r - The Hong Kong Polytechnic University

The present study is concerned with the fatigue cracks that may develop at the longitudinal welds located at the
conjunction of the web and the bottom flange at the mid-span, which falls into fatigue detail Category B
(AASHTO 2007). When the fatigue detail coefficient A is assumed to follow a lognormal distribution, the mean
and COV values can be calculated based on the test results of welded bridge details. Based on the tests
performed by Keating and Fisher (1986), the mean and COV are calculated as 7.83×10 10 and 0.34.
S w , the revised equivalent stress range, is calculated for given combinations of vehicle speed and road roughness
condition. In the present study, the chi-square goodness-of-fit test is used to check the distribution type of the
parameter S w for each combination of vehicle speed and road roughness condition. Based on Sturges’ rule, the
number of intervals is seven for a 50-data bin and the degree of freedom is four. If a 5% significance level is
chosen, the test limit for the Chi-Square test is calculated as C 0.95,4 =9.488. As demonstrations in the present
study, six out of the total thirty groups of cases are employed to verify the distribution type. Both normal and
lognormal are acceptable for revised equivalent stress range S w based on the Chi-Square test.
Based on the assumption that all the variables (i.e. A, D f and S wj ) follow a certain distribution, the fatigue
reliability index is obtained using the method in the literature (Estes and Frangopol 1998). Based on such a
method, an arbitrary initial design point can be chosen and the solving process for the complex equation of
g()=0 can be avoided. After several iterations, convergence can be achieved without forcing every design points
to fall on the original failure surface.
In order to investigate the effects of the vehicle speed and road roughness condition on the fatigue reliability, the
fatigue reliability are calculated for all the 30 combinations of 6 vehicle speeds from 10m/s to 60m/s and 5 road
roughness conditions from very good to very poor. The limit state function for a given combination of vehicle
speed and road roughness condition is simplified as:
m
ntr
⋅ Sw
g( X ) = Df
−
(11)
A
Based on the LSF function, the fatigue reliability index, β, can be derived, assuming that all random variables
(i.e. A, D f and S w ) are lognormal, as follows:
λD + λ ( ln )
f A
− m⋅ λS + n
w tr
β =
(12)
2 2 2
ζ + ζ + ( m⋅ζ
)
λ
D f
Df
A Sw
where , λ
A , λ
S and ζ
w
D , ζ
f A , ζ
S denote the mean value and standard deviation of ln(A), ln(D
w
f ) and ln(S w ),
respectively.
Generally, the fatigue reliability indices are found to decrease with the increase of vehicle speed and road
roughness coefficient. If a 5% failure probability, i.e., a 95% survival probability is assumed, the corresponding
reliability index is 1.65 (Kwon and Frangopol 2010). When the vehicle increase rate is 0%, the reliability index
in all the thirty cases is larger than the target index of 1.65. Accordingly, the survival probability of all the thirty
cases is larger than 95%. When the vehicle increase rate is 3% or 5%, seven or eight cases are found with a
fatigue reliability index less than 1.65, i.e., the probability of fatigue failure for the bridge is larger than 5%
during its 75 years life. In addition, when the vehicle increase rate is 5% and the road roughness condition is
very poor, the reliability index is negative, which suggests the bridge would be more likely to suffer fatigue
failure than to survive in its 75 years life. In general, the higher vehicle speed, the smaller reliability index and
the higher probability of failure the structure will have in most cases. The deteriorated road surface seems to
accelerate fatigue damages due to the dynamic effects from vehicles, which implies the importance of road
surface maintenances for existing bridges.
Α
Table 1 Fatigue reliability index and Fatigue life
β (for Fatigue life of 75
years)
Fatigue life for target β=1.65
(unit: years)
0 % 6.5 1906
3 % 6.0 136
5 % 4.4 93
In the real circumstances, vehicle speeds vary for different trucks and road surface conditions deteriorate with
time. By treating the vehicle speed and road roughness as random variables as discussed earlier, the fatigue
reliability index is calculated and listed in Table 1 for different traffic increase rate, i.e., α= 0%, 3% and 5%. The
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