Estimation of the mechanical behavior of CFRP-to-steel bonded joints with quantification of uncertainty

The strengthening and repair of existing infrastructures, a large portion of which is comprised of steel structures, is essential for sustainable material use and energy resource management. Bonded strengthening using Carbon Fiber Reinforced Polymers (CFRPs) offers great potential toward a sustainable infrastructure management. In establishing CFRP retrofitting as a reliable solution for steel strengthening, a solid understanding of the mechanical behavior of the CFRP-to-steel bonded joints is essential. Given the variability in the evidence attained by experiments, in this study, we tackle this challenge from an uncertainty quantification perspective by proposing a model based on Polynomial Chaos Expansion (PCE) to predict the load capacity of the bonded joints. A stochastic bond–slip model, featuring a parsimonious representation with one deterministic coefficient and one probabilistic coefficient, is further proposed. A Monte-Carlo (MC) simulation is used to demonstrate the efficacy of the bond–slip model in predicting the mechanical behavior such as load–displacement behavior, shear stress profile, and effective bond length of strengthened specimens. Results are compared with existing deterministic models.


Introduction
Circular economy, an idealized form of economy, is founded on three fundamental principles: (1) reduce (or even eliminate) the use of raw materials in producing and manufacturing process; (2) reuse or recycle used materials so that no waste is treated at landfills or incinerators; and (3) all materials remain in a loop of produce-(re)userecycle [1].According to a recent survey conducted by the European Commission (2020) [2], about half of the raw material extraction is used in construction and building, while circa one third of the total waste is attributed to the construction sector.Although the recycling rate of steel material is one of the highest in the world (around 80-90% globally), steel production still requires 70% of new raw material as input [3].Meanwhile, transporting and processing recycled material still requires a large amount of energy.Alternatively, strengthening existing structures requires comparatively very little material and energy.

CFRP strengthening steel structures
Some recent strengthening works on steel bridges exploited the Carbon Fiber Reinforced Polymer (CFRP) strengthening approach.The strengthening of steel structures via use of prestressed unbonded CFRP * Corresponding author at: Empa, Swiss Federal Laboratories for Materials Science and Technology, Überlandstrasse 129, 8600, Dübendorf, Switzerland.E-mail addresses: lingzhen.li@empa.ch(L.Li), niels.pichler@empa.ch(N.Pichler), chatzi@ibk.baug.ethz.ch(E.Chatzi), elyas.ghafoori@empa.ch(E.Ghafoori).is a relatively mature technique at this stage.Examples are the Münchenstein bridge [4] (Switzerland), the Diamond bridge [5] (Australia), and the Aabach bridge [6] (Switzerland); their maximum stress in the fatigue sensitive spots reduced approximately 30-50% after the prestressed unbonded CFRP strengthening.In these cases, the effectiveness of the strengthening highly relies on the prestress level and mechanical clamping system.Another approach, bonded strengthening, involves the direct bonding of CFRP strips on steel elements.Examples are the Acton bridge [7] (UK), Jarama bridge [8] (Spain), I-10 bridge [9] (USA), and Diamond bridge [10] (Australia); the local stress of steel girder underneath the bonded strengthening were reduced approximately 24-50%.In these cases, ensuring a reliable bond is vital for the strengthening procedure.

Studies of the bond behavior
The characterization of the bond behavior between CFRP and steel requires thorough experimentation.Lap-shear tests form a main means to this end; these are meant to mimic, as closely as possible, the on-site application of a CFRP repair/strengthening patch bonded to steel.Based on the outcomes of such tests, several models [11][12][13][14][15][16][17] https://doi.org/10.1016/j.engstruct.2022 were proposed to simulate and explain the experimentally observed behavior.These models typically assume specific shapes of the bondslip curve, e.g., a triangular shape for linear (i.e., brittle) adhesive, or a trapezoidal shape for the nonlinear (i.e., ductile) case.These models are capable of not only predicting the bond capacity and effective bond length, which are the two most important aspects for engineering design, but are further suited for estimation of mechanical details, such as the shear stress profile along the bond line and the load-displacement behavior.The underlying idea of these models is very similar: (1) select the influential variables based on engineering judgment, (2) combine these variables in a power-law expression, and (3) determine the corresponding coefficients via regression in order to minimize the discrepancy between the experimentally derived and estimated values.
This approach is widely exploited for creating analytical prediction models on the basis of experiments.It offers easily interpretable and parsimonious expressions, but can often prove overly simplified.Two aspects are of defining importance and should be addressed in constructing such predictive models.The first pertains to proper categorization of the adhesive type on the basis of brittle or ductile behavior, corresponding to a triangular or trapezoidal model.These models require the calibration of several parameters, including the maximum shear stress in the adhesive bond and the corresponding shear deformation, the maximum shear deformation, and, for a trapezoidal model, the shear deformation at the end of the plateau stage.The second defining aspect pertains to refinement and quantification of the prediction accuracy, as demonstrated by Yu et al. [18].
Recently, Yu et al. [18] proposed a modification based on adoption of two of the aforementioned models, i.e., the single-lap shear model [11] and the double-lap shear model [14].They plotted the correlation between variables and the prediction error, and used an exponential relationship to describe each correlation.A model factor was then assigned by multiplying the fitted set of exponential expressions, with the model factor further multiplied to the prediction of the original model, eventually enhancing the model accuracy.However, the samples collected failed under different modes; a trait which affects bond capacity and was not taken into account, as the samples were not categorized into different groups, while corresponding failure modes were not quantitatively taken into consideration.In related literature, the CFRP-to-concrete bond [19] and CFRP-to-masonry bond [20] were investigated considering the variability of the estimated bond capacity.In those works, a calibration factor was introduced in the theoretical model by adopting the median of the ratios of measured/estimated bond capacities.More recently, a unified deterministic bond-slip model, spanning over linear and nonlinear adhesives and considering different failure modes, were proposed by Jiang et al. [21].The bond-slip model has an exponential form, with two parameters determined by a nonlinear regression on the experimental data.A good fit between model prediction and experimental observation was observed.Yet, the predicting accuracy is worth further enhancement, even though the failure modes of specimens are known, indicating existence of uncertainties which are not involved in the analysis.
Emerging data-driven methods have been exploited to solve some similar bond problems.Deterministic machine learning (ML) models were built to predict the bond capacity of CFRP-to-steel bonded joints [22] and CFRP-to-concrete bonded joints [23] with fairly good accuracy.Probabilistic models based on Markov chain Monte-Carlo method and Bayesian theory were further utilized to quantify the uncertainty of shear strength of steel rebars embedded in concrete [24,25].

The current study
This study aims to introduce models, whose relationship is presented in Fig. 1, to estimate the mechanical behavior of CFRP-to-steel bonded joints with quantifying uncertainties.To do so, in Section 2, a Polynomial Chaos Expansion (PCE) is used to quantify the influence of each variable on the final estimation of the bond capacity of CFRPto-steel bonded joints.To the authors' knowledge, such a stochastic analysis tool has not so far been applied to the problem at hand, even though a considerable amount of experimental data is now available in existing literature.In Section 3, we further introduce a new bond-slip model, which is applicable to both linear and nonlinear adhesives, with two coefficients to be determined; one coefficient is deterministic and can be evaluated from Section 2, while the other is stochastic.When the suggested bond-slip model is coupled with a Monte-Carlo (MC) simulation, the mechanical behavior of CFRP-to-steel bonded joints, such as the bond-slip behavior, load-displacement behavior, shear stress profile, and effective bond length, are shown to be well approximated.Among all the mechanical features, the effective bond length is of vital importance for the engineering design.Therefore, in Section 4, a process based on MC simulation is suggested to estimate the effective bond length for engineering design considering the uncertainties.At the end (Section 5), the advantages and limitations of the current study are discussed.

Data collection & defining variables
In the past two decades, a considerable amount of single-lap and double-lap shear tests have been carried out.In a single-lap shear joint, CFRP is bonded to only one side of the steel element, whereas CFRP is bonded to both sides of the steel element in a double-lap shear joint.Compilations of these test results have been used to develop semi-analytical models (single-lap shear [11][12][13] and double-lap shear [14,15]), as described previously.Since the current study focuses on bonded strengthening of existing steel structures (e.g., steel girder and column), where typically only one side of the steel surface is accessible (e.g., outer side of steel columns), we limit the focus of this study to the single-lap shear problem.
Three are the main failure modes for the CFRP-to-steel adhesive bond that have been identified in the literature: cohesion failure, adhesion failure, and CFRP delamination.In cohesion failure, the crack path lies in the bulk material of the adhesive layer.It is the preferred failure mode in most applications [26][27][28], as (1) the full load-carrying capacity of the adhesive bond can be exploited, (2) the bond capacity can be guaranteed by choosing an appropriate adhesive, and (3) the design of the bonded joint is independent on the surface state.The second failure mode is adhesion failure, where damage occurs in the adhesive-CFRP interface or adhesive-steel interface.This is the least desired failure mode since the bond capacity is lower than the one corresponding to cohesion failure, leading to premature failure.This failure type can be prevented through the use of surface preparation methods ranging from surface roughening [29] and sand-blasting [30] to use of primers [31,32] and chemical etching or anodizing [33].The third and final failure case pertains to CFRP delamination.In this situation, cracks propagate inside the composite matrix or at the carbon-matrix interface.This failure type is brittle, whereas the first two failure modes represent ductile processes.Due to these, most bonded joints are designed to fail primarily in a cohesion mode, which therefore forms the focus of this study.
In the study presented herein, 97 test samples, which correspond to a cohesion failure mode, were gathered for development of an analytical model, which can simulate and reproduce the observed behavior.Among these, 89 are gathered from the open literature [11,12,29,[34][35][36][37][38][39][40][41][42][43][44], while two originate from internal Empa (Swiss Federal Laboratories for Materials Science and Technology) tests.The remaining six tests were recently carried out as part of parallel research work in Empa.
Since all tests pertain to cohesion failure, the surface state does not affect the failure process, which is influenced instead by the properties of the adherent, adhesive, and the geometry.Eight variables were determined as highly influential for bond capacity.Three variables pertain to the adhesive properties, namely the Young's modulus (  ), tensile strength (  , ), and ultimate tensile strain (  , ).One variable corresponds to the CFRP Young's modulus (  ), while four variables are tied to the geometry and correspond to the bond length (), bond width (  ), adhesive thickness (  ), and CFRP thickness (  ).The respective ranges of validity for these eight variables are listed in Table 1.The steel Young's modulus is here not considered as a variable to explore, since the steel element is much stiffer than the strengthening material and this value presents little variability.CFRP tensile strength is not considered, as critical loads are unlikely to be reached before the bond capacity.Between the estimated bond capacity and the CFRP tensile load capacity, the smaller one should be considered as the ultimate load-carrying capacity.Furthermore, the type of CFRP reinforcing materials (pultruded plates or sheets) can influence the bond capacity of CFRP-to-concrete bonded joints by affecting the interfacial adhesion and preventing/causing premature failure [19].Nevertheless, the majority of the existing studies with the CFRP bonded on a single side of steel elements utilized pultruded CFRP plates.As a result, the existing information is not enough to support a data-driven method investigating the effect of the CFRP type.In order to make this explicit, the CFRP material in this study is specifically referred to as CFRP plates (also known as CFRP strips).Other variables affecting the bond capacity, such as the loading rate, were either not reported or not well controlled, therefore, they were not considered in this study.A generic schematic of the bonded joints with all the geometric variables can be found in Fig. 2.

Stochastic bond capacity surrogate model
Polynomial Chaos Expansion (PCE) is a powerful tool commonly used for the purpose of surrogate construction and Uncertainty Quantification [45,46].In this study, PCE is implemented on the aggregated experimental data through use of the UQLab [47] platform, which is implemented in a MATLAB [48] environment.PCE serves here as a stochastic surrogate model for predicting the bond capacity of CFRPto-steel bonded joints.PCE offer a number of advantages: (1) it is straight-forward, since it essentially comprises a polynomial model with coefficients inferred by means of a simple method, such as leastsquare regression; (2) it is interpretable, since the computed coefficients can offer insights to the influence of the variables, as well as a quantification to the output (computed bond capacity) uncertainty.The PCE basis is built from the tensor product of univariate polynomials that are mutually orthonormal with respect to the probability measure  () [49], such that: Here,  () is the multivariate polynomial basis and   denotes the Kronecker delta symbol.The element   of the multi-index  indicates the degree of   in the th variable,  = 1, … .  has a total degree given by || = ∑    .There exist several commonly used polynomial families, depending on the assumption on the distribution that best characterizes the input variables.For example, the Legendre polynomial family corresponds to a uniform distribution assumption, while the Hermite polynomial family with respect to a standard Gaussian distribution.More details on the theoretical framework underpinning implementation of the PCE can be found in literature [47,49,50] etc.
Based on the above, a PCE model admits the following form, which eventually describes a linear combination of multivariate products of the basis polynomials, as expressed in Eq. (2).
where  () is the model output; in our case, the bond capacity;   describes the set of coefficients of the expansion   ().The goal of constructing a PCE model is to determine the expansion coefficients   .The finite number of polynomial terms (i.e., the number of coefficients) that are sufficient to retain in the expansion is calculated on the basis of the following expression: where  is the number of employed polynomial terms;  is the dimension of PCE model;  is the maximum basis degree of polynomial expansion.The corresponding dimensions adopted in this study are 45, 8, and 2, respectively.
In engineering applications, the values of the eight selected influencing variables (  ,   , ,   , ,   , ,   ,   , and   ) are determined on the basis of engineering judgment.Therefore, a uniform distribution was assumed in this case for all eight variables, since this problem is separate to a design problem where material and geometry properties are assumed centered around a specified mean value and distributed according to a normal or lognormal distribution.Correspondingly, Legendre polynomials were used as bases to build the PCE model, with Ordinary least-square (OLS) regression used to infer the expansion coefficients.The solution reads: where  is the vector of coefficients that is to be inferred;  is a vector of measured model outputs (e.g., via experimentation);  is the regression matrix which contains values of the basis polynomials evaluated at the experimental design points and constructed as Eq. ( 5): where  () is the input vector (variable set) of th specimen;    is the th term of base polynomial.The OLS regression requires a data set of dimension larger than 2 ∼ 3 times the number of polynomial terms, which means 90-135 data points for our retained 45 polynomial terms, with only 97 experimental points available in the assembled experimental database.To tackle this issues, we further generate a set of artificial data.Previous semi-analytical models [11,13,51] demonstrated that the bond capacity is proportional to the bond width, if ignoring the edge effect in the width direction.Thus, we here generate another 97 experimental points by simply halving the bond width and bond capacity, simultaneously.This results in 194 data points for training the PCE model.
The PCE computation procedure is as follows; the gathered data are split into a training set and a validation set.Variable values and experimental results from the training set are substituted in  and  in Eq. ( 4), respectively, to infer the coefficient vector .Variable values from the validation set and the trained coefficients () are then substituted into Eq.( 2) to estimate the results of the validation set.The estimated and experimentally obtained results for the validation set are finally compared to demonstrate the goodness of fit of the trained PCE model.
In this study, 10-fold validation was used to evaluate the performance of the derived PCE model.The collected data were randomly split into 10 folds, each comprising approximately 1/10th of the total experimental sample size.In each training-validation process, nine folds were selected as the training set and the remaining one as the validation set.The training-validation process was repeated 10 times, until every fold was used to validate the trained model.Eventually, 10 eightdimensional PCE models with a maximum degree of two were trained.The above process is a complete-run of the PCE models on the given data set.Fig. 3 shows the load-carrying capacity of the bonded joints estimated from a 10-fold validation process against the tested results, with a goodness of fit of  2 = 0.95.Since the 10 folds were randomly split in every complete-run of the model,  2 was changing during each complete-run.After 1000 complete-runs, the mean value of  2 is 0.94 (best score is 1), confirming the robustness and accuracy of the trained prediction model.Note here, so far, no assumption of the distribution of the output (i.e., the bond capacity) was made.The only assumption is that all eight input variables follow a uniform distribution, which is on the basis of engineering judgment.The moments (e.g., mean and variance) of the target can be easily computed via analyzing the polynomial coefficients [49,50], and the form of distribution of the target can well be approximated via Monte-Carlo simulation.
Since the performance of the procedure is confirmed, a final PCE model has been trained on the total data set (194 test specimens) without further validation.The model has a slightly higher goodness of fit, with  2 = 0.97, and will be used for design of the bonded joints.Furthermore, PCE models have been trained to a maximal degree of one and three.The model corresponding to a degree of one yields a rather poor goodness of fit, with  2 = 0.7, which indicates that linear approximations cannot capture all the effects at play.The model of degree three would require a larger dataset of 300-500 points.This is possible by artificially extending the current dataset, but since the ratio of artificial to real datapoints would be high, this can lead to biased and likely misleading results.
Attention should be raised to an issue of the model.As mentioned earlier, the CFRP tensile strength is not considered as a model variable.In reality, the estimated bond capacity cannot be reached if the tensile load capacity of the CFRP strip is lower.In such a case, the CFRP strip would fail under tension prior to failure of the adhesive bond.This situation may occur when a nonlinear adhesive is used in combination with a high CFRP modulus.Thus, the lowest value between estimated bond capacity and CFRP tensile load capacity should in practice be considered as the load-carrying capacity of the bonded joint.
The PCE method can further provide some insights into the model sensitivity, as the trained coefficients indicate the importance of each variable.Total Sobol indices can be obtained as Eq. ( 6): where   is the total Sobol index of the th variable; ∑ 8 =0  2  refers to the sum of square of coefficients related to th variable; denotes the sum of square of all coefficients.Indices  and  reflect the adopted maximum polynomial degree of two.
The computed total Sobol indices, outlined in Table 1, reflect the importance of each variable; the larger the index, the more influential the variable is on the bond capacity.Considering that the adhesive tensile strength is negatively correlated with the maximum tensile strain, only one of the two variables is treated as independent.The three most important variables, thus, are: (1) adhesive ultimate tensile strain (and adhesive tensile strength), ( 2) CFRP E-modulus, and (3) CFRP width, i.e., bond width.The remaining variables are less important as their indices range from 0.01 to 0.03.This is in accordance with existing literature.Examples of (1) can be found in studies [37,38]: when replacing a linear adhesive (  , = 0.29%) by a nonlinear adhesive (  , = 1.74%), the bond capacity was increased to 3-4 times; (2) Fernando et al. [52] increased the CFRP E-modulus from 150 GPa to 235 and 340 GPa, resulting in enhanced bond capacity by approximately 57% and 113%, respectively; (3) CFRP width [11,13,51]: if the edge effect in the width of the bond is not considered, the bond capacity is proportional to the bond width.

Safe design of bond capacity
Fig. 3 shows that the bond capacities of 49.5% samples were overestimated, while 50.5% were underestimated.To ensure that the bonded joints for strengthening are safe enough, which means the designed capacity is lower than the actual capacity, a factor lower than 1 was introduced.
When adopting the final PCE model, which was trained on all 194 data points (including the original 97 and duplicated 97), the ratio of estimated over tested bond capacities follows a normal distribution with a mean value of 1 and a standard deviation of 0.11, see Fig. 4. By evaluating the quantile of the normal distribution, the factor and the corresponding probability of survival were obtained and listed in Table 2. Fig. 5 shows an example of estimated bond capacity vs. tested capacity using a factor of 0.80, which brings a probability of survival of 97%.There, the majority of designed bond capacities are slightly lower than the actual values.Even though the bond capacity of six samples were slightly overestimated, which implies that they lie on the unsafe side, their values are very close to the tested ones.Different factors in Table 2 can be chosen when designing, depending on the importance of a structure on the economy and society etc.
Although the trained PCE model already contains the effect of bond length, more than half of the specimens yield a bond length that is higher than the effective bond length.For those specimens where the bond is relatively short, an increase can further enhance the bond capacity until the effective bond length is reached.Thus, the effective bond length forms a critical parameter, which is worthy of precise determination.In this work, a novel approach is proposed to estimate the effective bond length in the following two sections (Sections 3 and 4).

Stochastic bond-slip model
Prior to estimating the effective bond length, a good understanding of the bond-slip behavior, namely shear stress vs. shear deformation of the adhesive bond, is essential.A stochastic bond-slip model, with two coefficients to be determined (one being deterministic and the other being stochastic) and capable of quantifying uncertainties in the mechanical behavior, is proposed in this study.

Newly proposed bond-slip model
The newly proposed bond-slip model is expressed as Eq. ( 7) and schematically plotted in Fig. 6: where  is the slip of CFRP (relative displacement between CFRP and steel); () is the shear stress as a function of slip;   and   are Young's modulus and thickness of CFRP strip, respectively;  , is the maximum tensile strain of CFRP during the loading process;   is a shape parameter with a unit of length: the bond-slip curve features a narrow shape when   has a small value, while the curve is wider, featuring a lower peak when   has a large value;   does not change the area under the bond-slip curve, which is usually regarded as the fracture energy of the adhesive bond.More details regarding constructing the bond-slip model, please refer to Appendix A. This bond-slip model is similar in form to those appearing in relevant studies [21,53], but with inclusion of the aspect of uncertainty quantification, which is vital when handling experimental datasets (of inherent scatter) and will be further explained in the following sections.
There are only two unknown coefficients to be determined in this bond-slip model:  , and   . , can be evaluated from the maximum force carried by CFRP strip (i.e., bond capacity estimated from the model in Section 2) divided by cross sectional area (  ⋅  ) and   .  is a function of the   (maximum shear stress of the adhesive bond) and will be determined in the next section.

Determining the maximum shear stress
In the bond-slip curve (see Fig. 6),   is reached when the derivative of shear stress with respect to slip becomes zero, Eq. (8).Slip corresponding to   , as expressed in Eq. ( 9), is the solution of Eq. ( 8).Substituting Eq. ( 9) into Eq.( 7) yields Eq. (10), which indicates the dependence between   and   .Parameter   can thus be inferred upon determination of   along the bond line.The determination of the maximum shear stress is a challenging task, linked to determination of material plasticity and failure under multiaxial loading.A simplified 2D stress state (see Fig. 7(a)), along with a Mohr-Coulomb criterion (see Fig. 7(b) and Eq. ( 11)), were used to analyze the maximum shear stress.The reason for considering a mixed shear and tension/compression state rather than a pure shear state is that the normal stress, i.e.,   , does exist at the debonding tip, as a result of the local bending effect of the CFRP strip, even if the tensile force is perfectly aligned at the center of CFRP strips [54].The Mohr-Coulomb theory has been widely used in analyzing the failure of concrete and rock, examples see [55][56][57][58][59]. Nevertheless, its application on the adhesive failure is rather limited (a successful example see García and Leguillon [60]).The cohesion parameter () can be written as a function of the friction angle () and either the tensile strength (  , ) or the compressive strength ( , ) of adhesive, as expressed in Eq. (12).With the presence of the compressive and tensile stress, the maximum shear stress increases and decreases, respectively, according to Eq. (11).Further substituting Eq. ( 12) into Eq.( 11) yields Eq. ( 13).Since the adhesive tensile strength is a property that is easily obtainable, it is fair to represent the maximum shear stress (  ) by multiplying the adhesive tensile strength (  , ) by a factor, which is determined by the normal stress and the friction angle of adhesive or the adhesive compressive strength. or Among all 97 lap-shear specimens, adhesive tensile strength and maximum shear stress of 64 specimens are available; their maximum shear stresses are positively correlated with the adhesive tensile strength, see Fig. 8.For a specific adhesive material, when its tensile strength becomes zero, its shear strength should also be zero, which corresponds to the presence of a crack.Therefore, the correlation line should pass through the origin, where both tensile and shear strengths are zero.As a result, the best fit line of   = 0.93⋅  , is obtained, with    2 = 0.67.Statistically, this means that tensile strength can only explain 2/3 of the maximum shear stress.The remaining 1/3 could be related to vertical tension/compression at the debonding tip, different friction angles and/or compressive strength, and stress triaxiality which cannot be explained by 2D model etc. Fig. 9 shows that the ratio of   /  , follows a Lognormal distribution with a mean value of 0.96 and a standard deviation of 0.14, which can be used to determine   in the next step.On the other hand, studies claimed that   is 80% [11] or 90% [13,51] of   , .They each suggested a specific value rather than a range.This will further be discussed in Section 5.

Generating a bond-slip model according to the maximum shear stress
Thus far, for a specific CFRP-to-steel bonded joint, the  , can be assessed by substituting the eight features (listed in Section 2) into the trained PCE model.Although the exact value of   is still unclear, its distribution is known and the possible values can be randomly generated around   , following the Lognormal distribution shown in  Two lap-shear specimens comprising linear and nonlinear adhesives, recently tested by the part of the authorin team, were used to validate the newly proposed bond-slip model.1000   values were randomly generated using the Lognormal distribution described above.Correspondingly, a set of 1000 bond-slip curves, load-displacement curves, and shear stress profile along the bond line were generated, namely 1000 runs of MC simulation (the range and distribution from 1000 runs have no obvious difference compared to 10,000 runs or 100,000 runs).Figs.10-12 present the comparison between experimentally measured results and MC simulation based on the proposed bondslip model (expression and derivation of load-displacement and shear stress profile, please refer to Appendix A).The red and blue dots are the experimentally measured values using two different measurement methods.MC simulation is shown as green dots, whose density indicates the possibility of happening (the denser the higher possibility).It is noted that the MC simulation either fully covers the experimental behavior, or behaves very similarly to the experimental curves.In the specimen using a linear adhesive, although the predicted bond-slip curve cannot perfectly match the test result (Fig. 10(a)), it is capable of well-approximating the load-displacement behavior (Fig. 11(a)), and the predicted shear stress profile along the bond line fully covers the experimental behavior (Fig. 12(a)).In the specimen comprising a nonlinear adhesive, the experimental bond-slip curves (Fig. 10(b)) and shear stress profile (Fig. 12(b)) are mostly covered by the MC simulation, despite some outlier.The predicted load-displacement curves fully cover the real behavior (Fig. 11(b)).
The results of comparison suggest that it is possible to fairly accurately predict the range of actual behavior of the CFRP-to-steel bonded joints without conducting any lap-shear tests.The information needed for such an estimation is material properties and geometry, which can be easily obtained from the manufacturer's specifications etc.Furthermore, the effective bond length can be evaluated by means of a MC simulation, as elaborated next.

Effective bond length
The effective bond length (   ) corresponds to a threshold, below which the bond capacity increases with the bond length and above which the bond capacity no longer increases.Fig. 13 presents the schematic shear stress profile along the bond line.The area underneath the curve is the force per unit width carried by the adhesive bond.The definition of    therefore reflects a distance between two points, where the curve and the horizontal axis converge.However, mathematically this distance is infinitely long.Therefore, a truncation was made to ensure 95% area underneath the curve was covered; 96% [37] and 97% [51] were also used in the literature.The two small areas, each with 2.5% bond capacity, were ignored.The distance between two bounds, covering a 95% area under the shear stress profile, is equivalent to the distance between the 2.5% and 97.5% maximum strain in the tensile strain profile.The solution is written as Eq. ( 14), whose derivation is offered in Appendix A.

Safe design of effective bond length
A unique   value points out a specific    in a bonded joint.However, only the distribution of   is available, and the exact value is unknown.A solution relying on MC simulation is introduced in this section, and is further validated on 60 specimens, whose    are available, out of the 97 collected specimens.The collected    values are either reported in the literature or estimated by the authors of the current study using the reported bond-slip behavior and models [11,51,61] on estimating the effective bond length, depending on the ductility of the adhesive used, i.e., linear or nonlinear adhesive.
For each specimen, 1000 sets of lap-shear behavior were sampled, and 1000    values were calculated for each test specimen using Eq. ( 14).A strategy of  ±  ⋅  was used to estimate the bond length, where  and  are the statistical mean value and standard deviation from MC simulation.Fig. 14 shows the estimated    and the experimentally determined    .In this figure, specimens were rearranged using the mean values of estimated    in an ascending order.It can be observed that the experimentally measured    are mostly covered by  ± 2.This indicates good estimation of    , considering that this model does not have any    information as model input.It can further be observed that there are two distinct regions of    : (1) 40-110 mm and (2) 150-240 mm.In region (1), all adhesives used are linear adhesives, and the tested    values are distributed around the mean values estimated by the model.In region ( 2), all adhesives are nonlinear adhesives, and the model tends to overestimate the    as the experimentally determined    are mostly staying in the area between  − 0.5 and  − 2.At the end, the authors suggest using  + 2 of the estimated    for design.Meanwhile, an upper limit of    = 250 mm (could also be 300 mm) is suggested to save material, as the red dashed line shown in Fig. 14.

Discussion
Three novel models are proposed for estimating the bond capacity, the mechanical response, and the effective bond length CFRP-to-steel bonded joints, as plotted in Fig. 1.By multiplying the PCE-estimated bond capacity with a factor lower than unity (as provided in Table 2), the bond capacity is slightly underestimated, which lies on the side of safety.The use of a MC approach on the newly proposed stochastic bond-slip model results in a cluster of mechanical response, including load-displacement curves etc., of a bonded joint.This cluster indicates the possible range of actual mechanical response of a joint and provides much understanding to the engineers.Further scrutinizing the MC simulation leads to a suggested effective bond length, corresponding to a value of  + 2 ⋅  which is slightly larger than the experimental observation (as the dashed line plotted in Fig. 14).This value is eventually suggested for design, implying once again that we lie toward the side of Fig. 15.Evaluation of the model performance with 100 runs.Blue: Xia and Teng model [11], red: Fernando model [12], green: Wang and Wu model [13], and black: PCE model.Solid lines represent  2 scores, the higher the better; dashed lines refer to as mean absolute percentage error (MAPE), the smaller the better.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)safety.No lap-shear test is needed as explicit input to the models, which facilitates the use of the proposed model for prediction purposes.The necessary inputs are the material properties and the geometry of the bonded which can be easily Furthermore, model can be updated when new emerge, updated for bonded joints accordingly derived.

PCE model on predicting the bond capacity, comparison with other models
The PCE approach proposes a more flexible model than the classical power law approach, while further improving accuracy and also offering a quantification of the involved uncertainty.Three semi-analytical models focusing on cohesion failure are used for comparison; they are the Xia and Teng model (2005) [11], the Fernando model (2010) [12], and the Wang and Wu model (2018) [13], with the number of specimens used in regression (also known as training) being 5, 10, and 37, respectively.Since the PCE model was trained on 90% of data volume and validated by the remaining 10% data, which means a validation/training data ratio of 1/9, the validation/training data ratio of these three models should be kept roughly 1/9.Therefore, 1, 1, and 4 unseen specimens randomly selected from the datasets are used for the validation of the three models, respectively.Two scores are introduced to evaluate the performance of models:  2 (Eq.( 15)) with higher value indicating better performance and mean absolute percentage error (MAPE, Eq. ( 16)) with lower value suggesting better performance.Fig. 15 shows the performance of three semi-analytical models of 100 runs of validation using randomly selected unseen specimens; the performance of the PCE model is also shown by 100 runs of 10-fold validation.These three models have a  2 score of 0.88, 0.95, and 0.94, respectively, on the training (seen) data and a MAPE of 0.06, 0.08, and 0.08, respectively.When the unseen data is involved in the evaluation, the  2 of Xia and Teng model drops drastically to nearly 0.7 with MAPE almost unchanged.The two other models maintain  2 around 0.9, with MAPE increasing to 0.  b Trained by 70% of data and evaluated by all data, in Chen et al. [22].
good performance of the PCE model.
where   and ŷ are the experimental and prediction values of th specimen, while ȳ refers to the average of all   .Besides, Chen et al. [22] trained four ML models, i.e., artificial neural network (ANN), support vector machine (SVM), gradient boosting decision trees (GBDT), and classification and regression tree (CART), predicting the CFRP-to-steel shear capacity, with  2 scores of 0.87-0.98 and MAPE of 0.05-0.1 (see Table 3), which are at the same level of the PCE model in the current study.This further suggests the good performance of the PCE model, since ML has been proven being able to accurately predict the performance of the CFRP bond capacity [22,23] and the CFRP strengthening performance [63,64].
Compared with another recent semi-analytical model [18], which also simulates the uncertainty involved in the prediction of the bond capacity, the proposed PCE approach is simpler, even though the expression of the polynomial model looks complex.The PCE involves a single regression to infer the coefficients of the polynomial model.The output of the PCE is the prediction of the Quantity of Interest, in this case bond capacity, while it further allows calculation of the importance of each variable in shaping the model output.

Stochastic bond-slip model with 𝐴 𝑎 as uncertain parameter
The proposed bond-slip model is a unified model spanning over linear and nonlinear adhesives smoothly.In previous studies, the bondslip models are either assigned to a linear adhesive with accurate model [65] or bilinear model [11], or a nonlinear adhesive with trapezoidal model [12,17].However, the boundary between linear and nonlinear adhesives is not exactly defined.With the newly proposed bond-slip model, an explicit priori distinction in terms of the adhesive type (i.e., linear or nonlinear) is no longer necessary.A recent study by Jiang et al. [21] proposed a unified bond-slip model, which is also capable of simulating the mechanical behavior captured by our proposed our bond-slip model.Nevertheless, the bond-slip model from Jiang et al. [21] forms a deterministic model based on nonlinear regression of experimental data.Our bond-slip model intrinsically reflects a quantification of uncertainty that can extend to higher moments (than simply mean and variance).When coupled with MC simulation, the mechanical behavior and the corresponding confidence intervals (uncertainty bounds) can be estimated.Given the experimentally observed variability of the mechanical behavior of bonded joints, the proper quantification of uncertainties becomes critical.
The coefficient   used in this study has a twofold significance: (1) a model parameter (see Eq. ( 7)) and (2) a physical variable.Within a bonded joint, a group of   values, which follow a Lognormal distribution different from Fig. 9, can be evaluated, so as to generate the range of response behavior for the bonded joint.Therefore, if one only focuses on one bonded joint,   is simply a model parameter, which helps constructing the mechanical behavior of the adhesively bonded joints.However,   could become a physical variable when analyzing a set of bonded joints.Adhesive ductility is found highly positively correlated to effective bond length (   ), with a Pearson correlation coefficient of 0.81, and Eq. ( 14) mentions that    is proportional to   .Thus, if one considers a set of bonded joints,   becomes a variable further the of the adhesive, and thus has a clear physical connotation.
The bond-slip behavior of a CFRP-to-steel bonded joint is determined once   is known.However, unlike   , , which is a commonly available property,   is not easily obtainable.A strength criterion, such as the Mohr-Coulomb model, expressed as Eq. ( 13), creates a bridge connecting   and   , .A probabilistic assumption is necessary in view of the uncertainties involved in estimation of the variables defining our problem.More specifically, the ratio of   ∕  , defined in Eq. ( 13) and further noted as   ∕  , =   ∕  , ⋅()+1∕2⋅(  4 +  2 ) is not a deterministic value but instead assumes a probabilistic character, following Lognormal distribution, as described earlier.The distribution of this ratio is derived from 64 bonded joint samples, containing more than 10 types of adhesives in combination with different CFRP strips and specimen configurations.Two advantages exist in using a range of   ∕  , rather than a single value.The first pertains to comprehension of the uncertainties involved in the model.A simplified 2D Mohr-Coulomb model is here adopted, implying that uncertainty is related to the normal stress (  ) at the debonding tip and friction angle () of the adhesive, however further sources of uncertainty could relate to additional effects, such as stress triaxiality, which cannot be explained by a 2D model, as well as the form of the yield surface, which would though require adoption of a different criterion.The ratios following a Lognormal distribution, as plotted in Fig. 9, already involve the above-mentioned uncertainties.The second advantage pertains to a quantification of the involved uncertainty via use of a probabilistic approach, as opposed to relying in the prediction furnished by a single ratio assumption.If one overestimates the maximum shear stress,   , the expected response in the load-displacement curve will be more rigid than reality.This can lead to a general overestimation of the system rigidity.More importantly, overestimating   will lead to an underestimation of    possibly introducing major design flaws in the joint.Consequently, a group of ratios   ∕  , instead of a single value were used in this study.The two commonly adopted ratios of 0.8 [11] and 0.9 [13,51], which typically reflect an average over an observed range, are included in the parametric range assumed herein, however our proposed approach further reflect the effect of variability in this ratio.
In the meantime, this unified bond-slip model suffers a potential limitation, namely a poor approximation of the bond-slip behavior if the plateau between the initial elastic and final softening branch is very long.The 97 collected specimens already cover a wide range of the eight considered parameters, e.g., the adhesive ultimate tensile strain ranges from 0.19% − 4.37%, which spans from brittle (linear) adhesive to one of the most ductile (nonlinear) adhesives met in the civil engineering field (to the authors' knowledge).The unified bondslip model with an exponential shape fits the bond-slip curves of these specimens fairly well.However, one should be careful with extrapolation as the applicability of the model for highly ductile adhesives would be limited.Therefore, the authors do not recommend adoption of this model for adhesives with a maximal tensile strain greater than 4.4%, which though reflects a rare case.

Stochastic mechanical response of joints
The MC simulation, which relies on use of the proposed bondslip model and ratios   ∕  , , seems adequately capture the test behavior, see Figs. 10(a), 10(b), 11(a), 11(b), 12(a) and 12(b).It is worth noting that not all experimental curves are fully covered by MC simulation.The reason is that the aim of this study is not to provide a universal model that accurately predicts and describes the experimental behavior, which is very challenging.Rather, the authors wish to use the proposed model to predict the range of the actual behavior of adhesively bonded joints between CFRP and steel, and eventually use the predicted range to assist engineering design.

Estimating the effective bond length
The proposed stochastic bond-slip model and the design strategy of  ±  ⋅  of the MC simulation allow for a good approximation of the effective bond length, as shown in Fig. 14.Nevertheless, the model tends to overestimate the effective bond length when the joints comprise nonlinear adhesives.The reason is that the proposed bondslip model possesses an exponential form, whose tail is longer than an equivalent trapezoidal shape corresponding to the same area, as shown in Fig. 10(b).Therefore, a longer bond length is estimated to activate the full shear profile, which is not realistic.To tackle this issue, an upper limit of the estimated effective bond length of 250 mm, based on the experimentally determined    and shown in Fig. 14, is suggested in this study.

Strong data dependency
The proposed solutions for estimating bond capacity and effective bond length bear a strong dependency on the experimental data used to infer the respective data-driven models.Some artificial data were used to train the PCE model, which is used to estimate the bond capacity, from which some bias on the importance of each variable on the model output could have been introduced.Another possibility is when new data are input in the model, some results be e.g., Lognormal distribution could become Normal distribution, and  and  could also change.
Extending the proposed models to other types of bonded joints should be of extreme cautious, since as all features and results used for training models were obtained from CFRP-to-steel adhesively bonded joints.For CFRP-to-CFRP bonded joints (example see Shahverdi et al. [66]) and Shape Memory Alloy (SMA)-to-steel bonded joints (example see Wang et al. [67]) etc., their own data should be used to either validate the current models or train new models.Further notice should be drawn to the fact that the current modeling approach assumes that the steel bonded to CFRP should be limited to a piece of bulky cast-iron or mild steel in the modeling.This allows to ignore the deformation of steel substrate, which is though a reasonable in strengthening of most existing steel structures.When high strength or ultra-high strength steel is bonded with CFRP for strengthening, the steel element would be slender due to its high strength, and its deformation can no longer be ignored.The stress in the steel element would then be elevated, causing debonding [68,69].Therefore, the models in the current study should be modified to incorporate the use of high strength and ultra-high strength steel.

Application scenarios
The aim of the current study is to facilitate the design of CFRPto-steel bonded joints of the CFRP bonded strengthening system.If lap-shear tests (or other types of joint tests) comprising the selected materials for strengthening are available in the literature, or lap-shear tests are to be conducted, the design of bonded joints are to follow the test results; the proposed models can quantify the variability of the mechanical behavior.In the absence of any joint tests, our model predicts a cluster of mechanical behavior, which suggests a mean response and the variability of target joints.The process of application is thoroughly described in Appendix B.

Conclusion
In engineering design of an adhesively bonded joint between CFRP strip and steel member, the two most important features are (1) bond capacity and (2) effective bond length which guarantees the designed bond capacity.Meanwhile, good estimation on other mechanical behaviors such as load-displacement behavior is very helpful.In this study, two innovative methods were proposed to estimate the bond capacity and the effective bond length of CFRP-to-steel bonded joints, respectively.A newly proposed stochastic bond-slip model along with MC simulation were used to describe the mechanical behavior of the bonded joints.The following conclusions can be drawn:   7), (A.5), (A.12) and (A.15) to generate a mechanical response cluster, representing the possible range of mechanical behavior of the bonded joint.7. Substitute the values of   into Eq.( 14) to calculate a group of effective bond length (   ), whose statistical feature  + 2 ⋅  is adopted.The lower value between this value and 250 mm is set as the design value of    , which can be increased, if needed, to consider a safety margin.8. Check if the design value of bond length is greater than the bond length selected in step 2. If yes, the design is finished; if no, go back to step 2 and re-design.

Fig. 4 .
Fig. 4. Ratio of estimated bond capacity on tested bond capacity.

L
.Li et al.

Fig. 7 .
Fig. 7. Stress state and Mohr-Coulomb criterion: (a) Stress state with normal and shear stresses; positive and negative  refer to compressive and tensile stress, respectively.(b) Schematic plot of the Mohr-Coulomb criterion, with  denoting the friction angle and  the cohesion parameter;   , and  , are the adhesive tensile and compressive strengths, respectively.

Fig. 9 .
Fig. 9.This process is known as Monte-Carlo (MC) simulation, another commonly used method in Uncertainty Quantification.

Fig. 10 .Fig. 11 .
Fig. 10.Bond-slip curves for the case of linear and nonlinear adhesives.Red and blue dots reflect experimental measurements, while the green dots correspond to the MC simulation.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 12 .Fig. 13 .
Fig. 12. Shear stress along the bond line.Red and blue dots reflect experimental measurements, while the green dots correspond to the MC simulation.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14 .
Fig. 14.Effective bond length, specimens rearranged in ascending order using the estimated mean value.Only 60 experimental    values out of the 97 specimens are available. and  reflect the mean value and standard deviation from the MC simulation.Square marks denote the values obtained in the experiment; the colored solid lines reflect estimations of the model proposed in this study; the red dashed line denotes the suggested    .This figure is plotted using a function from Matlab community[62].(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) 2-0.25.This indicates overfitting of three models on training specimens and a lack of ability of accurately predicting unseen specimens.The PCE model, on the other hand, has a  2 score fluctuating around 0.94 and MAPE score keeping almost 0.1, indicating

Table 1
Variable ranges and Sobol indices.

Table 2
Factors and corresponding probability of survival.

Table 3
Comparison with four ML models.Trained by 90% of data and evaluated by all data, in Section 2. a 1.The proposed PCE model can fairly well estimate the bond capacity of a CFRP-to-steel bonded joint.A group of factors less than unity further ensures safety of the design.2. Sobol indices from the trained coefficients in the PCE model The newly proposed bond-slip model is a unified model spanning over linear and nonlinear adhesives smoothly.Along with MC simulation, it is capable of predicting the bond behavior, including bond-slip curve, load-displacement behavior, and shear stress profile along the bond line.No experiment is needed prior to the analysis.4. From the MC simulation, a group effective bond length can calculated.At the end, it was found  + 2 (namely mean value of these effective bond length plus two times standard deviation) can well estimate the effective bond length.An upper limit of 250 mm is suggested to prevent wasting material.5.The proposed models also have limitations.High dependency on experimental data make the models vary when new test results emerge.Only the interpolation is valid, and users should be cautious with extrapolation.All results are only suitable for CFRP-to-steel bonded joints.
interpret the importance of each variable.They suggest (1) using adhesive with higher ultimate tensile strain, (2) selecting CFRP strip with higher E-modulus, and (3) increasing CFRP width (bond width) to enhance the bond capacity.3.
[mm]   [MPa]  [mm]   [mm]   [MPa]   , [MPa]   , [%] Specimens 92-97 were tested by some of the authors of the current study.4. Check if the designed bond capacity is greater than the requirement estimated in step 1.If yes, continue; if no, go to step 2 and re-design.5. Calculate the maximum tensile strain of the CFRP strip(s) ( , ).Generate a group of maximum shear stress (  ) of the adhesive bond according to the adhesive tensile strength (  , ) and subsequently compute the parameter   .6. Substitute  , and   into Eqs.( a