m 300m

72m 72m 72m 72m

206.4m 206.4m

Anchorage

Tsing Yi IslandAnchorage

Ma Wan Island

78.58m

Tsing Yi TowerMa Wan Tower

M1 M2 T1T2T3

Fig. 1. Con¯guration of Tsing Ma Bridge.


model components into the whole model so that the modeling task can be ful¯lled by

several programmers individually and simultaneously to promote the project pro-

gress. The most important reason for choosing MSC/NASTRAN as the ¯nite

element solver is its function of parallel processing and multi-central processor unit

(CPU) since millions of degree-of-freedom (DOF) will be involved in the global

bridge model which may be out of the handling ability of other programs. Thanks to

the function of parallel processing and multi-CPU of MSC/NSATRAN, and

resorting to the hardware of the 64-bit Itanium Server with 8 CPUs (each of 1.5GHz,

under the HP-UX operating system) provided by the Highways Department of Hong

Kong, this project is becoming feasible and practical.

3. Finite Element Modeling

The work on establishment of a full 3D ¯nite element model of the Tsing Ma Bridge

are divided into four major parts: (1) modeling of bridge deck; (2) modeling of towers

and piers; (3) modeling of cable system and ¯xture components; and (4) modeling of

the global bridge structure.

3.1. Modeling of bridge deck

Since the Tsing Ma Bridge carries both highway and railway, both structural and

geometric con¯gurations of the bridge decks are very complicated and di®erent at

di®erent locations. Nevertheless, the bridge decks can be e®ectively modeled and

assembled by a number of modules: (1) deck module of main span, (2) deck module

at Ma Wan tower (3) deck module of Ma Wan approach span, (4) deck module at

Tsing Yi tower, and (5) deck module of Tsing Yi approach span.

3.1.1. Bridge deck module of main span

The bridge deck at the main span is a suspended deck and the structural con¯gur-

ation is typical for every 18-m segment. Therefore, the modeling of full span of the

bridge deck in the main span can be achieved by assembling a typical suspended deck

module of 18-m long. As shown in Fig. 2, a typical 18-m suspended deck module

consists of longitudinal trusses, cross frames, highway decks, railway tracks, and

bracings. Two longitudinal trusses link up the cross frames along the bridge longi-

tudinal axis, acting as the main girder of the bridge. Each longitudinal truss

is comprised of upper and lower chords and vertical and diagonal members. For the

18-m module, there are ¯ve cross frames, one main cross frame (in the middle) and

four intermediate cross frames, 4.5m apart from each other. Each cross frame is

comprised of upper and lower chords, inner struts, outer struts (also the vertical

members of longitudinal trusses), and upper and lower inclined edge members.

Through suspender units connected to the intersections of edge members of the main

cross frame, this deck module is suspended to the main cable. Two pairs of sway


bracings are connected from the suspension points at the main cross frame to the

outer ends of the upper chords of the two adjacent intermediate cross frames to

strengthen the structural stability. Two symmetric bays of top orthotropic highway

decks are supported by the upper chords of cross frames and longitudinal trusses.

Between them are a row of top center cross bracings stretching from neighboring

Cross bracings (top center)Orthotropic

deck (top)

Cross frames

(main & intermediate)

Corrugated sheets

Longitudinal trusses Cross bracings

(bottom centre)Cross bracings

(bottom outer)

Railway tracks

Orthotropic deck (bottom)

(a)

Longitudinal truss

Sway

bracings

Intermediate cross frame

Main cross frame

(b)

Tee diaphragm

Top viewBottom chord

(Cross frame) Rail waybeams

Track plate

Bottom view

(d)

Deck plate

Deck trough

(c) (e)

Fig. 2. A typical 18-m suspended deck module: (a) 3D view; (b) Frames and trusses; (c) Orthotropic deck;

(d) Railway tracks and (e) Connections among di®erent components.


cross frames. Another two symmetric bays of highway decks are supported by the

lower chords of cross frames, laterally between the inner and outer struts. Two

symmetric railway tracks are also supported by the lower chords of cross frames, but

laterally between the inner struts. One row of bottom center cross bracings are

between the two railway tracks, and two rows of bottom outer cross bracings are

between the bottom highway bridge decks and railway tracks. Corrugated sheets

covering the edge members of the cross frames are used to protect against wind, rain,

and other environmental factors. This deck module is symmetric to the middle

vertical plane along the longitudinal bridge axis, with a width of 2� 20:5m, a lateral

distance between the two suspension points at the main cross frame of 2� 18m, a

height of 8.0m and an inner clearance in the middle of 5.35m.

The highway decks and railway tracks are modeled in detail in order to obtain the

stress/strain states of structures to be compared with the measured results from the

strain gauges attached on the plates, troughs, and rail waybeams. The orthotropic

decks are made of steel deck plates sti®ened by deck troughs. The deck plates are

modeled as 20-DOF shell elements (QUAD4), and the troughs 12-DOF beam

elements (BAR2). Since the troughs are very closely spaced (at the intervals of about

0.6m), the deck plates using shared nodes with the troughs and the cross frame

chords connected to the troughs need to be meshed very ¯nely in accordance with the

location of troughs. Along the longitudinal bridge axis, the 18-m deck plates are

meshed as 16 divisions. As a result, 800 beam elements and 800 shell elements are

generated for the top highway decks, and 384 beam elements and 384 shell elements

for the bottom highway decks. The railway tracks are composed of track plates, rail

waybeams, and tee diaphragms. The track plates modeled as 20-DOF shell elements

(QUAD4) are supported by two pairs of rail waybeams modeled as 12-DOF beam

elements (BAR2) using shared nodes with the shell elements. The tee diaphragms are

also modeled as 12-DOF beam elements (BAR2) using shared nodes with shell

elements of track plates and beam elements of rail waybeams.

All the frame and longitudinal trusses are modeled as 12-DOF beam elements

(BAR2) that are appropriately meshed for connections with highway decks and

railway tracks. Since the highway decks and railway tracks are vertically at di®erent

levels with the cross frames and longitudinal trusses, multi-point constraints (MPCs)

are used to connect them to simulate the master�slave relationship. There are

totally 1922 nodes, 3028 elements, and 478 MPCs in this ¯nite element model of a

typical 18-m suspended deck module.

\Steel_Deck" (Table 1) is the material used for all the elements of this deck

module except that the edge members of main cross frame are modeled using

\Steel_Rigid" (Table 1) whose elastic modulus is 10 times that of steel due to the

heavily sti®ened conditions at the suspension points. The density of \Steel_Deck"

and \Steel_Rigid" is 9000 kg=m3, larger than that of real steel (7850 kg=m3),

accounting for the e®ect of pavements above the deck plates and many accessory

components; and its elastic modulus is 2:1� 1011 N=m2, the same as real steel.


3.1.2. Other bridge deck modules

The bridge deck module at Ma Wan tower is individually established because its

structural con¯guration is quite di®erent from the typical suspended deck module in

the main span as described above. This deck module is in a length of 108m and it is

symmetrical about a bearing cross frame which contacts with Ma Wan tower in

bearing connections. The completed FEM of the deck module at Ma Wan tower is

shown in Fig. 3. This deck module is mainly constructed using longitudinal trusses,

main and intermediate cross frames, orthotropic decks, bracings, railway tracks, and

corrugated fairing sheets. There are totally 25 cross frames in the deck module at Ma

Wan tower, and each cross frame is 4.5m apart from its neighboring ones. One major

structural di®erence of this deck module from the suspended deck module in the main

span is the additional provision of two inner longitudinal trusses which are in a span

of 108m. The top orthotropic deck is in full width and spans 81m long without

separation at the central part. Despite the di®erences in the outer longitudinal

trusses and the top orthotropic deck, their modeling is in a similar way to the

modeling of the suspended deck modules in the main span.

Inner longitudinal trusses

Outer longitudinal trusses

Bearing cross frame

Bracings

(bottom)

Bracings

(upper)

Fig. 3. Finite element model of deck module at Ma Wan tower.

Table 1. Properties of materials used in global bridge model.

Material classi¯cations Elastic modulus (N=m2) Density (kg=m3) Poisson ratio

Concrete 2:6� 1010 2500 0.2

Reinforced_Concrete 3:4� 1010 2500 0.2

Structural_Steel 2:1� 1011 7850 0.3

Steel_Cable (23�) 1:96� 1011 7850 0.3

Steel_Deck 2:1� 1011 9000 0.3

Steel_Rigid 2:1� 1012 9000 0.3

Steel_Saddle 2:1� 1011 9000 0.3


The bridge deck of Ma Wan approach span in a length of 197m is modeled as one

deck module and the completed FEM is shown in Fig. 4. This deck module is sup-

ported at three locations: (1) Ma Wan anchorage; (2) Pier M1; and (3) Pier M2.

This deck module is mainly constructed from longitudinal trusses, cross frames,

orthotropic decks, bracings, railway tracks, and corrugated fairing sheets. The

notable di®erence of deck module in the Ma Wan approach span is its edging shape.

Therefore, the edge members of typical cross frames and the bearing cross frames at

M2, M1, and anchorage are di®erent from the counterparts of cross frames in the

suspended deck modules.

The bridge deck module at Tsing Yi tower is established for connections to the

suspended deck module in the main span and the deck module in Tsing Yi approach

span. This deck module is in a length of 58.5m and contacts with Tsing Yi tower in

bearing connections. The completed FEM of the deck module at Tsing Yi tower is

shown in Fig. 5. This deck module is structurally composed of outer and inner

longitudinal trusses, main and intermediate cross frames, orthotropic decks, and top

and bottom deck bracings, railway tracks, and corrugated fairing sheets.

Pier M1

Pier M2

Ma Wan

anchorage

Suspended

segment

Fig. 4. Finite element model of Ma Wan approach span.

Bearing cross frame

Fig. 5. Finite element model of deck module at Tsing Yi tower.


The bridge deck in Tsing Yi approach span is in a total length of 288 m. The whole

span is supported by Tsing Yi tower, Piers T3, T2, T1, and Tsing Yi anchorage in

equal spans. The modeling of this part of deck is separated into two deck modules —

Module Tower/T3/T2 and Module T2/T1/Anchorage as shown in Fig. 6. For both

modules, there are one pair of outer longitudinal trusses and one pair of inner

longitudinal trusses acting as the main girders. Because the deck between Piers T2

and T1 is wider than the other three spans of the Tsing Yi approach span, there is an

additional pair of inclined trusses placed on the two outermost sides of this deck

segment.

3.2. Modeling of bridge towers and Piers

3.2.1. Bridge towers

Each of MaWan and Tsing Yi towers (Fig. 7) is composed of two reinforced concrete

legs built on massive reinforced concrete foundations and four deep pre-stressed

portal beams embedded with steel trusses. Each portal beam includes a steel truss

cast in the concrete enclosing a narrow corridor for access between legs. One special

Tsing Yi tower

Pier T2

Pier T3

(a)

Tsing Yi anchorage

Pier T1

Pier T2

(b)

Fig. 6. Finite element model of Tsing Yi approach span: (a) Tower/T3/T2 and (b) T2/T1/anchorage.


feature of the portal beams is that they are composite structures of reinforced con-

crete and embedded steelwork truss consisting of horizontal, diagonal, and vertical

members. The geometric and structural con¯gurations are almost the same for the

two towers except that the topmost portal beam in Ma Man tower is 150mm higher

than the counterparts in Tsing Yi tower. The reinforced concrete in tower legs,

foundations, and portal beams is modeled as 24-DOF solid elements (Hex8), and

steel trusses in portal beams are modeled as 12-DOF beam elements (BAR2).

Additionally, rigid elements are used for connection between the ends of horizontal

steel truss members and the surface of tower legs. The material properties of

\Reinforced_Concrete" and \Structural_Steel" as listed in Table 1 are used for the

reinforced concrete and steel trusses, respectively.

Previous researches1,4 indicated modal interactions between the deck, cable, and

towers; therefore, it is essential to calibrate the model of free standing towers using

the measurement data during construction stage.2 Modal analysis for the present

tower models has been carried out before assembling them into the entire bridge

model and a good agreement between the computed and measured dynamic

characteristics was achieved.26

3.2.2. Bridge Piers

The two side spans on the Ma Wan side and Tsing Yi side are supported by two and

three Piers (Fig. 1), respectively. As shown in Fig. 8, Piers M1, T2, and T3 are free-

standing piers of similar design but with di®erent heights, which only provide ver-

tical supports to the deck. Pier M2 provides lateral restraint to the bridge deck and

carries two saddles at its top above the deck. These two saddles de°ect the main

cables through a small angle. Pier T1 is part of the approach road and slip road

Fig. 7. Finite element model of bridge towers.


structure on the Tsing Yi side. It provides both vertical support and lateral restraint

to the bridge deck. All supporting piers in the side spans are reinforced concrete

structures and built on reinforced concrete pad footing supported on competent rock.

The ¯ve bridge piers are modeled by solid elements (Hex8 elements and Wedge6

elements) and shell elements (Quad4 elements and Tri3 elements). As noted that the

¯ve piers are not as heavily reinforced as the bridge towers, the property values of

\Concrete" listed in Table 1 are adopted. The value of elastic modulus 2:6�1010 N=m2 is lower than that of \Reinforced_Concrete," 3:4� 1010 N=m2, used for

the modeling of the tower, while the density and Poisson ratio are the same with

those for \Reinforced_Contrete."

Pier leg

Footing

Tie beam

(a)

Tie beam

(Upper)

Pier leg

Footing

Tie beam

(Lower)

(b)

Footing

Wall panel

(c)

Fig. 8. Finite element models of bridge piers: (a) Piers M1/T2/T3; (b) Pier M2 and (c) Pier T1.


3.3. Modeling of cable system and cable ¯xture components

3.3.1. Cable system

The cable system (Fig. 9) consisting of two main cables, 95 pairs of suspender

units, and 95 pairs of cable bands is the main supporting structure in the cable

suspension of the Tsing Ma Bridge. The two main cables are 36m apart, each with

91 strands of parallel galvanized steel wires in the main span and 97 strands in the

approach spans. The resultant cables have an overall diameter of 1.1m after

compacting, a cross-sectional area of 0:759m2 in the main span and 0:801m2 in

approach spans. The main cables are wrapped by the cable bands at the connection

locations to facilitate installing suspender units onto the main cables. Each sus-

pender unit consists of two pairs of wire ropes of 76mm diameter passing over the

clamps on the cable bands and then attached to main cross frames of the bridge

decks. The distance between neighboring suspender units is 18m along the longi-

tudinal bridge axis.

The main cables are modeled as cable elements: 14-DOF two-node beam element

(BAR2) considering di®erential sti®ness due to internal tensions. The pro¯les of the

main cables are taken as the geometry under dead load at design temperature (23�)from the design drawings. The horizontal tensions are 405838 kN for the main span,

Tsing Yi approach span, and Ma Wan approach span from Ma Wan Tower to Pier

M2, and 400013 kN for the other part of Ma Wan approach span from Pier M2 to Ma

Wan anchorage. The four wire ropes within each suspender unit are modeled by a

single 14-DOF two-node beam element (BAR2) considering the e®ect of internal

tensions, with an equivalent radius of 76mm. Two 4-DOF pipe elements (TUBE)

simulate each cable band. The connections among main cables, cable bands, and

suspender units are achieved using shared nodes in the ¯nite element model. The

total number of elements used in the Ma Wan approach span and the main span are

58 and 229, respectively. For the main cable in the Tsing Yi approach span, there are

only nine elements used. Five more elements are used to ¯x the main cable on the

tower saddle at the top of Ma Wan tower and Tsing Yi tower, respectively. The

material for the cable system is \Steel_Cable" (Table 1), of which the properties

follow design values.

Fig. 9. Finite element model of cable system.


3.3.2. Cable ¯xture components

The major cable ¯xture components of the Tsing Ma Bridge include two pairs of

tower saddles, one pair of pier saddles at Pier M2, the Ma Wan anchorage, and the

Tsing Yi anchorage.

The FEM modeling of the tower saddles is the same for either tower. As shown in

Fig. 10, the tower saddle is composed of three parts: the upper part is a U-shaped and

curved steel channel where the main cable is tightly clamped along the curved

trough; the lower part is underneath the steel channel as a supporting structure

consisting of a series of steel plates; and the bottom part is a bearing plate mounted

on the top of tower leg. The upper part of tower saddle is modeled by 24-DOF solid

elements (Hex8 elements). The upper part is relatively ¯ne meshed to better model

the curvature and a total of 250 elements are used. Since the lower part of tower

saddle and the bearing plate are constructed from pieces of steel plates, 20-DOF shell

elements (Quad4 elements) are used. The bearing plate is so meshed as to match the

element grid of the tower at its top surface. The upper and bearing part use the

material of \Steel_Saddle" in Table 1, while the lower part use \Steel _Rigid" in

Table 1.

Di®erent from the tower saddles, on Pier M2 are rotatable pier saddles. The cables

are ¯xed on the pier saddles, but the pier saddles are rotatable relative to the pier,

which allows the adaptation of the main cable con¯gurations. Solid element (HEX8,

and Wedge6) and shell element (QUAD4) are employed to model it. The material

\Structural_Steel" in Table 1 is used in the modeling.

The Ma Wan anchorage and Tsing Yi anchorage ¯x the lower ends of main cables

at Ma Wan and Tsing Ying approach spans, respectively. They are modeled using

solid (HEX8, Wedge6) and shell elements (QUAD4). The material \Concrete" in

Table 1 is used for all the anchorages and piers.

3.4. Modeling of global bridge structure

After the local components are ready as illustrated in the previous sections, they can

be assembled together to obtain the global model. The assembly procedures are: (1)

to assemble all the deck modules to form the whole deck in accordance with the

Fig. 10. Finite element model of tower saddles.


designed deck pro¯le; (2) to integrate the towers and piers into the model; (3) to

include the cable system and cable ¯xture components (saddles and anchorages); and

(4) to properly model all connections among di®erent components and (5) for proper

modeling boundary conditions for the global structure.

3.4.1. Integration of bridge deck

For the convenience in integrating the bridge deck components to form a complete

bridge deck model, a global coordinate system for the whole bridge and a pro¯le for

the bridge deck have been set up before building up these deck modules. In the global

coordinate system (x�y�z), the x-axis is along the longitudinal bridge axis (from

West to East), originating from the location of the Ma Wan abutment bearings

(Chainage 23 128.00) and ending at the location of the Tsing Ying abutment

bearings (Chainage 25 288.00) with a total length of 2160m; the y-axis is along the

lateral direction (perpendicular to the bridge axis) with a positive direction from the

Hong Kong side (South) to the New Territories side (North); the z-axis is along the

vertical direction initiating from Principal Datum Hong Kong. Since the bridge deck

is structurally formed by 481 cross frames interconnected by the longitudinal trusses,

the pro¯le of the deck can be geometrically illustrated by the locations of these cross

frames in terms of the upper freeway level. By using this route pro¯le datum line and

the global coordinate system, the above-mentioned deck modules are built in their

corresponding locations which are ready for the ¯nal integration. In consideration of

the similar structural con¯gurations, the suspended deck units at the Ma Wan

approach span and at the main span can be modeled using a basic 18-m deck module

with some small modi¯cation. Because the deck is cambered along the span, the 18-m

section is located using the suspension points, and then rotated by the right angle,

and ¯nally connected to the next 18-m section by merging the connection nodes. All

the nonsuspended deck units are directly modeled as parts of their corresponding

deck modules. After the ¯ve deck modules have been completed, the entire bridge

deck can be formed by integrating them together by merging the connection nodes

between neighboring deck modules with reference to the route pro¯le datum line and

the common global coordinate system.

3.4.2. Modeling of connections

The formation of a completed global bridge model also includes the connections

between di®erent bridge components and the boundary conditions (or supports).

Deck and tower connections

At Ma Wan tower, the bridge deck is connected to the bottom cross beam of the

tower through four articulated link bearings (or rockers) and to the tower legs

through four lateral bearings (rollers). The articulated link bearings restrict the

movement in the vertical direction (z). The lateral bearings are to restrain the lateral

movement (y) of the deck. Therefore, the deck is allowed to move along the


longitudinal direction of the bridge (x). At the Tsing Yi tower, there are also four

bottom bearings connecting the deck to the lowest cross beam of the tower and four

lateral bearings connecting the deck to the tower legs. The only di®erence of the

bearings at the Tsing Yi tower from those at the Ma Wan tower is that the four

bottom bearings at the Tsing Yi Tower are rollers rather than rockers as used at the

Ma Wan tower.

In modeling these connections, each of the bottom bearings (rollers) at the Tsing

Yi towers is modeled as an MPC connecting the tower cross beam and the deck cross

frame with the constraints that the z-direction (vertical) displacement of its upper

point attached to the deck cross frame is dependent on and equal to that of its lower

point attached to the tower beam. However, for the bottom bearings at the Ma Wan

tower, each of them is modeled as a rigid rod element pinned to the deck cross frame

and the tower cross beam for simulating the rocker bearings. The elasticity modulus

of the rigid rod is taken as 2:1� 1012 N=m2. For each of the lateral bearings at both

the MaWan and Tsing Yi towers, it is also modeled as an MPC connecting the tower

legs and the deck cross beam with the constraints that the y-direction (lateral)

displacement of its point on the deck cross frame is dependent on and equal to that of

its counterpart on the bridge legs.

Deck and pier connections

As shown in Fig. 1, there are two Piers (M1 and M2) in the Ma Wan approach span,

and three Piers (T3, T2, and T1) in the Tsing Yi approach span. Piers M1, T2, and

T3 are free-standing piers of similar design, which provide only bottom bearings

(rollers) as their connections with the decks. Pier M2 provides both bottom bearings

(rollers) and lateral bearings (rollers) to the bridge deck. Pier T1 is part of the

approach road and slip road structure on the Tsing Yi side. It also provides both

bottom and lateral bearings (rollers) to the bridge deck.

For each of Piers M1, T2, and T3, there are four bottom bearings (rollers) which

are modeled as MPCs with the constraint that the z-direction (vertical) displacement

of its upper point on the bridge deck cross frame is dependent on and equal to that of

it lower point on the piers. For Pier M2, there are four bottom bearings (rollers) and

four lateral bearings (rollers). For Pier T1, there exist six bottom bearings (rollers)

and four lateral bearings (rollers). Each of these bottom bearings is modeled as an

MPC with the constraint that the z-direction (vertical) displacement of its upper

point on the bridge deck cross frame is dependent on and equal to that of it lower

point on the pier. Each of the lateral bearings is modeled as an MPC with the

constraint that the y-direction (lateral) displacement of its point on the bridge deck

cross frame is dependent on and equal to that of its counterpart on the pier.

Cable and tower saddle connections

The tower saddles are one of the major bridge components used to ¯x the main cables

on the top of bridge towers and as guiders to change curvature of the main cables

between the main span and the approach spans. Since the cables are tightly ¯xed to


the tower saddles, in modeling the connections between the saddle and the main cable,

six pairs of rigid plates connected to the nodes of the U-shaped steel channel in the

saddlemodel are used to clamp themain cable by sharing nodes between themain cable

and rigid plates. The Young's Modulus of the rigid plates is 2:1� 1012 N=m2.

Deck/suspender and suspender/main cable connections

For the suspended deck units in the main span and in part of the Ma Wan approach

span, the deck is supported by the suspenders hung from the main cables. The

suspenders are connected to the main cross frames at the suspension points. In

modeling the connections between the deck and suspenders, the method of sharing

nodes is adopted. For each connection, the suspender is connected to the intersection

of the two inclined edge members of the main cross fame. The connections between

the main cables and suspenders are also achieved by sharing nodes.

3.4.3. Modeling of boundary conditions

Four sets of boundary conditions (or supports) are modeled:

(1) Fixed supports at the bottom of the foundations for all the Piers (M1, M2, T3,

T2, and T1) and towers (Ma Wan tower and Ting Yi tower)

Since the piers and towers including their foundations are modeled in details using

solid elements according to their geometric and structural con¯gurations, the ¯xed

supports provided from the ground to these structural components are applied to all

the nodes at the bottom of their foundations.

(2) Fixed supports at the ends of main cables

Within the anchorages, the main cables are split to bundles of strands with each

bundle ¯xed to the anchor block at di®erent inclinations. Realizing that the overall

e®ects of the anchorage on the main cables are to ¯x them at the locations where the

cables are entering into the anchorage, the models of anchorages are not included

into the global model and the cables are ¯xed at the cable ends which are originally

the connection points between the cables and the anchorages.

(3) Hinge supports at the deck end on the Ma Wan side

The hinge supports with constraints on the translational directions along the x-, y-,

and z-directions but without constraints on the rotations are adopted to replicate the

e®ects of Ma Wan anchorages on the bridge deck. This support condition is applied

to all the nodes of the lower cross beam of the bearing cross frame at the deck end on

the Ma Wan side by adding boundary conditions of constraints on the x-, y-, and

z-displacements.

(4) Sliding supports at the deck end on the Tsing Yi side

The bottom and lateral bearings at the Tsing Yi anchorage provide sliding supports

at the deck end on the Tsing Yi side. Since the Tsing Yi anchorage is not included in


the global bridge model, these supports are modeled as rollers which allow

the movement of deck along the longitudinal direction. The vertical roller supports

for modeling bottom bearings are achieved by applying the boundary conditions

of constraints on the y- and z-displacements to the nodes of the bottom cross beam of

the bearing cross frame. The horizontal roller supports for modeling lateral bearings

are achieved by applying the boundary conditions with constraint on the

y-displacement to the edge nodes of the bearing cross frame at the levels of the upper

and lower cross beams.

3.4.4. Main features of global bridge model

By integrating the local bridge components with the proper modeling of the con-

nections and boundary conditions, the entire global bridge model is established as

shown in Fig. 11. As a result, more than 300 thousand nodes, 450 thousand elements

including about 50 thousand MPCs, and 1.2 million DOFs are used to establish the

global bridge model.

The main features of this model can be summarized as: (1) the structural and

geometric con¯gurations of the original structures are well replicated; (2) the

damage of each of the structural members can be directly and precisely simulated;

and (3) the stress/strain state of structural components can be computed directly for

comparison with ¯eld measurements.

4. Calibration of Vibration Modes

Usually model updating is necessary for a newly established ¯nite element model.

Since the present 3D ¯nite element model is very ¯ne and the geometric, material,

and structural properties are well simulated, it is found that few e®orts are necessary

for the model updating. The present results are obtained from the established model

as described in previous sections.

Ma Wan Approach Span

Main Span

Tsing YiApproach Span

Fig. 11. Full 3D ¯nite element model of Tsing Ma Bridge.


4.1. Modal analysis

Considering the e®ects of o®sets and initial stresses, the modal analysis is carried out

by the modal analysis module (SOL103) in MSC/NASTRAN. It is found that the

modal frequencies are closely spaced with the ¯rst 100 modes between 0.071Hz and

1.309Hz as shown in Fig. 12.

Dynamic interactions among vertical, lateral, torsional, and longitudinal motions,

among deck, cables, and towers, and among main span and approach spans can be

found. Eight classi¯cations of mode shapes can be identi¯ed in the ¯rst 18 modes

(Table 2):

(1) Predominant in-phase lateralmotion of deck and cables inmain span: L1 (mode 1),

L2 (mode 4), and L3 (mode 13). The wave number is half, one, and one and a

half for L1, L2, and L3, respectively. The 3D isometric view of the mode shapes is

shown in Fig. 13.

(2) Predominant in-phase vertical motion of deck and cables in main span: V1

(mode 2), V2 (mode 3), V3 (mode 5), V4 (mode 9), andV5 (mode 17). As shown in

Fig. 14, the ¯rst vertical mode (V1) is anti symmetric, with a wave number of one,

while the second vertical mode (V2) is symmetric, with the wave number of a half.

The wave numbers for the third to ¯fthmodes are increasing from one and a half to

two and a half.

(3) Predominant torsion of deck and cables in main span: T1 (mode 12) and T2

(mode 15). As shown in Fig. 15, the wave number is a half for T1, and one for T2.

(4) Predominant out-of-phase lateral motion of two main cables in main span:

Cable_L1_out (mode 6), Cable_L2_out (mode 7), and Cable_L3_out (mode 18).

The lateral motion is in out-of-phase for the two main cables, and wave number

is a half, one, and one and a half for the these three modes, respectively.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20 40 60 80 100Mode order

Fre

quen

cy (

Hz)

Fig. 12. Distribution of modal frequency.


Tab

le2.

Frequency

andmod

eshap

e.

Mod

alno.

Classi¯cation

Frequency

Mod

eshap

e

FEM

Freq.(H

z)

Meas.

Freq.(H

z)

Relative

di®erence

(%)

Calibration

factor

(k)

Rootmean

square(r)

Normalized

di®erence

(e)

Modalassurance

criterion(M

AC)

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

1L1

0.07

10.06

92.9

−43

28.9

0.05

50.09

20.996

2V1

0.11

90.11

35.5

4910.1

0.03

70.05

510.998

3V2

0.14

60.13

95.2

−39

02.4

0.07

50.14

80.989

4L2

0.17

00.16

43.4

−42

14.8

0.11

10.19

00.982

5V3

0.19

20.18

44.5

−40

11.9

0.08

20.16

40.987

6Cab

le_ L

1_ou

t0.21

90.21

42.2

2968.2

0.14

00.20

20.979

7Cab

le_ L

2_ou

t0.22

70.22

60.3

−26

37.6

0.10

30.16

20.987

8Cab

le_ L

1_in

0.24

20.23

62.3

−28

28.9

0.14

60.31

70.949

9V4

0.24

90.24

13.5

3021.5

0.12

30.26

40.965

10Cab

le_ L

2_in

0.25

00.24

04.2

4407.5

0.08

60.14

80.989

11MW

_ V1

0.29

40.28

43.7

−26

84.1

0.05

50.17

00.986

12T1

0.29

60.26

710

.9−38

70.0

0.05

60.08

90.997

13L3

0.31

20.29

75.0

−41

14.5

0.16

60.29

70.955

14MW

_ Cab

le_ L

1_in

0.32

50.33

6−3.3

−16

42.0

0.03

90.16

60.922

15T2

0.33

20.32

03.6

3869.4

0.06

40.11

60.995

16MW

_ Cab

le_ L

1_ou

t0.33

60.34

7−3.2

1612.9

0.06

70.10

40.995

17V5

0.34

30.32

74.8

4967.7

0.21

00.40

00.944

18Cab

le_ L

3_ou

t0.35

30.35

20.2

−24

79.8

0.14

00.35

00.980


(5) Predominant in-phase lateral motion of main cables in main span: Cable_L1_in(mode 8) and Cable_L2_in (mode 10). The lateral motion is in the same phase for

the two main cables, while the lateral motion of deck in smaller amplitude is out

of phase with the motion of main cables.

(6) Predominant in-phase vertical motion of deck and cables in Ma Wan approach

span: MW_V1 (mode 11). The motion in MaWan approach span is predominant,

(a) (b) (c)

Fig. 13. The ¯rst three modes of in-phase lateral motion of deck and cables in main span: (a) L1-mode 1;

(b) L2-mode 4 and (c) L3-mode 13.

(a) (b) (c)

(d) (e)

Fig. 14. The ¯rst ¯ve modes of in-phase vertical motion of deck and cables in main span: (a) V1-mode 2;

(b) V2-mode 3; (c) V3-mode 5; (d) V4-mode 9 and (e) V5-mode 17.


interacted with the smaller amplitude vibration atmain span. The vertical motion

is in-phase for the deck and cables in a half wave.

(7) Predominant in-phase lateral motion of cables at Ma Wan approach span:

MW_Cable_L1_in (mode 14). The lateral motion is in the same phase for the two

main cables in a half wave.

(8) Predominant out-of-phase lateral motion of cables at Ma Wan approach span:

MW_Cable_L1_out (mode 16). The lateral motion is out of phase for the two

main cables in a half wave.

Therefore, the 18 vibration modes can be classi¯ed into three categories: (1) modes

dominated by in-phase vibrations of deck and cables in main span, including three

lateral (L1�L3), 5 vertical (V1�V5), and two torsional vibration modes (T1 and T2);

(2) modes dominated by vibrations of main cables in main span, consisting of two

out-of-phase (Cable_L1_out, and Cable_L2_out) and two in-phase (Cable_L1_in,and Cable_L2_in) vibration modes; (3) modes dominated by vibrations in Ma Wan

approach span, comprised of one in-phase vertical motion of deck and cables

(MW_V1), one in-phase lateral motion of main cables (MW_Cable_L1_in), and one

out-of-phase lateral motion of main cables (MW_Cable_L1_out).

4.2. Field measurement

The ambient vibration measurement of the Tsing Ma Bridge after the completion of

bridge deck, was carried out in 1997, by The Hong Kong Polytechnic University

(HKPU) under the auspices of the Hong Kong Highways Department.27 In that

measurement, for obtaining the global dynamic characteristics including frequencies

and mode shapes, the sensors were so located that the longitudinal, lateral, torsional,

and vertical motions of the bridge deck, the main cables, and the towers were

measured. The measurement cross sections at 18-m intervals are numbered as 1 to

108 from MaWan side to Tsing Yi side, among which 30 cross sections are chosen for

measurement. The cross sections No. 1�95 are corresponding to the 95 pairs of

(a) (b)

Fig. 15. The ¯rst two modes of torsion of deck and cables in main span: (a) T1-mode 12 and

(b) T2-mode 15.


suspenders. Because of the limited number of sensors, the measurement is carried out

cross section by cross section. Two reference cross sections No. 35 and 71 were

selected at approximately the quarter point and three-quarter point along the main

span, respectively. For the reference cross sections, accelerometers were placed at the

main cable and bridge deck, only on the Hong Kong Island side (South) of the bridge.

For other measurement cross sections, accelerometers were deployed at the deck and

main cables, on both the Hong Kong Island side (South) and the New Territories

Side (North). Measurement at each cross section involved synchronous acquisition of

signals at the main cables and bridge deck plus signals at one of the reference cross

sections. Signals from vibration in the vertical, lateral, and longitudinal directions

were acquired one at a time by reorientating the sensors. This arrangement allows

crossreference of all recorded signals through the measurements with sensors in the

reference cross sections.

The measurements at the reference cross section No. 35 were taken as reference for

allmeasurements on theMaWan side span and the halfmain span close to theMaWan

tower while the measurements at the reference cross section No. 71 were taken as

reference for those on the Tsing Yi side span and the half main span close to the Tsing

Yi tower. A separate measurement was made with sensors in both reference cross

sections serving as crossreference for all measurements. Each of the towers was also

measured with crossreferences to measurements at one of the reference cross sections.

Each of the sensors at the bridge deck is mounted onto the structural steelwork

inside the deck unit with a magnetic stand, close to the suspender of the corre-

sponding measurement cross section, and vertically at about the centroidal axis of

the deck unit. When monitoring the main cables, the accelerometer was mounted on

a magnetic stand ¯xed to the cable band at the measurement cross sections. The

signal cables were laid along the catwalks beneath the main cables and connected to

the acquisition station. When monitoring the towers, the accelerometer was located

at the top of each tower leg. The signal wires from the sensors were tied to an

adjacent anchor, running along the catwalk, hanging down along one of the sus-

penders and then connected to the acquisition station at the lower deck level.

Through experimental modal analyses, the ¯rst 18 modes of modal frequencies

and mode shapes were identi¯ed; and the results are shown and compared with the

computed results in the next section.

4.3. Correlations of computed and measured results

4.3.1. Modal frequency

The computed modal frequencies from the FEM model as presented in Table 2 are

compared with the measured results. The relative di®erence in modal frequency is

de¯ned as

d ¼ f FEM � fMeasure

fMeasure� 100%; ð1Þ


where f FEM and fMeasure are the computed FEM modal frequency and measured

modal frequency, respectively.

Table 2 compares the ¯rst 18 modal frequencies computed from the present model

with the measured results. A good agreement between them is found. The relative

di®erences are no more than 5.5% for most of them except that a relative di®erence of

10.9% is found for the ¯rst torsion mode (T1).

4.3.2. Mode shape

The measured mode shapes were normalized by normalizing the largest deformation

at measurement locations to \1." A calibration factor should be therefore obtained

for the computed results in comparing them with the measured results because that

the computed mode shapes are normalized by the generalized mass. The least square

method is used to determine these calibration factors by minimizing

Y¼

Xni¼1

½’Measurei � k’FEM

i �2 ð2Þ

whereQ

is the minimization objective; k is the calibration factor; n is the number of

measurement locations for obtaining the measured mode shape; and ’Measurei and

’FEMi are a set of n terms of the measured mode shape and a set of n terms of the

corresponding computed FEM mode shape.

The calibration factor is then given by

k ¼Pn

i¼1½’Measurei ’FEM

i �Pni¼1½’FEM

i ’FEMi � : ð3Þ

To evaluate the di®erences between the measured results with the calibrated results

by the present FEM, the root mean square r and the normalized di®erence e are

adopted:

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1 ½�Measurei � k�FEM

i �2n

sð4Þ

e ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1 ½’Measurei � k’FEM

i �2Pni¼1 ½’Measure

i �2

s: ð5Þ

In order to provide a measure of consistency between the measured and calculated

mode shape, modal assurance criterion (MAC) is also calculated:

MAC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1½’Measurei ’FEM

i �Pni¼1½’Measure

i ’FEMi �Pn

i¼1½’Measurei ’Measure

i �Pni¼1½’FEM

i ’FEMi �

s: ð6Þ

MAC takes values from zero, representing no consistent correspondence, to one,

representing a consistent correspondence. In this manner, if the two sets of data truly


exhibited a consistent relationship, a unity of MAC is approached and hence the

results from the present FEM are considered as reasonable ones.

The comparison between the computed mode shapes from FEM and the measured

mode shapes from ¯eld measurements for the ¯rst 18 modes are carried out and a

good agreement between them is observed for each mode. Due to the limited space

here, only the second lateral mode (L2, mode 4), second vertical (V2, mode 3), and

second torsional mode (T2, mode 15) are respectively shown in Figs. 16�18. The

horizontal coordinate in these ¯gures is the number of measurement cross sections

according to the numbering method in ambient vibration measurement, while the

vertical coordinate shows the vibration amplitude of the mode shapes. The calibrated

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 20 40 60 80 100

FEM South FEM North Measure South Measure North

(a)

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 20 40 60 80 100


(b)

Fig. 16. Second lateral mode (L2, mode 4): (a) Cable component and (b) Deck component.


computation results agree well with the measurement results, no matter for the north

and south sides and for the cable and deck components.

Table 2 summarizes all of the calibration factors (k), the root mean square (r), the

normalized di®erence (e), and the MAC which are involved in the computation and

analysis of the mode shapes. It can be seen that the values of modal assurance

criterion are greater than 0.92 and the averaged value reaches 0.98, showing that the

computed mode shapes are acceptable and highly consistent with the measured ones.

The maximum root mean square and normalized di®erences occurs for the mode 17,

the ¯fth vertical mode for the main span (V5). The averaged values of the root mean

square and the normalized di®erence for the ¯rst 18 modes are 0.097 and 0.191,

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 20 40 60 80 100


(a)

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 20 40 60 80 100


(b)

Fig. 17. Second vertical (V2, mode 3): (a) Cable component and (b) Deck component.


respectively. These di®erences are acceptable, considering the unavoidable errors in

the ¯eld measurements.

5. Calibration of Stresses/Strains

The present study aims to establish a ¯ne 3D ¯nite element model for structural

performance evaluation at stress/strain level, based on which a bridge rating system

can be developed. Therefore, calibration of the stresses/strains at monitoring

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50 60 70 80 90 100


(a)

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50 60 70 80 90 100


(b)

Fig. 18. Second torsional mode — z direction (T2, mode 15): (a) Cable component and (b) Deck

component.


locations installed with strain sensors is also conducted by comparing the computed

and measured results.

Due to the limited space, the calibration of strains at the cross frame °ange is

presented. The main span cross section 337.5m — about one-fourth of the main

span — to Tsing Yi tower (Figs. 1 and 19(a)) is selected as the monitoring cross

section. The strain sensor SSTLS04 is on the top of the cross frame beam °ange, as

shown in Fig. 19(b). The strains/stresses acquired by this sensor when individual

trains were passing the monitoring section are used for comparison with the com-

puted results. The con¯guration of the trains is shown in Fig. 20. Each train contains

several cars, and eight-car train and seven-car train are typical ones. Each car has

two axles, with a spacing of 14.350 m. For the neighboring cars, the front axle of the

back car is 6.125m from the back axle of the front car. When one individual train

passes the monitoring cross section, the variation of the measured strain is shown in

Fig. 21, for an eight-car train and a seven-car train, respectively. Using the developed

model and computation of moving loading e®ect, the computed strain is also shown

in Fig. 21. In order to quantify the agreement between the measured and computed

J

CL OF BRIDGE

(a)

SSTLS04SSTLS04

VIEW J1DETAIL J

740 500

J1

Unit : mm

(b)

Fig. 19. Location of the sensor for strain/stress comparisons: (a) Main-span cross section 337.5m to Tsing

Yi tower and (b) Location of Sensor SSTLS04.


Unit: m

14.350 6.125 14.350 6.125 6.125 14.350 6.125 14.350

Fig. 20. Con¯guration of trains.

84.5 85 85.5 86 86.5 87 87.5 88 88.5 89 89.50

5

10

15

20

25

30

35

Str

ain

(µε)

Time (s)

Measurement

Computation

(a)

1077.5 1078 1078.5 1079 1079.5 1080 1080.5 1081 1081.5 1082 1082.50

5

10

15

20

25

30

35

Str

ain

(µε)

Time (s)

Measurement

Computation

(b)

Fig. 21. Strain variation due to the passing of a train: (a) Eight-car train and (b) Seven-car train.


results, the relative di®erence r2 and correlation coe±cient R are de¯ned as

r2 ¼Pn

i¼1 ½"mi � "ci�2Pni¼1 "

2mi

; R ¼Pn

i¼1ð"mi � �"mÞð"ci � �"cÞj jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPni¼1 ð"mi � �"mÞ2

Pni¼1 ð"ci � �"cÞ2

p ; ð7Þ

where "mi is measured strain, "ci is computed strain, �"m is mean value of measured

strain, and �"c is mean value of computed strain. The closer to zero r2 is and the closer

to 1 R is, the better the agreement between the computed and measured results. The

values of r2 and R are 0.04 and 0.90 for the case of eight-car train, and 0.036 and 0.80

for the case of seven-car train. Therefore, a good agreement between the measured

and computed strain results is obtained.

6. Conclusions

A health monitoring oriented 3D ¯nite element model of the cable suspension of the

Tsing Ma Bridge has been established and calibrated using the dynamic character-

istics and stresses/strains. The details on modeling bridge components, integrating

all the bridge component models, modeling of the connections among the bridge

components, and modeling of the supports (or boundary conditions) of the global

bridge model have been presented. Due to the unique modeling requirement of

stress/strain level for structural performance analysis, more than 300 thousand

nodes, 450 thousand elements including about 50 thousand MPCs are used and 1.2

million DOFs are involved in the entire model. To the best knowledge of the authors,

this is the most detailed bridge model with the greatest number of elements ever

reported. Its success depends on the modern hardware and software development of

computation technologies.

The calibration of dynamic characteristics of the bridge model is carried out by

comparing the computation and ¯eld measurement results of modal frequencies and

mode shapes, and a good agreement is found. The ¯rst 18 modal frequencies and

mode shapes are computed using the modal analysis module SOL 103. For the mode

shapes, dynamic interactions among vertical, lateral, torsional, and longitudinal

motions, among deck, cables, and towers, and among main span and approach spans

are found. Eight classi¯cations of mode shapes are identi¯ed in the ¯rst 18 modes.

The results from ¯eld measurement of the Tsing Ma Bridge after the completion of

deck are used for the dynamic calibration. The 18 modal frequencies and mode

shapes obtained from the ¯eld measurement are compared with their counterparts of

computed results. The computed results agree well with the measured results. The

calibration of stresses/strains conducted by comparing the computed and measured

results of stresses/strains due to individual passing trains also shows a good agreement.

This structural healthmonitoring orientated ¯nite elementmodel can be used not only

for the simulation of damage scenarios for investigation of diagnosis and prognosis

algorithms, but also for the stress/strain analysis to establish the structural per-

formance relationshipbetween instrumented components/locations byWASHMSand

those that are not instrumented; and a bridge rating system will be further developed


for health monitoring, safety evaluation, fatigue life assessment, and decision making

for inspection and maintenance.

Acknowledgments

The ¯nancial support from the Highways Department of Hong Kong and The Hong

Kong Polytechnic University (PolyU Account No.: K-ZB43), the Hong Kong

Research Grants Council (Account No.: PolyU 5299/05E), and the National Natural

Science Foundation of China (Account No.: 90915008) is gratefully acknowledged.

All views expressed in this paper are entirely those of the authors.

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Advanced Finite Element Model of Tsing Ma Bridge for Structural Health Monitoring