Impact in highway prestressed concrete bridges

Compurers & Smcrures Vol. 44, No. 3, pp. 525-534. 1992

Printed in Great Britain.

0045-7949192 S5.00 + 0.00

0 1992 Pergamon Press Ltd

IMPACT IN HIGHWAY PRESTRESSED CONCRETE BRIDGES

T. L. WANG,? M. SHAHAWY~ and D. Z. HUAN@~

tDepartment of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, U.S.A.

SStructures Research and Testing Center, Florida Department of Transportation, Tallahassee, FL 32310, U.S.A.

§Department of Civil and Environmental Engineering, Florida International University, Miami, FL 33199, U.S.A.

(Received 28 June 1991)

Abstract-The purpose of this paper is to study the factors which affect the dynamic behavior of highway prestressed concrete bridges. An HS20-44 truck model with 12 degrees of freedom was developed and used in this analysis. Highway prestressed concrete bridge models with spans of 30, 60, 90, and 120 ft (1 ft = 0.305 m) were designed according to American Association of State Highway and Transportation Oflicials (AASHTO) standard bridge girders. Two percent of the critical damping was assumed for the bridge. The road surface roughness of the approach roadways and bridge decks was generated from power spectral density (PSD) function for very good, good, average, and poor roads in accordance with International Organization for Standardization (ISO) specification. The impact percentages in moment, deflection, and end shear of the bridge for different bridge span lengths, vehicle speeds, and road surface roughness were calculated and compared with the AASHTO specified values.

1. INTRODUCTION

The prediction of the dynamic response of a bridge resulting from the passage of live loads across the span is a significant problem in the bridge design area, because a vehicle moving across a bridge at a normal speed produces greater stresses than a vehicle that remains in a static position on the structure. The behavior of highway bridges under the passage of heavy vehicles has been the crucial subject of numerous investigations since the early 1950s. A brief review of the literature can be found in [l-9].

Since it is difficult to make an accurate analysis of the bridge dynamic response, the normal design procedure has been to assign an approximate quantity for the dynamic effect. According to the American Association of State Highway and Trans- formation Officials (AASHTO) specifications [lo], this quantity is calculated by using a simple empirical formula based on the length of the bridge. But realistic bridge dynamic response due to moving live loads depends on the following factors: (1) vehicle nonlinear characteristics such as suspension and damping; (2) the speed, weight, and type of the vehicle; (3) bridge characteristics such as damping and natural frequencies of vibration; (4) road surface roughness of the approach roadway and bridge deck; (5) traffic intensity; and (6) behavior of drivers. Among the aforementioned factors, the first four

1 Formerly at Department of Bridge Engineering, Tongji University, Shanghai, P.R. China.

factors are more influential in bridge dynamic response.

The objective of this study is to analyze the dynamic behavior of highway prestressed concrete bridges caused by vehicles moving over different classes of roads with various speeds.

A brief description of the vehicle and bridge models is given first, then bridge/vehicle interaction equations and road surfaces roughness are introduced. The impact percentages of moment, deflection, and end shear of the bridge for different bridge span lengths, vehicle speeds, and road surface roughness are finally presented and compared with values specified by AASHTO specifications [lo] and the Ontario Highway Bridge Design (OHBD) code [l 11.

2. VEHICLE MODELS

The nonlinear vehicle model, presented in Figs 1 and 2, with 12 degrees of freedom (DOF) was developed according to an HS20-44 truck which is a major design vehicle in AASHTO specifications [lo]. The vehicle model consists of five rigid masses as tractor, semi-trailer, steer wheel/axle set, tractor wheel-axle set, and trailer wheel-axle set. The tractor and semi-trailer were each assigned three DOFs, corresponding to the vertical displacement (y), rotation about the transverse axis (pitch or e), and rotation about the longitudinal axis (roll or 4). Each wheel/axle set is provided with two DOFs in the vertical (v) and roll (9) directions. The total degrees of freedom in this model are 12. The tractor and

525

526 T. L. WANG et al.

i- 14’ 14’ _--

I --1

Fig. 1. Side view of the vehicle model.

semi-trailer were interconnected at the pivot point (so-called fifth wheel point, see Fig. 1).

Sus~nsion force consists of the linear elastic spring force and the constant interleaf friction force [12]. Figure 3 shows the load-displacement relationship for friction force, suspension spring force, and combination of these two forces. The tire springs were assumed to be linear. In the suspension system, damping elements were assumed to be linear and to be viscous. Damper force is proportional to the velocity. Ten percent of the critical damping value was used for damping coefficient [13]. The damping forces in the tires were neglected. Provision is made in the model for wheel lift. Under this condition, the vertical tire stiffness was taken as zero.

The total potential energy, I/ = CV,, of the system is then computed from the spring stiffnesses and relative displacements, whereas the dissipation energy, I) = ,?Z&, of the system is obtained from the damping forces. The total kinetic energy, T = CTi, of

(a) Friction Force

I-- l __--I- 6’ -----7

Fig. 2. Front view of the vehicle model.

the system is calculated using the mass, mass moment of inertia, and translations as well as rotational velocities, of the system components. The moment of inertia of all components is assumed to be constant and the weight of each component is considered as the external force on that component.

The equations of motion of the system are derived, using Lagrange’s formulation, as follows:

(1)

where q, and di are the generalized displacements and velocities. Details of derivation are presented in [14].

3. BRIDGE MODEL

The highway prestressed concrete bridges with spans of 30,60,90, and 120 ft (9.15, 18.30,27.45, and

(b) Suspension Spring Force

(c) Combination of Friction and Suspension Spring Forces

Fig. 3. The relationship between forces and displacements in the vehicle suspension system.

Impact in highway prestressed concrete bridges 527

ype IV AASHTO

Fig. 4. Typical cross-section of a prestressed concrete bridge.

Center-to-center girder mpacing 7

(a) 30-ft Span Bridge

Cent*r-to-center girder spacing -7

(c) W-it Span Bridge

Centtr-to-oenter girder #pacing

Type Ill MSETO

(bf @O-ft Span Bridge

Ccatcr-to-cente girder spacing 7

(d) 12thft Span Bridge

Fig. 5. Girder cross-section for 30, 60, 90, and 120 ft (9.15, 18.30, 27.45, and 36.00 m) span prestressed concrete bridges.

528 T. I.,. WANG et al.

Fig. 6. Bridge span.

36.60 m), designed according to AASHTO specifications [IO], were used as bridge models. The typical bridge section is shown in Fig. 4. The girder cross- sections of these four spans are shown in Fig 5. The values of concrete ultimate strength (ff) were assumed to be 33OOpsi (22.7 MPa) for slab and 5000 psi (34.5 MPa) for girder. Based on the Ameri- can Concrete Institute (ACI) formula E = 57,000~ and the abovef; values, the modulus of elasticity of concrete E was calculated as 3.274 x IO6 psi (22.6 GPa} for slab and 4.031 x IO6 psi (27.8 GPa) for girder. The shear modulus G for concrete was obtained from E by using a Poisson’s ratio of 0.02. Two percent of the critical damping was assumed for bridges. Details of the bridge data are available in [14].

The bridge span was divided into eight segments (refer to Fig. 6). Nodes were introduced at the end of each segment. Each node was assigned six DOFs, corresponding to three translations and three rotations. The DOFs of lateral and longitudinal displacements and rotation about the vertical axis were condensed by using Guyan’s condensation method [15]. The deflection at a point within a segment between two nodes was obtained by linear interpolation. The first naturaf frequencies of the bridges were calculated as 16.353, 6.760, 3.618, and 2.701 Hz for bridge spans of 30, 60, 90, and 120 ft (1 ft = 0.305 m), respectively.

4. BRIDGE/VEHICLE INTERACTION

The equations of motion of the bridge are

in which [lw,] is the condensed bridge mass matrix; [Q] the condensed bridge stiffness matrix; {S>, {&I, {6j the bridge node displacement, velocity, and acceleration vectors, respectively; [Fe] the bridge node applied loading vector, due to the interaction between the bridge and vehicle; and [DB] the bridge damping matrix = 2w, C,[M], where w1 is the funda- mental natural frequency of the bridge and C, is the percentage of the critical damping for the bridge.

The interaction force of the ith wheel between the bridge and the vehicle is given as

F’, = Kt?i u,y: + D,yi o,y, t (3)

where Kpyi is the tire stiffness of the ith wheel, 1), the tire damping coefficient of the ith wheel, V,, the relative displacement between the ith wheel and

bridge = y,,.; - (- V;,) - ( -yb,), y,,, the vertical displacement of the ith wheel, U, the road surface roughness under the ith wheel (positive upwards), and yhi the bridge vertical displa~ment under the ith wheel (positive wheel).

The bridge vertical displacement under the ith wheel ybi was related to the displacements at neigh- boring nodes by linear interpolation. A superscript dot denotes differentiai with respect to time.

5. ROAD SURFACE ROUGHNE~

The typical road surface may be described by a periodically modmated random process. The power spectral density (PSD) is a useful tool for analyzing the periodically modulated random process. The PSD functions for highway surface roughness have been developed by Dodds and Robson [16]. They are shown as

where S(e) is the PSD (mz/cycle~m), di the wave number (cycle~m), A the roughness coefficient (m/cycle), and GO the discontinuity frequency = $r {cycle/m).

The values of MI] and w2 varied from 1.36 to 2.28 [16]. In order to simplify the description of road surface roughness, both w, and w2 are assumed to have a value of two. Equations (3) and (4) are converted as follows:

0 -2

S(@)=A f 0

It was found that the comparison between numerical and analytical PSDs agreed fairly well 1141.

The random numbers which have approximate white noise properties were generated first [17], then these random numbers were passed through the first order recursive filter [IS]. Finally, the output function will be the road surface roughness. The details of the procedure have been discussed by Wang and Huang[14]. In this study, the values of 5 x 10a6, 20 x 10w6, 80 x lo-“, and 256 x 10-6m3/cycle were used according to International Organization for Standardization (ISO) specification [19] as the roughness coefficient A for the classes of very good, good, average, and poor roads, respectively. The sample length was taken as 256 m (839.9 ft) and 2048 (2”)


w u Q -10

kz

2

L - g-20 ’

Q -30 -‘50 IAb-.-I.-e 64 126 192 $ -40

r 256 in 0 -. _L-.~ 64 L .--- _A__ 126 192 256

DISTANCE ALONG THE ROAD (M) DISTANCE ALONG THE ROAD (M)

(4 (4

DISTANCE ALONG THE ROAD (M) DISTANCE ALONG 'THE ROAD (M)

(b) (b)

Fig. 7. Vertical highway surface profiles in a very good Fig. 9. Vertical highway surface profiles in an average road; road; (a) right line, (b) left line. (a) right line, (b) left line.

data points were generated for this distance. The 6. DYNAMIC ANALYSIS AND RESULTS

average vertical highway surface profiles of right and left lines from five simulations are shown in Figs 7, A single HS20-44 truck was used to analyze the 8, 9, and 10 for very good, good, average, and poor bridge vibration based on the assumption that there roads, respectively. is no eccentricity between the center line of the vehicle

? -2 o

L-_____-_- ~__ I N __6 0

/ _._1___L . _i~ -! 0 64 126 192 256 > m 0 64 126 _~.__192_ 256 -.

DISTANCE ALONG THE ROAD(M)

(4

,, DIS’I’ANCI’ ALONG THE ROAD (M)

(4

$ -201. 0 I-I-,.-1-- 64 126 192 256 J 2 a -6 0 0 I _A-._-.__L. 64 126 192 256

DISTANCE ALONG THE ROAD (M) DlSTANCEALONGTHEROAD(M)

(b) (b)

Fig. 8. Vertical highway surface profiles in a good road; Fig. 10. Vertical highway surface profiles in a poor road; (a) right line, (b) left line. (a) right line, (b) left line.

CAS 44,3-c


Fig. 11. Amplification factor for the midspan bending

~.__. __~~L .~_~, ,

00 01

moment in a prestressed concrete bridge of 30-ft (9.15 m)

0.2 03 04 05 06 07 OR

SIMULATION TIME (SEC)

span under the conditions of good road surface and 55 mph (88.5 km/hr) vehicle speed.

-4 -021.___._‘.. ..j. a._,.

00 01 02 03 04 05 06 07 OH

SIMULATION TIME ('EC)

Fig. 12. Amplification factor of the midspan deflection in a prestressed concrete bridge of 30-ft (9.15 m) span under the conditions of good road surface and 55 mph (88.5 km/hr)

vehicle speed.

moving path and the centroid of interior girder in the transverse direction (see Fig. 4). The equations of motion for the vehicle and the bridge were solved by using a fourth-order Runge-Kutta integration scheme [20,21], with a time step of 0.0003 sec. This time step was found to be the optimum for the nonlinear vehicle and bridge system. Two percent of the critical damping for the bridge was assumed in this study.

Fig. 13. Amplification factor for the end shear in a

op._ ,..__ _.L~__.~_/,.~,. L_._l

00 01 02

prestressed concrete bridge of 30-ft (9.15 m) span under the

03 04 05

conditions of good road surface and 55 mph (88.5 km/hr)

06 07 OH

SIMULATION TIME (SEC)

vehicle speed.

In order to obtain the initial displacements and velocities of vehicle DOFs when vehicle entered the bridge, the vehicle was started to move at a distance of 140ft (42.67 m), i.e. a five-car length, away from the left end of the bridge and continued moving until the entire vehicle left the right end of the bridge. The same class of road surface was assumed for both the approach roadways and bridge decks.

Static and dynamic responses were defined as bending moment, deflection, or end shear of the bridge due to the static and dynamic loading. The amplification factors A, and Ad for static and dynamic responses are, respectively

in which R, is the absolute maximum static response and R, as well as R, are the static and dynamic responses, respectively.

Figures 11-13 show the typical results of the amplification factors for bending moment and deflection at midspan as well as end shear for the 30 ft (9.15 m) span length bridge under the conditions of

t

1.0 2.5 4.5 8.0

First Flexural Frequency, Hz

Fig. 14. Dynamic load allowance of Ontario Highway Bridge Design Code.


13o - hASHTO SPEC. -... DHBD CODE

I$ 110

o---O VRRY GOOD ROAD SURFACE

;m

b---b GOOD ROAD SURFACE L3 --o AVRRAGE ROAD SURFACE

ix

[

2 10.

;<;<>za

-10

- AASHTO SPEC. ---- OHRD CODE D-0 VERY GOOD ROAD SURFACE D-d GOOD ROAD SURFACE 0-O AVRRAGE ROAD SURFACE l --* POOR ROAD SURFACE

10 20 30 40 50 60 70 80

VEHICLE SP~ED(MPH~

20 70 60 30 40 50 60

VEHICLE SPEED (MPH)

Fig. IS. The variation of impact factors for the bending Fig. 18. The variation of impact factors for the bending moment at midspan of 30.ft (9.15 m) span prestressed moment at midspan of 60-ft (18.30m) span prestressed

concrete bridge with vehicle speeds. concrete bridge with vehicle speeds.

good road surface and 55 mph (88.5 km/hr) vehicle speed.

The AASHTO impact formula for bridge is given as

I= 50

- < 0.3 L + 125

in which L is the span length in feet. In OHBD code [1 11, the dynamic load allowance is

0.3 for simple span bridges not greater than 22 m (71.13 ft) in length and is obtained from Fig. 14 for simple span bridges greater than 22 m (72.13) in length. This dynamic load allowance is multiplied by

-10 ’ .-.-.-I 10 20 30 40 50 60 70 80

VEHICLE SPEED (MPH)

Fig. 16. The variation of impact factors for the deflection at midspan of 30-ft (9.15 m) span prestressed concrete

bridge with vehicle speeds.

130

yg 110

g 90

t; 70

$ 50

G

iz 30 10

10

- - AASRTO SPEC. -.-- OHBD CODE 0-o VERY GOOD ROAQ SURFACE A-O GOOD ROAD SURFACE O-0 AVERAGE ROAD SURFACE .--. POOR ROAD SURFACE

20 30 40 50 60 70 80

VEHICLE SPEED (MPH)

Fig. 17. The variation of impact factors for the end shear of 30.ft (9.15 m) span prestressed concrete bridge with

Fig. 21. The variation of impact factors for the bending moment at midspan of 90-ft (27.45 m) span prestressed

vehicle speeds. concrete bridge with vehicle speeds.

190 - A&Sill0 SPEC. -‘-- OHBD CODE

170 O-0 VERY GOOD ROAD SURFACE g 150 a--b COODROADSURFACE

0-O AVERAGE ROAD SURFACE e: 130 h--e POORROADSURFACE

10 20 30 40 50 60 70 80

VEHICLE SPEED (MPH)

Fig. 19. The variation of impact factors for the deflection at midspan of 60-ft (18.30m) span prestressed concrete


VEHICLE SPEED (MPH)

Fig. 20. The variation of impact factors for the end shear of 60-ft (18.30m) span prestressed concrete bridge with

vehicle speeds.

10

- MSHTO SPEC. -... OHBD CODE 0-O VERY GOOD ROAD SURFACE A - r? GOOD ROAD SURFACE O---o AVERAGE ROAD SURFACE 0-e POOR ROAD SURFACE

20 50 60

V:HICL:SPEED (MPH) 70 60


-1 - A.&SHY0 SPEC. -.T OHBD CODE 0-O VERY GOOD ROAD SURFACE &-A GOOD ROAD SURFACE O-- 0 AVERAGE ROAD SURFACE .-. POOR ROAD SURFACE

- AASHTO SPEC -. OHRD CODE o---o VERY GOOD ROAD SURTKE n-d COOD ROAD SURFACE o---o .t”ERAGE ROAD SURFACE .-. POOR ROAD SURFACE

.,o I._-L/_-- -10 IA--I-_L..l 10 20 30 40 50 60 70 60 10 20 30 40 50 60 70 60

VEHICLE SPEED (MPH) VEHICLE SPEED (MPH)

Fig. 22. The variation of impact factors for the deflection at midspan of 90-ft (27.45 m) span prestressed concrete


Fig. 26. The variation of impact factors for the end shear of 120-ft (36.60 m) span prestressed concrete bridge with

vehicle speeds.

~~~. AASHTO SPEC [ I = SO/(125 + I.) ] 0 ~0 “EHlCLE SPEED - 15 MPH (24.14 KM/HR) A --~A VEHlCLE SPEED = 35 MPH (56.32 KM/HR) i, ~--u VEHICLE SPEED = 55 MPH (SE.50 KY/HR) . . “EHlCLE SPEED 75 MPH (120 66 KN/HR)

.

90

R _ 70

0:

g 50

2 30

u’

2 10

3

-10

- AASHTO SPEC -... OHBD CODE 0 - 0 VERY GOOD ROAD SURFACE b-d GOOD ROAD SURFACE II--o AVERAGE ROAD SURFACE .-. POOR ROAD SURFACE

10 20 30 40 50 60 70 ii0

VEHICLE SPEED (MPH) Fig. 27. The variation of impact factors for the midspan bending moment with bridge spans under the conditions of good road surface and vehicle speeds varied from 15 to

75 mph (24.14 to 120.68 km/hr).

Fig. 23. The variation of impact factors for the end shear of 90-ft (27.45 m) span prestressed concrete bridge with

vehicle speeds.

-- AASHTO SPEC [ I = SO/(125 + 1.) ] 0 -0 VEHICLE SPEED 15 MPH (24.14 KY/RR) n h VEHICLE SPEKD = 35 MPH (56.32 KY/HR) II -I I “EHICLE SPEED i 55 MPH (88.50 KM/HR) . . VEHICLE SPEED 75 MPH (120.68 KU/HR)

0

0

I

30 60 90 120 150

LMDGE SPAN (F’T) IOL. .-I

10 20 30 40 50 60 70 60

VEHICLE SPEED (MPH) Fig. 28. The variation of impact factors for the midspan deflection with bridge spans under the conditions of good road surface and vehicle speeds varied from 15 to 75 mph

(24.14 to 120.68 km/hr).

Fig 24. The variation of impact factors for the bending moment at midspan of 120-ft (366Om) span prestressed

concrete bridge with vehicle speeds.

AASHTO SPEC 1 [ 50/1125 + Ll 1 110

2 90

2 70

G !2 50

g 30

g 10

0~ -0 VEHICLE SPiED <5’MPH (24 i4 KY/HR) A A VEHICLE SPEED = 35 MPH (56 38 KY/HR) II -0 VEHICLE SPEED = 55 MPH (86 50 KM/HR) . . VEHlCLE SPEED 75 MPH (120 68 KY/HR)

- AASHTO SPEC. --.- OHBD CODE O-0 VERY GOOD ROAD SURFACE b-b GOOD ROAD SURFACE O-0 AVERAGE ROAD SURFACE .-. POOR ROAD SURFACE ,.._._..-..-..- .-..-.._..._. -..-..-..-.. .-..-.. _.. _ .* +Jm+3L$

o--o- :: 10 1 -

0 30 60 90 120 1 NO

BRIDGE SPAN (FT)

Fig. 29. The variation of impact factors for u. end shear with bridge spans under the conditions of good road surface and vehicle speeds varied from 15 to 75 mph (24.14 to

120.68 km/hr).

20 30 40 50 60 70 60

VEHICLE SPEED(MPH)

Fig. 25. The variation of impact factors for the deflection at midspan of 120-ft (36.6Om) span prestressed concrete



Table 1. Comparison of impact factors between 0% and 2% of critical bridge damping for different span lengths

0% Bridge damping 2% Bridge damping

Bridge Maximum Maximum Impact Maximum Impact span static dynamic factor dynamic factor (ft) response response (%) response (%)

Moment 30 1700.3 2152.0 26.57 2102.7 2366 at middle 60 6109.2 6842.9 12.01 6683.8 9.41 span point 90 11,697.4 12,900,l 10.28 12,838.6 9.76 (k-in) 120 16,411.Z 18,513.7 12.81 I8,344.9 11.78

- Deflection 30 ‘-0.08405 0.10030 19.33 0.09837 17.04 at middle 60 0.22467 0.25586 13.88 0.25056 11.52 span point 90 0.47977 0.53575 11.67 0.53327 11.15 point (in) 120 0.56148 0.62860 11.95 0.62529 11.36

End 30 27.0007 28.4773 5.47 28.2393 4.59 shear 60 38.6922 38.8696 0.46 38.2525 -1.14 (kips) 90 46.9431 47.6240 1.45 47.3520 0.87

120 48.3027 54.0794 11.96 53.8064 11.39

Note: The vehicle operating at 55 mph and good road surface; 1 k-in = 0. I I3 kN m; 1 in = 2.54 cm; 1 kip = 4.45 kN; 1 ft = 0.305 m.

0.5 for a vehicle speed limit of 10 and 25 km/hr (6.22 can be seen that the impact factors reduced when the and 15.54 mph). bridge damping was considered.

The real percentage of impact acquired from this study is defined as 7. CONCLUSIONS

Imp(%) = +- 1 x 100%. [ 1 (9)

sm

Figures 15-26 illustrate the impact factors of bending moments and detlections at midspan as well as end shear for bridge spans of 30,60,90, and 120 ft (1 ft = 0.305 m) for different vehicle speeds and road surface conditions, together with a comparison with the AASHTO specifications [lo] and OHBD code [I 11. Most computed values for very good, good, and average roads were lower than the values specified by both AASHTO and OHBD. However, very high impact factors could happen when the road condition is poor. Even though the impact factors were not affected significantly by the vehicle speeds, the impact factors increased with vehicle speeds in very good, good, and average roads. The impact factors of end shear are lower than those of bending moments and deflections in the same running condition.

Based on good road condition and vehicle speeds varied from 15 to 75 mph (24.14 to 120.68 kmfhr), the variation of impact factors for bending moments, deflections, and end shear with bridge spans were shown in Figs 27-29. They were also compared with the values specified by AASHTO. The larger impact factors were obtained in the short span bridges, i.e. 30-60-ft span. The impact factors reduced when the bridge spans increased for most cases.

Finally, the comparison of impact factors between 0% and 2% of criticai bridge damping was given in Table 1 under the conditions of good road surface and a vehicle operating at 55mph (88.50 km/hr). It

The conclusions of this study are summarized as follows:

1. In very good, good, and average roads, most impact factors of bending moments and deflections at midspan as well as end shear are lower than the values specified by both AASHTO and OHBD. However, very high impact factors were found with poor road surfaces.

2. The impact factors increased with vehicle speeds in very good, good, and average roads, although they were not affected significantly by vehicle speeds.

3. The impact factors of bending moments and detlections are higher than those of end shear in the same running condition.

4. The impact factors in the short span bridges are larger than those in long span bridges. In most cases, the impact factors reduced when the bridge spans increased.

5. The impact factors reduced when the bridge damping was considered.

6. This study was limited to the dynamic analysis of a bridge, which was considered as a single longitudinal girder. The lateral impact load distribution and floor beam impact have not been studied; further research is necessary in these areas. However, the methodology developed in this study would be useful in impact analysis of highway prestressed concrete bridges.

~cknowle~ge~e~~-This research is sponsored by the Florida Department of Transportation under the grant No. WPIO510542. This support is greatly acknowledged.


REFERENCES 11.

1. J. M. Biggs, H. S. Suer and J. M. Louw, The vibration of simple span highway bridges. J. Struct. Div., ASCE 12. 83, Paper No. 1186 (1957).

2. T. H. T. Chan and C. O’Connor, Vehicle model for highway bridge impact. J. Struct. Engng, ASCE 116, 13. 1772-1793 (1990).

3. S. J. Fenves, A. S. Veletsos and C. P. Siess, Dynamic studies of bridge on the AASHO road test. Highway Research Board, Reports 71 and 73, 14. National Academy of Sciences, Washington, DC (1962).

4. R. K. Gupta, Dynamic loading of highway bridges. J. Engng Mech. Div., ASCE 106, 377-394 (1980). 15.

5. S. G. Hutton and Y. K. Cheung, Dynamic response of single span highway bridges. J. Earthquake Engng 16. Struct. Dyn. 7, 543-553 (1979).

6. M. Inbanathan and M. Wieland, Bridge vibration due 17. to vehicle moving over rough surface. 1. Struct. Engng, ASCE 113. 1994-2008 11987).

7. M. Kawatani and S. Komatsu, Nonstationary random response of highway bridges under a series of moving 18. vehicles. Proceedings of Japan Society of Civil Engineers, No. 398, Part I-10, pp. 81-88 (1988). 19.

8. C. O’Connor and R. W. Pritchard, Impact studies on small composite girder bridge. J. Struct. Engng, ASCE 111, 641-653 (1985).

9. A. S. Veletsos and T. Huang, Analysis of dynamic 20. response of highway bridges. J. Engng Mech. Div., ASCE 96, 593-620 (1970).

10. Standard Specifications for Highway Bridges, 14th Edn. 21. American Association of State Highway and Transpor- tation Officials, Washington, DC (1989).

Ontario Highway Bridge Design Code, Ministry of Transportation and Communication, Ontario, Canada (1983). T. Huang, Dynamic response of three-span continuous highway bridges. Ph.D. dissertation, University of Illinois, Urbana, IL (1960). A. P. Whittemore, J. R. Wiley, P. C. Schultz and D. E. Pollock, Dynamic pavement loads of heavy highway vehicles. National- Cooperative Highway- Research Program Report, Washington, DC (1970). T. L. Wang and D. Z. Huang, Computer modeling analysis in bridge evaluation, Interim Research Report, No. FL/DOT/RMC/O542-3394, Florida Department of Transportation, Tallahassee, FL (1992). R. J. Guyan, Reduction of stiffness and mass matrices. Am. Inst: Aeronaut. Astronaut. Jnl 3, 380 (1965). C. J. Dodds and J. D. Robson. The descriotion of road surface roughness. J. Sound iibr. 31, 17i-183 (1973). J. Moshman, Random number generation. In Math- ematical Methodr for Digital Computers, Vol. II (Edited by A. Ralston and H. S. Wilf), Chap. 12, pp. 249-263. John Wiley, New York (1967). R. K. Otnes and L. Enochson, Digital Time Series Analysis. John Wiley, New York (1972). C. J. Dodds, BSI proposals for generalized terrain dynamic inputs to vehicles. ISO/TC/108/WG9, Document No. 5, International Organization for Stan- dardization (1972). K. H. Chu, V. K. Garg and T. L. Wang, Impact in railway prestressed concrete bridges. J. Struct. Engng, ASCE 112, 1036-1051 (1986). T. L. Wang, Ramp/bridge interface in railway prestressed concrete bridges. J. Struct. Engng, ASCE 116, 1648-1659 (1990).

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Impact in highway prestressed concrete bridges