Probabilistic-based assessment for tensile strain-hardening potential of fiber-reinforced cementitious composites

Cement and Concrete Composites 91 (2018) 108–117

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Probabilistic-based assessment for tensile strain-hardening potential of fiberreinforced cementitious composites

T

Junxia Lia,b, En-Hua Yangc,∗ a

Interdisciplinary Graduate School, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore Residues & Resource Reclamation Centre, Nanyang Environment and Water Research Institute, Nanyang Technological University, 1 Cleantech Loop, 637141 Singapore c School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798 Singapore b

A R T I C LE I N FO

A B S T R A C T

Keywords: Tensile strain-hardening Fiber-reinforced cementitious composites (FRCC) Engineered cementitious composites (ECC) Strain-hardening cementitious composites (SHCC) Multivariate adaptive regression splines (MARS) First order reliability method (FROM)

This paper presents a novel probabilistic-based approach considering material heterogeneity to assess the tensile strain-hardening potential of fiber-reinforced cementitious composites (FRCC). Multivariate adaptive regression splines (MARS) method is used to explicitly express the performance indices governing tensile strain-hardening. First order reliability method (FROM) is then carried out to evaluate tensile strain-hardening potential of FRCC. Results show that strain capacity of FRCC has a negative correlation with failure probability and it increases exponentially with decreasing failure probability. Analysis of variance (ANOVA) decomposition of MARS model indicates increasing fiber strength and volume, reducing fiber modulus, and moderate interface frictional bond are effective means to improve tensile strain-hardening potential of FRCC. The proposed approach is thus able to consider uncertainty in evaluating tensile strain-hardening potential of FRCC by treating micromechanical parameters as random variables and taking heterogeneity into account in the probabilistic-based model.

1. Introduction Fibers have been used to address the brittleness of cement-based material. Through proper tailoring, studies have shown fiber-reinforced cementitious composites (FRCC) can even exhibit tensile strain-hardening behavior with high tensile ductility of several percent, e.g. engineered cementitious composites (ECC) [1,2] or strain-hardening cementitious composites (SHCC) [3,4]. Tensile strain-hardening of FRCC is fundamentally governed by two criteria developed by Li and coworkers [5] as shown below. δ0

Jb' = σ0 δ0 −

∫ σ (δ ) dδ ≥ Jtip 0

(1) (2)

σ0 ≥ σc ’

where Jb is the complementary energy which can be calculated from the bridging stress σ versus crack opening δ curve, σ0 is the maximum bridging stress corresponding to the opening δ0, Jtip=Km2/Em is the crack tip toughness where Km is the matrix fracture toughness and Em is the matrix Young's modulus, and σc is the matrix tensile cracking strength governed by the matrix fracture Km and flaw size a0. The bridging stress-crack opening relation σ(δ) can be determined with an analytical model, involving a set of ten micromechanical parameters

∗

Corresponding author. E-mail address: [email protected] (E.-H. Yang).

https://doi.org/10.1016/j.cemconcomp.2018.05.003 Received 17 July 2017; Received in revised form 13 March 2018; Accepted 1 May 2018 Available online 08 May 2018 0958-9465/ © 2018 Elsevier Ltd. All rights reserved.

describing the fiber and the fiber/matrix interface properties. A numerical solution for the analytical model was proposed by Yang et al. [27] to calculate the σ(δ) as well as Jb’. Eqn. (1) ensures that the steady state crack propagation prevails under tension, while Eqn. (2) is the criterion to allow development of multiple cracks. Satisfactory of both equations is necessary to achieve the tensile strain-hardening of FRCC. This model has been used to guide ingredients selection and component tailoring to achieve tensile strainhardening performance at minimum fiber content [2,5]. For instance, tailoring of fiber types and geometry [6], tailoring of fiber/matrix interface through fiber surface modification [7,8], and control of matrix flaw size and distribution [9] were guided through the tensile strainhardening assessment of the composite. It is taken as a deterministic model assuming uniform fiber, matrix, and interface properties and thus constant micromechanical parameters are used as inputs to evaluate the tensile strain-hardening potential. In reality; however, heterogeneous nature of cement-based material, FRCC in particular, is inevitable. Heterogeneity may originate from variation of properties of ingredients such as inconsistent fiber diameter and fiber strength, processing (variation of flaw size and distribution and non-uniform fiber distribution [10]), and curing. To ensure robust tensile strain-hardening taking into account of material variability, one approach is to propose large margins of Jb’/Jtip

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J. Li, E.-H. Yang

and σ0/σc [11,12]. It is analogy to the build-in safety factor of the prescriptive-based design of structures. It has been demonstrated experimentally that ECC reinforced with polyethylene (PE) fibers requires Jb’/Jtip > 3 and σ0/σc > 1.2 to ensure saturated multiple cracking and robust tensile strain-hardening [13], while polyvinyl alcohol (PVA) fiber-reinforced ECC requires an even higher σ0/σc of 1.45 and above [14]. This empirical-based approach requires large amount of testing and the values are only limited to the specific SHCC system of interest. Any changes to the ingredients and proportions of the SHCC may require re-establishment of such safety factor. Thus, there is a need to develop a holistic approach which can be applied universally and adopted for tailoring different SHCC system. This paper presents a probabilistic approach to achieve this goal. In the approach, heterogeneity is considered in the micromechanical model to capture the uncertainty and to evaluate the tensile strainhardening potential of FRCC. It is realized by taking all micromechanical parameters, including fiber properties (i.e. fiber volume Vf, fiber modulus Ef, fiber length Lf, fiber diameter df, and fiber tensile strength σfu), interface properties (i.e. chemical bond strength Gd, frictional bond strength τ0, slip hardening coefficient β, fiber strength reduction coefficient f’, snubbing coefficient f), and matrix properties (i.e. matrix modulus Em, matrix fracture toughness Km, flaw size c), as random variables to capture sources of randomness. It is noteworthy that the randomness of fiber orientation and fiber embedment length have been accounted for in the micromechanical model σ(δ) by adopting probability density functions that describe the spatial variability of the fibers [16] in 2D. Other randomness, such as variation of raw ingredients, processing induced variation of flaw size and distribution and curing induced variation of matrix and interface properties, can be captured by adopting suitable probability density functions that describe the variability. However, randomness of fiber distribution caused by the formwork geometry and mixing induced uncertainty of quality of fiber dispersion are not considered in the current approach. As such, fiber bridging σ(δ) as well as Jb’, Jtip, σ0, and σc would also become random variables, and thus the reliability of the two tensile strain-hardening criteria (Eqns. (1) and (2)), i.e. tensile strain-hardening potential of FRCC, can be quantitatively assessed. To implement this probabilistic-based assessment, multivariate adaptive regression splines (MARS) models are firstly created to obtain an explicit mathematical expression of Jb’ and σ0 instead of integral analytical form [15] and numerical form [16]. Probabilistic assessment is then conducted on FRCC mixes by means of first order reliability method (FROM) to assess the reliability.

Fig. 1. 3D joint PDF.

include estimating the deformation of asphalt mixtures, analyzing shaking table tests of reinforced soil wall and analysis of geotechnical engineering systems [19–21]. Jtip is already an immediate function of Km and Em. As for σc, it is simply approximated to be the matrix cracking strength σmu in the current study. Reliability is defined as the probability of the performance function G(x) or F(x) greater than zero, i.e. P{G(x) > 0} and P{F(x) > 0}. The tensile strain-hardening potential of FRCC is evaluated with the concept of failure probability Pf. The calculation of the probability of failure Pf is the integration of the probability density function (PDF) over the failure domain as indicated by the volume abcd in Fig. 1. Many methods can be used for probabilistic assessment, such as response surface method (RSM) and Monte Carlo Simulation (MCS). FORM is used in the current study, where random variables are assumed as standard normal distributions, and thus Pf can be calculated by

Pf 1 = P {G (x ) < 0} ≈ ϕ (−β1)

(5)

Pf 2 = P { F (x ) < 0} ≈ ϕ (−β2)

(6)

where Pf is the probability that the failure event occurs; ϕ is the distribution function of the standard normal distribution; and β is the reliability index computed as

x i − μi ⎞T −1 ⎛ x i − μi ⎞ [R] β = min ⎛ ⎝ σi ⎠ ⎝ σi ⎠ ⎜

2. Conceptual framework of the proposed method

(3)

F (x ) = σ0 − σc = 0  

(4)

⎜

⎟

(7)

where xi is the random variable; μi and σi are the mean value and the standard deviation of the random variable, respectively; and R is the correlation matrix.

Satisfactory of both Eqns. (1) and (2) is necessary for tensile strainhardening and therefore the boundary separating the safe and failure domain is the limit state surface (performance function [17]) which can be defined as

G (x ) = Jb' − Jtip = 0

⎟

3. Mathematical expression of Jb’ and σ0 through MARS 3.1. Database for Jb’ and σ0 In seek of MARS models for Jb’ and σ0, a sound and large enough database is created using the numerical solution developed by Yang et al. [16]. A number of groups of inputs and outputs are acquired for establishment of the prediction model by regression fitting method. In this study, ten relevant micromechanical parameters (denoting x1 to x10 as input variables) and two performance index (denoting y1 and y2 as outputs) are shown in Table 1, the selected values are typical ones in PE fiber [8], PVA fiber [22] and PP fiber reinforced cementitious composites [6]. All possible combinations of input variables are used to get corresponding outputs, except that some outliers are removed using the residual analysis because it may lead to crude model. Thereof, 527 groups (75%) of the observations were randomly selected for training

where x denotes all micromechanical parameters as random variables. Mathematically, G(x) > 0 and F(x) > 0 denote the ‘safe’ domain, while G(x) < 0 and F(x) < 0 denote the ‘failure’ domain. In this case, the safe domain represents the possession of tensile strain-hardening behavior of FRCC, and the failure domain stands for tensile strainsoftening behavior. The limit state surface G(x) and F(x) are required to be known explicitly. Thus, closed-form mathematical functions of Jb’ and σ0 are constructed through MARS, which is a nonparametric regression method to model the nonlinear responses between input variables and output introduced by Friedman in 1991 [18]. The main advantages of MARS are of its capacity to deal with high dimensional data and easy to interpret the model. Extensive applications of MARS 109

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Table 1 Summary of input variables and outputs. Inputs & outputs

Input variables

Outputs

Parameters

x1 x2 x3 x4 x5 x6 x7

Ef Lf df σfu Vf Gd τ0

x8 x9 x10

β f' f

y1 y2

Jb’ σ0

Observations

(GPa) (mm) (μm) (MPa) (%) (J/m2) (MPa)

PE

PVA

PP

120 12.7, 19 20, 38 2700 0.5, 1, 1.5, 2 0 0.3, 0.6, 0.9, 1.5 0 0.5 0.5, 0.8

22, 42.8 10 40 1060, 1620 1, 2, 3 0.5, 2.5, 5.0 0.5, 1.5, 3, 5 0.05, 0.5, 5 0.3 0.2, 0.5

11.6 8, 19 16.6 400, 928 2, 4, 6 0, 1.5 0.5, 1, 3 0.005 0.1 0.39

(J/m2) (MPa)

and the remaining 175 groups (25%) used as testing data.

3.2. MARS model of Jb’ and σ0 Based on the MATLAB platform, two sets of program code for MARS model of Jb’ and σ0 are written by virtue of ARESLab toolbox [23] as Table A.1 in the appendix. For the sake of best fitting MARS model of Jb’ and σ0, piecewise linear and piecewise cubic basis functions (BFs) with the maximum interaction of 2 are applied. By comparison, piecewise linear function is selected for the following studies because it can reach high accuracy with less complexity and computational time. High R2 approaching 1 for testing and training data are obtained as graphically illustrated in Fig. 2. A summary of basis functions (BFs) and the final equations of Jb’ and σ0 are listed in Tables A.2 and A.3. MARS model can automatically prunes the model by removing the extraneous variables with the lowest contribution and the relative importance assessment of all variables is processed using analysis of variance (ANOVA) procedure. Table A.4 in the appendix displays the ANOVA decomposition of the built MARS model for Jb’ and σ0. The first column lists the ANOVA function number. The second column gives an indication of the importance of the corresponding ANOVA function, by listing the GCV score for a model with all BFs corresponding to that particular ANOVA function removed. The GCV score can be used to evaluate whether the ANOVA function is making an important contribution to the model, or whether it just marginally improves the global GCV score. The third column gives the particular input variables associated with the ANOVA function. Fig. 3 gives the plots of the relative importance of the input variables for the two MARS models. It can be observed that variable 1 (Ef) and variable 7 (τ0) are the two most important parameters to affect the maximum complementary energy Jb’ of fiber bridging. Variable 5 (Vf), variable 4 (σfu), variable 3 (df) and variable 7 (τ0) are significantly important in determining the fiber bridging strength σ0. The relative importance of micromechanical parameters as above gives particularly useful information on component tailoring and ingredient selection for tensile strain-hardening FRCC design. As shown in Table 2, increasing σfu and Vf and reducing Ef are effective means to improve the tensile strain-hardening performance of FRCC. Furthermore, moderate interface frictional bond τ0 is extremely critical to achieve the optimum condition for tensile strain-hardening by ensuring a balance behavior between fiber rupture and fiber pullout otherwise either Jb’ or σ0 is compromised.

Fig. 2. MARS prediction for (a) Jb’ and (b) σ0 of training and testing data.

4. Probabilistic-based assessment for FRCC tensile strainhardening potential 4.1. Probabilistic distribution of micromechanical parameters Probabilistic distribution of each micromechanical parameter is determined from literature and experimental data. It has been reported that variation of fiber diameter df has minimal effect on the fracture energy and tensile strength [24] and thus uniform distribution of df is assumed in the current study. Fiber length Lf is assumed to be uniformly distributed [25]. Since there is no study reporting the variation of fiber elastic modulus Ef, fiber tensile strength σfu and fiber volume Vf, they are assumed to follow normal distribution with a relative large variation of ± 30% for conservative estimation. The mean values of all fiber related parameters are measured directly. Similarly, micromechanical parameters related to the fiber/matrix interface, including chemical bond strength Gd, interface friction bond strength τ0, and slip hardening coefficient β are assumed to follow normal distribution due to lack of information from published 110

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distribution. Mean values of f and f’ are taken from published literature as summarized in Table 3. Matrix modulus Em and matrix fracture toughness Km are also assumed as normal distribution. Mean values of these two parameters can be determined by means of the compressive test and the matrix toughness test. Matrix tensile cracking strength σmu is assumed to follow the two-parameter Weibull distribution according to [26]. Weibull analysis is a widely-used statistical tool for describing the strength behavior of brittle materials, which is based on the assumption that failure at the most critical flaw leads to total failure of the specimen. Weibull distribution is easy to interpret and very versatile. The cumulative distribution function for the Weibull distribution is

F (x , k , λ ) = 1 − e

k − x λ

()

(8)

where k is the shape parameter, λ is the scale parameter, which determines the varying range of the variable. Mean value of σmu can be determined experimentally. Matrix flaw size c is simply related to σmu according to Eqn. (9) [27]. 2

π Km ⎞ c=⎛ ⎝ 2 σmu ⎠ ⎜

⎟

(9)

where σmu is the tensile strength of matrix. 4.2. Tensile strain-hardening potential assessment of FRCCs In the current study, FORM is used to obtain the failure probability of tensile strain-hardening potential. The details of FORM can be found in Figure A.1. In the FORM procedure, the performance functions and MARS models are incorporated into an EXCEL spreadsheet using the approach developed by Low and Tang [28]. The Distribution column indicates the probabilistic distribution, such as Normal, Lognormal, Uniform, Weibull, and Gamma distribution, of micromechanical parameters. The Para 1 and Para 2 column are key characteristics describing the probability density function. For example, Para 1 and Para 2 are the mean value and the standard deviation of a Normal distribution, respectively. The MARS models of Jb’ and σ0 are formatted into the MARS column. The design point (xi*) is obtained using EXCEL's built-in optimization routine SOLVER with the objective of minimizing β in cell X2, subject to the constraint of the performance functions of G(x) and F(x) as Eqns. (3) and (4), respectively. Iterative numerical search for xi* starts from the mean values of the original random variables by setting ni = 0 initially, whereby xi* is a function of ni. Pf is the failure probability of the strain-hardening criteria of SHCC corresponding to G(x) < 0 and F(x) < 0. Different tensile strain-hardening FRCC mixes with strain capacity below 1% and up to above 4% are selected from literature [7,13,29]. Mixes 1 to 3 are types of lightweight FRCCs using cenosphere with incorporation of slag or fly ash in order to achieve high ductility, and PVA fiber with 1.2% coating is used in these mixtures [29]. Mixes 4 to 7 are mixtures only consist of cement, sand, water and PVA fibers with different coating of 0.3%, 0.5%, 0.8%, and 1.2% [7]. In mixes 8 to 11, PE fiber with different fiber volume of 0.5%, 0.75%, 1% and 1.25% are applied into the simple matrix with cement and silica fume [13]. Mix design and relevant micromechanical parameters of the 11 FRCC mixes are summarized in Tables A.5 and A.6. The failure probability of tensile strain-hardening behavior for the eleven FRCCs are calculated and listed in Table A.5. Pf1 and Pf2 denote the failure probability of the energy criterion and the strength criterion, respectively. Lower Pf1 means the composite has higher possibility of steady-state cracking mode in favor of tensile strain-hardening behavior, while lower Pf2 indicates the material has higher potential to produce more cracks because the fiber bridging strength is sufficient to take the ambient load once cracks occur. Satisfactory of both Eqns. (1) and (2) is necessary for tensile strain-hardening and therefore max.{Pf1, Pf2} is taken as the overall failure probability Pf of the composite (Table A.5).

Fig. 3. Relative importance of the input variables for (a) Jb’ and (b) σ0 according to the MARS model. Table 2 Critical micromechanical parameters to enhance Jb’ and σ0.

’

Complementary energy (J/m ) Jb ↑ Fiber bridging strength (MPa) σ0 ↑ 2

Fiber related

Interface related

σfu ↑ Ef ↓ σfu ↑ Vf ↑

τ0 ↓ τ0 ↑

Table 3 Reference values of f and f’. Fiber type

σfu (MPa)

f

f'

Reference

Polyvinyl alcohol (PVA) Polypropylene (PP) Polyethylene (PE)

1666 400 2400

0.5 0.39 0.8

0.3 0.1 0

Kanda & Li 1999 [30] Yang & Li 2010 [6] Kanda & Li 1999 [30]

literature. Mean values and standard deviation of Gd, τ0, β are determined from the single fiber pullout tests. The snubbing coefficient f and fiber strength reduction coefficient f’ are assumed to follow uniform 111

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and σ0 based on 702 groups of data with a total of ten random variables. Tensile strain-hardening potential of eleven FRCCs is assessed by FROM based on the established MARS model. ANOVA decomposition of MARS model indicates that increasing fiber strength and volume and reducing fiber modulus are effective means to improve the tensile strain-hardening performance of FRCC. Furthermore, moderate interface frictional bond is extremely critical to achieve the optimum condition for tensile strain-hardening by ensuring a balance behavior between fiber rupture and fiber pullout otherwise either Jb’ or σ0 is compromised. Results show that strain capacity of tensile strain-hardening FRCC has a negative correlation with failure probability and it increases exponentially with decreasing failure probability. This shows the proposed method is able to consider material heterogeneity in evaluating tensile strainhardening potential of FRCC by treating micromechanical parameters as random variables. It is noteworthy that mathematical expression of Jb’ and σ0 through MARS in the current study is based on a limited database. Only common fiber types and matrix types, and typical interface properties (Table 1) are used to create the database. Thus, Table 1 represents the domain in which the illustration in this paper is valid. Furthermore, probabilistic distribution of micromechanical parameters used in the illustration of this study is either assumed or derived from limited data which may not represent the exact distribution. Admittedly, the proposed approach for assessing tensile strain-hardening potential of FRCC was validated with only eleven selected mixes with strain capacity of 0.7–4.9% from literature (Fig. 4), which did not cover all possible combination scenarios and tensile behavior types. While this work aims at universal procedures to estimate the strainhardening potential of FRCCs, the current limitations are strong and this goal is far from being reached due to the huge complexity of the general problem. Further studies are needed: 1) to enrich the database covering different fiber types, matrix types, and interface properties, and to have a more general expression of Jb’ and σ0 in a wider domain; 2) to determine probabilistic distribution of micromechanical parameters originated from variation of raw ingredients, processing, and curing; and 3) to validate the proposed approach with FRCC mixes covering all possible combination scenarios and tensile behavior types.

Fig. 4. Strain capacity of FRCCs as a function of failure probability.

Fig. 4 plots stain capacity against Pf of the eleven FRCCs. As can be seen, strain capacity of FRCC has a negative correlation with failure probability and it decreases with increasing Pf. For FRCC with strain capacity less than 1%, the failure probability can be as high as approximate 50%. A failure probability of less than 10% is necessary to ensure a moderate strain capacity above 2% for all FRCCs. This shows the proposed approach is able to consider material heterogeneity in evaluating tensile strain-hardening potential of FRCC by treating micromechanical parameters as random variables and taking material variability into account in the probabilistic-based model. Further study is necessary to ensure accurate probabilistic distribution of micromechanical parameters so that the accuracy of the proposed probabilistic-based assessment can be confirmed.

Acknowledgement 5. Conclusions and outlook This research grant is supported by the Singapore National Research Foundation under its Environmental & Water Technologies Strategic Research Programme and administered by the Environment & Water Industry Programme Office (EWI) of the PUB.

A novel probabilistic-based method considering heterogeneity is developed to assess the tensile strain-hardening potential of FRCC. MARS method is used to explicitly express the performance indices Jb’ Appendix Table A.1 The algorithm adopted for the MARS models using MATLAB Program

Notes

a = load('Training.txt'); X2 = a(:,2); X3 = a(:,3); X4 = a(:,4); X5 = a(:,5); X6 = a(:,6); X7 = a(:,7); X8 = a(:,8); X9 = a(:,9); X10 = a(:,10); X = [X1, X2,X3,X4,X5,X6,X7,X8,X9,X10]; Y = a(:,11);

Input training data

(continued on next page)

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Table A.1 (continued) Program

Notes

params = aresparams(150, 0, false, [ ], [ ], 2); model = aresbuild(X, Y, params) aresanova(model, X, Y) Yq = arespredict(model,X); J = Yq-Y; J1 = sum(J.ˆ2); S1 = sum((Y-mu).ˆ2); K1 = 1-J1/S1 S = mean(J); tmse = sum((J-S).ˆ2); mse = tmse/length(Y); mae = sum(abs(J))/length(Y); rmse = sqrt(mse); K=(J-S)/rmse; b = load('Testing.txt'); Xt(:,1) = b(:, 1); Xt(:,2) = b(:, 2); Xt(:,3) = b(:, 3); Xt(:,4) = b(:, 4); Xt(:,5) = b(:, 5); Xt(:,6) = b(:, 6); Xt(:,7) = b(:, 7); Xt(:,8) = b(:, 8); Xt(:,9) = b(:, 9); Xt(:,10) = b(:, 10); Yt = b(:, 11); [MSE, RMSE, RRMSE, R2] = arestest(model, Xt, Yt) aresplot(model) areseq(model,5)

Build MARS model ANOVA decomposition analysis Residual analysis to remove outliers with K ɛ [-1, 1]

Input testing data

Assess MARS model with test data Output MARS model in an mathematical form

Table A.2 Basis functions and corresponding equations of MARS model for Jb’ Basis function

Equation

Basis function

Equation

BF1 BF2 BF3 BF4 BF5 BF6 BF7 BF8 BF9 BF10 BF11 BF12 BF13 BF14 BF15 BF16 BF17 BF18 BF19 BF20 BF21

max(0, x4-1620) max(0, 1620-x4) max(0, x1-22) max(0, 22-x1) max(0, x7-1.5) max(0, 1.5-x7) max(0, x10–0.5) max(0, 0.5-x10) max(0, x5-2) BF6 * max(0, x8-0.05) BF6 * max(0, x4-1060) BF6 * max(0, 1060-x4) max(0, x8-0.5) max(0, 0.5-x8) BF14 * max(0, x7-1.5) BF14 * max(0, 1.5-x7) BF6 * max(0, 0.5-x10) max(0, 2-x5) * max(0, x4-1620) max(0, 2-x5) * max(0, 1620-x4) BF2 * max(0, x1-22) BF2 * max(0, 22-x1)

BF32 BF33 BF34 BF35 BF36 BF37 BF38 BF39 BF40 BF41 BF42 BF43 BF44 BF45 BF46 BF47 BF48 BF49 BF50 BF51 BF52

BF8 * max(0, x7-3) BF8 * max(0, 3-x7) max(0, 2-x5) * max(0, x7-1.5) BF9 * max(0, x4-928) BF9 * max(0, 928-x4) max(0, x6-0.5) BF6 * max(0, x2-10) BF6 * max(0, 10-x2) BF13 * max(0, 2-x5) BF14 * max(0, x7-3) BF37 * max(0, 3-x5) max(0, 3-x7) BF5 * max(0, x1-22) BF8 * max(0, 0.5-x8) BF6 * max(0, 3-x5) BF43 * max(0, x4-1060) BF43 * max(0, 1060-x4) BF3 * max(0, x8-0.5) max(0, 0.5-x6) * max(0, x7-0.9) max(0, 0.5-x6) * max(0, 0.9-x7) BF9 * max(0, 1-x7) (continued on next page)

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Table A.2 (continued) Basis function

Equation

BF22 BF23 BF24 BF25 BF26 BF27 BF28 BF29 BF30 BF31

BF8 BF2 BF2 BF2 BF3 BF3 BF8 BF3 BF3 BF3

* * * * * * * * * *

max(0, max(0, max(0, max(0, max(0, max(0, max(0, max(0, max(0, max(0,

x4-1060) x7-1.5) 1.5 -x7) 0.5-x8) x7-3) 3-x7) 2-x5) x5-2) 2 -x 5) 0.8 -x 10)

Basis function

Equation

BF53 BF54 BF55 BF56 BF57 BF58 BF59 BF60 BF61 BF62

BF6 * max(0, x6-1.5) BF43 * max(0, x2-10) BF43 * max(0, 10-x2) max(0, 2-x5) * max(0, 12.7-x2) BF37 * max(0, 3-x7) BF37 * max(0, x2-8) BF2 * max(0, x2-8) max(0, 0.5-x6) * max(0, 19-x2) BF43 * max(0, x6-2.5) BF43 * max(0, 2.5-x6)

Table A.3 Basis functions and corresponding equations of MARS model for σ0 Basis function

Equation

Basis function

Equation

BF1 BF2 BF3 BF4 BF5 BF6 BF7 BF8 BF9 BF10 BF11 BF12 BF13 BF14 BF15 BF16 BF17 BF18 BF19 BF20 BF21 BF22 BF23 BF24 BF25

max(0, x5-3) max(0, 3-x5) max(0, 1.5-x7) max(0, x4-1620) max(0, 1620-x4) max(0, x7-1.5) * max(0, x5-3) max(0, x7-1.5) * max(0, 3-x5) max(0, x8-0.05) BF5 * max(0, x7-1) BF5 * max(0, 1-x7) BF2 * max(0, x7-0.5) BF2 * max(0, 0.5-x7) BF2 * max(0, x4-1620) BF2 * max(0, 1620-x4) max(0, x7-1.5) * max(0, x10–0.2) max(0, x3-20) BF2 * max(0, x8-0.05) BF2 * max(0, 0.05-x8) max(0, x2-12.7) max(0, 12.7-x2) BF5 * max(0, x8-0.05) BF5 * max(0, 0.05-x8) max(0, x7-1.5) * max(0, x8-0.05) max(0, x7-1.5) * max(0, 0.05-x8) max(0, 0.05-x8) * max(0, x7-0.6)

BF26 BF27 BF28 BF29 BF30 BF31 BF32 BF33 BF34 BF35 BF36 BF37 BF38 BF39 BF40 BF41 BF42 BF43 BF44 BF45 BF46 BF47 BF48 BF49 BF50

max(0, 0.05-x8) * max(0, 0.6-x7) BF16 * max(0, x5-2) BF16 * max(0, 2-x5) BF16 * max(0, x7-3) max(0, 0.5-x8) BF3 * max(0, x10–0.5) BF3 * max(0, 0.5-x10) max(0, 0.39-x10) BF2 * max(0, 0.39-x10) BF2 * max(0, x2-12.7) BF2 * max(0, 12.7-x2) BF30 * max(0, 1.5-x7) BF4 * max(0, x7-0.9) BF4 * max(0, 0.9-x7) BF30 * max(0, 2700-x4) BF3 * max(0, x6-1.5) BF30 * max(0, 3-x7) max(0, x10–0.39) * max(0, x8-0.5) max(0, x1-11.6) max(0, x10–0.39) * max(0, 1620-x4) BF44 * max(0, 1.5-x7) BF16 * max(0, x7-0.6) BF16 * max(0, 0.6-x7) BF2 * max(0, 3-x7) BF5 * max(0, x5-4)

Table A.4 ANOVA decomposition of MARS model Function

MARS model Jb’

1 2 3 4 5

σ0

GCV

Variable(s)

GCV

440106 36530 1019 13670 344792

x1 x4 x5 x6 x7

0.32 2.48 13.16 17.77 36.26

114

Variable(s) x1 x2 x3 x4 x5 (continued on next page)

Cement and Concrete Composites 91 (2018) 108–117

J. Li, E.-H. Yang

Table A.4 (continued) Function

MARS model Jb’

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

σ0

GCV

Variable(s)

GCV

Variable(s)

7851 5945 751525 3021 48959 218 4300 3892 2680 13134 5943 4935 182513 901 831 262 1045 249 413 330745 3678 758 234

x8 x10 x1, x4 x1, x5 x1, x7 x1, x8 x1, x10 x2, x4 x2, x5 x2, x6 x2, x7 x4, x5 x4, x7 x4, x8 x4, x10 x5, x6 x5, x7 x5, x8 x5, x10 x6, x7 x7, x8 x7, x10 x8, x10

11.19 4.72 1.04 0.73 0.81 1.79 8.15 5.53 2.99 1.03 0.18 7.69 2.60 0.30 0.19 2.26 0.39 0.18

x7 x8 x10 x1, x7 x2, x5 x3, x5 x3, x7 x4, x5 x4, x7 x4, x8 x4, x10 x5, x7 x5, x8 x5, x10 x6, x7 x7, x8 x7, x10 x8, x10

Table A.5 Mix design of eleven FRCCs Mix ID

Mix design Cement GGBS Fly ash

1 [29] 2 [29] 3 [29] 4 [7] 5 [7] 6 [7] 7 [7] 8 [13] 9 [13] 10 [13] 11 [13]

1 1 1 1 1 1 1 0.8 0.8 0.8 0.8

– 0.66 – – – – – – – – –

1.5

Silica fume

Cenosphere Sand Water SP

0.1 0.15 0.22 – – – – 0.2 0.2 0.2 0.2

0.45 0.76 1.14 – – – – – – – –

– – – 0.6 0.6 0.6 0.6 – – – –

0.3 0.49 0.73 0.45 0.45 0.45 0.45 0.27 0.27 0.27 0.27

115

0.008 0.006 0.006 0.02 0.02 0.02 0.02 0.04 0.04 0.04 0.04

Fiber type, vol Fiber % coating PVA 2% PVA 2% PVA 2% PVA 2% PVA 2% PVA 2% PVA 2% PE 0.5% PE 0.75% PE 1.0% PE 1.25%

1.2% 1.2% 1.2% 0.3% 0.5% 0.8% 1.2% – – – –

Strain capacity ε (%)

Pf1 (%)

Pf2 (%)

Pf (%)

1.5 0.93 3.0 1.55 2.73 3.81 4.88 0.73 1.4 3.8 4.0

21.5 43.8 8.2 25.3 11.2 6.9 0.02 0 3.6 1.13 4.87

0 0 0 0 9.4 0 0.12 47.4 22.7 7.8 1.26

21.5 43.8 8.2 25.3 11.2 6.9 0.12 47.4 22.7 7.8 4.87

116

Normal Uniform Uniform Normal Normal Normal Normal Normal Uniform Uniform Normal Weibull Normal

Ef (GPa) Lf (mm) df (μm) σfu (MPa) Vf (%) Gd (J/m2) τ0 (MPa) β f' f Em (GPa) σmu (MPa) Km (MPa·m1/2)

a: Assumed.

Distribution

Parameters

42 ± 4.2 12 39 1070 ± 107 2 ± 0.2 1.5 ± 0.7 2.3 ± 0.5 0.5 ± 0.2 0.3a 0.5a 20 ± 2a 2.4/1.1a 0.5 ± 0.05

Mix 1 [29]

0.6 ± 0.2 3.0 ± 0.8 0.4 ± 0.2

0.4 ± 0.04

0.5 ± 0.03

Mix 3 [29]

0.8 ± 0.4 2.3 ± 0.4 0.4 ± 0.2

Mix 2 [29]

Table A.6 Distributions and values of micromechanical parameters of eleven FRCCs Mix 5 [7]

42.8 ± 4.3 12 39 1092 ± 109 2 ± 0.2 3.2 ± 0.7 3.0 ± 0.8 2.2 ± 0.2 2.1 ± 0.2 2.3 ± 0.2 1.8 ± 0.2 0.3a 0.5a 20 ± 2 2.4/1.1a 0.8 ± 0.1a

Mix 4 [7]

2.2 ± 0.7 2.0 ± 0.1 1.2 ± 0.3

Mix 6 [7]

1.6 ± 0.6 1.1 ± 0.1 1.2 ± 0.2

Mix 7 [7]

Mix 9 [13]

117 ± 11.7 19.1 38 2400 ± 240 0.5 ± 0.05 0.75 ± 0.07 0 0.62 0 0 0.8a 23 ± 2.3 2.4/2.0a 0.8 ± 0.1a

Mix 8 [13]

1.0 ± 0.1

Mix 10 [13]

1.25 ± 0.12

Mix 11 [13]

J. Li, E.-H. Yang

Cement and Concrete Composites 91 (2018) 108–117

Cement and Concrete Composites 91 (2018) 108–117

J. Li, E.-H. Yang

Fig. A.1. First order reliability method.

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