Uncertainty quantification in mechanics

There are many questions where this is entirely true!

FEM softwares, particularly commercial ones, promise a lot.  Research is done on large scales, and engineers speak of "virtual prototyping" and "predictions".  This is completely out of reality, particularly for fatigue!

See a recent paper I posted to to some forums in Railways --- despite there have been large research projects, and big research centres have worked on railways corrugation for almost a century now, there is no understanding of the problem!

 

Rail corrugation, RCF, wear: a Tower of Babel?

The need for more collaboration, to prevent new disasters (perhaps in China) and win Global Challenges.

by M.Ciavarella
¹LMS-Ecole
Polytechnique, Palaiseaux, Paris. mciava@lms.polytechnique.fr

²Engineering Challenges, Cité
Universitaire, Paris, France. ciavarella@engineeringchallenges.eu

Railways: an established technology?

As recollected by Rod Smith recently (Smith, 2006), while some 20 or 30 years ago Railways were thought to be an old technology that had had its day, now their environmental advantages are now appreciated, and in many parts of the world, high-speed trains running on dedicated tracks are bringing high standards of speed, safety, comfort and punctuality to inter-city ground transport. Europe is leading so far the field and for example, ALSTON proudly announces (IRJ Editorial March 2008 [1]) the world's newest and fastest high-speed train, was unveiled at Alstom's plant in La Rochelle, France, on February 5 by President Nicolas Sarkozy of France. Alstom's faith in its new train has already been rewarded with a contract for 25 trains from the world's first open-access high-speed operator, reports David Briginshaw from France».

Fig.1
French President Sarkozy at the La Rochelle presentation, 5 Feb.2008

However, what is the real basic understanding of the rail-wheel contact, its induced damage, in the form of wear, RCF, corrugation, when only 8 years ago we had such a major disaster as the Hatfield accident, major not only for the 4 killed and the hundreds injured, but for the major shock it was for the industry. While on the technical side, the problem of gauge corner cracking (GCC) was in principle known and recommended checking procedures on tracks should perhaps have been sufficient to prevent the accident as declared in the court case by David Ventry, Former head of track at Railtrack [2], the following panic, which led to partial renationalization, suggests also that our knowledge is very limited on this problem. If knowledge were not limited, despite the resulting panic is much larger than justified if we look at the statistics of transport (Smith, 2003), some Engineer would have convinced the top managements involved, the politicians, and the public to take action immediately after the accident, and return rapidly to routine. Instead, already in May 2001 a record £534m loss was announced, and The Guardian in 2005 said « It is only now, nearly five years on, that punctuality has finally returned to its pre-Hatfield level among passenger train companies «[3] saying «Hatfield changed the rail industry... forever ». Wow!

However, has technical knowledge changed forever, or just the companies involved, returning incidentally closer to the (national) structure they had before (what a funny back-revolution)? And is such a big mess compatible with a good knowledge of the technicalities of the rail fatigue, wear, and corrugation. We believe the answer to both questions is a firm: NO!

It is only because the knowledge of friction, Rail Contact Fatigue (RCF), wear, noise production, and all their interactions, is extremely weak, despite research so far.

Of course, Railways companies are reluctant to admit any lack of knowledge, big research programs have been established for decades, and huge research centers are at work, so it may sound a little destructive to say that our understand is extremely weak. Companies do make huge computational models today, with state-of-the-art modeling of the contact (either elastic, or elasto-plastic), taking into account of the complex dynamics of the vehicle (with sufficient
accurate description of the vehicle, the track, the support, etc.), although at this stage already some assumptions have to be made as the number of possible conditions starts to grow without limit.

Promising techniques are those using direct methods instead of full computation, as developed in Top Engineering Schools (DangVan, 2008, DangVan and Maitournam, 2003). But this is still not enough, as the prediction of the plastic response is not the prediction of fatigue life, and so far we haven't considered any fatigue and wear interaction, on interaction with more phenomena, usually studied by separate research departments, like « watertight
compartments » and this is not good. Only recently, we start to hear of « holistic approaches ».

But the truth is: we know very little about damage induced by rail-wheel contact. As recollected also in a 1998 lecture by Prof. Rod Smith of Imperial College, one of the leading of the authorities in the field, warned that rolling contact fatigue (RCF) is much less understood than fatigue in general, which incidentally has its origin when railways were born. While in fatigue in general, we have made considerable progress, mostly for constant amplitude loading acting on the same parts, under mainly tensile stresses, the railways contact is nothing of this sort. If different trains pass over different locations of the rail, and with different loads (actually each wheel will have different loading), it is almost impossible to predict the loads really occurring there, let alone «predict» fatigue damage, and even less likely, wear (perhaps in the presence of some lubricated contact) which in turn is related to fatigue since crack mouths are worn out so that wear may beneficial in the end. Vice versa, corrugations tend to make dynamic loads, and only recently it has been suggested that these dynamic loads should be used for the computation of fatigue life, and not the quasi-static ones as everybody does. So, really people within the companies themselves do not talk each other enough (a first reference to the Tower of Babel).

Moreover, Smith warns us that railway industry has no comparable extensive prototypical testing with respect to, for example, car manufacturing and aeronautical industries where ultimately fatigue testing in service is conducted on a large number of vehicles. So, like an Editorial of conservative The Engineer against the change from iron to steel in 1867 said «there is no steel rail now down in England or elsewhere which is not an experiment in itself. The progress of deterioration is slow and track may be serviceable for years, and then begin to fail without warning»: we suggest situation has not changed, nor is likely to change: strictly speaking there is no prototypical testing in this industry.

If this is the state of affairs, each time we change technology a little, either to increase the speed, or for example, to adapt a certain existing technology to existing tracks for example in China and India, then do we risk a Hatfield accident? My answer is: yes!

Hence, our provocative title. We clearly have the sensation of the Tower of Babel. In other words, there is no collaboration as a result of every company saying they know everything --- as the Tower of Babel dedicated with a motive of making a 'celebrated name' for the
builders [4] [the Railways companies in this case], God seeing what the people were doing and sinning against him, confused
their languages and scattered the people throughout the earth!

Fig.2
-- the Tower of Babel in the background of a depiction of the Hanging Gardens of Babylon by Martin Heemskerck. Is Railways research the modern attempt to build the Tower?

In recent years, we have reviewed many concepts which are believed to be the basis for the design against RCF. We have suggested also that the early investigation on the basic mechanisms of RCF, corrugation and wear were very naïf, and later proved of less use than what the early investigations would suggest. However, it was too late for the industry and the research centers to change approach, and so we have some inertia in finding a really good approach. For example, the concept of shakedown maps, and of ratcheting as the mechanism dominating in RCF fatigue, as suggested by Prof. Johnson in Cambridge often in collaboration with BR research, and Kapoor’s “critical ratchet strain” hypothesis, was reviewed critically in Ponter et al (2003), and Afferrante et al (2004), finding that the original Merwin and Johnson work was naïf in many respects, the experimental ratcheting rate being apparently similar to a very simple calculation for a double coincidence: first, the calculation using the simple perfect-plasticity model was wrong by a large factor, and the yield strength in the material had been adapted a posteriori to find what seemed to be constant ratchet rate. Today, after 40 years, I cannot think of a simple calculation along these lines, and this is only for the basic problem. Forget about a realistic application.

Surprisingly, much simpler strategy would be to use design procedures similar to those used in gears design based on hardness (although of course the lubrication problem could be different) have, in the best of my knowledge, not been attempted by the Railways Industry. I would
not take a train if it were only based on these kind of calculations!

Rail corrugation.
Going to the point of rail corrugation, this phenomenon has been noticed at least for 100 years, but is still not fully understood (Grassie and Kalousek, 1993) particularly for short-pitch rail
corrugation ("roaring rails") in the range of 20÷80 mm wavelength. It has been the subject to extensive research programs both at single industries, in single Academia, and also of joint research efforts in the European Community, but (particularly short pitch one in the range 20÷80 mm) has been considered an enigma because measured corrugation wavelength did not relate well with wear-instability models for short-pitch rail corrugation ("roaring
rails") in the range of 20÷80 mm wavelength. Most available data seem to show a non-linearly increasing wavelength with speed, i.e. an almost fixed-wavelength and not fixed-frequency feature (see figure 3 adapted from figure 1 of Bhaskar et al. (1997), with quite a lot of data in BR old reports, David Harrison's Cambridge Ph.D. thesis and the Vancouver SkyTrain data – another case when change of technology in the lack of modeling capability led to very unexpected problems). Instead, most models generally obtain a resonance on the system, the most obvious being the vertical direction ones, the torsional ones of the wheel set, and the lateral ones. However, careful check of the phase between the possible corrugation ripple and the resulting dynamic amplification of load and of where needs to be done to suggest possible mechanisms.

Ideally, one would then compare between them to see most unstable under certain conditions (straight track, curve, etc.). It has been difficult to collect « clear evidence », and many tentative hypothesis have been advanced including very bizarre
ones, the majority of them contradict each other and are confusing.

Fig.3 – Some corrugation data in the « corrugation wavelength-train speed » plane from old BR report, from Cambridge group of Prof. Johnson in the late 1970s' and from the 1992 quite different Skytrain intercity track system show surprisingly
consistent results, not corresponding to any frequency in particular. Yet, while Prof. Johnson admit his attempts do not explain corrugation, many others have suggested mechanisms which correspond approximately to the 3 lines in the Figure!

Apart from the evidence collected from the beginning of 1900, up to the 1970's, it was the introduction of concrete sleepers and welded rails, when BR started to experience more serious corrugation problems at least on some lines [5], and accordingly seriously studied the problem (Frederick, 1986). Clearly, this is due to the change of the vertical receptance of the rail, and indeed, since then, use of resilient pads if often a
palliative solution (together with friction modifiers).

However, in the many attempts of modeling, BR's evidence is generally neglected, as it is probably disturbing to try to use it --- except by Prof. Johnson from Cambridge who in fact remained always very negative in his conclusions in every paper, up to the late ones of 1997 (Bhaskar et al 1997). Others, like even his former student, Stuart Grassie, do not return to discuss these data and seem to change their mind various times never getting to a clear single model. In particular, Grassie & Johnson (1985) fail to justify corrugation, and Grassie & Kalousek (1993) admit it is not fully understood. However, suddenly Grassie (2005) says quite aggressively that Johnson's Cambridge group was following a « chimera » of a wavelength-fixing mechanics, but doesn't explain why, while he puts forward the work of Prof. Knothe (Hempelmann 1994, Hempelmann and Knothe, 1996) as to prove «definitive evidence» of short-pitch corrugation as being due to pinned-pinned resonance, which corresponds to a fixed frequency of about 1100 Hz (but see Fig.3?). Grassie & Edwards (2006) however returns to a model without pinned-pinned resonance!

However, Ciavarella and Barber (2008) recently suggested that most of the confusion around, and in particular the failure of the early models like Grassie and Johnson (1985) comes from erroneously assuming constant longitudinal creepage: when the inertia of the wheel and the rotational dynamics of the system are considered, a much better qualitative agreement with experimental evidence is immediately found. We decided to look at the simplest possible model, given we are not in a position to easily check results in the literature with
often sophisticated numerical simulations. To us, a lot more validation is needed for even the simplest of these numerical models, and our own efforts to produce clean and closed form solutions is also along the lines of extending the possibilities to check these models. In Ciavarella and Barber (2008), we assumed a simple full stick Winkler contact mechanics (but ongoing studies submitted for publication permit to say that not much is changed with other
approximate treatments considering full continuum solutions, 3D effects, and non-circular contact). To simplify the discussion even further from Ciavarella and Barber, we shall report the result of an extremely simplified Euler beam vertical receptance model (neglecting the effects of supports), which is valid in most cases in the frequency range greater than 500 Hz (and certainly with the vertical receptance from Bhaskar et al. (1997), i.e. a continuous support typical rail for Intercity Track -- the paper refers to the Vancouver SkyTrain, but in part II seems to deal with BR track constants), of course under the assumption of neglecting pinned-pinned resonance due to discrete supports. This way, the vertical receptance high frequency tail has a single functional form, which depends on a single groups of parameters (mass per unit length, flexural rigidity of the rail), and therefore permits to present results of general validity in terms of frequency, naturally for this high frequency range.

This is extremely convenient, and perhaps corresponds also to the fact that the short pitch corrugation does not depend much on the particular system, as clearly seen in Fig.3, where, despite railways and intercity systems are included spanning very different technologies ( the old BR rails on wooden supports with unwelded rails, the newer continuously welded rails on concrete supports introduced in England by BR after 1966, the Vancouver Skytrain), the corrugation is seen to have more or less the same features. Hence, we produce results which are of interest for the frequency range f>450 Hz. In other cases, for particularly stiff pads for example, the frequency "crossover" may change, but these would be extreme cases which require separate investigations. By neglecting the effect of the pinned-pinned resonance at about 1 kHz, and also the difference between receptance mid-way between sleepers with that at a sleeper, we make the treatment extremely simple, a closed form equation, for the oscillatory part of the energy dissipation function (a complex function W1) in the contact area, in presence of an initial perturbation. In other words, if conditions are such that more energy is dissipated in the through of the contact, corrugation will continue, otherwise it will be suppressed. By plotting the minimum of the real part of W1 as a contour plot in the wavelength-speed plane, only taking the negative part of the function, and dividing the contour lines from zero to the minimum in 10 contour levels, we obtain the results of fig.4.

We see that most data fall in the correct negative region of the function W1 indicating there is no enigma to explain this phenomenon. Also, for every given speed, we compute the local minimum of the function W1 to suggest the wavelength most likely to grow, and this is indicated by big dots. This defines (at least within the strong assumptions in the model) a frequency of corrugation nearly constant, not too far from the Winkler results of Ciavarella and Barber (2008), and close anyway to 500 Hz.

Fig.4 – Our model compares to the corrugation data from old BR report, from Cambridge group of Prof. Johnson in the late 1970s' and from the 1992 quite different Skytrain intercity track quite consistently. The shadowed area indicates expected poor predictions since the vertical receptance is simplified. Ellipticity of contact 2b = 3a; normal load P = 50 kN ; τ = 0.1. For more details, see
Afferrante et al (2008)

Fig.5 – Comparison between zero inertia (constant tangential force, full black circles) and infinite inertia (constant creepage, full black triangles), and a realist inertia (empty circles)

Other studies, omitted here for brevity, show that using the full receptance of the model, there are other minima of the dissipation function in the low frequency regime, and the tractive ratio τ=Q/μP may have an important rôle in deciding which regime is picked up as most fast growing. Also, the minimum of the real part of W1 decreases with the speed V almost linearly, but increases with the tractive ratio τ. Details are too complex to be shown here, see Afferrante et al (2008).

A comparison between constant creepage (large inertia) and constant tangential force (low inertia) is proposed in figure 5, to show that the assumptions of constant creepage, or constant tangential load, very common in the Literature, are dramatically wrong, and explains why the Grassie-Johnson (1985) simplified model could not justify the observed corrugation, but also many other bizarre results in the models around. Notice not only the frequency is wrong, but also the modulus of the energy dissipation is erroneous, in particular, it is not suggesting corrugation for the constant creepage case (the Cambridge « enigma »), and it is over predicting corrugation in the constant tangential case. Recent examples assuming constant tangential load are Meehan et al. (2005), and Wu and Thompson, (2005).

Hence, we conclude that the original data collected by British Rail Research and Cambridge University, which are probably still the most reliable around, are very likely not explained at all in terms of pinned-pinned resonance, as it is clear from figure 3. Also, short-pitch corrugation appeared very strongly on Vancouver SkyTrain which has not a typical discrete support, and indeed did not change when the spacing of support changed. Indeed, we believe that the conclusions about pinned-pinned modes are affected by three unfortunate circumstances: first, the effect of parametric resonance attempt to introduce their effect, generally use local eigenvalue analysis, which means that they assume a steady state is reached like if the local receptance were valid for an infinite length. This exaggerates the amplification of normal load (or of lateral loads, or both) above sleepers easily by a factor between 2 and 5 (see Grassie et al 1982 one of the few papers which deals with this effect correctly). Also, models with continuum support have partly failed in quantitative prediction of wavelength only because of the erroneous tangential dynamics assumptions. Proper parametric resonance models appeared recently, but mostly they deal with noise rather than corrugation, and their effects are very difficult even to describe and to study.

Discussion

A general problem in RCF damage, wear and corrugation, was the unfortunate circumstance that there was never much discussion and comparison of models, perhaps for IPR reasons and fierce competition between railways. Also, many problems appeared "specific" and prone to various interpretations (the Tower of Babel punishment for having decided similar but slightly different systems in each country). To give an idea of the complexity of the problem which induces at time trivial errors, and on the other hand of the unfortunate lack of serious collaboration between industries and universities in the field, in the end of the 1990's, ERRI (European Railways Research Institute, which didn't survive very long) sponsored some research in order to make a comparison of predictive capabilities of the various models (Frederick' BR one, Berlin ones, Cambridge ones). This is hardly mentioned in the literature (perhaps for the prohibitive cost of these reports) but it is unfortunate, since they are extremely interesting and elucidating. In the report D185/RP1 (1993), various interesting facts are worth reporting:

b) There was a difference in the vertical receptance by a factor 10 or 20, which is certainly not a good start for any modeling!

In another such useful report (ERRI-D185WP2, 1997), not only more models are compared (Hempelmann's, Frederick in the extended 3D contact version of 1991, Bhaskar's model from Cambridge, a large creepage model from Clark at BR, and the Tassilly model by RATP and Vibratec), but an attempt is made to compare to experimental measurements on test sites. The 1993 findings were repeated; probably the 1500Hz regime difference was associated to difference in calculating wear in the contact area. The experimental validation was extremely confusing and even giving to the models the roughness input at the starting conditions, there was only vague qualitative agreement. But the most striking finding was that in the test site there was development of the 20mm roughness that everyone had considered to be "suppressed by contact filter"!

The risks of new Hatfields in China.
To make a long story short: we don’t know much about RCF, wear, corrugation and their interactions. If we go to China and use fast speed trains, even at slower speeds than what designed initially, but using the existing rail track, we don’t know nor we can predict the
outcome. More Hatfield will occur? Very likely, yes!

Way forward.
While we can make progress by more collaboration, the only solution is constant careful control of the lines, and trial and error in new design. Many privatizations in this business have been failures. In England, BR, before disappearing, had already given up the attempt to «understand» corrugation, for example, and moved later to «monitoring» and «corrective grinding». Most recently, after the Hatfield accident the companies responsible for the track are talking of «preventative grinding». The Dutch seem to have been between the first to use the idea of "preventative grinding" on rails within the first 6 months after their pose because "for reasons unknown" rail corrugation started only rarely on such grinded rails before they had seen significant traffic (Van der Bosch, 2002). Similar forms of "preventive grinding" or "regular grinding" is
also the routine solution for many railways (French never have admitted much corrugation problems probably because of a very serious such maintenance program), and while steels are harder and harder and wear in principle is reduced, regular grinding is nowadays conducted also as a means to reduce cracks anyway (Grassie and Baker, 2000) but this becomes then a matter hard to judge scientifically also because of lack of data and the presence obvious commercial, managerial and
political interests. This strategy is suggested more successful also by Zarembski (1997) and even convenient in Holland (van der Bosch, 2002), but it is not entirely clear why. Also, notice that grinding there is aimed not just at removing corrugation, but mostly avoid gauge corner cracking and in general contact fatigue, together with enhanced lubrication in certain areas (Middleton 2002). Speculations about future «proactive grinding» are not entirely well specified.

Holistic Approach?
A few people start to notice (Kuijpers et al 2007) that the «holistic approach» should be considered with a carefully-tailored maintenance program that combines acoustic and preventative grinding. However, without a good model, how to do that? The problems of corrugation and RCF are related, although in not too clear form. First of all, wear and RCF are related in general, since removal of material also means removal of cracks. With harder and harder steels, wear becomes less crucial than RCF --- In 1999 the European Rail Research Institute estimated the total cost of RCF to European railways at €300m a year. But it may be larger today. Too many and too independent efforts to generate models, many false beliefs, the difficulty to collect experimental evidence, and the lack of sufficient comparisons between models between different European and non-European companies and Research Centers. A very useful exercise was done only by the European Railways Research Institute, which was perhaps useful, and hence was shut in 2004. ERRI did find some extremely surprising results, and obvious problems in comparing different models, and the latter with experiments in test sites. By working each with his own means, there has been no attempt of « collective knowledge » as it the Wikipedia times, this would be called. Hence, more than an enigma, corrugation was the result of a « Tower of Babel » approach.

Andreotti or Moro? For his final slide of the 19th Jenkin Lecture, Smith (2006) showed us a cautionary quotation from Aldo Moro, late Prime Minister of Italy, that: "There are two types of madmen: those who believe they are Napoleon and those who think they can sort out the railways."

 

Very true, but there is a mis-quote here, Rod, it was Andreotti not Moro!

References

L.Afferrante,
M. Ciavarella, G. Demelio, A re-examination of rolling contact
fatigue experiments by Clayton and Su with suggestions for surface
durability calculations, Wear 256 (2004) 329–334

L.Afferrante,
M.Ciavarella, Rail corrugation: “one hundred years of solitude”?
or a true enigma? Submitted 2008.

A.
Bhaskar, K. L. Johnson, G. D. Wood and J. Woodhouse, Wheel-rail
dynamics with closely conformal contact Part 1: dynamic modelling and
stability analysis IMechE Proc Instn Mech Engrs Vol. 211 Part F
(1997), pp. 11-26.

M.
Ciavarella and J.R.Barber, Influence of longitudinal creepage and
wheel inertia onshort-pitch corrugation: a resonance-free mechanism
to explain the roaring rail phenomenon Proc. IMechE, Part J: J.
Engineering Tribology Vol. 222 (2008), pp. 1-11

Ky
Dang Van, Modelling of damage induced by contacts between solids,
Comptes Rendus Mécanique, 336 1-2, 2008, Pages 91-101

K.
Dang Van and H. Maitournam, Steady-state flow in classical
elastoplasticity: application to repeated rolling and sliding
contact, J. Mech. Phys. Solids 41 (11) (1993), pp. 1691–1710.

ERRI
D185/RP1. Reduction of rail corrugation. Rail corrugation models -
comparison of results obtained using the Berlin Technical university
and British rail methods, 1993.

ERRI-D185/RP3.
Theoretical modelling of rail corrugations and validation by
measurement, 1997.

C.
O. Frederick, A rail corrugation theory. Proceedings of the 2nd
International Conference on Contact Mechanics of Rail-Wheel Systems,
University of Rhode Island, (1986), pp. 181-211 (University of
Waterloo Press).

S.L.
Grassie and P Baker, Routine maintenance extends rail life and offers
long-term savings, Railway Gazette International, Feb 2000, pp.
88-90.

S.L.
Grassie and K.L. Johnson, Periodic microslip between a rolling wheel
and a corrugated rail, Wear, Vol. 101 (1985), pp. 291-305.

S.L.
Grassie and J. Kalousek, Rail corrugation: characteristics, causes
and treatments, J Rail Rapid Transit, Proc. Inst. Mech. Eng., Vol.
207F (1993), pp. 57-68.

S.L.
Grassie R.W. Gregory D. Harrison and K.L. Johnson, The dynamic
response of railways track to high frequency vertical excitation, Int
J Mech Sci, Vol. 24 (1982), pp. 77-90.

S.L.
Grassie, Rail corrugation: advances in measurement, understanding and
treatment, Wear, Vol. 258 (7-8) (2005), pp. 1224-1234.

S.L.
Grassie and J.W. Edwards, Development of corrugation as a result of
varying normal load, 7th Int Conf on Contact Mechanics and Wear of
Rail/Wheel Systems (CM2006), Brisbane, Australia, September 24-26,
2006.

K.
Hempelmann and K. Knothe, An Extended Linear Model for the Prediction
of Short Pitch Corrugation. Wear, Vol. 191 (1996), pp. 161-169.

K.
Hempelmann, Short Pitch Corrugation on Railway Rails -- A Linear
Model for Prediction. VDI Fortschritt-Berichte, Reihe 12, No. 231,
Dusseldort, 1994.

A
Kuijpers, JC Schaffner M Bekooy, Grinding programme combines noise
reduction and preventative maintenance Railways Gazette 01 Jun 2007

P.A.
Meehan, W.J.T. Daniel and T. Campey,. Prediction of the growth of
wear-type rail corrugation, Wear, Vol. 258 (7-8) (2005), pp.
1001-1013.

R
Middleton Railtrack invests in grinding capability Railways Gazette
01 Jan 2002

TG
Pearce The development of corrugations in rails of acid bessemer and
open hearth steele, Technical Report IMDA257, BR Research, 1976.

A.R.S.
Ponter, L. Afferrante, M. Ciavarella, A note on Merwin’s
measurements of forward flow in rolling contact. Wear 256 (2004)
321–328

R.A.
Smith, Fatigue in transport: problems, solutions and future threats,
Process Safe Environ. 76 (B3) (1998) 217–223.

R.A.
Smith, The 19th Jenkin Lecture, 23 September 2006: Railways: The
Technical Challenges of their Renaissance,
http://www.soue.org.uk/souenews/issue6/jenkinlect.html

R.
A. Smith (2003) The wheel-rail interface—some recent accidents,
Fatigue & Fracture of Engineering Materials and Structures 26
(10) , 901–907 doi:10.1046/j.1460-2695.2003.00701.x

R.A.
Van der Bosch, Les strategies de meulage des chemin de fer
neerlandais, Revue Generale des Chemins de Fer, nov., (2002), pp.
40-43.

T.X
Wu. and D.J. Thompson, An investigation into rail corrugation due to
micro-slip under multiple wheel/rail interactions. Wear, Vol. 258
(2005), pp. 1115-1125.

AM
Zarembski, Intelligent use of rail grinding, Railways Gazette 01 Feb
1997

1
http://www.railjournal.com/altfeature1.shtml

2
Hatfield crash 'was preventable',
http://news.bbc.co.uk/2/hi/uk_news/4325731.stm

3http://www.guardian.co.uk/uk/2005/sep/06/hatfield.transport

4
Genesis 11:4

5
Pearce (1976) noticed only corrugation in the West Main line up to
the border of Scotland where Acid Bessemer steel left place to Open
Heart Steele, but not on the East main line, an enigma perhaps due
to different gauge.

 
 

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  calibration and revision.

Upon completion of this course, attendees will have learned:

• The latest techniques for evaluating the accuracy of computational
  models over a range of parameter values;

• How to design validation experiments that will help determine the
  simulation range of validity;

• How to calibrate model parameters to reflect the measured response
  from experiments — even for non-linear models!

Course Outline

• Introduction

• Historical overview

• Feature extraction

• Code verification and solution convergence

• Uncertainty propagation and quantification

• Local and global sensitivity analysis

• Meta-modeling, surrogate modeling

• Design of validation experiment

• Test-analysis correlation and validation metrics

• Model revision and updating

• Applications for linear and non-linear structural dynamics

Course Instructor

Dr. François Hemez has been Technical Staff Member at the
Los Alamos National Laboratory (LANL) since 1997 where he is
contributing to the development of technology for solution
verification, model validation, uncertainty quantification, and
decision-making for engineering and weapon physics applications.
Before joining Los Alamos Dr. Hemez was a research associate of the
French National Center for Scientific Research, working in the area
of test analysis correlation and finite element model updating. He
has developed and taught undergraduate and graduate courses in
applied mathematics and structural dynamics. Dr. Hemez has taught
the first-ever graduate course offered in a U.S. University on
uncertainty quantification and Verification and Validation
(V&V) at the University of California San Diego (Spring 2006).
Dr. Hemez is serving as chair of the Society for Experimental
Mechanics technical division on model validation and uncertainty
quantification (2005-2009). In 2005 Dr. Hemez received the Junior
Research Award of the European Association of Structural Dynamics;
in 2006 he received two U.S. Department of Energy Defense Programs
Awards of Excellence for applying V&V to programmatic work at
LANL. Dr. Hemez has authored over 200 peerreviewed publications,
conference papers, and reports since 1992.

Sample course notes at www.la-dynamics.com
.

Please direct questions to hemez@la-dynamics.com
.

The instructors will assume a basic knowledge of structural
mechanics, dynamics, and mathematics, such as that obtained in a
bachelor’s level aerospace, civil, or mechanical engineering
program. To ensure top-quality content, the instructors reserve the
right to alter the course schedule. Los Alamos Dynamics is an
independent company, not affiliated with the Los Alamos National
Laboratory.

Course Outline (Somewhat More Detailed)

1. Introduction

• Description of the model validation process

• Definition of key terminology

• Sources of modeling errors, examples of validation studies

2. Historical Overview

• Discipline specific applications (aerospace, civil, automotive)

• Linear test/analysis correlation methods

• Examples in linear and non-linear structural dynamics

3. Code and Calculation Verification

• Code and solution verification

• Asymptotic convergence, estimators of convergence

• Extrapolation, solution error Ansatz, and self-convergence

4. Extraction of Response Features

• Desirable properties of response features

• Classification and selection of features

• Calculation of features, examples in linear and non-linear dynamics

5. Overview of Probability and Statistics

• Descriptive statistics and their properties

• Statistical sampling, design of experiments (DOE)

• Statistical testing, confidence intervals

6. Local and Global Sensitivity Analysis

• Local sensitivity analysis, derivatives and finite differences

• Global sensitivity analysis

• Analysis of variance (ANOVA) and effect screening

7. Meta-modeling

• Overview of surrogate modeling, meta-model forms

• Model regression and error estimation

• Calibration, optimization, and sensitivity studies using
  meta-models

8. Quantification of Uncertainty

• Classification of measurement, simulation uncertainty

• Forward propagation of uncertainty, sampling, DOE

• Inverse propagation of uncertainty, optimization under uncertainty

9. Design of Validation Experiments

• What makes validation experiments different from other tests?

• Experiment design, instrumentation selection and placement

• Replication, effect blocking, variable screening, experiment
  uncertainty

10. Test/Analysis Correlation

• Comparisons between predictions and measurements

• Definitions of fidelity metrics, correlation metrics

• How good is good enough?

11. Model Updating and Revision

• Revising conceptual model forms and assumptions

• Model parameter calibration, finite element model updating

• Independent parameter assessment

12. Examples of Model Validation Studies

• Explosively driven impulse on a threaded interface

• Modal response, delamination damage of composite plates

• Simulation of polymer foam impacts, shell buckling, Taylor anvil
  impacts

NAFEMS Travel Policy
: NAFEMS is not responsible for the purchase of non-refundable
airline tickets or the cancellation/change fees associated with
canceling a flight. Please call to confirm that the course is
running before purchasing airline tickets. NAFEMS retains the right
to cancel a course up to 3 weeks before scheduled presentation
date.

Members Price: £510 ( $1006 / €640 )
Non-Members Price: £715 ( $1411 / €898 )
Order Ref:VandV_NAFEMS_NA_2008
Event Type:Course
Location: Hampton, Virginia,USA
Date: October 27, 2008

Hi Dr. Biswajit

    As my research is related to uncertainity qualification, I can share some experiences that I had.

    
Uncertainity qualification is one of the most important issues in
structural mechanics now a days. The material property, Loading
condition  and material response is not deterministic in reality.
Theoritically we can define a deterministic form of the governing
equation but we can convert this deterministic form into random form by
adding a white noise to the uncertain parameter such as modulus of
elasticity, load or displacements. For uncertainity assesment, there
are several approaches are followed now a days. If we take the
Euler-Bernoulli beam equation as an example; we can define the load or
modulus as a vector of Random variables or more generally Stochastic
Process. So, at any probability state the modulus or load changes with
respect to a independent random variable such as time. Once the beam
equation contains a Stochastic process, usually it can be called as
Stochastic Differential Equation (SDE). But in practice we use white
noise or Brownian Process as the stochsatic process. Suppose the
modulus is random, so we can write: E (deterministic or Predictable
Process) + White Noise (or Brownian Motion as Stochastic Process). It
is well known that the white noise is nowhere differentiable. We can
not use usual solution technique for PDE or ODE in general (The exception is there such as my graduate research where I used the formal solution technique for PDEs).The
most well known method is Ito's formula or Ito's Process. This is a
very wide topic. I can give a simple concept here. The differential
equation is converted into differential form rather than keeping in
derivative form. Hence Ito's form is deduced and we can use Ito's
method to solve the SDE of the beam. The solution is analytical. But
once we go for simulating the SDE, usually we have to go for
approximation method. The finite difference form of the SDE is quite
re-known in literature. However, as far as large structure is
concerned, these SDEs are used in Stochastic Finite Element Method
(SFEM). There we can use perturbation technique as well as other method
also used. The main idea behind SFEA is to find/estimate Cummulative
Distribution function of failure. Hence we can find reliability of the
structure. The stiffnes or load and strength are defined to be
modulated with white or Gaussian noise. Another straight forward
apporach in SFEA is Monte carlo Simulation. As we know the probability
distribution is required for Monte Carlo Simulation, it is kind of a
drawback, sometimes. Also the computation time is a big factor too. In
my research, what I realized that the distributaion function is not
well defined in most of the practicle applications. For these
situations, we can do several things to overcome the problems such as:

-
Approximating the physical parameters such as load or modulus by a
Random Walk model based on some criteria of Random Walk models. (High
computational cost)

- Estimating the probability
distribution based on several methods such as EM, Maximum likelihood
algorithm, Bayesian Inference, Hypothesis Testing, Nonparametric
methods etc. Some practical data such as histogram of the events or
random variables is required.

- Using
Non-parametric Monte Carlo approach such as Markov Chain Monte Carlo
method (see the paper by Geyer: "Practical Markov Chain Monte Carlo").

Besides
these there are other techiques are coming up in order to reduce
computational cost. Another important issue iis generating random
numbers, more generally generating psudo-random numbers. This issue
truely affects the computaion cost and accuracy of the Monte Carlo
simulation.

   As far as validity of the outputs
from uncertainity qualification is concerned, you can not ensure 100%
of the validation. Because some cases may not arise in experiments or
testings. Usually reliability assessment is followed up, as the final
step in uncertainity qualification. Some hypothesis testing can be done
based on the experiments. The goal is to predict varience of the
outputs such as deflection of the beam at different locations as well
as covariance of the several outputs due to several probability states.

   The uncertainity qualification is becoming a
very important too in Structural engineering. As I have noticed that
the Non Destructive Evaluation helps you to aquire some current status
of the structure or Part of the structure. We may be concerned with
damages, outside conditions, vibrations, micro-structure change etc.
There is no way out to specify a global function about these
parameters. If 100 aircraft wings are manufactured, no one except GOD
can tell us when these wings are going to be failed. Hence the demand
for uncertainity qualification is increasing day by day. For further
reading I can recomend texts:

Probabilistic Methods in Structural Engineering: Giuliano Augusti, Alessandro Baratta, Fabio Casciati.
Stochastic Differential Equations: An Introduction with Applications by Bernt Karsten Øksendal  (One of the best books on SDE I have ever seen)

 

 Thanks

 Rezwan

    

I wish this of stochastic ODE were really a solution!

Often you simply are requested even MORE information on the input and your model, than if you had a deterministic model.  Now, you don't have only to know your input value, but its distribution.... You can easily see how this produces nice mathematics, but not a response to a real problem in mechanics, i.e. when you DON'T have a model....

In fatigue, you simply do NOT know this "equation" for the behaviour of material, and sometimes it is even already difficult to predict the elasto-plastic response without many FITTING parameters.

So I don't want to sound destructive, but this is even more smoke in the eyes, in most cases.  "Smoke and mirrors" as the English say....  This is to clarify the issue.  If you really have THE model, then statistics ODE is interesting.

One of the first examples and indeed the original interest of Weibull, was ceramic materials.

In that case, you find in experiment so much a deviation from a single "strength", that you need a distribution to model it.

The next step is that you recognize the shape of the distribution, and in some cases it is "mild" like a Gaussian, which means that more or less you are happy with the mean (and perhaps the standard deviation) as information to carry over.

However, in other cases you have a "wild" distribution, like a power law --- this is the case of many critical phenomena in nature, there is a huge literature, Mandelbrot, Sornette, to name but a few (the denomination "mild" vs "wild" is indeed Mandelbrot's recent contribution in finance).

For a wild distribution, the mean means little! You need to carry over some analysis also with the tails.

To return to Weibull statistics, it is not precisely power law, nor is a Gaussian, so it is somewhere intermediate. You are supposed to measure the strength, and as I said, having a brittle ceramic means you need MORE tests to make the distribution, than if you have tensile test of a steel, which is quite "deterministic", i.e. it has a narrow Gaussian distribution of which mean is ok, and you can even forget standard deviation.

 

The next problem becomes that people think the property of the statistical ceramic material can be measured easily and are "material properties". This is NOT entirely true, I wish life were easy....

Is Weibull’s modulus really a material constant? Example case with interacting collinear cracks
International Journal of Solids and Structures, Volume 43, Issue 17, August 2006, Pages 5147-5157
L. Afferrante, M. Ciavarella, E. Valenza

 

Anyway, the question becomes long.  Speaking of Weibull, however, I suggest to read this remarkable paper using some analogies between "failure" of material, and human death.

A statistical analogy between collapse of solids and death of living organisms: Proposal for a ‘law of life’
Medical Hypotheses, Volume 69, Issue 2, 2007, Pages 441-447
Nicola M. Pugno

 

Enough for now!  I wait for more input to give more output...

 

michele ciavarella
www.micheleciavarella.it

michele ciavarella
www.micheleciavarella.it

Please look at the paper:

A probabilistic approach to fatigue risk assessment in aerospace components : G. Cavallini , R. Lazzeri

It is a nice source of idea on typical uncertainity qualification method in structural mechanics. The schematic diagram (figure-1 in the paper) can give a nice instant idea about formal uncertainity qualification method.