Load Model for Bridge Design Code - Canadian Journal of Civil Engineering

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Load model for bridge design code

ARTICLE in CANADIAN JOURNAL OF CIVIL ENGINEERING FEBRUARY 2011

Impact Factor: 0.56 DOI: 10.1139/l94-004

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Andrzej S Nowak

University of Michigan

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oad model for bridge design code

A N D R Z E J. NOWAK

D ep ar tn le tl t o f C iv il a n d E ~ ~ v i r o r ~ n ~ e r ~ t a lng i neer i ng , Uni ver s i t y o f Mi ch i gan , At111 A r b o ~ ,M I 48109-2125 U.S.A.

Received January 8 1993

Revised manuscript accepted May 26, 1993

The paper deals with the development of load model for the Ontario Highway Bridge Design Code. Three com-

ponents of dead load are considered: weight of factory-made elements, weight of cast-in-place concrete, and bitu-

minous surface (asphalt). The live load model is based on the truck survey data. The maximum live load moments

and shears are calculated for one-lane and two-lane bridges. For spans up to about 40 m, one truck per lane governs;

for longer span s, two trucks follow ing behind the o ther provide the largest live load effect. For two lanes , two

fully correlated trucks govern. The dynamic load is modeled on the basis of simulations. The results of calcula-

tions indicate that dynamic load depends not only on the span but also on road surface roughness and vehicle

dynamics. Load combination including dead load, live load, dynamic load, wind, and earthquake is modeled using

Turkstra s rule. The maximum effect is determined as a sum of the extreme value of one load co mpon ent plus the

average values of other simultaneous load components. The developed load models can be used in the calculation

of load and resistance factors for the design and evaluation code.

Key words : bridge, dead load, live load, dynamic load, load combinations.

Cet article traite du dCveloppement d un modble de charge pour le Code de concep tion des po nts routiers de

I On tario (CC PRO ). Tro is caractkris tique s de la charg e permanente ont CtC CtudiCes le poids des ClCments fabriquCs

a I usin e, le poids du bCton coulC sur place ainsi qu e celui du revste men t bitumineux (asph alte). Le modble de

charg e mobile a CtC ClaborC en tenant com pte de certain es m esures et de donnCes d en qu ste sur les cam ions . Les

moments et les cisaillements maximums dus h la surch arge ont CtC calculCs pour des ponts une et deux voi es.

Pour les portCes d un e lo ngueur d e 40 m et mo ins, l effet maxim al est causC par un camion par voie; pour les

portCes plus long ues, deux camio ns qui se suivent crCent l effet le plus imp ortant en termes de charge mobile. La

modClisation de la charge dynam ique a CtC eff ectu ie en tenant c omp te de simu lation s. Les rCsultats des calculs

indiquent que la charge dynamique depend non seulement de la portCe, mais aussi de la rugositC de la surface de roule-

ment et de la dynamique des vChicules. Une combinaison de charges comprenant la charge permanente, la charge mobile,

la charge dynamique, le vent et les secousses sismiques a fait I objet d une modClisation l aid e de la rbgle de

Turkstra. L effet max imal est obtenu en additionn ant la valeur extrgm e d un e caractkristique de ch arge et les valeurs

moyennes des autres. Les modbles de charge ClaborCs peuvent servir au calcul des coefficients de rksistance et de charge

pour le code d e conception et dlCvaluation.

Mo ts cle s

pont, charge permanente, charge mobile, charge dynamique, combinaisons de charge.

[Traduit par la rCdaction]

Can. I.

Civ.

Eng.

21

3 6 4 9 (1994)

Introduction

Bridge loads p lay an increas ingly impor tant ro le in the

deve lopmen t of des ign and eva lua t ion c r i te r ia . Th e funda-

mental load combination includes dead load, l ive load, and

dynamic load. Th is paper dea ls wi th the der iva t ion of s ta -

t i s ti c a l m ode l f o r t he se l oa d c om pon e n t s . Th e p r e se n te d

research provided statistical mod els for the development of

load and res is tance fac tors in the Onta r io Highway Br idge

Design Code (OHBDC) 1991 edi t ion .

he analysis of bridge loads was performed in conjunction

with the development of two previous editions of the OHBDC

( Nowa k a nd L ind 1979 ; Gr oun i a nd Nowa k 1984) . Loa d

models were deve loped on the bas is of the ava i lable t ruck

s u r v e y s a n d o t h e r m e a s u r em e n t s . T h e m a x i m u m 5 0 - y e a r

l ive load was de te rmined by exponent ia l ext rapola t ion of

the extreme values obtained in the survey. AASHTO (1989)

girder distribution factors were used in the analysis. Dynam ic

load was modeled using the available test data .

The new deve lopments a f fec t dead load, l ive load, and

dynamic load. Dead load is based on the la tes t ava i lable

da t a . The l i ve l oa d m ode l i s de ve lope d f o r one - l a ne a nd

NOT E :

Written discussion of this paper is welcomed and will be

received by the Editor until June 30, 1994 (address inside front

cover).

two- l a ne b r idge s . An im por t a n t pa r t o f t h i s s tudy i s t h

dynam ic load analysis. Th e model is developed on the basi

of an analytical simulation of the actual br idge behavior .

The major load com ponents of h ighway br idges a re dea

load, l ive load, ( s ta t ic and dynamic) , environmenta l load

(temperature, wind, ear thquake) , and other loads (collision

emergency braking). The load models are developed using th

available statist ical data , surveys, and other observations

Loa d c om pone n t s a r e t r ea t e d a s r a ndom va r i ab l e s . The i

variation is described by the cumulative distribution function

the mean value, and the coeff icient of variation. The rela

tionship between load parameters is described by a coefficien

of correlation.

T h e b a s i c l o a d c o m b i n a t i o n f o r h i g h w a y b r i d g e s i s

simultaneous occurrence of dead load, live load, and dynami

loa d . The c om bina t ions i nvo lv ing o the r l oa d c om pone n t

(wind, earthquake, collision forces) require a special approac

which takes into account a reduced probabili ty of a simul

taneous occur rence of ex t reme va lues of severa l indepen

dent loads.

ead load

Dead load, D is the gravity load due to the self weight o

the structural and nonstructural e lements permanently con

nected to the bridge. Because of different degrees of varia

Printed

in Canada Innprime nu C ln;~d;l


3/15

N O W A K

TABLE

.

Statistical parameters of dead load

Compo nent Mean-to-nom inal Coefficient of vibration

Factory-made members 1.03

Cast-in-place members 1.05

Asphalt 90 mm*

Miscellaneous 1.03- 1.05

Mean thickness.

t ion, i t is convenient to con sider the following comp onents

of

D:

i )

D

the weight of fac tory-made e lements ( s tee l ,

precas t concre te m embers) ; ( i i )

Dz

the weight of cas t - in-

place concrete members; ( i i i)

D

the weight of the wearing

surface (asphalt) ; and ( iv)

D

miscellaneous weight (e .g. ,

r a i l ing , luminar ies) . Al l components of

D

a r e t r e a t e d a s

normal random var iables . The s ta t i s t ica l pa ramete rs used

in the c a l ib r a t ion a r e l i s t e d in Ta b le 1 . Th e b i a s f a c to r s

(mean-to-nominal ratios) are taken as in the previous cali-

bra tion w ork (Now ak and Lind 1979) . However , the coef -

f icients of variation are increased to include human errors,

as recommended by El l ingwood e t a l . (1980) .

The thickness of asphalt was f irst modeled on the basis

of s ta t i s t ica l da ta ava i lable f rom the Onta r io Minis t ry of

Transportation (MTO). Measurements were done in various

regions of the Province. The distributions of

D

(thickness of

asphalt) are plotted on normal probability paper in Fig. 1. The

average th ickness of aspha l t i s 75 m m. Th e coef fic ient o f

variation, calculated from the slope of the distr ibutions in

Fig. 1, is 0.25. However, fur ther information provided by

the MTO indica tes tha t the mean th ickness of aspha l t has

increased to 90 mm and the coefficient of variation is reduced

to 0.15 (Agarwal, yet unpublished).

For miscellaneous i tems (weight or rail ings, curbs, lumi-

nar ies , s igns , condui ts , p ipes , cables , e tc . ) , the s ta t i s t ica l

parameters (means and coefficients of variation) are similar

to those of

D

if the considered i tem is factory-made with

the h igh qua l i ty contro l measures , and

D2

if the i tem is

cast- in-place, with less str ic t quality control.

Live load data base

Live load, L, covers a range of forces produced by vehicles

moving o n the bridge. Traditionally, the static and d ynamic

effects are considered separately. Therefore, in this study,

L covers only the s ta t ic component . The dynamic compo-

nent is denoted by I.

T h e e f f e c t o f l i v e l o a d d e p e n d s o n m a n y p a r a m e t e r s ,

inc luding the span length , t ruck weight , axle loads , axle

configuration, position of the vehicle on the bridge (transverse

and longitudinal), number of vehicles on the bridge (multiple

presence), girder spacing, and stiffness of structural members

(slab and girders) .

Th e live load model is based on the truck survey in On tario

per formed by the MTO in 1975. The s tudy covered about

10 000 s e l e c t e d t r uc ks ( on ly t r uc ks t ha t a ppe a r e d t o be

he a v i ly l oa de d we r e m e a su r e d a nd inc lude d in t he da t a

base) . The results of the 198 8 truck survey including o ver

2000 trucks (Agarwal, yet unpublished) are also considered

to study the changes in l ive load over the years.

T h e u n c e r t a i n ti e s i n v o l v e d i n t h e a n a l y s i s a r e d u e t o

l i m i t a t io n s a n d b i a s e s i n t h e s u r v e y d a t a . E v e n t h o u g h

10 000 trucks is a large number, i t is very sm all compared

with the actual number of heavy vehicles in a 50-year l ife-

ctual sphalt

hickness 7

mm

FIG. 1. Cumulative distribution functions of asphalt thick-

oions.

ess by MTO re,'

t ime . I t i s a lso reasonable to expec t tha t some ext remely

heavy trucks purposefully avoided the weighing stations. A

c ons ide r a b le de g r e e o f unc e r t a in ty i s c a use d by unpr e -

dictability of the future trends with regard to the configuration

of axles and weights.

The 1975 Ontario survey included a total of 9250 heavy

t r u c k s ( A g a r w a l a n d W o l k o w i c z 1 9 7 6 ) . F o r e a c h t r u c k ,

bending mo ments and shear forces were calculated for a wide

range of simple spans. The cumulative distribution functions

are plotted on normal probability paper in Fig. 2 for moments

and F ig . 3 for shears , for spans f rom 9 to 6 0 m. The con-

struction and use of the normal probability paper is explained

in the fundam enta l textbo oks on probabi l i ty theory (eg . ,

Benjamin and Cornell 1970). Th e horizontal scale is in terms

of the OHBDC (1983) live load (truck or lane load, whichever

governs) , as shown in Fig. 4. The vertical scale , z, is

[ l ] z a- '[F,(x)]

where F,(x) is the cumu lative distr ibution function of

X, X


4/15

38

C A N .

J .

CIV. ENG VOL. 21. 1991

M o m e n t

OHBDC 1983

o m e n t

S h e a r

OHBDC 1983

h e a r

FIG

2 Cum ulative distribution functions of truck m oments

FIG

3. Cumulative distribution functions of truck shears from

from 19 75 survey in terms of the OHB DC 1983 m oment.

1975 survey in terms of the OHBDC 1983 shear.

OH D Lan e

Load

OH D

Truck

2 kN

I

FIG

4 OHBDC 1983 live load.

i4 m i4 m

6

kN

b e i n g t h e m o m en t o r t h e s h ea r ; an d a- i s the inver se o f

funct ions o f mom ents and shear s ar e p lo t ted in F ig . an

the s tandard normal d i s t r ibu t ion funct ion .

Fig. 6 r e s p ect i v e ly . T h e r e s u l t s d o n o t i n d i ca t e an y co n

The moments and shear s were a l so ca lcu la ted fo r the 1988 s iderab le change in the maxim um mom ents and shear s i

t r u ck s u r v ey d a t a . T h e r e s u l t i n g cu m u l a t i v e d i s t r i b u t io n t h e t w o s u r v ey s .

16 k


5/15

4

0 0 5 1 1 5

Moment

OHBDC 1983

Moment

FIG

Cumulative distribution functions of truck moments

from 1988 survey in terms of the OHBDC-1983 moment.

Maximum truck m oments and shears

The maximum moments and shears for various t ime ~e ri o d s

are determined by extrapolation of the distributions as shown

in Figs. 7 and 8. Let N be the total number of trucks in time

period

T.

I t is assumed that the surveyed trucks represent

about 2-week traffic on a class A highway. Therefore, in

T

=

50 years, the number of trucks, N, will be about 1000 times

larger than in the survey. This will result in N

=

10 mill ion

trucks. The probability level corresponding to N is 1/N; for

N = 10 million, the probability is 1/10 000 000

=

lo- , which

cor responds to z = 5.19 on the ver t ica l sca le , a s shown in

Figs . 7 and 8 . The number of t rucks (N) , the probabi l i ty

( l / N ) , and the inverse normal d is t r ibution v a lue (z) cor re -

sponding to various time periods

T),

from 1 day to 75 years,

a re shown in Table 2 . The l ines cor responding to the con-

s id er ed p ro ba bi li ty l ev el s a re a ls o shoin in ~ i ~ s .and 8.

The mean maximum moments and shears corresponding to

various periods of time can be read directly from the graph.

F or e xa m ple , f o r 15 m spa n a nd T = 50 ye a r s , t he m e a n

m a x im um m om e n t i s 1 . 2 t im e s the de s ign m om e n t . I t i s

e qua l t o t he ho r i z on ta l c oo r d ina t e o f i n t e r se c t ion o f t he

extrapolated distr ibution and z

=

5.19 on the vertical scale .

For comp ar ison, the number of t rucks pass ing throug h the

bridge in 75 years is 1500 times larger than in the survey.

This corresponds to

=

5.26 on the vertical scale (Figs. 7 and

8). Similar calculations can be performed for other periods

of t ime.

4

0 0 5 1 1 5

Shear

OHBDC 1983

Shear

FIG

Cumulative distribution functions of truck shears from

1988 survey in terms of the OHBD C-1983 shear.

4

0 0.5 1 1.5

M o m e n t OHBDC1983M o m e n t

75 Years

50

Years

5

Years

1

Year

6

M o n t h s

2 M o n t h s

1

M o n t h

2weeks

1Day

The m ean moments and shears calculated for t ime periods

FIG

Extrapolated cum ulative distribution functions of truck

f r om 1 da y to 75 ye a r s a r e p r e se n te d in F igs . 9 a nd 10 , m ome nt s.


6/15

40

C A N . I CIV. ENG. VOL.

2 1

1994

75 Y e a r s

50

Y e a r s

5 Y e a r s

1

ear

6 M o n t h s

2

M o n t h s

1

M o n t h

2 Weeks

Day

0 0 0 5 1O 1 5

Shear

OHBDG 983

Shear

FIG. 8 Extrapolated cumulative distribution functions of truck

shears.

respectively. For comparison, the means are also plotted for

an average truck. Th e coeff icients of variation for the max-

imum truck m oments and shears can be calculated by trans-

f o r m a t ion o f t he d i s t r i bu t ion f unc t ions i n F igs .

7

a nd 8 .

Each function can be raised to a certain power, so that the

calculated earlier mean maximum m oment or shear) becomes

the mean va lue a f te r the t ransformat ion. The s lope of the

transformed cumulative distr ibution function determ ines the

coefficient of variation. The results are plotted in Figs. 11 and

12 for moments and shears, respectively. For 50 years, the

bias fac tors a re a lso g iven in Table 3 .

One lane moments nd shears

F or one - l a ne b r idge s, t he m a x im um e f f e c t m om e n t o r

shear) is caused by a single truck or two or more) trucks

fol lowing behind each other . For a mul t ip le t ruck occur -

rence , the impor tant pa ramete rs a re the headway dis tance

and the degree of cor re la t ion be tween t ruck weights . The

maximum one- lane e f fec t i s de r ived as the la rges t of the

fol lowing cases :

a ) S ing le t r uc k ef f e ct e qua l t o the m a x im u m 50- y e a r

momen t or shear ) wi th the paramete rs mean and coef f i -

cient of variation) given in Figs. 9 and 11 for the moment and

in F igs . 10 and 12 for the shear ;

b) Two t rucks , each wi th the weight smal le r than tha t

of a single truck in case a) . Thre e degrees of correlatio n

between truck weights are considered: none

p

0), partia l

p 0 . 5 ) , a nd f u l l p l ) , whe r e p i s t he c oe f f i c i e n t o f

correlation.

I t is assumed that, on average, about every 50th truck is

fo l lowed by another t ruck wi th the headway dis tance less

than 30 m, about every 250th t ruck is fo l lowed by a par -

t ia lly correlated truck, and about every 500th truck is fol-

TA BL E Number of trucks vs. time period and probability

Time period Num ber of trucks Probability Inverse norma

T

N 1

IN

75 years

50 years

5 years

1 year

6

months

2

months

1 month

2 weeks

day

TAB LE. Bias factors atio of the maximum

50-year l ive load and OHBDC-1983 design

live load per lane)

One or two

Single truck trucks

Span

m) Moment Shear Moment Shear

lowed by a fully correlated truck. The two trucks are denote

by T I a n d T . Three cases a re considered:

i ) No correlation between T I and T? The parameters of T

are taken for every 50th truck, or the maximum of 200 00

1-year truck in Table 2). This corresponds to

4.42

on th

ve r t i c a l s c a l e i n F igs

7

a n d 8 . T h e p a r a m e t e r s o f T a r

taken for an average t ruck.

i i ) Par t ia l cor re la t ion be tween T I a nd T7. The parame

ters of T I are taken for every 250th truck, or the maximum

of 40 000 2-month truck in Table 2). This corresponds to

4.05 on the vertical scale in Figs.

7

and 8. The parameters o

T Z a r e t ake n f o r e ve r y 1000 th tr uc k , o r t he m a x im um o

1 0 0 0 I - d a y t r u ck i n T a b l e 2 ) , w h i c h c o r r e s p o n d s t

3 .09.

iii) Full correlation between T I and T z .The parameters o

T I and T are taken for every 500th truck, or the maximum

of 20 000 I - m on th t r uc k in Ta b le 2 ) , wh ich c o r r espond

to 3.8 9 on the vertical scale in Figs. 7 and 8 .

The truck effects are determined by simulation for variou

time periods, for a headway distance equal to 5 m bumper

to-bumper traffic). Th e results are presented in Figs. 13 an

14. For the 50-year period, the bias factors are also listed i

Table 3 . A compar ison wi th F igs . 9 and 10 indica tes tha

one t ruck governs for spans less than 30-40 m. For longe

spans, two fully correlated trucks govern. The headway dis

t a nc e o f 5 m i s a s soc i a t e d w i th non- m ov ing ve h ic l e s o


7/15

N O W A K

75

Ye a r s

50 Ye a r s

5 Ye a r s

Year

6

M o n t h s

2

M o n t h s

M o n t h

2

We e ks

D a y

Average Truck

Span m)

FIG Bias factors for various time periods: m oment for a single truck.

75

Ye a r s

50 Ye a r s

5 Ye a r s

Year

6 M o n t h s

2 M o n t h s

1 M o n t h

2 We e ks

D a y

Average Truck

Span

m)

FIG 10

Bias facto rs for various time periods: shear for a single truck.

trucks moving at reduced speeds. This is important in con-

s idera t ion of dynamic loads . In fur ther ca lcula t ions , i t i s

assumed, conservatively, that the headway distance is 5 m

even for normal speeds.

Two lane moments and shears

Th e a na lys i s i nvo lve s t he de t e r m ina t ion o f t he l oa d in

each lane and the load distribution to girders. The effect of

mul t ip le t rucks i s ca lcula ted by sup erposi t ion . The m axi-

mum moments are calculated as the largest of the following

cases:

a) One lane fully loaded and the other lane unloaded;

b) Both lanes loaded. Three degrees of correlation between

the lane loads are consid ered: no correlatio n p 0), partial

correlation p

0.5) , and full correlation p 1) .

I t has been observed that, on average, about every 10th

t ruck is on the br idge s imul taneously wi th another t ruck

side-by-side) . For each such a simultaneous occurrence, i t

is assumed that every 10th time the trucks are partially cor-

related and every 50th t ime they are fully correlated with

regard to w eight) . I t is a lso conservatively assumed that the

transverse distance between two side-by-side trucks is 1.2 m

whee l center- to-center) .

In case a) only one lane loaded ), the parameters mean

and coeff icient of variation) of the m aximum effects are as

given in Table 3. In case b) two lanes loaded), the param-

e te r s of m om e n t s a nd she a r s in e a c h l a ne de pe nd o n the

degree of co rrelation:

i)

No correlation p 0) . The maximum 50-year moment

is caused by a s imul taneou s occur ren ce of the maximum

5-year moment 2 4.75) in lane 1 and the averag e momen t

in lane 2 .

ii)

Partial correlation p 0.5) . Th e maxim um 50-yea r

moment is caused by a simultaneous occurrence of the max-


8/15

42

C A N .

J.

CIV.

ENG. VOL.

21

1994

Average

Truck

D a y

2

D a y s

2

We e ks

M o n t h

2 M o n t h s

6 M o n t h s

Year

5

Ye a r s

5

75Years

Span m)

FIG . 11. Coefficient of variation of the maximum moment for a single truck.

Average

Truck

D a y

2

D a y s

2 We e ks

M o n t h

2

M o n t h s

6 M o n t h s

Year

5 Ye a r s

5

7 5 Years

span

m)

FIG . 12. Coefficient of variation of the maximum s hear for a single truck.

imum 6-month moment z = 4.26) in lane and the max i-

mum da i ly moment

z=

3.09) in lane 2.

i i i ) F u l l c o r r e l a t io n

p =

0 ) . T h e m a x i m u m 5 0 - y e a r

moment is caused by a simultaneous occurrence of the max-

imum 1-month moment z = 3.89) in both lanes.

The structural analysis was performed using the finite ele-

ment method. T he mode l i s based on a l inear behavior of

girders and slabs. The maximum girder moments and shears

were calculated by superposition of truck loads in both lanes.

The results indicate that for inter ior girders, the case with

two fully correlated side-by-side trucks governs, with each

truck equa l to the maximum 1-month t ruck. However , for

some cases of exter ior girders, one truck may govern.

Th e bias fac tors a re ca lcula ted as the ra tios of the mean

maximum 50-year moments shears) and nomina l moments

shears) spec i f ied by OHBDC

(

1983). Th e ca lcula t ions a re

per formed for a s ingle lane and tw o lanes . The resul ts a

plotted vs. span in Figs. 15 and 16. For two lanes, the m ult

lane reduction factor 0.9) is included.

Recommended changes in design live load

On the basis of the performed load analysis, i t is recom

mended to increase the design load for spans less than 40 m

Therefore , the tandem axle load has been increased f ro

the cur rent OH BDC 1983) 140 kN to 160

kN

see Fig. 4

Th e bias factors, calculated using the new live load 160 k

pe r a x l e i n a t a nde m ) , a r e shown in F igs . 17 a nd 18 f o

mom ents and sh ears , r espec t ively .

ynamic load

Dyna m ic loa d e f f e c t , I i s c ons ide r e d a s a n e qu iva l e

s ta t ic load e f fec t added to the l ive load,

L.

Th e ob je c t iv


9/15

N O W A K

1.3

1.2

M

9 1 1

7 5 Ye a r s

1 0

50

Ye a r s

5

Ye a r s

1

Ye a r

0 . 9

6

M o n t h s

2 M o n t h s

6

0.8

1 M o n t h

0 2 We e ks

1 0 . 7

1 D a y

@ g

.2

0 6

0 1

2 0

30 4

5

6

Span

m)

FIG

13. Bias factors for various time periods: moment for one-lane bridges.

7 5 Y e ar s

50

Ye a r s

5

Ye a r s

1

Year

6

M o n t h s

2 M o n t h s

1

M o n t h

2 We e ks

1

D a y

Span m)

FIG

14. Bias factors for various time periods: shears for one-lane bridges

of th is ana lys is i s to de te rmine the paramete rs mean and

coeff icient of variation) of

I.

The dyna m ic b r idge t e s t s we r e c a r r i e d ou t by B i l l i ng

1984). Th e results are available for

22

bridges and 30 spans,

including prestressed concrete girders and slabs, steel girders

hot-rolled sections, plate girders, box girders), steel trusses,

a nd r ig id f r a m e s . The m e a su r e m e n t s we r e t a ke n f o r t e s t

vehicles and a normal traffic. The means and standard devi-

a t i ons , a s a f r a c t ion o f t he s t a t i c l i ve l oa d , a r e g ive n in

Table 4. Considerable differences between the distr ibution

functions for very similar structures point to the im portance

of other factors e .g. , surface condition) . Resu lts collected

f rom the weigh- in-mot ion s tudies Ghosn and Moses 1984)

indicate an average dynamic load factor of 0.11. This value

falls in the middle range of the data obtained from the MT O

tests Table 4) . However, interpretation of these results is

diff icult because the dynamic loads are separated from the

static live loads. It has been observed that the dynamic load,

as a f rac t ion of l ive load, decreases for heavie r t rucks . I t

i s expec ted tha t the la rges t dynamic load f rac t ions in the

survey cor respond to l ight -weight t rucks .

To ver i fy these observa t ions , a compu te r procedure was

deve loped for s imula t ion of the dynamic br idge behavior

H w a n gand Nowak 1991) . The d ynamic load is a func t ion

of three major parameters: road surface roughness, br idge

dynamics f requency of v ibra t ion) , and vehic le dynamics

suspe ns ion sys t e m ) . The de ve lope d m ode l i nc lude s t he

effect of these three parameters. Simulation of the dynamic

load requires the generation of a road profile, which is done

by using a Fourier transform of the power spectral density

function. The bridge is modeled as a prismatic beam. Modal

equa t ions of mot ion a re formula ted . In the ana lys is , each

truck is composed of a body, a suspension system, and tires.

The body is subjected to a r igid-body motion including the

vertical displacement and pitching rotation. Suspen sions are

assumed to be of mul t i -lea f type spr ings .

The dynamic load allowance

DLA)

is defined as the max-

im um dyna m ic de f l e c t ion ,

D

iv ided by the maximum
https://www.researchgate.net/publication/275188222_Simulation_of_Dynamic_Load_for_Bridges?el=1_x_8&enrichId=rgreq-1001a0e2-bbed-443b-abea-b7113856c903&enrichSource=Y292ZXJQYWdlOzIzNzE5MDc4NDtBUzoyNzExMDk0MDA1NTk2MTZAMTQ0MTY0ODkyNjA1OQ==https://www.researchgate.net/publication/275188222_Simulation_of_Dynamic_Load_for_Bridges?el=1_x_8&enrichId=rgreq-1001a0e2-bbed-443b-abea-b7113856c903&enrichSource=Y292ZXJQYWdlOzIzNzE5MDc4NDtBUzoyNzExMDk0MDA1NTk2MTZAMTQ0MTY0ODkyNjA1OQ==


10/15

CAN. 1 C I V . E N G . VOL. 21

1994

Span m)

FIG.

15.

Bias factors for various time periods: moments for one-lane bridges in terms of OHBDC-1983 model.

Span m)

FIG . 16. Bias factors for various time periods: shears for one-lane bridges in terms of OH BDC -1983 model.

static deflection,

D

as shown in Fig. 19. Static and dyn amic

deflections are calculated for typical girder br idges. I t has

been observed that the absolute value of the dynamic deflec-

t ion is a lmost a constant . There fore , a s the gross vehic le

weight is increased, the dynam ic load allowan ce is decreased.

Th e decrease of DLA is mainly due to the increase of static

deflection.

In most cases, the maximum live load is governed by two

trucks side-by-side. The corresponding DLAs are calculated for

two trucks by superposition of one truck effects as shown in

Fig. 20. The obtained average DLAs for one truck and two

trucks are presented in Fig. 21. Therefore, the resulting m ean

dynamic load is 0 .10 of the mean l ive load for two t rucks

and 0.15 for one truck. The coefficient of variation is 0.80.

In OHBD C 1983), the design values of DLA are specifie

as a function of the natural frequency of vibration, as show

in F ig . 22 . The r e su l t s o f s im u la t ions i nd ic a t e t ha t DL

values can be reduced and they a re lower for two t ruck

than for one truck. In general, dynamic load is reduced f

a la rger number of axles . Fur thermore , DLA is appl ied

the maximum 50-year l ive load. The ac tua l DLA is c los

to the mean. There fore , i t i s r ecommended to use a DL

equa l to 0 .25 for spans la rger than 6 m.

Load

combinations

The to t a l l oa d ,

Q

i s a combina t ion of severa l compo

ne n ts . Th e f o l lowing c om bina t ions a r e c ons ide r e d in t h

paper:


11/15

N O W A K

wo l nes

IG. 17. Bias factors for moments for one-lane and two-lane bridges in terms of OHBDC-1991 model.

FIG. 18. Bias factors for shears for one-lane and two-lane bridges in terms of OHBDC-1991 model.

(1) D

L I ;

( 2 ) D

L

I

W ;

(3) D

L

I EQ ;

where

W

is the wind load and Q is the earthquake load.

The maximum 50-year combina t ion of l ive load,

L,

a nd

dynamic load, I , is modeled using the statistical parameters

derived for

L

and I. It is assumed that live load is a product

of two paramete rs ,

L

and

P ,

where L is the static l ive load

and

P

is the live load analysis factor (influence factor). The

mean va lue of

P

is 1.0 and the coeff icient of variation is

0.12. The coeff icient of variation of

LP

can be ca lcula ted

using the fo l lowing formula :

[2]

v,, (v ; vp2) I2

where V, is the coef fic ient of va r iat ion of

L

an d V i s th e

coeff icient of variation of

P .

The m e a n m a x im um 50- ye a r

LP

I ,

nz

can be cal-

culated by multiplying the mean

L

by the mean value of

P

( e q u a l t o 1 . 0 ) a n d b y ( 1 m ,) , w h e r e m , i s t h e m e a n

d y n a m i c l o a d . T h e s t a n d a r d d e v i a t i o n o f t h e m a x i m u m

50-year

LP I ,

u,,+,, is

where a

V,,m,,;

in

i s t he m e a n

LP

a nd i s e qua l t o

mean

L ,

because mean

P

1; and

a V,m,

is the standard

deviation of the dynamic load. The coeff icient of variation

of

LP

I , V,,,,, i s


12/15

C A N .

J . CIV. ENG.

VOL. 21, 1994

TAB LE

.

Dynamic load factors from test results

Mean Standard deviation

Type of structu re Range Average Range Average

Prestressed concrete AASHTO girders 0.05-0.10 0 .09 0.03-0.07 0.05

Prestressed concrete box and slabs

0.10-0.15

0.14 0.08-0.40 0.30

Steel girders

0.08-0.20 0.14 0.05-0.20 0.10

Rigid frame, truss

0.10-0.25 0.17 0.12-0.30 0.26

ime

s)

FIG. 19. Time history for midspan deflection due to a single

truck on a bridge.

The statist ical parameters of L and

I

depend on the span

length, and they are different for a single lane and two lanes.

For a single lane, VLp , 0.19 for most spans , and 0 .205

for very short spans. For two-lane bridges,

yp ,

0.18 for

most spans, and 0.19 for very short spans.

T h e b a s i c l o a d c o m b i n a t i o n f o r h i g h w a y b r i d g e s i s a

simultaneous occurrence of dead load, live load, and dynamic

l o a d . T h e u n c e rt a i n t y i n v o l v e d i n t h e l o a d a n a l y s i s i s

expressed by the load analysis factor, E. The mean E is 1.0

and the coefficient of variation is 0.04 for simple spans and

0.06 for cont inuous spans .

Th e load, Q is given in the following form:

The mean Q in is equal to the sum of the means of the

componen ts D, , D,, D,, L, and I) . Th e coeff icient of vari-

ation of

Q V,,

is

where

and

st Truck Dynamic

2nd

Truck Dynamic

ime

s )

FIG.

20.

Time history for midspan deflection due to two truck

on a bridge.

the m ode l de pe nds on the c ons ide r e d t im e in t e r val . Th i

particularly applies to environmen tal loads, including wind

e a r thqua ke , snow, i c e , t e m pe r a tu r e , wa te r p r e s su r e , e t c

These load models can be based on the report by Ellingwood

e t al . 1980) or Nowak and Cur t i s 1980) . Th e bas ic da ta

h a v e b e e n g a t h e r e d f o r b u i l d i n g s t r u c t u r e s , r a t h e r t h an

br idge s . Howe ve r , i n m os t c a se s t he s a m e m ode l c a n be

u s e d . S o m e s p e c i a l b r i d g e - r e l a te d p r o b l e m s m a y o c c u

because of the unique des ign condi t ions , such as founda

t ion condi t ions , ext remely long span s , or wind exposure .

Load e f fec t i s a resul tant of severa l components . I t i

unl ike ly tha t a l l components take the i r maximum va lue

simulta~eouslv. here is a need for a form ula to calculate th

parameters of Q mean and coefficient of variation). In gen

eral, a l l load components are t ime-variant, except of dead

load. There a re sophis tica ted load combina t ion technique

available to calculate the distr ibution of the total load, Q

However, they involve a co nsiderable numerical effort. Som

of these methods are summarized by Madsen et a l . 1986)

The total load effect in highway bridge members is a joint

effect of dead load, D; live load, L

I

static and dynamic)

environmenta l loads ,

E

wind, snow, ice, ear thquake, ear th

p r e s su r e , a nd wa te r p r es su r e ) ; a nd o the r l oa ds , A e m e r

gency braking, coll ision forces) .

[9] Q = D + L + I + E - k A

The effect of a sum of loads is not always equal to th

The to ta l load e f fec t , Q is the result of dead load, l ive

sum of the effect s of single loads . In particular, this may

load, dynamic load, and other effects environmental, other).

apply to the nonlinear behavior of the structure. Nevertheless

The r e a r e s e ve r al l oa d c om bina t ions f o r c ons ide r a t ion in

it is further assumed that [9] represents the joint effect. Th

the reliabil i ty analysis of br idges. For t ime-varying loads,

d is t r ibut ion of the jo in t e f fec t can be ana lyzed us ing th


13/15

N O W A K

FIG . 21. Average dynamic load allowance in terms of span for one truck and two truck s.

so-ca l led Turks t ra s ru le . Turks t ra (1970) observed tha t a

combination of several load components reaches its extreme

0.40

when one of the components takes an ext reme va lue and

all other components are at their average (arbitrary-point-

in- t ime) leve l . For example , the combina t ion of l ive load

3

0.30

and earthquake produces a maximum effect for the l ifetime

T

when either (i) the earthquake takes its maximum expected

3 0.20

value for T and the l ive load takes i t s maximum expec ted

value corresponding to the duration of earthquake (i.e. , about

'

3 0 s), or (ii) the live load takes its maximu m expected valu e

0.10

for T and the earthquake takes i ts maximum expected value

cor responding to the dura t ion of th is maximum l ive load

(time of truck passage on the bridge) .

~~~~~~~~~. . . . .r

In prac t ice , the expec ted va lue of an ea r thquake in any

0 1 2 3 4 6 7

sho r t tim e in t e r val i s a lm os t z e r o . The e x ~ e c t e d a lue o f

irst lexural requency Hz)

truck load for a short t ime interval depends on the class of

the road. For a very busy highway, i t is l ikely that there is

som e t ra f f ic a t any point in t ime . There fore , the maximum

earthquake may o ccur simultaneously with an average truck

passing through the bridge.

I n a ge ne r a l c a se , Tu r ks t r a s r u l e c a n be e x p r e s se d a s

follows:

where

In a l l cases , the average load va lue i s ca lcula ted for the

period of time corresponding to the duration of the maximum

load. The formula can be extended to include various com-

ponents of

D

E a nd

A

Th e jo in t d is t r ibut ion can be mode led us ing the cent ra l

l imi t theorem of the theory of probabi l i ty (Benjamin and

Cornell 1970).

A

sum of several random variables is a nor-

mal random variable if the number of components is large,

and if the average values of the components are of the sam e

FIG . 22. Dynamic load allowance specified in

OHBDC-1983.

order . I f one variable dominates ( i ts average value is much

la r ge r t han a ny o the r ) , t he n the jo in t d i s t r i bu t ion c a n be

c lose to tha t of the domina t ing var iable .

F o r e a c h sum Q i i n [ l o ] , t he m e a n a nd va r i a nc e o f t he

sum a r e e qua l t o t he sum o f m e a ns a nd the sum o f va r i -

ances of components, respectively. The distr ibution of Q is

that which minimizes the overall structural reliability. Usually

it is Qi with the larg est mean value. T he identif ication of

the governing load combina t ion is impor tant in the se lec -

tion of the optimum load factors (including load combination

factors) .

F o r e a c h l o a d c o m p o n e n t , t h e m a x i m u m a n d a v e r a g e

va lues a re es t imated. Dead load does not va ry wi th t ime .

There fore , the maximum and average va lues a re the same.

For live load (including dynamic load), the maximum values

are calculated for 5 0 years and shorter periods. The statistical

paramete rs of wind and ea r thquake a re g iven in Table 5 .

Th e p r obab i l i ty o f a n e a r thqua ke EQ, o r he a vy wind

W,

occurring in a short period of time is very small. Therefore,

s imul taneous occur rence of E Q and

W

is not considered. In

the result, the number of load combinations considered in the

code can be reduced as fo l lows:


14/15

4 8

C A N . J .

CIV. ENG V O L . 21. 1994

TA BL E . Statistical param eters of wind and earthquake

Live load corresponding

Maximum 50- Coefficient Basic to basic time period

Load year value of variation, time

component

bias factor COV

period Bias factor

COV

Wind 0.875

0.20

4 h 0 .80-0 .90 0 .25

Earthquake 0.30 0.70 30 s

0-0.50 0.50

whe re (L I),,,, is the ma xim um 50-ye ar L I; (L

I ,,

i s t h e m a x i m u m 4 - h o u r L I ; W ,,,,, i s t h e m a x i m u m

50-y ear wind ; W,,,,, is the maxim um daily wi nd; and EQ,,,

i s t he m ax i m u m 5 0 - y ea r ea r t h q u ak e . T h e m ean m ax i m u m

4-ho ur l ive load mo me nt, (L I ), ., , can be read direc t ly

f r o m F i g . 7, f o r

z

=

2 .5 ( m a x i m u m o f 2 0 0 t r u ck s ) . T h e

parameters of

(L

I),., are also show n in Table 5.

Live load for evaluation of existing bridges

Exis t ing b r idges are evaluated to de termine the i r ac tual

s t r eng th a nd p red ic t the r emain ing l i f e . Th e major d i f f er -

ence between the load model for the design of new br idges

an d t h e ev a l u a t i o n o f ex i s t i n g s t r uc t u r e s i s t h e r e f e r en ce

time per iod. New br idges are designed for 50-year l i fet ime

and ex is t ing b r idges are checked fo r 5 - to 10-year per iods .

Load m odel depends on the reference t ime per iod. Maximum

mom ents and shear s ar e smal ler fo r 5 - to 10-year per iods

than for 50-year l i fet ime. However , the coeff icient of var i-

at ion is larger for shor ter per iods.

The load combination including dead load, l ive load, and

d y n am i c l o ad is co n s i d e red . T h e m ax i m u m 5 - o r 1 0 - y ea r

live loads, and the corresponding dynamic loads, are derived

using the tables and f igures included ear l ier in this paper .

Dead load model i s no t t ime-dependen t and the s ta t i s t i -

cal parameters are as given in Table 1. From Figs. 7 and 8 ,

the maxim um 5-year mom ent (o r shear ) i s abou t 5 less

than the maximum 50-year moment (or shear). The difference

b e t w een t h e 1 0 - y ea r m o m en t an d t h e 5 0 - y ea r m o m en t i s

a b o u t 3 . F o r a p o s t e d s t r u c t u r e , w i t h a r e d u c e d t r u c k

weigh t l imi t , the maximum l ive load values are lower than

for b r idges tha t a r e no t pos ted . However , the cor respond-

ing dynamic load allowance, DLA, is increased (as a fraction

o f l i v e l o a d ) . T h e r e f o r e , t h e D L A s s p e c i fi e d f o r p o s t e d

br idges are a l so increased by 0 .1 to 0 .6 , depend ing on the

value of the evaluation level and the number of axles .

Conclusions

Th e ob jec t ive o f the paper i s to p resen t the development

of load models for the br idge design code. The major br idge

load componen ts inc lude dead load , live load , and dynam ic

load.

Th e s ta t i st i ca l parameter s o f dead load are p resen ted fo r

f ac t o ry - m ad e co m p o n en t s , c a s t -i n - p lace c o m p o n en t s , an d

asphalt wear ing surface.

L ive load i s based on the Ontar io truck survey data . The

avai lab le s ta t i s t i ca l da ta base i s summar ized . The ex t r eme

effects (moment and shear) are determined for various periods

of t ime by extrapolat ion of the truck survey data. Mult iple

presence o f more than one t ruck i s cons idered by s imul

t ion. For one- lane br idges, a s ingle truck governs for spa

up to 30-40 m. For two- lane s t ruc tu res , two s ide-by- s i

trucks produce the largest moment and shear. The analysis

the des ign l ive load speci f ied by OH BD C (1983) ind ica t

the need for an increase for shorter spans. Therefore, it is re

ommended to increase the des ign t ruck , by increas ing th

axle loads in a tandem from the current 140 to 160 kN. Th

modi f ied des ign t ruck p rov ides a more un i fo rm mean- t

nominal r a t io fo r l ive load .

T h e d e r i v a t i o n o f d y n a m i c l o a d i s s u m m a r i z e d . T h

dynamic load a l lowance, expressed in te rms o f def lec t io

p r ac t i ca l l y d o es n o t d ep en d o n t r u ck w e i g h t . T h e r e f o r

dynam ic load as a f ract ion of l ive load decreases for heavi

t r u ck s . I t i s f u r t h e r r ed u ced f o r t w o t r u ck s s i d e - b y - s i d

Therefore , the r ecommend ed des ign value o f dynamic lo

is 0 .25, for al l spans larger than 6 m.

Th e load combination procedure is formulated for desi

formula including dead load, l ive load, dynamic load, win

and ear thquake.

Th e developed load model c an be used fo r the des ign

new br idges and the evaluation of exist ing s tructures.

cknowledgments

T h e p r e s en t ed r e s ea rch w as ca r r i ed o u t in co n j u n c t i o

w i t h t h e d ev e l o p m en t o f t h e t h i r d ed i t i o n o f t h e O n t a r

H i g h w a y B r i d g e D e s i g n C o d e . T h e a u t h o r a c k n ow l e d g

m an y f r u it f u l d i s cu ss i o n s , s u g g es t i o n s , an d c o m m en t s b

the MTO staff , in par t icular , Hid N. Grouni , Roger Dorto

B a i d a r B a k h t , A k h i l e s h A g a r w a l , J o h n B i l l i ng , a n

T . Tharmabala , as wel l as MTO consu l tan ts , Roger Gre

( U n i v e r s i t y o f W a t e r l o o ) , F r e d M o s e s ( U n i v e r s i t y

P i t ts b u r g h ), R o y S k e l t o n ( M c C o r m i c k , R a n k i n a n

A s s o c i a t e s ) , a n d D a v i d H a r m a n ( U n i v e r s i t y of W e s t e

Ontar io). Thanks are also due to former and current resear

assistants at the University of Michigan: Young-Kyun Hon

Hani Nass i f , Eu i -Seung Hwang , and Tadeusz Alber sk i .

AASHTO. 1989. Standard specifications for highway bridge

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