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7/24/2019 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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1 AUTHOR:
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
7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
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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
7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
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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
7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
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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.
7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
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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
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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-
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J.
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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
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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==7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
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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:
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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
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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
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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:
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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
14 th ed . Amer ican Associa t ion o f S ta te Highway an
Transportation Officials, Washington, D.C.
Agarwal, A.C., and Wo lkowicz , M. 1976. Interim report on 19
comm ercial vehicle survey. Research and D evelopment Divisio
Ministry of Transportation and Communications, Downsvie
Ont.
Ben jamin, J.R., and C ornell, C.A. 1970. Probability, statistics, a
decision for civi l engineers . McGraw-Hil l Book C o., Ne
York, p. 684.
Billing, J.R. 1984. Dyna mic loading and testing of bridges
Ontario. Canadian Journal of C ivil Engine ering, ll (4 ): 833-84
Ellingwood, B., et al. 1980. Development of a probability bas
load criterion for American National Standard A58. Nation
Bureau o f S tan dards , Wa sh ing ton , D .C. , NBS Spec i
Publication 577.
7/24/2019 Load Model for Bridge Design Code - Canadian Journal of Civil Engineering
15/15
N O W A K 4 9
Ghosn, M., and Moses, F 1984. Bridge load modeling and reli-
ability analysis. Department of Civil Engineering, Case Western
Reserve University, Cleveland, Ohio, Report No. R 84-1.
Grouni, H.N., and Nowak, A.S. 1984. Calibration of the Ontario
Highway Bridge D esign Code 1983 edition. Canadian Journal
of Civil Engineering, l l 4): 760-770.
Hwang, E-S . , and Nowak, A.S. 1 991. Simula t ion of dynamic
load for br idges. ASCE Journal of St ruc tura l Engineer ing,
117 5): 141 3-1434.
Madsen, H.O., K renk, S., and Lind, N.C. 1986. Methods of struc-
tural safety. Prentice-Hall, Inc., Englewood Cliffs, N.J., p. 403.
Nowak, A.S., and Curtis, J.D. 1980. Risk analysis computer pro-
g ram. Unive r s i t y o f Mich igan , Ann Arbor , Mich . , Repor t
UMEE 8OR2.
Nowak, A.S., and Hong, Y-K. 1991. Bridge load models. ASCE
Journal of Structural Engineering, 117 9 ): 2757-2767.
Nowak, A S . , and Lind, N.C. 1979. Pract ical br idge code ca li -
bration. ASCE Journal of the Structural Division, lOS ST12):
2497-25 10.
OH BD C. 1 979. Ontar io highway br idge design code . 1st ed.
Ministry of Transportation, Downsview, Ont.
OH BD C. 1983. Ontar io highway br idge design code . 2nd ed.
Ministry of Transportation, Downsview, Ont.
OH BD C. 1991. Ontar io highway br idge design code . 3rd ed.
Ministry of Transportation, Downsview, Ont.
Turkstra, C.J. 1970. Theo ry of structural design decisions. Solid
Mechanics Division, University of Waterloo, Waterloo, Ont.
Study No. 2, p. 124.
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