PSI - Issue 5

John Leander et al. / Procedia Structural Integrity 5 (2017) 1221–1228 Author name / Structural Integrity Procedia 00 (2017) 000–000

1225

5

where N c ( x ) represent the resistance as the number of cycles to failure and N is the total number of accumulated cycles. A state of failure is then defined by g ” 0 and the probability of failure as > @ f 0 P P g d (3) f ). The reliability, or equivalently the associated probability of failure, can be estimated using conventional reliability methods such as FORM, SORM or a simulation based method. However, it should be noted that the limit state equations can be strongly nonlinear, especially if LEFM is considered, and as such, the use of FORM may not be appropriate. For a performance model based on linear damage accumulation, a limit state based on accumulated damage is often preferred. Reformulating and expanding (2) gives The reliability index ȕ is related to the probability of failure as ȕ = – ĭ –1 ( P

1

1

m

¦

¦

m

, g N x

n C S

n C S

G

2

1

(4)

i

r, S i

j

r, S j

K

K

i

j

1

2

A description of the limit state equation (4) can be found in Leander el al. (2015). For a performance model based on LEFM, the limit state equation (2) is applicable with N c ( x ) determined by integration of the expected crack growth E [ da / dN ] from the initial crack depth a 0 to a critical crack depth a c . The expected crack growth rate can then be expressed as (JCSS, 2011)

ª º

m

m

S

S

a

b

a ª º ¬ ¼ m r

b ª º ¬ ¼ m r

a E da A E S «¬ »¼

S SIF C C a Y a M a S k

A E S

S SIF C C a Y a M a S k

ab

f

b

dN

S

S

th

ab

(5)

A complete description of the limit state equation for LEFM can be found in Leander el al. (2016). Stochastic variables used for the current case are listed in Table 1 together with their distributions, mean values and coefficients of variation (CoV). These properties are essentially as suggested by JCSS (2013).

Table 1. Stochastic variables. N~Normal, LN~Lognormal , DET~Deterministic.

Linear damage accumulation

Linear elastic fracture mechanics (LEFM) (Crack growth in mm/cycle and stress intensity in MPa(m) 1/2 )

Variable

Distribution

Mean

CoV

Variable

Distribution

Mean

CoV 0.04 0.07 1.70 1.70

į

C S

LN LN

1 1

0.3

LN LN LN LN

1 1

C S

C SIF

0.04 0.49

–18

ln K 1

A a A a m a m b K th

N

26.5

4.80×10 4.80×10

–18

Fully correlated to K 1

K2 m1 m2

DET DET

3 5

- -

DET DET

5.10 2.88 140 0.15 113

- -

LN LN

0.40 0.66

a 0 a c

DET

-

3.4. Knowledge content

Additional knowledge or information regarding the actual physical state of the bridge can be directly integrated with the reliability-based models described in Sec. 3.3. In this case study, a stress range spectrum determined by