bridge roughness index as an indicator of bridge dynamic amplification

www.elsevier.com/locate/compstruc

Computers and Structures 84 (2006) 759–769

Bridge roughness index as an indicator of bridge dynamic amplification

Eugene OBrien, Yingyan Li *, Arturo Gonzalez

School of Architecture, Landscape & Civil Engineering, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland

Received 19 July 2005; accepted 1 February 2006

Abstract

The concept of a road roughness index for bridge dynamics is developed. The international roughness index (IRI) is shown to be verypoorly correlated with bridge dynamic amplification as it takes no account of the location of individual road surface irregularities. It isshown in this paper that a bridge roughness index (BRI) is possible for a given bridge span which is a function only of the road surfaceprofile and truck fleet statistical characteristics. The index is a simple linear combination of the changes in road surface profile; the coef-ficients are specific to the load effect and span of interest. The BRI is well correlated with bridge dynamic amplification for bendingmoment due to 2-axle truck crossing events. A similar process can be used to develop a BRI for trucks with other numbers of axlesor combinations of trucks meeting on a bridge.� 2006 Elsevier Ltd. All rights reserved.

Keywords: DAF; DAE; BRI; Dynamic; Roughness; Bridge; Vehicle

1. Introduction

The dynamic amplification factor (DAF) for a bridge isdefined as the maximum total (dynamic plus static) loadeffect divided by the maximum static load effect. DAF isinfluenced by many factors. Speed is particularly impor-tant, as noted by many authors [1–6] and in particularthe ratio of speed to bridge 1st natural frequency [7]. Roadroughness is also a key factor [3,4,8], particularly for shortbridges.

In the highway industry, indices for the evaluation ofpavement surface evenness have been developed since the1960s. The most popular parameters are the internationalroughness index (IRI) [9–11] which was developed and rec-ommended by the World Bank to evaluate pavementroughness, and the power spectral density (PSD) [12].

Olsson [13], Lin [14], Henchi et al. [15], Liu et al. [3] andMajumder and Manohar [16] are just some of the authorsthat use finite element analysis with six degrees of freedom

0045-7949/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compstruc.2006.02.008

* Corresponding author. Tel.: +353 1 716 7281; fax: +353 1 716 7399.E-mail address: [email protected] (Y. Li).

per node to model bridges dynamically. Olsson [13] com-pared the results of finite element analysis with differentbeam models. Green and Cebon [17] modelled two bridgesin the U.K. with finite elements. Chompooming and Yener[18] present a finite element model of a beam and slabbridge. This type of section is also modeled by Kou andDewolf [19] who use plate elements for the deck and beamelements for the girders. Gonzalez [20] couples displace-ments and velocities at the bridge/vehicle contact pointsat each time step in an iterative formulation. Yau andYang [21] allow for instability and inertial effects in a finiteelement model of a cable stayed bridge.

The effect of road surface irregularities on bridge vibra-tion has been examined by DIVINE [1], Green et al. [4] ,Kou and Dewolf [19], Law and Zhu [22], Lei and Noda[23] and Chatterjee et al. [24]. Vehicle and bridge modelshave been used to simulate the vehicle-bridge interactionsystem and to determine the effect of profile unevenness.However, these papers investigate the influence of differentPSD levels. It has been found by Li et al. [25] that there aresubstantial differences in dynamic amplification betweenroad profiles with the same PSD level and the same IRIvalue.

mailto:[email protected]

760 E. OBrien et al. / Computers and Structures 84 (2006) 759–769

Michaltsos [26] and Pesterev et al. [27,28] have shownthat the position of an irregularity is important for bridgedynamic amplification. In a previous paper [25], theauthors confirm this and show that if bridge deflectionsare negligible [28], the principle of superposition appliesfor the dynamic response to individual road surface irregu-larities. This makes possible the estimation of dynamicamplification by adding together the DAFs due to eachirregularity that makes up the road surface profile. Verygood agreement with critical DAF values is reported fora range of ‘good’ road surface profiles. A significant prob-lem is that the resulting dynamic amplification estimate isspecific to the properties (spring stiffness etc.,) of thatvehicle.

Yang et al. [12], Zhu and Law [29] and Brady andOBrien [30] examine cases of two following loads and com-pare the effects to single load crossings. Axle spacing isidentified as being particularly important [30,31] and itquickly becomes clear that an amplification factor derivedfrom a 1-axle vehicle model is of limited value as an indica-tor of DAF for multiple-axle vehicles.

The goal of this paper is to develop the concept of a BRIwhich can be used as an estimator of DAF for a given vehi-cle class, and in particular in this paper, a BRI for 2-axlevehicles. The BRI will be a function of the road profileonly; it will not be dependent on the speeds or propertiesof particular vehicles. A BRI potentially constitutes anextremely useful measure of road surface roughness thatcould be used by bridge maintenance managers as an indi-cator of the level of dynamic amplification that might beexpected on a bridge.

2. Bridge vibration

2.1. Vehicle-bridge interaction model

A half-car model crossing a simply supported Bernoulli-Euler beam at a constant speed is used to simulate 2-axlevehicle events (Fig. 1). The motion controlling this systemis defined by the ordinary differential equations [32]:

� Id2uðtÞ

dt2þ ð�1ÞiDiðZiðtÞ þ ZbiðtÞÞ ¼ 0 ð1aÞ

CsKs

Kt

m3, I

y1(t)y2(t)

y3(t)

(t)

m2 m1

Ks Cs

Kt

D1D2

v(x,t)

x1

x2

E, J, u

r(x1)r(x2)

ϕ

Fig. 1. Schematic of the two-axle vehicle and bridge interaction system.

� m3

d2y3ðtÞdt2

�X2

i¼1

½ZiðtÞ þ ZbiðtÞ� ¼ 0 ð1bÞ

m3ig þ mig � mid2yiðtÞ

dt2þ ZiðtÞ þ ZbiðtÞ � RiðtÞ ¼ 0 i ¼ 1; 2

ð1cÞ

EJo4vðx; tÞ

ox4þ l

o2vðx; tÞot2

þ 2lxbovðx; tÞ

ot¼X2

i¼1

eidðx� xiÞRiðtÞ

ð1dÞ

where I and u(t) are the mass moment of inertia and rota-tional degree of freedom of the body mass respectively; Di

is the horizontal distance between the centroid of sprungmass and unsprung mass i; m1, m2, m3 are masses of thefront axle, rear axle and vehicle body respectively andy1(t), y2(t) and y3(t) are the corresponding vertical displace-ment of their centers of gravity; g is acceleration due togravity; v(x, t) is the displacement of the bridge at locationx and time t; E, J, l and xb are modulus of elasticity, iner-tia of the cross-section, mass per unit length and circularfrequency of damping of the bridge respectively; ei = 1 ifaxle i is present on the bridge and ei = 0 if not; d(x � xi)is the Dirac function. Therefore,

ZiðtÞ ¼ Ks½y3iðtÞ � yiðtÞ� ð2aÞis the force in the spring between the ith axle and thevehicle body, where Ks is the suspension spring stiffness.

ZbiðtÞ ¼ Csdy3iðtÞ

dt� dyiðtÞ

dt

� �ð2bÞ

is the damping force between the ith axle and the vehiclebody, where Cs is a suspension linear damper.

y3iðtÞ ¼ y3ðtÞ � ð�1ÞiDiuðtÞ ð2cÞis the displacement of the contact point between the ith axleand the vehicle body.

m31 ¼ m3D2

D1 þ D2

m32 ¼ m3D1

D1 þ D2

ð2dÞ

are the masses of the sprung part of the vehicle by axle.

RiðtÞ ¼ Kt½yiðtÞ � eivðxi; tÞ � rðxiÞ�P 0 ð2eÞis the tire force imparted to the bridge by the ith axle, whereKt is tire stiffness and r(xi) is height of road profile at locationof axle i. Negative values of the tire force, representing lossof contact with the road surface, are set to zero. Based on thework of Fryba [32], numerical results are found in the timedomain through the Runge–Kutta–Nystrom method [33].

2.2. Dynamic amplification due to 2-axle vehicle

In a previous study [25], the authors have proven thatthe effect of individual road irregularities can be super-posed for a quarter-car model traveling on a short-spanbridge with a good road profile. Bending moment is found

E. OBrien et al. / Computers and Structures 84 (2006) 759–769 761

by adding the moment due to the vehicle traveling on a per-fectly smooth surface and the moment due to each individ-ual irregularity that exists on the surface. The approach isaccurate for negligible bridge deflections and a goodapproximation when bridge deflections are small comparedto the road irregularities. Based on this assumption ofsuperposition, the total normalized midspan bending

0.1m0.1m

CsKs

Kt

mt

ms Speed

H2H1

-5 0 5 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

M

(

CsKs

Kt

mt

ms Speed

-5 0 5 10-0.2

0

0.2

0.4

0.6

0.8

1

1.2

x

M0

(b

Fig. 2. Concept of superposition of unit ramps: (a) total moment; (b) moment d(d) moment due to a unit ramp at the second location.

moment, M(c, t), for a vehicle speed, c, and position of1st axle x1, can be calculated as

Mðc; x1Þ ¼ M0ðc; x1Þ þXN

i¼1

½Muðc; i; x1Þ �M0ðc; x1Þ� � sðiÞ

ð3Þ

x

0 15 20 25x

a)

x

15 20 25

)

ue to a smooth profile; (c) moment due to a unit ramp at the first location;

0.1m

x0.001m

CsKs

Kt

mt

ms Speed

-5 0 5 10 15 20 25-6

-4

-2

0

2

4

6x 10

-3

x

Mu(

x=-0

.1) -

M0

0.001m

0.1m

x

CsKs

Kt

mt

ms Speed

-5 0 5 10 15 20 25-6

-4

-2

0

2

4

6x 10

-3

x

Mu(

x=0)

- M

0

(c)

(d)

Fig. 2 (continued)


where M0 is the normalized midspan bending momentcaused by the vehicle on a smooth profile and Mu is thenormalized midspan moment due to a unit ramp at loca-tion i. The profile is discretized into N ramps each100 mm long and the measured fall in the ith ramp inmm is s(i). All bending moments are normalized by divid-ing them by the maximum static bending moment. Thecalculation procedure is illustrated in Fig. 2 for a roadprofile made of two ramps of heights H1 and �H2 locatedat �0.1 and 0 m respectively. The total moment M due toa vehicle of given mechanical parameters and speed

(Fig. 2(a)) is the result of adding three moments: (1) M0

corresponding to a smooth profile (Fig. 2(b)), (2)[s(1) · (Mu(x=�0.1) �M0)], or moment due to a unit rampat x = �0.1 (Fig. 2(c)) scaled by the height of the firstramp (s(1) = �H1/0.001), and (3) [s(2) · (Mu(x=0) �M0)],or moment due to a unit ramp at x = 0 (Fig. 2(d)) scaledby its height (s(2) = H2/0.001). This approach involves theaddition of bending moment values corresponding to dif-ferent critical points near the center of the bridge. Then,the dynamic amplification factor due to a vehicle travelingat speed c can be estimated from the maximum normalized

-25 -20 -15 -10 -5 0 5 10 15 20 25

0

5

10

15

20

25

Distance along the bridge (m)

Ele

vatio

n (m

m)

Fig. 3. Road Profile adjacent to and on the bridge (0 corresponds to thestart of the bridge).

50 60 70 80 90 100 110 120 130 140 1500.95

1

1.05

1.1

1.15

1.2

1.25

Speed (km/h)

0.7DD 1.3D

DA

E

Fig. 4. Influence of axle spacing on dynamic amplification.

50 60 70 80 90 100 110 120 130 140 1500.85

0.9

0.95

1

1.05

1.1

Speed (km/h)

(a)

Position of ramp / Length of bridge-1 -0.5 0 0.5 1

50

60

70

80

90

100

110

120

130

140

150

-8

-6

-4

-2

0

2

4

6

8x 10

-3

(b)

M0

(0.5

L)S

peed

(km

/h)

Fig. 5. (a) Normalized moment for smooth surface and a range of speeds(front axle at mid-span). (b) Normalized moment for a range of unit ramplocations and speeds (front axle at mid-span).


total moment for any location x1 while the vehicle is nearthe center of the bridge:

DAEðcÞ ¼ max0:4L6x160:8L

½Mðc; x1Þ� ð4Þ

Table 1Vehicle and beam properties

Vehicle properties

Physical description Symbol Value

Sprung mass m3 18000 kgUnsprung mass m1,m2 1000 kgSuspension stiffness Ks 80 kN/mTire vertical stiffness Kt 1800 kN/mPassive damping coefficient C 7 kNs/mDistance between the axles to center of gravity Di 2.5 m

It becomes clear that the nature of a given road profilerelating bridge dynamics, can only be characterized as‘good’ or ‘poor’ linked to a given vehicle type.

Unfortunately, while DAE provides an excellent esti-mate of dynamic amplification for a particular vehicle, itis only accurate for that vehicle. For example, on the road

Beam properties

Physical description Symbol Value

Mass per unit length l 18358 kg/mMoment of inertia of beam cross-section J 1.3901 m4

Bridge length L 25 mYoung’s modulus of beam E 3.5 · 1010 N/m2

-1 -0.5 0 0.5 1-8

-6

-4

-2

0

2

4

6

8x 10

-3

Position of ramp / Length of bridge(a)

-1 -0.5 0 0.5 1-6

-4

-2

0

2

4

6

8x 10

-3

Position of ramp / Length of bridge(b)

(c)

Dyn

amic

Am

plifi

catio

nD

ynam

ic A

mpl

ifica

tion

Dyn

amic

Am

plifi

catio

n

-1 -0.5 0 0.5 1-5

-4

-3

-2

-1

0

1

2

3

4x 10-3

Position of ramp / Length of bridge

Fig. 6. (a) Contribution to dynamic amplification due to a range of unitramp locations for vehicle travelling at 80 km/h. (b) Contribution todynamic amplification due to each road irregularity in Fig. 3 for vehicle at80 km/h. (c) Contribution to dynamic amplification of each roadirregularity in 3(a) for vehicle at 120 km/h.


profile described in Fig. 3, the 2-axle estimator is illustratedfor three vehicles with axle spacings D (=5 m), 0.7D and1.3D in Fig. 4. Clearly the spacing used to generate theterms, M0(c) and Mu(c, i) in Eq. (3), has a most significantinfluence on the result. Further, there is no one spacing fora 2-axle vehicle that will provide good estimates of dynamicamplification for other vehicle spacings.

DAF and DAE are also strongly influenced by vehiclespeed. The normalized bending moment due to the two-axlevehicle defined by parameters in Table 1 and traveling on asmooth road profile, M0(c,x1), is illustrated in Fig. 5(a) forx1 = 0.5L, i.e., front axle at the center of the bridge. The cor-responding normalized moments for unit ramp excitationsof the vehicle, Mu(c, i, 0.5L) �M0(c, 0.5L), are given inFig. 5(b). This graph provides a valuable insight into thesource of the total bending moment. It can be seen that par-ticular locations on the bridge are especially important. Forexample a ramp at 0.4L is highly significant while a ramp at0.6L makes no contribution. It can also be seen that there arepositive and negative effects – light and dark zones in the fig-ure. Hence the effects of some ramps will cancel out whileothers will be additive and there will be particular combina-tions of ramps that will result in very high bending moment.

A section through Fig. 5(b) corresponding to a speed of80 km/h is illustrated in Fig. 6(a). The product of thisgraph and the road profile of Fig. 3 is illustrated inFig. 6(b). The sum of all ordinates in the latter graph rep-resents the total contribution of the road profile to the nor-malized bending moment (see Eq. (3)). It can be seen thatbending and hence dynamic amplification is highly sensi-tive to the location of road surface irregularities. The cor-responding graph for a speed of 120 km/h, illustrated inFig. 6(c), shows the sensitivity to speed.

3. Monte Carlo simulation

It has been shown that dynamic amplification and theestimate of dynamic amplification provided by DAE fora given vehicle, is strongly influenced by the inter-axle spac-ing and speed of that vehicle. Similar variability in resultscan be shown for other vehicle properties such as the ratioof the axle weights. A BRI does not need to be accurate butmust provide an indication of DAF for the range of vehi-cles and speeds that are likely to occur on the bridge.

To overcome the problem of multiple parameters, statis-tical data is used to determine the properties of ‘typical’vehicles from a truck fleet. For this paper, truck fleet statis-tics were obtained from a Weigh-in-Motion station on theA1 road near Ressons in France. Data was collected over aperiod of 105 h for 9498 trucks. Only 2-axle trucks of morethan 15 tonne gross vehicle weight were considered. Thehistograms for gross vehicle weight, spacing, axle weightratio and speed are illustrated in Fig. 7.

Monte carlo simulation [34] is used to generate 2-axletruck properties representative of the measured data. Thisis achieved by a simple bootstrapping from the measureddata, i.e., frequency is selected by generating a random

0 0.5 1 1.50

100

200

300

Axle weight ratio20 40 60 80 100

0

100

200

300

Speed (km/h)

15 20 25 300

50

100

150

200

250

Total weight (tonne)2 4 6 8

0

100

200

300

400

500

Spacing (m)N

umbe

r

Num

ber

Num

ber

Num

ber

Fig. 7. Histograms of gross vehicle weights, axle spacings, axle weight distribution and speeds for a two-axle truck fleet.

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.65

0.7

0.75

0.8

0.85

0.9

0.95

1

x1


x 1

-1 -0.5 0 0.5 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

-4

-2

0

2

4

6

x 10-3

Mo

(a)

(b)

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

x1


x 1

-1 -0.5 0 0.5 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

1

2

3

4

5

6

x 10-3

M0

SD

(c)

(d)

Fig. 8. Normalized mid-span moments in a 25 m bridge due to a truck fleet: (a) Mean normalized moment for smooth surface and a range of vehiclelocations. (b) Mean normalized moment for a range of vehicle and unit ramp locations. (c) Standard deviation of normalized moment for smooth surfaceand a range of vehicle locations. (d) Standard deviation of normalized moment for a range of vehicle and unit ramp locations.


0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

x1


x1

-1 -0.5 0 0.5 10.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

-6

-4

-2

0

2

4

6

8

10

x 10-3

Dyn

amic

Am

plifi

catio

n

(a)

(b)

-1 -0.5 0 0.5 1-4

-2

0

2

4

6

8

10

12x 10-3


0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.80.7

0.75

0.8

0.85

0.9

0.95

1

1.05

x1

Dyn

amic

Am

plifi

catio

n

Dyn

amic

Am

plifi

catio

n

(c)

(d)

Fig. 9. (a) Mean contribution to dynamic amplification due to road profile in Fig. 3 for a range of vehicle locations. (b) Mean contribution to dynamicamplification due to each road irregularity for a range of vehicle locations. (c) Mean contribution to dynamic amplification due to each road irregularitywhen front axle is at 0.68L. (d) Total mean plus the standard deviation normalized moment due to the road profile.


number between zero andP

ifi, where fi is the frequency ofinterval i; the property from the corresponding interval inthe histogram is then used in the simulation. While theremay be some correlation between vehicle properties, allproperties are assumed to be independent in this study.For each of 21 values of x1 in the interval 0.4L 6 x1 6

0.8L, 500 sets of property values were generated and themean and standard deviation for M0(x1) calculated. Theresults are illustrated in Fig. 8(a) and (c). Fig. 8(a) corre-sponds to some extent to a truck fleet equivalent ofFig. 5(a) – it represents the mean normalized moment ona smooth road profile for 500 trucks with speeds, axle spac-ings etc. that are typical of the fleet. However, whileFig. 5(a) provides the maximum normalized moment fora range of speeds when the front axle is at midspan,Fig. 8(a) allows the vehicle to be at a range of points, aver-aged over many speeds and other vehicle parameters.Hence it gives an indication of the mean dynamic amplifi-cation that might be expected of 2-axle vehicles, if the roadprofile were smooth. For example, when the front axle is

located at the center of the bridge (x1 = 0.5L), the meannormalized midspan moment, Mð0:5LÞ, is 0.8501, whichmeans that the mean midspan moment from the 500trucks/speeds considered is 0.8501 of the static value.

Fig. 8(b) is in some sense the truck fleet equivalent ofFig. 5(b). It gives the mean moment due to a unit ramponly at a given point from a representative sample of 500trucks/speeds for a range x1 of vehicle locations. Figs.8(c) and (d) give the corresponding standard deviationsof normalized bending moment: MSD

0 ðx1Þ and[Mu(x1) �M0(x1)]SD respectively.

Fig. 8 is used to evaluate the Bridge Roughness Index,BRI, for a given vehicle class, defined as:

BRI ¼ max0:4L6x160:8L

Mðx1Þ þMSDðx1Þ� �

ð5Þ

where,

Mrði; x1Þ ¼ Muði; x1Þ �M0ðx1Þ ð6aÞ

Mðx1Þ ¼ M0ðx1Þ þXN

i¼1

Mrði; x1Þ � sðiÞ� �

ð6bÞ

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1


x 1

-1 -0.5 0 0.5 1

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

Mo

(a)

(b)

Fig. 11. Normalized mid-span moments in a 15 m bridge due to a truck fleet:locations. (b) Mean normalized moment for a range of vehicle and unit ramp loand a range of vehicle locations. (d) Standard deviation of normalized momen

Fig. 10. (a) DAF versus IRI for a 25 m bridge. (b) DAF versus BRI for a25 m bridge.


MSDðx1Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMSD

0 ðx1Þ� �2 þ

XN

i¼1

MSDr ði; x1Þ

� �2 � ½sðiÞ�2n ovuut

ð6cÞ

The BRI represents a kind of mean estimate of dynamicamplification for vehicle properties and speeds representa-tive of site measurements of key vehicle properties. Ofcourse, considerable inaccuracy is introduced by takingthe maximum of many averaged components instead ofaveraging the maxima. However, this is necessary as it isonly in this way that the BRI can be evaluated directlyfrom a knowledge of Fig. 8 and the road profile, s(i).

4. Application and assessment of BRI

The application of the BRI is illustrated using the roadprofile of Fig. 3. The contribution of the different vehicleson a smooth profile can be seen to be varying from about0.85 to 0.99 in the range of 0.5L to 0.75L as described in

0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

x1


x 1

-1 -0.5 0 0.5 1

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.005

0.01

0.015

0.02

M0

(c)

(d)

SD

(a) Mean normalized moment for smooth surface and a range of vehiclecations. (c) Standard deviation of normalized moment for smooth surfacet for a range of vehicle and unit ramp locations.


Fig. 8(a). The influence of the road profile roughness isillustrated in Fig. 9(a), which gives a positive contributionof up to 0.02 at the critical position of 0.68L, and a nega-tive contribution at the position of 0.6L. Fig. 9(b) providesa breakdown of the contributions of the ramps that makeup the profile to the mean normalized moment, namely,Mrði; x1Þ � sðiÞ. It is interesting to note that for this partic-ular road profile, irregularities in the approach and at thestart of the bridge are not significant (this would of coursebe different if there were large irregularities due to damageat a joint). The most important contributions of the roadprofile to bending moment occur in the range of 0.3L to0.6L and depend on the location of the critical moment.At the time when x1 = 0.68L, the critical point for thisexample, the contributions are illustrated in Fig. 9(c). Tak-ing into account the standard deviation of the normalizedbending moment, the dynamic amplification in Eq. (5) isdescribed as Fig. 9(d), where the maximum value of 1.04at the critical position of 0.7L represents BRI.

The accuracy of the BRI as a measure of profile rough-ness is assessed through comparison with DAF for 500,000vehicle/speed/profile combinations. For this purpose, onethousand random road profiles are simulated of Class ‘B’[20,35]. Together, the profiles represent a wide range ofconditions with different positions of the ramps. For eachroad profile, 500 sets of vehicle properties are generatedby Monte Carlo simulation. In all cases, the exact dynamicamplification, DAF, is calculated for comparison. TheDAF is related to IRI in Fig. 10(a). Each point in this fig-ure corresponds to one road profile, the mean of DAF plusthe standard deviation is plotted against the IRI. It can beseen that there is no discernable correlation between IRIand DAF. (However, it is of note that better correlationis evident if a wider range of road roughnesses is consid-ered). The coefficient of correlation for this set of Class

Fig. 12. (a) DAF versus IRI for a 15 m bridge. (b) DAF versus BRI for a15 m bridge.

‘B’ profiles is 0.185. The ratio of DAF to BRI is illustratedin Fig. 10(b). While there is considerable scatter, there is aclear correlation and the correlation coefficient is 0.762.

The effect of road roughness is strongly influenced bythe span of bridge considered. Figs. 11(a) and (b) givethe mean normalized moment on a smooth profile anddue to a unit ramp for a 15 m bridge respectively (equiva-lent of Figs. 8(a) and (b)). While Fig. 11(a) is similar toFig. 8(a), considerable differences are evident in Figs. 8(b)and Fig. 11(b) in the magnitudes of the effects of unitramps. The corresponding standard deviation of normal-ized moment caused by the 2-axle truck fleet when ignoringand taking into account the road roughness are shown inFigs. 11(c) and (d) respectively. Nevertheless, the principleof superposition still applies and the BRI has a good corre-lation with DAF as can be seen in Fig. 12.

5. Discussion and conclusions

The concept of a bridge roughness indes, BRI, is devel-oped in this paper. It is shown that IRI is poorly corre-lated with dynamic amplification for roads of averageroughness. While there is considerable scatter in the rela-tionship between the proposed BRI and dynamic amplifi-cation, this index is well correlated and considerably moreso than IRI.

The index is independent of individual vehicle proper-ties such as speed and axle spacing but is a function ofthe vehicle fleet properties, represented by histograms(Fig. 8). For a given bridge, it is calculated from the pro-file information only using factors derived from the fleethistograms, represented here by Fig. 7. Once the fleet-spe-cific factors are known, the calculation of the BRI is quitesimple (Eq. (5)).

The BRI factors developed here are applicable to 2-axlevehicles over 15 tonnes and to mid-span bending momentin 25 m simply supported bridges. It is shown that the cor-responding factors for a 15 m bridge are quite different.Clearly the same procedure can be used to develop similarfactors for other fleets, spans and load effects. For exam-ple, it would be straightforward to develop factors for 5-axle trucks representative of Western European highwaytraffic. This could be repeated for a number of spansand load effects. However, this would still only representan indicator of dynamic amplification for a single truckcrossing event and would not be accurate for two or moretrucks meeting or passing on the bridge. The great varietyof combinations of truck speeds, properties and otherparameters is such that there appears to be no simpleroughness measure of a road surface that is universallyapplicable and has a strong correlation with dynamicamplification. Nevertheless, the BRI is valuable in that itgives insight into the contribution that road roughnessmakes to dynamics. The importance of irregularity loca-tion and the sensitivity of dynamic amplification to irreg-ularities at particular points both on and in the approachto the bridge are identified.


References

[1] DIVINE Programme, OECD, Dynamic interaction of heavy vehicleswith roads and bridges. DIVINE concluding conference. Ottawa,Canada, 1997.

[2] Olsson M. On the fundamental moving load problem. J Sound Vib1991;154(2):299–307.

[3] Liu C, Huang D, Wang TL. Analytical dynamic impact study basedon correlated road roughness. Comput Struct 2002;80:1639–50.

[4] Green MF, Cebon D, Cole DJ. Effects of vehicle suspension design ondynamics of highway bridges. J Struct Eng, ASCE 1995;121(2):272–82.

[5] Santini Greco A. Dynamic response of a flexural non-classicallydamped continuous beam under moving loadings. Comput Struct2002;80:1945–53.

[6] Fafard M, Bennur M, Savard M. A general multi-axle vehicle modelto study the bridge-vehicle interaction. Eng Comput 1997;15(5):491–508.

[7] Brady SP, OBrien EJ, Znidaric A. The effect of vehicle velocity on thedynamic amplification of a vehicle crossing a simply supportedbridge. J Brid Eng, ASCE 2006;11(2):241–9.

[8] Coussy O, Said M, Van Hoove JP. The influence of random surfaceirregularities on the dynamic response of bridges under suspendedmoving loads. J Sound Vib 1989;130(2):313–20.

[9] Gillespie TD, Sayers MW, Segel L. Calibration of response-type pavement roughness measuring systems. National CooperativeHighway Research Program. Res. Rep. No. 228, Transporta-tion Research Board. National Research Council. Washington DC,1980.

[10] Gillespie TD, Karimihas SM, Cebon D, Sayers M, Nasim MA,Hansen W, Ehsan N. Effects of heavy-vehicle characteristics onpavement response and performance. National Cooperative HighwayResearch Program. Res. Rep. No. 353, Transportation ResearchBoard, National Research Council. Washington, DC, 1993.

[11] Marcondes JA, Snyder MB, Singh SP. Predicting vertical accelerationin vehicles through pavement roughness. J Trans Eng, ASCE1992;118(1):33–49.

[12] Yang YB, Lin BH. Vehicle-bridge analysis by dynamic condensationmethod. J Struct Eng, ASCE 1995;121(11):1636–43.

[13] Ollsson M. Finite element, model co-ordinate analysis of structuressubjected to moving loads. J Sound Vib 1985;99(1):1–12.

[14] Lin YH. Comments on dynamic behaviour of beams and rectangularplates under moving loads. J Sound Vib 1997;200(5):721–8.

[15] Henchi K, Fafard M, Dhatt G, Talbot M. Dynamic behavior ofmulti-span beams under moving loads. J Sound Vib 1997;199(1):33–50.

[16] Majumder L, Manohar CS. A time-domain approach for damagedetection in beam structures using vibration data with a movingoscillator as an excitation source. J Sound Vib 2003;268:699–716.

[17] Green MF, Cebon D. Dynamic response of highway bridges to heavyvehicle loads: Theory and experimental validation. J Sound Vib1994;170(1):51–78.

[18] Chompooming K, Yener M. The influence of roadway surfaceirregularities and vehicle deceleration of bridge dynamics using themethod of lines. J Sound Vib 1995;183(4):567–89.

[19] Kou JW, Dewolf JT. Vibration behavior of continuous span highwaybridge-influencing variables. J Struct Eng, ASCE 1997;123(3):333–44.

[20] Gonzalez A. Development of accurate methods of weighing trucks inmotion, PhD Thesis. Department of Civil Engineering, TrinityCollege, Dublin, Ireland, 2001.

[21] Yau JD, Yang YB. Vibration reduction for cable-stayed bridgestraveled by high-speed trains. Finite Elem Anal Des 2004;40:341–59.

[22] Law SS, Zhu XQ. Bridge dynamic responses due to road surfaceroughness and braking of vehicle. J Sound Vib 2005;282(3–5):805–30.

[23] Lei X, Noda NA. Analyses of dynamic response of vehicle and trackcoupling system with random irregularity of track vertical Profile. JSound Vib 2002;258(1):147–65.

[24] Chatterjee PK, Datta TK, Surana CS. Vibration of continuousbridges under moving vehicles. J Sound Vib 1994;169(5):619–32.

[25] Li YY, OBrien EJ, Gonzalez A. The Development of a dynamicamplification estimator for bridges with good road profiles. J SounVib, in press.

[26] Michaltsos GT. Parameters affecting the dynamic response of light(Steel) bridges. Facta Universitatis, Series Mechanics, AutomaticControl and Robotics 2000;2(10):1203–18.

[27] Pesterev AV, Bergman LA, Tan CA. A novel approach to thecalculation of pothole-induced contact forces in MDOF vehiclemodels. J Sound Vib 2004;275(1–2):127–49.

[28] Pesterev AV, Bergman LA, Tan CA, Yang B. Assessing tire forcesdue to roadway unevenness by the pothole dynamic amplificationfactor method. J Sound Vib 2005;279(3–5):817–41.

[29] Zhu XQ, Law SS. Dynamic load on continuous multi-lane bridgedeck from moving vehicles. J Sound Vib 2002;251(4):697–716.

[30] Brady SP, OBrien EJ. The effect of vehicle velocity on the dynamicamplification of two vehicles crossing a simply supported bridge.J Bridge Eng, ASCE 2006;11(2):250–6.

[31] Yang YB, Yau JD, Wu YS. Vehicle-bridge interaction dynamics withapplications to high-speed railways. Singapore: World ScientificPublishing Co. Pte. Ltd; 2004.

[32] Fryba L. Vibration of solids and structures under moving loads.Groningen, The Netherlands: Noordhoff International Publishing;1972.

[33] Kreyszig E. Advanced engineering mathematics. New York: JohnWiley & Sons Inc; 1983.

[34] Metcalfe AV. Statistics in civil engineering. New York: John Wiley& Sons Inc; 1997.

[35] ISO 8608. Mechanical vibration-road surface profiles-reporting ofmeasure data, 1995.

bridge roughness index as an indicator of bridge dynamic amplification

Documents