Application of Probabilistic Approach to Investigate Influence of Details in Time History of Temperature Changes on the HCF Life of Integrated Bridge Steel Piles Installed on Water

Abstract

This research estimates the high-cycle fatigue (HCF) life of integrated concrete bridge installed on water due to temperature changes. To this end, CATIA software was used to geometrically model of a real-scale bridge. Next, thermal–structural coupling analysis was performed by finite element (FE) simulation in ANSYS WORKBENCH software. The comparison technique with experimental data was used to validate the simulation. Afterward, thermal analysis was performed due to air temperature changes in different modes, including the average monthly temperature changes (large variations) as well as the maximum and minimum monthly temperature changes (small variations). The results showed that the most changes in deck length and subsequent maximum deviation in the upper part of steel piles were related to the three warm seasons in the presence of the water. Eventually, a probabilistic approach was employed to find variable amplitude fatigue lifetime of the component based on the number of annual loading blocks. To achieve the high-accuracy response, the effective parameters of the proposed probabilistic approach, including order of Fourier series and the stress range, were optimized automatically. In addition, to obtain HCF behavior of raw material, axial tension–compression fatigue tests were performed on the standard specimens fabricated from steel piles. The results revealed that considering small variations in the calculation of structural fatigue life led to a 550% reduction in life compared to structural analysis due to large variations. In addition, the obtained results were compared with the finite element results.

1. Introduction

Bridges play a major role in a country’s transportation system. In general, bridges are considered as communication nodes of the country’s transportation arteries and are of particular importance in this regard. On the other hand, these concrete structures are old in some cases and are gradually approaching their design life according to the regulations of civil engineering. Therefore, it is necessary to evaluate the strength of concrete structures to estimate the current condition and make a decision to repair, improve, or dismantle them [1]. Extensive studies have been conducted to assess the strength of concrete structures based on laboratory results and the use of various techniques, such as finite element simulation, data mining, and machine learning [2,3,4,5,6,7]. Moreover, some scholars have sought to optimize raw materials of concrete and their compounds [8,9,10,11,12] or production process parameters [13,14,15,16] to increase the strength of such structures. The scientists have given most attention to the instantaneous strength due to different loads, including tension, compression, bending, and shear loading conditions or a combination of two or more loads. In fact, these calculations are carried out by engineers at the design stage. However, one of the most destructive phenomena that leads to sudden failure due to cyclic loading is fatigue [17,18,19,20]. Therefore, it is necessary to evaluate the service life and fatigue behavior of engineering structures, especially civil structures such as bridges, which are subjected to various and complex cyclic loads such as temperature variations in thermal cyclic loading and moving vehicle load in structural cyclic loading. In addition to the study on concrete, welding joints are one of the critical points in engineering structures, where cracks are usually formed in the heat-affected zone (HAZ) and the system fails from the weld area. One of the conventional methods to increase the fatigue life of metallic materials is surface treatment by creating compressive residual stress (CRS) on the surface and subsurface and also refinement of grain size. In this regard, different types of shot peening process are successful and sometimes multifold the fatigue life of metals [21,22,23,24,25,26]. This process also has an effect on the welding joints and leads to the elimination of tension residual stress (TRS) caused by the welding process [27]. However, the Almen intensity as one of the most effective parameters of shot peening treatment is very important because the martensitic phase is formed in the HAZ (i.e., material is extremely brittle in this area) [28,29]. In this way, the incorrect selection of shot peening parameters causes the formation of cracks in the welding area and ultimately reduces the life of the structure. In other words, in this case, shot peening has a negative effect. Recently, in order to prevent from this happening, some scientists have suggested water jet peening treatment for welded joints [30,31]. Apart from employing various techniques to improve the welding joints that exist in bridges, both metal and others, nowadays, the use of integrated bridges to address the shortcomings of previous structures in its implementation and efficiency during operation has become very common all around the world. In addition, to prevent corrosion of steel piles in the vicinity of water, reinforced concrete barriers are used around the piles or completely composite concrete bridges are implemented. In this regard, integrated bridges are those concrete bridges in which the sub-structure and the main structure are executed continuously. The most important advantage of this issue is that the expansion interface between the deck and the wall of other parts is removed with integrated concreting. Accordingly, various behaviors of such structures (static strength, vibrational properties, fatigue strength, corrosion resistance, and other damaging phenomena) should be identified considering different conditions, including the type of application (human, car, and cargo), environmental conditions (temperature variations, wind intensity, mounting steel piles in stagnant water, or investigating the effect of wave impact due to running water), and working conditions. In this paper, studies conducted by other scholars and experts in this field are reviewed. Additionally, their important achievements are expressed and finally, the innovation of the current research is presented in comparison with the previously published papers.

The impact of thermal elongation of the integrated bridge deck on the pressure applied to the bridge has been examined [32]. In this research, the analyses were performed for one steel pile in the FE software. The FE results indicated that the pressure value on the walls had a direct relationship with the displacement amplitude and the number of thermal cycles. Furthermore, the impact of deck height as a geometric parameter of the bridge on the behavior of piles in clay soils and rock shallow floors has been studied [33]. The most important achievement of this study is that on the bridges with taller piles, the bending moment may be reaching its final level before the piles are fatigued and the bridge fails. In other words, as the height of the bridge increases, the application of the fatigue criterion decreases, and the use of the bending moment criterion to investigate the main cause of failure will be appropriate. Moreover, Ehteshami and Shooshtari studied the interaction of soil and structure on the seismic behavior of integrated bridges [34]. They considered six different modes of soil–structure interaction, and compared the analyses results to the non-interacting structural response. The effects of thermal-induced flexural strain cycles on the low-cycle fatigue (LCF) life of steel H-piles have been studied [35]. In the research conducted by Karalar and Dicleli, a new cycle counting technique was proposed. They also developed a mathematical formula for estimating the LCF life of H-shaped steel piles based on the measurement data in North America. In addition, a novel sinusoidal function-based model has been proposed to obtain bridge displacements due to the daily and seasonal temperature changes [36]. Abdollahnia et al. predicted fatigue life of H cross-section steel piles of an integrated concrete bridge under different conditions of motionless water and sea waves clash [37]. They reported that motionless water could not led to the fatigue phenomenon in bridges. Furthermore, the impact of sea waves clash could trigger a great deal of deformation in the steel piles and eventually cause fatigue damage and failure to bridges. Afterward, they predicted the fatigue life of steel piles considering the effect of water waves clash using two methods, including finite element simulation and probabilistic approach. In addition, the authors evaluated the LCF behavior of steel piles with H cross-section due to temperature changes [38]. In this study, the strain-life ($\epsilon -N$) criterion was employed to assess the fatigue lifetime of the component under displacement control loading (zero ratios: R = 0 and frequency of 1 Hz). Finally, it was shown that the simulation results presented with 6.5% of the discrepancies compared to the fatigue test results were suitable for future studies. Recently, different well-known equivalent stress criteria have been utilized to assess the multi-axial fatigue life of integrated concrete bridge due to the simultaneous effects of temperature changes and sea waves clash [39]. In this research, the history of temperature changes was considered as a monthly average. The analysis results indicated that the Liu–Zenner criterion is more conservative than other equivalent stress-based fatigue criteria (i.e., Findley, McDiarmid, Carpinteri-Spagnoli, von Mises, Dang Van, and Modified-Findley and Modified-McDiarmid proposed by Shariyat). Similar conclusions have been repeatedly reported for industrial components [40,41,42,43]. Moreover, they reported that the multi-axial fatigue life of the bridge due to the simultaneous effects of temperature variations and sea waves clash is reduced by a minimum 66% compared to the fatigue life of the structure due to the effect of each factor. In summary, extensive studies have been carried out in the field of fatigue life prediction of integrated concrete bridges and their H-shaped steel piles subjected to temperature changes. Meanwhile, most of them have been conducted analytically [41,42,43,44,45,46,47,48]. Finite element simulations are rarely found in this area. Nevertheless, valuable results from costly and time-consuming cyclic experiments have also been published. However, in none of the previous studies, have the details of the time–temperature diagram and their effects on the system response and fatigue life prediction been discussed. In fact, to predict the fatigue life of integrated concrete bridges and also in the design stage of structures under cyclic thermal loads caused by temperature changes, there is no general instruction to clearly define the quality of temperature measurement (the time interval of temperature events i.e., daily, weekly, monthly, or seasonal). Furthermore, based on the above-mentioned literature review, further details about them are not recommended. For example, if daily temperature events are considered, the average temperature is sufficient to achieve the most accurate assessment of the structure, or the minimum and maximum temperature should be taken into account. To study this challenge, for the first time, the detailed influences in time history of temperature changes (i.e., how to consider loading conditions) on the fatigue life of these structures were investigated. In other words, herein, the difference of the fatigue life of a structure installed on water considering the average and the max–min temperature conditions throughout a year and monthly was detected. To this end, thermal analysis was performed due to air temperature changes in different modes, including the mean values of monthly temperature changes (large variations) as well as the maximum and the minimum values of monthly temperature changes (small variations). Next, thermal–structural coupling analysis was performed to extract displacements, stresses, and strains in the structure, especially in the critical region prone to fail. Eventually, by utilizing a valid probabilistic approach with automatic optimization of effective parameters, the variable amplitude fatigue lifetime of component was estimated. In addition, the obtained results were compared with the finite element results.

2. Simulation Method

2.1. Geometry of Bridge Structure

In the current study, the bridge structure and its accessories, including a concrete deck, concrete anchors (walls), I-shaped concrete beams (Ribs), and steel piles, were modeled three-dimensionally and separately. Afterward, they were assembled in CATIA software and inserted into the ANSYS WORKBENCH software as a well-known finite element software (using *.stp file). This model consists of two middle pedestals, each of which includes 10 steel piles. Moreover, this model comprises 44 steel piles (type IPE-220). In addition, four rectangular concrete beams were considered for connecting steel piles to concrete material under the bridge’s deck. Between each of these beams, I-shaped cross-section concrete beams (Ribs) with IPE-600 specifications were integrated into the concrete deck. Dimensional specifications and geometrical parameters of piles are given in

Table 1. Dimensional specifications and geometrical parameters of piles used in the current study.

	Symbol	Unit	Value
			IPE-220	IPE-600
	h	mm	220	600
	B	mm	110	220
	s	mm	5.9	12
	t	mm	9.2	19
	r	mm	12	24
	c	mm	21.2	43
	h-2c	mm	177	514
	A	${\mathrm{cm}}^2$	33.4	156
	G	$\mathrm{kg}/\mathrm{m}$	26.2	122
	$J_x$	${\mathrm{cm}}^4$	2770	92,080
	$W_x$	${\mathrm{cm}}^3$	252	3070
	$I_x$	cm	9.11	24.3
	$J_y$	${\mathrm{cm}}^4$	205	3390
	$W_y$	${\mathrm{cm}}^3$	37.3	308
	$I_y$	cm	2.48	4.66

[37].

These beams are divided into three sections: between the first wall and the first middle pedestals, between the first and second middle pedestals, and between the second middle pedestals and the second wall. Five Ribs at equal intervals were considered for each section (it means that 15 Ribs were modeled as integrated with the bridge’s deck) [37,39]. Figure 1 presents the real-scale geometric model of an integrated concrete bridge in details. The length of steel piles is 6 m. Additionally, width and length of the bridge deck, regardless of the concrete anchors and its accessories, are 11.688 and 104.644 m, respectively.

To consider the integrated behavior for concrete parts of the bridge, one group, including the geometries of the deck, middle beams, and I-shaped concrete beams (IPE 600), was defined by non-separation and continuous deformation in the FE software. However, the edges and surfaces of these parts are identifiable, and boundary and loading conditions could be applied on them separately.

2.2. Materials Used for FE Simulation

In the current study, structural steel and ordinary concrete were used as the raw materials. The mechanical characteristics and thermal properties for both materials are given in

Table 2. Mechanical characteristics of the materials used in the current study [37,39].

Parameter	Unit	Value
		Structural Steel	Ordinary Concrete
Density	$\mathrm{Kg}/{\mathrm{m}}^3$	7850	2300
Young Modulus	GPa	200	30
Poisson’s Ratio	---	0.3	0.18
Bulk Modulus	GPa	166	---
Shear Modulus	GPa	76.92	---
Tensile Yield Stress	MPa	250	---
Compressive Yield Stress	MPa	250	---
Tensile Ultimate Strength	MPa	460	5
Compression Ultimate Strength	MPa	---	41

and

Table 3. Thermal properties of the materials used in the current study.

Parameter	Unit	Value
		Structural Steel	Ordinary Concrete
Reference temperature	$℃$	22	22
Coefficient of thermal expansion (CTE)	${10}^{-5}\times \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$℃$}\right.$	1.3	0.95

, respectively [37,39]. Since temperature changes are studied as the main cause of failure in this research, the thermal properties of the material, i.e., coefficient of thermal expansion (CTE) in terms of temperature for steel and concrete were considered in accordance with Figure 2a,b, respectively (red color lines) [49]. It is obvious that the CTE changes are noticeable by increasing temperature over 200 degrees, and in the current case study where the maximum ambient temperature does not exceed 35 degrees, there are no significant changes in thermal coefficients. Accordingly, most scholars consider constant value for this parameter and perform steady-state thermal analysis, but in this research, to achieve the most accurate response, temperature-dependent CTE was defined in the software and transient thermal analysis was performed.

The current study aimed to predict the fatigue life of steel piles as one of the most important and vulnerable components in the bridge structure due to the temperature variations and also sea waves clash [32,33,34,35,36,37,38,39]. Most research in this area has identified intermediate steel piles as the most critical part of the bridge under cyclic loading. Therefore, the focus of the present study is solely on assessing the fatigue life of this component. Accordingly, the fatigue properties of concrete material were neglected [37,38,39]. In addition, to obtain HCF behavior of structural steel, one IPE-220 steel beam was provided, and standard axial high-cycle fatigue test specimens were fabricated according to Figure 3. To this end, a CNC wire-cutting machine was used to prepare samples based on dimensional recommendations and surface roughness stated in ASTM E466 standard [50]. The force controlled axial fatigue test was performed at room temperature; loading ratio and loading frequency were considered to zero and 10 Hz, respectively. Moreover, fatigue tests were conducted on six applied stress levels and in order to check the accuracy and repeatability of the response, four tests were performed in each stress level and the average number of cycles to failure was entered in the FE software as the failure cycle corresponding to the applied stress (based on the recommendations of fatigue standards, at least three repetitions are required in each fatigue test). Figure 4 demonstrates the stress-life diagram as the results of fatigue tests performed in the present research.

2.3. Meshing and Convergence Analysis

The element type and meshing method directly affect the FE responses. Accordingly, after selecting the appropriate type of element related to the desired analysis, it is necessary to check the independence of the response to the meshing method and the number of elements. Furthermore, this sensitivity analysis helps to achieve a reliable response by reducing computational costs and having sufficient accuracy [51,52,53]. Hence, mesh sensitivity analysis was conducted considering different sizes of the element in each component. To this end, the maximum deformation in the upper part of steel piles was assumed as the target (for further details of the process of the mesh convergence, the readers refer to [37]). The final FE model (Figure 5) contains 121,655 elements; the element size of the deck, wall, concrete beam under the deck, concrete I-shaped beam (IPE-600), and H-shaped steel piles (IPE-220) is 1, 0.5, 0.2, 0.2, and 0.2 m, respectively.

3. Finite Element Analysis

The validation process of the presented FE simulation was thoroughly examined in the authors’ previous research [38]. However, for the LCF analysis due to temperature variations, the accuracy of the FE results in comparison with the real-scale laboratory results was reported to be approximately 93.5%. Therefore, to prevent the repetition of statements and also to provide more important content, this section was ignored in this article. Next, to predict variable amplitude fatigue life of the structure, we described different types of analysis using this valid FEM.

3.1. Thermal Analysis

Daily changes in the air temperature led to an increase or decrease in component length of the structure, which results in thermal strain and stress in the structure. These changes are known as the non-uniform cyclic thermal stresses applied to the structure, which after a while, leads to damage and eventually failure in the structure. Hence, one of the most important steps in calculating and predicting the thermal fatigue life of a structure is to extract the history of temperature variations, which must be collected depending on the geographical zone of the bridge.

3.1.1. Extraction of the Temperature Variations History for One Year

In the present research, the temperature information of Tehran city (the capital of Iran) was collected as the maximum and minimum daily temperature. Afterward, the average temperature of each day was calculated (Figure 6). As can be seen, these temperature changes do not have a specific trend; in other words, they are random in nature. On the other hand, each color represents the temperature changes in one month. To draw this graph, the average temperature of each day was considered, and the data were discontinuous. To simplify the matter, the temperature data in one month were connected to each other. In other words, the data were considered as a continuously data. So, there is a data jump between the connection of daily temperature and consecutive months with each other. In fact, this task is known as a prerequisite for conducting transient thermal analysis. Nevertheless, it is not economical for engineers since the data on daily temperature information is very large and transient thermal analysis using FE simulation requires a significant amount of time and high computational costs. To overcome this challenge, the mean value of temperature was calculated monthly and the maximum and minimum temperatures were extracted for each month to be used in future analyses.

In general, temperature variations are applied to the bridge deck (top surface of the structure) and cause changes in the dimensions, specifically the length of deck. However, these temperature changes do not significantly change the dimensions of other bridge components, such as infrastructure. This is because these parts are usually located in the soil and will not be affected by these temperature changes, or they will not be of considerable length to be sensitive to temperature changes. The thermal energy of the sun is the main factor in changing the length of the deck due to air temperature variations and external loads do not play a role in creating it. This change in length does not impose any force or stress on the structure if it does not limit, but if this change in length is not possible (this movement is prevented by fixed supports), large forces and stresses are created in the structure.

To apply the initial, boundary, and loading conditions with the aim of thermal analysis, the temperature history was considered as two modes, namely, large variations (mean values) and small variations (maximum and minimum values) on the concrete deck. Figure 7 displays the comparison of temperature loading conditions in the two different modes. In this diagram, the daily temperature changes are ignored, and the focus is on the monthly temperature changes. So, we lose a significant amount of data in this way. Importantly, as a representative of the daily temperature, it is enough to know the average temperature of that day or it is necessary to consider the maximum and minimum temperature of each day. Furthermore, it was assumed that the temperature of steel piles is equal to the ambient temperature (22 °C). In addition, a temperature of 7 degrees Celsius was considered for the part of the steel piles that was located in the water (height of 75 cm from the floor). In the present research, it is assumed that the water temperature is constant throughout the year. Moreover, in the simulation, the time interval of temperature variations was considered to be constant. Transient thermal analysis was performed to extract the temperature distribution in various parts of the bridge in terms of months.

3.2. Thermal–Structural Coupling Analysis

The temperature distribution obtained from the previous step was entered into the structural analysis. In other words, in the new structural analysis, the temperature values are considered as the initial boundaries of nodes. As the boundary conditions, the end of steel piles was fixed in different directions (all DOF). In addition, water pressure flow ($P=51.2\times K\times V^2$), was uniformly distributed on the middle steel piles [37]. In this equation, K coefficient was assumed to be 0.78 based on the geometric shape of the pile cross-section. In addition, the speed of water flow (V) is 0.3 m/s. This pressure distribution is in the form of a triangle with a height of 75 cm and its base, which is the highest pressure, corresponds to the end of the steel pile. Furthermore, transient dynamic analysis was performed to obtain the deck deformation in the length direction and consequently, the skew of the upper part of steel piles was compared to its initial state.

3.3. Fatigue Life Assessment by Utilizing Probabilistic Approach

The results of thermal and structural analyses in different working conditions showed that load histories applied to the structure had time-dependent values. In other words, the type of load is variable amplitude loading. In the previous section, the most critical region of integrated bridge (i.e., H cross-section steel piles) was determined, which is apt to fail under cyclic loading based on the results of the thermal–structural coupling analysis [52]. Afterward, time histories of stress tensor components were extracted in the critical element for large and small variations [42]. Next, the time history of von Mises equivalent stress was calculated on the critical element for both loading modes. This well-known equivalent criterion is appropriate, since the time histories of the stress tensor components showed that proportional loading prevails in the critical element [42,52]. On the other hand, the maximum and minimum values of all the normal and shear stresses occurred at an equal time [54,55,56]. Eventually, service lifetime of IPE-220 steel pile was assessed using an algorithm based on the probabilistic approach presented by Reza Kashyzadeh [40]. Additionally, two variables influencing the accuracy of the algorithm, including the curve fitting order and stress counting ranges, were optimized so that the statistical responses could be examined more accurately. Moreover, the results obtained from the statistical algorithm were compared with the fatigue analysis results in the finite element software. Because both of them used the same criterion (i.e., von Mises) for equivalent stress, only the evaluation method is different.

4. Results and Discussion

4.1. Thermal Analysis

The temperature distribution contour in the entire structure of the integrated concrete bridge with steel piles was extracted separately in different months; for example, the temperature contour for the first month of the year is shown in Figure 8. The details of the thermal analysis results for all months are also given in

Table 4. Thermal analysis results of the integrated concrete bridge subjected to temperature variations.

Month No.	Temperature of Bridge Members (℃)
	Maximum Value	Minimum Value
1	31.3467	12.3399
2	33.8716	9.7182
3	41.2061	1.9898
4	47.2158	−5.3795
5	40.7483	0.4667
6	28.6778	12.6523
7	30.9703	13.4125
8	39.8413	3.9106
9	40.3069	3.6578
10	31.0620	13.1651
11	27.9941	13.4379
12	40.9562	1.1799

.

The obtained results indicated that in the warm seasons of the year, the temperature of the concrete deck is significantly higher than other seasons and in other parts. Moreover, in these seasons, the difference between the maximum and minimum temperatures is more than other seasons. Accordingly, temperature changes and temperature gradients of the structure were extracted.

4.2. Thermal–Structural Coupling Analysis

The longitudinal deformation contour of the deck in the integrated concrete bridge with steel piles was extracted due to the mean values of monthly temperature changes as the results of the thermal–structural coupling analysis (Figure 9).

As Figure 9 shows, the system deformation is completely symmetric. This is because both the initial model of the symmetrical bridge and the temperature distribution in different parts, water conditions, as well as the number of steel piles and their location were symmetric, which indicates the accuracy of the response obtained from the FE analysis [37]. In addition, the maximum deviation of the upper part of steel piles with IPE-220 characteristic is equal to 2.45 mm (the results extracted at the end of the 12th month of the year). This deformation was also observed in the area of 1.3 m above the steel piles. Moreover, the area of 1.25 m from the bottom of the steel pile was without deformation (in real, the deformation was so small that it can be ignored). Afterward, the deformation history via months for the critical area, including both modes of large and small variations, was extracted and compared to each other in Figure 10.

From this Figure, deformation changed suddenly with the month, and it is not continuous, because, as stated in Section 3.1.1, temperature changes were considered discontinuously and suddenly without the passage time from one event to other. Rather, only the duration of the novel model staying at any temperature is equal to half a day, i.e., 12 h. This is the reason why there are several jumps in the stress or deformation, and these jumps indicate the change in the day, month, or season. The system response related to the loading conditions of large variations indicated that a deformation of 3.3157 mm also occurred in the upper part of the bridge, which is associated with the passing of the assumed ambient temperature equal to 22 degrees compared to the temperature gradient changes in the first month. On the other hand, the results of FE simulation showed that the effect of the minimum and maximum temperatures applied to the structure in comparison with the effect of mean temperature applied to the structure was different in the deformation created in the upper part of the steel pile. In other words, the relationship between temperature changes and deformation variations was completely linear, which indicates the correctness of the FE response considering thermal length change in metals (it has been proven in physics that the change in metal’s length at high temperatures directly relates to the temperature, initial length, and the longitudinal expansion coefficient of the metal). It is clear that each of these deformation histories and their repetition would result in their unique damage that is effective in calculating and predicting the fatigue life of a structure.

4.3. Fatigue Analysis

Figure 11 presents the time histories of stress tensor components, including normal and shear stresses in terms of various months, in the critical element for small variations mode. In addition, the results showed that the trend of stress changes is the same in both modes of large and small variations and only the stress values shifted. Therefore, presenting the stress components for the large variations mode was omitted in this manuscript. However, the stress levels in the large temperature variations mode are much smaller than the stress levels in the small temperature variations mode, and this behavior was observed in all normal stress components, but not in the case of shear stress components. Although, the difference in the values of stress components leads to the creation of different von Mises equivalent stress histories. Therefore, the von Mises stress history for both studied modes is demonstrated in Figure 12.

Level counting was carried out by considering various stress ranges including 5, 10, and 20 MPa. Additionally, normalization of the values was carried out to plot Probability Distribution of Stress (PDS). Figure 13 illustrates the PDS of von Mises for small temperature variations.

It is clear that as the stress count range decreases, the accuracy of statistical operations increases, but this is not always true, and as the data shrinks and their number increases, calculation errors will occur, especially in statistical parameters. Moreover, the results of previous studies have shown that the effects of natural resources mostly have Gaussian probability distribution (normal). Therefore, in this issue, probably the 20 MPa stress range is more suitable compared to the others. However, different orders of Fourier series were used to fit the PDS data in all three states of the stress range. The general equation is written as follows:

$$y=a_0+{{\displaystyle \sum }}_{n=1}^{n=\infty }{a}_n\times \cos \left(nxw\right)+{{\displaystyle \sum }}_{n=1}^{n=\infty }{b}_n\times \sin \left(nxw\right)$$

(1)

in which $a_0,a_n, \mathrm{and} b_n$ are the Fourier coefficients, and w represents the non-linear system frequency. Additionally, index n indicates the order of the Fourier series. The initial data related to the stress counting curves (Figure 13) were entered as inputs in MATLAB software. Then the curve fitting module was selected and the Fourier method was set. The various coefficients obtained for this approximation as well as the accuracy measurement criteria for different ranges of stress levels are reported in

Table 5. Fourier coefficients and different types of fitting errors considering various orders of Fourier series for stress leveling of 5 MPa.

	Symbol	First-Order	Second-Order	Third-Order
Fourier coefficients	$a_0$	0.0778	0.08021	0.05739
	$a_1$	−0.08372	−0.06899	0.002429
	$b_1$	−0.06318	0.04705	−0.03576
	$a_2$	------	−0.01307	0.06272
	$b_2$	------	−0.1736	−0.04106
	$a_3$	------	------	−0.03809
	$b_3$	------	------	−0.05169
System frequency	w	0.3781	0.4246	0.4442
Assessment criteria	SSE	0.1188	0.1226	0.1206
	R-square	0.2973	0.4197	0.429
	RMSE	0.09948	0.1107	0.1228

,

Table 6. Fourier coefficients and different types of fitting errors considering various orders of Fourier series for stress leveling of 10 MPa.

	Symbol	First-Order	Second-Order	Third-Order
Fourier coefficients	$a_0$	0.1075	0.1229	0.1226
	$a_1$	−0.04235	−0.05927	−0.06913
	$b_1$	0.1041	0.08669	0.08115
	$a_2$	------	−0.03531	−0.04746
	$b_2$	------	0.02659	0.02173
	$a_3$	------	------	0.03297
	$b_3$	------	------	0.01008
System frequency	w	0.06986	0.07981	0.08205
Assessment criteria	SSE	0.02351	0.01667	0.01121
	R-square	0.6922	0.7818	0.8533
	RMSE	0.06857	0.07454	0.1059

and

Table 7. Fourier coefficients and different types of fitting errors considering various orders of Fourier series for stress leveling of 20 MPa.

	Symbol	First-Order	Second-Order	Third-Order
Fourier coefficients	$a_0$	0.2418	0.1982	0.1372
	$a_1$	−0.2103	0.1792	0.01288
	$b_1$	0.1693	−0.2073	−0.2311
	$a_2$	------	−0.06963	−0.1287
	$b_2$	------	−0.02105	0.007925
	$a_3$	------	------	0.05035
	$b_3$	------	------	0.01083
System frequency	w	0.085508	2.435	2.273
Assessment criteria	SSE	0.002005	2.778 × 10−32	2.136 × 10−32
	R-square	0.9883	1	1
	RMSE	0.04478	NaN	Nan

. In this regard, Equations (2) and (3) were used to calculate the error criteria [1,57].

$$R^2=\frac{{{\displaystyle \sum }}_{i=1}^n\left(f_{EXP,i}-{\overline{F}}_{EXP}\right)\left(f_{fitting,i}-{\overline{F}}_{fitting}\right)}{\sqrt{{{\displaystyle \sum }}_{i=1}^n\left({\left(f_{EXP,i}-{\overline{F}}_{EXP}\right)}^2{\left(f_{fitting,i}-{\overline{F}}_{fitting}\right)}^2\right)}}$$

(2)

$$\mathrm{RMSE}=\sqrt{\frac{1}{n}{\displaystyle \sum }_{i=1}^n{\left(f_{fitting,i}-f_{EXP,i}\right)}^2}$$

(3)

in which, n is the number of fed samples, $f_{EXP}$ and $f_{fitting}$ represent the experimental values and predicted values by fitting method, respectively. The values of ${\overline{F}}_{EXP}$ and ${\overline{F}}_{fitting}$ were determined as follows [58]:

$${\overline{F}}_{EXP}=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{f}_{EXP,i}$$

(4)

$${\overline{F}}_{fitting}=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{f}_{fitting,i}$$

(5)

Considering different error criteria reported in the above tables and the shape of the PDS function, the stress leveling range of 20 MPa and the first-order Fourier series were selected for future research. Next, fatigue life of the system on the critical region was calculated using statistical parameters, which included the number of peak and Fourier curve fitting data [40,42].

$$\mathrm{E}\left[\mathrm{D}\right]=\mathrm{E}\left[\mathrm{P}\right]\times \frac{\mathrm{N}}{\mathrm{K}}{{\displaystyle \int }}_0^{{\mathsf{\sigma }}_{\mathrm{eq},\max }}{\mathrm{S}}^m.\mathrm{F}\left(\mathrm{S}\right)\mathrm{dS}$$

(6)

where E[D] and E[P] are the mathematical expectation of fatigue damage and the number of upload history peaks, respectively [40,42]. In addition, F(S) is the function of PDS (F(S) = 0.2418–0.2103 × Cos (0.085508 × S) + 0.1693 × Sin (0.085508 × S)). The m and k are material constant, which are obtained experimentally and based on the Basquin equation for HCF regime as $\mathrm{N}\times {\mathrm{S}}^m=\mathrm{k}$. In summary, the prediction algorithm of fatigue life due to variable amplitude multi-axial loading based on the statistical parameters used in this research is given in the Appendix A.

As expected, the fatigue life of steel piles due to small temperature variations was much shorter than that of steel piles due to the large temperature variations. In this regard, loading conditions of small temperature variations contain the maximum and minimum temperature values per month, which led to higher temperature gradients applied to the system. The results of fatigue analysis indicated that by considering the mean temperature of each month, the middle steel piles of the bridge will last over 610 years. In this case study, if the maximum and minimum temperatures are considered for every month, its lifetime would be approximately 108 years. This difference is almost more than 550%. To compare the results obtained from this methodology with the routine one for fatigue life prediction (i.e., cycle counting by rain-flow technique and employing the linear damage accumulation rule), the results of variable domain fatigue analysis in Ncode-Design-Life module of ANSYS WORKBENCH software are given below. From Figure 14, the routine fatigue prediction method has reported the fatigue lifetime of the structure equal to 98.427 and 705.12 years, respectively, to consider small and large temperature variations. The difference between both methodologies for predicting the fatigue life of the structure is less than 16% and 10%, respectively, according to the type of loading including large and small temperature variations. However, in calculating the fatigue life of the structure using the present algorithm and also using Miner’s linear damage rule, it does not consider the precedence and latency of the amplitudes, and this issue can have a significant impact on the fatigue life of the structure [27,59]. In summary, to perform fatigue life calculations, it is necessary to consider the small daily changes to obtain a more accurate and realistic response. This issue is in the future plans of the authors.

5. Conclusions

In the current study, the fatigue life of H cross-section steel piles of the integrated concrete bridge installed on water subjected to the temperature variations was predicted using a probabilistic approach. In this regard, the effect of details in the temperature history was considered as an influential variable in the calculation of fatigue life. Temperature data were primarily collected over a one year period, and two types of loading, called large variations (i.e., monthly mean temperature variations) and small variations (i.e., maximum and minimum monthly temperature values), were considered. Thermal analysis of the real-scale bridge model was conducted to extract the temperature distribution and temperature gradient. Furthermore, the previous results were coupled to the structural analysis, water pressure was applied as a triangular loading, and transient dynamic analysis was performed to extract the longitudinal deformation of the bridge and its members, particularly, in the concrete deck and middle steel piles. Next, the deformation history in the critical region was extracted considering both types of loading (large and small variations). Eventually, the fatigue life of the H cross-section steel pile was assessed. To increase the prediction accuracy in the used algorithm, two parameters of the stress counting range and the order of the Fourier series were investigated and the best one was selected. Finally, the findings revealed that the lifetime of steel piles subjected to the monthly temperature variations was about 610 and 108 years for the loading conditions of large and small changes, respectively. These values were obtained using fatigue analysis in the software, respectively equal to 705.12 and 98.427. One of the most important achievements of this research is that it could recommend computational engineers to consider small temperature changes as much as possible to obtain the fatigue life of the bridge due to the air temperature variations. It is also better to enter temperature changes in the calculations daily to achieve an accurate and close to reality response.