Enhanced assessment rule for concrete fatigue under compression considering the nonlinear effect of loading sequence

International Journal of Fatigue 126 (2019) 130–142

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Enhanced assessment rule for concrete fatigue under compression considering the nonlinear effect of loading sequence

T

⁎

Abedulgader Baktheer , Josef Hegger, Rostislav Chudoba Institute of Structural Concrete, RWTH Aachen University, Mies-van-der-Rohe-Straße 1, 52074 Aachen, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Concrete fatigue Numerical modeling Loading sequence effect Damage accumulation

In this paper, a refined engineering rule for the assessment of remaining fatigue life of concrete under compressive cyclic loading with varying amplitudes is proposed. The rule has been derived based on a combined numerical and experimental investigation of the loading sequence effect. The applied modeling approach is based on a damage model using the equivalent tensile strain rate to govern the fatigue damage evolution upon loading and reloading at subcritical load levels. A systematic calibration and validation procedure of the numerical model was performed based on the available experimental results. The prediction of the numerical model was compared with existing damage accumulation rules for the assessment of the concrete fatigue life exposed to varying loading ranges. Based on these studies, an enhancement of the Palmgren-Miner rule is proposed and validated for several loading sequence scenarios.

1. Introduction Concrete fatigue behavior has been usually studied for loading scenarios with constant amplitudes. However, in reality the loading can have variable amplitudes ordered in different sequences. During its service life any structure would be subjected to various randomly varying loading patterns [1,2]. For example, highway and railway bridges are subjected to millions of load cycles at low amplitudes interleaved with occasional stress peaks and overloads [3,4]. Under seismic action concrete structures are exposed to a multistage loading consisting of several cyclic loading blocks with constant amplitudes [5,6]. In offshore structures, like wind power plants wave movements and stochastic wind loading impose recurring load pulses of variable magnitude and amplitude [7]. Without studying the effect of load sequencing and variable amplitudes, the safety factors may be set uneconomically large or dangerously low. The well-known Palmgren-Miner (P-M) [8,9] rule applied in the current design codes postulates, that fatigue life for a loading scenario consisting of two different cycling load ranges can be estimated by linearly interpolating the fatigue life observed for each of the two cycling load ranges individually. According to this rule, if the fatigue loading history consists of n number of different loading ranges, then the fatigue failure occurs when

⁎

n

η=

∑ i=1

Ni = 1, Nif

(1)

Nif

is the where Ni is the applied number of cycles of the range i, and number of cycles to failure under a constant amplitude of the range i. Apparently, this hypothesis assumes linear accumulation of fatigue damage with increasing number of cycles on the lifetime scale of the corresponding load range. Even though widely used [10], an experimental evidence of this rule does not exist. On the contrary, the few available published experimental results show that the P-M rule can result in a highly unsafe prediction of fatigue life [11–15]. Currently available experimental results show qualitatively different fatigue behavior for concrete exposed to compression and tension with varying load ranges. The situation is qualitatively summarized in Fig. 1 showing the response to compressive cyclic loading measured by Holmen et al. [11] and Petkovic et al. [12] in Fig. 1 left and the response to bending cyclic loading reported by Hilsdorf et al. [14] is in Fig. 1 right. The axes of these diagrams represent the relative fatigue lifetime observed for two cycling loading ranges: ηH with a higher (H) and ηL with a lower (L) upper bound. The lower bound is equal for both ranges. A combined loading scenario is represented by two linked arrows, one in vertical and the other one in horizontal direction. A vertical arrow corresponds to a cycling with the upper bound H and a horizontal one with the lower bound L. Thus, the H-L scenario shown using the red arrows in (Fig. 1 left), started with the load range H and was applied for a number of cycles consuming 40% of lifetime as shown

Corresponding author. E-mail address: [email protected] (A. Baktheer).

https://doi.org/10.1016/j.ijfatigue.2019.04.027 Received 6 March 2019; Received in revised form 17 April 2019; Accepted 23 April 2019 Available online 25 April 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

ηi ̃

(.)+ α Si β β↓ β↑ ω ε I Y Δηmax Δηi ΔS max ΔS max η η↓ η↑ ηrem ηx λ  () tr μ ω ϕ

ε̃ A a B b C c C0 C1 f f (ε, ω) fc g I2 K k (ω) n n Nif

positive part of a tensor material parameter of Alliche model mean loading level within two loading ranges material parameter of Alliche model material parameter of the proposed rule material parameter of the proposed rule damage tensor strain tensor identity tensor energy release rate tensor y coordinate of the knee point of the response surface correction term of the P-M rule lower jump between two loading ranges upper jump between two loading ranges cumulative fatigue life material parameter of the proposed rule material parameter of the proposed rule remaining fatigue life x coordinate of the knee point of the response surface first Lamé constant response surface of parameters trace of the tensor second Lamé constant damage parameter free energy potential

Ni S max S min Sim

cumulatively consumed fatigue life prior to the loading jump equivalent strain material parameter of the proposed rule constant for the cumulative damage rule by Shah material parameter of the proposed rule constant for the cumulative damage rule by Shah material parameter of the proposed rule constant for the cumulative damage rule by Shah material parameter of Alliche model material parameter of Alliche model material parameter of Alliche model yield function of Alliche model compressive strength of the concrete material parameter of Alliche model second invariant of the energy release rate tensor material parameter of Alliche model damage threshold material parameter of Alliche model number of loading ranges the number of cycles to failure under a constant amplitude ofthe range i number of loading cycles of the range i the upper level of the loading range the lower level of the loading range mean value of individual loading range

Fig. 1. Qualitative diagrams of the opposite effect of loading sequence (H-L) and (L-H) on the fatigue lifetime in relation to the Palmgren-Miner rule shown as a solid line; left: compression load; right: tensile bending load.

vertical cracks, whereas if the high stress does not occur until later in the programme the effect is more evenly distributed among the greater number of cracks then present.”

by the vertical arrow starting from zero. Then, in the second phase, cycling with the lower upper bound L was applied until failure as marked by the red horizontal arrow. Apparently, the failure for the (HL) scenario was observed significantly earlier than predicted by the P-M rule. The phenomenology behind the load sequence effect under compression is not fully understood. An interesting hypothesis has been provided by Bennet [13]. In contrast, the reverse order of load levels (LH) resulted in a fatigue life that was much longer than the P-M rule estimation. A blue and red envelopes indicate the failure points for all possible H-L and L-H scenarios. Thus, for compressive loading, the P-M rule delivers unsafe results for (H-L) and conservative results for (L-H) scenarios.

On the other hand, an inverse picture has been observed for tensile cyclic loading as shown in (Fig. 1 right). Here, the P-M rule delivered unsafe lifetime estimation for L-H scenario and conservative estimation for H-L scenario. Another results presented by Klausen [16], and Tepfers et al. [7] show a dependency of the fatigue behavior on the changed order of loading ranges, but due to the large scatter of the results clear tendency for the (L-H) and (H-L) sequences cannot be concluded [17,18]. Even though the experimental evidence is still not sound, the mentioned observations document that the postulated linear damage accumulation hypothesis behind the P-M rule should be critically scrutinized. As an alternative to the P-M rule assuming linear damage accumulation a few nonlinear damage accumulation rules have been proposed [21,22]. These rules are derived based on a small number of tests with specific load scenarios so that their range of validity can be assumed very narrow and they cannot be expected to realistically capture the effect of load order on a broad scale as discussed in detail later on in Section 3.

”It is difficult to explain the significantly reduced Miner number for the one-step programmes of tests which commenced with a high stress. The physical damage due to repeated compressive loading of concrete may be observed in the form of a system of fine cracks, mainly of vertical orientation contrasting with the smaller number of vertical cracks which lead to failure under static loading. It is possible that an early occurrence of a high stress in a fatigue loading programme causes damage by the propagation of a few major

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2. Applied concrete fatigue model

In this paper, we propose a refined rule which takes the effect of the order of loading ranges on the remaining fatigue life into account. This rule can be considered as an enhanced P-M rule taking the following form n

η=

∑ i=1

Ni + Nif

The modeling approach proposed by Alliche [30] has been chosen in this study for several reasons: It is formulated in an incremental form which implies its applicability to any fatigue loading history. The model can be written in a stress driven formulation to efficiently simulate the concrete cylinder test under compressive loading cycle by cycle assuming a uniform stress state. Moreover, the model is able to reflect the non-linearity of the fatigue damage accumulation and the effect of loading sequence in the framework of viscoplasticity by replacing the traditional yield limit with an irreversibility condition to properly reflect the loading-unloading behavior. It should be emphasized that other models reflecting the dissipative effects during the loading cycles, e.g. [39], might be used as well. In compression, both types of inelastic models can be used to interpret the available experimental observation of the load sequence effect on fatigue to provide a basis for the proposal of an engineering assessment rule.

n−1

∑

Δηi = 1,

(2)

i=1 n−1

The introduced additional term ∑i = 1 Δηi accounts for the loading sequence effect by correcting the deviation of the remaining fatigue life with respect to the P-M rule presented in Fig. 1. This term depends on the loading jump between the ranges, the middle load level value of the two loading ranges and on the consumed fatigue life up to the load jump. The aim of the present paper is to explain and justify the proposed enhancement. The rule is described and exemplified in detail in Section 5. The proposed rule is based on a combination of numerical and experimental investigations. The applied approach to the construction of the assessment rule aims to exploit the low amount of available data on fatigue behavior to provide a sound theoretical interpretation of the fatigue phenomenology. The experimental studies used in this paper to calibrate and validate the modeling approach are summarized in Table 1. Advanced numerical models validated for selected configurations are used to obtain the data needed for the definition of the improved assessment rule for fatigue life under varying loading ranges. The applied combined numerical and experimental approach is summarized in Fig. 2. To choose a suitable model for the numerical analysis of the loading sequence effect let us briefly review the existing models of fatigue damage propagation in concrete. Pragmatic approaches to fatigue damage modeling use the number of performed loading cycles as a damage driving variable [23–28]. It is not possible to use them for realistic prediction of tri-axial stress redistribution in a material point. They can neither be used to simulate the fatigue response for non-periodic loading scenarios occurring in reality. On the other hand, these models do not require explicit simulation of loading cycles so that they can serve for efficient fatigue analysis of large structures to study the fatigue-induced stress redistribution. More advanced approaches to simulation of fatigue damage process cycle by cycle at subcritical load levels have been proposed based on assumption that the damage is linked either to the total strain [29–33], or to the inelastic strain [34–38]. To reflect the opening/closure and growth of microcracks and/or the frictional sliding along their lips, the formulation of the fundamental dissipative mechanisms related to fatigue has been refined by introducing the internal sliding strain as a damage driving variable [39]. An approach relating the dissipative terms owing to fatigue damage even closer to the observable disintegration mechanisms within the material structure appeared recently in [40]. The key idea of this model is to relate the damage evolution to the cumulative measure of volumetric strain. In the present paper, the modeling approach introduced by Alliche [30] is applied to analyze the loading sequence effect for concrete under compressive loading. This numerical modeling approach is briefly explained in Section 2. Its feasibility for modeling of fatigue behavior under loading with variable amplitudes is studied using elementary examples followed by the calibration and qualitative validation. Then, existing assessment rules for concrete fatigue under compression are summarized in Section 3. Based on the validated model, further numerical investigations of the loading sequence effect are performed and evaluated in Section 4. Finally, in Section 5, an enhanced engineering rule for predicting the remaining fatigue life under varying load ranges is proposed and validated for several loading scenarios.

2.1. Model formulation The model is based on the free energy potential proposed by Halm and Dragon [41], and Dragon et al. [42] in the following form

1 ϕ ⎛⎜ε, ω⎞⎟ = λ [tr(ε )]2 + μ tr ⎛⎜ε·ε ⎞⎟ + g tr ⎛⎜ε·ω⎞⎟ + α tr(ε ) tr ⎛⎜ε·ω⎞⎟ 2 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ + 2 β tr ⎜⎛ε·ε·ω⎞⎟ ⎝ ⎠

(3)

where λ, μ are the Lamé constants. α, β , g are material parameters. The thermodynamic forces, i.e. the stress tensor and energy release rate tensor, are obtained as derivatives of the free energy potential with respect to the state variables, i.e. the strain tensor ε , and the damage tensor ω , respectively. The stress tensor is obtained as

σ=

∂ϕ (ε, ω) = λ tr(ε ) I + 2με + gω + α [tr(ε·ω) I + tr(ε ) ω] ∂ε + 2β ⎜⎛ε·ω + ω·ε ⎟⎞, ⎝ ⎠

(4)

and the second order energy release rate tensor reads

Y=−

∂ϕ (ε, ω) = −gε − α ⎛⎜trε ⎞⎟ ε − 2β ⎛⎜ε·ε ⎞⎟. ∂ω ⎝ ⎠ ⎝ ⎠

(5)

To distinguish the compressive and tensile loading regimes, the strain tensor is split into positive part ε + and negative part ε −. The damage evolution is governed by the positive part of the strain tensor. The elastic domain is defined by a yield function f ⩽ 0 which takes the following form Table 1 Considered loading scenarios with available experimental observation. Fatigue loading scenario

132

Figure

Purpose

Performed by

LS1: Constant amplitudes

Calibration

Schneider et al., 2018 [19]

LS2: Cyclic increased loading

Validation

Baktheer et al., 2018 [20]

LS3: Two loading levels with different order

Qualitative validation of loading sequence effect

Holmen, 1982 [11]

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Fig. 2. Combined experimental and numerical methodology.

f (ε, ω) =

I2∗ − k (ω),

f n [ε +: ε +̇ ] ε+ · ω̇ = ⎛ ⎞ · . +) K C tr( ε 2 tr( ε +·ε +) 1 ⎝ ⎠

(6)

where I2∗ is the second invariant of the positive part of the energy release rate tensor Y + given as

(14)

2.2. Dimensional reduction to uniaxial stress state

I2∗ =

1 ⎛ + +⎞ 1 tr ⎜Y ·Y ⎟ = g 2 tr ⎜⎛ε +·ε +⎞⎟ 2 ⎝ 2 ⎠ ⎠ ⎝

(7)

To simulate the concrete fatigue behavior under compressive loading, the problem can be simplified by assuming a uniform stress and strain state within the specimen, i.e. (ε1 < 0 ), and (ε2 = ε3 > 0 ). The damage depends only on the positive part of the strain tensor. Therefore, the stress, strain and damage tensors can be written as

and k (ω) is the damage threshold

k (ω) = C0 − C1 tr (ω)

(8)

where g , C0, C1 are material parameters. Therefore, the yield function can be written as

g ε ̃ − [C0 − C1 tr (ω)], f ⎛⎜ε, ω⎞⎟ = 2 ⎝ ⎠

⎡ ε1 0 0 ⎤ ⎡0 0 0 ⎤ ⎡ σ1 0 0 ⎤ σ = ⎢ 0 0 0 ⎥, ε = ⎢ 0 ε2 0 ⎥, ω = ⎢ 0 ω2 0 ⎥. ⎢ 0 0 ε3 ⎥ ⎢ ⎣ 0 0 0⎦ ⎣ 0 0 ω2 ⎥ ⎦ ⎦ ⎣

(9)

With regard to the constitutive Eq. (4) the stress components in (15) can be expressed as

where ε ̃ is the equivalent strain proposed by Mazars [43] given as

ε ̃=

( ε1 + )2 + ( ε2 + )2 + ( ε3 + )2 ,

σ1 = (λ + 2μ) ε1 + 2(λ + α ω2) ε2

(10)

⎨ if ⎩

∂f (ε , ω) ·Ẏ + ∂Y + ∂f (ε , ω) ·Ẏ + ∂Y +

From (16) the strain component ε2 can be resolved as follows

ε2 =

(λ + α ω2) σ1 + g ω2 (λ + 2μ) . (λ + 2μ)[2(λ + μ) + 4(α + β ) ω2] − 2(λ + αω2)2

(17)

The yield function can then be reduced to

> 0, loading stage (ω̇ > 0) ⩽ 0, unloading stage (ω̇ = 0).

f (ε, ω) = g ε2 − (C0 + 2C1 ω2) (11)

∂f (ε, ω) , ∂Y +

ω̇ 2 =

g ⎛ f ⎞n λ + α ω2 σ1̇ − 2C1 ⎝ K ⎠ κ

+

,

(19)

where κ is given as

g2 g ⎛ ⎞ κ = ⎜⎛λ + 2μ⎞⎟ 2 ⎛⎜λ + μ⎞⎟ + 4 ⎛⎜α + β ⎞⎟ ω2 − α ⎜2ε2 + ε1⎟ − 2C1 ⎝ 2C1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎠ 2 − 2(λ + α ω2) .

(12)

where λ ω̇ is the damage multiplier which can be determined from the consistency condition f ̇ (ε, ω) = 0 delivering the equation

1 f n ∂f (ε, ω) ̇ +⎤ λ ω̇ = ⎛ ⎞ ⎡ ·Y h ⎝ K ⎠ ⎣ ∂Y + ⎦

(18)

so that the damage rate can be obtained as

Thus, damage grows upon loading and reloading without having to achieve a defined damage threshold. The damage evolution equation can be obtained by differentiating Eq. (6) with respect to energy release rate tensor as

ω̇ = λ ω̇

(16)

σ2 = 0 = [2(λ + μ) + 4(α + β ) ω2 ] ε1 + (λ + α ω2) ε1 + g ω2

where ε1 + is the positive part of the eigenvalue of the strain tensor ε . The extension of the model to consider the fatigue loading proposed by Marigo [29] allows for an evolution of the damage parameter even inside the damage yield surface. The yield limit concept is replaced by irreversibility loading-unloading criterion given as

⎧ if

(15)

(20)

In this form, the model is prepared for an implementation in a standard time-stepping algorithm representing the state of a compressive specimen using two state variables and a single degree of freedom. As a result, a cycle-by-cycle simulation of a high-cycle fatigue loading can be performed within a few minutes on a standard computational platform. The incremental form of the model formulation makes it applicable for

(13)

where h, n, K are material parameters. From Eqs. (12), (13) and (9), the damage evolution equation is obtained as 133

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equal to 0.05 and 0.2 respectively. The obtained material parameters are summarized in Table 2. The simulation of these Wöhler curves involved up to several millions of cycles. The computation time needed for 105 loading cycles with 100 increments in each cycle was 920 s using a computer with 16 GB of memory. One million cycles simulation take around 153.3 min. Even though this can be regarded as a highly efficient analysis of fatigue behavior it is still not applicable in engineering practice and further simplification are required.

arbitrary loading scenarios. 2.3. Elementary numerical study Numerically simulated concrete fatigue behavior with compressive strength fc = 84 MPa [44] using the described model is depicted in Fig. 3. The stress-strain curve and the damage evolution for the whole loading history of the loading range with S max = 0.9 and S min = 0.25 are plotted in Fig. 3ab, respectively. To examine the model behavior under variable amplitudes with different loading sequences, an example with two upper load levels is presented in Fig. 3cd showing the fatigue creep curves and the corresponding damage evolution. The fatigue creep curve represents the increasing displacement during the cycling. The effect of inherent, time-dependent creep of concrete is not included in the present studies. For both sequences, a consumed fatigue life equal to 50% is applied both for the first and for the second loading range. The damage evolution depicted in Fig. 3d, shows a different level of damage obtained for each sequence. Thus, the model predicts different state of fatigue for switched order of the loading ranges which contradicts with the P-M rule.

2.5. Model validation under varying loading ranges The experimental results of the test program with step-wise increased maximum load level presented in [20] and denoted as LS2 in Table 1 have been used to validate the model. The predicted response is shown as a red stress-strain curve in Fig. 5a together with the experimental results plotted as black curves. The corresponding fatigue creep curves for the upper and lower loading levels are depicted in Fig. 5b. Apparently, the results obtained by the numerical model show a reasonable match between the prediction and the test results.

2.4. Model calibration under constant loading ranges

2.6. Qualitative validation for loading sequence effect

The model parameters have been calibrated using the test program presented recently by Schneider et al. [19]. In this test program, the compressive fatigue behavior of concrete class C100/115 has been investigated under constant loading ranges corresponding to the loading scenario LS1 in Table 1 using cylinder specimens with the dimensions of 100 × 300 mm and loading frequency of 5 Hz.The Lamé constants λ, μ were obtained from the elastic properties i.e. Young’s modulus E = 49, 000 MPa and Poisson’s ratio ν = 0.2 . The other six parameters were identified by the curve fitting of the obtained fatigue test results as shown in Fig. 4ab. The comparison of the experimentally obtained Wöhler curves with the model is performed for lower loading levels

The aforementioned experimental results by Holmen et al. [11] investigating the effect of loading sequence for concrete class C40 are presented Fig. 6ab in form of fatigue creep curves for the (H-L) and (LH) loading scenarios corresponding to LS3 in Table 1, respectively. The value of the consumed fatigue life of the first loading range was set to the values (15, 20, 25, 27, 30%). The lifetime on the horizontal axis is normalized with respect to the lifetime prediction by the P-M rule. As already discussed in the introduction, the failure for the (H-L) scenario (Fig. 6a) was observed significantly earlier than predicted by the P-M rule. On the other hand, the reverse order of load levels (L-H) resulted in a fatigue life that was in some cases longer than the P-M rule

Fig. 3. Example of concrete fatigue behavior using Alliche model: (a) stress-strain curve under cyclic loading; (b) corresponding damage evolution during the loading history; (c) fatigue creep curves under constant and variable amplitudes; (d) corresponding damage evolution. 134

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Fig. 4. Calibration of the fatigue model under constant amplitudes using the test program presented in [19]: (a) Wöhler curve with Smin = 0.05; (b) Wöhler curve with Smin = 0.2 . Table 2 Parameters of the Alliche fatigue model for the studied concrete class C100/115. Parameter

λ

μ

g

K

C0

C1

α

β

n

Value

12500

18750

−10.0

0.00485

0.0

0.0019

2237.5

−2116.5

10

a)

S

Fig. 5. Validation of the fatigue model using the performed loading scenario (LS2): (a) stress-strain curve – comparison between the test and the model; (b) the corresponding fatigue creep curves for the upper and lower loading.

Fig. 6. Qualitative comparison of the loading sequence effect between the numerical and the experimental results presented by Holmen [11] in terms of fatigue creep curves: (a) H-L loading scenario – experimental results; (b) L-H loading scenario – experimental results; (c) H-L loading scenario – numerical results; (d) L-H loading scenario – numerical results. 135

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Then, after additional number of cycles N2 applied with the loading level S2max the damage becomes

estimation (Fig. 6b). The prediction of the numerical model for the studied concrete class C110/115 using the parameters from Table 2 exposed to the same loading scenarios is shown in Fig. 6cd. Even though the concrete classes used in the test (C40) and in the simulation (C110/115) were different, the experimentally observed relative reduction of lifetime for (H-L) loading, and relative extension of lifetime for (L-H) loading could be qualitatively reproduced by the numerical model. The validity of the model certainly cannot be proved using a single experiment series conducted with a non-matching concrete class. Still, the model delivers plausible results and reproduces the qualitative trend observed in the test. With regard to the lack of experimental data it can be used for a systematic study of the loading sequence effect on a broad scale of loading configurations as a basis for the refinement of concrete fatigue damage accumulation rule.

∗ N1∗ + N2 ⎧⎛ max ⎞ N1 + N2 ⎪⎜1.7 − S2 ⎟ ⎛ N2f ⎞, for ⎛ N2f ⎞ ⩽ 0.6 ⎝ ⎠ ⎝ ⎠ ⎠ ω2 = ⎝ ⎨ N ∗ + N 1.6 S2max ∗ N + N2 ⎪⎛ 1 f 2 ⎞ , for ⎛ 1 f ⎞ > 0.6, N ⎝ N2 ⎠ ⎩⎝ 2 ⎠

where max

N1∗

Besides the P-M rule several fatigue damage accumulation rules for varying loading ranges exist for homogeneous materials like steel [45–50]. Only a few fatigue damage accumulation rules, however, have been proposed for concrete compressive fatigue behavior. To set up the context for the rule proposed later on in Section 5 the existing rules are reviewed in the following paragraphs.

(21)

(22)

3.2. Nonlinear damage rule by Grzyboski and Meyer Another nonlinear damage rule has been proposed by Grzybowski and Meyer [22]. In this rule, the damage growth is dependent on the loading range defined by the upper stress level S max as follows

( )

N

Nf

for for

⎞ ⩽ 0.6 ⎠

S max

1 ⎨ max ⎪⎛ N1f ⎞ S2 N2f , N 1 ⎝ ⎠ ⎩

for ⎛ ⎝

N1∗ + N2 N2f

⎞ > 0.6 ⎠

( ) ⩽ 0.6 N

Nf

( ) > 0.6. N

Nf

(23)

The nonlinear damage rule is plotted in Fig. 7 for different values of the upper load level S max . To illustrate how this rule can be used to predict the fatigue life under varying loading ranges, let us take two loading levels S1max and S2max . The damage developed after applying N1 cycles of the first level S1max reads

⎧⎛ N1 max ⎞ N1 ⎪⎜1.7 − S1 ⎟ ⎛ N1f ⎞, for ⎛ N1f ⎞ ⩽ 0.6 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ω1 = ⎨ 1.6 S1max N ⎪⎛ N1f ⎞ , for ⎛ 1f ⎞ > 0.6. N1 ⎠ N1 ⎠ ⎝ ⎝ ⎩

(26)

For the two loading ranges scenario, a comparison between the model described in Section 2 and the existing fatigue damage accumulation rules reviewed in Section 3 is shown in Fig. 8. The diagram represents again the relative fatigue lifetime observed for the two cycling loading ranges. As already discussed in Section 2.4, the results obtained from the numerical model plotted in red color show a similar trend as observed in the experiments by Holmen et al. [11]. The nonlinear damage accumulation rule proposed by Shah shows an equal reduction of the fatigue life for both loading sequences (H-L) and (L-H). This implies that this rule does not account for the loading sequence effect. On the other hand, the result obtained by the rule proposed by Grzybowski is close to the P-M rule. A summary of the fatigue aspects that are reflected by the considered approaches to fatigue evaluation is presented in Fig. 9. The first aspect of the concrete fatigue behavior discussed by many authors is the nonlinear accumulation of the fatigue damage [44,56–60]. The second aspect is the consideration of the loading sequence effect. The P-M rule assumes a linear damage accumulation and does not account for the

2

( ),

N2f

4.1. Comparison of fatigue damage accumulation rules

where a, b and c are constants to be determined from the experimentally observed shape of the damage growth during fatigue life. In the present comparison the values proposed in [21] are taken as a = 1.14, b = −2.4, c = 2.26. The shape of the nonlinear damage rule is depicted as the solid curve in Fig. 7. For the prediction of the fatigue life under varying loading ranges, the cumulative damage rule has been proposed in the form

⎧⎛ max ⎞ ⎪⎜1.7 − S ⎟ ⎝ ⎠ ω= max ⎨ N 1.6 S ⎪ f , ⎩ N

N1∗ + N2

In consistence with other papers addressing the effect of loading sequence [11,51–55], the following studies are performed using two loading ranges with switched order, i.e. L-H and H-L, with the goal to provide a deeper insight both into the fatigue phenomenology and into the behavior of the model.

Based on the observable shape of the damage growth in concrete during fatigue life presented by Holmen et al. [11], a nonlinear damage rule has been proposed by Shah [21] as follows

3

for ⎛ ⎝

4. Numerical investigation of the loading sequence effect

3.1. Nonlinear damage rule by Shah

N N N a ∑ ⎛⎜ if ⎞⎟ + b ∑ ⎛⎜ if ⎞⎟ + c ∑ ⎛⎜ if ⎞⎟ = 1. ⎝ Ni ⎠ ⎝ Ni ⎠ ⎝ Ni ⎠

=

⎧ (1.7 − S1 ) N f ⎛ N1 ⎞, ⎪ (1.7 − S2max ) 2 ⎝ N1f ⎠

These accumulation rules are constructed empirically and do not provide any mechanical interpretation of the fatigue development. They will be confronted with the numerical prediction delivered by Aliche model in the following numerical studies.

3. Existing fatigue damage accumulation rules for concrete

N 3 N 2 N ω = a ⎛ f ⎞ + b ⎛ f ⎞ + c ⎛ f ⎞, ⎝N ⎠ ⎝N ⎠ ⎝N ⎠

(25)

Fig. 7. Fatigue damage accumulation rules.

(24) 136

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amount of Δη1− . which is equal to the reduction of the fatigue life of this (H-L) loading sequence with respect to the P-M rule. Similar observation can be done for the (L-H) sequence depicted in Fig. 10c), where the horizontal distance of the shift now represents the increase of the fatigue life with respect to the P-M rule. These observations lead us to the proposal of an improved engineering rule taking into account the loading sequence effect in a simple way: The key principle applied in the refined fatigue accumulation rule exploits the possibility to horizontally shift the segments of the curves obtained for constant loading ranges at the corresponding level of fatigue strain. 5. Enhanced rule for the loading sequence effect 5.1. Correction term for the P-M rule Fig. 8. Fatigue life prediction of the two loading ranges scenario – comparison between the different numerical models and damage accumulation rules.

The expression for the consumed fatigue life under a loading scenario consisting of n varying loading ranges is proposed in the form n−1

n

η=

∑ i=1

ηi +

∑

Δηi .

i=1

(27)

Similarly to the P-M rule, the fatigue failure occurs when the cumulative fatigue damage η = 1. The consumed fatigue life of the i-th loading level applied for the whole lifetime is given as ηi = Ni / Nif . The extension of the P-M rule is introduced by the correction terms Δηi reflecting the effect of a change between the cyclic loading ranges i and i + 1 on the response of a compressive specimen. The correction term Δηi is introduced as a response function of the parameters describing the load jump def

Δηi =  (Si, ΔSimax , ΔSimin, ηi )̃

Fig. 9. Comparison between the different approaches in terms of the reflection of the fatigue aspects.

(28)

The individual parameters of the response function have the following meaning (Fig. 11):

effect of applied loading sequences. On the other hand, the accumulation rules proposed by Shah [21], and Grzybowski et al. [22] assume nonlinear damage accumulation but do not consider the loading sequence effect. As already shown, the chosen numerical model has the ability to reproduce the qualitative trends in the experimental observations with a nonlinear damage accumulation and reflects the loading sequence effect.

• The mean load level S • •

4.2. Parametric study The effect of loading sequence has been studied for several values of the jump ΔS max between two loading ranges. As shown in Fig. 10a the results exhibit a significant deviation of the numerical prediction from the P-M rule that grows with increasing loading jump ΔS max . A more detailed view to the evaluation of the fatigue creep curves is provided in Fig. 10bc in form of the fatigue creep curves. The considered level of loading jump was ΔS max = 0.1. The horizontal axis represents the consumed fatigue life for each loading range. For example, for the case depicted in Fig. 10b with the red line, the higher loading range (H) has been applied first and consumed 20% from the fatigue life under constant loading. Then, the lower loading range (L) has been applied until failure. The graphical representation of the fatigue creep curve reveals an interesting aspect of fatigue damage evolution. Considering again for example the red line in Fig. 10b we observe that after the loading jump from the higher to the lower level (point 1), the remaining part of the fatigue creep curve applied with lower range is equivalent to the fatigue creep curve for constant loading range (L) starting at the same level of strain after the jump (point 2). To construct the fatigue creep curve for the (H-L) sequence, we can simply shift the curve obtained for the constant loading range (L) starting at the point 2 to the point 1 by the

i within the two loading ranges is given as m m Si = (Sim + Sim + 1)/2 where Si , Si + 1 are the mean values of the individual loading ranges i, i + 1 expressed as Sim = (Simax + Simin )/2 max min and Sim + 1 = (Si + 1 + Si + 1 )/2 , respectively. max The jumps ΔSi and ΔSimin between the upper and lower bounds of min , Simin the two subsequent loading ranges Simax , Simax + 1 are + 1 and Si min min min − S Δ S = S − Simin , reintroduced as ΔSimax = Simax and +1 i i i+1 spectively. The parameter ηi ̃ represents the cumulatively consumed fatigue life prior to the loading jump between the loading ranges i and i + 1.

To construct the response function, the numerical model introduced and validated in Section 4 has been used with the values of the load jump parameters covering the relevant range of values. The analysis was focused on loading scenarios with the lower load level kept constant, i.e. ΔSimin = 0 . The first view to the response surface function is provided along the axis aligned with the consumed fatigue life prior to the jump ηi ̃ shown in Fig. 12a. The depicted isolines of lifetime were calculated with the mean load level set to Si = 0.475 and with the upper jump parameter varied as follows ΔS max ∈ [± 0.05, ± 0.1, ± 0.2, ± 0.3]. In agreement with the studies performed previously in Section 4 lifetime extension is obtained for positive jump Simax > 0 representing an (L-H) sequence and lifetime reduction for negative jump Simax < 0 corresponding to an (H-L) sequence. The view on the response surface along the η ̃ axis shown in Fig. 12a for several values of Simax reveals that it can be sufficiently well represented using a bilinear approximation. This feature of the response surface has been exploited to construct the response surface around the line connecting the knee points with the coordinates (ηx , Δηmax ) as follows 137

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Fig. 10. Numerical investigations of the loading sequence effect using two loading level High (H) and Low (L): (a) fatigue failure prediction with comparison to P-M rule for different loading jump; (b) fatigue creep curves for the (L-H) scenario; (c) fatigue creep curves for the (H-L) scenario.

Δηmax (S , ΔS max ) = [f1 (ΔS max ) + f2 (S )] sign(ΔS max )

(30)

where

f1 (ΔS max ) = A (ΔS max )2 + B ΔS max sign(ΔS max )

(31)

f2 (S ) = C (0.475 − S ).

(32)

Here, A, B, C are material parameters identified using the numerical simulation of the fatigue behavior under two loading ranges and, for the studied concrete mixture, can be taken as

A = −10.66, B = 6.1, C = 2. Each of the 12 points in Fig. 12b has been performed for 11 levels of consumed fatigue life η .̃ Further, each level of η ̃ has been simulated for 5 levels of the mean value of the load S . Thus, 12·11·5 = 660 simulations have been performed to obtain the parameters A, B , and C defining the response surface. From Fig. 12a we can further conclude that the horizontal value ηx can be assumed linearly dependent on Δηmax with different linearity coefficients for positive and negative signs of the load jump, i.e.

Fig. 11. Illustration of the loading parameters.

ηx = ⎧ ⎪ Δηmax ⎪ Δηi = ⎨ ⎪ Δηmax ⎪ ⎩

⎛1 − ⎝

⎜

ηx − ηi ̃ ⎞ ηx

⎛ ηi ̃ − 1 ⎟⎞, ⎜ ⎝

ηx − 1

⎠

⎟

,

0 < ηi ̃ ⩽ ηx

⎠ ηx < ηi ̃ < 1,

⎧ η↑ + Δηmax /tan(β↑), ⎨ η↓ + Δηmax /tan(β↓), ⎩

Δηmax > 0 Δηmax < 0.

(33)

Here, η↑ , β↑, η↓ , β↓ are material parameters that for the concrete matrix at hand evaluate to

η↑ = 0.74, β↑, =74.7∘, η↓ = 0.59, β↓ = 60.5∘.

(29)

With the identified material parameters we have obtained an approximation of the refined fatigue accumulation rule (28). The seven parameters have been obtained from the fatigue tests with two constant load ranges (Section 2.4) and from the numerical simulations of the two-level loading sequence (H-L) and (L-H). Let us now briefly address

To cover the whole domain of possible load scenarios with constant minimum load, the response function has been developed around the slice Si = 0.475 as indicated in Fig. 12bc. Based on the numerical results shown as black circles, we propose to approximate Δηmax as a function of the jump parameters with step index omitted for brevity as 138

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Fig. 12. Analyzing of the two loading levels results: (a) idealized bi-linear envelope of Δη for different values of the loading jump ΔS max ; (b) the relation between the loading jump ΔS max and the horizontal value of the knee point Δηmax ; (c) the relation between the middle load level Si and the horizontal value of the knee point Δηmax for different values of the loading jump ΔS max .

Fig. 13. Validation of the proposed rule for predicting the remaining fatigue life under two loading ranges scenario; (a) results for different values of the loading jump ΔS max ; (b) results for different values of the mean value of the loading levels Si . 139

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from the numerically obtained results that grows with the increasing loading jump ΔS max as indicated by the three different patterns corresponding to the three studied values of the loading jump. The influence of the mean load level value has been examined in Fig. 13b showing a strong deviation of the P-M rule from the simulation results. Summarizing, for the two-level loading sequences the prediction obtained by the proposed rule shows a good agreement with the numerically simulated results in comparison to the prediction of the P-M rule in the whole range of the parameters S , ΔS max and η .̃

the question if the constructed engineering model can capture also more complex loading scenarios. 5.2. Validation of the engineering rule using the simulation Consider a two loading ranges scenario after consuming a certain percentage of the fatigue life of the first range η1. The prediction of the remaining fatigue ηrem life of the second range then reads

ηrem = 1 − η = 1 − η1 − Δη1.

(34)

In the evaluation of the correction term Δη1 using the approximation of the response surface function (29) the consumed fatigue life prior to the first jump is set to

η1̃ = η1.

5.2.2. Multiple loading levels scenario To check the rule validity under multiple loading levels, two scenarios consisting of three loading ranges have been used i.e. (H-L-H) and (L-H-L). In particular, several combinations of the consumed fatigue life η ̃ of the first and the second loading ranges have been used. The remaining fatigue life of the third loading range was then predicted. The results of the (H-L-H) and (L-H-L) loading scenarios are depicted in Fig. 14ab, respectively. These results show that even though the proposed rule was calibrated based on the two loading levels it can also reproduce the fatigue response for loading scenarios with multiple loading ranges.

(35)

For three loading ranges, similar procedure can be applied. The remaining lifetime is given as

ηrem = 1 − η = 1 − η1 − Δη1 − η2 − Δη2.

(36)

The consumed lifetime prior to the second loading jump between the levels (2, 3) is given as

η2̃ = η1̃ + Δη1 + η2.

(37)

Thus, ηi ̃ represents a path dependent state variable capturing the fatigue life consumption during the history up to the load change i, i + 1. In the two following studies, the remaining fatigue life obtained the approximated correction terms is compared with the full scale numerical evaluation and with the P-M rule.

6. Conclusions The validity of the well-known (P-M) rule for predicting the fatigue life under variable amplitudes has been studied for changed order of loading ranges. The study was performed using a fatigue model that couples the damage evolution with the positive rate of the total strain. Thus, the damage grows upon loading and reloading steps within the individual cycles at subcritical load levels. The model was calibrated using two test series with constant loading ranges. Subsequently, the model prediction for non-uniform cyclic loading was qualitatively compared with available experimental observations. The results obtained using the numerical model for the loading scenario with two loading ranges agree well with the experimental observations presented

5.2.1. Two loading levels scenario The prediction of the remaining fatigue life obtained by the proposed fatigue assessment rule is plotted with the green points and compared with the numerically calculated results simulating the whole fatigue process cycle by cycle. In Fig. 13a, three different values of the amplitude jump ΔS max are presented. The results predicted using the PM rule are plotted as the red points. They show a significant deviation

Fig. 14. Validation of the proposed rule for predicting the remaining fatigue life under three loading ranges scenario; (a) results for (H-L-H) loading scenario; (b) results for (L-H-L) loading scenario. 140

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by Holmen et al. [11]. These results show an unsafe prediction of (P-M) rule for the (H-L) sequence and more conservative prediction for the (LH) sequence. The combined numerical and experimental methodology provides both efficient and accurate approach to studying the concrete fatigue behavior. It can serve as a basis for developing a refined engineering rule taking into account the loading sequence effect. The modeling framework will be used to design an experimental program systematically covering the relevant range of loading scenarios with the goal to validate the proposed engineering rule.

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