Advances in Steel Structures - part 56 doc

FLEXIBLE PARAMETRIC STUDY ON CABLE-STAYED BRIDGES The present system focuses the design objectives on three parts for the preliminary design of the cable- stayed bridges: 1 the variation

530 B Zhou and M Hoshino

designs, which also may be decomposed into the hypothetical designs from recorded designs as well as from candidate designs recommended by the system or specified by the designer In this system, a learning, representing and storing method of Object-Oriented Multiple Regression Model (OOMRM) (Zhou and Hoshino 1998; 1999) is introduced to explain and represent the design situation This paper focuses on the process of the system realisation and not on the explanation of the multiple regression method (Balakrishnan et al 1965; Eric and John 1977; Hald 1952; Harald 1966; Hwang et al 1994; John 1967; Robb 1980; Samuel 1962; Warren 1976; William and Douglas 1980), the object-oriented programming technologies (Booch 1994; Rumbaugh et al 1991), OOMRM (Zhou and Hoshino 1998) and the parametric studies of the cable-stayed bridges (Agrawal 1997; Hegab 1989; Krishna et al 1985)

SIMPLE CONCEPT OF OOMRM

The Object-Oriented Multiple Regression Model (OOMRM) (Zhou and Hoshino 1998; 1999) is a method of knowledge engineering that integrates the object-oriented programming (OOP) and the multiple regression analysis for explaining and predicting (inference mechanism) the comple x engineering designs with the ability of adequacy and learning of the design situation The method deals CRK with an adding-MRM-overriding process through a number of temporary views for the design

situation to build up the knowledge base more completely and accurately, which can bring together knowledge from different domains about the design situation, by the process of trial-and-use It allows

evolution of the knowledge representation and storage with the change of CRK or with the change of design situations Both the numerical and qualitative knowledge are represented as STATE, RECOMMEND and RULE, which we will discuss later, and the refined knowledge is organised by OOP

The MRM general class, which is used to create different objects as instances for different purposes and to apply the adding-MRM-overriding process, involves following items:

LIll

CRK:

Condition:

Explanation:

Evaluation:

Prediction:

Source:

Condition: Girder continuous at tower

0.45>=Span_ratio>=0.35

-

~ n : Girder continuous at towel ~ - E ~ MRM (13, P )

O 40>=Span_ratio>=O.~ 35 f:~valuation: Mcc, SSn, Sn

~xplanation: M R M (]3, P ) [ Prediction: yo - ~' < y0 ": Y0 +

Mcc, SSR, SR [ Source: design, expert, site

Prediction: Y0 -~' < Yo ": Y0 § ~' Source: design, expert, site m

Figure 1" MRM general class and its derived objects of cable_ tension temporary views

(1) object name which describes the substance of the situation;

(2) CRK which is relative to the object;

(3) given conditions which should be satisfied;

(4) possible actions which may be performed:

(a) explanation of the situation which describes the relationship between CRK (from MRM); Co) evaluation of the situation which indicates the degree of strength and validity of the actions

(from MRM);

(c) prediction of the situation which is based on stored knowledge (from MRM);

Expert System of Flexible Parametric Study on Cable-Stayed Bridges

(5) source which the knowledge comes from

531

Figure 1 shows the representation of MRM (the MRM general class) and its derived objects of cable_ tension temporary views The properties of Explanation, Evaluation and Prediction are not explicitly

stored as static attribute values, and may be defined by functional expression or dynamic data

established by MRM that having the attributes of CRK, Condition and Source as arguments

FLEXIBLE PARAMETRIC STUDY ON CABLE-STAYED BRIDGES

The present system focuses the design objectives on three parts for the preliminary design of the cable- stayed bridges:

(1) the variations of topology arrangement of the side span length, the middle span length and the tower height above deck;

(2) the variations of cable arrangement with respect to the spacing of girder-cable connections and the unsupported spacing in middle span;

(3) the variations of cross section of girder, cable and tower

The parametric studies on the cable-stayed bridges have been reported on several papers (Agrawal

1997; Hegab 1989; Krishna et al 1985) However, a common feature of these papers is that the range

of the parameters is restricted in detailed values for widely interpolating by programming reuse and the variety of the parameters are fixed in several situations for applying to different design situations This study is tried to make the parameters representation space as flexible as possible by the introduction of the derived design parameters that could adjust the design parameters each other with the preliminarily stored knowledge, and represent functional explanations according to the design situation

(span ratio ) 1 (span ratio ) "[" (span ratio ) "1

/ / / / / I \ \ ~ ~ [~nsupp~ / / / / X 2 / / I \ \ ~ ~

I ~ [cable number~4] @ [lenght side x cable ratio s] (caote~gsnlyness) " ," I

J [cable number~4] @ [(lenght mid x cable ratio m] I_ r'

Figure 2: General arrangement: some parametric properties and derived parametric properties Table 1 shows some design properties used in the system for describing the candidate designs A candidate design is one of the hypothetical designs that is temporarily expected to explain and represent a certain design situation by the design properties according to the experience The table contains two groups of properties: parametric properties and derived parametric properties The parametric properties (specified by the designer or suggested by the system) describe the candidate design, while the derived parametric properties describe the relationships between the parametric properties To make the system more flexible, the parametric properties are restricted within the

interpolation range of the derived parametric properties (interpolation properties), not within the range

of the parametric properties themselves The range of the interpolation properties is designed flexibly

that it may be modified or extended when new knowledge is learned E.g., at the beginning, we would

learn the knowledge for the span_ratio range between 0.35~0.4; while the training examples increase, the range would be extended to 0.35~0.45 The derived parametric properties are purposely defined in

the form of ratios, which aim to be perceived easily by the designer to select and compare between the candidate designs Figure 2 shows some parametric properties and derived parametric properties in the

532

general arrangement

B Zhou and M Hoshino

TABLE 1 DESIGN PROPERTIES Properties

Side span length

Middle span length

Middle span unsupported length

Tower height above desk

Girder inertia moment

Tower inertia moment

Girder area

Total cable cross-sectional area

Tower area

Cable number

] Acronym Parametric Properties

side_length mid_length unsupportlength tower_height girder_inetria tower inetria girder_area cable area

t o w e r a r e a

cable nwnber

Restriction

span_ratio span_ratio unsupportratio tower ratio a cable_stiffness, tower_stiffness tower_stiffness

cable_stiffness, tower_stiffness cable_sttffness

tower_stiffness

4-100 Derived Parametric Properties

I Acronym

span_ratio unsupportratio tower_ratio cable_g_stiffness tower g_stiffness

Properties Side span to main span ratio

Middle span unsupported spacing to total span ratio

Tower height above desk to total span ratio

Cable to girder stiffness

Tower to ~irder stiffness

Interpolation Range

INTERPOLATION PROPERTIES

TABLE 2 PRODUCTIVE PROPERTIES Properties I Acronym

Productive Properties Maximum girder moment at middle span ]girder m max

Minimum girder moment at support I girder m m in

Maximum cable tension [cable tension

Tower base moment [tower_m

Maximum girder deflection Igirder deflection

Tower tip deflection I tower~deflection

Derived Productive Properties Conversion weight of girder

Weight of cable

Conversion weight of tower

girdercweigh cableweight tower cweight

I Expected Range

OBJECTWE PROPERTIES

OBJECTWE PROPERTIES

Instead of dealing with the complete results from the structural analysis program, only six important productive properties and three derived productive properties are stored for evaluating the candidate design The productive properties describe the structural behaviour due to the specified design in terms

of stress, deflection and weight They are sufficient for evaluating the behaviour in the preliminary design stage Both the productive properties and the derived productive properties are expected to be

within the range of the objective properties (given specifications or design constraints)

DEVELOPING STATIC KNOWLEDGE AND DYNAMIC KNOWLEDGE

As mentioned before, systematic and general knowledge of the cable-stayed bridges is hard to find for

a variety of design situations Fragmental design recommendations can be abstracted from experts or limited descriptions in documentary materials, which usually play a conceptual control or value-

restricted role in the design process; here, we call it static knowledge In the present system, the static knowledge is obtained from guidelines, books and papers (Agrawal 1997; Carl et al 1992; Troitsky 1988; Hegab 1989; Hunt et al 1997; Krishna et al 1985; Starossek 1996; Xanbakos 1993), and is represented as STATE, RECOMMEND and RULE

Expert System of Flexible Parametric Study on Cable-Stayed Bridges 533 (1) STATE that cable_tension decreases rapidly with the increase of cable_number (Agrawal,

1997)

(2) RECOMMEND unsupportlength 20-30% larger than supportlength (Troitsky 1988, pp.181)

(3) IF the mid_length is in the range of 140-150m,

THEN RECOMMEND supportlength of 20m (Troitsky 1988, pp.181)

However, the static knowledge, which is represented as the pure abstract statement of the general

recommendation (1), is difficult for designers to make accurate and convincing decisions in the practical designs Eventually, the detailed numerical design specification and evaluation should be mainly depended on the designers' experience and heuristic judgement, i.e on subjective decisions Similarly, the other two value-restricted recommendations (2), (3), which are abstracted from past experience of existing design comparisons, can not be easily adapted to the variations of upcoming design situations Therefore, only the static knowledge may be insufficient in explaining and

representing the design situation for practical designs and satisfying the improvement for future designs

In contrast to the world static, if the knowledge is stored in the form of an organised database of

evaluated designs with the design properties and the productive properties, and is processed and represented by OOMRM, the recommendations can be updated and re-represented at any time and given functional expressions, in cases of the representation of the design situation is not complete; CRK that is relative to the design situation is changed; and adjustment is necessary to match the change of the design situation Accordingly, we introduce the concept of dynamic knowledge to

remedy the defect of the static knowledge by means of continuous acquiring and improving the

knowledge with the adding-MRM-overriding method for forming the functional expressions E.g.,

omitting the conditions in the rule, the dynamic knowledge of cable_tension and its derived objects of

the value-restricted temporary views can be represented as follows (training in a particular design

situation that having the total cable area per plane kept constant for all the candidate designs)

(4) THEN RECOMMAND the influence on cable_tension IS

cable_tension[span_ratio, cable_number, cable_area]

AND The prediction of cable_tension IS

Y o - ~P < cable _ tension ~spanratio, cable_nu mber, cabl e_area ], a ] < Y o + ~P

(5) THEN RECOMMAND the influence on cable_tension IS

span_ratio(-0.8995) > cable_area(0.7587) > cable_number(O.lO02)

AND The prediction of cable_tension IS

84.7976 <= cable_tension[[0.40,20,O.500]',95] <= 106.0321 (95% prediction interval)

(training the system with 36 set of examples)

(6) THEN RECOMMAND the influence on cable_tension IS

span_ratio(-0.9230) > cable area(0.7603) > cable_number(O.1272)

AND The prediction of cable_tension IS

87.1251 <= cable_tension[[0.40,20,O.500]',95] <= 106.1621 (95% prediction interval)

(training the system with 60 set of examples)

(7) THEN RECOMMAND the influence on cable_tension IS

span_ratio(-0.8233) > cable_area(0.5799) > cable_number(O.lO08)

AND The prediction of cable_tension IS

70.9726 <= cable_tension[[0.40,20,O.500]',95] <= 112.0235 (95% prediction interval)

(training the system with 108 set of examples)

534 B Zhou and M Hoshino

The coefficients in the parentheses indicate the influence of the parameters on cable-tension varying one parameter with others held constant (Zhou and Hoshino 1999) The dynamic knowledge is regarded as the lower hierarchy of the static knowledge, that the static knowledge represents the abstraction of the dynamic knowledge, while the dynamic knowledge represents the value-unrestricted situation in the functional expression Often the dynamic knowledge can be translated into the static knowledge represented as the abstract form to play a conceptual control or value-restricted role in the design process, usually at the sacrifice of the value-unrestricted and the numerical prediction effects Accordingly, in the above training situation, the dynamic knowledge (5), (6) and (7) can be translated into the following abstract rule

(8) THEN RECOMMAND the influence on cable_tension IS

span_ratio > cable_area > cable_number

Different from many hierarchical knowledge classifications that have many relative hierarchies (Reich and Fenves 1995; Kushida et al 1997), both the static knowledge and the dynamic knowledge are processed, organised and represented with the relationship between CRK (Zhou and Hoshino 1998; 1999) As a simply example, in investigating the effect of cable stiffness on the behaviour of the structure, instead of facing enormous raw data obtained from structural analysis software arranged by the relational order or internally organised by the hierarchical classification tree, we can just link the name of cable_area to an object of a predefined general class that explains the design situation and predicts its productive properties within its CRK established by OOMRM

SYSTEM GENERAL STRUCTURE

Figure 3 illustrates the architecture and the flow of the general system The static knowledge, which is learned by the designer from documentary materials or experts who are in the fields of application, has conceptual or value-restricted influence on the candidate design or the hypothetical design Influenced

by the static knowledge or specified by the designer, sometimes by a heuristic selection, the parametric properties of the candidate design are specified as a temporary view for the design situation If the derived parametric properties of the specified parametric properties are within the range of the pre- stored interpolation properties, the specified design is then submitted to OOMRM to give the explanation of the design situation and give the prediction as the productive properties The explanation is describing the situation of the specified design, and the prediction is submitted to the evaluation decision process for evaluating If the derived parametric properties are exceeding the interpolation properties, the specified design is then submitted to the traditional structural analysis program to give the productive properties for evaluating

As a matter of fact, every engineering design may be regarded as an estimation or prediction of a certain specified design situation, which we call it the design situation temporary view, including explanation and problem solving Because, no matter how many times the design has been confirmed

in the past, it is always subject to the future confirmation by different design situations, different design methods or different practical uses It is useful to explain and predict what will happen when changes are performed on the any of the structural parameters The evaluation decision is simply made from the objective properties using the IF-THEN rules If the productive properties are within the

range of the objective properties, the specified parametric properties of the candidate design are regarded as the acceptable design and are then stored into the dynamic knowledge as improved knowledge, sometimes being accompanied with the change of the interpolation properties

However, the productive properties that are generated from the specified design usually exceed the expected range of the objective properties, and should be adapted Frequently, for a complex structure, several alternatives are always available for consideration Comparing the productive properties with

Expert System of Flexible Parametric Study on Cable-Stayed Bridges 535 the objective properties, sometimes redesigns should be carried out to converge the productive properties on the objective properties The back propagation method (Guo and Xiao 1991), which is a structural back propagation optimum method, is integrated into the system for providing possible design properties according to the objective properties by fixing some design properties within the

range in advance The redesign is a process of giving recommendations that may be adopted by the designer If the designer adopts the recommendations, the design properties will be within the interpolation range in Table 1 The process of the redesign iterates until the candidate design satisfies the desired requirements Because of the adjustable ability of the cable-stayed bridge in later designs and erection stages, usually the designer selects a partial set of the recommendations for redesigning the candidate design and the final design selection is mostly based on the subject decision Finally, the acceptable design for the dynamic knowledge can be translated into the static knowledge and both of them can be used for future candidate designs

Figure 3: Architecture and flow of the system

CONCLUSIONS AND FUTURE DIRECTIONS

The flexible parametric study in the present system includes two meanings: the flexible range of the design properties and the flexible representation of the knowledge The flexible representation of both

static and dynamic knowledge based on STATE, RECOMMEND and RULE with conceptual and functional expressions is clearer and more convenient for designers to make decisions than the representation of only numerical or only qualitative knowledge Especially, when the knowledge is vague or the design situation is changed, it is difficult for designers to make accurate and convincing decisions

In contrast to the restricted and narrowed parametric knowledge in the cable-stayed bridges, the system intends to acquire, store and represent the knowledge that is relevant to the design situation incrementally and continually, and adapts it to the change of the design situation The system is mainly based on the traditional structural analysis program for the knowledge extension, on MRM for the knowledge analysis and on OOP for the knowledge engineering This method can be used very efficiently for the knowledge acquisition, storage and representation to the expert systems and very

536 B Zhou and M Hoshino

economically for the optimum designs using past experience

In consideration of the variety of the cable-stayed systems, future directions should be turned to integrate three additional groups of properties into the system: construction properties describing the variety of erection methods; under construction productive properties describing the control situation during the erection phases; and the final cost properties

ACKNOWLEDGEMENTS

The first writer should greatly appreciate the financial support for this research from Kameda Gumi Co., Ltd., Japan

REFERENCE

1 Agrawal T P (1997) "Cable-Stayed Bridges- Parametric Study" J Struct Engrg., ASCE, 2:2, 61-67

2 Balakrishnan A., George V D and Lotfi Z (1965) Probability, Random Variables, and Stochastic Processes Mcgraw-Hill Book Comp

3 Booch G (1994) Object-Oriented Analysis and Design with Applications 2nd ed Addison Wesley Longman, Inc CA

4 Carl C et al (1992) Guidelines for Design of Cable-Stayed Bridges ASCE Committee on Cable-Stayed Bridges

5 Eric H A and John E J (1977) Statistical Methods for Social Scientists Academic Press Inc., New York

6 Guo W F and Xiao R C (1991) Liner and Non-Liner Bridge Structural Analysis Program System

Shanghai Institute of Urban Construction

7 Hald A (1952) Statistical Theory with Engineering Application John Wiley & Sons, New York

8 Harald C (1966) Mathematical Methods of Statistics Overseas Publications, Ltd Tokyo

9 Hegab H I A (1989) "Parametric Investigation of Cable-Stayed Bridges" J Struct Engrg., ASCE, 114:8, 1917-1928

10 Hunt I (1997) "Initial Thought on the Design Cable-Stayed Bridge" Proc Instn Civ Engrs Structures & Bridges 1997, 112, May, 218-226

11 Hwang J N., Lay S R., Martin R D and Schimert, J (1994) "Regression Modelling in Back-Propagation and Projection Pursuit Learning" Translations on Neural Networks, IEEE, 5:3, 342-353

12 John R W (1967) Prediction Analysis D Van Nostrand Company Inc., Toronto

13 Krishna P., Arys A S and Agrawal T P (1985) "Effect of Cables Stiffness on Cable-Stayed Bridges" J

Struct Engrg., ASCE, 111:9, 2008-2020

14 Kushida M., Miyamoto A and Kinoshita K (1997) "Development of Concrete Bridge Rating Prototype Expert System with Machine Learning" J Comput Engrg., ASCE, 11:4, 238-247

15 Reich Y and Fenves S J (1995) "System that Learns to Design Cable-Stayed Bridges" J Struct Engrg

12:7, 1090-1100

16 Robb J M (1980).Aspects of Multivariate Statistical Theory John Wiley & Sons, New York

17 Rumbaugh J., Blaha M., Premerlani W., Eddy F and Lorensen W (1991) Object Oriented Modelling and Design General Electric Research and Development Centre Schenectady, New York

18 Troitsky M S (1988) Cable-Stayed Bridges: an Approach to Modern Bridge Design Second Ed., Van Nostrand Reinhold, New York, N.Y

19 Samuel S W (1962) Mathematical Statistics John Wiley & Sons, New York

20 Starossek U (1996) "Cable-Stayed Bridge Concept for Longer Spans" J Bridge Engrg., ASCE, 1:3,99-

103

21 Warren G (1976) Statistical Forecasting John Wiley & Sons, New York

22 Xanbakos P P (1993) Theory and Design of Bridges John Wiley & Sons, Inc New York, N.Y

23 William W H and Douglas C M (1980) Probability and Statistics in Engineering and Management Science John Wiley & Sons, New York

24 Zhou B and Hoshino M (1998) "OOMRM: Object-Oriented Multiple Regression for Complex Engineering Designs" Advances in Engrg Computational Tech Civil-Comp Ltd., UK 249-255

25 Zhou B and Hoshino M (1999) "Knowledge-Based Multiple Regression Model for Complex Engineering Designs" J Struct Engrg., JSCE, 45A 501-510

PARAMETER STUDIES OF MOVING FORCE

IDENTIFICATION IN LABORATORY

Tommy H T CHAN, Ling YU, S S LAW, and T H YUNG Department of Civil and Structural Engineering The Hong Kong Polytechnic University Hunghom, Kowloon, HONG KONG

ABSTRACT

The parameters of both vehicle and bridge play an important role in moving force identification This paper aims to investigate the effect of various parameters on Time Domain Method (TDM) and Frequency-Time Domain Method (FTDM) For this purpose, a steel bridge model and a vehicle model were constructed in laboratory Bending moment and acceleration responses of the bridge were simultaneously measured when the model vehicle moved across the bridge at different speeds The moving forces were identified using the TDM and FTDM and rebuilt responses were calculated from the identified forces for comparison of identification accuracy Assessment results show that both the TDM and FTDM are effective and acceptable with higher accuracy but the TDM is better than the FTDM Further work includes enhancement of the two methods and merging them into a Moving Force Identification System (MFIS)

KEYWORDS

Moving Force Identification, Bridge-Vehicle Interaction, Bending Moment, Acceleration Response, Measurement, System Identification

INTRODUCTION

Force identification or force reconstruction from dynamic responses of bridges is an important inverse problem Many methods have been presented for its prediction in recent years (Fryba 1972, Moses

1984, Hoshiya and Maruyama 1987, Brigges and Tse 1992,) Stevens (1987) gave an excellent survey

of the literature on the force identification problem as well as an overview However, some of the above mentioned methods measure only static axle loads O'Connor and Chan (1988) suggested an advanced force identification method - Interpretive Method I (IMI) to interpret the force history, which is an advancement of the weight-in-motion methods mentioned above and is able to measure the dynamic axle forces of multi-axle system Based on system identification theory, the authors have developed another two moving force identification methods, namely Time Domain Method (Law, Chan and Zeng 1997) and Frequency-Time Domain Method (Law, Chan and Zeng) respectively

537

538 T.H.T Chan et al

Recently, a new method similar to IMI, called Interpretive Method (IMII), has also been published (Chan, Law and Yuan 1999) The preliminary and comparative studies showed that all these four methods could identify moving forces with acceptable accuracy However, each method has its merits, limitations and disadvantages They should be improved and strengthened for practical application in field tests In order to enhance the four methods and merge them into a Moving Force Identification System (MFIS), the effects of various parameters on two of the methods, namely the TDM and FTDM had been critically investigated in laboratory The parameters include mode numbers of bridge, sampling frequencies, vehicle speeds, computational time, sensor numbers and locations Acceptable results were obtained and some observations had been made in this paper

BRIEF DESCRIPTION OF THEORY

Referring to Figure 1, the bridge deck is considered as a simply supported beam with a span length L and constant flexural stiffness El, constant mass per unit length p , and viscous proportional damping

C, and the effects of shear deformation and rotary inertia are not taken into account (Bernoulli-Euler beam) The force P moves from left to fight at a speed c

L

Figure 1 Moving force on a steel beam bridge

An equation of motion in term of the modal displacement q, (t) can be given as

2

Where

(1)

co" = L -7- ' (" = 2pco, ' pn (t)= P ( t ) s i n ( ~ ) (2) are the modal frequency of the nth mode, the damping ratio of the nth mode and modal force respectively If the time-varying force P(t) is known, the equation (1) can be solved to yield q, (t) then the dynamic deflection v(x,t) can be found from the q,(t)and the nth mode shape function O, (x) This is called the forward problem The moving force identification is an inverse problem, in which the unknown time-varying force P(t) could be identified based on measuring the displacements, bending moments or accelerations of practical structures Two methods are developed for the purpose

Time Domain Method (TDM)

As mentioned above, the equation (1) can be solved in time domain by the convolution integral and the dynamic deflection v(x,t) of the beam at point x and time t can be obtained as

2 sinnntzt t

Where co', = co ~/1 - ( 2 , therefore, the bending moment of the beam at point x and time t is

Parameter Studies of Moving Force Identification in Laboratory 539 The acceleration at the point x and time t is

,=I pL

Where

(O n

Assuming that both the time-varying force P(t) and the bending moment m(x,t)or the acceleration

terms and rearranged into a set of equations as follows

Where, P is the time series vector of time-varying force P(t), R is the time series vector of the measured response of the bridge deck at the point x, such as the bending moment m(x,t) or acceleration a(x,t) The system matrix B is associated with the system of bridge deck and the force The subscripts N B = L / cAt and N are the numbers of sample points for the force P(t) and measured response R respectively when the force goes through the whole bridge deck

Frequency-Time Domain Method (FTDM)

Equation (1) can also be solved in the frequency domain Performing the Fourier Transform for Equations (1) and v(v,t) = s ap, (x)q n (t), the Fourier Transform of the dynamic deflection v(x,t) is

n=l

2

V ( x, (O ) = ~.~ 7-j- O , ( x ) H, ( (O ) P ( (O ) ( 8 )

Where H,((O) and P((O)are the Fourier Transform of q,(t)and P(t) respectively Similarly, the relationships between bending moment or acceleration and dynamic deflection can also be used to execute the corresponding Fourier Transform Finally, a set of N-order simultaneously equations can

be established in the frequency domain The force P((O)consisted of the real and imaginary parts can

be found by solving the N-order linear equations The time history of the time-varying force P(t) can then be obtained by performing the inverse Fourier Transformation From the procedures mentioned above, initially, the governing equations are formulated in the frequency domain However, the solution is obtained in the time domain Therefore this method is named frequency-time domain method

The above procedure is derived for a single force identification in TDM and FTDM methods They can be modified for multi-force identification using the linear superposition principle

EXPERIMENTAL DESIGN

The model car and model bridge deck were constructed in the laboratory An Axle-Spacing-to-Span- Ratio (ASSR) is defined as the ratio of the axle spacing between two consecutive axles of a vehicle to the bridge span length Here, the ASSR was set to be 0.15 The model car had two axles at a spacing

of 0.55 m and it ran on four rubber wheels The static mass of the whole vehicle was 12.1 kg in which the mass of rear wheel was 3.825 kg The model bridge deck consisted of a main beam, a leading beam and a trailing beam as shown in Figure 2 The leading beam was used to achieve acquired constant speed of vehicle when the model car approached the bridge The trailing beam was for the slowing down of the car The main beam with a span of 3.678 m long and 101 mm x 25 mm uniform cross-section, was simply supported It was made from a solid rectangular mild steel bar with a