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Structural Safety13 (1993) 53-66 53Elsevier

ive loa d m od el for highw ay br idgesAndrzej S Nowak

Department of Civil an d Environmental Engineering Universi ty of Michigan An n Arbor M148109 U

Abstract Load models are developed for highway bridges. The models are based on the available statistical data ondead load, truck loads and dynamic loads. The paper deals mostly with the static live load. The model is derivedfrom truck surveys, weigh-in-motion measurements and other observations. Simple span moments, shears andnegative moments are calculated for various spans. Extreme 75 year loads are determined by extrapolation. Theimportant parameters also include girder distribution factors and multiple presence (more than one truck on the

bridge). Multiple presence is considered in lane and side-by-side with various degrees of correlation between truckweights. The maximum load is calculated by simulations. The developed live load model served as a basis for thedevelopment of new design provisions in the United States (LRFD AASHTO) and Canada (Ontario Highway BridgeDesign Code).

Key words : highway bridges; live load; multiple presence; truck load.

1 I n t r o d u c t i o n

The major load co mpon ents of highway bridges are de ad load, live load (static and dynamic),environmental loads ( temperature, wind, earthquake) and other loads (coll ision, emergencybraking). Load components are random variables. Their variation is described by the cumula-t ive distr ibution function (CDF), and/or parameters such as the mean value, bias factor(mean -to-n ominal ratio) and coefficient of variation. The relationship am ong various loadpara mete rs is described in terms of the coefficients of correlation.

The basic load comb inatio n for highway bridges is a simultan eous occurre nce of dead load,live load and dyn amic load. Therefor e, these thr ee load com ponen ts are consider ed in thepresent study. It is assumed that the economic life time for newly designed bridges is 75 years.The extreme values of load are extrapolated from the available data base. Nominal (design)values of load components are calculated according to AASHTO [1].

Dea d load is the gravity load due to the self weight of the structural an d non structuraleleme nts per man ent ly conn ecte d to the bridge. Because of differen t degrees of variation, it isconvenient to consider three components of dead load: weight of factory made elements (steel ,precast concrete members), weight of cast-in-place concrete members, and weight of the

* Discussion is open until June 1994 (please submit your discussion paper to the Editor, Ross B. Corotis).

0167-4730/93/ 06.00 1993 - Elsevier Science Publishers B.V. All rights reserved

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54

TABLE 1

Statistical parame ters of dead load

Com ponent Bias Coefficient of variationFactory-mad e mem bers 1.03 0.08Cast-in-place m em bers 1.05 0.10

Asphalt 90 mm * 0.25 mean thickness

w e a r i n g s u r f a c e (a s p h a lt ) . A l l c o m p o n e n t s o f d e a d l o a d a r e t r e a t e d a s n o r m a l r a n d o m v a r i aT h e s ta ti st ic a l p a r a m e t e r s w e r e d e r i v e d in c o n j u n c ti o n w i t h t h e d e v e l o p m e n t o f th e O H B[ 2 - 4 ] a n d t h e y a r e l i s t e d in Ta b l e 1 . T h e b i a s f a c t o r s a r e t a k e n a s u s e d i n t h e p r e v i o u s b rc o d e c a l i b r a t i o n w o r k , h o w e v e r , t h e c o e f f i c ie n t s o f v a r i a t i o n a r e i n c r e a s e d t o i n c l u d e h ue r r o r a s r e c o m m e n d e d b y E l l in g w o o d , G a l a m b o s , M a c G r e g o r a n d C o r n e ll [5].

L i v e lo a d c o v e r s a r a n g e o f f o rc e s p r o d u c e d b y v e h i c le s m o v i n g o n t h e b r i d g e . T h e e f f el iv e lo 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 i n c l u d i n g t h e s p a n l e n g t h , t r u c k w e i g h t , ax le l o a d s,

c o n f i g u r a t i o n , p o s i t i o n o f t h e v e h ic l e o n t h e b r i d g e ( t r a n s v e rs e a n d l o n g i t u d in a l ) , n u m b ev e h i c le s o n t h e b r i d g e ( m u l t i p l e p r e s e n c e ) , g i r d e r s p a c in g , a n d s t if f ne s s o f s t r u c t u r a l m e m( sl ab a n d g i rd e r s) . T h e e f f e ct o f t h e s e p a r a m e t e r s is c o n s i d e r e d s e p a r a t e l y.

T h e d y n a m i c l o a d m o d e l w a s d e v e l o p e d b y H w a n g a n d N o w a k [ 6] . I t is a f u n c t i o n o f tm a j o r p a r a m e t e r s : r o a d s u r f a c e r o u g h n e s s , b r i d g e d y n a m i c s ( f r e q u e n c y o f v i b r a t io n ) a n d v ed y n a m i c s ( s u s p e n s i o n s y s te m ) . I t w a s o b s e r v e d t h a t d y n a m i c d e f l e c t i o n is a l m o s t c o n s t a n t ad o e s n o t d e p e n d o n t r u c k w e i g h t . T h e r e f o r e , t h e d y n a m i c l o a d , a s a f r a c t io n o f li ve ld e c r e a s e s f o r h e a v i e r t r u ck s . F o r t h e m a x i m u m 7 5 y e a r v a lu e s , t h e c o r r e s p o n d i n g d y n a m i c d o e s n o t e x c e e d 0 . 1 5 o f l i v e l o a d f o r a s i n g l e t r u c k a n d 0 . 1 0 o f l i v e l o a d f o r t w o t r us i d e - b y -s i d e . T h e c o e f f i c i e n t o f v a r i a t i o n o f d y n a m i c s lo a d i s a b o u t 0 . 80 . T h e r e s u l t s o fs i m u l a t i o n s i n d i c a t e t h a t D L F ( d y n a m i c l o a d f a c to r ) v a l u e s a re a l m o s t e q u a ll y d e p e n d e nr o a d s u r f a c e r o u g h n e s s , b r i d g e d y n a m i c s a n d v e h i c l e d yn a m i c s . T h e a c t u a l c o n t r i b u t i o n o f tt h r e e p a r a m e t e r s v a r i e s f r o m s i t e to s i t e a n d i t i s v e r y d i ff i c u l t t o p r e d i c t . T h e r e f o r e , r e c o m m e n d e d t o s p e ci fy D L F a s a c o n s t a n t p e r c e n t a g e o f l iv e l o ad .

T h e d e v e l o p m e n t o f li ve lo a d m o d e l is e s s e n t ia l f o r a r a t io n a l b r i d g e d e s i g n a n de v a l u a t i o n c o d e . G h o s n a n d M o s e s [ 7] p r o p o s e d s t a ti s ti c al p a r a m e t e r s f o r t r u c k l o ad , i n c l uw e i g h t , ax le c o n f i g u r a t i o n , d y n a m i c l o a d a n d f u t u r e g r o w t h . T h e i r v a l u e s w e r e b a s e d o nw e i g h - i n - m o t i o n d a t a [ 8] . T h e l iv e l o a d m o d e l f o r t h e f ir s t t w o e d i t i o n s o f O H B D C [2]d e v e l o p e d b y N o w a k a n d L i n d [ 3]. T h e m o d e l w a s r e vi s ed b y N o w a k a n d H o n g [ 9]. T h e p r es t u d y is a c o n t i n u a t i o n a n d e x t e n s i o n o f t h e p r e v i o u s w o r k .

2 D a t a b a s e

T h e a v ai la b le w e i g h - i n - m o t i o n d a t a w a s a n a l y ze d b y L i u a n d I m b s e n a s p a rt o f N C HP r o j e c t 1 2 -2 8( 11 ) ( u n p u b l i s h e d ) . H o w e v e r , i t w a s f o u n d t h a t a l a rg e p o r t i o n o f t h e dc o l l e c t e d i n m i d 1 9 8 0 's w a s n o t r e l ia b l e , w i t h e r r o r s e s t i m a t e d a t 3 0 - 4 0 . T h e r e f o r e , i ns t u d y, t h e d a t a b a s e c o n s is t s o f t h e r e s u l ts o f t r u c k s u r ve y p e r f o r m e d b y t h e O n t a r i o M i n i s tTr a n s p o r t a t i o n [1 0]. T h e s t u d y c o v e r e d 9 ,2 5 0 s e l e c te d t r u c k s ( o n ly tr u c k s w h i c h a p p e a r e d th e a v il y l o a d e d w e r e m e a s u r e d a n d i n c l u d e d i n t h e d a t a b a s e ) .

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(a) Stand ardHS20Truck

~35 kN ~ 142 kN4.25 m 4.25 m

(b) H,S, 20 = -e; ~rltqg

I 80 kN (formoment

116 kN (for shear)

~ 142 kN

9.35 kN/m

55

F i g . 1 . H S 2 0 L o a d i n g A A S H TO [ 1] .

T h e u n c e r t a i n t i e s i n v o l v e d i n t h e a n a l y si s a re d u e t o l im i t a t i o n s a n d b i a s e s i n t h e s u r v ey.a v a il a b le d a t a b a s e i s s m a l l c o m p a r e d t o t h e a c t u a l n u m b e r o f h e a v y v e h i c l e s i n a 75 y e at i m e . I t is al s o r e a s o n a b l e t o e x p e c t t h a t s o m e e x t r e m e l y h e a v y t r u c k s p u r p o s e f u l l y a v o i d ew e i g h i n g s t a ti o n s . A c o n s i d e r a b l e d e g r e e o f u n c e r t a i n t y is c a u s e d b y u n p r e d i c t a b i l i t y o ff u t u r e t r e n d s w i t h r e g a r d t o c o n f i g u r a t i o n o f a x le s a n d w e i g h t s .

T h e d a t a b a s e i n c l u d e s t r u c k c o n f i g u r a t i o n ( n u m b e r o f a x le s a n d a x le s p a ci n g ) a n d w e( ax le l o a d s a n d g r o s s v e h i c l e w e i g h t ) . F o r e a c h t r u c k i n t h e s u r v ey, b e n d i n g m o m e n t s a n d sf o r c e s a re c a l c u l a t e d f o r a w i d e r a n g e o f s p an s . S i m p l e s p a n s a n d c o n t i n u o u s tw o e q u a l sa r e c o n s i d e r ed . T h e m o m e n t s a n d s h e a rs a r e c a l c u l a te d i n t e r m s o f t h e s t a n d a r d H S 2 0 t r ul a n e l o a d i n g , w h i c h e v e r g o v e r n s [ 1] , a s s h o w n i n F ig . 1 . T h e C D F s a r e p l o t t e d o n n op r o b a b i l i t y p a p e r i n F i g . 2 , 3 a n d 4 , f o r s i m p l e s p a n m o m e n t s , s h e a r s a n d n e g a t i v e m o m( c o n t i n u o u s s p an s ) , re s p e c ti v e l y, f o r sp a n s f r o m 9 to 6 0 m . F o r a r a n d o m v a r i a b le X ( m o mo r s h e a r ) , a n d C D F,Fx x) ,t he ve r t i ca l sca le , z , i n F ig . 2 to 4 , i s

Z = d P - I[ F x ) ] (1)

w h e r e ~ - 1 = i n v e r s e o f t h e s t a n d a r d n o r m a l d i s t r i b u t i o n f u n c t i o n .

3 M a x i m u m t ru c k m o m e n t s an d s h e a r s

T h e m a x i m u m m o m e n t s a n d s h e a rs f o r v a r io u s t i m e p e r i o d s a re d e t e r m i n e d b y ex t r apt io n . T h e e x t r a p o l a t e d d i s t r i b u t i o n s a re a l s o s h o w n i n F i g . 2 t o 4. L e t N b e t h e t o t a l n u m bt r u c k s in t i m e p e r i o d T. I t is a s s u m e d t h a t t h e s u r v e y e d t r u c k s r e p r e s e n t a b o u t t w o w e e k t rT h e r e f o r e , i n T - - 7 5 y e a r s, t h e n u m b e r o f tr u c k s , N , w i ll b e a b o u t 2 ,0 0 0 ti m e s l a rg e r t h at h e s u r v e y. T h i s w i l l r e s u l t i n N = 2 0 m i l l i o n t r u c k s . T h e p r o b a b i l i t y l e v e l c o r r e s p o n d i n g t o5 . 1 0 - 8 , w h i c h c o r r e s p o n d s t o z = 5 .3 3. T h e n u m b e r o f tr u c ks , N , p r o b a b il i ti e s, l / N ,

i n v e r s e n o r m a l d i s t r i b u t i o n v a l u e s , z , c o r r e s p o n d i n g to v a r io u s t i m e p e r i o d s T f r o m 1 d a y y e a r s , a r e s h o w n i n Ta b l e 2 .

T h e m e a n m a x i m u m m o m e n t s a n d s h e ar s c o r r e s p o n d i n g t o v ar io u s p e ri o d s o f t im e c ad e t e r m i n e d f r o m C D F s . T h e r e s ul t s fo r t i m e p e r i o d s f r o m 1 d a y to 7 5 y e a rs a r e p r e s e n tF i g . 5 , 6 a n d 7 , f o r s i m p l e s p a n m o m e n t s , s h e a r s a n d n e g a t i v e m o m e n t s , r e sp e c t iv e l y.c o m p a r i s o n , t h e m e a n s a r e a l s o g i v en fo r a n a v e r a g e tr u ck . T h e m o m e n t s a n d s h e a rsd i v i d e d b y t h e c o r r e s p o n d i n g H S 2 0 d e s i g n v a lu e s [ 1].

T h e c o e ff i ci e n ts o f v a r i a ti o n f o r t h e m a x i m u m t r u c k m o m e n t s a n d s h e a r s c a n b e c a l c ub y tr a n s f o r m a t i o n o f C D F. E a c h f u n c t i o n c a n b e r a i s e d t o a c e r t a in p o w e r , s o t h a t

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56

>o

3

2

I

0

I

;5 Years

Montlas

2 w ~ I6

18 _

-4 i i ..... i i i i

o 2T r u c k M m n e ta t / H S 2 0 M o m e n t

F i g . 2 . C D F o f m o m e n t s f o r s i m p l e s p a n s .

calculated earlier mean maximum moment or shear) becomes the mea n value after the

transformation. The slope of the tra nsform ed CDF determ ines the coefficient of variation. Theresults are plotted in Fig. 8 and 9 for moments and shears, respectively.

4 O n e l a n e m o m e n t s a n d s h e a r s

The maximum one lane m omen t or shear is caused either by a single truck or two or more)trucks following behind each other. For a multiple truck occurrence, the important parametersare the headway distance and degree of correlation between truck weights. The maximum one

lane eff ect mom ent or shear) is derived as the lar gest of the following two cases:a) One truck effect, equal to maximum 75 year momen t or shear) with the paramet ers given

in Fig. 5 to 7 for the mea n and in Fig.8 and 9 for the coeffici ent of variation;b) Two trucks, each with the weight smaller than tha t of a single truc k in a). Various headway

distances are considered, from 5 to 30 m. Headway distance is measu red from the r ear axleof one vehicle to the front axle of the following vehicle, therefore 5 m means bumper tobumper traffic. Three degrees of correlation between truck weight are considered: p = 0no correlation), p = 0.5 partial) and p = 1 full correla tion), wher e p is the coeff icient of

correlation.

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75 Ye a r s

oU yeaFs H ~ / /5 = Ye~ ' s ]

I Y I ~

2 ; ; ~ :z

off : l ay ii l

d . B l ~ m(m) 9 -

-2

-3 j f

-4 i io I 2

Truck Shear HS20 Sh, ar

Fig. 3. CD F o f shears.

57

O b s e r v a t i o n s i n d i c a t e t h a t , o n a v e r a g e , a b o u t e v e r y 5 0 t h t r u c k i s f o l l o w e d b y a n o t h e r t

w i t h t h e h e a d w a y d i s t a n c e l e ss t h a n 3 0 m [ 11 ]. I t a ls o a s s u m e d t h a t a b o u t e v e r y 1 5 0 th t r uf o l l o w e d b y a p a r t i a ll y c o r r e l a t e d t r u c k , a n d a b o u t e v e r y 5 0 0 t h t r u c k i s f o l l o w e d b y a c o r r e l a t e d t r u c k . T h e p a r a m e t e r s o f t h e t w o tr u c k s in l a n e , i n c l u d i n g N t h e c o n s i d e r e d t r ua m a x i m u m o f N tr u c ks ) , c o r r e s p o n d i n g z = - ~ - I 1 / N ) , a n d T t h e c o n s i d e r e d t r u c k ism a x i m u m f o r t i m e p e r i o d T ) a r e g i v e n i n Ta b l e 3 .

T h e m a x i m u m v a lu e s o f m o m e n t s a n d s h e a rs a r e c a l c u l a t ed b y s i m u l at i on s . T h e p a r a m ec o n s i d e r e d i n c l u de t r u c k c o n f i g u r a ti o n , w e i g h t , h e a d w a y d i s ta n c e a n d f r e q u e n c y o f o c c u r r eT h e m e a n 7 5 y e a r v a l u e s a r e s h o w n i n F ig . 1 0, 11 a n d 1 2 f o r s im p l e s p a n m o m e n t s , s h e a r sn e g a t i v e m o m e n t s , r e s p e c t i v e l y.

5 G i r d e r d i s t r i b u t i o n f a c t o r s

T h e a n a ly s i s o f t w o l a n e l o a d i n g i n v ol v e s t h e d i s t r i b u t i o n o f t r u c k l o a d t o g i r d e rs . s t r u c t u r a l a n a l ys i s i s c a r r i e d o u t u s i n g th e f i n i t e e l e m e n t m e t h o d . T h e m o d e l is b a s e d l i n e a r b e h a v i o r o f g ir d e r s a n d s l a b. T h e c a l c u l a t i o n s a r e p e r f o r m e d f o r s p a n s r a n g i n g f r o m6 0 m . F i v e c a s e s o f g i r d e r s p a c i n g a r e c o n s i d e r e d : 1 .2 , 1 .8 , 2 .4 , 3 . 0 a n d 3 . 6 m . F o r e a c h c as p a n a n d g i r d e r s p a c i n g , g i r d e r d i s t r i b u t i o n f a c t o r s a r e c a l c u l a t e d f o r v a r io u s t r u c k p o s i t i o n

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8

7 Ye a r s

l ~ l I

: ' i t Ig

~ - i

/ / / / iI I ] / /

40 1 2

T r u c k o m e n t s / H 20 o m e n t

Fig. 4. CD F of negative mome nts for cont inuous spans.

m o v i n g t h e t r u c k t r a n sv e r s e ly b y 0 . 3 m a t a t i m e . T h e r e s u l t i n g t r u c k i n f l u e n c e l in e s a r e u sf o r c a l c u l a t i o n o f t h e j o i n t e f f e c t o f t w o t r u c k s i n a d j a c e n t l a n e s , b y s u p e r p o s i t i o n .

T h e r e s u l t i n g g i r d e r d i s t r i b u t i o n f a c t o r s ( G D F ) a r e c o m p a r e d w i t h t h e A A S H T O ( 1 9 8v a l u e s a n d t h o s e r e c o m m e n d e d b y Z o k a i e e t a l . [ 1 2 ] .

TABLE 2

Numb er of trucks vs. time period and probability

Time period Num ber of trucks Probability Inverse normalT N 1/ N z

75 years 20 000 000 5. 10 - s 5.3350 year s 15 000 000 7.10 8 5.27

5 years 1 500 000 7. 10 - 7 4.831 year 300 000 3. 10 - 6 4.506 mont hs 150 000 7. 10 - 6 4.362 mont hs 50 000 2. 10 - s 4.111 month 30 000 3. 10 - s 3.992 weeks 10 000 1. 10 -4 3.711 day 1 000 1. 10 - 3 3.09

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2 50 yeara5 yearsl y ~ t r6 mon ths

2 ~o l

average

00 30 60

S p a n m )

Fig. 5 . Mean maximum moment s for s imple spans.

9

Q

o3

O

qJ

2.0

1 O

O.O0

I

75 year

lday

average

g

3 0S P a n ( r a )

Fig. 6. Mean maximum shears.

50 years5 yearsI year6 months2 months1 month2 weeks

6 0

O

0

75 year

l d ~

average

50 ye=m5 y e a mZyear6 mon ths2 mol t thsI m m l t b

2 w e e l m

3 0 6 0S l ~ a ( m )

Fig. 7 . Mean maximum negat ive mome nts for cont inuous spans.

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6 0

0.5

0 . 4 'o

> 0 3

0.2

Q) 0.1

0.00

75 & 50 years

3 0 6 0S p a n m }

F i g . 8 . C o e f f i c i e n t o f v a r i a t i o n o f th e m a x i m u m m o m e n t .

/ 2 aays2 ~ . e k n1 m o n t h

m o n t h s6 mon ths

\ l y e a r5 years

Co

t~

>

oo

0,5

0 .4

0 . 3 '

0 .2

0 . I

0.0

7 5 & 5 0 y e a ~

i i

0 3 0 6 0

S p a n m )

F i g . 9. C o e f f i c i e n t o f v a r i a t i o n o f t h e m a x i m u m s h e a r .

I m o n t h

m o n t s

m o n t h s

T A B L E 3

T r u c k p a r a m e t e r s f o r t w o t ru c k s i n o n e l a n e

O n e / t w o t ru c k s N

O n e 2 0 0 0 0 0 0 0Tw o : p = 0 Tr u c k 1 3 0 0 0 0 0

T r u c k 2 1p - - 0 . 5 Tr u c k 1 1 5 0 0 0 0

T r u c k 2 1 0 0 0p = 1 Tr u c k 1 3 0 0 0 0

T r u c k 2 3 0 0 0 0

Z

5 . 3 34 . 5 00 . 0 04 . 3 63 . 0 93 . 9 93 . 9 9

T

7 5 y e a r s1 y e a r

a v e r a g e6 m o n t h s1 d a y1 m o n t h1 m o n t h

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61

~ 2.0

1 0

0.0

2 t rucks

1 t ruck

0 3 0 6 0S p a n m )

Fig 10 Mea n maximum 75 year simple span moments

2

I t r u c k

0 3 0 6 0pan m }

F ig . 1 1. M c a n m a x i m u m 7 5 y c a r s hc a r s .

c

c

e

1

2 t r u c k s

~ 2 2 . . . . . . .. . . . . . . . .. . . . _

0 30 60S p a n m }

Fig 12 Mea n maximum 75 year negative moments

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62

e~

O

O

0 ,

0 1 .2

j

2 . 4 3 . 6

G i r d e r S p a c i n g m }

A A S H T ~

S p a n m} :

- - - -o - - - - 9a 8

- - - - -o - - - 27 3 6

6 0

F i g . 1 3. G i r d e r d i s t r i b u t io n f a c to r s : c a l c u l a t e d a n d s p e c i f i e d by A A S H T O [1 ].

F o r m o m e n t i n a n i n t e ri o r g i rd e r , A A S H T O [1] s p e c if ie s a G D F a s f ol lo w s .

G D F = s / D (2)

w h e r e s i s t h e g i r d e r s p a c i n g ( ft ; f t = 0 . 3 0 5 m ) a n d D i s a c o n s t a n t , e q u a l t o 5 . 5 f o r s t e e l g i ra n d p r e s t r e s s e d c o n c r e t e g i r d e rs , a n d D = 6 .0 f o r r e i n f o r c e d c o n c r e t e T - b e a m s . T h e d em o m e n t i n a g i r d e r is e q u a l t o th e p r o d u c t o f ( 0.5 s / D ) a n d H S 2 0 m o m e n t . F o r s h eA A S H T O [1 ] s p e c i f ie s G D F g i v e n b y e q n . ( 2) , e x c e p t o f t h e a x l e d i r e c t ly o v e r t h e s u p p o r t . a s s u m e d t h a t o v e r s u p p o r t t h e s l a b is s i m p l y s u p p o r t e d b y t h e g i rd e r s .

Z o k a i e e t a l. [1 2] p r o p o s e d G D F a s a f u n c t i o n o f g i r d e r s p a c in g , s ( ft ; f t = 0 .3 0 5 m ) , a n d sl e n g t h , L ( ft ; f t = 0 .3 0 5 m ) . F o r m o m e n t i n i n t e r i o r g i r d e r s (s t e e l, p r e s t r e s s e d c o n c r e t e r e i n f o r c e d c o n c r e t e T- b e a m s ) t h e f o r m u l a i s

G D F = 0 . 1 5 + ( s / 3 ) 6 ( s /L ) 2 (3)

a n d f o r s h e a r ,

G D F = 0 .4 + ( s / 6 ) - ( s / 2 5 ) 2 ( 4)

T h e r e s u l t s o f c a l c u l a ti o n s , p e r f o r m e d a s a p a r t o f th i s st u d y, a r e i n a g o o d a g r e e m e n t wv a l u e s o b t a i n e d u s i n g e q n . (3 ) a n d (4 ). H o w e v e r , i n m o s t c a s es , G D F s s p e c i fi e d by A A S H[1 ] a r e t o o c o n s e r v a t i v e . I n F ig . 1 3, c a l c u l a t e d G D F s a r e p l o t t e d a s a f u n c t i o n o f g i r d e r s p af o r s p a n s 9 t o 6 0 m . F o r c o m p a r i s o n , A A S H T O [1] G D F s a r e a ls o s h o w n .

6 T w o l a n e m o m e n t s an d s h e a r s

T h e a n a l y s is in v o lv e s t h e d e t e r m i n a t i o n o f th e l o a d i n e a c h l a n e a n d l o a d d i s t r i b u t io ng i r d e rs . T h e e f f e c t o f m u l t i p l e tr u c k s i s c a l c u l a t e d b y s u p e r p o s i t io n . T h e m a x i m u m e f f e c t sc a l c u l a t e d a s t h e l a r g e s t o f t h e f o l l o w i n g ca s e s :(1 ) O n e l a n e f u ll y l o a d e d a n d t h e o t h e r l a n e u n l o a d e d .(2 ) B o t h l a n e s l o a d e d ; t h r e e d e g r e e s o f c o r r e l a ti o n b e t w e e n t h e l a n e l o a d s a r e c o n s i d e r e d

c o r r e l a t i o n ( p = 0 ), p a r t i a l c o r r e l a t i o n ( p = 0 .5 ) a n d f u l l c o r r e l a t i o n ( p = 1 ).

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TABLE 4

Lane load parameters for two lane traffic

63

On e/ tw o lanes loaded N z T

ne

Two:20,000,000 5.33 75 yea rs

p = 0 Lane 1 1,500,000 4.83 5 year s

Lan e 2 1 0.00 averagep = 0.5 Lane 1 150,000 4.36 6 mon ths

Lane 2 1,000 3.09 1 dayp = 1 Lan e 1 50,000 4.11 2 mon th

Lan e 2 50,000 4.11 2 mon th

I t h a s b e e n o b s e r v e d t h a t , o n a v e r a g e , a b o u t e v e r y 1 5 t h t r u c k i s o n t h e b r i d g e s i m u l t a n e ow i t h a n o t h e r t r u c k ( s id e - b y -s i d e) . F o r e a c h s u c h a s i m u l t a n e o u s o c c u r r e n c e , i t i s a s s u m e de v e r y 1 0 th t i m e t h e t r u c k s a r e p a r t i a ll y c o r r e l a t e d a n d e v e r y 3 0 t h t i m e t h e y a r e f u ll y c o r r e( w i t h r e g a r d t o w e i g h t ). I t is al s o a s s u m e d t h a t t h e t r a n s v e r s e d i s t a n c e b e t w e e n t w o s i d e -b y

t r u c k s i s 1 . 2 m ( w h e e l c e n t e r - t o - c e n t e r ) .T h e p a r a m e t e r s o f l a n e l o a d, i n c lu d i n g N ( t h e c o n s i d e r e d l an e l o a d is t h e m a x i m u m oo c c u r re n c e s ) , z = _ ~ - 1 ( l / N ) , a n d T ( th e c o n s i d e r e d l a n e l o a d is t h e m a x i m u m i n p e r i o d T ) a r e g i v e n i n Ta b l e 4 .

T h e r e s u l t s o f s im u l a t i o n s i n d i c a t e t h a t f o r i n t e r i o r g i r d e rs , t h e c a s e w i t h t w o f u ll y c o r r es i d e -b y - s id e t ru c k s g o v e r n s , w i t h e a c h t r u c k e q u a l t o t h e m a x i m u m 2 m o n t h t r u c k . T h e r a ta m e a n m a x i m u m 7 5 y e a r m o m e n t a n d a m e a n 2 m o n t h m o m e n t is a b o u t 0 .8 5 fo r al l t h e s

7 M u l t i l a n e l iv e lo a d f o r v a r i o u s A D T T

A D T T ( a v e r a g e d a i ly t r u c k t ra f fi c ) is an i m p o r t a n t p a r a m e t e r o f l iv e l o a d . T h e o b s e r v ai n d i c a t e a c o n s i d e r a b l e s i t e -s p e c i fi c v a r i a t i o n o f t h e n u m b e r o f v e h i c l e s a n d p e r c e n t a gt r u ck s . T h e l iv e l o a d p a r a m e t e r s d e r i v e d i n t h is p a p e r c o r r e s p o n d t o A D T T = 1 ,0 00 (tt r a f f i c i n a l l l a n e s i n o n e d i r e c t i o n ) . T h e l i v e l o a d m o m e n t s f o r m u l t i l a n e b r i d g e s w i t h v aA D T T s a r e d e r i v e d b y si m u l a ti o n s .

T h e m a x i m u m 7 5 y e a r m o m e n t f or a s i n g le la n e i s c o n s i d e r e d a s a r e f e r e n c e v a l ue .m u l t i p l e l a n e s , m o m e n t p e r l a n e i s l o w e r t h a n t h a t v a l u e . F o r t w o l a n e s, t h e r a t i o o f t h el a n e m o m e n t a n d t h e m a x i m u m s i n gl e l a n e m o m e n t is 0.8 5 (a s c a lc u l a te d e a r li er ). F o r tl a n e s, t h e p r o b a b i l i t y o f a s i m u l t a n e o u s o c c u r r e n c e o f v e r y h e a v y tr u c k s i n a ll t h r e e l a n

TABLE 5

Per lane mome nt ratios for multilane loads

A D Trin one direction)

Number of lanes

1 2 3 4

100 0.95 0.80 0.55 0.451,000 1.00 0.85 0.70 0.505,000 1.05 0.90 0.75 0.55

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6

O

1

0

o

i

LRFD

3 0 6 0S p a n m )

F i g . 14 . B ia s f a c to r s f o r si m p l e s p a n m o m e n t s : H S 2 0 a n d L R F D .

e v e n s m a l l e r. A c c o r d i n g l y, t h e r a t io o f p e r l a n e m o m e n t a n d r e f e r e n c e v a l u e i s 0 .7 0 . F o r f o

l anes , t he r a t i o i s 0 .50 .F o r o t h e r A D T T ' s ( in o n e d i r e c ti o n ) , t h e p r o b a b i l it i es o f s i m u l t a n e o u s o c c u r r e n c e

m u l t i p l e t r u c k s a r e d i f f e r e n t t h a n f o r A D T I = 1 , 0 0 0 . T h e r e f o r e , t h e p e r l a n e m o m e n t s a r e a ld i f fe r e n t . T h e m o m e n t r a t io s d e t e r m i n e d b y s im u l a t i o n s o f m u l t i l a n e t r a ff ic a r e l is t ed Ta b l e 5 .

8 Des ign l ive load

T h e o b j e c t i v e i n t h e s e l e c t i o n o f t h e d e s i g n l iv e l o a d m o d e l i s a u n i f o r m b i a s fa c t o r. B

f a c to r s c a l c u la t e d f o r t h e c u r r e n t A A S H T O [1 ] a r e p lo t t e d i n F i g . 1 0 to 1 2 . To i m p r o v e tu n f o rm i t y, a n e w l iv e l o a d is p r o p o s e d ( L R F D A A S H T O C o d e , y e t u n p u b l i s h e d ). L R F D s t af o r l o a d a n d r e si s t a n ce f a c t o r d e s ig n . T h e p r o p o s e d L R F D l o a d is a su p e r p o s i t i o n o f H St ru c k , s h o w n i n F ig . 1 , a n d u n i f o r m l o a d o f 9 . 3 5 k N / m . F o r s h o r t e r s p a n s a t a n d e m i s s p e c i fi

0

H S 2 0

P

0 3 0S p i n , - )

6 0

F i g . 1 5 . B i a s f a c t o r s f o r s h e a r s : H S 2 0 a n d L R F D .

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u

2

o - . o - o . - o ~ ~ . _ _

0 3 0 6 0S p a n - , )

F i g 1 6 B i a s f a c t o r s f o r n e g a t i v e m o m e n t s : H S 2 0 a n d L R F D

65

F o r n e g a t i v e m o m e n t s t w o H S 2 0 t r u c k s a r e u s e d , b u t t h e t o t a l e f f e c t i s r e d u c e d b y 1 0 p e r c

F o r c o m p a r i s o n , t h e c o r r e s p o n d i n g b i a s f a c t o r s f o r l i v e l o a d s p e c i f i e d b y t h e p r o p o s e d L Ra n d A A S H TO [ 1 ] a r e p l o t t e d i n F i g . 1 4 , 1 5 a n d 1 6 f o r s i m p l e s p a n m o m e n t , s h e a r a n d n e g am o m e n t , r e s p e c t i v e l y.

T h e p r o p o s e d l i v e l o a d m o d e l w a s a l s o u s e d i n t h e d e v e l o p m e n t o f d e s i g n l o a d s f o r t h e e d i t io n o f O H B D C [ 2] . T h e p r o c e d u r e s a r e s u m m a r i z e d b y N o w a k [ 13 ].

9 C o n c l u s io n s

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 id g e s i s d e a d l o a d , li ve lo a d a n d d y n a m i c lS t a ti s ti c a l m o d e l s a r e s u m m a r i z e d f o r d e a d l o a d a n d d y n a m i c l o a d . T h e m o d e l i s d e v e l o p e dl iv e l o a d . T h e s t at i st ic a l p a r a m e t e r s a r e d e r i v e d u s i n g t h e a v a i l ab l e t r u c k s u rv e y d a t a . M o ma n d s h e a r s a r e c a l c u l a t e d f o r s u r v e y e d t r u c k s . T h e r e s u l t i n g C D F s a r e e x t r a p o l a t e d f o r l ot ime per iods (up 75 years ) .

M u l t i p l e p r e s e n c e o f t r u c k s i s c a l c u l a t e d i n l a n e a n d s i d e - b y - s i d e . T h e p a r a m e t e r s c o ne r e d i n c l u d e h e a d w a y d i s t a n c e a n d d e g r e e o f c o r re l a t i o n (w i th r e g a r d t o w e i g h t) . T h e f r e q uo f o c c u r r e n c e i s a s i t e - s p e c i f i c p a r a m e t e r. I t i s m o d e l e d o n t h e b a s i s o f t h e a v a i l ao b s e rv a t io n s . T h e m a x i m u m v a l u e s a r e d e t e r m i n e d b y s i m u l at io n s .

I t w a s f o u n d t h a t t h e l a n e l i v e l o a d m o m e n t i s g o v e r n e d b y a s i n g l e t r u c k u p t o a b o u t 4s p a n , s h e a r u p t o a b o u t 3 5 m , a n d n e g a t i v e m o m e n t ( c o n t i n u o u s s p a n s ) u p t o a b o u t 1 5 m .

t w o l a n e b r i d g e s , t h e m a x i m u m 7 5 y e a r e f f e c t i s c a u s e d b y t w o s i d e - b y - s i d e m a x i m u m m o n t h t r u c k s , w i t h f u l ly c o r r e l a t e d w e i g h t s . T h e t w o m o n t h t r u c k i s a b o u t 0 .8 5 o f t h e 7 5 t ruck .

F o r t w o l a n e b r id g e s , g ir d e r d i s tr i b u ti o n f a c t o r ( G D F ) is v e r y i m p o r t a n t . G D F d e p e n dg i r d e r s p a c i n g a n d s p a n l e n g t h . T h e a n a l y s i s i n d i c a t e s t h a t t h e c u r r e n t A A S H TO [ 1c o n s e r v a t iv e i n m o s t c a s e s , i n p a r t i c u l a r f o r l a rg e r g i r d e r s p a c in g .

F o r m u l t i l a n e b r id g e s , t h e m a x i m u m 7 5 y e a r e f f e c t is c a u s e d b y a s u p e r p o s i t i o n o f l a n e l oP e r l a n e m o m e n t s a r e l o w e r t h a n t h e m a x i m u m 7 5 y e a r s i n g l e l a n e m o m e n t s . T h e p e r m o m e n t r a t io s a r e p r o v i d e d a s a f u n c t io n o f n u m b e r o f l o a d e d l a n e s a n d A D T T.

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The proposed l ive load mode l i s used in the deve lopment o f new LRFD codes in the Uni tedStates and Canada.

cknowledgments

Th e presen ted research was perform ed in conjun ct ion with NC H R P Project 12-33 withCo-Principal Invest igators John M. Kul icki and Den nis Mertz . NC H R P was represented by IanFriedland. The author thanks mem bers of the Cal ibrat ion Task Group in par ticular C. Al l inCorne l l Ted . V. Ga lambos Dan M. Frangopol and Fred M oses for the i r d i scuss ions andcomments . Comments and d i scuss ion were a l so rece ived f rom the Ca l ib ra t ion Commit teework ing on OH BD C in pa rt icu la r Hid N. Groun i Roger Dor ton Akh i lesh Agarwal and T.Tharmabala .

Thanks are a lso due to the doctoral s tudents a t the Univers i ty of Michigan Hani Nassi fYoung-Kyun Hong and Eui -Seung Hwang fo r the i r he lp in computa t ions .

References

[1] AASHTO, Standard specif icat ions for highway br idges , American Associat ion of State Highway and Tportation Officials, Washington, DC, 1989.

[2] OHB DC , O ntar io highway br idge design code, Minis try of Transp orta t ion, Downsview, O ntar io , CanadEdition 1979, 2nd Editio n 1983, 3rd Edition 1991.

[3] A.S. Nowak and N.C. Lind, Practical bridge code calibration,Journal of the Structural Division of ASCEDec em ber 1979) 2497-2510.

[4] H.N. Grou ni an d A.S. Nowak, Cal ibrat ion of the Ontar io highway br idge design code,The Canadian Journal ofCivil Engineering 11 4) De cem ber 1984) 760-770.

[5] B. El l ingwood, T.V. Galambos, J .G. MacGregor and C.A. Cornel l , Development of a probabi l i ty based

cr i ter ion for Ame rican Nat ional S tandard A58, N at ional Bu reau of Standards , NBS Special Publ icationWashington, DC, 1980.

[6] E-S. Hw ang and A.S. N owak, Simulat ion o f dynamic load for br idges ,ASCE Journal of Structural Engineering117 5 ) 1991) 1413-1434.

[7] M. G hosn and F. Moses , A M arkov renewal m odel for maxim um bridge loading,Journal of EngineeringMechanics ASC E, 111 9) Sep tem ber 1985).

[8] F. Moses e t a l. , Loading spectrum exper ienced by br idge s t ructures in the Un i ted States , Repo rtFHWA/RD-85/012 Case We stern R eserve Univers i ty, Cleveland, OH, 1985.

[9] A.S. Nowak and Y-K. Hong, Bridge live load models,ASCE Journal of Structural Engineering 117 9) 1991)2757-2767.

[10] A.C. Agarwal an d M . Wolkowicz, Inter im report on 1975 comm ercial vehicle survey, R esearch and Devm ent Division, M inistry of Tran spo rtation , Downsview, Ontario , Canad a, 1976.

[11] A.S. N owak and H. Nassif , Effect of t ruck loading on br idges , Rep ort U M CE 91-11, De part me nt o fEngineer ing, U nivers ity of Michigan, Ann Arbo r, M I, 1991.

[12] T. Zokaie , T.A. O sterka m p and R .A. Im bsen, Dis t r ibut ion of wheel loads on highway br idges , NC12-26/1, Proposed changes in AASHTO, Imbsen and Associates , Sacramento, CA, 1992.

[13] A.S. Nowak, Cal ibrat ion of the O H BD C 1991,Canadian Journal of Civil Engineering to appear) .