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

Abstract

In the whole lifetime of structures, fatigue damage accumulation will exist in the shear connector of steel–concrete composite beams. It is essential to determine the residual mechanical properties of shear connectors under long-term fatigue loads, e.g., the vehicle load on bridges. In this regard, a shear-strength degradation model is proposed for shear connectors. The Bayes theorem is used to develop posterior estimates of the unknown parameters in the degradation model based on the collected pushout test data of pre-damaged stud connectors caused by high-cycle fatigue loads. In addition, according to the proposed shear-strength degradation model, the service reliability assessment is performed with a composite bridge beam. The results indicate that (1) There is a large diversion in the traditional strength degradation model under the action of fatigue cumulative damage. More importantly, this proposed physics-based degradation model can effectively reduce uncertainty. (2) The effects of steel type and test specimen size can be well considered in the proposed shear-strength degradation model, which is beneficial for improving the reliability of risk assessment for fatigued bridges.

1. Introduction

The steel–concrete composite beams are widely adopted in buildings and infrastructures, in which stud connectors are commonly used to transfer the shear force across the steel–concrete interface [1,2,3,4]. The fatigue damage accumulation may cause shear capacity degradation of stud connectors. Much effort has been devoted to the fatigue behavior of stud connectors in steel–concrete composite beams by the pushout experiments [5,6]; however, only a few studies on investigating the residual mechanical properties of pre-fatigued stud connectors have been presented [7].

The experimental approach is commonly used to investigate the residual strength degradation of materials and components under high-cycle fatigue loads. Bro et al. [8] studied the fatigue life of the intact stud connector and the residual strength of pre-fatigued connectors. The results indicated that shear strength will decrease as the fatigue damage degree goes. Wang et al. [3] investigated the mechanical property degradation law of pre-fatigued stud connectors and indicated that the shear strength of pre-damaged stud connectors will nonlinearly decrease along with the loading ratio (i.e., the ratio between the cyclic stress amplitude and tensile strength of intact material) and the cycle ratio (i.e., the ratio between cycle times under given loading stress and the fatigue life corresponding to the same stress level). Other work also focused on the residual strength of pre-damaged stud connectors under high-cycle fatigue loads [9,10,11,12]. Wang et al. [3,13] investigated the residual bearing capacity of steel–concrete composite beams considering the effect of high-cycle fatigue cumulative damage through experiments and theory analysis. Furthermore, researchers developed vibration-based models to identify the residual strength capacity, damage level, and potential failures of composite steel-reinforced concrete structures (components), such as Fang et al. [14] and Pranno et al. [15].

Significant efforts indicated that the strength degradation model of fatigued components can be expressed as a function of loading ratio, cycle ratio, and other variables [16]. Based on the traditional strength degradation model, Wang et al. [17] presented a modified model with respect to the loading ratio and cycle ratio. However, the uncertainty associated with the unknown model parameters is commonly neglected in the widely used strength degradation model under high-cycle fatigue loads. In addition, the size effect has been verified to have a significant impact on the shear capacity of shear connectors [18], which is commonly ignored in the traditional strength degradation model. In light of these shortcomings, this study develops a physics-based shear-strength degradation model of fatigued stud connectors, considering the effects of fatigue load mechanism and diameter of the stud.

The rest of this paper is organized as: Section 2 presents 26 monotonic pushout tests of pre-fatigued stud connectors collected from previous publications. Section 3 discusses the widely used traditional strength degradation model of materials and components under fatigue loads. In Section 4, a physics-based shear-strength degradation model is proposed for fatigued stud connectors, in which, the Bayesian updating rule is used to develop the posterior probability distribution of the unknown model parameters based on the collected experimental data. In Section 5, this physics-based strength degradation model is applied for the reliability analysis of a representative composite beam under fatigue loads. The main conclusions drawn from this study are summarized in the final section.

2. Experimental Data Collection

The static strength of stud connectors that have been loaded to different cycle ratios under a given stress level was determined by experiments [7,19]. In this study, the coupled pre-fatigue test and static test approach were used to obtain the residual strength of the pre-fatigued stud connector. The testing procedure was conducted in the following steps:

(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 × 10⁴, 5 × 10⁴, 10 × 10⁴ and 25 × 10⁴) 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.

The collected experimental data are summarized in

Table 1. Summarization of experimental data.

Sources	Variable k	Loading Ratio	Fatigue Life N (×103)	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	–	–

, in which, P_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

A Chinese specification [21] indicates that the shear strength of stud connectors is proportional to the strength of the material for fabricating the stud. Thus, the shear-strength degradation model of stud connectors, considering the fatigue cumulative damage, can be determined based on the strength degradation model of the materials [17]. Brountman and Sahu [22] proposed that the residual strength of the materials under fatigue loads can be directly calculated by the loading ratio and cycle ratio, given by

$$\frac{S(n)}{S(0)}=1-(1-\frac{S_{\max }}{S(0)})\frac{n}{N}$$

(1)

where S_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

$$\frac{P(n)}{P(0)}=\frac{P_{\max }}{P(0)}+\left[1-\frac{P_{\max }}{P(0)}\right]{(1-\frac{n}{N})}^{\theta }$$

(2)

in which, P(n) denotes the residual strength of the stud connector after loading n cycles; P(0) is the static strength of the stud connector obtained by the pushout test; P_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.

Referring to the characteristics of the fatigue cumulative damage and the existing strength degradation models, Zhang et al. [24] proposed a modified strength degradation model under high-cycle fatigue loads, written as

$${\left[\frac{P(n)}{P(0)}\right]}^{1/\theta }=1-\left[1-{\left(\frac{P_{\max }}{P(0)}\right)}^{1/\theta }\right]\frac{n}{N}$$

(3)

It should be stressed that the frequency inference (e.g., least square method) is commonly used to determine the unknown coefficients involved in the strength degradation model. In other words, the traditional degradation model is treated as deterministic without considering the epistemic uncertainty in the unknown model parameters. Thus, this paper applies the Bayes theorem to develop the posterior probability distribution of the unknown model parameters, and the uncertainty can be considered in the structural reliability analysis based on the full probability theory.

4. Physics-Based Shear-Strength Degradation Model

As previous discussions, various variables, such as the size of the testing specimen, fatigue load mode, and stress ranges, may have significant impacts on the strength degradation law of pre-damaged stud connectors under fatigue loads. It might cause vast errors in evaluating the residual strength of the pre-fatigued stud connector without the uncertainties associated with these variables, and the probability model of the variables can well quantify their corresponding uncertainties. Referring to the basic concept of the capacity model of the bridge piers for the seismic fragility estimates presented in the study of Gardoni et al. [25], a physics-based shear-strength degradation model of fatigued stud connectors can be expressed as

$$\frac{P(n|\Theta )}{P(0)}=\mathrm{f}(X|{\theta }_{\mathrm{m}})+\gamma +\sigma \epsilon =\mathrm{f}(X|{\theta }_{\mathrm{m}})+{\displaystyle \sum _{i=1}^{N_{\mathrm{h}}}{\mathsf{\theta }}_{\mathrm{h},i}{\mathrm{h}}_i}+\sigma \epsilon$$

(4)

where the first term in the right of this equation is the deterministic term that can be expressed as Equations (1), (2), or (3); Vector 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_m, θ_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, h₁ 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]

$$\mathrm{f}(\Theta )=\mathrm{cL}(\Theta )\mathrm{p}(\Theta )$$

(5)

in which, the normalized coefficient c = [∫L(Θ)p(Θ)dΘ]⁻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

$$\mathrm{p}(\Theta )\propto \frac{1}{\sigma }$$

(6)

For the n-dimensional model, the correlation between the unknown model parameters should be considered in the prior PDF of Θ, given by [25]

$$\mathrm{p}\left(\Theta \right)\propto {\left|R\right|}^{-\left(n+1\right)/2}\prod \frac{1}{\sigma }$$

(7)

where R = [ρ_ij] denotes the correlation matrix among Θ; n is the number of variables in the model.

Herein, only the specimen size is considered in the collected experimental data. Equations (1)–(3) are, respectively, selected as the deterministic term, and this physics-based model is defined as Model I, Model II and Model III, respectively. This paper uses a Markov Chain Monte Carlo (MCMC) sampling [28] to obtain the posterior estimates of the unknown parameters in the strength degradation models, as shown in Figure 1.

Since there are no unknown parameters in the deterministic term of Model I, Figure 1a shows θ₁ and θ₂ of the correction term, conversely, Figure 1b,c illustrate the unknown parameter of the deterministic term and θ₂ 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. Posterior statistics of unknown parameters in the strength degradation model.

Models	Parameters	Mean	Standard Deviation
I	θ1	−0.125	0.222
	θ2	0.067	0.122
	σ	0.260	0.038
II	θm	0.025	0.038
	θ1	−0.248	0.136
	θ2	0.025	0.074
	σ	0.156	0.024

.

A comparison between the prior and posterior PDFs of the model errors (σ) is shown in Figure 2. It can be observed that there are significant differences between different model errors. Additionally, Figure 2 shows that compared to the prior distribution of the model errors, there are narrow ranges of the posterior probability distribution. This result indicates that information in the experiment data can concentrate the distributions of the unknown model parameters and reduce their corresponding uncertainty.

To further illustrate the decay attenuation law of the shear strength of stud connectors, the degradation distributions of Model I and Model II under different cases are shown in Figure 3 and Figure 4.

When the cycle ratio is 0.3, the contours of Model I concerning the loading ratio and variable k are shown in Figure 3a. It can be observed that the loading ratio and k both have significant influences on the strength degradation of stud connectors. Figure 3b shows the strength degradation distribution versus the loading ratio and cycle ratio with k equaling 1.5. It is shown that the shear strength of stud connectors will vastly decrease as the cycle ratio increases.

In addition, when the loading ratio equals 0.3, Figure 4a illustrates the contours of the shear-strength degradation model concerning the cycle ratio and variable k. Similarly, the cycle ratio and k both have significant impacts on the degradation model, and the strength degradation coefficient increases as k increases. In contrast, the distribution of Model II versus the loading ratio and cycle ratio with k of 1.5 is shown in Figure 4b. It indicates that when the loading ratio is less than about 0.9, the shear-strength degradation model of stud connectors is insensitive to the loading ratio, which is inconsistent with the actual situation. Thus, the physics-based strength degradation model with a deterministic term of Equation (1) will be used in the service reliability analysis of the deteriorated girder bridges caused by high-cycle fatigue cumulative damage.

It should be stressed that variable k (=d/d_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

In this study, a cantilever I-shape steel-concrete composite bridge beam is used to assess the reliability considering the shear-strength degradation of fatigued stud connectors, as detailed in Figure 5. In the composite beam with a length (l) of 6 m, the interval (Δl) between the adjacent studs is designed as 150 mm, and there are 80 (=2 × 6000/150) studs. The diameter (d) of the stud is 20 mm, which is treated as a random variable following a Normal distribution with a coefficient of variation (COV) of 0.1 [29,30]. For the cross-section of I-shape steel, the thicknesses of the flange and web are 10 mm and 6 mm, respectively. In addition, the steel types of Q345 and ML-15 are, respectively, selected for I-shape steel and studs. The reinforced concrete deck of this composite beam is formed with the C50 concrete and HPB300 reinforcement bars. The mechanical properties of the steel and concrete materials are treated as random variables herein, and their corresponding detailed information is summarized in

Table 3. Probabilistic description of the materials.

Name	Mean (MPa)		COV/%	Distribution	Upper Level	Lower Level	References
Q345	fy1	352	5	Lognormal	1.1 fy1,mean	0.9 fy1,mean	Zheng et al. [31]
	fu1	495	5	Lognormal	1.1 fu1,mean	0.9 fu1,mean	Zheng et al. [31]
	E1	2.06 × 105	3.3	Lognormal	1.1 Ey1,mean	0.9 Ey1,mean	Barbato et al. [32]
C50	fc	44.8	20	Lognormal	1.4 fc,mean	0.6 fc,mean	Barbato et al. [32]
	Ec	4733√fc	12	Normal	1.2 fc,mean	0.8 fc,mean	Xu et al. [33]
ML-15	fy2	442	5	Lognormal	1.1 fy2,mean	0.9 fy2,mean	Melchers [34]
	fu2	525	5	Lognormal	1.1 fu2,mean	0.9 fu2,mean	Zheng et al. [35]
	E2	2.0 × 105	3.3	Lognormal	1.1 Ey2,mean	0.9 Ey2,mean	Barbato et al. [32]
HPB300	fy3	300	5	Lognormal	1.1 fy2,mean	0.9 fy2,mean	Zheng et al. [36]
	E2	2.0 × 105	3.3	Lognormal	1.1 Ey2 mean	0.9 Ey2,mean	Zheng et al. [37]

. It should be pointed out that the shear strength of this composite bridge beam is only related to the strength of the corresponding materials. Thus, the strain values of the steel and concrete materials are not presented in Table 3.

In Table 3, f_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).

$$N_{\mathrm{c}}=-N_{\mathrm{a}}=N$$

(8)

and

$$M(x)=M_{\mathrm{c}}(x)+M_a(x)+\frac{Nh}{2}$$

(9)

where h represents the total depth of the beam equaling the height (h_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

$$\{\begin{array}{l}{\epsilon }_1=\frac{M_{\mathrm{c}}(x)h_{\mathrm{c}}}{2E_{\mathrm{c}}{I}_{\mathrm{c}}}-\frac{N}{E_{\mathrm{c}}{A}_{\mathrm{c}}}\\ {\epsilon }_2=\frac{N}{E_{\mathrm{a}}{A}_{\mathrm{a}}}-\frac{M_{\mathrm{a}}(x)h_{\mathrm{a}}}{2E_{\mathrm{a}}{I}_{\mathrm{a}}}\end{array}$$

(10)

In addition, considering the deformation coordination condition (i.e., ε₁ = ε₂), the different layers of the composite beams have the same axial elongation, which is expressed as

$${\displaystyle {\int }_{l-i\cdot \mathrm{\Delta }l}^{l-(i-1)\cdot \mathrm{\Delta }l}{\mathbf{\epsilon }}_1\mathbf{d}\mathit{x}}={\displaystyle {\int }_{l-i\cdot \mathrm{\Delta }l}^{l-(i-1)\cdot \mathrm{\Delta }l}{\mathbf{\epsilon }}_2\mathbf{d}\mathit{x}}$$

(11)

Because the i-th section of the different layers have the same curvature (i.e., M_c(x)/E_cI_c = M_a(x)/E_aI_a), combined with Equation (9), the moments at the i-th section of the different layers are calculated as

$$\{\begin{array}{l}{M}_{\mathrm{c}}(x)=\frac{E_{\mathrm{c}}{I}_{\mathrm{c}}}{E_{\mathrm{c}}{I}_{\mathrm{c}}+E_{\mathrm{a}}{I}_{\mathrm{a}}}\left[M(x)-\frac{Nh}{2}\right]\\ M_{\mathrm{a}}(x)=\frac{E_{\mathrm{a}}{I}_{\mathrm{a}}}{E_{\mathrm{c}}{I}_{\mathrm{c}}+E_{\mathrm{a}}{I}_{\mathrm{a}}}\left[M(x)-\frac{Nh}{2}\right]\end{array}$$

(12)

in which, E_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.

For a cantilever beam under the uniform load q, the moment caused by the uniform load at the i-th section is written as

$$M(x)=\frac{q{(l-x)}^2}{2}$$

(13)

Combined with Equations (11)–(13), the axial force at the i-th section is calculated as

$$N_i=\frac{(3i^2-3i+1)q\mathrm{\Delta }{l}^2}{3h+\frac{12(E_{\mathrm{c}}{I}_{\mathrm{c}}+E_{\mathrm{a}}{I}_{\mathrm{a}})(E_{\mathrm{c}}{A}_{\mathrm{c}}+E_{\mathrm{a}}{A}_{\mathrm{a}})}{E_{\mathrm{c}}{A}_{\mathrm{c}}{E}_{\mathrm{a}}{A}_{\mathrm{a}}h}}$$

(14)

where A_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 q₁ and live load q₂, and for the highway of Class I, q₂ is set to 10.5 kN/m [38]. Similarly, the axial force at the (i − 1)-th section can be expressed as

$$N_{i-1}=\frac{\left[3{(i-1)}^2-3(i-1)+1\right]q\mathrm{\Delta }{l}^2}{3h+\frac{12(E_{\mathrm{c}}{I}_{\mathrm{c}}+E_{\mathrm{a}}{I}_{\mathrm{a}})(E_{\mathrm{c}}{A}_{\mathrm{c}}+E_{\mathrm{a}}{A}_{\mathrm{a}})}{E_{\mathrm{c}}{A}_{\mathrm{c}}{E}_{\mathrm{a}}{A}_{\mathrm{a}}h}}$$

(15)

Thus, the shear force on a single stud at the i-th section under the uniform load is given by

$$V_{s,i}=\frac{N_i-N_{i-1}}{2}=\frac{iq\mathrm{\Delta }{l}^2}{h+\frac{4(E_{\mathrm{c}}{I}_{\mathrm{c}}+E_{\mathrm{a}}{I}_{\mathrm{a}})(E_{\mathrm{c}}{A}_{\mathrm{c}}+E_{\mathrm{a}}{A}_{\mathrm{a}})}{E_{\mathrm{c}}{A}_{\mathrm{c}}{E}_{\mathrm{a}}{A}_{\mathrm{a}}h}}$$

(16)

If the failure of this composite beam is controlled by the concrete crush, the design shear load at a single stud is calculated as V_s,d = 0.43ηA_s√f_cE_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_sf_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.

The calculated failure probability results under the different cases are illustrated in Figure 7. It can be observed that the stud connector is not always safe considering the effects of the shear-strength degradation caused by the fatigue cumulative damage, even though this composite beam is just subject to the combined action of the self-weight and the design live load. This proposed physics-based model can be used to estimate the residual strength of the studs under fatigue loads given the loading ratio and cycle ratio. Furthermore, similar to the degradation law of the shear strength of the stud connector, the loading ratio, cycle ratio, and specimen size have significant impacts on the failure probability. Thus, in the performance assessment and design of the composite beam bonded with the stud connector or other shear connectors, the shear capacity degradation effects should be carefully considered.

6. Conclusions

This paper proposed a physics-based shear-strength degradation model of stud connectors, and the Bayes theorem is used to determine the posterior probability distribution of the unknown model parameters based on the collected experimental data. The main conclusions are as follows:

(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.

It should be stressed that this physics-based strength degradation model of stud connectors was developed based on the experimental data under shear loading conditions. Thus, this proposed strength degradation model may not be well suitable for the torsional loading condition. However, this degradation model can be updated if new data under various loading conditions are obtained.