1. JCSS Model Code Fatigue, inspection and reliability
Ton Vrouwenvelder
TNO / TU-Delft
The Netherlands

2. JCSS Probabilistic Model Code
Part 1 Basis of Design
Part 2 Modeling of loads
Part 3 Modeling of structural properties
select “publications”
select “jcss model code”

3. Part 3 Resistance models
3.0 General
3.1 Concrete
3.2 Reinforcement
3.3 Prestr steel
3.4 Steel
3.5 Timber
3.6 Aluminium
3.7 Soil
3.8 Masonry
3.9 Model uncert.
3.10 Dimensions
3.11 Imperfections

4. C and m material parameters
fatigue crack growth based on Paris-Erdogan relation:
stress range
crack size a
As already stated above, in most fatigue reliability calculations no or little attention is being paid to the effects of spatial correlation as well as correlation in time. However, in order to assess the overall fatigue system reliability for a structure, the correlation between the material properties in the various hotspots in the structure has to be included. This correlation is also essential to draw conclusions from the inspection of one hotspot with respect to the reliability of others (Straub, 2002). This correlation may be referred to as a macro scale spatial fluctuation.
Further it is important to consider the correlation in the cause of the time. The stress, for instance, is not a random variable but a random process with both ergodic and nonergodic uncertainties. The material properties may also change in the course of the process as the crack enters other material every cycle when it grows. These material properties will not be the same all over the thickness of a plate. This in fact is a micro scale spatial correlation.
Let us consider the correlation matter more in detail, looking to the growth of a crack in a steel bridge for road traffic. According to the theory of fracture mechanics the crack growth per cycle (da/dN) is depending on the range of the stress intensity factor (DK). The stress intensity factor is a measure for the stress near the crack tip in a linear elastic analysis and is therefore the governing parameter for fatigue crack growth. The stress intensity factor is depending on the geometry the crack dimensions and the loading. The most well known crack growth model is the Paris-Erdogan 1963 relation:
da/dN = C (DK)m (1)
where a = crack depth, K = stress intensity factor, m = material parameter, C = material parameter and N= number of cycles. Integration of the crack growth law from initial defect to a final defect gives the fatigue life of a joint. For the sake of discussion we will consider a simple example where:
K = (2)
Y being a geometry dependent factor and  the stress, including the stress concentration factor (SCF). Accordingly, the Limit State Function Z for a crack through failure is then given by:
(3)
where d is the plate thickness. By definition of limit state function the structure fails for Z<0 and is okay for Z>0. If we elaborate the integral for Y=1 and m=3 we arrive at:
(4)
The properties of the various stochastic variables (mean values and coefficients of variation V) may be defined as (see the according to the JCSS Probabilistic Model Code , JCSS 2000):
symbol
variable
distribution
mean value V C
material parameter
lognormal
[N, mm]
0.50 m
deterministic 3 -
a i
initial crack
0.15 mm d
wall thickness
normal
28 mm
0.05 
cycle rate
/ year
0.20 
stress range
40 N/mm2
C and m material parameters

5. Limit state function for crack through failure:
Elaboration for constant amplitude, m=3 & Y=1 :

6. Crack growth model da/dn = C (DK ) m K = U Y M S ( a) /Q
a = crack size
C = material parameter
m = material parameter
K = stress intensity factor
S = stress, including SCF
U = crack closure factor
Y = crack geometry factor
M = weld geometry factor
Q = elliptic shape factor
Paris, 1962

7. Crack growth model
(C1, m1) and (C2, m2)

8. crack growth model

9. Crack growth model – 2 dimensional
A semi-elliptical crack in a steel plate at/near the weld toe

10. Limit state formulations
Fixed critical crack size ( plate thickness)
Fatigue-fracture
g (X, t) = min { R -  (Kr2 + Lr2) }
Kr = Ks (a) / Kmat
Lr = S (a) / Sy

11. Fatigue fracture
Dijkstra (1991)

12. Statistical properties

13. Statistical properties
Statistical properties

14. Statistical properties
Random variable
- average value
- standard deviation
- distribution type
Vectors/fields/processes
- correlation in time - correlation in space   
x, t
1.0

15. relevant types of correlation
member to member
(system reliability)
before and after inspection
(updating)

16. Correlation in space Correlation in time C material property
ai initial crack size
d wall thickness
S stress level
Correlation in time
 cycle rate
S stress level

17. Correlation from hotspot to hotspot
Paris Law parameter:
V(C) = 0.50 (global scatter)
V(C) = 0.20 (internal scatter)
ρ = 1 – (0.202 / 0.502) = 0.85
initial crack size:
same order of magnitude
wall thicknes: ρ = 0.7 (JCSS Model Code, ch 2.11)
stress (systematic part) ρ = 0.6 (estimate)

18. Reliability index (one year period)
The basic problem is the correlation between the various variables before and after the inspection. Of course, the initial crack does not change because of the inspection, so this variable is fully correlated. The load parameters on the other hand may be different before and after the inspection. This is a matter of correlation in time. In a similar way the material parameter C need not be fully correlated. After the inspection the crack enters fresh material and there is no need for the steel to be entirely homogeneous. So here we may have to consider a micro scale spatial correlation. In figure 3 the effect of three different options are presented as well as the line without inspection as a kind of reference. This latter one is the almost horizontal line starting at  = 4.4 at 10 years and corresponding to the upper line in Figure 1. The assumptions for the other three lines are respectively:
upper curve: full correlation
middle curve: correlation of 0.85
lower curve: no correlation
It can be shown that the measured crack size of 0.3 mm is larger than the expected value after 10 years. Nevertheless we see in Figure 3 that the reliability (yearly basis) increases just after the inspection, regardless the degree of correlation. The point of course is that the observed crack is larger then expected, but that the inspection at least has proven that no extreme bad situation are present. The degree in which one may profit from this positive observation depends on the degree of correlation. In the case of full correlation the reliability index is still above 5 a decade after the inspection. If on the other hand zero correlation is assumed the positive effect of inspection already disappears after two years. It is interesting that even the relatively large degree of correlation present in the middle line leads to a modest improvement in reliability compared to the reference line without inspection. After about 7 years the updated line becomes lower.

19. Reliability index (one year period) given crack found at tins = 10 yr
The basic problem is the correlation between the various variables before and after the inspection. Of course, the initial crack does not change because of the inspection, so this variable is fully correlated. The load parameters on the other hand may be different before and after the inspection. This is a matter of correlation in time. In a similar way the material parameter C need not be fully correlated. After the inspection the crack enters fresh material and there is no need for the steel to be entirely homogeneous. So here we may have to consider a micro scale spatial correlation. In figure 3 the effect of three different options are presented as well as the line without inspection as a kind of reference. This latter one is the almost horizontal line starting at  = 4.4 at 10 years and corresponding to the upper line in Figure 1. The assumptions for the other three lines are respectively:
upper curve: full correlation
middle curve: correlation of 0.85
lower curve: no correlation
It can be shown that the measured crack size of 0.3 mm is larger than the expected value after 10 years. Nevertheless we see in Figure 3 that the reliability (yearly basis) increases just after the inspection, regardless the degree of correlation. The point of course is that the observed crack is larger then expected, but that the inspection at least has proven that no extreme bad situation are present. The degree in which one may profit from this positive observation depends on the degree of correlation. In the case of full correlation the reliability index is still above 5 a decade after the inspection. If on the other hand zero correlation is assumed the positive effect of inspection already disappears after two years. It is interesting that even the relatively large degree of correlation present in the middle line leads to a modest improvement in reliability compared to the reference line without inspection. After about 7 years the updated line becomes lower.
Correlation is related to C and  before and after inspection

20. spotlight on C

22. model Ferry Borges Castanheta model V(C) = 0.2 dC = 40 mm

24. proposal for correlations

25. Fatigue assessment of a welded detail
25 m m m
25m
30m
25m 5m
5 m
Detail location
Cover plate detail

26. Narrow band Gaussian process; Rayleigh amplitudes
Loading data
40 MPa r s =
Narrow band Gaussian process; Rayleigh amplitudes
 = 106 cycles per year

27. Calculation procedure PF(t)
Fixed critical crack size:
1. P( a(t)>d )
Fatigue fracture approach: g = R –  ( Kr2 + Lr2)
1. PF(t) = P(g(a(t), max S()) < 0 )
2. Add the annual probabilities
3. Outcrossing approach
4. Conditional failure rate approach

28. x XX Sorensen Result of Sudret Righiniotis
general case (g1(X),g2(X) and g3(X))
Sorensen
1.0E-04
1.0E-03
1.0E-02
1.0E-01
1.0E+00 2 4 6 8
log(N) Pf x
Result of
Righiniotis
g1,2(X), 10**4 simulations
g3(X), 10**4 simulations
simple case XX
Sudret

29. Conclusion:
This talk is over,
The work is not.