Mathematical Problems in Engineering

Volume 2014, Article ID 407193, 10 pages

http://dx.doi.org/10.1155/2014/407193

## Fatigue Damage Assessment for Concrete Structures Using a Frequency-Domain Method

^{1}State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China^{2}Key Laboratory of Coast Civil Structure Safety, Tianjin University, Ministry of Education, Tianjin 300072, China^{3}School of Civil Engineering, Tianjin University, Tianjin 300072, China

Received 22 October 2014; Revised 15 December 2014; Accepted 18 December 2014; Published 29 December 2014

Academic Editor: Yang Tang

Copyright © 2014 Hongyan Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A fatigue damage assessment for concrete was carried out according to Eurocode 2. Three frequency-domain methods, the level crossing counting (LCC) method, the range counting (RC) method, and a new proposed method, were used for the damage assessment. The applicability of these frequency-domain methods was evaluated by comparison with the rainflow counting method in the time domain. A preliminary numerical study was carried out to verify the applicability of the frequency-domain methods for stress processes with different bandwidths; thus, the applicability of the LCC method and the new method was preliminarily confirmed. The fatigue strength of concrete had a minor effect on the fatigue damage assessment. The applicability of the LCC and the new methods deteriorated for relatively low coefficients of variance of the stress process because the ultimate number of constant amplitude cycles was sensitive to the range of the cycles. The validity of the joint probability functions of the two methods was proven using a numerical simulation. The integration intervals of the two frequency-domain methods were varied to estimate the lower and upper bounds on the fatigue damage, which can serve as references to evaluate the accuracy of the time-domain method results.

#### 1. Introduction

Fatigue is the process of gradual damage to materials that are subjected to continually changing stresses. Concrete fatigue is primarily a problem for offshore structures, railway sleepers, and bridges, which are often exposed to alternating loadings [1]. Unlike steel structures, both the range and level of stress affect fatigue damage in concrete structures [2]. In this study, the European Standard, Eurocode 2 [3], was applied to assess the fatigue damage of concrete structures.

A well-established procedure for fatigue damage assessment involves the time-domain analysis of the stress processes that a structure will be subjected to over its lifetime [4–6]. Stress processes in structures that are induced by wind, waves, or road irregularities often occur on a fairly irregular and random basis; however, only the ultimate number of stress cycles for a constant stress range and the mean stress that can be sustained up to failure is known. Thus, a description of the cycles in the stress records in terms of parameters such as the number of counted cycles, the distribution of cycle amplitudes, and the means is required. The counting method [7] was applied in this study. Level crossing counting (LCC), peak counting (PC), simple range counting (RC), and rainflow counting (RFC) are the counting methods most commonly used in engineering practice. The RFC method is widely accepted as the most efficient counting method available [8].

The fatigue damage of structures can also be assessed in the frequency domain. The frequency-domain method is faster and more cost efficient than the time-domain method [9–11]. Variations in parameters and optimization can be rapidly executed for a properly functioning frequency-domain method. The stress processes in the frequency domain are represented using a spectral formulation, such as a power spectral density (PSD) function, which provides information on the power distribution over a range of frequencies. The first* n* (typically 4) moments of the PSD function are used to find the probability density function (PDF) for the peaks and valleys of the stress process (i.e., the range of the stress and mean stress) [12]. The fatigue damage can be calculated by integration using the PDF and Palmgren-Miner’s rule.

In this study, PDFs that are available in the literature and a PDF developed by the authors were applied to calculate the fatigue damage of concrete structures. The RFC method in the time domain is the standard method for fatigue assessment. The results of different frequency-domain methods were compared with the results of the RFC method to select the most effective frequency-domain methods. A preliminary numerical study was carried out to verify the applicability of the frequency-domain methods for stress processes with different bandwidths; thus, the applicability of the LCC method and the new method was preliminarily confirmed. The integration intervals of the two frequency-domain methods were varied to estimate the lower and upper bounds on the fatigue damage, which can serve as references to evaluate the accuracy of the time-domain method results. Few published reports on fatigue assessment for concrete in the frequency domain are currently available; thus, the results of this study can serve as a useful reference in this field [13].

#### 2. Basic Theory

##### 2.1. Frequency-Domain Process

A stress process is assumed to be a stationary Gaussian process. The autocorrelation function of is defined as follows [14]:

The Fourier transform of the autocorrelation function is expressed as where the new function, , is the PSD of . can also be written in terms of :

Equations (2) and (3) are known as the Wiener-Khintchine relations.

We are typically not interested in distinguishing a PSD that is associated with a negative frequency from one that is associated with a positive frequency. A single-sided PSD can be defined as follows:

Variables that are encountered in engineering analysis, such as the stress, are real. The PSD is an even function for real variables; thus,

The th moment of the PSD function can be obtained from the characteristic of the PSD function. The spectral moments are defined as follows:

The first four moments (, , , and ) are the most commonly used moments of the PSD function.

The bandwidth of a signal is taken to be the frequency interval over which most of the signal power is concentrated. The bandwidth can be defined in terms of the spectral moments [15]:

These two bandwidth parameters tend to unity for a “narrow-band” signal and zero for a “wide-band” signal.

##### 2.2. Fatigue Damage Assessment Method for Concrete Structures from Eurocode 2

In Eurocode 2, Palmgren-Miner’s rule [16] is applied to calculate the total fatigue damage, as given in (8). A satisfactory fatigue resistance may be assumed for concrete under compression if : where is the total fatigue damage; is the number of intervals with a constant amplitude; is the actual number of constant-amplitude cycles in the interval “”; is the ultimate number of constant-amplitude cycles in the interval “” that can be carried out before failure: where is the stress ratio; is the minimum stress compression level; is the maximum stress compression level; is the lowest stress in a cycle; is the highest stress in a cycle; and is the design fatigue strength of the concrete. is a function of both the peaks and valleys of the stress process.

##### 2.3. Counting Methods

LCC, PC, RC, and RFC are the four most commonly used counting methods in engineering practice.

In the LCC method, every crossing of the predetermined stress levels is counted. Then, the most damaging level cycle is counted for fatigue by constructing the largest possible cycle, followed by the second largest cycle, and so on, until all of the level crossings have been considered.

In the PC method, the occurrence of a relative maximum or minimum stress value is identified. Peaks (valleys) that are above (below) the reference level are counted.

In the RC method, a range is defined as the difference between two successive reversals: the range is positive when a valley is followed by a peak and negative when a peak is followed by a valley.

The RFC method is the most frequently used cycle counting method in engineering practice. Starting from a local load maximum, , two minima are identified before and after , that is, and . The point with the smallest deviation from is chosen as the rainflow minimum, , thus producing the* k*th rainflow cycle (, ). The aforementioned procedure is repeated for the entire stress process.

#### 3. Fatigue Damage Assessment for Concrete in the Frequency Domain

##### 3.1. Joint Probability Density Applied to Fatigue Damage Assessment

As described in Section 2.2, is a function of both the peaks and valleys in the stress process; thus, the fatigue damage for concrete structures is also a function of the peaks and valleys in the stress process. The joint probability density of the counted cycles , which is a function of peak “” (corresponding to in (8)) and valley “” (corresponding to in (8)), is constructed by applying the spectral moment parameters of the PSD function. As the RFC method is the “standard” counting method, we determine the joint probability density of counted cycles using . However, no analytical solutions of are currently available, and only approximate approaches can be used to obtain an accurate fatigue damage assessment.

Tovo [17, 18] calculated the joint probability density of counted cycles for a zero-mean Gaussian process from the distribution of level crossing counted cycles as follows:where is the Dirac delta function and and are the cumulative distributions of the peaks and valleys, respectively: where is the stress, is the standard deviation, is the cumulative Gaussian distribution, and is the bandwidth parameter, as defined by (7).

Tovo [17, 18] also calculated a joint probability density for the cycles in a Gaussian process, where the peaks and valleys distributions are given by (11) and (12), respectively, from the distribution of the counted cycles over the stress range:

Equation (13) can be rewritten as a function of the mean stress “” and stress range “”: where and .

Equation (14) shows that the joint probability density is a product of and ; thus, it is reasonable to assume that the mean stress “” and stress range “” are two independent variables. Then, other expressions for can replace that in (14) to obtain a new joint probability density. The Dirlik method [19] is considered the most accurate approach for constructing an empirical expression for the PDF over the stress range “” [20] and is used to obtain a new proposed : where

##### 3.2. Expressions for Fatigue Damage

In this section, (10), (14), and (15) are used to calculate the fatigue damage in the frequency domain.

The expected peak occurrence frequency is defined as [21]

The fatigue damage over a given time can be calculated as follows:

When (see (10)) is used, the component related to may be neglected in the calculation because this function implies that ; that is, there is no effect on the damage. When , ; thus,

From the definition of , ; thus, , and the component related to clearly has no effect on the damage.

Equations (10)–(15) and (21) are all based on a zero-mean process; for a process with a non-zero-mean , these equations can easily be modified using a variable shift. In this study, we perform a fatigue damage assessment for concrete under compression; thus, all of the stresses have the same sign (and are hereafter assumed to be positive). The following equations are given for a non-zero-mean process.

The fatigue damage can be calculated by applying as follows:

Because all of the stresses are positive, the lower limit of the valley is zero. When , the valley of a cycle is set to zero.

Similarly, the fatigue damage can be calculated by the RC method, and can be calculated by the new method developed by the authors, that is, by applying (14), (15), and (23):

As the stress record is assumed to be Gaussian, almost all of the values will fall in the interval ; thus, the following integration intervals can be used to calculate and , . (The upper bound on the integration interval for is estimated as . For , the lower bound on the integration interval is set to zero.)

#### 4. Numerical Simulation

Numerical simulations were performed to investigate the applicability of the aforementioned frequency-domain methods. The fatigue damage was assessed using the time-domain method with RFC and Palmgren-Miner’s rule to obtain the “standard” results. The accuracy of the results obtained from the frequency-domain methods was evaluated by comparison with the accuracy of these “standard” results.

The aforementioned joint PDFs were all constructed from spectral moments and bandwidth parameters. Preliminary numerical simulations were carried out to investigate the effect of the bandwidth parameters, particularly , on the applicability of the frequency-domain methods. Some PSDs with simple shapes (see Figure 1) were applied in the simulation. The random stress processes that were used in the RFC methods in the time domain can be derived from these PSDs (see Figure 2) [22, 23].