Probabilistic approaches in evaluating structural fatigue
Probabilistic approaches in evaluating structural fatigue Kesireddy Rakesh Reddy 001086539
Problem Statement:
The evaluation of fatigue failure is basic and very important in design structures, because it is becoming determinant and cause of failure of high number of mechanical components and structural elements in real practice. Fatigue failure is due to progressive propagation of flaws in steel under cyclic loading. Analyzing a fatigue failure in structure is a highly complex task due to the significant number of parameters associated with it. Fatigue is a many facetted phenomenon influenced by many diverse factors as surface condition, temperature, applied stress, geometry, crack initiation, crack width, geometry, internal voids in material, manufacturing technology internal defects, material properties et al. Such a varied complex model is analyzed in mechanics by considering many assumptions. All these factors have an enormous statistical variation in their determined magnitude when undergoing fatigue process. These variations in parametric values has led to the development of the reliability analysis techniques for predicting the useful life of the component. Fatigue failure, as an ultimate limit state, requires a probabilistic framework allowing relating damage levels to probabilities of occurrence (Castillo et al., 2007c). Due to the presence of significant involvement of intrinsic randomness and uncertainty in relevant variables inherent to fatigue phenomenon, and due to their effect on the fatigue safety margin, probabilistic approaches are used to calculate the fatigue failure taking into the account of the effect of almost all deign parameters.
Background:
The probabilistic approach of the fatigue failure model have demonstrated the ability to deliver more promising prediction. Classic S-N (stress-number of cycles) curves, strain-N curves and crack growth curves which are modelled using Linear elastic fracture mechanics model (LEFM) were used for fatigue loaded structures, and a Weibull or Gumbel distribution are used to postulate the probability models for the structures. Most of these S-N curves were modelled in laboratory and are available for various material types. The Gumbel –Wohler field, is taken in all these approaches. When fatigue tests have been with alternating maximum and minimum load on specimens, the obtained stress data is fitted using Gumbel distribution. These stress based curves are also called High cycle fatigue (HCF) models. These curves developed for various materials are reference, to provide mean lifetime N as a function of stress range for constant stress level. A family of percentiles S-N curves are formulated with fatigue life as a random variable. When local plastic deformations are present during the fatigue process, strain based curves are applied. These strain based approach needs ε-N curves, which represents mean lifetime N as a function of strain range. Instead of a single ε-N curve (mean strain), a family of percentiles ε-N curves are formulated with fatigue life as a random variable. Initially probabilistic S-N curves are developed using stress or strain based for the material. Based on the model proposed by Noroozi, we can compute p-S-N-R curves can be computed for the crack propagation in structures. Where p is the probability of failure, S is the mean stress, N number of cycles required for element to failure, R is the stress ratio. All these methods available for the reliability assessment can be broadly classified into two types. The first one is the frequency-domain approaches [3–6], modeling the stress-time variability with a random process defined by its Power Spectral Density (PSD) allowing, with a reduced mechanical time evaluation, to assess the expectation of the time to failure using closed-form expressions. This approach is very interesting and very efficient in the design stage. The second approach is the Stress–strength approach. It is a more general time-domain method based on probabilistic concepts for designing new structures with a reliability objective or assessing the failure probability of already designed structures. It consists of the comparison of two PDFs, the Stress (S) and the Strength (R) of the structure. This engineering approach is more general and very convenient to use.
Stress probability curves
Strain probability curves
Crack length curves
Regular S-N curves
Probabilistic Approach:
As defined earlier fatigue is a many facetted problem. Realistic probabilistic methods are used to define the characteristic response of an entire structure. Such a model to be accurate we need to develop and apply engineering models based on understanding the behavior and failure modes and the statistical distributions of the controlling parameters. Probabilistic framework for fatigue assessment closely follows the classical deterministic fatigue failure approach. The end result is the probabilistic definition of the cumulative damage D (f) which is now a random variable. It is possible to compute the mean value, the standard deviation as well as the complete probability density function (PDF) of D (f). Reliability analysis may be performed in order to get the probability that D is greater than 1 as a function of the total number of cycles. The following types Of uncertainties should be taken into account in the analysis: 1. A family of percentiles S-N curves are formulated with fatigue life as a random variable. 2. The data which we obtain is a scattered data, which is assessed as distribution to fit into a curve to obtain certain values. 3. Many uncertainties involved like size effects, surface finish are considered as random variables. The PDF’s of such variables are obtained from literature, and usually are assumed uniform PDF’s. 4. Uncertainties in mechanical model like material parameters, geometrical parameters, are again modelled as random variables using appropriate density functions. Almost all uncertainties raising in regular fracture fatigue model are modelled as pdf’s are accounted on a certain level using probabilistic approaches. So that they are not completely ignored and still have effect on the model failure.
Assumptions/Limitations:
The various approaches used for assessment of failure criteria have many assumptions and certain limitations. 1. The frequency domain approach does not give any idea of the time to failure dispersion essential for the structural probability of failure assessment. 2. The failure probability is very sensitive to the PDFs selected for R and S, in stress(S) – Strength (R) approach, since the number of mechanical evaluations must be well controlled to have an adequate computation time. Also the influence of each random variable on reliability cannot be determined since the uncertain parameters characterizing geometry, material properties and loads are gathered in the Stress PDF. 3. All of the approaches used are based on LEFM, using linear mechanical behavior and the graphs developed based on the same criteria. 4. The PDF’s defining stress and strength are taken independent. 5. The S-N, strain-N curves are developed using carefully selected samples in laboratory, those values may not exactly represent the outside phenomenon. Unknown model parameters are determined by statistical means from the database generated using above procedure. Such a data and its relevance to all problems is highly debatable. 6. Extrinsic conditions like corrosion, environmental degradation, high temperature et al, are not considered as parameters when determining the PDF’s. 7. The stress applied to calculate the S-N fields are constant, but in general it is varying stress.
Future work:
1. The various available statistical models for fatigue failure analysis are based on linear elastic fracture mechanics (LEFM) technique by Inglis and Griffith, Developed by Irwin. It postulates a major assumption that the cracks at which the stress concentrations occur in the material have smooth surfaces, the material for which LEFM method was developed are based on brittle solids with known defects (Inglis theory) following a linear propagation. This is a major assumption and we are using the S-N and crack width curves developed based on these models. However a new theory since has been developed called fractal fracture mechanics (Mandelbrot), which states that the crack surfaces are rough and the stress at such crack tip can cause new modes of fracture and crack propagation which follows bilinear propagation. It has an irregular set of scaling transformations. These surfaces have non regular geometric shapes. Stresses in such models can be normal, radial and hoop stresses, which are formulated using mechanics. Such stress equations can be used to calculate the new S-N curves. These newly developed curves include the real life situation of crack geometry. Crack propagation models are developed and these basis can be used to formulate power law, which can include uncertainties like combining load, stress intensity factors et al. The use of such advanced models to formulate the Probability density functions of operating stress and failure strength to calculate the probabilities of failure of fatigue which are caused by the stresses above. The calculation of the stresses using newly developed formulae on various specimens under different loading, can be used to formulate the PDF’s.
Linear and bilinear crack propagation models
2. The second proposal is the load time graph used to calculate the stresses. The loads are applied as maximum and minimum, a constant load difference is applied again and again to create the effect of fatigue and stresses are calculated. However in real life the load step will not be constant, it will be a varying load function. To simulate such a function we can take a random load response spectrum to create a real life situation in lab. Stresses them can be calculated depending on how the load is being applied. However we have to Step and ramp function here as there is no constant cyclic load to have a clear number of cycles. Instead a response spectrum is used and number of cycles are hypothesized and an S-N curve is formulated. Also the stress curve then generated need not necessarily follow Gumbel distribution an assumed in regular case. A better model is flitted and analyzed.
3. One of the major assumption in fatigue formulation is that the material is homogeneous and follow continuum mechanics, but in real life there are many flaws included in the material such as discontinuities, material differences et al. To accommodate all such flaws a Radom equally distributed Pdf is taken in all the approaches scene so far, considering the
material as equally distributed voids. But flaws in materials can be known by certain NDT methods. Which gives us the distribution of various voids in samples, with which we can formulate a pdf for certain batch sample, produced at a manufacturing unit. Which may not be exact but will be as near as possible as we are accommodating the randomness of voids in to different samples. A similar case can be done with the case of irregularity of material properties and Geometric properties of different samples.
Load response spectrum and the stress following a Gaussian distribution 4. All these uncertainties are considered interdependent, but this may not be the case always as some of them may be independent of each other but still affect the fatigue failure. A study can be proposed to calculate the effect of the various uncertainties contributing t to fatigue. 5. A method to incorporate thermal loading, corrosion et al depending on the environment changes and the conditions prevailing at the various sites can also be postulated as uncertainties and can be incorporated into the model. 6. We can still use different simulation to find better model to define he randomness of uncertainties involved.
References: 1. J. Maljaars, H.M.G.M. Steenbergen, A.C.W.M. Vrouwenvelder, Probabilistic model for fatigue crack growth and fracture of welded joints in civil engineering structures, International Journal of Fatigue, Volume 38, May 2012, Pages 108-117, ISSN 01421123. 2. Y. Prawoto, M.N. Tamin, A new direction in computational fracture mechanics in materials science: Will the combination of probabilistic and fractal fracture mechanics become mainstream?, Computational Materials Science, Volume 69, March 2013, Pages 197-203, ISSN 0927-0256. 3. B. Sudret, Z. Guédé, Probabilistic assessment of thermal fatigue in nuclear components, Nuclear Engineering and Design, Volume 235, Issues 17–19, August 2005, Pages 1819-1835, ISSN 0029-5493. 4. B. Echard, N. Gayton, A. Bignonnet, A reliability analysis method for fatigue design, International Journal of Fatigue, Volume 59, February 2014, Pages 292-300, ISSN 0142-1123. 5. José A.F.O. Correia, Abílio M.P. De Jesus, Alfonso Fernández-Canteli, Local unified probabilistic model for fatigue crack initiation and propagation: Application to a notched geometry, Engineering Structures, Volume 52, July 2013, Pages 394-407, ISSN 0141-0296. 6. Enrique Castillo, Alfonso Fernández-Canteli, María Luisa Ruiz-Ripoll, A general model for fatigue damage due to any stress history, International Journal of Fatigue, Volume 30, Issue 1, January 2008, Pages 150-164, ISSN 0142-1123 7. H. Khezrzadeh, M. Mofid, Tensile fracture behavior of heterogeneous materials based on fractal geometry, Theoretical and Applied Fracture Mechanics, Volume 46, Issue 1, August 2006, Pages 46-56, ISSN 0167-8442.
Problem Statement:
The evaluation of fatigue failure is basic and very important in design structures, because it is becoming determinant and cause of failure of high number of mechanical components and structural elements in real practice. Fatigue failure is due to progressive propagation of flaws in steel under cyclic loading. Analyzing a fatigue failure in structure is a highly complex task due to the significant number of parameters associated with it. Fatigue is a many facetted phenomenon influenced by many diverse factors as surface condition, temperature, applied stress, geometry, crack initiation, crack width, geometry, internal voids in material, manufacturing technology internal defects, material properties et al. Such a varied complex model is analyzed in mechanics by considering many assumptions. All these factors have an enormous statistical variation in their determined magnitude when undergoing fatigue process. These variations in parametric values has led to the development of the reliability analysis techniques for predicting the useful life of the component. Fatigue failure, as an ultimate limit state, requires a probabilistic framework allowing relating damage levels to probabilities of occurrence (Castillo et al., 2007c). Due to the presence of significant involvement of intrinsic randomness and uncertainty in relevant variables inherent to fatigue phenomenon, and due to their effect on the fatigue safety margin, probabilistic approaches are used to calculate the fatigue failure taking into the account of the effect of almost all deign parameters.
Background:
The probabilistic approach of the fatigue failure model have demonstrated the ability to deliver more promising prediction. Classic S-N (stress-number of cycles) curves, strain-N curves and crack growth curves which are modelled using Linear elastic fracture mechanics model (LEFM) were used for fatigue loaded structures, and a Weibull or Gumbel distribution are used to postulate the probability models for the structures. Most of these S-N curves were modelled in laboratory and are available for various material types. The Gumbel –Wohler field, is taken in all these approaches. When fatigue tests have been with alternating maximum and minimum load on specimens, the obtained stress data is fitted using Gumbel distribution. These stress based curves are also called High cycle fatigue (HCF) models. These curves developed for various materials are reference, to provide mean lifetime N as a function of stress range for constant stress level. A family of percentiles S-N curves are formulated with fatigue life as a random variable. When local plastic deformations are present during the fatigue process, strain based curves are applied. These strain based approach needs ε-N curves, which represents mean lifetime N as a function of strain range. Instead of a single ε-N curve (mean strain), a family of percentiles ε-N curves are formulated with fatigue life as a random variable. Initially probabilistic S-N curves are developed using stress or strain based for the material. Based on the model proposed by Noroozi, we can compute p-S-N-R curves can be computed for the crack propagation in structures. Where p is the probability of failure, S is the mean stress, N number of cycles required for element to failure, R is the stress ratio. All these methods available for the reliability assessment can be broadly classified into two types. The first one is the frequency-domain approaches [3–6], modeling the stress-time variability with a random process defined by its Power Spectral Density (PSD) allowing, with a reduced mechanical time evaluation, to assess the expectation of the time to failure using closed-form expressions. This approach is very interesting and very efficient in the design stage. The second approach is the Stress–strength approach. It is a more general time-domain method based on probabilistic concepts for designing new structures with a reliability objective or assessing the failure probability of already designed structures. It consists of the comparison of two PDFs, the Stress (S) and the Strength (R) of the structure. This engineering approach is more general and very convenient to use.
Stress probability curves
Strain probability curves
Crack length curves
Regular S-N curves
Probabilistic Approach:
As defined earlier fatigue is a many facetted problem. Realistic probabilistic methods are used to define the characteristic response of an entire structure. Such a model to be accurate we need to develop and apply engineering models based on understanding the behavior and failure modes and the statistical distributions of the controlling parameters. Probabilistic framework for fatigue assessment closely follows the classical deterministic fatigue failure approach. The end result is the probabilistic definition of the cumulative damage D (f) which is now a random variable. It is possible to compute the mean value, the standard deviation as well as the complete probability density function (PDF) of D (f). Reliability analysis may be performed in order to get the probability that D is greater than 1 as a function of the total number of cycles. The following types Of uncertainties should be taken into account in the analysis: 1. A family of percentiles S-N curves are formulated with fatigue life as a random variable. 2. The data which we obtain is a scattered data, which is assessed as distribution to fit into a curve to obtain certain values. 3. Many uncertainties involved like size effects, surface finish are considered as random variables. The PDF’s of such variables are obtained from literature, and usually are assumed uniform PDF’s. 4. Uncertainties in mechanical model like material parameters, geometrical parameters, are again modelled as random variables using appropriate density functions. Almost all uncertainties raising in regular fracture fatigue model are modelled as pdf’s are accounted on a certain level using probabilistic approaches. So that they are not completely ignored and still have effect on the model failure.
Assumptions/Limitations:
The various approaches used for assessment of failure criteria have many assumptions and certain limitations. 1. The frequency domain approach does not give any idea of the time to failure dispersion essential for the structural probability of failure assessment. 2. The failure probability is very sensitive to the PDFs selected for R and S, in stress(S) – Strength (R) approach, since the number of mechanical evaluations must be well controlled to have an adequate computation time. Also the influence of each random variable on reliability cannot be determined since the uncertain parameters characterizing geometry, material properties and loads are gathered in the Stress PDF. 3. All of the approaches used are based on LEFM, using linear mechanical behavior and the graphs developed based on the same criteria. 4. The PDF’s defining stress and strength are taken independent. 5. The S-N, strain-N curves are developed using carefully selected samples in laboratory, those values may not exactly represent the outside phenomenon. Unknown model parameters are determined by statistical means from the database generated using above procedure. Such a data and its relevance to all problems is highly debatable. 6. Extrinsic conditions like corrosion, environmental degradation, high temperature et al, are not considered as parameters when determining the PDF’s. 7. The stress applied to calculate the S-N fields are constant, but in general it is varying stress.
Future work:
1. The various available statistical models for fatigue failure analysis are based on linear elastic fracture mechanics (LEFM) technique by Inglis and Griffith, Developed by Irwin. It postulates a major assumption that the cracks at which the stress concentrations occur in the material have smooth surfaces, the material for which LEFM method was developed are based on brittle solids with known defects (Inglis theory) following a linear propagation. This is a major assumption and we are using the S-N and crack width curves developed based on these models. However a new theory since has been developed called fractal fracture mechanics (Mandelbrot), which states that the crack surfaces are rough and the stress at such crack tip can cause new modes of fracture and crack propagation which follows bilinear propagation. It has an irregular set of scaling transformations. These surfaces have non regular geometric shapes. Stresses in such models can be normal, radial and hoop stresses, which are formulated using mechanics. Such stress equations can be used to calculate the new S-N curves. These newly developed curves include the real life situation of crack geometry. Crack propagation models are developed and these basis can be used to formulate power law, which can include uncertainties like combining load, stress intensity factors et al. The use of such advanced models to formulate the Probability density functions of operating stress and failure strength to calculate the probabilities of failure of fatigue which are caused by the stresses above. The calculation of the stresses using newly developed formulae on various specimens under different loading, can be used to formulate the PDF’s.
Linear and bilinear crack propagation models
2. The second proposal is the load time graph used to calculate the stresses. The loads are applied as maximum and minimum, a constant load difference is applied again and again to create the effect of fatigue and stresses are calculated. However in real life the load step will not be constant, it will be a varying load function. To simulate such a function we can take a random load response spectrum to create a real life situation in lab. Stresses them can be calculated depending on how the load is being applied. However we have to Step and ramp function here as there is no constant cyclic load to have a clear number of cycles. Instead a response spectrum is used and number of cycles are hypothesized and an S-N curve is formulated. Also the stress curve then generated need not necessarily follow Gumbel distribution an assumed in regular case. A better model is flitted and analyzed.
3. One of the major assumption in fatigue formulation is that the material is homogeneous and follow continuum mechanics, but in real life there are many flaws included in the material such as discontinuities, material differences et al. To accommodate all such flaws a Radom equally distributed Pdf is taken in all the approaches scene so far, considering the
material as equally distributed voids. But flaws in materials can be known by certain NDT methods. Which gives us the distribution of various voids in samples, with which we can formulate a pdf for certain batch sample, produced at a manufacturing unit. Which may not be exact but will be as near as possible as we are accommodating the randomness of voids in to different samples. A similar case can be done with the case of irregularity of material properties and Geometric properties of different samples.
Load response spectrum and the stress following a Gaussian distribution 4. All these uncertainties are considered interdependent, but this may not be the case always as some of them may be independent of each other but still affect the fatigue failure. A study can be proposed to calculate the effect of the various uncertainties contributing t to fatigue. 5. A method to incorporate thermal loading, corrosion et al depending on the environment changes and the conditions prevailing at the various sites can also be postulated as uncertainties and can be incorporated into the model. 6. We can still use different simulation to find better model to define he randomness of uncertainties involved.
References: 1. J. Maljaars, H.M.G.M. Steenbergen, A.C.W.M. Vrouwenvelder, Probabilistic model for fatigue crack growth and fracture of welded joints in civil engineering structures, International Journal of Fatigue, Volume 38, May 2012, Pages 108-117, ISSN 01421123. 2. Y. Prawoto, M.N. Tamin, A new direction in computational fracture mechanics in materials science: Will the combination of probabilistic and fractal fracture mechanics become mainstream?, Computational Materials Science, Volume 69, March 2013, Pages 197-203, ISSN 0927-0256. 3. B. Sudret, Z. Guédé, Probabilistic assessment of thermal fatigue in nuclear components, Nuclear Engineering and Design, Volume 235, Issues 17–19, August 2005, Pages 1819-1835, ISSN 0029-5493. 4. B. Echard, N. Gayton, A. Bignonnet, A reliability analysis method for fatigue design, International Journal of Fatigue, Volume 59, February 2014, Pages 292-300, ISSN 0142-1123. 5. José A.F.O. Correia, Abílio M.P. De Jesus, Alfonso Fernández-Canteli, Local unified probabilistic model for fatigue crack initiation and propagation: Application to a notched geometry, Engineering Structures, Volume 52, July 2013, Pages 394-407, ISSN 0141-0296. 6. Enrique Castillo, Alfonso Fernández-Canteli, María Luisa Ruiz-Ripoll, A general model for fatigue damage due to any stress history, International Journal of Fatigue, Volume 30, Issue 1, January 2008, Pages 150-164, ISSN 0142-1123 7. H. Khezrzadeh, M. Mofid, Tensile fracture behavior of heterogeneous materials based on fractal geometry, Theoretical and Applied Fracture Mechanics, Volume 46, Issue 1, August 2006, Pages 46-56, ISSN 0167-8442.
