A1_2_JAFOCorreia

DURABLE STRUCTURESLNEC Lisbon 31 May - 1June 2012

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J. A. F. O. Correia1, A. M. P. De Jesus1 and A. Fernández‐Canteli2

1 Unit of Design and Experimental Validation, IDMEC, LAETA, Engineering Department, School of Sciences and Technology

University of Trás‐os‐Montes and Alto Douro, 5001‐801 Vila Real, Portugal.

E‐mail: [email protected]; [email protected]

2 Department of Construction and Manufacturing Engineering, University of Oviedo, 33203 Campus Viesques, Gijón, Spain.

E‐mail: [email protected]

PROBABILISTIC FATIGUE BEHAVIOUR OF A STRUCTURAL DETAIL OF PUDDLE IRON FROM THE EIFFEL BRIDGE


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1. INTRODUCTION

2. OVERVIEW OF THE DETERMINISTIC UNIGROWMODEL

3. PROBABILISTIC SWT‐N FIELDS

4. PROCEDURE TO GENERATE P‐S‐N‐R FIELDS

5. PROBABILISTIC FATIGUE DATA OF THE PUDDLE IRON FROM THE EIFFEL(VIANA) BRIDGE

6. PREDICTION OF PROBABILISTIC S‐N FIELDS FOR A NOTCHED DETAIL

7. CONCLUSIONS

ACKNOWLEDGMENTS

ICDS12 International Conference DURABLE STRUCTURES: from construction to rehabilitation LNEC • Lisbon • Portugal • 31 May - 1 June 2012

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1. INTRODUCTION (1/2)

• Probabilistic fatigue models are required to account conveniently for theseveral sources of uncertainty arising in the prediction procedures, such as thescatter in material behaviour.

• In this paper, a probabilistic approach is proposed to generate S‐N curves forstress ratio R=0, applied to a structural detail of puddle iron from the EiffelBridge, using local approaches based on probabilistic SWT‐N fields, in order tomodel both fatigue crack initiation and fatigue crack propagation.


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1. INTRODUCTION (2/2)

• This study suggests a progress of the model proposed by Noroozi et al. to modelfatigue crack propagation, based on the local strain approach to fatigue, todetermine the crack propagation for a structural detail, and is applied to deriveprobabilistic fatigue crack propagation fields (P‐S‐Np‐R fields) for a notcheddetail, for distinct stress R‐ratios.

• The probabilistic fatigue crack initiation fields (P‐S‐Ni‐R fields), for notched plate,are determined using an elastoplastic approach together with the P‐SWT‐N fieldsto calculate the fatigue damage of the first elementary material block.

• The global prediction (crack initiation and propagation) of the probabilistic S‐Nfields is presented as a unified approach. The predictions are compared withavailable experimental S‐N fatigue data for the notched plate underconsideration.


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2. OVERVIEW OF THE DETERMINISTIC UNIGROWMODEL (1/3)The UniGrow model was proposed by Noroozi et al. based on

the following assumptions:

• The material is composed of simple particles of a finitedimension * that represents the elementary materialblock size, below which material cannot be regarded as acontinuum.

• The fatigue crack tip is considered equivalent to a notchwith radius *.

• The fatigue crack growth process is considered as beingsuccessive crack increments due to crack re‐initiations overthe distance *.

• The fatigue crack growth rate can be determined as:

• This equation may be used in an inverse way, to estimatethe * parameter, using constant‐amplitude fatigue crackgrowth data for a material at various R‐ratios.

Noroozi, AH, Glinka, G, and Lambert S., International Journal of Fatigue 2005; 27: 1277‐96.


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2. OVERVIEW OF THE DETERMINISTIC UNIGROWMODEL (2/3)

The failure of the elementary material block may be computed using the Smith‐Watson‐Topper (SWT) relation:

where max and are the average values of the maximum elastoplastic stress andstrain range, through the first elementary material block ahead the crack tip.

This work suggests an extension of the model proposed by Noroozi et al. to modelfatigue crack propagation:

• The stress intensity factors are determined for the detail under investigationbased on a finite element analysis, using the J‐integral method. Then, the elasticstress fields are estimated ahead of the crack tip.

• The actual elastoplastic stresses and strains, are computed using the Neuber’sapproach.

• The residual stress distribution ahead the crack tip is computed using the actualelastoplastic stresses computed at the end of the first load reversal andsubsequent cyclic elastoplastic stress range, σr=σmax‐σ.


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2. OVERVIEW OF THE DETERMINISTIC UNIGROWMODEL (3/3)This work suggests an extension of the model proposed by Noroozi et al. to model fatigue

crack propagation (continuation):

• The residual stress intensity factor, Kr, is computed using the weight function methodaccording to the following general expression:

• The weight function was found using the following expression:

• The applied stress intensity factor (maximum and range values) is corrected using theresidual stress intensity value, resulting the total values, Kmax,tot and Ktot. For positiveapplied stress ratios, Kmax,tot and Ktot may be computed as follows:

where Kr takes a negative value corresponding to the compressive stress field.

• Using the total values of the stress intensity factors, the first and second steps above arerepeated to determine the corrected values for the maximum actual stress and actualstrain range at the material representative elements.


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3. PROBABILISTIC SWT‐N FIELDS (1/2)

• The UniGrow model needs a fatigue damage law to compute the number of cyclesto fail the elementary material block.

• In this study, a probabilistic fatigue model is proposed rather than thedeterministic SWT‐Nmodel.

• The probabilistic SWT‐N model is developed as an extension of the P‐ε‐N modelproposed by Castillo and Fernández‐Canteli.

• The SWT (=σmax.εa) parameter was proposed by Smith‐Watson‐Topper to take intoaccount the mean stress effects on fatigue life. Any combination of maximumstress and strain amplitude that leads to the same SWT parameter predicts thesame fatigue life. The SWT‐N and εa‐N fields exhibit similar characteristics.


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3. PROBABILISTIC SWT‐N FIELDS (2/2)

where p is the probability of failure, N0 andSWT0 are normalizing values, and λ, δ and βare the non‐dimensional Weibull modelparameters. Their physical meanings are:N0: threshold value of lifetime;SWT0: fatigue limit of SWT;λ: parameter defining the position of thecorresponding zero‐percentile curve;δ: scale parameter;β: shape parameter.

• Therefore the P‐ε‐N field proposed by Castillo and Fernández‐Cantelimay be extended to represent the P‐SWT‐N field:

• The parameters log N0 and log SWT0 of the P‐SWT‐N model may be estimated usingthe least squares method. The Weibull parameters are estimated by the maximumlikelihood method.


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4. PROCEDURE TO GENERATE P‐S‐N‐R FIELDS (1/3)

PREDICTION OF PROBABILISTIC S‐Ni FIELDS FOR A STRUCTURAL DETAIL


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PREDICTION OF PROBABILISTIC S‐Np

FIELDS FOR A STRUCTURAL DETAIL


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GLOBAL PREDICTION OF PROBABILISTIC S‐N FIELDS FOR A STRUCTURAL DETAIL


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5. PROBABILISTIC FATIGUE DATA OF THE PUDDLE IRON FROMTHE EIFFEL (VIANA) BRIDGE (1/2)

The elastic, static strength, cyclic and strain‐life properties of the puddle iron from theEiffel Bridge are summarized in Tables 1 and 2.

Table 1

Table 2

A notched detail consisting plate with a circular symmetric hole, as illustrated in Figure,was considered in this investigation.


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5. PROBABILISTIC FATIGUE DATA OF THE PUDDLE IRON FROMTHE EIFFEL (VIANA) BRIDGE (2/2)

This geometry was fatigue tested underremote stress controlled conditions, forstress ratio, R=0.0.

The probabilistic SWT‐N model and therespective parameters identified for thismaterial are applied to model both crackinitiation and crack propagation lives of anotched detail.


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6. PREDICTION OF PROBABILISTIC S‐N FIELDS FOR A NOTCHED DETAIL (1/10)

The P‐SWT‐N model is used to model the fatigue crack initiation (failure of the firstelementary material block) at the notch root of the detail. Since the detail was testedunder elastoplastic conditions, elastoplastic stress analysis must be used to assess the realstress conditions at the notch root.

An elastoplastic stress analysis is proposed based on Seeger‐Heuler’s model, which is ageneralization of the Neuber’s model. This model is used in conjunction with the Ramberg‐Osgood description of the material. The initial loading is modelled using appropriatemonotonic properties of the material. The following set of equations was used to describethe monotonic loading:

Modelling fatigue crack initiation


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Modelling fatigue crack initiation (continuation)

For cyclic loading, the following equations were used to assess the local stress range:

Seeger‐Heuler’s model are used to compute the maximum stress and the stress range atthe notch root, respectively, and relate them to the nominal or remote stresses, appliedto the notch.

Using the probabilistic SWT‐N model for the puddle iron from the Eiffel Bridge, it ispossible to relate the nominal or remote stresses applied to the notched detail, withthe number of cycles to failure, for a given probability, p.


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The notch detail shows an elastic stress concentration factor, kt, equal to 2.43, which is thelimit value of the fatigue notch reduction factor to be used in the elastoplastic fatigueanalysis.


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The fatigue reduction factor, kf, was calibrated in order the resulting P‐S‐Ni field to give asatisfactory description of all available experimental fatigue data. The resulting fatiguenotch reduction factor is equal to 1.45. The fatigue reduction factor effects were onlyaccounted in modelling fatigue crack initiation. Figure shows the P‐S‐Ni fieldscorresponding to the fatigue crack initiation (dominating fatigue damage) for the notcheddetail, using the P‐SWT‐Nmodel for R=0.0.


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6. Prediction of probabilistic S‐N fields for a notched detail (5/10)

The fatigue crack propagation modelling was supported by a bi‐dimensional finiteelement model of the notched detail, which was used to compute both stress intensityfactor and residual elastoplastic stress distribution.

Taking into account the existing symmetry plane, only half of the geometry is modelled.Plane stress quadratic triangular elements were used in the analysis due to the limitedspecimen thickness.

The stress intensity factors were determined based on a linear‐elastic finite elementanalysis using the J‐integral method.

Concerning the residual elastoplastic stress computation, a highly refined mesh at thecrack tip region was used in order to model the crack tip notch radius, *. The residualstress intensity factor, Kr, was determined using the weight function method.

A Von Mises yield model, with multilinear kinematic hardening, was used in simulationsusing ANSYS® 12.0 code. The plasticity model was fitted to the cyclic curve of thematerial.

Modelling fatigue crack propagation


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Bi‐dimensional finite element model ‐ residual elastoplastic stress distribution


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Bi‐dimensional finite element model ‐ stress intensity factor and weight function


22Stress intensity evolution with the crack length, for a unitremote stress, which was used to determine the Kapplied.

Weight functions used to determine the Kr fornotched detail, for the distinct crack lengths.

The P‐S‐Np field was predicted using the P‐SWT‐N model of puddle iron from the EiffelBridge together with the UniGrow model proposed by Noroozi et al. to structural detail,for R=0, and using the * determined by José A.F.O. Correia et al (Correia JAFO, Jesus AMP,

Fernández‐Canteli A. A procedure to derive probabilistic fatigue crack propagation data”. International Journal

Structural Integrity, ISSN 1757‐9864, Issue 2, Vol. 3, 2012, p. 158‐183).




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Evolution of Kr with the applied stress intensity factor range




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Global S‐N field predictions (crack initiation and crack propagation S‐N field)

Fatigue crack propagation had a small contribution to the total fatigue life for this detail.



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Conclusions

The global P‐S‐N field prediction for the notched detail (R=0.0), taking into accountthe fatigue crack initiation and propagation, shows satisfactory results.

The adaptation of the UniGrow model allows us to reproduce satisfactorily crackpropagation prediction using residual compressive stress estimation, based onelastoplastic finite element analysis of the notched detail, and the P‐SWT‐N damagemodel.

In this study, crack initiation is the dominating damaging process, while the fatiguecrack propagation represents a small influence on global predictions of the P‐S‐Nfield. However its importance increases with the fatigue life, enhancing the slope ofthe P‐S‐N field.

The procedure proposed to derive the probabilistic S‐N curves for structural detailsproved to be quite efficient, since it can be used to reduce the need for extensivetesting and takes into account material variability.


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ACKNOWLEDGMENTS

The authors acknowledge the Portuguese Science Foundation (FCT) for thefinancial support through the doctoral grant SFRH/BD/66497/2009.

José Correia

E‐mail: [email protected]

http://www.utad.pt/~jcorreia

http://www.utad.pt/~pontesfct


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