Marine Structures 15 (2002) 599–613
A probabilistic damage tolerance concept for
welded joints. Part 1: data base and stochastic
modelling
Tom Lassena,*,1, John D S
rensenb
a
Agder University College, 4876 Grimstad, Norway
b
Aalborg University, Denmark
Received 1 February 2001; received in revised form 10 December 2001; accepted 20 February 2002
Abstract
The present paper presents the necessary crack growth statistics and suggests stochastic
models for a reliability analysis of the fatigue fracture of welded steel plate joints. The
reliability levels are derived from extensive testing with fillet-welded joints for which the entire
crack growth history has been measured, not only the final fatigue life. The statistics for the
time to reach given crack depths are determined. Fracture-mechanics-derived crack growth
curves are fitted to the measured experimental curves and the best fit defines the growth
parameters involved for each test specimen. The derived statistics and distribution function for
these parameters are used as variables in a Monte Carlo simulation (MCS). In addition a
Markov model is developed as an alternative stochastic model. It is a Markov chain for which
the discrete damage states are related to chosen crack depths in the material. This model works
directly with the experimental time statistics. It is a ‘‘stochastic bulk approach’’ not involving
any random variables or fracture mechanics modeling. Both models are fitted to the data base
and scaled to in-service conditions. Both methods are compared and discussed. The aim is to
provide data for the variables used in a MCS and to develop a Markov chain for fast reliability
calculation, especially when predicting the most likely influence of numerous future
inspections. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Welded joints; Fatigue life; Reliability
*Corresponding author.
E-mail addresses: (T. Lassen), (J.D. S
rensen).
1
Affiliated with Advanced Production and Loading (APL), Arendal, Norway.
0951-8339/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 5 1 - 8 3 3 9 ( 0 2 ) 0 0 0 2 0 - 5
, 600 T. Lassen, J.D. S
rensen / Marine Structures 15 (2002) 599–613
1. Introduction
Fatigue durability and inspection planning have long been important issues in the
design and scheduled inspection of welded offshore structures. For welded structures
subjected to cyclic repetitive loading, the admissible stresses in the vicinity of welded
joints are relatively low due to the joints vulnerability to fatigue damage. Usually,
structures have to be designed for a finite life with an accepted probability of failure
based on the S–N approach. Hence, cracks may propagate and become critical
during the predicted ‘‘safe-life’’, unless discovered in time and repaired. If fracture is
unacceptable, additional safety measures must be taken through in-service inspection
requirements specifying non-destructive inspection (NDI) methods, inspection
intervals and repair procedures. This leads to the damage tolerance concept; a joint
containing a crack has to withstand the service loading for some time. During this
time there must be a large probability that the crack be detected before it becomes
critical. To verify this probability, the reliability against such a failure must be
calculated explicitly as a function of service time in support of the inspection
strategy. A combination of elements from fracture mechanics and stochastic
modeling provides the necessary tools for these reliability calculations. This so-called
probabilistic fracture mechanics approach has lately been widely used on welded
tubular joints which often are the primary structural joints in fixed structures in the
North-Sea. The analysis is often carried out using Monte Carlo simulation (MCS) or
the first order reliability method (FORM), which have become standard methods in
structural reliability, Refs. [1,2]. A limit state function is formulated by applying
linear elastic fracture mechanics (LEFM). The uncertainties of all the influencing
parameters can be taken into account by treating them as basic random variables.
One problem which usually arises is that the necessary statistical informations (mean
value, standard deviation and distribution type) for these variables are not known. It
is therefore of interest to obtain the necessary data and to carry out an investigation
on some supplementary stochastic methods which are more direct and more easily
applicable. An alternative is to assume an evolutionary probabilistic structure from
the start, based directly on test results, thus avoiding any fracture mechanical
modeling. One of these methods is the Markov chain approach where crack depths
are treated as damage states. The model has to some extent been presented earlier,
Refs. [3,4], but has now matured to a stage where it can provide a practical tool for
the designer, without demanding extensive knowledge of fracture mechanics and
stochastic modeling. The first part of this paper is concerned with the underlying
statistics and the development and corroboration of stochastic models based
on experimental results, whereas the rule-based practical application is presented in
Part 2.
2. The fracture mechanics approach to crack propagation
The fatigue propagation is often predicted using a LEFM approach. It is assumed
that the crack growth rate at a macroscopic average level may be described by the
A probabilistic damage tolerance concept for
welded joints. Part 1: data base and stochastic
modelling
Tom Lassena,*,1, John D S
rensenb
a
Agder University College, 4876 Grimstad, Norway
b
Aalborg University, Denmark
Received 1 February 2001; received in revised form 10 December 2001; accepted 20 February 2002
Abstract
The present paper presents the necessary crack growth statistics and suggests stochastic
models for a reliability analysis of the fatigue fracture of welded steel plate joints. The
reliability levels are derived from extensive testing with fillet-welded joints for which the entire
crack growth history has been measured, not only the final fatigue life. The statistics for the
time to reach given crack depths are determined. Fracture-mechanics-derived crack growth
curves are fitted to the measured experimental curves and the best fit defines the growth
parameters involved for each test specimen. The derived statistics and distribution function for
these parameters are used as variables in a Monte Carlo simulation (MCS). In addition a
Markov model is developed as an alternative stochastic model. It is a Markov chain for which
the discrete damage states are related to chosen crack depths in the material. This model works
directly with the experimental time statistics. It is a ‘‘stochastic bulk approach’’ not involving
any random variables or fracture mechanics modeling. Both models are fitted to the data base
and scaled to in-service conditions. Both methods are compared and discussed. The aim is to
provide data for the variables used in a MCS and to develop a Markov chain for fast reliability
calculation, especially when predicting the most likely influence of numerous future
inspections. r 2002 Elsevier Science Ltd. All rights reserved.
Keywords: Welded joints; Fatigue life; Reliability
*Corresponding author.
E-mail addresses: (T. Lassen), (J.D. S
rensen).
1
Affiliated with Advanced Production and Loading (APL), Arendal, Norway.
0951-8339/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 5 1 - 8 3 3 9 ( 0 2 ) 0 0 0 2 0 - 5
, 600 T. Lassen, J.D. S
rensen / Marine Structures 15 (2002) 599–613
1. Introduction
Fatigue durability and inspection planning have long been important issues in the
design and scheduled inspection of welded offshore structures. For welded structures
subjected to cyclic repetitive loading, the admissible stresses in the vicinity of welded
joints are relatively low due to the joints vulnerability to fatigue damage. Usually,
structures have to be designed for a finite life with an accepted probability of failure
based on the S–N approach. Hence, cracks may propagate and become critical
during the predicted ‘‘safe-life’’, unless discovered in time and repaired. If fracture is
unacceptable, additional safety measures must be taken through in-service inspection
requirements specifying non-destructive inspection (NDI) methods, inspection
intervals and repair procedures. This leads to the damage tolerance concept; a joint
containing a crack has to withstand the service loading for some time. During this
time there must be a large probability that the crack be detected before it becomes
critical. To verify this probability, the reliability against such a failure must be
calculated explicitly as a function of service time in support of the inspection
strategy. A combination of elements from fracture mechanics and stochastic
modeling provides the necessary tools for these reliability calculations. This so-called
probabilistic fracture mechanics approach has lately been widely used on welded
tubular joints which often are the primary structural joints in fixed structures in the
North-Sea. The analysis is often carried out using Monte Carlo simulation (MCS) or
the first order reliability method (FORM), which have become standard methods in
structural reliability, Refs. [1,2]. A limit state function is formulated by applying
linear elastic fracture mechanics (LEFM). The uncertainties of all the influencing
parameters can be taken into account by treating them as basic random variables.
One problem which usually arises is that the necessary statistical informations (mean
value, standard deviation and distribution type) for these variables are not known. It
is therefore of interest to obtain the necessary data and to carry out an investigation
on some supplementary stochastic methods which are more direct and more easily
applicable. An alternative is to assume an evolutionary probabilistic structure from
the start, based directly on test results, thus avoiding any fracture mechanical
modeling. One of these methods is the Markov chain approach where crack depths
are treated as damage states. The model has to some extent been presented earlier,
Refs. [3,4], but has now matured to a stage where it can provide a practical tool for
the designer, without demanding extensive knowledge of fracture mechanics and
stochastic modeling. The first part of this paper is concerned with the underlying
statistics and the development and corroboration of stochastic models based
on experimental results, whereas the rule-based practical application is presented in
Part 2.
2. The fracture mechanics approach to crack propagation
The fatigue propagation is often predicted using a LEFM approach. It is assumed
that the crack growth rate at a macroscopic average level may be described by the
