Equivalent fatigue load approach for fatigue design of uncertain structures

Journal Pre-proofs Equivalent fatigue load approach for fatigue design of uncertain structures Ida Raoult, Benoit Delattre PII: DOI: Reference:

S0142-1123(20)30047-5 https://doi.org/10.1016/j.ijfatigue.2020.105516 JIJF 105516

To appear in:

International Journal of Fatigue

Received Date: Revised Date: Accepted Date:

9 October 2019 2 January 2020 28 January 2020

Please cite this article as: Raoult, I., Delattre, B., Equivalent fatigue load approach for fatigue design of uncertain structures, International Journal of Fatigue (2020), doi: https://doi.org/10.1016/j.ijfatigue.2020.105516

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Equivalent fatigue load approach for fatigue design of uncertain structures Ida Raoulta,∗, Benoit Delattrea a

Science and Future Technologies Department, Groupe PSA - Centre Technique de V´elizy, CC VV1415, Route de Gisy, Parc Inovel Sud, 78943 V´elizy-Villacoublay Cedex, France

Abstract Industrial structures undergo complex loading, often unsuitable for the design and validation phases of their development. Engineers seek simplified signals to replace them, yet equivalent in terms of fatigue. A general framework for the construction of equivalent fatigue loads is presented, intended to take into account some uncertainty on the structure of interest. The guiding idea is to describe this uncertainty through parameters of the structure model, and then to assure the equivalence of damage or failure behavior over the whole range of expected parameters. The approach is applied in the case of structures undergoing multiple channels of concentrated forces. Keywords: Fatigue design, Fatigue load spectra, Multiaxial fatigue, Uncertainty 1. Introduction 1.1. Loads for tests and simulation Engineers wish to design a mechanical structure against fatigue so that it ensures its function for a given time and in most use cases. Here, the structure is considered defective as soon as a crack initiates, even if, in practice, it can often continue to fulfill its function with a crack. The choice of an initiation failure criterion is a conservative choice, often made in the automotive industry, because there is no obligatory maintenance except for the legal vehicle inspection. It also greatly simplifies the simulations since the geometry of the components does not change during its lifetime. Loads on the structure can be measured directly in service, or in conditions representative of the use. A vehicle is equipped with multiple sensors and then driven on an open road or a test track. The forces applying to the component of interest are recorded; there are usually several channels of them, either that the part has several interfaces with the rest of the structure, or that there are several distinct components of forces, acting on different directions, at these Corresponding author, tel : (+33) 9 66 66 67 71 Email addresses: [email protected] (Ida Raoult), [email protected] (Benoit Delattre) ∗

Preprint submitted to International Journal of Fatigue

interfaces (see figure 1). The resulting record is then used as a design requirement. The structure must withstand this sequence of forces, repeated as many times as needed to guaranty its target strength. This target behavior can be determined by the stress-strength interference method [1]. The choice for such a load sequence used as a strength requirement is the responsibility of the carmaker since no external certification exists for fatigue strength, although there were various attempts to establish some standard loads for the automotive suspension system [2, 3]. The strength of the components is estimated by simulation and demonstrated by physical tests. Unfortunately, the previously mentioned recording is generally unsuitable for both, because of its complexity and duration. First, consider the case of simulation. When the structure of interest responds linearly (linear elasticity in small deformations, absence of contact, etc.), a reduced number of simulations, corresponding to the unit response of the structure under each independent degree of freedom of the loading (e.g. for each force channel) allows to reconstruct the solution under the complete loading by linear combination. When the structure has a non-linear response, like a car suspension, this operation is no longer possible and each instant of the signal must be calculated. In practice, the necessary computation times are excessively long. The tests present even worse difficulties. First, January 2, 2020

Nomenclature F

Load vector

σL

Unit localization tensor

D

Material damage function

L

Localization function

D ◦ L Structural damage function pL

Material damage function parameter vector

pD

Localization function parameter vector

p

Structural damage function parameter vector

p ˜

Structural damage function parameter vector verifying D ◦ L = 1

λeq

Equivalent signal parameter vector

λ∗eq

Equivalent signal parameter vector, solution of the equivalence equation

10

−2 −4

FZ [kN ]

12

0

8 0

20

40

60

80

100

6

10

4

5

2

FZ [kN ]

FY [kN ]

2

0

0 0

20

40

60 time [s]

80

−4

100

−3

−2

−1

0

1

2

FY [kN ]

Figure 1: A few minutes record of transversal and vertical force measured at the center of a passenger car wheel on a test track. Moments are not shown.

the signal is difficult to reproduce on a test bench. Each channel must be controlled correctly regardless of the value of the other channels, a difficult task since each structure exhibits its own couplings. Moreover, the signal can contain high frequencies that hydraulic machines cannot reproduce correctly. Secondly, the signal is long (usually, several weeks) in comparison with the typical times of a development project. When the part response is independent of the loading rate, it is possible to accelerate the signal by a simple compression of the time base. Unfortunately, for large structures, the machines capabilities prevent a significant increase of the loading rate. The resulting duration and complexity of the tests usually imply high costs. In both cases, it is desired to substitute this com-

plex load by a simpler one; simpler, but nevertheless equivalent - in the sense of the strength criterion to the original signal. Let us call it the equivalent signal. The objective of this paper is to propose a general framework for constructing these equivalent signals. Note that even if this work was performed in the framework of the automotive industry, the question addressed here is of interest for many other industries, especially in the transportation domain [4] but not only [5]. 1.2. Dealing with uncertainty When the structure is fully designed, the damage under the original signal may be assessed by simulation, using some damage model. The process can 2

be repeated with any other simplified signal until it induces the same damage. This is a procedure used, for example, to choose the testing conditions for the validation of wind turbines [6]. However this method is not applicable in many cases. Firstly, the calculations may be excessively long, especially when the structure exhibits a non linear behavior. Secondly, the imperfection of models, in terms of geometry as well as material behavior, or uncertainty on some fatigue model parameters, may induce a significant error [6]. This error can be reduced by multiplying the configurations of the simulations, so as to take into account a variety of cases [7], but then again, the cost of simulation might be too high. To workaround the calculation time problem, it is possible to replace the detailed model by a simplified one, built on the basis of a statistical model and a set of simulations, as in [8, 9]. Nonetheless, even those sophisticated methods cannot solve the main problem encountered in many design projects: when the validation test signal is required, the component is not designed yet, or only in an early version. This situation happens, in particular, when the component is bought from a supplier. The signal to withstand has to be known in advance; it is part of the specification. Even in a fully internal design process, the first stages of design need simplified specifications, robust to small variations of geometry or material. The equivalent signal should be described by few parameters. Those parameters are easy to compare from one project to the other, which helps classifying technical solutions. Moreover, the distribution of these parameters for a given application can be used for building specification standards, very useful in the first stages of design. The parameters of the equivalent signal can also be used for reliable design as a stress variable in the stress-strength interference method [10].

with σL the local stress tensor experienced at point M per unit force F . The material is given and we assume that its SN curve can be simply modeled with the Basquin law:  −β σa (2) N= σ1 with σ1 and β some known material parameters and σa the amplitude of the principal stress (or any invariant of the local stress tensor). Although the material behavior usually features a knee point [12] around a few million cycles, it is seldom considered in the equivalent fatigue process. The sequence effect is also ignored. The force signal is decomposed into unit cycles using the Rainflow procedure [13]. Since the invariant exhibits a multiplicative scaling behavior, a cycle of the local variable responsible for damage (here, the principal stress) is just scaled from the force cycle and their amplitude is: σa (M, j) = σL (M )Fa (j)

(3)

with Fa the amplitude of the force cycle, σL the principal stress of σL and j the index of the Rainflow cycle. The linear Palmgren-Miner accumulation law gives the local damage d such as:   σL (M ) β X β Fa (j) (4) d(M ) = σ1 j

A fatigue damage equivalent signal is searched in the form of a constant amplitude signal with a given number of cycles Neq (usually close to the apparent endurance limit or knee-point of the SN curve, e.g. 106 cycles). The signal is said to be equivalent if it yields the same damage as the original signal:     σL (M ) β X β σL (M ) β β Fa (j) Fa eq = ∀M, Neq σ1 σ1 j

(5) In a very convenient way, the localization factor σL (M ) and the Basquin constant σ1 vanish, so that the condition can indeed be satisfied for every point of the structure with a unique Faeq , given that the exponent β is known: 1  β X 1 β   . (6) Fa (j) Faeq = Neq

1.3. Equivalent fatigue load approach in the literature 1.3.1. Linear elastic quasi-static structure, 1D load The equivalent fatigue load approach is mainly used with linear elastic structures undergoing high cycle fatigue under a quasi-static 1D concentrated force F [11]. The loading is 1D in the sens that it is described by a single scalar, but it varies over time. Then, the stress at some time t and point M writes simply as: σ(M, t) = σL (M )F (t) (1)

j

A mean stress correction such as Goodman’s or Gerber’s can easily be introduced in these equations,

3

but an additional information on the structure will be required: the ultimate force Fu of the structure. It cannot be simply deduced from the material parameter σu because in this case the unknown localization factor does not vanish. The equivalent signal can then also have an additional parameter since its mean value influences its damage. Problems arise when the exponent β (or Fu ) is somewhat uncertain or known to be multiple (in the case when the structure is composed of several materials or suffers from several damage mechanisms). Each β will then yield a different Faeq . Many authors have performed some sensitivity study of the equivalent fatigue load to material parameters (see [14] for example) to conclude that this was significant. Thomas [15] proposes a method to circumvent the problem of the uncertainty on the mean stress correction parameter σu . The localization factor is supposed to be such that the structure fails exactly at the end of the original signal (σL such that d = 1). Thus, the fatigue equivalence is realized for the structures of interest: the ones that might fail. As will be stated later in this document, the meaning of the term ”equivalence” is different in this case. Freebury and Musial [6] propose to choose the parameters of the equivalent load so as to minimize its sensitivity to the uncertain parameters (Basquin’s exponent and ultimate load of the structure), estimated on a whole given range of possible parameters for their wind turbine blades application. This solution is not completely satisfying, because some error will remain; however it is possible to quantify this error on the range of the uncertain parameters.

The above equivalence method (without any mean stress correction) can then be applied frequency by frequency and the so-called ”Fatigue Damage Spectrum” is built. A Gaussian synthetic signal can be found with the appropriate spectral content to ensure damage equivalence over a whole frequency range. It should be noted that: • The behavior of the structure is approximated; the quality of the approximation (damping characteristics, for instance) should be verified a posteriori ; • The equivalence is given on a selected frequency range; it should also be checked that it encompasses the actual characteristics of the service load of the component; • The frequency space is discrete. Indeed, each considered frequency gives an equation for the synthetic signal to satisfy. It must then have as many degrees of freedom (parameters) to calibrate. This method is an improvement over the quasistatic case with unknown parameters since the equivalence is ensured for a whole range of the unknown parameter (the natural frequency), the cost of it being a high number of parameters for the equivalent signal (the power spectral density). 1.3.3. Non-linear behavior, 1D load When the structure exhibits a non linear behavior, the determination of a localization factor can dangerously impair the equivalent load approach. However, it is sometimes possible to approximate the response of a family of similar components. Szmytka et al. [18] describes this approach for cylinder heads made of aluminum and subject to thermomechanical fatigue. The load is a thermal flux and the local damage a dissipated energy density. In this case, failure always occurs at the same location and the damage at this point can be obtained from simulations on multiple components, previously designed. In this study, the approximate surrogate model was proposed in an empirical way. Techniques such as parametric model order reduction [19] might help to automatize the approach. Anyway, this approach strongly relies on the perpetuation of the design of the component. Once the component is designed, the validity of the approximate response should be checked. This also helps improving iteratively the quality of the surrogate model.

1.3.2. Linear elastic vibrating structure, 1D load The case of a linear elastic structure expected to vibrate under some 1D load is extensively documented, especially for accelerated vibration testing. The approach exposed above no longer holds because the localization stress is no longer independent of time. It depends on the dynamic response of the structure. To solve the problem, Lalanne [16, 17] uses an approximation of the typical response of the structure: it is assumed to behave like a mass-spring system with a given damping coefficient and some unknown natural frequency, but on a given frequency range. Thus, for each frequency f of this range, the local stress is proportional to the output force signal obtained by filtering the input force with the corresponding mass-spring system transfer function. 4

basic assumption of proportional stresses (equation 7) was deemed too strong for most industrial applications. A close look at critical points on many structures yet often reveals notched areas with strongly oriented local stresses, indicating a possible application of the method. Anyway, this hypothesis can be relaxed, as will be seen further in this paper. Recently, Roux[23] proposed an alternative to avoid this hypothesis. The equivalence is sought for every point of a finite number of similar structures already in use; in this case, train wheels. The local stress tensors are taken at each integration point of a finite element model of each wheel. He shows that the equivalent signal obtained for each wheel is very similar, and can serve as a base for a specification test for future wheels. The advantage of this method is that it takes into account the effective response of the structure, but a major drawback is that it applies only to a family of structures with very similar geometries, excluding innovative design. An improvement of the method could be to precise the scope of application of the method, using parameterized geometries in the library of structures used for determining the equivalent fatigue load. This introduction reviewed several methods for simplifying the loads used either for testing or as an input for simulation, with a special focus on their ability to take into account some uncertainty on the structure of interest. Some of these methods are practiced everyday in the industry, while others are still in their infancy, trying to take into account more complexity in the structure behavior or the loads themselves. They address various situations, in which the uncertainty on the structure is either related to its geometry or its material parameters. Even the concept of equivalence appears to be slightly different from one study to the other. However, the authors believe that there is a general philosophy of equivalent fatigue load and a need for a proper mathematical description. The next section is a first attempt at giving a general framework for equivalent fatigue load methods. Then, an application is proposed for the case of multichannel input, which can be considered as a extension of Genˆet’s work. The last section discusses the assets and drawbacks of the method and suggests further investigations.

1.3.4. Linear elastic quasi-static, nD load The case of components subjected to multiple inputs of loads is considered as difficult (see [20], Chapter 6), even in the simple case where the linear superposition principle holds, because of the unknown interactions of the different loads at the local scale. Industrial practice is not well documented but to the authors’ knowledge usually consists in analyzing each channel independently. Sometimes the average correlation is taken into account by considering the principal directions of the forces. In 2006, Genˆet [21] establishes the first basis for a multi-input equivalent approach in the case of a linear elastic structure undergoing a finite number nF of forces (Fi )i∈[1,nF ] . She makes a major assumption about the local behavior of the structure under these loads: each force Fi induces at the potential critical point M the same form of stress tensor. Hence the local stress writes: σ(M, t) = σL (M )C(M )

nF X

ai (M )Fi (t)

(7)

i=1

with σL a unit localization tensor (i.e. kσL k = 1 for a given norm), (ai )i∈[1,nF ] a unit vector (i.e. such that kak = 1) of coefficients giving the relative importance, locally, of each input force Fi , and C an average intensity factor. Then, using either a coarse damage model (such as Basquin’s model) or a more sophisticated one (a simplified version of Morel’s model [22]), the damage of the original signal and that of an equivalent signal can be written. When equating those two damages, the localization stress tensor σL vanishes, so that the only point (M ) dependent terms remaining in the equation are C and the ai . The author then proposes to browse the ai (i.e. the different possible combinations of Fi ) and to look for an equivalent fatigue load such that its damage would be as close as possible to the damage induced by the original signal for all the ai . With Basquin’s model, the intensity factor C vanishes (as in the 1D case described in paragraph 1.3.1) as opposed with Morel’s criterion. There, the model describes a fatigue limit, so that the intensity factor C greatly impacts the calculated damage by selecting the threshold above which the signal induces damage. The author then proposes to reduce the damage equivalence to special structures of interest, with C factors such that the damage of the original signal is exactly one. This interesting approach was not given much more attention in the following years, probably because the 5

The structural damage function is then the composition of the localization function and the material damage function, written as D ◦ L. Note that the explicit decomposition in two steps as previously described is not necessary. One may search directly for the function D ◦ L through the construction of a parametric meta-model, for example. For simpler notations the structural and material parameters pL and pD can be grouped in a single vector p.   (F(t))t∈[0,T ] D◦L −−−→ d (10) p(M ), M ∈ Ω

2. Formalization of the approach The method described below is a generalization of the one exposed in [24] in the case of a linear elastic structure. 2.1. Position of the problem Consider a structure, submitted to a finite (usually small) number nF of global loads F = (Fi )i∈[1,nF ] . These global loads are usually forces, hence the choice of notation, but can be any kind of variable applied on the structure at a global scale, such as displacement, acceleration or even temperature. These loads vary over a time interval denoted [0, T ].

The function D ◦ L describes the behavior (in terms of damage) of a family of structures among which can be found the structure that will indeed be designed. The space of parameters p might be bounded in some directions by physical requirements. For example, if the Young modulus is involved, it should be positive. Moreover, this space might be restricted by the engineer, because he has some prior information about the structure to be designed like a choice of materials, fabrication process or general ideas about the shape of the structure. For example, the Young modulus of the material of an automotive structure is likely to be lesser than 210 GPa, since steel is the stiffer material widely used in this field. However, theses assumptions have to be clearly stated because they restrict the applicability of the equivalent signal. Note that if two points M1 and M2 have the same parameter p, their damage will be equal. All the local information is contained in p. Consequently, it is equivalent to consider two points from a unique structure or two points from two different structures.

2.2. Structural damage model The structure Ω fails if it does so in at least one location. A damage variable d is evaluated in each point M . Failure happens when at least one point M can be found such that d(M ) = 1. A model is needed to relate the damage in each point to the loading undergone by the structure. This model is usually composed of two elements: a localization function and a material damage function. The localization function L associates, for each point M , the history of global loads F(t), t ∈ [0, T ] and structural parameters pL to the history of local variables σ(t), t ∈ [0, T ].   (F(t))t∈[0,T ] L −→ (σ(t))t∈[0,T ] (8) pL (M ), M ∈ Ω These local variables are typically stress tensors, but can also be strains or temperatures, whatever is needed to calculate the damage. The parameters pL depend on the point M and the global behavior of the structure. For example, in the case of a linear elastic structure submitted to concentrated forces F, under which its response can be considered quasi-static, the parameters pL can be seen as the nF localization tensors σLi (M ), i ∈ [1, nP F ] such that the stress at point M writes σ(M, t) = i σL i (M )Fi (t). The material damage function D associates the history of local variables σ and material parameters pD to the scalar variable of damage, d, in each point.   (σ(t))t∈[0,T ] D −→ d (9) pD

2.3. Equivalent fatigue signal model The equivalent fatigue signal is a parametric model for a history of global loads F(t), t ∈ [0, T ] suited for either simulation or physical tests purpose. The parameters of the equivalent signal are to be tuned to obtain the equivalence with the original signal. They are denoted λeq in the following. The notion of equivalence will be detailed further in this section. The equivalent fatigue signal can be built as a simplification of the original signal; for example, by removing windows of the signal that do not (or little) damage the structure [25]. This technique is usually referred to as damage editing and the parameters λeq of the reduced signal are the indexes of the kept points. This shortens the test duration, but does not

It is a local function that does not depend on the shape of the structure. It usually comprises the calculation of some variable associated with the constant amplitude life and a cumulative damage rule. 6

solve the problem of controlling the test rigs. Moreover, this type of signal often remains too complex for simulation. One therefore may prefer a synthetic signal with few parameters. A popular choice for quasi-static design is block programs, consisting of a repeated sequence of several stages, each stage having a different waveform (usually sinus), mean value and amplitude, frequency and number of cycles. Other propositions, such as Gaussian processes or Markov Chains [21] can also be found in the literature. Gaussian signals are frequently used, especially for frequency response analysis and shaker experiments [16]. It may be interesting to constrain some parameters of the equivalent signal a priori, in order to meet requirements that would not be included in the considered damage models or to confine the equivalent signal in the relevant domain of the damage model. For example, one may wish to keep the extreme values of the original signal to avoid exciting other (not described by the model) unwelcome mechanisms, such as plasticity. The frequency might also be limited in order to avoid any dynamic response if the model relies on a quasi-static assumption. Some constraints on the mixing sequence of small and large amplitudes can also reduce the sequence effect that is not taken into account in the simplest (and mostly used) Palmgren-Miner accumulation law. In the case of an expected resonance of the structure, the extreme response spectrum is used to constrain the amplitude of the equivalent Gaussian signal [16].

sidering multiple points, or more precisely, multiple p: Find λ∗eq such as ∀p,

D◦L(Feq (λ∗eq ), p) = D◦L(F, p) (12) with p the parameters for any point of any structure in the scope of the design possibilities. Unfortunately this condition is often almost impossible to satisfy. The one case when it is known to hold true is the 1D case described in section 1.3.1, without any mean stress correction and a known Basquin’s exponent. Indeed, in this case, the unknown structural parameter σL vanishes in the equivalence equation. With the same model, one can consider the situation when the structure is composed of two different materials, or suffer from two different damage mechanisms, so that the equivalent signal should be relevant for two different Basquin’s exponents β1 and β2 . It is still possible to find a solution to the equation 12 because fortunately the equivalent signal has two parameters (its amplitude and number of cycles). But this time, these parameters are completely determined by the two equations (5) given by β1 and β2 :

Faeq =

P

P

and

2.4. Strong formulation of the damage equivalence condition The general idea is to find values of the parameters of the equivalent signal, λeq , such that in every point of any structure, the damage caused by the equivalent load is the same as the damage caused par the original load. The vector of parameters which satisfies this condition is denoted λ∗eq . This condition writes:

β1 j Fa (j) j

Faβ2 (j) Neq =

!β P

1 1 −β2

j

Faβ1 (j)

Faeq β1

=

P

j

Faβ2 (j)

Faeq β2

(13)

The user cannot choose the approximate number of cycles of the equivalent signal, which might be out of the relevant physical range of interest or reasonable test duration. Increasing the number of parameters of the equivalent signal can then give additional degrees of freedom to constrain the equivalent signal to stay in some given boundaries, or to take into account even more values of the Basquin’s exponent β. A blockprogram with several (enough) different amplitudes can satisfy the equation 12 for a finite number of β. 2.5. Weak formulation of the damage equivalence condition

Find λ∗eq such as ∀p, ∀M,

Now, consider the case when the precise material of the structure is not yet decided, or uncertain, so that this β exponent is uncertain. It might take any value between β1 and β2 . In this case, the equivalent fatigue signal would have to fulfill an infinite number of conditions, with a finite and preferably small number of parameters λeq (here, amplitudes and numbers of cycles), which is not possible.

D ◦ L(Feq (λ∗eq ), p, M ) = D ◦ L(F, p, M ) (11)

As was pointed out before, the damage depends on the point M only through the vector p (which includes all the parameters for all the points), so that considering multiple structures comes down to con7

• the points that do not fail under the original signal should not fail either under the equivalent one;

However, one can remark that first, the damage is calculated following a model with some amount of imprecision and second, that the fatigue life is highly scattered. Hence, a rigorous equivalence condition is not really needed and one can be content with an approximate condition. This condition, here referred to as the ”weak formulation of damage equivalence” (”weak” as ”approximate”) writes:

• and reciprocally, the points that do fail under the original signal should also fail under the equivalent one. This condition writes: Find λ∗eq such as ∀p,

λ∗eq = argmin kD ◦ L(F, p) − D ◦ L(Feq (λeq ), p)kp λeq

(14) The notation k•kp is a norm (to be defined) on the space of parameters and points of the structure. A simple choice is to take the maximum error over all the parameters; a quadratic norm (the variance of the error) could be another one (as in [21]). One could even imagine to weight the error with a probability density function of the unknown parameters (in the - unlikely?- situation when the design engineer has some knowledge about their expected occurrence). The optimal parameters λ∗eq are usually found thanks to some numerical minimization algorithm. It is usually necessary to discretize the parameter space in order to evaluate the error function. The error can be calculated a posteriori and some additional parameters λeq can be added if it is considered as too high. This weak condition is less demanding than the strong formulation. However, in many cases, the variation of the damage error over ”possible structures” (parameters p) is so large (possibly infinite) that there is no way finding an equivalent fatigue signal with a reasonable (small) number of parameters. This case can happen when some parameters vary over a too large space. Moreover, a large size of the unknown parameter space leads to high computational times because the error function has to be evaluated on it. 2.6. Strong formulation of the failure equivalence condition Following Thomas [15], reducing the space of ”possible structures” can be done considering that the damage itself is of no interest to the design engineer, and that only the failure behavior matters. This typically happens when the fatigue requirement is expressed as a pass or fail test. Then, the equivalent fatigue signal must provoke the same failure behavior as the original signal: 8


sign 1 − D ◦ L(Feq (λ∗eq ), p) = sign (1 − D ◦ L(F, p)) (15) If D ◦ L(F, p) defines sufficiently regular isosurfaces in the neighborhood of 1 that the implicit function theorem applies in the unknown parameter space, the condition becomes: Find λ∗eq such as ∀˜ p/D ◦ L(F, p ˜) = 1,

D ◦ L(Feq (λ∗eq ), p ˜) = 1 (16)

The isosurface of the parameter space such that D = 1 is the frontier between the structures that would fail and the structures that would not fail under the original signal (see figure 2). It represents the strength requirement in the parameter space. The equivalent signal is required to yield the same isosurface D = 1 as the original signal. Hence a structure designed to fall on one side of this isosurface with the equivalent signal, falls necessarily on the same side for the original signal. However, since the other isosurfaces (such that D is constant but different from 1) are not required to be the same (see solid and dashed lines on figure 2), the damage induced by the two signals might be different if the effective design is such that the structure’s strength is different from the exact strength requirement (a common situation since structures are usually over-designed because of safety margins). For example, if a component is tested under some equivalent signal and happens to break after twice the required time, it can be inferred that it would also have withstood the original signal. However, since the structure is stronger than required (in other words, its parameters are such that D < 1), one cannot conclude that the component would have failed exactly after twice the original signal.

p2

Genet’s work. They are the unknown parameters of the structural damage function. As in Genˆet’s thesis, Morel’s model is here chosen to describe the fatigue damage. Only the main elements are stated here; for more details and physical arguments, see [22, 21]. Morel’s model is based on a micro-macro approach. A scale transition model is proposed and the local damage is associated with the accumulated plastic strain on a critical glide system. The microscopic yield limit evolves in 3 phases (hardening,saturation, and softening) but here the saturation phase is supposed to be predominant so that the other phases are not taken into account. In a spherical coordinate system defined by the angles θ, φ, ψ a plane is defined by its normal unit vector n(θ, φ) and a direction on this plane by its unit vector m(ψ). The resolved shear stress on the glide system (n, m) is then:

structures with excessive strength, D < 1 D = 0.8 D = 0.9

D=1 D = 1.1 D = 1.2 structures with unsufficient strength, D > 1

p1

Figure 2: Schematic representation of damage isovalues in a 2D parameter space for the original signal (solid lines) and the equivalent signal (dashed lines) in the case of failure equivalence. They coincide for D = 1 only.

2.7. Weak formulation of the failure equivalence condition Here again, the strong equality is relaxed to get an easier condition to fulfill:

τ (θ, φ, ψ, t) = m.σ(t).n

The critical plane (θc , φc ) maximizes the mean value (over directions m) of the variance of the resolved 2 (θ, φ, ψ): shear stress τrms s Z 2π 2 (θ, φ, ψ)dψ τrms (20) Trms = max

λ∗eq = argmin k1 − D ◦ L(Feq (λeq ), p ˜)kp˜ /D◦L(F,˜p)=1 λeq

(17)

(θ,φ)

3. Multi-input equivalent load for a linear elastic quasi-static structure

σLi (M )Fi (t)

(21)

with Pm and Prms the mean value and the standard deviation of the hydrostatic pressure, α and β some material parameters (identical as those of the Dang Van criterion [27]) and Crms the maximum value over directions m of the variance of the resolved shear 2 stress τrms on the critical plane: Crms = max τrms (θc , φc , ψ) ψ

The damage then writes: P (τi − τlim (θc , φc , ψ))+ d = max i ψ 2qτlim

3.1. Damage model The superposition principle applies: nF X

0

The mesoscopic yield limit is computed as: √ √ Trms −α πPm + β π Trms τlim = √ Crms Prms α π + PTrms rms

In this section, an application of the general approach exposed above is developed in the case of a linear elastic structure, undergoing infinitesimal strain and under quasi-static loading. The loading consists of a finite number of concentrated forces. It is an extension of Genˆet’s work [21] to the general case when no specific hypothesis is needed about the form of the local stress induced by each force [26]. The method is illustrated here in the case of only two input forces F1 and F2 , and inducing a state of plane stress (cracks are supposed to initiate on the surface).

σ(M, t) =

(19)

(22)

(23)

with (•)+ denoting the positive part, q a material parameter, and τi the amplitude of the resolved shear stress on the glide system (θc , φc , ψ) obtained by Rainflow counting on the variable τ .

(18)

i=1

Here the localization stress tensors σLi can take any form; they are not assumed to be proportional as in 9

so that, with the previously described parameterization: |||(σL1 , σL2 )||| = K. The space of unit stress tensors couples can then be explored by a variation on the angles α1 , α2 , αrot and γ. Those angles are evenly discretized for calculation purpose. The damage function is monotonic in K, hence there is only ˜ such that d (|||(σL1 , σL2 )|||) = 1, easily found one K by dichotomy, for example.

Note that any other criteria could be used as well. Here, this simplified version of Morel’s criterion is chosen for its interesting balance between accuracy in case of complex loading and reasonable cost of evaluation thanks to the a priori calculation of the critical plane. The material parameters are given; they are not part of the unknown parameters of the structural damage function.

3.2.2. Equivalent fatigue model The equivalent signal is chosen as a block-program, By combining the equations 18 to 23, one can evaluate the structural damage function d(F1 , F2 , σL1 , σL2 ). each block a sinusoidal wave with 6 parameters: number of cycles n, mean values Feq,m 1 and Feq,m 2 , amThe weak formulation of the failure equivalence is applitudes Feq,a 1 and Feq,a 2 , and the phase ϕ between plied: the two input forces. However, many sets of theses λ∗eq = argmin max |1 − d(Feq (λeq ), σ˜Li )| (24) parameters will give the same damage: for example, σ˜L i λeq increasing the amplitudes or the mean value is equivalent to reducing the number of cycles. Some of these with the σ˜Li such that d(F, σ˜Li ) = 1. The infinite parameters can then be set to predefined values. The norm (i.e. the maximum) is chosen to evaluate the ratio between Feq,a 1 and Feq,a 2 , as well as the phase residue. In practice, the space of the σ˜L i is disϕ, allow to represent the correlations in the signal. cretized and the maximum of |1 − d(Feq (λeq ), σ˜L i )| These correlations can be complex, and will be better is calculated over this finite number of σ˜L i . represented in the equivalent signal when increasing the number of blocks. 3.2.1. Discretization of the unknown parameter space The procedure starts with a one-block signal. The In a first step, the space of the unknown paramoptimal set of parameters for the equivalent signal is eters σ˜Li such that d = 1 is searched. The space of get thanks to minimization algorithms, with initial stress tensors is parameterized as follows: conditions given by the user. Several initial condi  tions are tested to avoid local minima. Using some sin α1 0 0   more sophisticated minimization algorithm could probσL1 = K cos γ 0 cos α1 0 (25) ably improve this delicate process. The maximum er0 0 0 B 1 ror over the angles α1 , α2 , αrot and γ (in other words,   over the localization tensors σ˜L i ) is checked and if sin α2 0 0 it is superior to a given threshold, another block is σL2 = K sin γ  0 cos α2 0 (26) added and the minimization procedure starts all over 0 0 0 B 2 again. Blocks are thus added until the error criterion with R the rotation matrix between the principal diis reached. Usually few are required. rections basis B1 and B2 of σL1 and σL2 respectively:   3.3. Example cos αrot − sin αrot 0 The above described procedure is applied to a R =  sin αrot cos αrot 0 (27) two-input signal recorded on a car component on a 0 0 0 test track. The measured sequence is composed of nearly 6000 time points and is repeated 333 times. K is a positive real number, (α1 , α2 ) ∈ [π/4, 3π/4]2 , The component is made of steel with Morel’s model αrot ∈ [0, π/2] (thanks to some symmetry of the dammaterial parameters (α = 0.3; β = 150 MPa; q = age function) and γ ∈ [0, 2π]. The following norm is 45000 cycles ). The unknown parameter space (i.e. defined over (σL1 , σL2 ): couples of localization tensors (σL1 , σL2 )) is discretized q 2 2 in 662 points, which can be interpreted as 662 struc(28) |||(σL1 , σL2 )||| = kσL1 k + kσL2 k tures, using a linear discretisation of the four angles √ with kσk = σ : σ 3.2. Fatigue failure equivalency

10

ior of the structure, and yet simple enough to keep down the cost of an evaluation. The size of the input loads nF has a great influence on the duration and convergence of the minimization procedure, so that the formulation of the structural damage model is even more critical for large input channels problems. One might consider increasing the constraints on the equivalent signal parameters (and possibly their number) in order to reproduce in the equivalent signal some characteristics of the original signal that might have an influence on the lifetime but that are numerically costly to describe (e.g. phase between channels).

F2 [kN ]

10 5

2 3

1

0 0

5

10

15

F1 [kN ]

Number of (σ˜L1 , σ˜L2 )

Figure 3: Two-input signal and 3 blocks equivalent signal. Block 1 is repeated 14518 times, block 2, 19585 times and block 3, 12113 times.

4.2. Implementation The question of a sufficient discretization of the parameter p space is an open question. It should ensure a sound evaluation of the error over the unknown parameter space. All the parameter space should be explored with a precision related to the regularity of the structural damage model over the parameter space. A local variation of the damage functions should not be missed. In the case presented here, the (σL1 , σL2 ) space is discretized following some given parameterization of the stress tensors, which is not demonstrated to be optimal. The problem is similar to the question of an even discretization of directions in critical plane fatigue criteria [28], although here the dimension of the space to browse is much larger. Some additional work on this topic is needed.

200

100

0 −0,002

0

0,002

0,004

0,006

0,008

D−1

Figure 4: Distribution of the damage error for the equivalent signal.

4.3. Amount of uncertainty The size of the uncertain parameters p, in addition to weighting on the calculation time, could overconstrain the problem, so that the expected simplification does not occur. This problem has not been encountered in the authors experience. However, the user should be aware that some knowledge about the structure is needed in order to restrain the size of p, and that the hypothesis under which the equivalent signal is indeed equivalent should be checked afterwards, that is, when the design has been completed.

(α1 , α2 , αrot and γ) used to parameterize the localization tensors. The threshold on damage is 1%, thus a small value compared to the usual lifetime scatter for automotive steel sheets. The mean value of the sinus are constrained to be equal to the mean value of the original signal. The equivalent signal is shown on figure 3. Each block of sinus appears as an ellipse because of the phase between the two inputs. The quality of the equivalence can be checked on figure 4.

4.4. Equivalent signal shape Another point of interest is the way to choose an equivalent load form such that it has the relevant degrees of freedom to satisfy the equivalence condition. On the one hand, the degrees of freedom should reproduce the principal features of the original signal that act on the damage parameter. For example, in the chosen example given in this paper, it will not be

4. Discussion 4.1. Cost versus accuracy of the structural damage model The method proposed here needs a structural damage model precise enough to take into account the expected local mechanical variables and fatigue behav11

possible to describe the 2D signal with a 1D signal. On the other hand, the shape of the equivalent signal is also important in constraining the solution in an acceptable physical domain. The structural damage model is the element of the method that allows compression of the information. The model is usually imperfect, because of the above mentioned compromise needed to restrain the computation duration, but also because the material behavior might still be poorly understood. Thus, the process might lead to unwanted features in the equivalent signal, exciting the damage model in a range where it does not apply, or where it is a coarse approximation of the effective material behavior. For example, in the 2D-equivalence example presented here, the phase parameters allow to respect the desired amplitudes in all the directions of the input force space. Depending on the material behavior under out-of-phase loading, another fatigue model might be more relevant: integral approaches, for instance. However, such a model is very computationally expensive. An alternative could be to constraint the relationship between the channels by choosing, instead of a synthetic signal such as a block of sinus, a set of short measurements (possibly idealized) representing typical driving events [29]. In this case, the method to choose a relevant set, acting as a base of the signal, remains to be investigated.

too be a very good approximation of a two-channel load-time history recorded on a car. Many other applications can be imagined following the same framework. Acknowledgements This research was entirely funded by groupe PSA and did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] M. Lemaire, Structural reliability, 1 edition ed., WileyISTE, 2009. [2] D. Sch¨ utz, H. Kl¨ atschke, P. Heuler, Standardized multiaxial load sequences for car wheel suspension components - CAr LOading Standard multiaxial, Technical Report, Fraunhofer - Institut f¨ ur Bestriebsfestigkeit (LBF), Darmstadt, 1994. [3] P. Heuler, H. Kl¨ atschke, Generation and use of standardised load spectra and load–time histories, International Journal of Fatigue 27 (2005) 974 – 990. doi:10.1016/j.ijfatigue.2004.09.012, cumulative Fatigue Damage Conference - University of Seville 2003. [4] C. Roux, X. Lorang, H. Maitournam, M. L. NguyenTajan, Fatigue design of railway wheels: a probabilistic approach, Fatigue & Fracture of Engineering Materials & Structures 37 (2014) 1136–1145. doi:10.1111/ffe.12194, fFEMS-5471.R3. [5] P. Tibbits, Fatigue load equivalent to distribution of loads from the US population, in: ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, 2007. doi:10.1115/DETC2007-35678. [6] G. Freebury, W. Musial, Determining equivalent damage loading for full-scale wind turbine blade fatigue tests february 2000 nrel / cp-500-27510, in: 19th American Society of Mechanical Engineers (ASME) Wind Energy Symposium, Reno, Nevada, 2000. [7] M.-H. Herman Shen, Reliability assessment of high cycle fatigue design of gas turbine blades using the probabilistic goodman diagram, International Journal of Fatigue 21 (1999) 699 – 708. doi:10.1016/S0142-1123(99)00033-X. [8] B. Echard, N. Gayton, A. Bignonnet, A reliability analysis method for fatigue design, International Journal of Fatigue 59 (2014) 292 – 300. doi:10.1016/j.ijfatigue.2013.08.004. [9] R. Teixeira, M. Nogal, A. O’Connor, J. Nichols, A. Dumas, Stress-cycle fatigue design with Kriging applied to offshore wind turbines, International Journal of Fatigue 125 (2019) 454 – 467. doi:10.1016/j.ijfatigue.2019.04.012. [10] C. Lipson, N. J. Sheth, R. L. Disney, Reliability Prediction-Mechanical Stress/Strength Interference, Technical Report RADC-TR-66-710, University of Michigan, Ann Arbor, 1967.

5. Conclusion In this paper was presented a general framework for building some simplified load, equivalent, in terms of fatigue, to some complex loads, taking into account some uncertainty on the target structure. The proposed method consists of the following steps: • the space of possible expected structures is to be defined and described thanks to some model or library; • some fatigue equivalence is to be chosen, either in terms of damage or failure; • the parameters of a synthetic signal are sought such as to get the equivalence criterion, on the whole space of the expected structures, possibly in an approximate manner. The case of multi-channel input, when the uncertainty lies in the way the different channels might interact at the local scale in the structure, was taken as an example. A simple 3-block signal was shown 12

[24] B. Delattre, Analyse des chargements en service pour le dimensionnement fiabiliste ` a la fatigue polycyclique, in: Colloque M´ecamat - Fatigue des structures et des mat´eriaux, Aussois, 2017. [25] B. Weber, C. Montero, S. Bergamo, R. Rennert, A. W¨ unsche, S. Budano, I. Aranguren, Load spectrum lightening of fatigue tests data for time reduction of design validation - Speefat, Technical Report EUR 242006, European Commission - Research fund for coal and steel, 2010. doi:10.2777/86474. [26] R. Jossic, G´en´eralisation d’une m´ethode de construction ´ d’´equivalents fatigue multi-entr´ee, Master’s thesis, Ecole Centrale de Paris, 2007. [27] K. Dang Van, B. Griveau, O. Message, On a new multiaxial fatigue limit criterion: theory and application, in: M. W. Brown, K. J. Miller (Eds.), Biaxial and Multiaxial Fatigue, EGF 3, Mechanical Engineering Publications, London, 1989, pp. 479–496. [28] B. Weber, B. Kenmeugne, J.-C. Clement, J.-L. Robert, Improvements of multiaxial fatigue criteria computation for a strong reduction of calculation duration, Computational Materials Science 15 (1999) 381 – 399. doi:10.1016/S0927-0256(98)00129-3. [29] P. Lelan, Accelerated tests design for vehicle carried systems, in: 60th Annual Technical Meeting of the Institute of Environmental Sciences and Technology (ESTECH 2014), Institute of Environmental Sciences and Technology, San Antonio, Texas, United States of America, 2014.

[11] J.-J. Thomas, A. Bignonnet, G. Perroud, Fatigue design and experimentations with variable amplitude loadings in the automotive industry, in: P. C. McKeighan, N. Ranganathan (Eds.), Fatigue Testing and Analysis Under Variable Amplitude Loading Conditions, STP1439EB, ASTM International, West Conshohocken, 2005, pp. 381 – 394. [12] C. M. Sonsino, Fatigue testing under variable amplitude loading, International Journal of Fatigue 29 (2007) 1080 – 1089. doi:10.1016/j.ijfatigue.2006.10.011. [13] M. Matsuishi, T. Endo, Fatigue of metals subjected to varying stress, Technical Report, Japan Society of Mechanical Engineers, Fukuoka, Japan, 1968. [14] H. Hendriks, B. H. Bulder, Fatigue equivalent load cycle method - A general method to compare the fatigue loading of different load spectrums, Technical Report ECN-C–95-074, Netherlands Energy Research Foundation (ECN), Petten, Netherlands, 1995. URL: https://www.osti.gov/etdeweb/biblio/191445, this study has been carried out within the framework of the programmes JOULE and TWIN. [15] J. Thomas, G. Perroud, A. Bignonnet, D. Monnet, Fatigue design and reliability in the automotive industry, in: G. Marquis, J. Solin (Eds.), Fatigue Design and Reliability, volume 23 of European Structural Integrity Society, Elsevier, 1999, pp. 1 – 11. doi:10.1016/S1566-1369(99)800259. [16] C. Lalanne, Mechanical Vibration & Shock: Fatigue damage, Mechanical Vibration & Shock, Hermes Penton Science, 2002. [17] C. Lalanne, Specification Development, Mechanical Vibration & Shock, Hermes Penton Science, 2002. [18] F. Szmytka, A. Oudin, A reliability analysis method in thermomechanical fatigue design, International Journal of Fatigue 53 (2013) 82 – 91. doi:10.1016/j.ijfatigue.2012.01.025, proceedings of the 2nd International Workshop on Thermo-Mechanical Fatigue 2011. [19] C. Farhat, A. Bos, P. Avery, C. Soize, Modeling and quantification of model-form uncertainties in eigenvalue computations using a stochastic reduced model, AIAA Journal 56 (2018) 1198–1210. [20] M. K¨ ohler, S. Jenne, K. P¨ otter, H. Zenner, Load assumption for fatigue design of structures and components, Springer-Verlag Berlin Heidelberg, 2017. doi:10.1007/9783-642-55248-9, translation from the German language edition: Z¨ ahlverfahren und Lastannahme in der Betriebsfestigkeit. [21] G. Genet, A statistical approach to multi-input equivalent fatigue loads for the durability of automotive structures, Ph.D. thesis, Chalmers University of Technology and G¨ oteborg University, 2006. [22] F. Morel, A critical plane approach for life prediction of high cycle fatigue under multiaxial variable amplitude loading, International Journal of Fatigue 22 (2000) 101 – 119. doi:10.1016/S0142-1123(99)00118-8. [23] C. Roux, X. Lorang, H. Maitournam, M.-L. NguyenTajan, B. Quesson, Multi-parameter fatigue equivalence loadings for specification applications, Procedia Engineering 66 (2013) 393 – 402. doi:10.1016/j.proeng.2013.12.093, fatigue Design 2013, International Conference Proceedings.

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Graphical Abstract Equivalent fatigue load approach for fatigue design of uncertain structures Ida Raoult, Benoit Delattre F2

F1 F2

D F1

F1

Deq

D

F2 F1

Feq 2 F2

Feq 1

p

p∈Ω

∀p ∈ Ω, D(p) ≃ Deq (p)

Highlights Equivalent fatigue load approach for fatigue design of uncertain structures Ida Raoult, Benoit Delattre • A general framework to build simplified loads for fatigue design. • The equivalent signal is insensitive to some amount of uncertainty on the structure. • The case of a multi-channel input is developed and illustrated with road load data.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: