# Kalman Filter-Based Multitype Measurement Data Fusion for Stress Intensity Factor Evaluation in Marine Structures

Authors: Tong-Tong Dai; Zi-Guang Jia; Liang Ren; Xing Fu
Journal: Structural Control and Health Monitoring
 (2023)

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## Abstract

Fracture is the main form of marine structural failure due to the action of waves and oceans. It is necessary to monitor the initiation and extension of cracks in a timely and accurate manner, and the stress intensity factor (SIF) is an important indicator for analysing the propagation of cracks. Based on existing research, this paper combines the calculation method of the SIF based on a single strain gauge (SSG) and maximum crack opening displacement (CMOD) and proposes a method based on the Kalman filter (KF). Numerical simulation and experimental verifications of this method have been carried out. The results show that the method can be used for a variety of configurations of precracked specimens. The KF-based method obtains more accurate results than the SSG-based and CMOD-based methods, and the relative error (RE) is not greater than 1.2%.


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## Body

## 1. Introduction

The service status and failure modes of metallic marine structures are usually complex due to the action of alternating loads such as waves. The frequency and cost of maintenance greatly increase as the structure ages, and performance significantly decreases [1–6]. For example, during the inspection and investigation of a semisubmersible platform in the South China Sea, cracks were observed in the corner of the living building, the connecting structure of the living building, and the related transverse brace, as shown in Figure 1. Cracks were detected in the four corners of the living building shortly after the platform was put into operation and still appeared and spread despite treatment. According to the investigation and research, the crack was caused by alternating loads or extreme loads. The specific causes need further analysis. Failure to detect and evaluate in time may reduce the local and overall bearing capacity of the structure. Therefore, it is necessary to monitor the structure, assess the structural condition, and take effective measures in a timely manner for the maintenance and extension of the service life of marine structures [7–10]. Fatigue fracture is the main form of metal structure failure, accounting for more than 90% of the failure modes. The impact on structural performance is small when the crack size is small, but with the cyclic action of loads, a crack may further expand. To predict the crack growth rate and fracture strength of cracked members, it is necessary to accurately analyse the stress state.Figure 1
Cracks of a semisubmersible drilling platform caused by alternating or extreme ocean loads.Irwin [11] proposed the concept of the SIF (K) based on Griffith’s work [12] in 1957, which is a metric parameter of crack severity. The SIF is related to the size, stress, and geometry, but is independent of the material. In addition, the SIF represents the strength of the stress-strain field at the crack tip and shows the ability of crack propagation, which is widely utilized in the calculation of fatigue strength and crack propagation of cracked structures [13–15]. In the definition of (K), it is assumed that the mechanical behaviour of the material is linear elastic, and Hooke’s law is followed. Linear elastic fracture mechanics are adopted. The SIF can also be applied to estimate the crack growth life. Paris proposed an empirical formula for the crack growth rate: da/dN=C∆Km, where, a is the crack size, C and m are constants related to materials, N is the number of stress cycles, and ∆K is the variation range of the SIF. Therefore, determining the value of the SIF is necessary to assess the current state of the structure and predict the crack life. The calculation of the SIF includes numerical methods, analytical methods, and direct measurement methods. The analytical solution is the basis for the development of fracture mechanics, which is based on the basic equations of the stress field and displacement field near the crack tip and is derived by using mathematical and mechanical knowledge. Numerous numerical methods for the SIF, such as the stress method, finite element method (FEM) [13–18], boundary element method [19], and meshless method [20], have been proposed after years of research.However, in practical engineering, the analytical method and numerical method will greatly reduce the calculation accuracy and cannot achieve long-term continuous monitoring of cracked structures due to the limitations of the structural form and boundary conditions. Therefore, it is necessary to develop a method to determine the SIF based on easily measured physical quantities, such as strain and displacement, to realize real-time online monitoring of structures.For the SIF calculation based on the strain field near the crack tip, Dally and Sanford [21] derived the four-parameter solution of the SIF according to the Westergaard stress function and proposed the experimental method of measuring the SIF with a strain gauge. Some scholars have conducted further studies on the technology. Wei and Zhao [22] discovered that the SIF could be measured by two strain gauges via numerical and experimental studies. Sarangi et al. [23, 24] carried out theoretical research and experimental verification on the optimal strain gauge position and its importance in the accurate determination of the SIF. In terms of the calculation of the SIF based on the crack opening displacement (COD), Masksimenko and Tyagnii [25] proposed an experimental method to calculate the SIF and stress distribution along the crack line by using discrete displacement data of several points on the crack edge. Chen et al. [26] proposed a method to determine the SIF according to the CMOD and determined the applicable range of the method via finite element simulation. Then, the method for calculating the SIF with collinear double cracks and impulse load based on the CMOD was investigated [27, 28]. Saucedo-Mora et al. [29] proposed using digital image correlation (DIC) to measure the SIF of crack tips. The stress-strain field was obtained by identifying the crack tip and by measuring the full-field displacement; thus, the SIF and the energy release rate of each crack during shear and tensile tests were calculated. Chen and Qian [30] proposed a method to quantify the J integral of the through-thickness of planar strain samples by DIC and tested the constraint effect of DIC measurement. Then, the basic relationship between the average J value of the free surface and the full thickness is theoretically and experimentally established and verified [31]. However, the SIF determination method based on the strain is greatly affected by the position of the strain measurement points due to the large strain gradient near the crack tip. The CMOD-based method is proposed under the assumption of ideal boundary conditions, which will deviate in practical engineering due to the complex load form.Currently, structural health monitoring (SHM) systems are deployed on large civil structures and typically contain various types of sensors to measure different types of responses of structures. Numerous studies relating to data fusion have investigated SHM in time and space [32–40]. Wu and Jahanshahi [41] reviewed data fusion methods for SHM and system identification. Park et al. [34] realized data fusion between vision-based displacement and acceleration by a complementary filter, which reduces vision-based displacement error and expands the feasible frequency range. This method accurately measures the displacement response in terms of noise level and viable frequency range. Jin et al. [39] adopted the traditional, multifidelity data fusion framework based on Gaussian regression to investigate the complementary data fusion of point strain sensors and distributed sensors and obtained an accurate strain distribution by combining their advantages. Qi et al. [40] proposed a model-based, inverse finite element algorithm by using two kinds of sensors to identify the crack length and carried out experimental verification. However, according to the investigation, there is no precedent for applying data fusion technology to SIF calculation.The paper is aimed at combining the determination methods of SIF based on strain and CMOD and at adopting the data fusion framework based on KF to improve the estimation accuracy of the SIF. To verify the feasibility of the proposed method, numerical and experimental studies were carried out. As the study is aimed at the current state of cracks in the living building part of an offshore platform, the test pieces are steel plates. The structure of this paper is described as follows: the calculation method of the SIF based on the SSG and CMOD is deduced in Section2; in Section 3, the principle of KF and the specific process employed in this study are introduced; in Section 4, numerical simulation is carried out for plates with different ratios of crack length to plate width, and the SIF is obtained by SSG-based, CMOD-based and KF-based methods and compared with real values; experiments on plates with prefabricated cracks in different configurations are introduced to verify the effectiveness of the proposed method.


## 2. Methodology

In this section, the method of SIF calculation by the SSG and CMOD is discussed, the calculation formula of SIF is derived, and then the basic method of classical KF modelling is introduced. According to the stress form of the crack, the propagation mode is divided into three types (Figure2): opening crack (mode I), sliding crack (mode II), and tearing crack (mode III). In ocean engineering, the crack form is mainly the opening mode. Therefore, in this section, the calculation of the SIF of mode I is introduced.Figure 2
Three modes of crack propagation: (a) Mode I, (b) Mode II, and (c) Mode III.
(a)(b)(c)


### 2.1. SIF Determination Method Based on the SSG

The plane problem of elasticity, can be solved by the stress functionUx,y, which is expressed as Ur,θ in polar coordinates and expressed as Ur,θ=r1+λFλθ in the form of separated variables, satisfying the coordination equation ∇2U=0. At one end of the finite crack shown in Figure 3, the origin of the coordinate system is selected at the crack tip. The x axis is collinear with the crack. The strain gauge is placed on the line (“line θ”) with angle θ with the x axis, and the included angle between the strain gauge and the x-axis is α.Figure 3
Coordinate system of the crack tip.Considering the boundary condition, that is, the force on the crack surface is zero as follows:(1)σθ=τrθ=0,θ=±π.The general solution is obtained as follows:(2)Fλθ=Acosλ+1θ+Bsinλ+1θ+Ccosλ−1θ+Dsinλ−1θ.The stress function is obtained as follows:(3)Ur,θ=∑n=1∞Cnr1+n/2cosn2+1θ−n/2+−1nn/2+1cosn2+1θ+∑m=1∞Dmr1+m/2sinm2−1θ−m/2−−1mm/2+1sinm2+1θ.The SIF is obtained by(4)KI=C12π.The three terms of strain componentsεx and εy are obtained by Hooke’s law and the geometric equation(5)2Gεx=C1r−1/2cosθ21−v1+v−sinθ2sin3θ2+2D11+v+C2r1/2cosθ21−v1+v+sin2θ2,2Gεy=C1r−1/2cosθ21−v1+v+sinθ2sin3θ2−2vD11+v+C2r1/2cosθ21−v1+v−sin2θ2.The strain componentεx′ at the strain gauge position satisfies the following equation:(6)2Gεx′=C1r−1/21−v1+vcosθ2−12sinθ2sin3θ2cos2α+12sinθcos3θ2sin2α+C2r1/2cosθ21−v1+v+sin2θ2cos2α−12sinθ2sin2α+D11−v1+v+cos2α.D1 and C2 can be eliminated when(7)cos2α=−1−v1+v,tanθ2=−cot2α.Whenv=0.3,(8)2Gεx′=0.64434C1r−1/2,(9)εx′=Cr−1/2,C is a constant when the material and load are determined. Taking the log of both sides of equation (9)(10)lnεx′=−12lnr+lnC,that is, within the range of radial length satisfying equation (10), the SIF can be expressed as(11)KI=22πrGεx′0.64434.


### 2.2. SIF Determination Method Based on the CMOD

The stress function indicated by Westergaard is adopted in the model of an infinite plate with a penetrating crack under uniform tensile stressσ as follows:(12)ϕ=ReZ¯¯+yImZ¯,where,dZ¯¯dZ=Z¯;dZ¯dZ=Z;dZdZ=Z′.The displacement is obtained as(13)u=1E′1−νReZ¯−1+νyImZ,v=1E′2ImZ¯−1+νyReZ.IfZz=σz/z2−a2 and z=reiθ+a, then(14)Z¯z=σr2cos2θ+isin2θ+2arcosθ+isinθ=σx02+y02⋅cosα+isinα.Among them,(15)x0=r2cos2θ+2arcosθ,y0=r2sin2θ+2arsinθ,α=arctany0x0,that is,(16)Z¯z=σx02+y021/4cosα2+isinα2,Zz=σrcosθ+a+ir⋅sinθx02+y021/4⋅cosα/2+isinα/2.The polar coordinates of the CMOD arer=a,θ=±π, where x0=−a2,y0=0,α=±π.Therefore, the CMOD isumax=2⋅ν=4/E′σa.For an infinite plate with a central penetrating crack, the SIF at the crack tip isKI=σπa under uniform tensile loading. Therefore, the SIF expressed by the CMOD is(17)KI=umax⋅E′4a/π.


## 2.1. SIF Determination Method Based on the SSG

The plane problem of elasticity, can be solved by the stress functionUx,y, which is expressed as Ur,θ in polar coordinates and expressed as Ur,θ=r1+λFλθ in the form of separated variables, satisfying the coordination equation ∇2U=0. At one end of the finite crack shown in Figure 3, the origin of the coordinate system is selected at the crack tip. The x axis is collinear with the crack. The strain gauge is placed on the line (“line θ”) with angle θ with the x axis, and the included angle between the strain gauge and the x-axis is α.Figure 3
Coordinate system of the crack tip.Considering the boundary condition, that is, the force on the crack surface is zero as follows:(1)σθ=τrθ=0,θ=±π.The general solution is obtained as follows:(2)Fλθ=Acosλ+1θ+Bsinλ+1θ+Ccosλ−1θ+Dsinλ−1θ.The stress function is obtained as follows:(3)Ur,θ=∑n=1∞Cnr1+n/2cosn2+1θ−n/2+−1nn/2+1cosn2+1θ+∑m=1∞Dmr1+m/2sinm2−1θ−m/2−−1mm/2+1sinm2+1θ.The SIF is obtained by(4)KI=C12π.The three terms of strain componentsεx and εy are obtained by Hooke’s law and the geometric equation(5)2Gεx=C1r−1/2cosθ21−v1+v−sinθ2sin3θ2+2D11+v+C2r1/2cosθ21−v1+v+sin2θ2,2Gεy=C1r−1/2cosθ21−v1+v+sinθ2sin3θ2−2vD11+v+C2r1/2cosθ21−v1+v−sin2θ2.The strain componentεx′ at the strain gauge position satisfies the following equation:(6)2Gεx′=C1r−1/21−v1+vcosθ2−12sinθ2sin3θ2cos2α+12sinθcos3θ2sin2α+C2r1/2cosθ21−v1+v+sin2θ2cos2α−12sinθ2sin2α+D11−v1+v+cos2α.D1 and C2 can be eliminated when(7)cos2α=−1−v1+v,tanθ2=−cot2α.Whenv=0.3,(8)2Gεx′=0.64434C1r−1/2,(9)εx′=Cr−1/2,C is a constant when the material and load are determined. Taking the log of both sides of equation (9)(10)lnεx′=−12lnr+lnC,that is, within the range of radial length satisfying equation (10), the SIF can be expressed as(11)KI=22πrGεx′0.64434.


## 2.2. SIF Determination Method Based on the CMOD

The stress function indicated by Westergaard is adopted in the model of an infinite plate with a penetrating crack under uniform tensile stressσ as follows:(12)ϕ=ReZ¯¯+yImZ¯,where,dZ¯¯dZ=Z¯;dZ¯dZ=Z;dZdZ=Z′.The displacement is obtained as(13)u=1E′1−νReZ¯−1+νyImZ,v=1E′2ImZ¯−1+νyReZ.IfZz=σz/z2−a2 and z=reiθ+a, then(14)Z¯z=σr2cos2θ+isin2θ+2arcosθ+isinθ=σx02+y02⋅cosα+isinα.Among them,(15)x0=r2cos2θ+2arcosθ,y0=r2sin2θ+2arsinθ,α=arctany0x0,that is,(16)Z¯z=σx02+y021/4cosα2+isinα2,Zz=σrcosθ+a+ir⋅sinθx02+y021/4⋅cosα/2+isinα/2.The polar coordinates of the CMOD arer=a,θ=±π, where x0=−a2,y0=0,α=±π.Therefore, the CMOD isumax=2⋅ν=4/E′σa.For an infinite plate with a central penetrating crack, the SIF at the crack tip isKI=σπa under uniform tensile loading. Therefore, the SIF expressed by the CMOD is(17)KI=umax⋅E′4a/π.


## 3. Determination of the SIF Based on the KF

This section introduces the classical KF algorithm flow, and then introduces the specific process for determining SIF in the subsequent numerical simulation and experiment.


### 3.1. Kalman Filtering Model

The KF optimizes the state estimation of the system by two processes of “prediction” and “update.” The basic model of the KF will be introduced in this section [42].Suppose that the discrete dynamic system is modelled as(18)Xk=AXk−1+BUk−1+Wk−1a,Zk=HXk+Vkb,where Xk is the system state at moment k, and Zk is the observed value at moment k, that is, the measured value. A is the state transition matrix, B is the control input matrix, and H is the observation matrix.Since most systems are not strictly linear and time-varying and the structural system parameters may be uncertain, there will be a deviation in the estimated state value, which is characterized by process noiseWk−1. For the state observation equation (18)-b, due to the existence of noise and interference signals or the error of the observation matrix, there will be a deviation between the observed value and the real value, which is characterized by Vk. Wk−1 and Vk are uncorrelated white noise with mean value of 0 and variances of Q and R, respectively. The principle of the KF is to modify the state prediction value so that it is similar to the real value using the Kalman gain value.The a priori estimate and measurement of the state as follows:(19)X^−k=AXk−1+BUk−1,Zk=HXk⟶X^k=H−Zk.Superscripts “^” and “−” denote estimates and priors, respectively.The posterior value is calculated from the prior value and measured value as follows:(20)X^k=X^−k+KkZk−HX^−k.The goal is to obtain the Kalman gain valueKk, such that X^k⟶Xk.The posterior state error and prior state error are(21)Ek=Xk−X^k,E−k=Xk−X^−k.Then, the variance matrix of the estimation error is(22)Pk=EEkETk,TheKk with the smallest variance is obtained as follows:(23)∂trPk∂Kk=0⟶Kk=P−kHTHP−kHT+R.Then, the error variance matrix is(24)Pk=I−KkP−k,P−k=APk−1AT+Q.Figure4 summarizes the flow of KF.Figure 4
Flowchart of the KF.


### 3.2. SIF Determination Method Based on Data Fusion

The calculation process based on the KF in this study will be introduced in this section. This method includes the determination method of the SIF based on the SSG and CMOD. Therefore, the model of the system contains two observation equations, and the system variable is one-dimensional.The model of the system is(25)xk=Axk−1+Buk−1+wk−1a,z1,k−1=H1xk+v1,kb,z2,k−1=H2xk+v2,kc,where wk−1 is the process noise, and v1,k and v2,k represent the observation noise of two observation values, which are uncorrelated white noise with a mean value of 0 and variances of Q, R1, and R2. As there is no way to model the noise term during the modelling process, the Kalman filter process is generally divided into two parts: a priori estimation and correction.(1)
The SIF at the current time is estimated based on the value of the previous step(26)x^k−=Ax^k−1+Buk−1.(2)
The error covariance of the a priori estimation is calculated(27)P1,k−=AP1,k−1AT+Q,P2,k−=AP2,k−1AT+Q.(3)
The prior estimate is corrected, and the Kalman gain of the two measured values is calculated(28)K1,k=P1,k−H1TH1P1,k−H1T+R1,K2,k=P2,k−H2TH2P2,k−H2T+R2.(4)
The a priori estimation, Kalman gain, and two measured values are employed for optimal estimation(29)x^k=x^k−+K1,kz1,k−H1x^k−+K2,kz2,k−1−H2x^k−.(5)
The error covariance is updated(30)P1,k=1−K1,kH1P1,k−,P2,k=1−K2,kH2P2,k−.(6)
If the termination conditions are met, the filtering process is ended. Otherwise, steps (1)–(5) are repeated (the termination condition is to reach the maximum filtering times in this study).


## 3.1. Kalman Filtering Model

The KF optimizes the state estimation of the system by two processes of “prediction” and “update.” The basic model of the KF will be introduced in this section [42].Suppose that the discrete dynamic system is modelled as(18)Xk=AXk−1+BUk−1+Wk−1a,Zk=HXk+Vkb,where Xk is the system state at moment k, and Zk is the observed value at moment k, that is, the measured value. A is the state transition matrix, B is the control input matrix, and H is the observation matrix.Since most systems are not strictly linear and time-varying and the structural system parameters may be uncertain, there will be a deviation in the estimated state value, which is characterized by process noiseWk−1. For the state observation equation (18)-b, due to the existence of noise and interference signals or the error of the observation matrix, there will be a deviation between the observed value and the real value, which is characterized by Vk. Wk−1 and Vk are uncorrelated white noise with mean value of 0 and variances of Q and R, respectively. The principle of the KF is to modify the state prediction value so that it is similar to the real value using the Kalman gain value.The a priori estimate and measurement of the state as follows:(19)X^−k=AXk−1+BUk−1,Zk=HXk⟶X^k=H−Zk.Superscripts “^” and “−” denote estimates and priors, respectively.The posterior value is calculated from the prior value and measured value as follows:(20)X^k=X^−k+KkZk−HX^−k.The goal is to obtain the Kalman gain valueKk, such that X^k⟶Xk.The posterior state error and prior state error are(21)Ek=Xk−X^k,E−k=Xk−X^−k.Then, the variance matrix of the estimation error is(22)Pk=EEkETk,TheKk with the smallest variance is obtained as follows:(23)∂trPk∂Kk=0⟶Kk=P−kHTHP−kHT+R.Then, the error variance matrix is(24)Pk=I−KkP−k,P−k=APk−1AT+Q.Figure4 summarizes the flow of KF.Figure 4
Flowchart of the KF.


## 3.2. SIF Determination Method Based on Data Fusion

The calculation process based on the KF in this study will be introduced in this section. This method includes the determination method of the SIF based on the SSG and CMOD. Therefore, the model of the system contains two observation equations, and the system variable is one-dimensional.The model of the system is(25)xk=Axk−1+Buk−1+wk−1a,z1,k−1=H1xk+v1,kb,z2,k−1=H2xk+v2,kc,where wk−1 is the process noise, and v1,k and v2,k represent the observation noise of two observation values, which are uncorrelated white noise with a mean value of 0 and variances of Q, R1, and R2. As there is no way to model the noise term during the modelling process, the Kalman filter process is generally divided into two parts: a priori estimation and correction.(1)
The SIF at the current time is estimated based on the value of the previous step(26)x^k−=Ax^k−1+Buk−1.(2)
The error covariance of the a priori estimation is calculated(27)P1,k−=AP1,k−1AT+Q,P2,k−=AP2,k−1AT+Q.(3)
The prior estimate is corrected, and the Kalman gain of the two measured values is calculated(28)K1,k=P1,k−H1TH1P1,k−H1T+R1,K2,k=P2,k−H2TH2P2,k−H2T+R2.(4)
The a priori estimation, Kalman gain, and two measured values are employed for optimal estimation(29)x^k=x^k−+K1,kz1,k−H1x^k−+K2,kz2,k−1−H2x^k−.(5)
The error covariance is updated(30)P1,k=1−K1,kH1P1,k−,P2,k=1−K2,kH2P2,k−.(6)
If the termination conditions are met, the filtering process is ended. Otherwise, steps (1)–(5) are repeated (the termination condition is to reach the maximum filtering times in this study).


## 4. Numerical Simulation

As most marine structures are plates and shells, plates are selected as the research object in this paper. In this section, a plate with prefabricated cracks is numerically simulated, and the applicable radial length range of the measuring points in the SSG-based method is calculated. The SIF values are obtained by SSG-based, CMOD-based, and KF-based methods and compared with the real values.


### 4.1. Determination of the Radial Position of the Strain Gauge

As described in Section2.1, the determination method of the SIF based on the SSG is not applicable to the whole “line θ” that has angle θ with the x-axis. The relationship between the strain component εx′ and the radial length r satisfies equation (10), that is, the slope of curve lnεx′−lnr is equal to −0.5.In this section, the plate with prefabricated cracks shown in Figure5 is simulated, where b=100mm,c=150mm, the plate thickness t=3mm, the elastic modulus E=206GPa, Poisson’s ratio v=0.3, and α=61.29∘,θ=65.16∘ can be calculated by equation (7).Figure 5
Configuration of the simulated plate.Usinga=40mm, that is, a/b=0.4, as an example, to obtain the exact value of the applicable range, the element on “line θ” at both ends of the crack is refined, as shown in Figure 6. Figure 7 is the enlarged view of the crack tip. The applicable radial length of the SIF determination method based on the SSG for different values of a/b is calculated.Figure 6
Overall meshing.Figure 7
Refine mesh on “lineθ.”A uniform tensile load is applied to the model, the coordinates and strain components of each node on “lineθ” are obtained, and the curve of lnεx′−lnr is obtained. As shown in Figure 8, there is an obvious linear segment in the curve. A straight line with slope = −0.5 is inserted into the curve, and the range of radial length with an error less than 0.5% is obtained (i.e., the area between two dotted lines in Figure 8).Figure 8
Applicable radial length range of different crack lengths: (a)a = 10 mm, (b) a = 20 mm, (c) a = 30 mm, (d) a = 35 mm, and (e) a = 40 mm.
(a)(b)(c)(d)(e)The minimum and maximum of the applicable radial length are listed in Table1.Table 1
Applicable radial length range of different crack lengths (b=100mm,c=150mm,t=3mm,v=0.3).
aa/brmin (mm)rmax (mm)100.101.105.58200.201.3012.00300.302.5013.02350.354.0719.10400.403.8121.40Research by Rosakis and Ravichandar [43] shows that the radial length from the strain measuring point to the crack tip should be greater than half of the plate thickness. Therefore, the minimum radial length should be larger between t/2 and rmin.


### 4.2. Calculation of the SIF Based on the SSG and CMOD

Using the condition ofa/b=0.4 as an example, the strain values within the applicable radial length (3.81–21.40 mm) are substituted into equation (11) for the SIF calculation. Selecting the SIF obtained by the extended finite element method (XFEM) as the real value, Figure 9 shows that the strain value within the applicable range can be used to obtain a more accurate result. During the whole loading process, the relative error (RE) of the SIF obtained by the single strain gauge at r = 12.58 mm is concentrated at 3.18%, while the result obtained by r = 23.05 mm is 15.00%, indicating that it is necessary to calculate the applicable radial length.Figure 9
Comparison of the SIF obtained by the SSG-based method with differentr.Next, the plates with prefabricated cracks of differenta/b are simulated, and the SIF by SSG-based and the CMOD-based methods are calculated. Table 2 shows the SIF for σ=8.33MPa. The RE of the two methods is not greater than 8.5%.Table 2
Comparison of SIF forσ=8.33MPa.
a/bSIF (XFEM) (MPa∗ mm0.5)SSGCMODRadial lengthSIF (MPa∗ mm0.5)RE (%)SIF (MPa∗ mm0.5)RE (%)0.1052.164.2755.817.0056.508.320.2073.906.9277.785.2578.426.120.3093.398.5291.931.5697.684.590.35102.9512.57107.844.75105.392.370.40112.7412.58116.333.18117.093.86Table3 shows the RE of the SIF obtained by the two methods during the whole loading process when a/b=0.4. The results obtained by the SSG-based method are concentrated at 3.2%, while those obtained by the CMOD-based method are concentrated at 3.8%.Table 3
RE of the two methods for different stresses whena/b=0.4.
σ (MPa)SIF (XFEM) (MPa∗ mm0.5)SSGCMODSIF (MPa∗ mm0.5)RE (%)SIF (MPa∗ mm0.5)RE (%)0.4175.645.823.195.853.722.0828.1829.083.1929.273.874.1756.3758.173.1958.553.876.2584.5587.253.1987.823.878.33112.74116.333.18117.093.86To further improve the accuracy of SIF estimation, KF is used to fuse the SSG-based method and CMOD method, which will be introduced in Section4.3.


### 4.3. Determination Method of SIF Based on the Kalman Filter

In this section, the KF is used for data fusion of the SSG-based method and CMOD-based method, and the principle of the KF is introduced in Section3. The KF process contains two observations, and the number of iterations is 100. Similarly, using a/b=0.4 as an example, the SIF for each stress state is obtained by data fusion (subgraph of Figure 10). Figure 10 shows that in the initial stage of the iterative process, the RE of the SIF obtained by data fusion is larger, which gradually decreases as the number of iterations increases. The RE with σ=8.33MPa is 0.14%.Figure 10
Variation trend of RE during the loading process.The RE of the SIF obtained by the three methods (SSG, CMOD, and KF) withσ=8.33MPa is listed in Table 4. The RE of the SIF estimation by the KF is not greater than 1%. For the SSG-based method, the RE may be caused by the influence of ambient noise, strain gradient, and the omission of higher order terms in the derivation process. For the CMOD-based method, the error may be caused by the sensor accuracy, the notion that the location of the sensor is not the CMOD position due to human operation and the omission of the higher order terms. For the KF-based method, since the method is based on the SSG and CMOD methods, the error is caused by the inaccuracy of the two methods and the selection of the noise variance.Table 4
RE of the SIF obtained by three methods withσ=8.33MPa.
a/bRE (%)SSGCMODKF0.107.008.320.610.205.256.120.100.301.564.590.670.354.752.370.010.403.183.860.11


### 4.4. Noise Influence

In the previous research, no noise is added to introduce additional errors to the strain data and crack opening displacement data. However, the influence of human operation and environmental noise cannot be disregarded in practical engineering. Therefore, in this section, different levels of Gaussian white noise are added to the original data to evaluate the robustness of KF.Similarly, usinga/b=0.4 as an example, Gaussian white noise with a level of 1%–5% is added to the simulated data. Figure 11 shows a comparison of the SIF obtained by the KF, SSG, and CMOD-based methods under the influence of a 5% noise level. During the whole process of loading, there is a large deviation between the SSG-based method and CMOD-based method with the real value due to the influence of noise. The result of the KF has an error in the initial stage, but with an increasing number of iterations, it basically coincides with the real value and is not affected by noise. Figure 12 shows the RE of the three methods during the whole process.Figure 11
Comparison of the SIF obtained by three methods under the influence of 5% noise.Figure 12
RE of the three methods during the whole process under the influence of 5% noise.Table5 shows the RE of the SIF obtained by the three methods under the influence of different levels of noise when σ=8.33MPa. The RE of the SIF obtained by the KF is still at a low level, which indicates that the method based on data fusion can still obtain a more accurate SIF under the influence of noise.Table 5
RE of the three methods at different noise levels whenσ=8.33MPa.
Noise level (%)RE (%)SSGCMODKF13.984.490.1425.894.370.2233.531.120.2845.893.620.8352.305.121.73


## 4.1. Determination of the Radial Position of the Strain Gauge

As described in Section2.1, the determination method of the SIF based on the SSG is not applicable to the whole “line θ” that has angle θ with the x-axis. The relationship between the strain component εx′ and the radial length r satisfies equation (10), that is, the slope of curve lnεx′−lnr is equal to −0.5.In this section, the plate with prefabricated cracks shown in Figure5 is simulated, where b=100mm,c=150mm, the plate thickness t=3mm, the elastic modulus E=206GPa, Poisson’s ratio v=0.3, and α=61.29∘,θ=65.16∘ can be calculated by equation (7).Figure 5
Configuration of the simulated plate.Usinga=40mm, that is, a/b=0.4, as an example, to obtain the exact value of the applicable range, the element on “line θ” at both ends of the crack is refined, as shown in Figure 6. Figure 7 is the enlarged view of the crack tip. The applicable radial length of the SIF determination method based on the SSG for different values of a/b is calculated.Figure 6
Overall meshing.Figure 7
Refine mesh on “lineθ.”A uniform tensile load is applied to the model, the coordinates and strain components of each node on “lineθ” are obtained, and the curve of lnεx′−lnr is obtained. As shown in Figure 8, there is an obvious linear segment in the curve. A straight line with slope = −0.5 is inserted into the curve, and the range of radial length with an error less than 0.5% is obtained (i.e., the area between two dotted lines in Figure 8).Figure 8
Applicable radial length range of different crack lengths: (a)a = 10 mm, (b) a = 20 mm, (c) a = 30 mm, (d) a = 35 mm, and (e) a = 40 mm.
(a)(b)(c)(d)(e)The minimum and maximum of the applicable radial length are listed in Table1.Table 1
Applicable radial length range of different crack lengths (b=100mm,c=150mm,t=3mm,v=0.3).
aa/brmin (mm)rmax (mm)100.101.105.58200.201.3012.00300.302.5013.02350.354.0719.10400.403.8121.40Research by Rosakis and Ravichandar [43] shows that the radial length from the strain measuring point to the crack tip should be greater than half of the plate thickness. Therefore, the minimum radial length should be larger between t/2 and rmin.


## 4.2. Calculation of the SIF Based on the SSG and CMOD

Using the condition ofa/b=0.4 as an example, the strain values within the applicable radial length (3.81–21.40 mm) are substituted into equation (11) for the SIF calculation. Selecting the SIF obtained by the extended finite element method (XFEM) as the real value, Figure 9 shows that the strain value within the applicable range can be used to obtain a more accurate result. During the whole loading process, the relative error (RE) of the SIF obtained by the single strain gauge at r = 12.58 mm is concentrated at 3.18%, while the result obtained by r = 23.05 mm is 15.00%, indicating that it is necessary to calculate the applicable radial length.Figure 9
Comparison of the SIF obtained by the SSG-based method with differentr.Next, the plates with prefabricated cracks of differenta/b are simulated, and the SIF by SSG-based and the CMOD-based methods are calculated. Table 2 shows the SIF for σ=8.33MPa. The RE of the two methods is not greater than 8.5%.Table 2
Comparison of SIF forσ=8.33MPa.
a/bSIF (XFEM) (MPa∗ mm0.5)SSGCMODRadial lengthSIF (MPa∗ mm0.5)RE (%)SIF (MPa∗ mm0.5)RE (%)0.1052.164.2755.817.0056.508.320.2073.906.9277.785.2578.426.120.3093.398.5291.931.5697.684.590.35102.9512.57107.844.75105.392.370.40112.7412.58116.333.18117.093.86Table3 shows the RE of the SIF obtained by the two methods during the whole loading process when a/b=0.4. The results obtained by the SSG-based method are concentrated at 3.2%, while those obtained by the CMOD-based method are concentrated at 3.8%.Table 3
RE of the two methods for different stresses whena/b=0.4.
σ (MPa)SIF (XFEM) (MPa∗ mm0.5)SSGCMODSIF (MPa∗ mm0.5)RE (%)SIF (MPa∗ mm0.5)RE (%)0.4175.645.823.195.853.722.0828.1829.083.1929.273.874.1756.3758.173.1958.553.876.2584.5587.253.1987.823.878.33112.74116.333.18117.093.86To further improve the accuracy of SIF estimation, KF is used to fuse the SSG-based method and CMOD method, which will be introduced in Section4.3.


## 4.3. Determination Method of SIF Based on the Kalman Filter

In this section, the KF is used for data fusion of the SSG-based method and CMOD-based method, and the principle of the KF is introduced in Section3. The KF process contains two observations, and the number of iterations is 100. Similarly, using a/b=0.4 as an example, the SIF for each stress state is obtained by data fusion (subgraph of Figure 10). Figure 10 shows that in the initial stage of the iterative process, the RE of the SIF obtained by data fusion is larger, which gradually decreases as the number of iterations increases. The RE with σ=8.33MPa is 0.14%.Figure 10
Variation trend of RE during the loading process.The RE of the SIF obtained by the three methods (SSG, CMOD, and KF) withσ=8.33MPa is listed in Table 4. The RE of the SIF estimation by the KF is not greater than 1%. For the SSG-based method, the RE may be caused by the influence of ambient noise, strain gradient, and the omission of higher order terms in the derivation process. For the CMOD-based method, the error may be caused by the sensor accuracy, the notion that the location of the sensor is not the CMOD position due to human operation and the omission of the higher order terms. For the KF-based method, since the method is based on the SSG and CMOD methods, the error is caused by the inaccuracy of the two methods and the selection of the noise variance.Table 4
RE of the SIF obtained by three methods withσ=8.33MPa.
a/bRE (%)SSGCMODKF0.107.008.320.610.205.256.120.100.301.564.590.670.354.752.370.010.403.183.860.11


## 4.4. Noise Influence

In the previous research, no noise is added to introduce additional errors to the strain data and crack opening displacement data. However, the influence of human operation and environmental noise cannot be disregarded in practical engineering. Therefore, in this section, different levels of Gaussian white noise are added to the original data to evaluate the robustness of KF.Similarly, usinga/b=0.4 as an example, Gaussian white noise with a level of 1%–5% is added to the simulated data. Figure 11 shows a comparison of the SIF obtained by the KF, SSG, and CMOD-based methods under the influence of a 5% noise level. During the whole process of loading, there is a large deviation between the SSG-based method and CMOD-based method with the real value due to the influence of noise. The result of the KF has an error in the initial stage, but with an increasing number of iterations, it basically coincides with the real value and is not affected by noise. Figure 12 shows the RE of the three methods during the whole process.Figure 11
Comparison of the SIF obtained by three methods under the influence of 5% noise.Figure 12
RE of the three methods during the whole process under the influence of 5% noise.Table5 shows the RE of the SIF obtained by the three methods under the influence of different levels of noise when σ=8.33MPa. The RE of the SIF obtained by the KF is still at a low level, which indicates that the method based on data fusion can still obtain a more accurate SIF under the influence of noise.Table 5
RE of the three methods at different noise levels whenσ=8.33MPa.
Noise level (%)RE (%)SSGCMODKF13.984.490.1425.894.370.2233.531.120.2845.893.620.8352.305.121.73


## 5. Experimental Study

An experiment was designed to verify the reliability of the proposed method, including two working conditions: (1) central cracks subjected to uniform loads and (2) central cracks under nonuniform loads. The test was carried out on a universal testing machine using strain gauges and a displacement metre to obtain the changes in strain and maximum crack opening displacement during the loading process.


### 5.1. Experimental Device

The specimen used in the experiment is a steel plate with a prefabricated crack, the modulus of elasticity is 206 GPa, and Poisson’s ratio is 0.3. The width and length of the specimen are 200 mm and 300 mm. The test device is shown in Figure13. The two ends of the plate are installed on the testing machine using T-shaped adapters. The position and direction of the strain gauge paste refer to the results calculated in Section 4.1. The loading method is hierarchical loading and the loading speed is 100 N/s. The load is maintained for 10 s every 4 kN, and the maximum loading force is 24 kN. The radial lengths of the strain gauges in the specimens are shown in Table 6, which are within the effective range calculated in Section 4.1.Figure 13
Diagram of experimental device.Table 6
Radial length of the specimens.
a/br (mm)0.104.000.2012.000.2512.000.3012.00


### 5.2. Central Crack under Uniform Loads

The strain value and maximum crack opening displacement during the loading process were recorded, and the SIF obtained by the two methods was calculated. The empirical value is used as the real value and compared with the experimental results, whose calculation refers to equation as follows:(31)KI=σπaFab,where Fa/b is the boundary correction coefficient, which is calculated using the empirical formula given by Tada et al. [44] as follows:(32)Fab=1+0.128ab−0.288ab2+1.525ab3.The value of SIF during the loading process with differenta/b is calculated and listed in Table 7. An approximate range of the force value in the holding phase is selected due to the fluctuation in the force value during the loading process.Table 7
SIF curve of cracks with different lengths during loading.
Loading force (kN)SIFMPa·mm0.5a/b=0.1a/b=0.2a/b=0.25a/b=0.3True valueKFTrue valueKFTrue valueKFTrue valueKF4±0.138.40631.33854.23849.44861.25837.01369.36847.3638±0.175.43765.794108.401101.015122.549101.524137.956111.63812±0.1112.381103.773162.757148.281183.944164.540201.856184.47216±0.1150.168144.244216.870205.604245.308234.317271.659259.36920±0.1187.834185.400271.245266.088306.450303.470339.939333.86524±0.1224.312226.208325.216326.992367.691371.713406.255407.069The SIF obtained by the SSG, CMOD, and KF is compared with the true value, and the RE is calculated according to equation.(33)RE%=SIFi−SIFeSIFe×100%,where SIFi is the result obtained by the SSG, CMOD, and KF, and SIFe is the true value. The RE of cracks with different a/b during the entire loading process is listed in Table 8. As the number of iterations increases, the error obtained by KF gradually decreases and is lower than that obtained by the other two methods. However, in the middle stage of the iteration, accurate results may be obtained using the CMOD-based and SSG-based methods. When the error increases, KF processing can still keep the error within a lower range, that is, for the KF-based method, a sufficient number of iterations is necessary to obtain accurate results.Table 8
RE of the SIF of different specimens under varying loading forces.
Loading force (kN)RE (%)SSGCMODKFSSGCMODKFa/b=0.1a/b=0.24±0.114.0633.5218.409.046.918.838±0.113.6928.8712.7813.457.926.8112±0.19.8522.607.6617.1711.218.8916±0.17.0718.733.9413.2111.965.1920±0.14.8015.211.307.9412.611.9024±0.13.0212.220.854.3912.750.55a/b=0.25a/b=0.34±0.122.7344.9639.5859.925.3031.728±0.111.7018.9017.1642.731.9319.0712±0.19.948.9010.5526.336.058.6116±0.18.792.764.4818.455.604.5220±0.17.651.240.9714.037.031.7924±0.16.963.601.0911.188.500.20Figure14 shows the RE of the SIF of cracks with different a/b values obtained by the KF when the loading is completed. The RE does not exceed 1.1%.Figure 14
RE of the SIF obtained by the KF when the loading is complete.


### 5.3. Central Cracks under Nonuniform Loads

In practice, the load borne by a structure may be nonuniform. Therefore, the validity of the calculation method for the SIF based on the KF for specimens under nonuniform loads is verified by experiments. The stress of the plate in this condition is shown in Figure15. Experimental studies were carried out on the specimens with a/b=0.1,0.2,and0.25.Figure 15
Diagram of the forces acting on the plate.Figure16 shows the RE during the whole loading process. The RE at the completion of the iteration is expressed in Table 9, and the KF-based method can still obtain a very accurate result even though the error obtained by the SSG and CMOD methods is large. The error does not exceed 1.2%.Figure 16
RE of the SIF calculated by three methods during the loading process.Table 9
RE of the results obtained by the three methods when the loading force is approximately 24 kN.
Loading force (kN)RE (%)a/b=0.1a/b=0.2a/b=0.25SSGCMODKFSSGCMODKFSSGCMODKF24±0.19.2111.601.1824.026.560.5328.2916.310.96


## 5.1. Experimental Device

The specimen used in the experiment is a steel plate with a prefabricated crack, the modulus of elasticity is 206 GPa, and Poisson’s ratio is 0.3. The width and length of the specimen are 200 mm and 300 mm. The test device is shown in Figure13. The two ends of the plate are installed on the testing machine using T-shaped adapters. The position and direction of the strain gauge paste refer to the results calculated in Section 4.1. The loading method is hierarchical loading and the loading speed is 100 N/s. The load is maintained for 10 s every 4 kN, and the maximum loading force is 24 kN. The radial lengths of the strain gauges in the specimens are shown in Table 6, which are within the effective range calculated in Section 4.1.Figure 13
Diagram of experimental device.Table 6
Radial length of the specimens.
a/br (mm)0.104.000.2012.000.2512.000.3012.00


## 5.2. Central Crack under Uniform Loads

The strain value and maximum crack opening displacement during the loading process were recorded, and the SIF obtained by the two methods was calculated. The empirical value is used as the real value and compared with the experimental results, whose calculation refers to equation as follows:(31)KI=σπaFab,where Fa/b is the boundary correction coefficient, which is calculated using the empirical formula given by Tada et al. [44] as follows:(32)Fab=1+0.128ab−0.288ab2+1.525ab3.The value of SIF during the loading process with differenta/b is calculated and listed in Table 7. An approximate range of the force value in the holding phase is selected due to the fluctuation in the force value during the loading process.Table 7
SIF curve of cracks with different lengths during loading.
Loading force (kN)SIFMPa·mm0.5a/b=0.1a/b=0.2a/b=0.25a/b=0.3True valueKFTrue valueKFTrue valueKFTrue valueKF4±0.138.40631.33854.23849.44861.25837.01369.36847.3638±0.175.43765.794108.401101.015122.549101.524137.956111.63812±0.1112.381103.773162.757148.281183.944164.540201.856184.47216±0.1150.168144.244216.870205.604245.308234.317271.659259.36920±0.1187.834185.400271.245266.088306.450303.470339.939333.86524±0.1224.312226.208325.216326.992367.691371.713406.255407.069The SIF obtained by the SSG, CMOD, and KF is compared with the true value, and the RE is calculated according to equation.(33)RE%=SIFi−SIFeSIFe×100%,where SIFi is the result obtained by the SSG, CMOD, and KF, and SIFe is the true value. The RE of cracks with different a/b during the entire loading process is listed in Table 8. As the number of iterations increases, the error obtained by KF gradually decreases and is lower than that obtained by the other two methods. However, in the middle stage of the iteration, accurate results may be obtained using the CMOD-based and SSG-based methods. When the error increases, KF processing can still keep the error within a lower range, that is, for the KF-based method, a sufficient number of iterations is necessary to obtain accurate results.Table 8
RE of the SIF of different specimens under varying loading forces.
Loading force (kN)RE (%)SSGCMODKFSSGCMODKFa/b=0.1a/b=0.24±0.114.0633.5218.409.046.918.838±0.113.6928.8712.7813.457.926.8112±0.19.8522.607.6617.1711.218.8916±0.17.0718.733.9413.2111.965.1920±0.14.8015.211.307.9412.611.9024±0.13.0212.220.854.3912.750.55a/b=0.25a/b=0.34±0.122.7344.9639.5859.925.3031.728±0.111.7018.9017.1642.731.9319.0712±0.19.948.9010.5526.336.058.6116±0.18.792.764.4818.455.604.5220±0.17.651.240.9714.037.031.7924±0.16.963.601.0911.188.500.20Figure14 shows the RE of the SIF of cracks with different a/b values obtained by the KF when the loading is completed. The RE does not exceed 1.1%.Figure 14
RE of the SIF obtained by the KF when the loading is complete.


## 5.3. Central Cracks under Nonuniform Loads

In practice, the load borne by a structure may be nonuniform. Therefore, the validity of the calculation method for the SIF based on the KF for specimens under nonuniform loads is verified by experiments. The stress of the plate in this condition is shown in Figure15. Experimental studies were carried out on the specimens with a/b=0.1,0.2,and0.25.Figure 15
Diagram of the forces acting on the plate.Figure16 shows the RE during the whole loading process. The RE at the completion of the iteration is expressed in Table 9, and the KF-based method can still obtain a very accurate result even though the error obtained by the SSG and CMOD methods is large. The error does not exceed 1.2%.Figure 16
RE of the SIF calculated by three methods during the loading process.Table 9
RE of the results obtained by the three methods when the loading force is approximately 24 kN.
Loading force (kN)RE (%)a/b=0.1a/b=0.2a/b=0.25SSGCMODKFSSGCMODKFSSGCMODKF24±0.19.2111.601.1824.026.560.5328.2916.310.96


## 6. Prospects

The work of this paper was carried out in view of cracks on a floating offshore platform in the South China Sea. Cracks were observed in the living building of the platform shortly after it was put into operation. As an indicator of crack severity, the SIF can be used in the calculation of fatigue strength and crack propagation of cracked structures; therefore, it is necessary to calculate the SIF of the crack tip of marine structures. Fatigue is one of the most important failure modes for marine structures; therefore, the proposed method will be applied in the evaluation of fatigue life assessment methods in future research.A more accurate fatigue crack propagation model can be established using the data fusion-based method proposed in this paper during the propagation process. However, the specific parameters need to be determined and verified by subsequent numerical simulations and experiments.


## 7. Conclusion

Based on the single strain gauge and maximum opening displacement, a novel calculation method of the SIF based on a Kalman filter is proposed in this paper. Numerical simulations and experimental verifications were carried out on specimens with precracks in different configurations. The following conclusions are drawn:(1)
It is necessary to calculate the effective radial range. Using the condition ofa/b=0.4 as an example, the effective radial length ranges from 3.81 to 21.4 mm. Radial lengths of 12.58 mm and 23.05 mm were used for calculation, and the results were more accurate when r = 12.58 mm.(2)
Simulation and experimental results show that the KF-based method will yield more accurate results as the number of iterations increases. For the simulation, the number of iteration steps is 100, while for the experiment, the number of iteration steps varies due to the different collection times. Sufficient sample size is necessary to obtain accurate SIF values.(3)
The method is effective for calculating the SIF of specimens with prefabricated cracks of different lengths, and the RE does not exceed 1.1% when the iteration is completed.(4)
The proposed method is also applicable to central cracks under nonuniform loads, and the RE of the results is not greater than 1.2%.



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*Source: 2743309-2023-02-09.xml*