# Physics-Based Shear-Strength Degradation Model of Stud Connector with the Fatigue Cumulative Damage

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Data Collection

- (1)
- the static strength, P(0), of the stud connector specimen was obtained by the static pushout test;
- (2)
- the loading ratio (=P
_{max}/P(0)), a random variable, was used to determine the maximum value of the fatigue load P_{max}; - (3)
- the fatigue life (N) of the stud connector specimen under a certain loading level was got by the fatigue test;
- (4)
- the intact stud connector was loaded to different cycles n (such as 1 × 10
^{4}, 5 × 10^{4}, 10 × 10^{4}and 25 × 10^{4}) under the same loading level that corresponds to the fatigue life N, in which, n/N was defined as the cycle ratios. Upon completing the loading process, there will be pre-fatigued damage existing in the stud connector, which was called a pre-fatigued specimen. - (5)
- finally, the static pushout test was performed to obtain the residual strength of the pre-fatigued stud connectors.

_{max}denotes the amplitude of cyclic load, n ≤ N is the loading times and P(n) is defined as the residual strength when a given stress level attains a certain loading time. Moreover, n/N = 0, P(n) represents the strength of the undamaged stud connector specimen; n/N = 1, P(n) denotes the fatigue strength of the specimen. In Table 1, a variable k = d/d

_{max}is introduced to normalize the specimen size in the collected data, which can account for the impacts of different specimen sizes of a stud on the proposed degradation model. d

_{max}denotes the maximum diameter of a stud in the collected experimental data. In this study, 26 experimental data have been collected from publications.

## 3. Traditional Strength Degradation Models

_{max}denotes the maximum value of the fatigue loads or fatigue stresses; S(n)/S(0) is defined as the strength degradation coefficient, namely, the ratio between the residual strength and the strength of the intact material. As indicated in the Chinese specification [21], the strength degradation coefficient of stud connectors is the same as the degradation coefficient of the steel material for fabricating the stud. In the following text, the strength degradation coefficient of stud connectors is directly written as P(n)/P(0) = S(n)/S(0). Li [23] proposed a strength degradation model with respect to the loading ratio and cycle ratio, which is applied to depict the strength decay attenuation law of 45# steel under the action of high-cycle fatigue cumulative damage, that is

_{max}is the maximum value of the fatigue load; P

_{max}/P(0) denotes the loading ratio; and n/N is the loading cycle ratios.

## 4. Physics-Based Shear-Strength Degradation Model

**X**represents variables in the deterministic term, e.g., the loading ratio and cycle ratio, and

**θ**

_{m}is a vector of coefficients in the deterministic term; the second term γ is the error correction term, θ

_{h,i}is the coefficient in the correction term, and h

_{i}denotes the normalized variables that can impact the residual strength of stud connectors under fatigue loads, e.g., material type, specimen size, and loading mechanism; N

_{h}is the size of variables in the correction term; the unknown model parameters

**Θ**= (

**θ**, θ

_{m}_{h,1}, θ

_{h,2}, ∙∙∙, θ

_{h,i}, σ); σε denotes the model error following a normal distribution with a mean value of 0 and a standard deviation of σ. To capture a potential bias in the model that is independent of the variables

**X**, h

_{1}is set to 1 [25]. Applying the Bayesian updating rule [25,26], the posterior probability density function (PDF) of

**Θ**, f(

**Θ**), is written as [27]

**Θ**)p(

**Θ**)d

**Θ**]

^{−1}is used to guarantee the integration of f(

**Θ**) equaling 1; L(

**Θ**) = ∏f(Data|

**Θ**) denotes the likelihood function, and f(Data|

**Θ**) is the PDF of the observations given

**Θ**; p(

**Θ**) is the prior PDF of

**Θ**. When only one variable is in the model, the prior PDF of

**Θ**can be written as

**Θ**, given by [25]

**R**= [ρ

_{ij}] denotes the correlation matrix among

**Θ**; n is the number of variables in the model.

_{1}and θ

_{2}of the correction term, conversely, Figure 1b,c illustrate the unknown parameter of the deterministic term and θ

_{2}in the correction term. It can be observed that the posterior estimates of the unknown parameters in the shear-strength degradation models are centered around their mean values, and there is a large variation in the posterior estimates. Thereby, uncertainties associated with the unknown model parameters should be carefully considered in the service reliability assessment of the fatigued bridges. In addition, as shown in Figure 1c, the mean values of the unknown parameters in Model III are contradicting the actual situation. The mean value of θ

_{m}in the deterministic term of Model III is a negative value indicating a rising trend in the shear strength of the connectors as the loading ratio increases. Thus, Model I and Model II will be used in the following analysis. The posterior statistics of the unknown model parameters in the shear-strength degradation models are listed in Table 2.

_{max}) is not considered in the traditional strength degradation model, which is one of the main defects of the traditional model compared to this proposed physics-based strength degradation model of the fatigued stud connectors.

## 5. Analytical Derivation for Service Reliability of Composite Girder

_{y,i}, f

_{u,i}and E

_{i}denote the yield strength, ultimate strength, and Young’s modulus of different types of steel materials, respectively. Moreover, f

_{c}and E

_{c}are the compressive strength and Young’s modulus of concrete. By the internal force analysis, the beam’s inner force at the i-th section corresponding to the i-th stud is shown in Figure 6. It can be observed that the axial forces and moments at the reinforced concrete deck and steel girder meet the following Equations (8) and (9).

_{c}) of the reinforced concrete deck plus the height (h

_{a}) of the steel girder. To compute the beam inner force, the strain at the i-th section of the reinforced concrete deck and steel girder is calculated by

_{1}= ε

_{2}), the different layers of the composite beams have the same axial elongation, which is expressed as

_{c}(x)/E

_{c}I

_{c}= M

_{a}(x)/E

_{a}I

_{a}), combined with Equation (9), the moments at the i-th section of the different layers are calculated as

_{c}and I

_{c}are the elastic modulus and section moment of inertia for the reinforced concrete deck, respectively. Conversely, E

_{a}and I

_{a}are the elastic modulus and section moment of inertia for an I-shape steel girder, respectively.

_{c}and A

_{a}are the cross-section area of the reinforced concrete deck and steel girder, respectively. Uniform q includes the design dead load q

_{1}and live load q

_{2}, and for the highway of Class I, q

_{2}is set to 10.5 kN/m [38]. Similarly, the axial force at the (i − 1)-th section can be expressed as

_{s,d}= 0.43ηA

_{s}√f

_{c}E

_{c}= 148.0 kN, in which, η (=0.016Δl/d + 0.8) denotes the reduction factor of the group stud effect [39], and A

_{s}is the cross-area of a stud. In contrast, when the failure is due to the cut of the studs, the design force is computed as V

_{s,d}= 0.7A

_{s}f

_{u}= 115.4 kN [38]. Therefore, the failure mode of the stud connector is controlled by the stud cutting. For the intact stud connector, the maxima of the mean shear force at the fixed end of this beam, which is the most dangerous section, is computed as 93.7 kN < 115.4 kN. In other words, the composite beam is safe without considering the shear-strength degradation of stud connectors. To investigate the impacts of fatigue-induced strength degradation, the Monte Carlo simulation is used to calculate the failure probability of stud connectors. The elastic modulus of the concrete and steel, as well as the ultimate strength and diameter of the stud, are treated as random variables.

## 6. Conclusions

- (1)
- There is a large variation in the traditional strength degradation model under the fatigue load, and the epistemic uncertainty in the unknown model parameters should be carefully considered;
- (2)
- For the same test results, there are significant differences among various strength degradation models and lack of necessary mathematical and physical background. The proposed physics-based degradation model can well fill up this shortcoming and consider the effects of various variables, such as the specimen size and loading mechanism;
- (3)
- Considering the shear-strength degradation of stud connectors, the composite beam may fail under the combined action of the self-weight and the design live load, which should be accounted for in the structural design phase.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Saleh, S.M.; Majeed, F.H. Shear Strength of Headed Stud Connectors in Self-Compacting Concrete with Recycled Coarse Aggregate. Buildings
**2022**, 12, 505. [Google Scholar] [CrossRef] - Fang, Z.; Fang, S.; Liu, F. Experimental and Numerical Study on the Shear Performance of Short Stud Shear Connectors in Steel-UHPC Composite Beams. Buildings
**2022**, 12, 418. [Google Scholar] [CrossRef] - Wang, B.; Huang, Q.; Liu, X.; Ding, Y. Study on stiffness deterioration in steel-concrete composite beams under fatigue loading. Steel Compos. Struct.
**2020**, 34, 499–509. [Google Scholar] [CrossRef] - Qin, X.; Yang, G.T. Elastic stiffness of stud connection in composite structures. Steel Compos. Struct.
**2021**, 39, 419–433. [Google Scholar] [CrossRef] - Xu, X.; Liu, Y. Analytical prediction of the deformation behavior of headed studs in monotonic push-out tests. Adv. Struct. Eng.
**2019**, 22, 1711–1726. [Google Scholar] [CrossRef] - Xu, C.; Sugiura, K.; Su, Q. Fatigue Behavior of the Group Stud Shear Connectors in Steel-Concrete Composite Bridges. J. Bridge Eng.
**2018**, 23, 04018055. [Google Scholar] [CrossRef] - Hanswille, G.; Porsch, M.; Ustundag, C. Resistance of headed studs subjected to fatigue loading—Part II: Analytical study. J. Constr. Steel Res.
**2007**, 63, 485–493. [Google Scholar] [CrossRef] - Bro, M.; Wilde, M. Influence of Fatigue on Headed Stud Connectors in Composite Bridges. Master’s Thesis, Lulea University of Technology Sweden, Luleå, Sweden, 2004. [Google Scholar]
- Veljkovic, M.; Johansson, B. Residual static resistance of welded stud shear connectors. In Proceedings of the 5th International Conference on Composite Construction in Steel and Concrete V, Berg-En-Dal, South Africa, 18–23 July 2004; p. 524. [Google Scholar]
- Feldmann, M.; Hechler, O.; Hegger, J.; Rauscher, S. New investigations on the fatigue behavior of composite beams made of high strength materials with two different kinds of shear connection. Stahlbau
**2007**, 76, 826–844. [Google Scholar] [CrossRef] - Kim, S.H.; Kim, K.S.; Park, S.; Aim, J.H.; Lee, M.K. Y-type perfobond rib shear connectors subjected to fatigue loading on highway bridges. J. Constr. Steel Res.
**2016**, 122, 445–454. [Google Scholar] [CrossRef] - Alsharari, F.; El-Sisi, A.E.D.; Mutnbak, M.; Salim, H.; El-Zohairy, A. Effect of the Progressive Failure of Shear Connectors on the Behavior of Steel-Reinforced Concrete Composite Girders. Buildings
**2022**, 12, 596. [Google Scholar] [CrossRef] - Wang, B.; Liu, X.L.; Zhuge, P. Residual bearing capacity of steel-concrete composite beams under fatigue loading. Struct. Eng. Mech.
**2021**, 77, 559–569. [Google Scholar] [CrossRef] - Fang, L.; Zhou, Y.; Jiang, Y.; Pei, Y.; Yi, W. Vibration-Based Damage Detection of a Steel-Concrete Composite Slab Using Non-Model-Based and Model-Based Methods. Adv. Civ. Eng.
**2020**, 2020, 8889277. [Google Scholar] [CrossRef] - Pranno, A.; Greco, F.; Lonetti, P.; Luciano, R.; De Maio, U. An improved fracture approach to investigate the degradation of vibration characteristics for reinforced concrete beams under progressive damage. Int. J. Fatigue
**2022**, 163, 107032. [Google Scholar] [CrossRef] - Philippidis, T.P.; Passipoularidis, V.A. Residual strength after fatigue in composites: Theory vs. experiment. Int. J. Fatigue
**2007**, 29, 2104–2116. [Google Scholar] [CrossRef] - Wang, B.; Huang, Q.; Liu, X.L. Deterioration in strength of studs based on two-parameter fatigue failure criterion. Steel Compos. Struct.
**2017**, 23, 239–250. [Google Scholar] [CrossRef] - He, J.; Lin, Z.; Liu, Y.; Xu, X.; Xin, H.; Wang, S. Shear stiffness of headed studs on structural behaviors of steel-concrete composite girders. Steel Compos. Struct.
**2020**, 36, 553–568. [Google Scholar] [CrossRef] - Oehlers, D.J. Deterioration in strength of stud connectors in composite bridge beams. J. Struct. Eng.
**1990**, 116, 3417–3431. [Google Scholar] [CrossRef] - Ahn, J.-H.; Kim, S.-H.; Jeong, Y.-J. Fatigue experiment of stud welded on steel plate for a new bridge deck system. Steel Compos. Struct.
**2007**, 7, 391–404. [Google Scholar] [CrossRef] - GB50017-2017; Standard for Design of Steel Structures. China Machine Press: Beijing, China, 2017.
- Broutman, L.J.; Sahu, S. A New Theory to Predict Cumulative Fatigue Damage in Fiberglass Reinforced Plastics. In Composite materials: Testing and Design (Second Conference), Proceedings of the Second Conference on Composite Materials: Testing and Design, Anaheim, PA, USA, 20–22 April 1971; ASTM International: West Conshohocken, PA, USA, 1972. [Google Scholar]
- Li, L. Research on the Law of the Stiffness Degradation and Strength Degradation of the Bolt under Fatigue Loads. Master’s Thesis, Northeastern University, Shenyang, China, 2013. [Google Scholar]
- Zhang, L.; Ji, W.; Zhou, W.; Li, W.-L.; Ren, C.-X. Fatigue cumulative damage models based on strength degradation. Trans. Chin. Soc. Agric. Eng.
**2015**, 31 (Suppl. S1), 47–52. [Google Scholar] - Gardoni, P.; Der Kiureghian, A.; Mosalam, K.M. Probabilistic capacity models and fragility estimates for reinforced concrete columns based on experimental observations. J. Eng. Mech.
**2002**, 128, 1024–1038. [Google Scholar] [CrossRef] - Zheng, X.-W.; Li, H.-N.; Gardoni, P. Probabilistic seismic demand models and life-cycle fragility estimates for high-rise buildings. J. Struct. Eng.
**2021**, 147, 04021216. [Google Scholar] [CrossRef] - Sobieraj, J.; Metelski, D. Quantifying Critical Success Factors (CSFs) in Management of Investment-Construction Projects: Insights from Bayesian Model Averaging. Buildings
**2021**, 11, 360. [Google Scholar] [CrossRef] - Gelman, A.; Carlin, J.B.; Stern, H.S.; Dunson, D.B.; Vehtari, A.; Rubin, D.B. Bayesian Data Analysis, 3rd ed.; Chapman & Hall: New York, NY, USA, 2020. [Google Scholar]
- Papakonstantinou, K.G.; Shinozuka, M. Probabilistic model for steel corrosion in reinforced concrete structures of large dimensions considering crack effects. Eng. Struct.
**2013**, 57, 306–326. [Google Scholar] [CrossRef] - He, Z.Q.; Ou, C.X.; Tian, F.; Liu, Z. Experimental Behavior of Steel-Concrete Composite Girders with UHPC-Grout Strip Shear Connection. Buildings
**2021**, 11, 182. [Google Scholar] [CrossRef] - Zheng, X.-W.; Li, H.-N.; Li, C. Damage probability analysis of a high-rise building against wind excitation with recorded field data and direction effect. J. Wind. Eng. Ind. Aerodyn.
**2019**, 184, 10–22. [Google Scholar] [CrossRef] - Barbato, M.; Gu, Q.; Conte, J.P. Probabilistic Pushover analysis of structural and soil-structure systems. J. Struct. Eng.
**2010**, 136, 1330–1341. [Google Scholar] [CrossRef] [Green Version] - Xu, Q.H.; Shi, D.D.; Shao, W. Service life prediction of RC square piles based on time-varying probability analysis. Constr. Build. Mater.
**2019**, 227, 116824. [Google Scholar] [CrossRef] - Melchers, R.E. Structural Reliability Analysis and Prediction; John Wiley: New York, NY, USA, 1999. [Google Scholar]
- Zheng, X.-W.; Li, H.-N.; Gardoni, P. Life-cycle probabilistic seismic risk assessment of high-rise buildings considering carbonation induced deterioration. Eng. Struct.
**2021**, 231, 111752. [Google Scholar] [CrossRef] - Zheng, X.-W.; Li, H.-N.; Gardoni, P. Reliability-based design approach for high-rise buildings subject to earthquakes and strong winds. Eng. Struct.
**2021**, 244, 112771. [Google Scholar] [CrossRef] - Zheng, X.-W.; Li, H.-N.; Yang, Y.-B.; Li, G.; Huo, L.-S.; Liu, Y. Damage risk assessment of a high-rise building against multihazard of earthquake and strong wind with recorded data. Eng. Struct.
**2019**, 200, 109697. [Google Scholar] [CrossRef] - JTG-D64-2015; Specifications for Design of Highway Steel Bridge. China Communications Press: Beijing, China, 2015.
- GB50917-2013; Code for Design of Steel and Concrete Composite Bridges. China Planning Press: Beijing, China, 2013.

**Figure 1.**The posterior estimates of the unknown model parameters. (

**a**) Model I. (

**b**) Model II. (

**c**) Model III.

**Figure 5.**Schematic diagram of steel–concrete composite girder. (

**a**) Evaluation view. (

**b**) Cross-section profile.

**Figure 7.**Failure probability of this composite bridge beam. (

**a**) Cycle ratio = 0.3. (

**b**) Parameter k = 1.5.

Sources | Variable k | Loading Ratio | Fatigue Life N (×10^{3}) | Loading Cases | |||||
---|---|---|---|---|---|---|---|---|---|

Oehlers [19] | 0.59 | 0.3 | 1379 | n/N | 0.18 | 0.36 | 0.54 | 0.74 | 0.91 |

P(n)/P(0) | 0.85 | 0.80 | 0.74 | 0.55 | 0.49 | ||||

Hanswille et al. [7] | 1.0 | 0.3 | 6400 | n/N | 0.19 | 0.73 | – | – | – |

P(n)/P(0) | 0.59 | 0.6 | – | – | – | ||||

0.44 | 6200 | n/N | 0.32 | 0.70 | – | – | – | ||

P(n)/P(0) | 0.75 | 0.63 | – | – | – | ||||

0.44 | 5100 | n/N | 0.24 | 0.69 | – | – | – | ||

P(n)/P(0) | 0.66 | 0.61 | – | – | – | ||||

0.71 | 3500 | n/N | 0.29 | 0.72 | – | – | – | ||

P(n)/P(0) | 1.0 | 0.86 | – | – | – | ||||

0.71 | 1200 | n/N | 0.32 | 0.7 | – | – | – | ||

P(n)/P(0) | 0.95 | 0.84 | – | – | – | ||||

Wang et al. [17] | 0.59 | 0.6 | 2705 | n/N | 0.19 | 0.37 | 0.56 | 0.75 | 0.93 |

P(n)/P(0) | 0.98 | 0.91 | 0.83 | 0.77 | 0.64 | ||||

Ahn et al. [20] | 0.73 | 0.25 | 2495 | n/N | 0.2 | 0.4 | 0.6 | – | – |

P(n)/P(0) | 0.909 | 0.875 | 0.787 | – | – | ||||

Bro et al. [8] | 1.0 | 0.138 | 4900 | n/N | 0.082 | 0.204 | 0.245 | – | – |

P(n)/P(0) | 0.929 | 0.905 | 0.893 | – | – |

Models | Parameters | Mean | Standard Deviation |
---|---|---|---|

I | θ_{1} | −0.125 | 0.222 |

θ_{2} | 0.067 | 0.122 | |

σ | 0.260 | 0.038 | |

II | θ_{m} | 0.025 | 0.038 |

θ_{1} | −0.248 | 0.136 | |

θ_{2} | 0.025 | 0.074 | |

σ | 0.156 | 0.024 |

Name | Mean (MPa) | COV/% | Distribution | Upper Level | Lower Level | References | |
---|---|---|---|---|---|---|---|

Q345 | f_{y1} | 352 | 5 | Lognormal | 1.1 f_{y1,mean} | 0.9 f_{y1,mean} | Zheng et al. [31] |

f_{u1} | 495 | 5 | Lognormal | 1.1 f_{u1,mean} | 0.9 f_{u1,mean} | Zheng et al. [31] | |

E_{1} | 2.06 × 10^{5} | 3.3 | Lognormal | 1.1 E_{y1,mean} | 0.9 E_{y1,mean} | Barbato et al. [32] | |

C50 | f_{c} | 44.8 | 20 | Lognormal | 1.4 f_{c,mean} | 0.6 f_{c,mean} | Barbato et al. [32] |

E_{c} | 4733√f_{c} | 12 | Normal | 1.2 f_{c,mean} | 0.8 f_{c,mean} | Xu et al. [33] | |

ML-15 | f_{y2} | 442 | 5 | Lognormal | 1.1 f_{y2,mean} | 0.9 f_{y2,mean} | Melchers [34] |

f_{u2} | 525 | 5 | Lognormal | 1.1 f_{u2,mean} | 0.9 f_{u2,mean} | Zheng et al. [35] | |

E_{2} | 2.0 × 10^{5} | 3.3 | Lognormal | 1.1 E_{y2,mean} | 0.9 E_{y2,mean} | Barbato et al. [32] | |

HPB300 | f_{y3} | 300 | 5 | Lognormal | 1.1 f_{y2,mean} | 0.9 f_{y2,mean} | Zheng et al. [36] |

E_{2} | 2.0 × 10^{5} | 3.3 | Lognormal | 1.1 E_{y2 mean} | 0.9 E_{y2,mean} | Zheng et al. [37] |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zheng, X.-W.; Lv, H.-L.; Fan, H.; Zhou, Y.-B.
Physics-Based Shear-Strength Degradation Model of Stud Connector with the Fatigue Cumulative Damage. *Buildings* **2022**, *12*, 2141.
https://doi.org/10.3390/buildings12122141

**AMA Style**

Zheng X-W, Lv H-L, Fan H, Zhou Y-B.
Physics-Based Shear-Strength Degradation Model of Stud Connector with the Fatigue Cumulative Damage. *Buildings*. 2022; 12(12):2141.
https://doi.org/10.3390/buildings12122141

**Chicago/Turabian Style**

Zheng, Xiao-Wei, Heng-Lin Lv, Hong Fan, and Yan-Bing Zhou.
2022. "Physics-Based Shear-Strength Degradation Model of Stud Connector with the Fatigue Cumulative Damage" *Buildings* 12, no. 12: 2141.
https://doi.org/10.3390/buildings12122141