ICSV18, Rio de Janeiro, Brazil, 10-14 July 2011 1

AMBIENTAL MODAL ANALYSIS OF 4B SG. MALAYSIAN BRIDGE

Cláudio Martins Department of Civil Engineering, Cefet Centro Federal de Educação Tecnológica de Minas Gerais, 30510-000, Belo Horizonte, Brazil. cjmartins @civil.cefetmg.br

Alberto Ortigão, Tiago Soares Terratek Ltd, Av. Érico Veríssimo, 901, 22621-000, Rio de Janeiro, Brazil

Rosli Bujang, Fadly Nor Azman SGS (M) Sdn Bhd, Kuala Lumpur, Malaysia

This paper presents the dynamic measurement and analyses of Bridge no 4 between the cities of Sebiu and Bintulu, province of Serawak, Malaysia. The bridge is 57 m long and consists of a mixed system with steel girders and a reinforced concrete deck. Reinforced concrete pillars support the structure, while large concrete blocks support its ends. The dynamic loading test was carried out with natural vibrations caused by wind and traffic, without inducing vibrations. Six independents measurements were carried out on the sides of the main deck in order to avoid traffic interruption. Signal analysis consisted of decomposition

The next step consisted of assembling a structural finite element model, followed by its validation. This step consisted of introducing damage to the model until the output eigenvalues agree with measured ones. This is a trial and error process aimed at to obtain agreement for the first five modes. At the end, the model is considered validated. Then it is used to analyse structural behaviour under design loading conditions.

1. Introduction

Identification of dynamic response of civil engineering structures through its vibrations is an important step to analyse their behaviour. The procedure is named Experimental Modal Analysis (Maia et al., 1998) is aimed at obtaining natural frequencies, vibration mode shapes and damping parameters of a structure.

Albeit this technique is well developed mainly in aeronautic and mechanical engineering, it has seldom been applied extensively in large civil engineering structures. Indeed, most aeronautical and mechanical structures are small sized and can be easily be excited with vibrators and impact hammers (He & Fu, 2001).

On the other hand, civil engineering structures can be large enough to render induced excita-tion to be inappropriate: very large and expensive vibrators may be necessary which in turn may

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lead to ricks of collapse of fragile structures. Additionally, the structure should not be used during the tests.

Until the 90’s electronics was not developed enough to enable precise ambient vibration measurements. This was about to change by late 90’s with the development of 24 bit analogue to digital converters. This made possible Environmental Modal Analysis (EMA) to take place with structures naturally excited by wind, waves, traffic and other equipment which turns out to be an effective and less costly method to test structures (Crawford & Ward, 1964, Trifunac, 1972, Rodri-gues, 2004).

One important characteristic of EMA is that excitation forces are unknown. Therefore, it is im-portant to assume non-stationary behaviour for the excitation source, i.e., its presents the character-istics of blank noise which has constant spectral density and a nil average. Therefore, vibration modes can be excited and well identified even in very rigid structures (Giraldo et al., 2009). In-deed, recent developments in electronics has lead to very accurate recording systems with sensitiv-ity as accurate as 10-8 g.

EMA tests usually lead to large quantities of recorded data, therefore, which are only possible to analyse with fast computers and high quality software employing state-of-the-art digictal signal analysis techniques (Rodrigues, 2004).

Modal identification techniques can classified into frequency or time domain. The first, are based on the analysis of the response spectral density obtained from Fast Fourier Transform (FFT) processing of time history data. (Brincker et al, 2001 and Rodrigues, 2004). The time domain method needs a numerical algorythm to simulate the dynamic behaviour of the structure, followed by the identification of modal parameters, until there is an agreement between measured and calcu-lated results. This method can be applied to the correlation functions or directly to the response time series.

Several methods based on correlation functions adopt parametric techniques with or without con-trolled loading. They are: the Ibrahim method ((Ibrahim and Mikulcik, 1977, the Minimum Squares with Complex Exponential method (Brown et al., 1979) and Subspace Stochastic Identification (SSI) method (Peeters, 2000). Rodrigues (2004) and Giraldo et al. (2009) present details of time domain techniques.

This paper present the results of the dynamic monitoring of Bridge no 4B located on the road be-tween the cities of Sebiu and Bintulu, province of Serawak, Malaysia. The measurements consist of precise vibration recording on six positions on the bridge deck, which as opened to the traffic.

Time series data were analysed using frequency domain decomposition techniques rendering natural vibration resonances. This was followed by a numerical model of the structure which was then calibrated by applying damage to the model until an agreement with measured spectrum was obtained. At this stage the model was said to be calibrated and could be used to analyse structure behaviour under several loading scenarios.

2. Mathematical background

In Finite Element Method (FEM) the continuum is modelled by a number of small finite ele-ments connected by nodes and faces. Variables are then described by their nodal values which yield a discrete system of ordinary differential equations (Zienkiewicz and Taylor, 1989, Hughes, 2000) given by:

FKUUDUM =++ &&& (1)

Where M , D and K represent respectively mass, damping and stiffness matrices, while U and F correspond to displacement and loading vectors. The dot over a variable means time derivative.

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The dynamic structural behaviour can be understood by means of vibration modes and natural frequencies of the model. This is chieved by solving the eigenvalues: eigenvalues and eigenvectors. Neglecting damping affects:

0UMK =− jj )( 2

ω (2)

where jω is the natural frequency of the j order and jU the corresponding mode shape.

The second order differential equation can be described in the following way, taking the vector X given by:

=

U

UX

& (3)

The dynamic equilibrium equation given by equation (1) is

CXY

BFAXX

=

+=&

(4)

where C is the system observation matrix and Y the observed response. A and B are the state and input matrices and are given by:

−−

=−−

DMKM

I0A 11 e

=

−1M

0B (5)

3. Signal decomposition in frequency domain

Frequency domain decomposition (FDD) also known as “Peak Picking” was applied to analyse the measurements. This technique enables the identification of the Frequency Response Function (FRF) (Baptista, 2004).

The estimation of the modal parameters is carried out by Singular Value Decomposition (SVD) of each spectral density matrix in single degree of freedom (SDF) systems. Single values represent estimates of the SDF and single vectors are modal shape estimates.

In general, this method presumes that the vibration modes can be estimated directly from spectral density functions, calculated from random input excitation. The peak picking method is recom-mended to slightly damped structures and the output is very sensible to frequency change (Baptista, 2004).

4. Structural characteristics and field measurements

The bridge structure is 57 m long and consists of a mix system of steel girders and a reinforced concrete main deck. The internal span is 29 m long supported by solid reinforced concrete pillars, while reinforced concrete pillars and walls support the 14 m external spans.

4.1 Measurements

Figure 1 presents a general view of the bridge, the vibration recorder and measurement positions on the structure.

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General view

Accelerometer

Points S1, P1, S2, R8, P2 e S3

Figure 1 Bridge general view, vibration recording equipment and measurement location

Figure 2 presents the spectral density matrix and Table 1 summarises the modal parameters after applying the SSI procedure.

.

Figure 2 Spectral density matrix

Table 1 Experimental modal parameters

Mode 1 Mode 2 Mode 3 Mode 4 Mode 5 Mode 6 Freq (Hz) 3.8303 4.36802 9.289 10.150 10.5527 11.4629

Damping (%) 0.0083118 0.0035362 0.0018078 0.0026077 0.001793 0.00113

4.2 Numerical model

Figure 3 presents a view of the FEM numerical model of the bridge including 169 bar elements and 192 thin plate elements and 1064 nodes.

The next step was calibration of the undamaged model in order to obtain agreement with the measured spectrum. This step consisted of a trial and error procedure in which damage was intro-duced to the model selected elements. The following damage types were introduced: discontinuity, section reduction, plastic hinges, chance to the elastic parameters of the materials and realeasing fixities of selected structural members.

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Table 2 presents the final results of this calibration procedure and the final model is said to be validated. Figure 4 presents the corresponding vibration modes.

Figure 3 Numerical model 3D view

Table 2 Natural frequencies (Hz)

Mode Experimental values Undamaged model values Damaged model values 1 3.8303 3.995 3.807 2 4.36802 4.340 4.190 3 9.289 9.811 9.143 4 10.150 10.777 9.762 5 10.5527 11.350 10.625 6 11.4629 15.399 12.043

5. Analysis of the results

The resulting natural frequencies and damping values indicate the bridge presents high stiffness and low damping, which is a strong indication of low damage level.

Accordingly, the initial (undamaged) numerical model and the final one show little change, as a result of minor changes to obtain the final model.

The accuracy of the calibrated model increase as the model is calibrated and, at final stage, the maximum error was less than 5% for the seven initial modes. Therefore, the model is considered to be validated for further structural analyses.

The validated model indicates the maximum midspan vertical displacement was 37 mm, corre-sponding to a distortion of L/783, which are within limits of BS-5400-2. The model also predicted that stresses and strains in the steell girders are within acceptable limits. On the other hand, the span supports steel ratios are too low, although still within the standard limits.

6. Conclusions

This paper presented a modal analysis of a bridge which was naturally excited by the effect of traffic and wind. Only six measurement positions on the bridge were necessary to model this struc-ture and to analyse its dynamic and static behaviour.

Signal processed used advanced digital techniques and yielded natural frequencies and damping values, even for the low amplitude signal data.

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A trial and error procedure was used to calibrated the model and this resulted in less than 5% er-ror in the first seven modes, resulting in a validated model.

This validated model was than used under different loading scenarios and, according to BS-5400-2, the damage level was found to be low

Acknowledgements JKR The Malaysaian Highways Department sponsored this investigation. Authors wish to thank Fapemig and CNPq, respectively the Minas Gerais State and National Re-

search Centres of Brazil

REFERENCES

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2. BS-5400-2- Steel. concrete and composite bridges – Part 2 – Specification for loads 3. Giraldo D F, Song W, Dyke S J, Caicedo J M 2009. Modal Identification through Ambient

Vibration: Comparative Study. Journal of Engineering Mechanics. Vol. 135. No. 8. pp. 759-770.

4. Brown D L, Allemang R J, Zimmerman R, Mergeay M. 1979. Parameter Estimation Tech-niques for Modal Analysis. SAE Technical Paper Series. No. 790221.

5. DIN-4150-3 - Structural vibration – Part 3 - Effects of vibration on structures 6. He J & Fu Z. 2001. Modal analysis. Butterworth-Heinemann. Oxford. England. 7. Rodrigues J 2004 Stochastic Modal Identification – Methods and Applications in Civil Engi-

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Mode 1 - Undamaged MEF–3.995 Hz

Mode 1 - Damaged MEF- 3.807 Hz

Mode 1 - Undamaged MEF- 4.340 Hz

Mode 1 - Damaged MEF- 4.190 Hz

Mode 1 - Undamaged MEF- 9.811 Hz

Mode 1 - Damaged MEF- 9.143 Hz

Mode 1 - Undamaged MEF- 10.777 Hz

Mode 1 - Damaged MEF- 9.762 Hz

Mode 1 - Undamaged MEF- 11.350 Hz

Mode 1 - Damaged MEF- 10.625 Hz

Mode 1 - Undamaged MEF- 15.399 Hz

Mode 1 - Damaged MEF- 12.043 Hz

Figure 4 Vibration modes