# Fatigue Damage Evaluation of Pile-Supported Bridges under Stochastic Ice Loads

Authors: Tianyu Wu; Wenliang Qiu; Guangrun Wu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

The Bohai Sea is the sea area with the worst ice condition in China, and the ice loads significantly threaten the safety of structures in the sea. The intense vibrations of the pile-supported bridge under stochastic ice loads will increase the fatigue damage of a bridge structure and reduce the fatigue life of a bridge structure. In the present study, a comprehensive analysis model is presented to study fatigue damage for pile-supported bridges under ice loads in Bohai Sea. On the basis of measured statistical data of ice parameters and stochastic ice loads spectrum of Bohai Sea, the time histories of the stochastic ice loads of Bohai Sea are simulated. Fatigue damage analysis is carried out in time domain utilizing the finite element method considering soil and bridge structure interaction. The effect of soil conditions and water depth on the cumulative fatigue damage of the pile-supported bridges is studied. Numerical results indicate that in comparison with stiff soil conditions, pile-supported bridges in soft oil conditions can increase the cumulative fatigue damage substantially; pile-supported bridges in deep water also can increase the cumulative fatigue damage obviously. The study presented the first danger position of cumulative damage of the pile cross section under stochastic ice loads. The findings of this study can be used to fatigue damage evaluation and bridge construction in the ice-covered sea area.


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## Body

## 1. Introduction

As a natural hazard, sea ice has seriously threatened the safety of offshore structures [1–3]. The Bohai Sea is the sea area with the worst ice condition in China. There is a lot of drifting sea ice on the surface of the Bohai Sea in every winter. Ice loads occur continuously during the whole lifetime of a bridge, which will lead to excessive vibration of structures. A pile-supported bridge is the most widely used type of bridge structure in sea-crossing engineering. The stress cycles caused by ice-induced vibration can directly lead to the fatigue damage of a bridge structure. The first event of ice-induced vibration happened on the oil platform of Cook Gulf [4, 5]. After that, many vibration events were found in different frozen seas of the world [6–8]. It is notable that some offshore structures such as oil drilling platform have experienced serious ice-induced vibration events and even collapsed in the past few decades [9]. Duan et al. [10] investigated the major accident that Bohai No. 2 platform was pushed down by the sea ice in 1968 and described the whole process of platform destruction. It is pointed out that the collapse of the platform is caused by the comprehensive factors of the three aspects. The essential reason is that the fatigue damage caused by the ice-induced vibration.The crushing failure of sea ice is the most common phenomenon in the ice failure modes when sea ice impacts on a vertical offshore structure. Yue et al. [11] carried out the field measurement on a compliant monopod platform under ice loads in Bohai Sea. The three modes of ice failure were found, and the corresponding structure vibration modes were obtained, as shown in Figure 1. The fast drifting ice is the main challenge for the offshore structures in Bohai Sea. Many researchers and engineers have made great efforts to investigate the ice loads when ice impacts on the offshore structures. Kärnä et al. [12] proposed a spectrum model of ice forces based on the field-measured data of Bohai Sea. This ice force spectrum is applied to the generation of stochastic ice loads at fast ice speed. Wu and Qiu [13] simulated the stochastic ice loads process based on ice force spectrum of Bohai Sea and verified the simulation performance. It was found that the ice load spectrum model can accurately simulate the stochastic ice force of vertical structures. In the present study, the stochastic ice loads spectrum of Bohai Sea will be used to generate the stochastic ice loads process of the pile-supported bridge.Figure 1
Ice loads and structure responses [11]. (a) Quasistatic. (b) Steady-state. (c) Stochastic.
(a)
(b)
(c)In order to maintain the safety of offshore structures, the dynamic structural characteristics should be precisely calculated under external loads. A comprehensive analysis model of a bridge structure is essential for understanding the dynamic characteristics of a structure. The pile-supported bridges are primarily constructed as a foundation type of cross-sea bridges in the deep-water region [14]. However, the soil-pile interaction can reduce the vibration frequency of the structure and even change the vibration mode of the structure, which will further affect the dynamic characteristics of the offshore structures [15]. Currently, American Petroleum Institute (API) [16] based p-y curve under cyclic load is usable to the small diameter piles, and it is the most widely used design guidelines of the offshore structures [15, 17–19]. In the present study, the p-y curves of API will be applied in the nonlinear spring of soil-pile interaction.The fatigue damage of structural components under environmental loading is a cumulative process. Fatigue life prediction and reliability evaluation is still a challenging problem despite significant progress made in the past decades [20–25]. Sun and Jahangiri [26] studied the fatigue damage mitigation of offshore wind turbines under real wind and wave loads. The authors utilized the rain-flow cycle counting method and Miner’s rule in calculating the fatigue damage. It was found that the damper of 3d-PTMD can improve the fatigue life of the offshore wind turbine effectively. Rezaei et al. [27] investigated the effect of damping on the sensitivity of the fatigue life of monopile-supported offshore wind turbines and utilized the S-N curves of DNV standard to access the fatigue damage life. This study shows that the fatigue life of the offshore structure is markedly improved due to the reduction in bending stress caused by the increased damping. Bisoi and Haldar [28] studied the effect of soil conditions on fatigue damage evaluation of offshore wind turbines. The authors considered soil-structure interaction and utilized S-N curves of DNV to access the fatigue life of offshore wind turbines. Research results show that the fatigue life is observed to be governing design criteria for offshore wind turbines at stiff clay.In this paper, the commercial finite element software ABAQUS is used to establish the finite element model of the offshore bridge in Bohai Sea. On the basis of measured statistical data of ice parameters and stochastic ice loads spectrum of Bohai Sea, the time histories of the stochastic ice loads of Bohai Sea are simulated. The finite element model to investigate the effects of soil conditions and water depths on the fatigue damage under stochastic ice loads is utilized. The findings of this study can be used to fatigue damage evaluation and bridge construction in the ice-covered sea area.


## 2. Numerical Model

### 2.1. Bridge Model

This paper considers a primary bridge design of the Qinhuangdao-Dalian sea-crossing bridges in Bohai Sea. The Bohai Sea is the sea area with the worst ice condition in China. There is a lot of drifting sea ice on the surface of the Bohai Sea in every winter. Figures2 and 3 show the layout of the bridge and the main dimensions of the pier and pile sections. The sea-crossing bridges are composed of (5 × 120) m continuous beam with box cross section, as shown in Figure 2. The widths of the top and bottom plate are 33.1 m and 15.0 m, and the depth of the girder is 4.5 m. The shape of the pier is round-ended cross sections, and the type of the foundations is steel pipe driven pile. The pier is designed with a round-ended hollow cross section, which has a plane size of 3.5 × 10.0 m with a round wall thickness of 0.8 m. The size of the pile cap is 13 × 21 × 6 m. 13 steel pipe piles support the pile cap with a diameter of 2.0 m, as shown in Figure 3.Figure 2
Configuration of the (5 × 120) m continuous bridge (unit: m).Figure 3
Configuration of bridge pier and foundation (unit: cm).The finite element model of the sea-crossing bridges is developed by utilizing the ABAQUS codes. All members of the bridge, including the main girders, piers, cap, and piles are modelled by the three-dimensional spatial beam element. The connections between the cap and piles are treated as a master-and-slave system. To consider the coupling effect between the bridge spans, three continuous girder bridges are analyzed. The total length of the simulated bridge is 1800 m which is composed of three continuous girder bridges. Each bridge has the same span arrangement of 5 × 120 m. In the current study, the second bridge is taken as the representative bridge, as shown in Figure2. Structural damping was modelled by Rayleigh damping in such way that Rayleigh coefficients were determined to correspond to a 1% damping ratio at the lowest natural frequency.


### 2.2. Added Water Mass

When the vibration of steel pipe piles in seawater occurs, the acceleration of surrounding seawater will be affected [29]. In this study, the pile-water interaction was simulated by the additional mass method, in which the effective mass me of the pile can be shown as(1)me=mp+ma,where mp denotes the mass of pile and is the additional mass. The extra mass ma can be expressed as(2)ma=CaApρw,where Ap is the pile cross-sectional area, ρw = 1030 kg/m3 denotes the density of seawater, and Ca denotes the additional mass coefficient.The various water depths are investigated to evaluate the effect of water depths on the fatigue damage of bridges subjected to the ice loads. The water depths ofH = 10, 20, and 30 m represent the water depths conditions of shallow, moderate, and deep water, as shown in Figure 4.Figure 4
Exhibition of bridge configurations with 10 m, 20 m, and 30 m water depth under ice loads.


### 2.3. Pile-Soil Interaction

The interaction of the pile with the surrounding soil can have a significant impact on the dynamic characteristics of pile-supported bridges. To investigate fatigue damage precisely, soil-pile interaction is considered in the present study. In this study, the calculation methods of nonlinear soil spring given in API [16] and DNV [18] codes are adopted.Thep-y springs denote the horizontal soil resistances and the t-z and Q-z spring was set in the vertical direction to simulate the soil-pile interaction [30]. Figure 5 shows a pile foundation model with a soil spring. The API [16] and DNV [18] recommended the parameters of p-y, t-z, and Q-z soil spring. Nonlinear cyclic p-y curves for undrained clay (su < 100 kPa) for Z > ZR is described as(3)p=0.5puyyc1/3,fory≤3yc,0.72pu,fory≤3yc.Figure 5
Schematic diagram of a soil-pile interaction model.WhenZ ≤ ZR, the p-y curve is described as(4)p=0.5puyyc1/3,fory≤3yc,0.72pu1−1−ZZRy−3yc12yc,for3yc<y≤15yc,0.72puZZR,fory>15yc,where pu is the soil strength per unit pile length, defined as(5)pu=3su+γ′Z+JsuZD,forZ<ZR,9su,forZ≥ZR,where γ′ stand for unit weight of soil γ′=8kN/m3, su denotes the undrained shear strength of soil (unit: kPa), Z stands for the the below the seabed (unit: m), D denotes the pile diameter (unit: m), J represents an empirical constant (J = 0.5) [9], and ZR stand for the depth below the mudline to the bottom.(6)ZR=6suDγ′D+Jsu.The deflection at one-half the ultimate soil resistance is estimated as(7)yc=2.5εcD,where εc is the strain corresponding to one-half of the maximum stress in laboratory undrained compression tests of undisturbed soil.The various soil conditions are investigated to evaluate the effect of soil strength on the fatigue damage of bridges subjected to the ice loads. The soil strengths ofsu = 25 kPa, su = 50 kPa, and su = 100 kPa represent the typical soft, moderate, and stiff soils. The corresponding values εc are 0.02, 0.008, and 0.006 [16]. Figure 5 shows the schematic diagram of the soil-pile interaction model and draws the lateral soil resistances (p-y springs), friction resistances in the vertical direction (t-z springs), and end bearing resistances in the vertical direction (Q-z springs). As an example, when soil undrained shear strengths su = 25 kPa (soft soil condition), the typical plots of nonlinear springs are drawn in Figure 6.Figure 6
Typical plots of nonlinear springs: (a)p-y curve; (b) t-z curve; (c) Q-z curve.
(a)
(b)
(c)


## 2.1. Bridge Model

This paper considers a primary bridge design of the Qinhuangdao-Dalian sea-crossing bridges in Bohai Sea. The Bohai Sea is the sea area with the worst ice condition in China. There is a lot of drifting sea ice on the surface of the Bohai Sea in every winter. Figures2 and 3 show the layout of the bridge and the main dimensions of the pier and pile sections. The sea-crossing bridges are composed of (5 × 120) m continuous beam with box cross section, as shown in Figure 2. The widths of the top and bottom plate are 33.1 m and 15.0 m, and the depth of the girder is 4.5 m. The shape of the pier is round-ended cross sections, and the type of the foundations is steel pipe driven pile. The pier is designed with a round-ended hollow cross section, which has a plane size of 3.5 × 10.0 m with a round wall thickness of 0.8 m. The size of the pile cap is 13 × 21 × 6 m. 13 steel pipe piles support the pile cap with a diameter of 2.0 m, as shown in Figure 3.Figure 2
Configuration of the (5 × 120) m continuous bridge (unit: m).Figure 3
Configuration of bridge pier and foundation (unit: cm).The finite element model of the sea-crossing bridges is developed by utilizing the ABAQUS codes. All members of the bridge, including the main girders, piers, cap, and piles are modelled by the three-dimensional spatial beam element. The connections between the cap and piles are treated as a master-and-slave system. To consider the coupling effect between the bridge spans, three continuous girder bridges are analyzed. The total length of the simulated bridge is 1800 m which is composed of three continuous girder bridges. Each bridge has the same span arrangement of 5 × 120 m. In the current study, the second bridge is taken as the representative bridge, as shown in Figure2. Structural damping was modelled by Rayleigh damping in such way that Rayleigh coefficients were determined to correspond to a 1% damping ratio at the lowest natural frequency.


## 2.2. Added Water Mass

When the vibration of steel pipe piles in seawater occurs, the acceleration of surrounding seawater will be affected [29]. In this study, the pile-water interaction was simulated by the additional mass method, in which the effective mass me of the pile can be shown as(1)me=mp+ma,where mp denotes the mass of pile and is the additional mass. The extra mass ma can be expressed as(2)ma=CaApρw,where Ap is the pile cross-sectional area, ρw = 1030 kg/m3 denotes the density of seawater, and Ca denotes the additional mass coefficient.The various water depths are investigated to evaluate the effect of water depths on the fatigue damage of bridges subjected to the ice loads. The water depths ofH = 10, 20, and 30 m represent the water depths conditions of shallow, moderate, and deep water, as shown in Figure 4.Figure 4
Exhibition of bridge configurations with 10 m, 20 m, and 30 m water depth under ice loads.


## 2.3. Pile-Soil Interaction

The interaction of the pile with the surrounding soil can have a significant impact on the dynamic characteristics of pile-supported bridges. To investigate fatigue damage precisely, soil-pile interaction is considered in the present study. In this study, the calculation methods of nonlinear soil spring given in API [16] and DNV [18] codes are adopted.Thep-y springs denote the horizontal soil resistances and the t-z and Q-z spring was set in the vertical direction to simulate the soil-pile interaction [30]. Figure 5 shows a pile foundation model with a soil spring. The API [16] and DNV [18] recommended the parameters of p-y, t-z, and Q-z soil spring. Nonlinear cyclic p-y curves for undrained clay (su < 100 kPa) for Z > ZR is described as(3)p=0.5puyyc1/3,fory≤3yc,0.72pu,fory≤3yc.Figure 5
Schematic diagram of a soil-pile interaction model.WhenZ ≤ ZR, the p-y curve is described as(4)p=0.5puyyc1/3,fory≤3yc,0.72pu1−1−ZZRy−3yc12yc,for3yc<y≤15yc,0.72puZZR,fory>15yc,where pu is the soil strength per unit pile length, defined as(5)pu=3su+γ′Z+JsuZD,forZ<ZR,9su,forZ≥ZR,where γ′ stand for unit weight of soil γ′=8kN/m3, su denotes the undrained shear strength of soil (unit: kPa), Z stands for the the below the seabed (unit: m), D denotes the pile diameter (unit: m), J represents an empirical constant (J = 0.5) [9], and ZR stand for the depth below the mudline to the bottom.(6)ZR=6suDγ′D+Jsu.The deflection at one-half the ultimate soil resistance is estimated as(7)yc=2.5εcD,where εc is the strain corresponding to one-half of the maximum stress in laboratory undrained compression tests of undisturbed soil.The various soil conditions are investigated to evaluate the effect of soil strength on the fatigue damage of bridges subjected to the ice loads. The soil strengths ofsu = 25 kPa, su = 50 kPa, and su = 100 kPa represent the typical soft, moderate, and stiff soils. The corresponding values εc are 0.02, 0.008, and 0.006 [16]. Figure 5 shows the schematic diagram of the soil-pile interaction model and draws the lateral soil resistances (p-y springs), friction resistances in the vertical direction (t-z springs), and end bearing resistances in the vertical direction (Q-z springs). As an example, when soil undrained shear strengths su = 25 kPa (soft soil condition), the typical plots of nonlinear springs are drawn in Figure 6.Figure 6
Typical plots of nonlinear springs: (a)p-y curve; (b) t-z curve; (c) Q-z curve.
(a)
(b)
(c)


## 3. Stochastic Ice Loads Process

### 3.1. Probability Density of Ice Parameters in Bohai Sea

Based on the available sea ice observed data of JZ20-2 in Bohai Sea from 1996 to 2000 four winter seasons, the ice parameters of the ice thickness, ice velocity, and ice strength in the Bohai Sea were analyzed. The effective ice period in four winter seasons from 1996 to 2000 in the Bohai Sea was 44, 39, 19, and 47 days [31]. Utilizing the data samples of ice loads, the sea ice parameters were analysed statistically.The statistical results of ice thicknessh show that it obeys Log-normal distribution well. The K-S test with significant level α=0.05 is also passed. The distribution parameters μ=1.9057, σ=0.5732 are obtained, and the probability density of ice thickness h can be written as [31](8)Fh=10.5732h2πexp−12lnh−1.90470.57322.The statistical results of ice velocityv show that it obeys Rayleigh distribution well. The K-S test with significant level α=0.05 is also passed, and the probability density of ice velocity v can be written as [31](9)Fv=v826.5512exp−v21653.1024.The statistical results of ice strengthσc show that it obeys Gaussian distribution well. The K-S test with significant level α=0.05 is also passed. The distribution mean value μ=1.0348, the mean square error σ=0.2641 is obtained, and the probability density of ice strength σc can be written as [31](10)Fσc=10.26412πexp−σc−1.034820.1395.As described above, the ice thickness, ice velocity and ice strength can be expressed by Log-normal, Rayleigh, and Gaussian distributions. And Figure7 shows the probability density of ice parameters (ice thickness, ice velocity, and ice strength). Due to the ice thickness, ice velocity and ice strength are relatively independent [31], the joint probability can be calculated from each two independent random variables. Figures 8(a)–8(c) show the three-dimensional data of the joint probability of ice thickness and ice velocity, ice thickness and ice strength, ice velocity and ice strength, respectively.Figure 7
Probability density of ice parameters of JZ20-2 sea area in Bohai Sea: (a) probability density of ice thickness; (b) probability density of ice velocity; (c) probability density of ice strength.
(a)
(b)
(c)Figure 8
The joint probability density for (a) ice thickness-ice velocity; (b) ice thickness-ice strength; (c) ice velocity-ice strength.
(a)
(b)
(c)


### 3.2. Spectral Model of Stochastic Ice Loads

The spectral model of ice loads was proposed by Kärnä et al. [12] based on field measurement of JZ9-3 MDP2 in Bohai Sea. The spectral model of stochastic ice loads includes auto-spectral functions and cross-spectral functions. The following is a detailed simulation process.The auto-spectral density function of stochastic ice loads can be expressed as(11)σi2=∫0∞Giifdf.The nondimensional spectrum of stochastic ice loads can be shown as(12)G˜iif=fGiifσi2.The nondimensional auto-spectral function was proposed by Kärnä et al. [12]:(13a)G˜iif=af1+ksa1.5f2,(13b)a=b×v−0.6,in which v stands for ice velocity and ks and b are fitting parameters. The data analysis of stochastic ice loads in JZ9-3 MDP2 indicates that the mean value of parameter ks = 3.24 and b = 1.34 [32].The auto-spectral density function of local ice loads is expressed as(14)Giif=a1+ksa1.5f2σi2.The cross-spectral density function denotes the interaction of adjacent load pointsi and j. The coherence function γij is used to calculate the relations between two local points of rij distance. The coherence function γij is expressed by auto-spectral and cross-spectral density functions as(15)γij2f=Gijf2GiifGijf.The coherence function was proposed by Kärnä et al. [32]:(16)γij=11+ρ+αξijρ+e−βξijf,ξij=rijh,in which ξij denotes the nondimensional distance of two locations rij divided by ice thickness h. The other test parameters ρ, α, and β can be selected as ρ=0.1, α=0.2, and β=3 [32].As is shown in Figure9, the stochastic ice loads consist of the incidence angle and shear force on the structure. The local stochastic ice loads can be expressed as(17)Fi′t=cosθ+μsinθFit,where i = 1, 2, …, n, denotes the number of parts, μ stands for the friction coefficient, and Fi (t) stands for the ice loads on the contact surface.Figure 9
Ice forces distribution on an offshore structure.The auto-spectral density functions of the frequency domain can be described as(18)Gii′f=Giif×cos2θ+μ2sin2θ.The spectral matrices are summed to obtain the spectrum of stochastic ice loads.(19)GFf=C+μSΤGfffC+μS,where C and S stand for vectors for cosθ and sinθ at each local load point. Gff (f) consist of auto-spectral and cross-spectral functions.The power spectrum of the frequency domain can be converted to the time domain.(20)Ft=∑2×GFf×Δfcos2πft+φf,where 2×GFf×Δf is the ice loads amplitude.The standard deviationσF and mean value FM of the time histories of ice loads can be expressed as(21)σF=∫0∞GFfdf1/2,FM=σFI.The total ice forceFG (t) against a vertical structure can be expressed as(22)FGt=FM+Ft.Therefore, the estimation value of maximum ice loads can be expressed as(23)FGmax=FM+kσF,where σF denotes the standard deviation of stochastic ice loads and k is a selected exceedance probability. According to the existing references [12, 32], the assumption that the value of crushing intensity I = 0.4 and k = 4.0 is reasonable.


### 3.3. The Process of the Stochastic Ice Loads

The sample function of the stochastic ice loads according to the above Section3.2 was simulated by utilizing the classical fast Fourier transforms. The upper cutoff frequency fn was taken as 50 Hz and the total period T0 was taken as 500 s. The ice loads time histories based on the spectral model were generated at an ice velocity of 36 cm/s, ice thickness of 8 cm, and ice strength of 1.0 MPa. For brevity, one of the ice loads time histories was displayed in this paper, as shown in Figure 10.Figure 10
The time history of the stochastic ice force for the present bridge (T = 500 s).


## 3.1. Probability Density of Ice Parameters in Bohai Sea

Based on the available sea ice observed data of JZ20-2 in Bohai Sea from 1996 to 2000 four winter seasons, the ice parameters of the ice thickness, ice velocity, and ice strength in the Bohai Sea were analyzed. The effective ice period in four winter seasons from 1996 to 2000 in the Bohai Sea was 44, 39, 19, and 47 days [31]. Utilizing the data samples of ice loads, the sea ice parameters were analysed statistically.The statistical results of ice thicknessh show that it obeys Log-normal distribution well. The K-S test with significant level α=0.05 is also passed. The distribution parameters μ=1.9057, σ=0.5732 are obtained, and the probability density of ice thickness h can be written as [31](8)Fh=10.5732h2πexp−12lnh−1.90470.57322.The statistical results of ice velocityv show that it obeys Rayleigh distribution well. The K-S test with significant level α=0.05 is also passed, and the probability density of ice velocity v can be written as [31](9)Fv=v826.5512exp−v21653.1024.The statistical results of ice strengthσc show that it obeys Gaussian distribution well. The K-S test with significant level α=0.05 is also passed. The distribution mean value μ=1.0348, the mean square error σ=0.2641 is obtained, and the probability density of ice strength σc can be written as [31](10)Fσc=10.26412πexp−σc−1.034820.1395.As described above, the ice thickness, ice velocity and ice strength can be expressed by Log-normal, Rayleigh, and Gaussian distributions. And Figure7 shows the probability density of ice parameters (ice thickness, ice velocity, and ice strength). Due to the ice thickness, ice velocity and ice strength are relatively independent [31], the joint probability can be calculated from each two independent random variables. Figures 8(a)–8(c) show the three-dimensional data of the joint probability of ice thickness and ice velocity, ice thickness and ice strength, ice velocity and ice strength, respectively.Figure 7
Probability density of ice parameters of JZ20-2 sea area in Bohai Sea: (a) probability density of ice thickness; (b) probability density of ice velocity; (c) probability density of ice strength.
(a)
(b)
(c)Figure 8
The joint probability density for (a) ice thickness-ice velocity; (b) ice thickness-ice strength; (c) ice velocity-ice strength.
(a)
(b)
(c)


## 3.2. Spectral Model of Stochastic Ice Loads

The spectral model of ice loads was proposed by Kärnä et al. [12] based on field measurement of JZ9-3 MDP2 in Bohai Sea. The spectral model of stochastic ice loads includes auto-spectral functions and cross-spectral functions. The following is a detailed simulation process.The auto-spectral density function of stochastic ice loads can be expressed as(11)σi2=∫0∞Giifdf.The nondimensional spectrum of stochastic ice loads can be shown as(12)G˜iif=fGiifσi2.The nondimensional auto-spectral function was proposed by Kärnä et al. [12]:(13a)G˜iif=af1+ksa1.5f2,(13b)a=b×v−0.6,in which v stands for ice velocity and ks and b are fitting parameters. The data analysis of stochastic ice loads in JZ9-3 MDP2 indicates that the mean value of parameter ks = 3.24 and b = 1.34 [32].The auto-spectral density function of local ice loads is expressed as(14)Giif=a1+ksa1.5f2σi2.The cross-spectral density function denotes the interaction of adjacent load pointsi and j. The coherence function γij is used to calculate the relations between two local points of rij distance. The coherence function γij is expressed by auto-spectral and cross-spectral density functions as(15)γij2f=Gijf2GiifGijf.The coherence function was proposed by Kärnä et al. [32]:(16)γij=11+ρ+αξijρ+e−βξijf,ξij=rijh,in which ξij denotes the nondimensional distance of two locations rij divided by ice thickness h. The other test parameters ρ, α, and β can be selected as ρ=0.1, α=0.2, and β=3 [32].As is shown in Figure9, the stochastic ice loads consist of the incidence angle and shear force on the structure. The local stochastic ice loads can be expressed as(17)Fi′t=cosθ+μsinθFit,where i = 1, 2, …, n, denotes the number of parts, μ stands for the friction coefficient, and Fi (t) stands for the ice loads on the contact surface.Figure 9
Ice forces distribution on an offshore structure.The auto-spectral density functions of the frequency domain can be described as(18)Gii′f=Giif×cos2θ+μ2sin2θ.The spectral matrices are summed to obtain the spectrum of stochastic ice loads.(19)GFf=C+μSΤGfffC+μS,where C and S stand for vectors for cosθ and sinθ at each local load point. Gff (f) consist of auto-spectral and cross-spectral functions.The power spectrum of the frequency domain can be converted to the time domain.(20)Ft=∑2×GFf×Δfcos2πft+φf,where 2×GFf×Δf is the ice loads amplitude.The standard deviationσF and mean value FM of the time histories of ice loads can be expressed as(21)σF=∫0∞GFfdf1/2,FM=σFI.The total ice forceFG (t) against a vertical structure can be expressed as(22)FGt=FM+Ft.Therefore, the estimation value of maximum ice loads can be expressed as(23)FGmax=FM+kσF,where σF denotes the standard deviation of stochastic ice loads and k is a selected exceedance probability. According to the existing references [12, 32], the assumption that the value of crushing intensity I = 0.4 and k = 4.0 is reasonable.


## 3.3. The Process of the Stochastic Ice Loads

The sample function of the stochastic ice loads according to the above Section3.2 was simulated by utilizing the classical fast Fourier transforms. The upper cutoff frequency fn was taken as 50 Hz and the total period T0 was taken as 500 s. The ice loads time histories based on the spectral model were generated at an ice velocity of 36 cm/s, ice thickness of 8 cm, and ice strength of 1.0 MPa. For brevity, one of the ice loads time histories was displayed in this paper, as shown in Figure 10.Figure 10
The time history of the stochastic ice force for the present bridge (T = 500 s).


## 4. Fatigue Damage Evaluation

In this paper, the rain-flow counting methodology was utilized to evaluate the fatigue damage of the pile-supported bridge under stochastic ice loads. To investigate the fatigue damage life of the pile-supported bridge, the connection section of the piles and the cap (S1 in Figure 4) and the piles and the seabed (S2 in Figure 4) were calculated.The design fatigue life of an offshore structure is determined based on the S-N curve and cumulative fatigue damage of pile-supported bridge is evaluated based on DNV [18].(24)log10N=log10a−mlog10Δσttrefk,where N denotes the predicted fatigue damage in cycles; Δσ denotes the stress range; m denotes the negative slope of S-N curve; loga is the intercept of logN axis; t is plate thickness; tref is the reference thickness and k denotes the thickness exponent.According to DNV standard, the S-N curve parameters for a pile are shown in Table1. After rain-flow counting for each stress range Δσ is processed, Miner’s sum can calculate a cumulative fatigue damage:(25)Dj=∑i=1IniNi,in which ni denotes the number of stress cycles in a given stress block, and Ni stands for the number of stress cycles to fatigue failure in a given stress block.Table 1
S-N curves parameters, according to [18].
N ≤ 106
N ≥ 106
Fatigue limit at 107 cycles
Thickness ratio
Thickness component
Stress concentration factor
Log (a1)
m
1
Log (a2)
m
2
σ (MPa)
t/t
ref
k
SCF
11.61
3.0
15.35
5.0
46.78
1.0
0.2
1.13The total cumulative fatigue damage is calculated as follows:(26)D=∑j=164PjDj,where Pj is the probability value of each ice load case. Dj is the fatigue damage value of each ice load case.Fatigue damage evaluation was applied by utilizing stochastic ice loads histories. The fatigue damage at the pile top and seabed cross section was evaluated by utilizing the rain-flow counting methodology. Figures7(a)–7(c) show a dominant range of ice parameters: ice thickness below 20 cm, ice velocity below 80 cm/s, and ice strength below 1.6 MPa, respectively. The joint probability of ice thickness, ice velocity, and ice strength are determined for the ice thickness range from 3 to 20 cm, the ice velocity range from 10 to 80 cm/s, and the ice strength range from 0.4 to 1.6 MPa. To reduce the computational cost, the ice thickness, ice velocity, and ice strength are divided into 4 bins, respectively. 64 ice loads cases are selected in this paper. Ice thickness is chosen as 4 cm, 8 cm, 12 cm, and 16 cm. The ice velocity values are selected as 16 cm/s, 36 cm/s, 56 cm/s, and 76 cm/s. Four representative ice strengths are used: 0.5 MPa, 1.0 MPa, 1.5 MPa, and 2.0 MPa.Considering that sea ice impact on the pile cap of the pile-supported bridge and largest total stress is expected to be at pile top cross section or seabed cross section. The selected bending stress points are shown in Figure11. The overall bending stress of the selected stress point can be expressed as(27)σi=NA+MyIyrcosφi,in which N denotes the axial force and A stand for the section area, r denotes the section radius, My stands for the bending moments, and Iy denotes the moments of cross-section area.Figure 11
Selected stress points distributed along the pile cross section.


## 5. Results

Various soil conditions are investigated to evaluate the effect of soil strength on the fatigue damage of bridges subjected to the ice loads. The soil strengths ofsu = 25 kPa, su = 50 kPa, and su = 100 kPa represent the typical soft, moderate, and stiff soils. Based on the determined number of stress cycles and stress ranges, the fatigue damage is evaluated based on the S-N curve and Miner’s rule.The total cumulative damage of the five selected points is evaluated by utilizing the rain-flow cycle counting methodology to calculate the stress cycles of a pile-supported bridge under each ice load case. In this paper, the effect of soil strength conditions and water depth on the fatigue damage of a bridge structure is investigated systematically. The selected stress point is investigated by utilizing equation (27) within 500 s to reduce the computational cost. For fatigue damage estimation, the response of the structure is calculated for all selected ice parameters. Due to the limitation of paper length, the response of the structure corresponding to ice velocity v = 36 cm/s, ice thickness h = 8 cm, and ice strength σ = 1.0 MPa, are shown as an example in Figures 12–15.Figure 12
Comparison of stress at the pile top cross section for different soil conditions: (a) time histories of stress atS11; (b) time histories of stress at S15 (v = 36 cm/s, h = 8 cm, σ = 1.0 MPa).
(a)
(b)Figure 13
Comparison of stress on seabed cross section for different soil conditions: (a) time histories of stress atS11; (b) time histories of stress at S15 (v = 36 cm/s, h = 8 cm, σ = 1.0 MPa).
(a)
(b)Figure 14
Comparison of stress at the pile top cross section with different water depth: (a) time histories of stress atS11; (b) time histories of stress at S15 (v = 36 cm/s, h = 8 cm, σ = 1.0 MPa).
(a)
(b)Figure 15
Comparison of stress on seabed cross section with different water depth: (a) time histories of stress atS11; (b) time histories of stress at S15 (v = 36 cm/s, h = 8 cm, σ = 1.0 MPa).
(a)
(b)Figures12 and 13 show the bending stress time histories for different soil conditions at S1 and S5 on the pile top and seabed cross section, respectively. Table 2 tabulates the cumulative fatigue damage results of the selected 5 locations at the pile top and seabed cross section with a different undrained shear strength of the soil. It can be found that S1i denotes the selected points distributed along the pile top cross section and S2i denotes the selected points distributed along the seabed cross section.Table 2
500 s cumulative fatigue damage with different soil conditions (water depth: 30 m).
Position
Position along pile section
Cumulative damage (10−4)
S
u = 25 kPa (soft)
S
u = 50 kPa (medium)
S
u = 100 kPa (stiff)
Pile-top
S
11
2.492
1.353
0.773
S
12
1.604
0.937
0.354
S
13
0.101
0.054
0.021
S
14
1.479
0.793
0.216
S
15
2.221
0.962
0.614
Seabed
S
21
0.157
0.111
0.041
S
22
0.095
0.068
0.017
S
23
0.007
0.003
0.001
S
24
0.079
0.052
0.011
S
25
0.132
0.102
0.034As shown in Figures12 and 13, the range of stress amplitude of pile top cross section are higher than seabed cross section. This phenomenon is due to the sea ice impact on the pile cap directly, which leads to higher stress cycles than the latter. Due to the increase of undrained shear strength, the lateral stiffness of soil increases, which leads to the decrease of stress amplitude range of pile and seabed significantly. This phenomenon is a reason that the piles insert into the seabed and is surrounded by the different stiffness of soil, which makes the pile-supported bridge dissipates less energy compared with the stiff soil. As shown in Figure 12(a) and Table 2, the cumulative fatigue damage at S11 of the pile top cross section are 2.492e−4 and 1.353e−4, respectively, when su = 25 and 50 kPa, which increase by 222.3% and 75.0% compared to soil strength su = 100 kPa. Figure 13(a) and Table 2 also show that when soil strength su = 25 and 50 kPa, cumulative fatigue damage at S21 of the seabed cross section are 0.157e−4 and 0.111e−4. The results of the cumulative fatigue damage increased by 282.9% and 170.7%, respectively, compared to the case of stiff soil condition. It is notable that the different soil conditions should be considered to more accurately predict the fatigue damage of the pile-supported bridge.The various water depths are investigated to evaluate the effect of water depths on the fatigue damage of bridges subjected to the ice loads. The water depths ofH = 10, 20, and 30 m represent the water depths conditions of shallow, moderate, and deep water.Figures14 and 15 show the bending stress time histories for different water depths at S1 and S5 on the pile top and seabed cross section, respectively. Table 3 tabulates the cumulative fatigue damage results of the selected 5 locations at the pile top and seabed cross section with different water depths. According to equation (27), we can see that the total bending stress of selected stress points can be influenced by bending moments, and the distance between the neutral axis and stress points. Consequently, the fluctuation of stress amplitude decreases from stress point S11 to S13 and then increases from stress point S13 to S15. Larger stress amplitudes can result in greater fatigue damage. As tabulated in Table 3, the cumulative fatigue damage of stress points of ice impact direction is more than that of any other positions of the pile cross section. Due to the direct effect of ice impact direction, the larger bending stress amplitude happened at stress points S11 and S15. However, the bending stress of the stress point S13 is minimum. This reason is the stress point S13 is perpendicular and far away from the action direction of ice loads, i.e., stress point S13 is on the neutral axis. The numerical result indicates that the cumulative fatigue damage decreases from stress point S11 to S13 and then increases from stress point S13 to S15. The variation tendency of the cumulative fatigue damage of stress points along the pile section was similar on the seabed cross section. As shown in Figures 14 and 15, the stress amplitude range of the pile top cross section is much larger than seabed cross section. This phenomenon is due to the cross section of the pile top experiences higher stress amplitudes than the cross section of the seabed when sea ice impacts on the pile cap directly.Table 3
500 s cumulative fatigue damage with different water depths (soil strength: 25 kPa).
Position
Position along pile section
Cumulative damage (10−4)
H = 30 m (deep)
H = 20 m (medium)
H = 10 m (shallow)
Pile-top
S
11
2.492
1.659
0.863
S
12
1.604
1.055
0.475
S
13
0.101
0.084
0.044
S
14
1.479
0.993
0.389
S
15
2.421
1.426
0.822
Seabed
S
21
0.157
0.122
0.049
S
22
0.095
0.078
0.028
S
23
0.007
0.004
0.002
S
24
0.079
0.066
0.019
S
25
0.132
0.093
0.026The range of the stress amplitude of the pile top and seabed cross sections increases with the water depth. As shown in Figure14(a) and Table 3, when the water depth is 30 m and 20 m, the cumulative fatigue damage at S11 of the pile top cross section are 2.492e−4 and 1.659e−4. The corresponding results are 188.7% and 92.2% higher than that of water depth of 10 m. Figure 15(a) and Table 3 also show that when the water depth is 30 and 20 m, the cumulative fatigue damage at S21 of the seabed cross section is 0.157e−4 and 0.122e−4. The results of the cumulative fatigue damage increased by 220.4% and 148.9%, respectively, compared to the case of shallow water depths. Note that the effect of water depth on the fatigue damage of the pile-supported bridge is essential and the different water depth should be considered to predict the fatigue damage reasonably.


## 6. Conclusions

A pile-supported bridge is the most widely used type of bridge structure in sea-crossing engineering. The stress cycles caused by ice-induced vibration can directly lead to the fatigue damage of the bridge structures. In this paper, the fatigue damage of the pile-supported bridge was analyzed based on the bending stress induced by the bridge vibrations under ice loads. The effects of soil conditions and water depths on the fatigue damage of the pile-supported bridge were investigated by utilizing the rain-flow counting method.The statistical data of the ice parameters of Bohai Sea including the ice thickness, ice velocity, and ice strength are investigated. The statistical ice parameters of Bohai Sea are essential for evaluating fatigue damage of pile-supported bridges in this paper and are greatly significant for the construction project of the pile-supported bridges in Bohai Sea.By calculating the fatigue damage caused by the ice load of each bin and considering its probability, the total fatigue damage within 500 s is calculated. The results show that the cumulative fatigue damage of piles with soft soil condition is much larger than those with stiff soil condition and that the deep-water condition also can increase the fatigue damage markedly. Furthermore, it can be seen that the whole bending stress of selected stress points can be influenced by bending moments and the distance between the neutral axis and stress points. The larger stress amplitudes can result in greater fatigue damage. The cumulative fatigue damage of stress points of ice impact direction is higher than any other position of the pile cross section. The stress amplitude range of the pile top cross section is much larger than the seabed cross section. This phenomenon is due to the cross section of the pile top experiences higher stress amplitudes than the cross section of the seabed when sea ice impacts on the pile cap directly. This provides significant design guidance for the construction and evaluation of the pile-supported bridge in the ice-covered sea area.



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*Source: 1853963-2020-02-21.xml*