Can structures recover from fatigue?

I like cycling. When the weather is fine (or not so fine) I really like to make a nice tour through the countryside. Sometimes easy going, but sometimes it will be a real work-out. So after the ride I’ll have a nice beer and a good rest. And next morning I’m fully recovered!

Does that work for my bicycle?

You could always try to give your bicycle a good rest as well. Unfortunately, fatigue in metallic structures is irreversible. Once a crack is initiated, it will not heal anymore. So my bicycle will not recover from a bumpy ride just be sleeping.

Is that bad?

Not necessarily. Despite of not recovering, my bicycle may still be able to reach a reputable age. How? Well, avoiding fatigue in a structure starts with a good design. But what do we need for a good design. In short: we need knowledge and experience. It takes time to obtain this. To speed this up, we might help you. Just look and see how…..


 

Fatigue in the Merwede Bridge

During autumn 2016 the Merwede bridge on the A27 in the Netherlands was closed for heavy traffic for 2 months. The reason was the presence of fatigue cracks close to a critical length. The firm Berenschot did investigate the process of closing the bridge. Based on their evaluation report, 2 professors claimed that The Netherlands just escaped from a disaster; the Merwede bridge had a residual life of only 6 days. A little later another professor claimed that there was no problem; the critical cracks were detected in time.

Since I’m working already some time (35 years) in the field of fatigue and because I was curieus what was going on, I have tried to make an attempt to clarify one and the other.

What is fatigue

Let’s see what fatigue actually is. Fatigue occurs when a structure is subjected to cyclic loading. If the stress amplitude exceeds a threshold value, microscopic cracks will initiate at locations with high stresses (stress concentrations). At first, the cracks propagate very slowly and remain undetectable for the bare eye for most of the fatigue life. Gradually the crack propagation rate increases and the cracks will become visible (detectable, point A in the figure below). Eventually the crack will reach a critical size (point B in the figure below) and the structure does not function as required anymore (e.g. too much deformation, partial failure, etc.) or even the full structure will fail. The critical crack length is determined by the fracture toughness of the material. At a given load the fracture toughness yields a critical crack length, and vice versa, at a given crack length the fracture toughness yields an allowable load.

Crack growth curve with inspection interval

Determination of the time (number of cycles) until the moment a crack becomes visible (i.e. fatigue initiation) is difficult. There are too many factors affecting initiation to perform an accurate analysis. Moreover, those factors cause a lot of scatter. Put two identical bridges next to each other with the same usage: The time to detectable crack lengths may differ significantly (orders of magnitude). On the other hand, the crack growth period from detectable to critical length is much easier to determine. For such an analysis, good calculation models are available. These models are used to determine inspection intervals. Such an analysis assumes that if no crack is found, a crack just a bit shorter than the detectable crack size may be present. The calculated inspection interval is the period a crack needs to grow to a length just below the critical size. Often, this interval is divided by a safety factor. In civil aerospace this factor equals 2.

What does this mean for the Merwede bridge

The Berenschot report states: “Vanaf half juli 2016 is belastingonderzoek gedaan naar de Merwedebrug door GPO en RWS WNZ in samenwerking met TNO en meetploegen. Bij belastingonderzoek doen meetploegen metingen en rekent TNO deze vervolgens door en toetst deze aan een model. De meetresultaten volgen de modellering; dit zegt iets over de betrouwbaarheid en voorspellende waarde hiervan”. [In short: Starting half July 2016 research on the Merwede bridge has been performed by GPO and RWS WNZ supported by TNO. Measurements are done and checked with a model by TNO. Measurements are in agreement with calculations]. Since the report also states that measurements are used for crack growth calculations I will assume that the measurements are crack length measurements. To my knowledge acoustic emission techniques were used to locate cracks and ultrasonic techniques were used to determine cracks lengths. In the end, cracks were found at the riveted joints in the bottom of both longitudinal beams. In one of the joints, the situation was critical.

It is reasonable to assume that a critical crack length has been determined prior to inspections. This requires the fracture toughness of the material. The fracture toughness can easily be established by testing, but is often derived from Charpy V notch test data. This derivation is not very accurate, so some conservatism will be used in the critical crack size calculation. Further, the fracture toughness itself shows scatter. Usually a minimum value for the fracture toughness is used, to reduce the probability of failure.

Apparently the measured crack length at the critical location was close to the critical crack size. Crack growth calculations indicated that the critical crack length would have been reached after 6 days. That does not mean that the structure would fail for sure after 6 days, it merely means that the propability of failure after 6 days would be equal to the propability of failure used for the calculation of the critical crack length, i.e. very small.

It looks like, based on the above, that the claim The Netherlands were close to a disaster is rather far-fetched. The probability of failure of the riveted joints after 6 days was very small, moreover the bridge would not have been completely collapsed but heavily damaged.

The decision to close the bridge for heavy traffic was a logical decision. The load on the bridge decreases, allowing a longer critical crack length.

Structural integrity

Open questions

Although a lot of information can be derived from the Berenschot report and other publications, I keep having a number of questions:

  1. What would have been happened if there was no study into expansion of the traffic capacity of the Merwede bridge. The way I look into it, there would have been no investigation into the condition of the bridge and cracks would have been remained undetected until failure of the joint. That would have been resulted in considerable damage of the bridge and possibly closure for all traffic.
    Although the bridge is already 55 years old and has covered a large part of that life without fatigue damage it does not mean there will never be any damage. Especially the increase of traffic intensity (more and heavier traffic) may lead to a situation that the fatigue life of a structure will shift from infinite to finite during its life. It seems to me that checking if boundary conditions used for making maintenance programs are still valid is not done systematically but only ad hoc basis. If my impression is correct, it is something to worry about.
  2. What was the goal of the measurement campaign started in July 2016?
    a. Regular inspection of cracks with an inspection interval as explained above? Or
    b. Extensive measurement campaign to obtain insight in the crack growth process in the bridge.
    The way I understand, it is option b. For both options you may ask: How often are the crack lengths measured in the riveted joints. Usually, the inspection interval is determined by the crack growth rate and divided by a safety factor. If the actual inspection interval was (much) smaller than the calculated interval, it is remarkable that the crack length in the critical joint was so close to the critical crack length.
  3. The critical crack length is calculated using the fracture toughness. From safety point of view this is of course fine. From economical point of view it is not. An aircraft being checked on cracks in a maintenance workshop is out of operation at that time. If cracks are found, they will be repaired inmediately. A bridge however is continuously in operation. If cracks are found close to the critical length it means restrictions to traffic until cracks are repaired. Taking the bridge out of operation is not acceptable. In case of the Merwede bridge, the traffic restrictions implied closure for heavy traffic. It seems covenient to me to use a kind of operational critical crack length, point B1 in the figure below, that is shorter than the calculated critical crack length. Such a crack length allows to use the period from point B1 to B for repairs etc. without restrictions to traffic.

Crack growth curve with inspection interval

  1. According to the Berenschot report the measurements agree with the calculation model. If that is correct, the question arises why there hasn’t been intervened earlier. Based on the model it should have been seen that crack lengths were close to critical sizes.
  2. Above questions are based on the assumption that inspections in October 2016 were not the first inspections on the riveted joints. The question is if this assumption is correct. The assumption seems very reasonable. A riveted joint is often a critical location in a structure. It is not logical to assume that such a location will inspected only after 3 months after beginning of the measurement campaign. The Berenschot report states that measurement results are in agreement with modeling results. This implies that there should be multiple measurements. However, the fact that in October 2016 cracks are found close to the critical size together with the startled reactions of the involved people suggests that this could be the first measurements anyway.

Used references


 

Principal Stresses vs. Equivalent Stresses in Fatigue

For those who are not too familiar with fatigue it seems always be attractive to use equivalent stresses like Von Mises for fatigue life analysis. I am sorry, but I have to disappoint you. An equivalent stress has nothing to do with fatigue, it is merely a calculation number to estimate the onset of yielding on macro scale in a multiaxial stress situation. Often for shear a factor of (1/3)√3 is applied on the fatigue strength for tensional fatigue. This factor gives a nice estimation, but that is pure coincidence.

Fatigue damage is propagating perpendicular to the largest principal stress range, therefore this stress range determines the fatigue behaviour. Principal stresses in tension in the other directions have hardly any influence on the crack growth, these stresses do not affect the shear stress in the activated slip planes. In case of shear (2D stress state with bi-axiality ratio of, or close to, -1), fatigue data for shear should be used.

3D stress states are hardly ever relevant for fatigue, cracks always start at a free surface. Even with sub-surface initiation it can be argued that the stress state is 2D, cracks start at inclusions or voids.

If there is a 2D or 3D stress state with varying largest principal stress direction, it is sometimes thought that using equivalent stresses is at least in this case attractive. This is still not true, equivalent stresses have no direction and certainly not a varying one. The best approach would be the “Critical Plane Approach”, i.e. analysing different crack growth directions (planes) and using for each plane the stress components (determined using the Mohr circles) perpendicular to that plane. Note that such an approach must be performed twice; viz. also for shear stresses.

What is the Difference between Low & High Cycle Fatigue?

The difference between low cycle fatigue (LCF) and high cycle fatigue (HCF) has to do with the deformations. LCF is characterized by repeated plastic deformation (i.e. in each cycle), whereas HCF is characterized by elastic deformation. The number of cycles to failure is low for LCF and high for HCF, hence the terms low and high cycle fatigue.
Transition between LCF and HCF is determined by the stress level, i.e. transition between plastic and elastic deformations. That implies that there is no fixed transition life, e.g. 103, but that transition life depends on the ductility of the material.

S-N Curve - Low Cycle Fatigue and High Cycle Fatigue

Large numbers of small cycles

Small cycles (i.e. cycles with a small amplitude) lead to longer fatigue lives than large cycles (i.e. cycles with a large amplitude). This fact may lead to the incorrect conclusion that small cycles can be neglected in a fatigue life analysis. However, in a spectrum the number of small cycles is often much larger than the number of large cycles. If so, the small cycles do give a significant contribution to the damage accumulation.

It is sometimes even thought that small cycles can be neglected, resulting in a small number of large cycles, which situation then erroneously is interpreted as LCF. Note that LCF corresponds with cyclic plastic deformations, not with small number of occurrences of large (elastic) cycles.

Stress peaks exceeding the yield limit

In some stress spectra a peak stress may incidentally exceed the yield limit. Does that make the fatigue process LCF? As long as the cycles are dominated by elastic strains, the fatigue mechanism will be typical for the HCF process. An incidental cycle with increased contribution of plastic strain does not change this.

If stresses close to a notch exceed the yield limit, some local plastic deformation will occur. Subsequent elastic unloading leads to an inhomogeneous stress distribution. At the edge of the notch there will be compressive stresses, which are in fact favourable for fatigue.

Damage Mechanism

The HCF mechanism is determined by cyclic elastic strains. Important parameters are the stress concentration factor (presence of a stress gradient), surface roughness and conditions and mean stress levels. The LCF mechanism is determined by cyclic plastic deformations. The parameters that are important for HCF have no impact on LCF.

Since the mechanisms are so different, different methods should be used for fatigue life estimation for HCF (using S-N data) and LCF (using e-N data). However, despite of the difference, it happens often that LCF methods are used for HCF analysis using local stresses (or strains). In a previous blog post (Fatigue Analysis using Local Stresses) and in the free eBook Common Mistakes in Fatigue Analysis it is explained why this approach is not correct.

 

Fatigue Crack in a Bicycle Frame

A friend of mine showed some time ago her bicycle to me and she asked if the crack in the seat tube would be a fatigue crack. Well, the answer is definitely yes and here is why:
In a classic diamond frame, the frame exists of two triangles: one formed by the top tube, seat tube and down tube and another one by the seat tube, seat stays and chain stays. Those two trangles make the frame very stiff. The tubes and stays are loaded in tension or compression, but there is hardly any bending.
The bicycle in the picture below is a typical Dutch ladies bicycle. It is a modification of the classic diamond frame; the top tube is lowered to allow women wearing a skirt or dress to mount and ride the bicycle in a convenient way.

fatigue crack in a bicycle

Due to lowering the top tube, this tube is not connected to the seat tube at the same location (just below the saddle) as the seat stays anymore. As a consequence, the top tube will introduce a bending moment in the seat tube and with that: increased stress levels. Further, the torsional stiffness of the frame is reduced considerably. The picture below the fatigue crack at the weld between seat tube and top tube. This is a fatigue crack that could have been easily avoided with proper design.

fatigue crack in a bicycle

 

Fatigue Analysis using Local Stresses (part 2)

In part 1 of this blogpost it was discussed that also for fatigue analysis using local stresses, the gradient of the S-N curve depends on the stress concentration.
But also the stress ratio R=Smin/Smax of a stress cycle affects the gradient. Often S-N curves are given for R=-1. To obtain the S-N curve for a different value of R some corrections are made. The upper asymptote of the S-N curve is determined by the condition that Smax≤Rm. With increasing mean stress level, the upper asymptote will decrease by the mean stress value.
The lower asymptote, i.e. the fatigue limit, will also change with changing R, but to a lesser extend. For R≠-1, the fatigue limit gets a mean stress correction according to Goodman, Gerber or some other method. The correction depends on the R value, but also on the ductility of the material. Since the corrections on upper and lower asymptote are different, S-N curves tend to become flatter with increasing R or mean stress. This applies to S-N curves based on nominal stresses, but also on S-N curves based on local stresses (the difference between both is just the factor Kt).

Many codes and FE fatigue tools use a fixed gradient with k=5 in de equation (Sa^k)*N=c. In reality, the value of exponent depends on the  stress concentration, the stress ratio and the material. Using k=5 is too much of a simplification, in practice values of k vary between 3 and 15.
Welded joints can be considered as an exception; using k=3 is acceptable because fatigue of welded joints is independent of mean stress levels and the stres concentrations are that severe that a change in welding detail has very small impact on the gradient.

Fatigue Analysis using Local Stresses (part 1)

The common approach for fatigue life analysis is using S-N curves based on nominal stresses (i.e. average stresses in the net section). Depending on the severity of the stress concentration (Kt) in the structure, a S-N curve is then selected that corresponds with the actual structure. In the diagram below a few S-N curves are shown for different Kt values. The larger the Kt, the lower the fatigue limit and the steeper the S-N curve (smaller value of k in the S-N curve equation (Sa^k)*N=c).

Very often, structural analysis is performed using Finite Element Analysis (FEA). Such an analysis gives local strains and stresses. Quite often, FEA is used because of the complexity of the structure. If so, nominal stresses and Kt are difficult (or impossible) to determine. Under those conditions, fatigue analysis using local stresses looks attractive, using the peak stress at a stress concentration as local stress.
In case the fatigue analysis is performed using local stresses, the S-N curves based on nominal stresses cannot be applied, instead S-N curves based on local stresses must be used. In the diagram below, a set of S-N curves is shown. These curves are derived from the S-N curves based on nominal stresses shown above, by multiplying the nominal stresses by Kt to obtain the local stresses. By doing so, an analysis using local stresses will give the same results as an analysis using nominal stresses, under the condition that for both analyses the S-N curve for the same Kt value is used.

This condition however gives a problem. The attraction of using local stresses in the fatigue analysis is based on not knowing the Kt. The actual problem is the fact that a local stress can be any combination of nominal stress and Kt, each giving a different result in a fatigue analysis. Just knowing the local stress is unfortunately insufficient input for the fatigue analysis.
The S-N curves based on local stresses show different fatigue limits and different gradients for different Kt values. The difference in fatigue limit can be accounted for by making corrections for notch sensitivity. But such a correction is not possible for the gradient. When looking into the S-N curves, the gradients for Kt=2 and Kt=4 may look at first sight not so different, the S-N curves however are plotted on log-log scale. Using the wrong Kt (i.e. wrong gradient) in the analysis may lead to significant errors in the outcome.

Many codes and FE fatigue tools use a fixed gradient with k=5 in de equation (Sa^k)*N=c. This is oversimplified. The value of k depends on the stress concentration, but also on the stress ratio of the stress cycle as will be discussed in part 2 of this blogpost.

Description of a S-N Curve

Fatigue properties of materials are often described using the fatigue limit or the S-N curve (fatigue curve, Wöhler curve). The S-N curve describes the relation between cyclic stress amplitude and number of cycles to failure. The figure below shows a typical S-N curve. On the horizontal axis the number of cycles to failure is given on logarithmic scale. On the vertical axis (either linear or logarithmic) the stress amplitude (sometimes the maximum stress) of the cycle is given.
S-N curves are derived from fatigue tests. Tests are performed by applying a cyclic stress with constant amplitude (CA) on specimens until failure of the specimen. In some cases the test is stopped after a very large number of cycles (N>10^6). The results is then interpreted as infinite life.
Fatigue curves are often given for Kt=1 (unnotched specimens). Those curves describe the fatigue properties of a material. Actual structures are better described with S-N curves for Kt>1 (notched specimens).

Fatigue S-N Curve
The S-N curve above has some characteristic features which are discussed below.
Fatigue Limit: For some materials (steel and titanium) there is a stress level (lower asymptote in the S-N curve) below which the material will not fail. This stress level is known as the fatigue limit, endurance limit or fatigue strength. For materials like aluminium, magnesium, austenitic steel, etc. the fatigue limit is not very distinct.
The level of the fatigue limit depends on many factors, such as geometry (stress concentration factor Kt), mean stress (stress ratio), surface conditions, corrosion, temperature, and residual stresses.
High Cycle Fatigue (HCF): In this region the material behaviour is fully elastic. On log-log scale the S-N curve can be considered to be linear.
Low Cycle Fatigue (LCF): If the maximum stress level in a cycle is exceeding the yield strength, the material  behaviour in the net section will be predominantly plastic. Number of cycles to failure will be very small, hence the term LCF. Usually a strain-life curve instead of the S-N curve is used to described fatigue behaviour.
Note that the actual distinction between HCF and LCF is not defined by a certain number of cycles but by the amount of plasticity in the net section, i.e. the stress level.

What is Fatigue?

Metal fatigue is about the predominant cause of failure of structures. But what is fatigue? Fatigue occurs when a structure is subjected to cyclic loading. If the stress amplitude exceeds a threshold value, microscopic cracks will initiate at locations with high stresses (stress concentrations). At first, the cracks propagate very slowly and remain undetectable for the bare eye for most of the fatigue life. Gradually the crack propagation rate increases and the cracks will become visible. Eventually the crack will reach a critical size and the structure will fail. Due to the nature of the fatigue process, fatigue failure can lead to safety issues.

The stress levels that cause fatigue damage are much lower than the static strength of the material, i.e. ultimate tensile strength and yield strength. Decisive for fatigue damage propagation are stress amplitudes; it is cyclic loading that determines fatigue.

Many factors play a role in fatigue, such as incorrect choice of material, rough finish or damaged metal surface, poor maintenance, including failure to timely replace a part. The shape of the structure will significantly affect the fatigue life; square holes or sharp corners will lead to high local stresses where fatigue cracks easily can initiate. Round holes and smooth transitions or fillets will increase the fatigue strength of the structure.

Some fatigue characteristics

  • Fatigue is a structures issue, not just a material issue.
  • Stress concentrations (holes, keyways, fillets) and locations with secondary bending are common locations at which fatigue cracks initiate.
  • Fatigue often shows significant scatter.
  • The larger the stress amplitude, the shorter the fatigue life.
  • No large scale plastic deformation.
  • Damage is cumulative. Unlike humans, materials do not recuperate from fatigue.