# Fatigue Crack Propagation and Life Analysis of Stud Connectors in Steel-Concrete Composite Structures

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Fatigue Propagation Analysis of Stud Connectors

#### 2.1. Analysis Method of LEFM

#### 2.2. Finite Element Modeling

^{5}MPa. The plastic damage model is adopted for the material properties of concrete (C50), and the parameters and calculation formulas of uniaxial tension and compression are given in the literature [27]. The uniaxial tension parameters include the parameter value α

_{t}of concrete uniaxial tensile stress–strain curve drop pair. The representative value f

_{t}

_{,r}of uniaxial tensile strength of concrete; The peak tensile strain ε

_{t}

_{,r}of concrete corresponds to the representative value of uniaxial tensile strength. The uniaxial compression parameters include elastic modulus E

_{c}of concrete, parameter value α

_{c}of the descending section of the stress–strain curve of concrete under uniaxial compression, and representative value f

_{c}

_{,r}of uniaxial compressive strength of concrete. The peak tensile strain ε

_{c}

_{,r}of concrete corresponds to the representative value of uniaxial compressive strength. Table 1 shows the values of the above parameters.

#### 2.3. Analysis of SIFs

_{eff}) is used to evaluate the SIFs of complex fatigue crack, which is calculated by Equation (1), where the v represents Poisson’s ratio and was taken as 0.3. Figure 4 shows the distribution of SIFs with an a/c ratio of 0.2 and 1, respectively. Two characteristics are presented. One is the convex distribution with the maximum value at the center of the crack tip as shown in Figure 4a, and the other is the concave distribution with the maximum value at the two endpoints as shown in Figure 4b. It indicates that the driving force of fatigue crack propagation at the crack tip is different depending on the initial defect.

_{eff}combining the stud length, diameter, and crack shape ratio is shown in Figure 5, which are compared to the values of K

_{eff}/K

_{eff}

_{,max}, the K

_{eff}

_{,max}represents the extreme value in each set of data. It can be seen from the figure that the K

_{eff}at the midpoint of the crack front is larger than the endpoint value for a crack with a small shape (long and narrow crack). The influence of crack shape ratio on different sizes of the stud is obvious, and the K

_{eff}of the initial crack is almost not affected by the length of the stud.

#### 2.4. Analysis of Fatigue Crack Propagation

_{0}in LEFM theory should not be less than 0.1 mm. By defining the fatigue crack as the most unfavorable case with a = c = 0.5 mm, Zhang et al. [29] analyzed the fatigue crack propagation in the double-sided welding details of steel bridge decks effectively. In this paper, this typical shape of the initial crack was introduced for calculation. In addition, other parameter settings of fracture mechanics in Zhang’s study were referred to. The fatigue crack propagation is determined by the threshold of SIFs (ΔK

_{th}). The crack propagates when the amplitude of effective SIF is greater than the threshold (ΔK > ΔK

_{th}). Otherwise, the expansion stops, and the threshold was taken as 63 MPa·mm

^{1/2}. The critical value of SIFs is the fracture toughness (K

_{IC}) of the material. The crack propagation is in the stage of rapid expansion when the maximum value of SIF (K

_{max}) is close to the K

_{IC}, and until the K

_{max}at the crack tip reaches the fracture toughness, and the material appears unstable fracture. The Paris formula is usually used to describe the law of fatigue crack propagation. However, there is a lack of data related to the fatigue properties of the stud material. C and m are both material parameters characterizing crack propagation characteristics. Referring to the study performed by Xu et al. [30], C and m were set to 4.74 × 10

^{−14}MPa·mm

^{1/2}and 3, respectively.

_{I}always plays a dominant role in the extended process, while the contribution of the other two SIFs types to the K

_{eff}amplitude is almost negligible. The K

_{I}is changing rapidly at the crack propagation initial stage, and as the crack expands the SIFs of the open type are stable. In the end, K

_{I}tends to decrease at the shear capacity critical position of the stud. Similarly for the model in which cracks were assumed to propagate along the plane, the K

_{I}is dominant in the early stage of crack propagation and tends to be stable. However, when the midpoint of the fatigue crack tip extended to the center of the stud section, the K

_{II}began to surpass the K

_{I}and the increase rate was fast. Meanwhile, the amplitude of the K

_{eff}also kept increasing gradually.

_{III}and ΔK

_{II}to ΔK

_{I}with the change of crack propagation depth is summarized in Figure 8a. It directly shows the dominant relationship of complex SIFs with different propagation models and initial crack sizes, and the case where a

_{0}= 0.2 and a

_{0}/c

_{0}= 1.0 was also calculated. As can be seen from the figure that when the expansion depth is close to the size of the stud radius, the ΔK

_{II}/ΔK

_{I}both exceeds 1.0 under the plane expansion model in the case of a

_{0}= 0.2 and a

_{0}= 0.5. The effective SIF amplitudes of the two different extended models were compared, as shown in Figure 8b. It is found that the ΔK

_{eff}along the depth direction differs little in the early stage of fatigue crack propagation. However, the ΔK

_{eff}of the 2D model increases gradually when the crack propagation depth is greater than the radius of the stud, which could lead to a certain difference in the calculation of fatigue life values of the assumed propagation model.

_{eff}

_{,diff}) between the two propagation models with a

_{0}= 0.5 mm and the curve of crack depth versus the number of cycles is presented in Figure 9. Referring to Figure 9a, the ΔK

_{eff}of the mixed-type crack propagation model is larger than that of the I-type crack propagation model before the fatigue crack depth reaches the stud radius. After that, the situation was reversed. In the final form of the extension, the ΔK

_{eff}of the I-type crack propagation model is 32.2% higher than the mixed-type crack propagation model. The fatigue life can be obtained by accumulative action times of each crack propagation step with crack depth, as shown in Figure 9b. It can be found that the calculation result of the I-type crack propagation model is 7.2% higher than that of the mixed-type crack propagation model. The fatigue life of the model with an initial crack depth of 0.2 is 3.14% higher than that of the model with an initial crack depth of 0.5 under the same I-type crack propagation model. From the above analysis, it can be concluded that both the initial crack and the fatigue propagation mode have a significant influence on the fatigue life of the stud connector.

## 3. Simplified Calculation Method for Fatigue Life of Stud Connectors

_{f}) of the stud in this research was obtained by calculating the sum of the initiation life (N

_{stage I}) and the stable propagation life (N

_{stage II}) of crack. The Smith–Watson–Topper (SWT) critical plane damage method [33] for multiaxial fatigue was applied to calculate the crack initiation life N

_{stage I}, and the stable extension life N

_{stage II}was calculated by the LEFM method. The flow chart of stud fatigue life assessment is presented in Figure 10.

_{stage I}of the stud. The SWT parameters are used to analyze multiaxial stress problems as shown in Equation (3). The product in Equation (3) is a characterization of strain energy which is considered that the critical plane direction is the direction in which the SWT parameter value of a point reaches its maximum. In general, for the critical plane method, the position of the critical plane is determined by the stress and strain parameters of the integral point of the element, it is usually located in the stress–strain concentrated area, for members subjected to complex stresses, and the fatigue crack propagates in the most unfavorable plane of the element.

_{max}represent the normal strain amplitude and maximum normal stress of each fatigue load cycle, respectively, and at the critical plane, the damage parameter reaches its maximum value.

_{x}, n

_{y}, and n

_{z}represent the cosine of the angle between the post-transformation coordinate system and the pre-transformation coordinate system, respectively. The calculation formula is as follows. Firstly, the stress and strain components in Equation (5) were extracted from the calculation results of the finite element model, and the angle parameters θ and φ in Equation (6) were successively evaluated from 0° to 180° which was separated by 10°. Then, the direction cosine value was calculated through Equation (6) and substituted into Equation (5). Finally, the normal stress and strain after coordinate transformation were obtained, which were used for the subsequent calculation of SWT parameters and fatigue life.

_{stage II}) of the stud. Based on the above finite element calculation of fracture mechanics, the SIF was simplified and obtained by Equation (5), it was assumed that a linear correlation with the fatigue crack depth, and then the fatigue life was calculated through Equation (6). The values of C and m were taken as 4.74 × 10

^{−14}and 3.0, respectively [32]. The initial crack a

_{0}is generally obtained by non-destructive testing, and the initial crack a

_{0}of the stud was set as 2 mm in this study.

_{0}represents the initial fatigue crack depth, and a

_{f}represents the crack depth under fatigue failure.

_{f}corresponding to the upper limit of the fatigue load amplitude P

_{max}(Equation (9)). Therefore, the fatigue crack depth a

_{f}in fatigue failure can be calculated by Equation (10), and f

_{u}is the ultimate tensile strength of the stud material.

## 4. Analysis of Calculation Examples

#### 4.1. Model Establishment

^{5}MPa. The yield strength and ultimate strength of stud and I-shaped steels were 345 MPa and 430 MPa, respectively, the yield strength and ultimate strength of steel were 335 MPa and 400 MPa, respectively, and the Poisson’s ratio of steel was set to 0.3. The compressive strength of the UHPC was taken as 129.1 MPa, and the initial elastic modulus was set to 42.6 GPa. It is similar to the finite element model for fatigue crack progradation, the example model consists of a UHPC plate, short stud, I-shaped steel plate, and structural steel bar. The modeling method was consistent with the previous model and not described here, the difference lies in the definition of constitutive relations for UHPC materials, which was explained here. The tension and compression stress–strain curves of UHPC were referred to in Figure 12. The specific parameters were set according to the experimental results in the literature [35,36,37], and the calculation formula is shown in Figure 12. Based on the principle of energy equivalence, the damage factor D of UHPC under tension and compression was defined by Equation (11). The parameter meanings of the formula in the Figure 12 are as follows. Ε

_{t}

_{0}represents strain at peak tension, f

_{t}represents average stress during strain hardening, ε

_{tp}represents ultimate strain under tension, l

_{c}represents the extension distance measured by the specimen, w

_{p}represents crack width parameter, p represents the parameters obtained by axial tension test fitting, f

_{c}represents the compressive strength, $\xi $ represents the ratio of compressive strain value to compressive peak strain, and ε

_{0}represents peak strain in compression.

#### 4.2. Verification of Load-Slip Curves

#### 4.3. Calculation of SWT Parameters

_{max}is 166.8 MPa, and the normal stress amplitude is 135.1 MPa.

#### 4.4. Calculation of Fatigue Life

_{stage I}) and the stable propagation life (N

_{stage II}) of crack can be calculated by Equations (4) and (8). As shown in Table 2, the fatigue test results (N

_{e}) and calculation results (N

_{f}) of specimens N1~N5 are summarized, the relationship between stress amplitude and fatigue life of stud is usually expressed by logarithmic relation. The error between logN

_{e}and logN

_{f}ranges from 0.5% to 7.7%, and the results of logN

_{e}/logN

_{f}range from 0.98 to 1.04, which shows that the calculated results have a good correlation with the experimental data. By further calculation, the correlation coefficient between the calculated results in this paper with the experimental results is 0.9955. The above analysis shows that the method of predicting the fatigue life by considering the fatigue initiation life and stable extension life of the stud is feasible. Among all the specimens, the predicted results of N1 were closest to the experimental values.

## 5. Conclusions and Observations

- (a)
- It belongs to the complex crack of the stud connector which is dominated by the open type SIF. The distribution of SIFs is different under the different crack shapes, and the extreme value of K
_{eff}is greatly affected by the depth and width of the crack, but not significantly affected by the length of the stud. - (b)
- The fatigue crack surface of the stud connector can be obtained by three-dimensional fatigue propagation. It is semi-elliptical in the early stage of the fatigue crack propagation, and the crack front gradually develops into a straight line in the later stage. The midpoint of the fatigue crack front grows faster, and the fatigue crack surface tends to incline toward the I-beam.
- (c)
- There is little difference between the ΔK
_{eff}of the I-type crack propagation and that of the mixed-type crack propagation in the early stage of fatigue propagation. However, the ΔK_{eff}of the I-type crack propagation model increases gradually after the crack propagates through half of the stud section, which leads to a certain difference in the number of effects calculated by the two methods in the late fatigue propagation period. The fatigue life of the I-type crack propagation model is 7.2% higher than that of the mixed-type crack propagation model. - (d)
- The calculation method of crack initiation life is considered by simplifying SIF and combining the SWT critical damage plane method. Compared with the experimental values, it is proved that the calculated values of fatigue life of stud connectors provide better predictive values. In addition to the stress amplitude of the traditional S-N curve, different material sizes, material properties, and contact characteristics of structures can also be taken into account by the calculation method in this paper combined with finite element modeling. It can effectively evaluate the fatigue life of stud connectors in steel-concrete composite structures.
- (e)
- The proportion of crack initiation life in the fatigue life of stud connectors in UHPC is different under different stress amplitudes, and the compressive strength of UHPC also affects the fatigue life of stud connectors. In the whole process of fatigue failure, the initiation life of fatigue crack accounts for a large proportion, up to more than 90%. The fatigue stress amplitude is the key parameter to determine the fatigue crack propagation life of stud connectors.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Comparison of K

_{eff}/K

_{eff}

_{,max}under different sizes of stud and crack. (

**a**) Ratio of stud height and crack shape; (

**b**) ratio of stud diameter and crack shape.

**Figure 7.**Crack propagation length-SIFs curve. (

**a**) MTS extended model (3D); (

**b**) planar extended model (2D).

**Figure 9.**The difference between mixed-type crack propagation and I-type crack propagation. (

**a**) ΔK

_{eff}

_{,diff}; (

**b**) crack depth versus the number of cycles.

**Figure 12.**Stress–strain curve of UHPC material. (

**a**) Tensile stress–strain curve; (

**b**) compression stress–strain curve.

Strength Grade | α_{t} | f_{t}_{,r} | ε_{t}_{,r/}10^{−4} | α_{c} | f_{c}_{,r} | ε_{c}_{,r}/10^{−3} |
---|---|---|---|---|---|---|

C50 | 1.95 | 2.5 | 1.07 | 2.48 | 50 | 1.92 |

Number | F_{max} | F_{min} | ΔF | τ_{max} | τ_{min} | Δτ | N_{e} | N_{stage I} | N_{stage II} | N_{f} | logN_{e}/logN_{f} |
---|---|---|---|---|---|---|---|---|---|---|---|

N1 ^{1} | 122 | 22 | 100 | 115 | 21 | 94 | 11,787,000 | 9,883,700 | 1,299,136 | 11,182,836 | 1.00 |

N2 | 151 | 27 | 124 | 143 | 26 | 117 | 1,130,000 | 1,206,000 | 398,725 | 1,604,725 | 0.98 |

N3 | 162 | 29 | 133 | 152 | 27 | 125 | 1,688,000 | 600,000 | 392,224 | 992,224 | 1.04 |

N4 | 175 | 31 | 143 | 165 | 30 | 135 | 441,000 | 243,000 | 224,643 | 467,643 | 0.99 |

N5 | 175 | 21 | 154 | 165 | 20 | 145 | 620,000 | 221,000 | 185,308 | 406,308 | 1.03 |

N6 ^{1} | 122 | 22 | 100 | 115 | 21 | 94 | / | 12,590,000 | 892,000 | 13,482,000 | / |

^{1}N1 and N6 were consistent except that the material parameters of UHPC were set differently.

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## Share and Cite

**MDPI and ACS Style**

Wang, D.; Tan, B.; Xiang, S.; Wang, X.
Fatigue Crack Propagation and Life Analysis of Stud Connectors in Steel-Concrete Composite Structures. *Sustainability* **2022**, *14*, 7253.
https://doi.org/10.3390/su14127253

**AMA Style**

Wang D, Tan B, Xiang S, Wang X.
Fatigue Crack Propagation and Life Analysis of Stud Connectors in Steel-Concrete Composite Structures. *Sustainability*. 2022; 14(12):7253.
https://doi.org/10.3390/su14127253

**Chicago/Turabian Style**

Wang, Da, Benkun Tan, Shengtao Xiang, and Xie Wang.
2022. "Fatigue Crack Propagation and Life Analysis of Stud Connectors in Steel-Concrete Composite Structures" *Sustainability* 14, no. 12: 7253.
https://doi.org/10.3390/su14127253