Focussed on Multiaxial Fatigue and Fracture Structure service life assessment under combined loading using probability approach

This article presents a basic research approach for probabilistic assessment of vehicle structure fatigue endurance under multiaxial loading. One representative welded structural detail of the bottom joint plate of an articulated bus was selected for the case study. The character of the loading on the joint during a service test is analyzed and discussed. The methodology for estimation of the damage level in the joint builds on deterministic procedures for railway vehicle assessment based on shear, longitudinal and transverse stress components. This approach based on the equivalent stress and particular load capacity factors was expanded to include the probabilistic perspective.


INTRODUCTION
n-service loads acting on components and structural details in vehicles tend to be complex and, in most cases, multiaxial and time-variable.The stress response to such loads can (but need not) be multi-axial.Critical locations in vehicle structures are subjected to fatigue loading which can lead to initiation of fatigue cracks, fatigue crack propagation and, sometimes, to final failure.Even today, sizing and assessment of such critical locations often involves the deterministic approach.This means that both loading and strength are represented by their mean values for which adequate safety factors are sought.In the following, a description is given of fatigue lifetime assessment of a welded joint operating under relatively complex loading in an articulated bus.The engineering approach which is normally used in technical practice has been expanded I into a reliability analysis which gives a more accurate description of the risk of fatigue failure in the critical location of the structural detail.

SERVICE LIFE ASSESSMENT USING PROBABILITY APPROACH
aterials subjected to time-varying loads suffer damage caused by the fatigue process.Their load response can be described in terms of time-varying strain and stress, and the cumulative fatigue damage can be found using an appropriately chosen rule.In order to evaluate the service fatigue life of a structure, one needs not only information about its service loading but also some data on the structure's fatigue strength.The resistance of a structure to high-cycle fatigue damage is described by the S-N curve.The S-N curve can be constructed using fatigue data from a sufficient number of test pieces representing the structural detail under examination.However, it can also be determined by estimate or obtained from one of the standards for design of structures.Fatigue life assessment is often based on the deterministic approach which relies on mean values of load-bearing capacity and load, and a set of factors of safety.If, however, the values of the safety factors are not chosen correctly, the load and the fatigue resistance of the material may prove to be mismatched under real-world service conditions, which may lead to fatigue damage or even failure of the part.Probabilistic approach, on the other hand, uses distributions of random input variables for finding the fatigue life distribution function, i.e. the probability that the material enters its limit state with respect to strength after a certain period in service.Fatigue life estimates are based on cumulative fatigue damage rules.Conformity to the life requirement may be formulated as the part's reliability.This means that over the required life treq the probability of fatigue failure does not exceed the allowable value P allow .Statistical characteristics of input variables are obtained from measurement and tests (service loading measurement, fatigue testing), determined from experience, or derived from standards and codes.Important to the life estimates are the relationships between input variables, e.g. in the form of cumulative fatigue damage rules, and the mean stress of loading cycles.Comparison can be done using several different rules.This, however, enlarges the variance of the lifetime estimate.Structures are normally designed to the guaranteed design life tg and failure probability Pg.These values are guaranteed with certain margins Δt g and ΔP g , and therefore t g >t req , P g < P allow (Fig. 1).
Figure 1: Schematic illustration of the probabilistic approach to life estimation -position of the point of design life of the structural detail with respect to the FLDF While material properties of parts of equipment are given before it is put into operation, the actual service loads can only be estimated for an already-known steady-state process.Changes to the loading conditions or out-of-standard load states affect the form and position of the fatigue life distribution function (FLDF).They may shorten the life margin Δt even lead to a loss of reliability at Δtg < 0. Such occurrences must be prevented by identifying and signaling the onset of hazardous states.The way to achieve that is to continuously monitor the service loads on critical locations of the structure and track (calculate) the fatigue damage in them.
To find the fatigue life distribution function, one needs the following input information.a) Service loads on the critical location.Continuous stress-time histories can be obtained from appropriate monitoring devices, for instance.Typical input into fatigue damage calculation is represented by a two-parameter histogram of cycle frequencies, reflecting cycle amplitudes and mean stress levels.Most often, this histogram (also referred to as "load spectrum", "stress spectrum", and otherwise) is obtained by the rain flow method but other techniques can be used as well.Signals obtained from the monitoring sometimes show trends (changes in variance with time, shifting mean value, and others) which can be accounted for in the fatigue damage predictions.b) Material properties.The input into the calculation is an experimentally measured and statistically evaluated S-N curve, or more precisely, an implicit representation of its left-sided tolerance limits for various failure probabilities.
The probabilistic evaluation of service fatigue life of a structural detail of a road vehicle which was carried out in the present case study was based on the segmentation of random loading processes proposed by Kliman [5,6].

Random loading process
The loading process of interest is analysed by appropriately segmenting a sufficiently long in-service measurement record from a critical location of the relevant part of a vehicle.A load record of certain length represents a random portion of a vehicle's service.Obviously, another measurement carried out at another time will be different due to the random nature of the load.If a loading process record of adequate length σ(t) is segmented appropriately, it can be substituted for repeated measurement runs.In-service loads will thus be represented by a set of records -process segments σ(t).
In order to calculate a faithful FLDF, one has to find the minimum acceptable length of the process segment.In practice, this means finding such length, at which the standard deviation and mean value of the calculated FLDF become stable (i.e. they do not change substantially with further increases of the segment length).The segment length is understood as a certain portion of the service run represented by time, mileage or otherwise.The procedure is as follows.The stress-time history is analyzed using the rain flow method applied to a series of segments of a constant length where the overlap between these consecutive segments ("windows") is 95 %.For each "window", the fatigue life is calculated.Mean value and standard deviation are calculated for each set of lifetimes obtained with a certain segment length.The procedure starts with the shortest segment length and continues by increasing the length.An example of such procedure is illustrated in Fig. 2 which also shows the chosen optimal segment length.The set of lifetimes obtained for the optimal segment length determined by the above procedure will be input into the FLDF calculation.Lifetime is calculated using an appropriate cumulative fatigue damage rule and the theory of accounting for the effect of the mean component of cycles.

Random nature of materials properties
Low-cycle or high-cycle fatigue properties of materials are described by the cyclic stress-strain curve, the lifetime curve or the S-N curve.These (or their coefficients) can be determined on the basis of fatigue tests.However, one can also estimate their values using various empirical formulas or find them in certain standards and codes.Statistical evaluation of fatigue tests can provide confidence intervals and tolerance limits for the chosen probability of a particular curve and the corresponding coefficients.Such limits are often difficult to define analytically by means of simple equations.They can be somewhat simplified by using linear relationships.This is common practice with fatigue curves for typical structural details listed in various categories in standards.There, the left-sided limits of tolerance are simply expressed in the form of the residual standard deviation and the d quantile for the probability in question.Kliman's methods requires that the parameters of the left-sided limits of tolerance of the lifetime curve are calculated for the probabilities P = 1 through 99 % in steps of 1 % (Fig. 3).

USED METHODOLOGY
he assessment of the endurance of the welded detail in question relied on a methodology which is used for rail vehicle testing in Germany.Allowable stresses were determined on the basis of nominal values and estimated notch factors according to IIW [1] and FKM [2] standards.Using the measured stress-time series in the location of the strain gauge rosette, stress components perpendicular to the weld σ, parallel to the weld σII, and shear stress components τ xy were calculated.A similar approach is used for demonstrating the fatigue life of railway vehicles on test rigs [3,4].Decomposition of these series by means of the rainflow technique yields a matrix of stress amplitude rates and mean values of the cycle.The resulting matrices are combined into a single macro block on the basis of representative samples of service runs.In our case, the matrices were combined from four half-hour runs of an empty vehicle and four runs of a vehicle fully-loaded with a 1:1 passenger-equivalent load.The resulting matrix is converted to a single-parameter histogram of amplitude rates, accounting for the sensitivity to the mean value by using the coefficient M = 0.15 for normal stresses and M = 0.09 for shear stresses, as expressed in Eq. (1) Using notch factors, the measured stress values are transformed to notch stress values.Notch factors can be found in various sources.Here, they follow the principle used in the FEMFAT software for the notch radius of r = 1 mm.The slope of the S-N curve for weldments is set as m = 3 for normal stresses and m = 5 for shear stresses.The characteristic stress amplitude for N = 2•106 and for the base material S355, which is the material of the structural detail in question, and for normal stresses and a completely reversed cycle is σ a,R=-1 = 150 MPa, whereas for shear stresses and a completely reversed cycle it is τa,R=-1 = 87 MPa (√3•σa,R=-1).These values are derived with respect to the IIW standard which has been developed for the stress ratio of R = 0.5.The S-N curve has no inflection point.It has been extended all the way to half the amplitude achieved upon N = 5 million cycles.This approach is more conservative than that used by the FKM standard.The allowable fatigue damage is D = 1.The above approach relies on the calculation of load capacity factors a. Fig. 7 explains the calculation.An aggregate macro block is determined from individual load matrices.This macro block is converted to equivalent stress amplitude which would cause the same amount of damage as the entire macro block.The ratio of this equivalent stress amplitude σa,ef and the allowable stress amplitude σa,BK for the given number of cycles Nd is the load capacity factor a.
For each stress amplitude component σ a,, σ a,II , τ xy , the local load capacity factor a is calculated using Eqs.( 2) through (4).Finally, a complex load capacity factor involving all three components (5) is determined in order to account for the multiaxial state of stress in a manner similar to yield criteria.If this factor is less than 1, sufficient fatigue strength of the detail in question has been demonstrated.

T
Figure 7: The relations between the effective stress amplitude σ a,ef , the limit stress amplitude σ a,BK and the load capacity factor a with respect to the S-N curve and the total number of cycles Nd.

PROBABILITY APPROACH FOR EVALUATION OF COMBINED LOADING
he methodology outlined above represents the deterministic approach.In a probability-oriented assessment, the random nature of the loading, as well as the random nature of materials properties must be taken into account.A load record of certain length represents a random portion of a vehicle's service.Obviously, another measurement carried out at another time will be different due to the random nature of the load.If a process record of adequate length σ(t) is segmented appropriately, it can be substituted for repeated measurement runs.In-service loads will thus be represented by a set of records -process segments [5].By this method, the segment length for a representative lifetime assessment and the variance of load spectra used in the lifetime calculation can be determined.In Fig. 8, this procedure is applied to the complex load capacity factor based on (5) which is calculated within a moving window of L size.In the figure, mean values and standard deviations for the set of values atot obtained are plotted for the expanding interval L. The segment length, at which the value of a tot ceases to change significantly, is used for calculating the load capacity factor distribution function.In the present case, a distance of 21 km has been used.The random nature of materials properties is reflected in the tolerance limits of the S-N curve.The variability of the number of cycles to failure N is expressed through the standard deviation s(log N) for the quantile d, depending on failure T probability.For this purpose, a median S-N curve must be obtained first -by shifting the curve defined in the standards for 97.5 % failure probability.The S-N curve equation is as follows The actual probabilistic assessment of fatigue strength is carried out using an MS Excel sheet.First, the distributions of individual components of the load capacity factor for corresponding tolerance limits of S-N curves are found.Then, the distribution of the combined load capacity factor is calculated.The real distributions are approximated by lognormal distributions

OBTAINED RESULTS DISCUSSION
he above procedure was applied to the measured location of the articulated joint (Fig. 5).Distributions of individual components of the load capacity factor were determined.Fig. 9 shows example plots of their distribution functions using the median S-N curve.In this case, the random nature has been accounted for on the part of the load only.

Figure 2 :
Figure 2: Example of determination of minimum segment length

Figure 3 :
Figure 3: Schematic illustration of the method of finding materials parameters -standardized fatigue curves.

Figure 6 :
Figure6: Absolute maximum principal stress against the angle to the x direction, absolute maximum principal stress vs. biaxiality ratio.

Figure 8 :
Figure 8: Determination of the moving window for the probabilistic characterization of the vehicle run.

Figure 9 :
Figure 9: Probability distribution of particular loading factor components ax, ay and aτxy.