The bond between carbon nanotube s (CNTs) and cementitious materials is the key characteristic for predicting the flexural strength . However, the dispersio n qu ality dominan tly changes bond mechanisms. A pro babilistic approach can benefit a ro bust model development to c a p t u r e v a r i o u s r e l a t i o n s . This study prop oses a probabilistic model based on the deterministic Kelly -Tyson theory to predict the flexural stren gth of CNT reinforced cementitious materials. Th e proposed m odel considers th e influ ences of multiple experimental variables and th eir interactions on CNT dis persion quality and the flexural streng th. A Bayesian methodology is adopted to calibrate the unknown model parameters and t h e i r s t a t i s t i c a l u ncertainty using extensive experimental test results from the literature including the author s’ work. The proposed model can b e reliably used to predict the flexural strength with reasonab le accuracy. Keywords: Carbon nanotubes ; Cementitious composites; Flexural s trength; Probabilistic model; Kelly - Tyson theory 1 Introduction Carbon nanotubes (CNTs) may sign ificantly improve the flexural strength of cemen titious materials because of their unique ph ysical and mechanical properties (Brown et al., 2019; Hajilounezhad a n d M a s c h m a n n , 2018; Hajilounezhad et al. , 2019) . For example, CNTs possess outstanding elastic modulu s and tensile strength of around 1 TPa (Demczyk et al. , 2002) and 30 GPa (Thostenson and Chou, 2003) , respectively. To yield the highest flexural strength, th e key factor is the bond stren gth b e t w e e n CNTs and the cementitious matrix which is dominan tly governed by CNT dispersion quality . The dispersion quality, itself, depends on multiple factors such as physical characteristics o f CNTs, dispersion p rocedure and mix proportio ns. Because of these compl exities , there is a concern that CNTs might no t disperse well within th e cement matrix . As a result, inconsistent improvement or even degradation in th e flexural streng th of the composite has been reported in literature (Ramezan i et al. , 2019a) . Therefore, it is vital to understand the relatio nship between CNT d ispersion quality and th e mechanical properties of the composite in ord er to facilitate their application in the co nstruction industry. The bond mechanism between CNTs and the hostin g matri x starts to engage when the stress is exerted . Therefore, the physical characteristics of CNTs are highly importan t for both th e dispersion quality and the bond mechanism . If CNTs are well dispersed within the matrix, their high aspect ratio (length -to-diameter ratio) along with their nanoscale sizes reduces the distance between adjacent nanotubes, delaying the crack propagation. On th e other hand, CNT high aspect ratio also makes them prone to agglomerate, reducing their reinforcement potential (Kafi et al. , 2016). If CNTs lack proper dispersion, the frictional forces become minimal within the ag glomerated CNTs and they easily debo nd from the matrix (Bakshi and Agarwal, 2011) , degrading the flexural st rength. To overcome this, vario us dispersion techniques have b een employ ed to uniformly disperse nanomaterials including CNTs within the cement matrix (Ramezani et al. , 2019a); amongst which an ultrasonication procedure is the most common practice (Ramezani et al. , 2016; Sadeghi -Nik et al. , 201 7). Despite extensive research effort s, the contribution of CNTs to the flexu ral strength of cementitious materials is not yet fully understood. Therefore, in 182 INTERNATIONAL CONFERENCE ON “CEMENT /hyphen.cap BASED MATERIALS TAILORED FOR A SUSTAINABLE FUTURE " IN HONOUR OF PROF . SURENDRA P . SHAH AND PROF . TURAN ÖZTURAN May 27 /hyphen.cap29, 2021 • ISTANBUL / TURKEY 2 the authors’ previous wo rk, ex tensive literature data was analyzed to identify the interactio ns between multiple experimental variables affecting the mechanical properties including the flexural strength (Rame zani et al., 2019a). Then , three empirical expressions were introduced to include the identified interactions which enabled t h e predictio n of the mechanical properties of CNT -cement nan ocomposites within 20% error of the experimental test data using a simple regression mo del (Ramezani et al. , 2019b). However, the proposed regressio n model has a limitation of the application: it does not reflect th e bond mechanisms of CNTs affecting the flexural strength. This study aims to develop a reliable predict ion model for the flexural strength of cemen titiou s materials (pastes and mortars) reinforced with CNTs . To this end , the Kelly -Tyson model (Kelly and Tyson, 1965) which is extensively used to predict the tensile strength of brittle and ductile materials containing CNTs is selected as the base model. The Kelly -Tyson model is then modified by adding CNT orientation, length and volume fraction factors. Finally , additional critical relations are added to inclu de different mechanisms of CNTs within cementitious materials. Besides, because incorporating CNTs is correlated with high degree of uncertainty, a probabilistic approach is adopted to consider various sources of uncertainty to affect the flexural strength . The proposed model provides v aluable information for eng ineers and desig ners to determine and optimize the most important variables and the interactions between them to achieve the desired flexural strength. 2 Modified Kelly- Tyson Model Different theoretical models are used to predict the tensile/flexural strength of CNT reinforced composites (e.g., polymer, metal and ceramic ) (Chen et al. , 2017; Lachman et al. , 2012 ; Zare, 2016). Most of these models, however, assume a homogen ous dispersion a n d p e r f e c t a d h e s i o n between CNTs and the matrix, resultin g in inaccurate predictions. Despite the very high i mportance of these factors in determining the flexu ral strength, very few researchers tried to include them in prediction models. Besides, concerning cementitious comp osites, CNTs mig ht also influ ence the hydration process of cement which needs to be consi dered (Ramezani et al. , 2019b). Therefore , there are very few models available to predict the flexural strength of CNT -cement nanocomposites. To address this, the current study adopts the Kelly -Tyson mod el as the base model to co nsider the bond mechanisms of CNTs in cementitious materials . The Kelly -Tyso n model is then modified to include the mechanisms of CNTs within cementitious materials. For condition of equal elastic strain in fiber and matrix, the rule of mixtures is considered as the simplest way to model the tensile strengt h of composites reinforced with aligned continuous fibers (Agarwal et al. , 2017): ߪ௖=ߪ௙ݒ௙+ߪ௠൫1െݒ௙൯ (1) where ߪ௖ is the ultimate tensile strength of the composite (MPa ), ߪ௙ is the tensile strength of the fiber (MPa ), ߪ௠ is the tensile strength of the matrix at the ultimate stra in of the composite (MPa ) and ݒ௙ is the volume fraction of the fiber (unitless). To consider the contribution of the semi -aligned short discrete fibers with lengths (L) smaller than the critical length , Eq. (1) can be modified to include the fiber length and orientation factor s as follows (Kelly and Tyson, 1965) : ߪ௖=16߬௜൬ܮ݀൰ݒ௙+ߪ௠൫1െݒ௙൯ (2) where ߬௜ is the interfacial shear stress (MPa ) and d is the fiber diameter ( mm). However, due to the very low volume fraction of CNTs used in cementitious materials (ݒ௙= 0.00004 -0.00128; based on the literature), the predicted flexural strength of CNT -cement nanocomposites using Eq. ( 2) is almost the same as the flexural strength of the matrix. The authors’ preliminary study has show n that ݒ௙ଶ/ଷ is appropriate to fit the experimental data. Therefore, t he modified form of the Kelly -Tyson model to predict the flexural strength of CNT -reinforced cementitious materials can be expressed as follows: ߪ௖=16߬௜൬ܮ݀൰ݒ௙ଶ/ଷ+ߪ௠൫1െݒ௙ଶ/ଷ൯ (3) Note that the modified Kelly -Tyson model (Eq. ( 3)) assumes uniform CNT dispersion and perfect bonding between CNTs and cement matrix . Therefore, the flexural strength increases as the CNT aspect ratio ( AR = ܮ݀ൗ) and/or ݒ௙ increases. Nevertheless, experimental studies exhibit nonlinear behavior: the flexural strength incre ases as AR 183 INTERNATIONAL CONFERENCE ON “CEMENT /hyphen.cap BASED MATERIALS TAILORED FOR A SUSTAINABLE FUTURE " IN HONOUR OF PROF . SURENDRA P . SHAH AND PROF . TURAN ÖZTURAN May 27 /hyphen.cap29, 2021 • ISTANBUL / TURKEY 3 and/or ௙ increases up to certain limits, beyond which the flexural strength degrades. The interfacial shear stress can be estimated by the link between the Pukan szky and the Kelly-Tyson models (Lazzeri and Phuong, 2014). 3 Proposed Critical Relations To understand the mechanisms of CNTs within cementitious materials, three critical relations were proposed in the authors’ previous work (Ramezani et al. , 2019b ) to capture the interacti ons between the important experimental variables to pred ict the mechanical properties in cluding the flexural strength. 1)dispersion relation ( ᢡ஽) to correlate CNT dispersion as a function of AR, CNT volume fraction in aqueous solution ( ݒ௙ି௪), superplasticizer -to-CNTs ( SP/CNTs ) ratio, and Ultrasonication Energy Indicator ( UEI) 2)matrix relation (ᢡெ) to include the interactions between water -to-cement (w/c) and sand-to-cement ( s/c) ratios 3)hydration age relation (ᢡு) to consider the interactions between CNT concentration ( ț) and hydration age of cement ( t) A brief description of these critical relations is provided in Table 1. More details are provided in the authors’ previous study (Ramezani et al. , 2019b) . The applicability of the proposed critical relations to predict the flexural strength is validated in Section 5. Table 1. Critical relations affecting mechanical properties . Critical Relation Equation Description s ᢡࡰ 1െ൥ܴܣ×ݒ௙ି௪ ܫܧܷ×(ܲܵݏܶܰܥൗ)൩If ݒ௙ି௪ or ܴܣ= 0 (no CNTs added), ᢡ஽= 1 (perfect dispersion). As ݒ௙ି௪ and/or ܴܣ increases, ᢡ஽ gets smaller. As UEI and/or SP/CNTs ratio increases, ᢡ஽ gradually reaches 1. Beyond certain UEI and/or SP/CNTs ratio, ᢡ஽ gets constant. ᢡࡹ (1+ݏܿΤ) (ݓܿΤ)As w/c ratio decreases, the value of ᢡெ increases. As s/c ratio increases, the value of ᢡெ increases. ᢡࡴ ൬ݐ 4 + 0. 85ݐ൰சIf CNT is not added (ț = 0 c-wt%), ᢡு= 1. At low CNT concentrations, the influence of t on ᢡு is minimal. As ț increases, ᢡு reaches 1 at later ages (i.e., a higher value of t). In this study, three critical relations (ᢡ஽,ᢡு,and ᢡெ) are added to Eq. ( 3) to fit the model t o the experimental test results as follows: ߪ௖=ᢡ஽ᢡுᢡெ൤16߬௜(ܴܣ)ݒ௙ଶଷൗ+ߪ௠൫1െݒ௙ଶ/ଷ൯൨ (4) In addition, a Bayesian methodology is adopted to formulate the proposed model in a probabilistic manner to consider different sources of uncertainty such as the statistical uncertainty in the estimation of the unknown model parameters, the model errors associated with the inaccurate form of the model and missing variables (Gardoni et al., 2002) that might influence the flexural strength. 4 Formulation of the Probabilistic Model The probabilistic approach is used based on the modified deterministic Kelly -Tyson model (Eq. ( 4)). Then, additional correction terms (herein, ᢡ஽, ᢡு, ᢡெ) and model error are introduced to the deterministic expression to account for inherent bias and to capture the remaining variability in re siduals, respectively (Gardoni et al. , 2002) . The probabilistic model relates the quantity of interest (herein, the flexural strength; ߪ௖) to a set of measurable variables (defined as ࢞(=ݔଵ,ݔଶ,ݔଷ,…) ) such as AR, ț, UEI, SP/CNTs ratio, w/c ratio, s/c ratio, t, etc. The probabilistic capacity model can be written as follows: 184 INTERNATIONAL CONFERENCE ON “CEMENT /hyphen.cap BASED MATERIALS TAILORED FOR A SUSTAINABLE FUTURE " IN HONOUR OF PROF . SURENDRA P . SHAH AND PROF . TURAN ÖZTURAN May 27 /hyphen.cap29, 2021 • ISTANBUL / TURKEY 4 ܥ(࢞,દ)=ܥ(࢞)+ߛ(ݔ,៭)+ݏߝ (5) where ܥ(࢞,દ) is the capacity model of interest (herein, ߪ௖), ࢞ is a set of measurable variables, દ=(ߠ,ߪ),ߠ= (ߠଵ,ߠଶ,ߠଷ,…) denotes a set of unknown model parameters used to fit the data, ܥ(࢞) is the selected deterministic model, ߛ(ݔ,ߠ) is the correction term (herein, ᢡ஽, ᢡு, and ᢡெ), ݏ is a standard deviation of the error term taken from Var [ܥ(࢞,દ)]=ݏଶ, ߝ is a random variable with zero mean and unit variance and ݏߝ is the error term of the model. The final form of the probabilistic model used to predict the flexural strength is as follows : ߪ௖=ᢡ஽(ݔ஽,ߠ஽)×ᢡு(ݔு,ߠு)×ᢡெ(ݔெ,ߠெ)×ቂ1/6߬௜(ܴܣ)ݒ௙ଶଷൗ+ߪ௠൫1െݒ௙ଶ/ଷ൯ቃ+ ݏߝ( 6) where ᢡ஽(ݔ஽,ߠ஽)=ቈ1െቆ஺ோഇభ×௩೑షೢഇయ ௎ாூഇమ×൫ௌ௉஼ே்௦ൗ൯ഇరቇ቉, ᢡு(ݔு,ߠு)=ቀ௧ ସା଴.଼ହ௧ቁசఏఱ, and ᢡெ(ݔெ,ߠெ)=(ଵା௦௖Τ)ഇల (௪௖Τ)ഇళare critical relations after adding the unknown model parameters . To estimate the unknown model parameters , a Bayesian updating rule is used (Box and Tiao, 2011) : ݂(ࢨ)=ߣܮ(દ)݌(દ) (7) where દ is a vector of unknown model parameters, ݌(દ) is prior distribution of દ based on knowledge about દ that represents the information available before collecting the data, ܮ(દ) is the likelihood function that captures the information on દ from data, ߣ=[׬ܮ(દ)݌(દ)݀(દ)]ିଵ is a normalizing factor and ݂(દ) is a posterior distribution. Once ݂(દ) is known, the posterior mean vector and the covariance matrix are obtained. Because there is no prior information about દ before collecting data , a non -informative prior distribution ݌(દ)ןଵ ௦ is used (Gardoni et al. , 2002) . If new data becomes available in the future, Eq. (7) can still be used to update the estimates of the unknown model parameters by using the current ݂(દ) as a new previous distribution . This approach has been extensively applied to various applications such as structural engineering and materials science (Choe et al. , 2007; Ga rdoni et al. , 2013; Gardoni et al. , 2009) . 5 Probabilistic Model Assessment A total of 105 published data was collected by the authors from several research papers to calibrate the unkn own model parameters in Eq. ( 6) using a Bayesian methodology. The dataset is reported in (Ramezani, 2019). To evalu ate the robustness of the proposed model , the total database is not used for the development of the probabilistic model due to two fundamental issues: overfitting and being overly optimistic about the model error. Therefore, the database is split into two sets of data using random selection; 80% of the data (84 data points) is used to develop the models (Dataset I) and the remaining 20% (21 data points) is used to validate the proposed probabilistic model (Dataset II). Table 2 show s the posterior statistics of Ĭ estimated using Dataset I. Table 2. Posterior statistics of the unknown model parameters using dataset I Unknown Parameter Mean Standard Deviation Correlation Matrix ࣂ૚ ࣂ૛ ࣂ૜ ࣂ૝ ࣂ૞ ࣂ૟ ࣂૠ ࢙ ࣂ૚ 0.393 0.0280 1 -0.539 0.998 0.950 0.861 -0.568 -0.977 0.628ࣂ૛ 0.280 0.0086 1 -0.571 -0.697 -0.658 -0.040 0.608 -0.428ࣂ૜ 0.481 0.0299 1 0.948 0.854 -0.567 -0.979 0.628ࣂ૝ 0.132 0.0369 1 0.946 -0.388 -0.963 0.675ࣂ૞ 1.059 0.0575 1 -0.213 -0.884 0.598ࣂ૟ 0.173 0.0046 1 0.423 -0.529ࣂૠ 0.127 0.0146 1 -0.601࢙ 0.149 0.0008 1The probabilistic model after estimating the unknown model parameters is shown below: 185 INTERNATIONAL CONFERENCE ON “CEMENT /hyphen.cap BASED MATERIALS TAILORED FOR A SUSTAINABLE FUTURE " IN HONOUR OF PROF . SURENDRA P . SHAH AND PROF . TURAN ÖZTURAN May 27 /hyphen.cap29, 2021 • ISTANBUL / TURKEY 5 ߪ௖= ቎1െቌܴܣ଴.ଷଽ×ݒ௙ି௪଴.ସ଼ ܫܧܷ଴.ଶ଼×൫ܲܵݏܶܰܥൗ൯଴.ଵଷቍ቏ ×൬ݐ 4+0 . 85ݐ൰ଵ.଴଺ச ×(1+ݏܿΤ)଴.ଵ଻ (ݓܿΤ)଴.ଵଷ× ቂ1/6߬௜(ܴܣ)ݒ௙ଶଷൗ+ߪ௠൫1െݒ௙ଶ/ଷ൯ቃ (8) Figure 1 show s the perform ance of the proposed probabilistic model versus the experimental test data using Eq. (8). The 1 :1 line (see the dashed line in Figure 1) indicates that the predicted flexural strength is identical to the experimental test data. The solid circles are the data used to develop the model (Dataset I) and the open triangles are the data used to validate the model (Dataset II). The robustness of the proposed probabilistic model is validated using Dataset II which is not used for model development, indicating the accuracy of the propos ed probabilistic model by showing the desirable performance (i.e., within ± standard deviation of the error term in the proposed model; ݏ = 15%). Note that three data points inscribed within the oval in Figure 1 are related to the flexural strength of cem ent mortar s reinforced with mechanical functionaliz ed CNTs at different ages (Konsta -Gdoutos et al. , 2017) . The mechani cal functionalized CNTs provide superior interfacial bond strength , result ing in under - prediction of the flexural strength. Figure 1. Predicted values vs. experimental test results for flexural strength. Figure 2 shows the ratio of the predicted to the experimental value of the flexural strength ( ܴி=ߪ௖௣௥௘ௗ. ߪ௖௘௫௣. ൘; y- axis) versus different influential variables ( x-axis; AR, ݒ௙, ț, t, w/c ratio, s/c ratio, UEI, and SP/CNTs ratio) when using the unknown m odel parameters shown in Table 1 . The solid lines show ±15% of the model error. The mean and median values of ܴி are 1.01 with the standard deviation of 0.11. In addition, ܴி exhibits a normal distribution with respect t o the reference line of ܴி= 1 (i.e., the predicted flexural strength is identical to the experimental flexural strength) , for all ranges of the investigated variables ( x-axis). This suggests that the proposed critical relations can be reliably used to predict the flexural strength without any bias. Figure 2 . Influence of the experimental variables on ܴி. 186 INTERNATIONAL CONFERENCE ON “CEMENT /hyphen.cap BASED MATERIALS TAILORED FOR A SUSTAINABLE FUTURE " IN HONOUR OF PROF . SURENDRA P . SHAH AND PROF . TURAN ÖZTURAN May 27 /hyphen.cap29, 2021 • ISTANBUL / TURKEY 6 6 Conclusions This paper proposes a bias-free probabilistic mo del based on the available deterministic Kelly -Tyson model to predict the flexural stren gth of CNT -cement nanocomposites (p astes and mortars). The prop osed model considers the influences of CNT prop erties, disp ersion state, matrix compositio n and cement hydration using a Bayesian approach. T he proposed model is able to predict the flexural strength within ± 1 5 % e r r o r o f the experimental test data. When additional data are available in the future, the propo sed model can be upd ated with appro priate values or additional critical relation s which can be discovered b y oth er researchers. Notations ߪ௖: flexural strength of CNT-cement nanocomposite ߪ௠: flexural strength o f the cement matrix without CNTs ௙: tensile streng th of the fiber : CNT volume fraction ݀ :CNT diameter L: CNT length ߬௜: interfacial shear stress between CNTs and matrix AR: CNT aspect ratio w/c: water -to-cement ratio s/c: sand-to-cement ratio t: hydr ation age of cement SP/CNTs : superplasticizer -to-CNTs ratio UEI: ultrasonication energy indicator ț: CNT concentration ᢡ஽: dispersion relation ᢡு: hydration age relation ᢡ: matrix relation ܥ(࢞,દ): probabilistic capacity model of interest (herein, ߪ௖) ࢞: set of measurable v ariables ܥ(࢞) :deterministic model દ: vector of unknown model parameters; દ=(ߠ,ܥ) ߠ௜: set of unknown model parameters used to fit the data ߛ(ݔ,ߠ) :correction term (herein, ᢡ஽, ᢡு and ᢡ) ܥ: standard deviation of the error term ߝ: random variable with zero mean a nd unit variance : error term of the model ݌(દ): prior distributio n of દ ܮ(દ): likelihood function ߣ :normalizing factor; ߣ=[׬ܮ(દ)݌(દ)݀(દ)]ିଵ ௙(દ): posterior distrib ution of દ