# Measurements and statistical analysis of fillet weld geometrical parameters for probabilistic modelling of the fatigue capacity

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel

► To cite this version:

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel. Measurements and statistical analysis of fillet weld geometrical parameters for probabilistic modelling of the fatigue capacity. Marine Structures, 2013, 34, pp.226-248. 10.1016/j.marstruc.2013.10.002. hal-01159165

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https://hal.science/hal-01159165

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<figure><img src="image_1.png" /></figure>

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# Measurements and statistical analysis of fillet weld geometrical parameters for probabilistic modelling of the fatigue capacity

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel

► To cite this version:

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel. Measurements and statistical analysis of fillet weld geometrical parameters for probabilistic modelling of the fatigue capacity. Marine Structures, 2013, 34, pp.226-248. 10.1016/j.marstruc.2013.10.002. hal-01159165

HAL Id: hal-01159165

https://hal.science/hal-01159165v1

Submitted on 3 Jun 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel

► To cite this version:

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel. Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis. Proceeding of 6th Conference on Network for Integrating Structural Analysis, Risk and Reliability, (ASRANet 2012), 2012, London, United Kingdom. pp.226-248, 10.1016/j.marstruc.2013.10.002. hal-01008949

HAL Id: hal-01008949

https://hal.science/hal-01008949

Submitted on 23 Jun 2019

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archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
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abroad, or from public or private research centers.

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# Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel

► To cite this version:

Olivier Pasqualini, Franck Schoefs, Mathilde Chevreuil, M. Cazuguel. Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis. Proceeding of 6th Conference on Network for Integrating Structural Analysis, Risk and Reliability, (ASRANet 2012), 2012, London, United Kingdom. pp.226-248, 10.1016/j.marstruc.2013.10.002. hal-01008949

HAL Id: hal-01008949

https://hal.science/hal-01008949

Submitted on 23 Jun 2019

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Please cite this paper as: Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M., "Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis", Proceeding of 6th Conference on Network for Integrating Structural Analysis, Risk and Reliability, (ASRANet 2012), Session 7 Reliability based Design and Optimisation (contd.), 6 pages, 2-4 July 2012, Croydon-London, UK (2012).

And Journal paper: Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M., "Measurements and statistical analysis of fillet welded joints geometrical parameters for probabilistic modelling of the fatigue capacity", Marine Structures, Final version published online: 6-NOV-2013, 34, 226-248, doi: 10.1016/j.marstruc.2013.10.002

# STATISTICAL ANALYSIS OF WELDED JOINTS GEOMETRY FOR STOCHASTIC MODELING AND RELIABILITY ANALYSIS

O. Pasqualini, F. Schoefs, M. Chevreuil, LUNAM University, GeM, Institute for Research in Civil and Mechanical Engineering, CNRS UMR 6183, University of Nantes – Ecole Centrale Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France.

M. Cazuguel, DCNS, Ingénierie Sous-Marins, rue Choiseul, 56311 Lorient Cedex, France.

## ABSTRACT

Welded joints are commonly used for various structures such as Civil Engineering infrastructures or marines and submarines structures. It is well known that the geometry of the joints has an important influence on the stress concentration factor and thus on fatigue lifetime. Non-Destructive Controls allow us to keep parameters inside bounds. However, the effect of the geometry characteristics within these bounds on the structural lifetime needs for a statistical analysis on the one hand and for a specific computational method on the other hand. When considering the first point, only few works have been carried out on the statistical analysis of the geometrical parameters of a welded joint. The measurement of the different parameters of this geometry is a long and scrupulous work. Recently, some laser process allows us to obtain a significant quantity of trajectories along a welded joint for these geometrical parameters. This paper aims at analyzing these trajectories for reliability purpose.

## NOMENCLATURE

*E*: expectation of a law

*h*: mean step between two points of measures

*L*<sub>c</sub>: correlation length

*L*<sub>s</sub>: length of the measurement after the statistical pre-processing

*m*<sub>X</sub>: mean of one trajectory of the variable X

*N*: number of points of measures

*N*<sub>Δx</sub>: number of points of measures on a length Δx

*n*<sub>e</sub>: number of trajectories used for the re-sampling

*n*<sub>f</sub>: number of trajectories after the data pre-processing

*n*<sub>s</sub>: number of trajectories after the statistical pre-processing

*n*<sub>t</sub>: number of trajectories at the beginning

*s*<sub>X</sub>: standard deviation of one trajectory of the variable X

*X*: random field analyzed
Please cite this paper as: Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M., "Statistical analysis of welded joints geometry for stochastic modeling and reliability analysis", Proceeding of 6th Conference on Network for Integrating Structural Analysis, Risk and Reliability, (ASRANet 2012), Session 7 Reliability based Design and Optimisation (contd.), 6 pages, 2-4 July 2012, Croydon-London, UK (2012).

And Journal paper : Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M., "Measurements and statistical analysis of fillet welded joints geometrical parameters for probabilistic modelling of the fatigue capacity", Marine Structures, Final version published online: 6-NOV-2013, 34, 226-248, doi: 10.1016/j.marstruc.2013.10.002

# STATISTICAL ANALYSIS OF WELDED JOINTS GEOMETRY FOR STOCHASTIC MODELING AND RELIABILITY ANALYSIS

O. Pasqualini, F. Schoefs, M. Chevreuil, LUNAM University, GeM, Institute for Research in Civil and Mechanical Engineering, CNRS UMR 6183, University of Nantes – Ecole Centrale Nantes, 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France.

M. Cazuguel, DCNS, Ingénierie Sous-Marins, rue Choiseul, 56311 Lorient Cedex, France.

## ABSTRACT

Welded joints are commonly used for various structures such as Civil Engineering infrastructures or marines and submarines structures. It is well known that the geometry of the joints has an important influence on the stress concentration factor and thus on fatigue lifetime. Non-Destructive Controls allow us to keep parameters inside bounds. However, the effect of the geometry characteristics within these bounds on the structural lifetime needs for a statistical analysis on the one hand and for a specific computational method on the other hand. When considering the first point, only few works have been carried out on the statistical analysis of the geometrical parameters of a welded joint. The measurement of the different parameters of this geometry is a long and scrupulous work. Recently, some laser process allows us to obtain a significant quantity of trajectories along a welded joint for these geometrical parameters. This paper aims at analyzing these trajectories for reliability purpose.

## NOMENCLATURE

* *E*: expectation of a law

* *h*: mean step between two points of measures

* *L*<sub>c</sub>: correlation length

* *L*<sub>s</sub>: length of the measurement after the statistical pre-processing

* *m*<sub>X</sub>: mean of one trajectory of the variable *X*

* *N*: number of points of measures

* *N*<sub>Δx</sub>: number of points of measures on a length Δx

* *n*<sub>e</sub>: number of trajectories used for the re-sampling

* *n*<sub>f</sub>: number of trajectories after the data pre-processing

* *n*<sub>s</sub>: number of trajectories after the statistical pre-processing

* *n*<sub>t</sub>: number of trajectories at the beginning

* *s*<sub>X</sub>: standard deviation of one trajectory of the variable *X*

* *X*: random field analyzed
$x$: abscissa

$\theta$: event

$\mu_X$: mean of the variable $X$ at a given abscissa

$\rho_{X,Y}$: general definition of the correlation coefficient between fields $X$ and $Y$

$\rho_X$: spatial correlation coefficient of the variable $X$

$\sigma$: standard deviation of a law

$\sigma_X$: standard deviation of the variable $X$ at a given abscissa

$\psi_{X,Y}$: correlation coefficient between $X$ and $Y$ at a given abscissa

# 1 INTRODUCTION

Welding joints are usually used in various buildings such as bridges or marine and submarine structures. It is well known that the geometry of the joint is important for realizing a fatigue computation. For some years, some Non-Destructive Controls Method have allowed us to get a quality control and obtain precise and discrete measurements of the geometry in order to make a fatigue calculus on the welded joints. A device named WISC based on a process of measurement by laser is used by DCNS to get a great number of trajectories of welded joints. Some geometrical parameters given in figure 1 can be measured with this device.

According with some studies, the parameters measured by the WISC are sufficient if we want to make a fiabilist calculus [1, 3].

<figure><img src="image_2.png" /><figcaption>Figure 1: Description of outputs of the WISC</figcaption></figure>

The study driven in this paper is a statistical analysis of one of the parameters. We give figure 2 the organigram which describes the different blocks and steps of the statistical analysis. Only the height of the joint S1 is studied in this paper. In the first two blocks, we only keep the most relevant measures by removing aberrant measurements. In the third block, we validate the stationary hypothesis of the second order. We also look for a length for which a hypothesis of non-correlation can be assumed. This distance can then be used in the last part of the work. It is useful to build some greater data to search the best probability law to represent the data distribution.

# 2 STATISTICAL PROCESSING METHOD

For the given data analysis, a statistical treatment is used, this methodology is detailed in figure 2.
$x$: abscissa

$\theta$: event

$\mu_X$: mean of the variable $X$ at a given abscissa

$\rho_{X,Y}$: general definition of the correlation coefficient between fields $X$ and $Y$

$\rho_X$: spatial correlation coefficient of the variable $X$

$\sigma$: standard deviation of a law

$\sigma_X$: standard deviation of the variable $X$ at a given abscissa

$\psi_{X,Y}$: correlation coefficient between $X$ and $Y$ at a given abscissa

# 1 INTRODUCTION

Welding joints are usually used in various buildings such as bridges or marine and submarine structures. It is well known that the geometry of the joint is important for realizing a fatigue computation. For some years, some Non-Destructive Controls Method have allowed us to get a quality control and obtain precise and discrete measurements of the geometry in order to make a fatigue calculus on the welded joints. A device named WISC based on a process of measurement by laser is used by DCNS to get a great number of trajectories of welded joints. Some geometrical parameters given in figure 1 can be measured with this device.

According with some studies, the parameters measured by the WISC are sufficient if we want to make a fiabilist calculus [1, 3].

<figure><img src="image_2.png" /><figcaption>Figure 1: Description of outputs of the WISC</figcaption></figure>

The study driven in this paper is a statistical analysis of one of the parameters. We give figure 2 the organigram which describes the different blocks and steps of the statistical analysis. Only the height of the joint S1 is studied in this paper. In the first two blocks, we only keep the most relevant measures by removing aberrant measurements. In the third block, we validate the stationary hypothesis of the second order. We also look for a length for which a hypothesis of non-correlation can be assumed. This distance can then be used in the last part of the work. It is useful to build some greater data to search the best probability law to represent the data distribution.

# 2 STATISTICAL PROCESSING METHOD

For the given data analysis, a statistical treatment is used, this methodology is detailed in figure 2.
<figure><img src="image_3.png" /><figcaption>Figure 2: Organigram of the used statistical analysis</figcaption></figure>

We note *X* the variable studied here, *n*<sub>t</sub> the number of trajectories initially given, *n*<sub>f</sub> the number of trajectories after the data pre-processing, *n*<sub>s</sub> the number of trajectories after the statistical pre-processing and *n*<sub>e</sub> the number of trajectories used for the distribution analysis. The step between two

measures is noted *h*. The abscissa is noted *x* and *θ* is the event. *X*(*x*,*θ*) is therefore the random one-dimensional variable depending of the event and the abscissa. The usable measured length is noted *L*<sub>s</sub>. The number of measurements done on length *L*<sub>s</sub> is denoted *N*. The following equations define
<figure><img src="image_3.png" /><figcaption>Figure 2: Organigram of the used statistical analysis</figcaption></figure>

We note *X* the variable studied here, *n*<sub>t</sub> the number of trajectories initially given, *n*<sub>f</sub> the number of trajectories after the data pre-processing, *n*<sub>s</sub> the number of trajectories after the statistical pre-processing and *n*<sub>e</sub> the number of trajectories used for the distribution analysis. The step between two

measures is noted *h*. The abscissa is noted *x* and *θ* is the event. *X*(*x*,*θ*) is therefore the random one-dimensional variable depending of the event and the abscissa. The usable measured length is noted *L*<sub>s</sub>. The number of measurements done on length *L*<sub>s</sub> is denoted *N*. The following equations define
different notations and discrete definitions of mean, standard deviation correlation coefficient used in this paper.

* The spatial mean of a particular trajectory i is given by:

$$m_X(\theta_i) = \frac{1}{N} \sum_{j=1}^{j=N} X(x_j; \theta_i) \quad (1)$$

* The spatial standard deviation of the trajectory i is given by:

$$s_X(\theta_i) = \sqrt{\frac{1}{N} \sum_{j=1}^{j=N} (X(x_j; \theta_i) - m_X(\theta_i))^2} \quad (2)$$

* The mean of the trajectories at a particular abscissa $x_j$ is given by:

$$\mu_X(x_j) = \frac{1}{n_S} \sum_{i=1}^{i=n_S} X(x_j; \theta_i) \quad (3)$$

* The standard deviation of the trajectories is given by:

$$\sigma_X(x_j) = \sqrt{\frac{1}{n_S} \sum_{i=1}^{i=n_S} (X(x_j; \theta_i) - \mu_X(x_j))^2} \quad (4)$$

In the paper, we also use notations $\mu_X$ and $\sigma_X$ without precising the abscissa. In this case, these are the spatial means of $\mu_X(x)$ and $\sigma_X(x)$.

In the following, the term “expectation” will be used to designate the expectation of a probabilistic law.

We also need to define the correlation coefficient

$\rho_{X,Y}$ of two particular vectors

$$X(x) = (X(x; \theta_i))_{i \in \{1, \dots, n_S\}} \quad \text{and}$$

$$Y(x + \Delta x) = (Y(x + \Delta x; \theta_i))_{i \in \{1, \dots, n_S\}} \text{ as :}$$

$$\rho_{X;Y}(x; x + \Delta x) = \frac{\frac{1}{n_S} \sum_{i=1}^{i=n_S} (X(x; \theta_i) - \mu_X(x))(Y(x + \Delta x; \theta_i) - \mu_Y(x + \Delta x))}{\sigma_X(x) \sigma_Y(x + \Delta x)} \quad (5)$$

From this equation, we can compute 2 different kinds of correlations. The first one is the spatial correlation. It corresponds to the mean of the different values obtained for different abscissa $x_j$ and using $X=Y$. We finally have:

$$\rho_X(\Delta x) = \frac{1}{N - N_{\Delta x} + 1} \sum_{k=1}^{k=N - N_{\Delta x} + 1} \rho_{X;X}(x_k; x_k + \Delta x) \quad (6)$$

where $N_{\Delta x}$ is the number of measurements on length $\Delta x$.

The second correlation coefficient is the coefficient between 2 variables at a given abscissa. It is given by

$$\psi_{X;Y}(x) = \rho_{X;Y}(x; x) \quad (7)$$

The process can be divided into 4 blocks (figure 2). In the first one, the data of the chosen variable X are filtered. The mean step $h$ between two points of measurement is estimated. To start with, we have $n_t$ trajectories of various lengths. The trajectories are removed under 2 conditions:

* The ones where the step is too variable along the trajectories;

* The ones where $h$ is too different of the mean step.

In order to simplify the treatment, the studies will be done using the number of points of measurement. The knowledge of this number and the mean step give us immediately the length $L_s$ in meters. The following process is the smoothing of the trajectories by overwriting the aberrant measures. We finally obtain $n_f$ trajectories filtered,
different notations and discrete definitions of mean, standard deviation correlation coefficient used in this paper.

• The spatial mean of a particular trajectory i is given by:

$$m_X(\theta_i) = \frac{1}{N} \sum_{j=1}^{j=N} X(x_j; \theta_i) \quad (1)$$

• The spatial standard deviation of the trajectory i is given by:

$$s_X(\theta_i) = \sqrt{\frac{1}{N} \sum_{j=1}^{j=N} (X(x_j; \theta_i) - m_X(\theta_i))^2} \quad (2)$$

• The mean of the trajectories at a particular abscissa $x_j$ is given by:

$$\mu_X(x_j) = \frac{1}{n_S} \sum_{i=1}^{i=n_S} X(x_j; \theta_i) \quad (3)$$

• The standard deviation of the trajectories is given by:

$$\sigma_X(x_j) = \sqrt{\frac{1}{n_S} \sum_{i=1}^{i=n_S} (X(x_j; \theta_i) - \mu_X(x_j))^2} \quad (4)$$

In the paper, we also use notations $\mu_X$ and $\sigma_X$ without precising the abscissa. In this case, these are the spatial means of $\mu_X(x)$ and $\sigma_X(x)$.

In the following, the term “expectation” will be used to designate the expectation of a probabilistic law.

We also need to define the correlation coefficient

$\rho_{X,Y}$ of two particular vectors

$$X(x) = (X(x; \theta_i))_{i \in \{1, \dots, n_S\}} \quad \text{and}$$

$$Y(x + \Delta x) = (Y(x + \Delta x; \theta_i))_{i \in \{1, \dots, n_S\}} \text{ as :}$$

$$\rho_{X;Y}(x; x + \Delta x) = \frac{1}{n_S} \frac{\sum_{i=1}^{i=n_S} (X(x; \theta_i) - \mu_X(x))(Y(x + \Delta x; \theta_i) - \mu_Y(x + \Delta x))}{\sigma_X(x) \sigma_Y(x + \Delta x)} \quad (5)$$

From this equation, we can compute 2 different kinds of correlations. The first one is the spatial correlation. It corresponds to the mean of the different values obtained for different abscissa $x_j$ and using $X=Y$. We finally have:

$$\rho_X(\Delta x) = \frac{1}{N - N_{\Delta x} + 1} \sum_{k=1}^{k=N-N_{\Delta x}+1} \rho_{X;X}(x_k; x_k + \Delta x) \quad (6)$$

where $N_{\Delta x}$ is the number of measurements on length $\Delta x$.

The second correlation coefficient is the coefficient between 2 variables at a given abscissa. It is given by

$$\psi_{X;Y}(x) = \rho_{X;Y}(x; x) \quad (7)$$

The process can be divided into 4 blocks (figure 2). In the first one, the data of the chosen variable $X$ are filtered. The mean step $h$ between two points of measurement is estimated. To start with, we have $n_t$ trajectories of various lengths. The trajectories are removed under 2 conditions:

• The ones where the step is too variable along the trajectories ;

• The ones where $h$ is too different of the mean step.

In order to simplify the treatment, the studies will be done using the number of points of measurement. The knowledge of this number and the mean step give us immediately the length $L_s$ in meters. The following process is the smoothing of the trajectories by overwriting the aberrant measures. We finally obtain $n_f$ trajectories filtered,
smoothed and truncated at a length $L_f$. This length
is the shortest of the $n_f$ trajectories.

In the second block, some trajectories can also be
removed if there are some irregularities in the
evolutions of the mean and the standard deviation.
We plot the evolution of the mean $\mu_X(x)$
(equation (3)) and the standard deviation $\sigma_X(x)$
(equation (4)) from $n_f$ trajectories. We finally
obtain $n_s$ trajectories of length $L_s$.

In the third block, we analyze the one-dimensional
field $X(x; \theta)$ where $X$ is the analyzed parameter, $x$
is the abscissa and $\theta$ is the event and we try to
answer to the following questions:

* Is it a second order stationary?

* Does it have a length named “correlation length” denoted $L_c$ for which the measures are independent?

In the fourth block, we now consider $n_e$ trajectories that correspond to the addition of the $n_s$ previously considered and some other trajectories that had been removed because they were too short. If a correlation length $L_c$ exists, we can build some new distribution by merging the values of the variable $X$ at the abscissa $x$ with the values at the abscissa $x+L_c$. We then look for the most adapted probabilistic law to represent the obtained distribution. The choice of this law is made among the Gamma law, the normal law and the log-normal law by calculating and using the maximum likelihood.

# 3 RESULTS ON THE CORRELATIONS AND DISTRIBUTION

## 3.1 PRE-PROCESSING OF THE DATABASE

We initially have $n_f=50$ trajectories. Each of them
has a step of measurement $h_i$ and a length which
are all different.

During the first step of the pre-processing of data,
we only keep the trajectories which have a step of
measurement near the mean value $h$ in order to
avoid a re-sampling process for the purpose of
spatial study.

We plot the length of the trajectories as a function
of the step number of measurement on the figure
3.

<figure><img src="image_4.png" /><figcaption>Figure 3: Plotting of the length of the trajectories in function of step number of measurement</figcaption></figure>

Many trajectories plotted have a mean step of
measurement nearly equal to the mean step $h$. To
decide which trajectories will be kept, we estimate
the mean step of each trajectory. First, we observe
two of them with a mean step too different of the
others and few measures. Those two trajectories
are removed before all analysis.

Only the trajectories with a mean step $h$ near 6mm
are kept. We finally obtain $n_s=n_f=22$ trajectories
with a mean step $h=6,2mm$ with a mean
coefficient of variation equal to 41.4%. The
analysis will be done on a length of $L_s=3.62m$.

## 3.2 ANALYSIS OF THE CORRELATION BETWEEN PARAMETERS

Before beginning the analysis of the random
fields, we first verify if the physical coherence of
the given database. To this purpose, a study on the
correlation between fields is made (equation (7)).
Preliminary to this study, we first assess the
correlation for two different cases: a first one
between fields that are presumably correlated and
smoothed and truncated at a length $L_f$. This length
is the shortest of the $n_f$ trajectories.

In the second block, some trajectories can also be
removed if there are some irregularities in the
evolutions of the mean and the standard deviation.
We plot the evolution of the mean $\mu_X(x)$
(equation (3)) and the standard deviation $\sigma_X(x)$
(equation (4)) from $n_f$ trajectories. We finally
obtain $n_s$ trajectories of length $L_s$.

In the third block, we analyze the one-dimensional
field $X(x; \theta)$ where $X$ is the analyzed parameter, $x$
is the abscissa and $\theta$ is the event and we try to
answer to the following questions:

* Is it a second order stationary?

* Does it have a length named “correlation length” denoted $L_c$ for which the measures are independent?

In the fourth block, we now consider $n_e$
trajectories that correspond to the addition of the
$n_s$ previously considered and some other
trajectories that had been removed because they
were too short. If a correlation length $L_c$ exists, we
can build some new distribution by merging the
values of the variable $X$ at the abscissa $x$ with the
values at the abscissa $x+L_c$. We then look for the
most adapted probabilistic law to represent the
obtained distribution. The choice of this law is
made among the Gamma law, the normal law and
the log-normal law by calculating and using the
maximum likelihood.

# 3 RESULTS ON THE CORRELATIONS AND DISTRIBUTION

## 3.1 PRE-PROCESSING OF THE DATABASE

We initially have $n_f=50$ trajectories. Each of them
has a step of measurement $h_i$ and a length which
are all different.

During the first step of the pre-processing of data,
we only keep the trajectories which have a step of
measurement near the mean value $h$ in order to
avoid a re-sampling process for the purpose of
spatial study.

We plot the length of the trajectories as a function
of the step number of measurement on the figure
3.

<figure><img src="image_4.png" /><figcaption>Figure 3: Plotting of the length of the trajectories in function of step number of measurement</figcaption></figure>

Many trajectories plotted have a mean step of
measurement nearly equal to the mean step $h$. To
decide which trajectories will be kept, we estimate
the mean step of each trajectory. First, we observe
two of them with a mean step too different of the
others and few measures. Those two trajectories
are removed before all analysis.

Only the trajectories with a mean step $h$ near 6mm
are kept. We finally obtain $n_s=n_f=22$ trajectories
with a mean step $h=6,2mm$ with a mean
coefficient of variation equal to 41.4%. The
analysis will be done on a length of $L_s=3.62m$.

## 3.2 ANALYSIS OF THE CORRELATION BETWEEN PARAMETERS

Before beginning the analysis of the random
fields, we first verify if the physical coherence of
the given database. To this purpose, a study on the
correlation between fields is made (equation (7)).
Preliminary to this study, we first assess the
correlation for two different cases: a first one
between fields that are presumably correlated and
a second between fields that are not correlated. Doing so, it will provide us an estimation of the correlation value for which we can assume that fields are not correlated.

The first case is that of known high correlation. We analyze the correlation between the area of the welded joint Area and the product between the height and the width of the joint S1*S2 and denoted $\psi_{Area,S1*S2}$. The second one is the case of low correlation between the overlap O1 and the height of the joint S1 and denoted $\psi_{O1,S1}$. The evolutions are given in figure 4 and 5.

We can observe a high correlation (over 0.9) and a weak noise for the correlation coefficient $\psi_{Area,S1*S2}$ in figure 4, which is in accordance with a functional relationship between the fields.

<figure><img src="image_5.png" /><figcaption>Figure 4: Analysis of the case of high correlation</figcaption></figure>

On the contrary, we can observe a low mean value of the correlation (over -0.1) between the height of the joints and the overlap. This could be due to the great noise or to the hardness of measurement of one of these parameters (O1 in our example).

<figure><img src="image_6.png" /><figcaption>Figure 5: Analysis of the case of low correlation</figcaption></figure>

## 3.3 ANALYSIS OF THE CORRELATIONS

Once we have limited the length of measurement, kept only the most homogeneous trajectories and verified the coherence of the database, we can analyze the trajectories retained.

This part of the analysis corresponds to the third block of the organigram (figure 2). We try to see if the hypothesis of the stationary of the second order is verified. We also search if a correlation length $L_c$ exists, distance that could be used to build some new samples with values with low correlation (following Bravais-Pearson). These values could therefore be considered as independent.

In order to validate the stationary hypothesis, some equalities given below have to be verified:

$$ \begin{gathered} \forall x, x', \mu_X(x) = \mu_X, \sigma_X(x) = \sigma_X, \\ \rho_{X;X}(x, x') = \rho_X(x'-x) \end{gathered} \qquad (8) $$

We plot in figures 6 and 7 the different evolutions of mean, standard deviation and correlation coefficient with the length.
a second between fields that are not correlated. Doing so, it will provide us an estimation of the correlation value for which we can assume that fields are not correlated.

The first case is that of known high correlation. We analyze the correlation between the area of the welded joint Area and the product between the height and the width of the joint S1*S2 and denoted $\psi_{Area,S1*S2}$. The second one is the case of low correlation between the overlap O1 and the height of the joint S1 and denoted $\psi_{O1,S1}$. The evolutions are given in figure 4 and 5.

We can observe a high correlation (over 0.9) and a weak noise for the correlation coefficient $\psi_{Area,S1*S2}$ in figure 4, which is in accordance with a functional relationship between the fields.

<figure><img src="image_5.png" /><figcaption>Figure 4: Analysis of the case of high correlation</figcaption></figure>

On the contrary, we can observe a low mean value of the correlation (over -0.1) between the height of the joints and the overlap. This could be due to the great noise or to the hardness of measurement of one of these parameters (O1 in our example).

<figure><img src="image_6.png" /><figcaption>Figure 5: Analysis of the case of low correlation</figcaption></figure>

## 3.3 ANALYSIS OF THE CORRELATIONS

Once we have limited the length of measurement, kept only the most homogeneous trajectories and verified the coherence of the database, we can analyze the trajectories retained.

This part of the analysis corresponds to the third block of the organigram (figure 2). We try to see if the hypothesis of the stationary of the second order is verified. We also search if a correlation length $L_c$ exists, distance that could be used to build some new samples with values with low correlation (following Bravais-Pearson). These values could therefore be considered as independent.

In order to validate the stationary hypothesis, some equalities given below have to be verified:

$$ \begin{gathered} \forall x, x', \mu_X(x) = \mu_X, \sigma_X(x) = \sigma_X, \\ \rho_{X;X}(x, x') = \rho_X(x'-x) \end{gathered} \qquad (8) $$

We plot in figures 6 and 7 the different evolutions of mean, standard deviation and correlation coefficient with the length.
<figure><img src="image_7.png" /><figcaption>Figure 6: Analysis of mean and standard deviation of the field S1</figcaption></figure>

We can observe that the values of mean and standard deviation are constant with respect to the abscissa $x$. Indeed, we can note $\mu_{S1} = 15.00$ and its coefficient of variation with respect to $x$ is only 1.1%. Concerning the standard deviation, the mean value is $\sigma_{S1} = 0.78$ with a coefficient of variation of 16%. Having only $n_s=22$ trajectories for the analysis, we can consider that the standard deviation is also constant with the abscissa.

Now, we decide to analyze the spatial correlation coefficient (equation (6)). We obtain the presented curve on figure 7. The bold curve corresponds to the evolution of the mean of the coefficient, the two thinner curves around the bold one correspond to the coefficient with more or less one standard deviation.

<figure><img src="image_8.png" /><figcaption>Figure 7: Spatial correlation coefficient of S1</figcaption></figure>

The coefficient is decreasing until a length equal to 1.5m. This evolution helps us conclude that the stationary hypothesis of the second order can be validated. Nevertheless, after the decrease of the coefficient until a value $\rho=0.3$ at $\Delta x=1.5m$, an

increase of the coefficient is observed. This latter phenomenon is not explained in this paper, but some complementary studies are currently in advancement.

To determine the distance $L_c$ for which the values of $X$ are non-correlated, we first have to estimate the threshold value of correlation for which the hypothesis of uncorrelation is valid. As mentioned in section 3.2, we choose two physically non-correlated variables. In this case, we consider the variables S1 and O1 previously presented and we give the repartition of the value of the correlation between these two variables $\psi_{S1,O1}$ in figure 8. Although the correlation should be equal to 0 in theory, in practice the deviation with this theoretical value is important.

<figure><img src="image_9.png" /><figcaption>Figure 8: Distribution and approach with a normal law of the repartition of the correlation between O1 and S1</figcaption></figure>

By approaching this distribution with a normal law with the maximum likelihood estimator, the parameters of the law are $E=0.13$ for the expectation and $\sigma=0.22$ for the standard deviation. Considering the confidence interval at 95% ($[E - 2\sigma; E + 2\sigma]$ for a normal law), the interval of non-correlation is over $[-0.5; 0.3]$. Finally non correlation is assumed for $|\rho(\Delta x)| \le 0.3$. Looking at figure 7, we can therefore consider a correlation distance $L_c=1.5m$.

## 3.4 ANALYSIS OF THE RE-SAMPLINGS
<figure><img src="image_7.png" /><figcaption>Figure 6: Analysis of mean and standard deviation of the field S1</figcaption></figure>

We can observe that the values of mean and standard deviation are constant with respect to the abscissa $x$. Indeed, we can note $\mu_{S1} = 15.00$ and its coefficient of variation with respect to $x$ is only 1.1%. Concerning the standard deviation, the mean value is $\sigma_{S1} = 0.78$ with a coefficient of variation of 16%. Having only $n_s=22$ trajectories for the analysis, we can consider that the standard deviation is also constant with the abscissa.

Now, we decide to analyze the spatial correlation coefficient (equation (6)). We obtain the presented curve on figure 7. The bold curve corresponds to the evolution of the mean of the coefficient, the two thinner curves around the bold one correspond to the coefficient with more or less one standard deviation.

<figure><img src="image_8.png" /><figcaption>Figure 7: Spatial correlation coefficient of S1</figcaption></figure>

The coefficient is decreasing until a length equal to 1.5m. This evolution helps us conclude that the stationary hypothesis of the second order can be validated. Nevertheless, after the decrease of the coefficient until a value $\rho=0.3$ at $\Delta x=1.5m$, an

increase of the coefficient is observed. This latter phenomenon is not explained in this paper, but some complementary studies are currently in advancement.

To determine the distance $L_c$ for which the values of $X$ are non-correlated, we first have to estimate the threshold value of correlation for which the hypothesis of uncorrelation is valid. As mentioned in section 3.2, we choose two physically non-correlated variables. In this case, we consider the variables S1 and O1 previously presented and we give the repartition of the value of the correlation between these two variables $\psi_{S1,O1}$ in figure 8. Although the correlation should be equal to 0 in theory, in practice the deviation with this theoretical value is important.

<figure><img src="image_9.png" /><figcaption>Figure 8: Distribution and approach with a normal law of the repartition of the correlation between O1 and S1</figcaption></figure>

By approaching this distribution with a normal law with the maximum likelihood estimator, the parameters of the law are $E=0.13$ for the expectation and $\sigma=0.22$ for the standard deviation. Considering the confidence interval at 95% ($[E - 2\sigma; E + 2\sigma]$ for a normal law), the interval of non-correlation is over $[-0.5; 0.3]$. Finally non correlation is assumed for $|\rho(\Delta x)| \le 0.3$. Looking at figure 7, we can therefore consider a correlation distance $L_c=1.5m$.

## 3.4 ANALYSIS OF THE RE-SAMPLINGS
Once having validated the second order stationary hypothesis and determined the length $L_c$ for which the hypothesis of non-correlation between values is acceptable, we can identify a probabilistic law to represent the distributions using the maximum likelihood.

The length of measurement $L_s$ being much longer than the correlation length $L_c$, we can merge 22 values at abscissas $x=0.46m$, $x=1.96m$ and $x=3.46m$ thus providing 66 samples. The corresponding distribution of S1 is depicted in figure 9.

<figure><img src="image_10.png" /><figcaption>Figure 9: plotting of three chosen laws on the distribution of S1</figcaption></figure>

As told earlier in section 2, we identify three different laws, the Gamma law, the normal law and the log-normal law. The corresponding values of expectation, standard deviation and maximum of likelihood are summarized in table 1.

<table><caption>Table 1: Values of expectation, standard deviation and maximum likelihood for each law</caption><thead><tr><th></th><th><i>E</i></th><th><i>σ</i></th><th>Maximum likelihood</th></tr></thead><tbody><tr><td>Gamma</td><td>14.92</td><td>0.76</td><td>-75.61</td></tr><tr><td>Normal</td><td>14.92</td><td>0.77</td><td>-75.38</td></tr><tr><td>Log-normal</td><td>14.92</td><td>0.77</td><td>-76.17</td></tr></tbody></table>

Although the values of maximum of likelihood are similar, we retain the log-normal law as the best of the three laws to approach the distribution given by the 66 values of S1. This choice is only based on the value of maximum log-likelihood and allows only positive values of the geometrical parameter S1.

## 4 CONCLUSION

The measurement of the geometry of welded joints is made easier thanks to devices using laser process. In particular, the device WISC is used for the measurement in this article. Some errors in the measurement or some experimental difficulties are present in the given data. These problems come in addition of the randomness on the geometrical parameters that appear during the welding process. In this paper, we have to use statistical processing method using different steps to achieve the analysis. But, for this purpose, we can only use less of the half of the given database. This hypothesis allows us to increase the number of samples by re-sampling using the correlation length.

## ACKNOWLEDGMENTS

Authors wish to thank Direction Générale de l'Armement for financing these works by the Programme d'Etudes Amont COMENAV (market n°2007 90 0907)

## REFERENCES

1. [Huther I.], [1996] ‘[Prévision du comportement à la fatigue des assemblages soudés – prise en compte de l’effet macro, micro-géométrique]’, [Mémoire de thèse de l'Université Blaise Pascal (Clermont-Ferrand 2), soutenue le 11 juin 1996 à Clermont-Ferrand]

2. [Vanmarcke E.], [1983] ‘[Random fields, analysis and synthesis]’, [The MIT Press Cambridge, 1983]
Once having validated the second order stationary hypothesis and determined the length $L_c$ for which the hypothesis of non-correlation between values is acceptable, we can identify a probabilistic law to represent the distributions using the maximum likelihood.

The length of measurement $L_s$ being much longer than the correlation length $L_c$, we can merge 22 values at abscissas $x=0.46m$, $x=1.96m$ and $x=3.46m$ thus providing 66 samples. The corresponding distribution of S1 is depicted in figure 9.

<figure><img src="image_10.png" /><figcaption>Figure 9: plotting of three chosen laws on the distribution of S1</figcaption></figure>

As told earlier in section 2, we identify three different laws, the Gamma law, the normal law and the log-normal law. The corresponding values of expectation, standard deviation and maximum of likelihood are summarized in table 1.

<table><caption>Table 1: Values of expectation, standard deviation and maximum likelihood for each law</caption><thead><tr><th></th><th><i>E</i></th><th><i>σ</i></th><th>Maximum likelihood</th></tr></thead><tbody><tr><td>Gamma</td><td>14.92</td><td>0.76</td><td>-75.61</td></tr><tr><td>Normal</td><td>14.92</td><td>0.77</td><td>-75.38</td></tr><tr><td>Log-normal</td><td>14.92</td><td>0.77</td><td>-76.17</td></tr></tbody></table>

Although the values of maximum of likelihood are similar, we retain the log-normal law as the best of the three laws to approach the distribution given by the 66 values of S1. This choice is only based on the value of maximum log-likelihood and allows only positive values of the geometrical parameter S1.

## 4 CONCLUSION

The measurement of the geometry of welded joints is made easier thanks to devices using laser process. In particular, the device WISC is used for the measurement in this article. Some errors in the measurement or some experimental difficulties are present in the given data. These problems come in addition of the randomness on the geometrical parameters that appear during the welding process. In this paper, we have to use statistical processing method using different steps to achieve the analysis. But, for this purpose, we can only use less of the half of the given database. This hypothesis allows us to increase the number of samples by re-sampling using the correlation length.

## ACKNOWLEDGMENTS

Authors wish to thank Direction Générale de l'Armement for financing these works by the Programme d'Etudes Amont COMENAV (market n°2007 90 0907)

## REFERENCES

1. [Huther I.], [1996] ‘[Prévision du comportement à la fatigue des assemblages soudés – prise en compte de l’effet macro, micro-géométrique]’, [Mémoire de thèse de l'Université Blaise Pascal (Clermont-Ferrand 2), soutenue le 11 juin 1996 à Clermont-Ferrand]

2. [Vanmarcke E.], [1983] ‘[Random fields, analysis and synthesis]’, [The MIT Press Cambridge, 1983]
3. [Schoefs F., Chevreuil M., Clément A., Cazuguel M., Pasqualini O.], [2010] '[Reliability of marine welded joints subjected to stochastic geometry : from measure to computation]', [*Proceeding of 5th Conference on Network for Integrating Structural Analysis, Risk and Reliability*, 7 pages, 14-16 June 2010, Edinburgh, Scotland, UK]

4. [Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M.], [2011] '[Analyse de la variabilité spatiale géométrique pour la fiabilité des cordons de soudure]', [*20ème Congrès Français de Mécanique – Incertitudes, Fiabilité et Maîtrise des Risques*, 5 pages, 28 Août au 2 Septembre 2011, Besançon, France]
3. [Schoefs F., Chevreuil M., Clément A., Cazuguel M., Pasqualini O.], [2010] '[Reliability of marine welded joints subjected to stochastic geometry: from measure to computation]', [*Proceeding of 5th Conference on Network for Integrating Structural Analysis, Risk and Reliability*, 7 pages, 14-16 June 2010, Edinburgh, Scotland, UK]

4. [Pasqualini O., Schoefs F., Chevreuil M., Cazuguel M.], [2011] '[Analyse de la variabilité spatiale géométrique pour la fiabilité des cordons de soudure]', [*20ème Congrès Français de Mécanique – Incertitudes, Fiabilité et Maîtrise des Risques*, 5 pages, 28 Août au 2 Septembre 2011, Besançon, France]