Size effect on tensile strength of parallel CFRP wire stay cable

Accepted Manuscript Size effect on tensile strength of parallel CFRP wire stay cable Chengming Lan, Jingyu Wu, Nani Bai, Dan Qiang, Hui Li PII: DOI: Reference:

S0263-8223(16)32659-9 http://dx.doi.org/10.1016/j.compstruct.2017.08.039 COST 8797

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

27 November 2016 18 June 2017 9 August 2017

Please cite this article as: Lan, C., Wu, J., Bai, N., Qiang, D., Li, H., Size effect on tensile strength of parallel CFRP wire stay cable, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct.2017.08.039

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Size effect on tensile strength of parallel CFRP wire stay cable Chengming Lan a, Jingyu Wu b, Nani Bai a, Dan Qiang a, Hui Li a,b, a

School of Civil & Resource Engineering, National Center for Materials Service Safety, University of Science &

Technology Beijing, Beijing, 100083, China. b

Research Center of Structural Monitoring and Control, School of Civil Engineering, Harbin Institute of Technology, Harbin, 150090, China.

Abstract The paper studies the size effect on the tensile strength of parallel carbon fiber reinforced polymer (CFRP) wire stay cable in the macroscale. First, an asymptotic weakest-link Weibull model that incorporates a statistical length effect for the tensile strength of longitudinal composites is proposed in this research. For the proposed model, the weakest-link effect gradually becomes dominant and causes a decrease in strength that increases along the length of the longitudinal elements. The asymptotic threshold length Lρ on the strength analysis can be evaluated by the asymptotic weakest-link Weibull model. The strength data of single carbon fibers and impregnated bundles with different lengths obtained by previous studies are employed to validate the proposed model and the log-likelihood ratio test and the Bayesian information criterion (BIC) are used as the criteria for the validation of the proposed model. The tensile strengths of CFRP wires with different lengths are obtained and analyzed by using the proposed model, and the results obtained from the random strength field model proposed by Vořechovský and Chudoba are illustrated and discussed in detail. Finally, the actual strengths of two parallel CFRP wire cables are compared with the simulated results to illustrate the Daniels’ effect on the strengths of parallel CFRP wire cables. To evaluate the strengths of parallel CFRP wire stay cables, the length effect and Daniels’ effect should be 1

considered in the design of the CFRP cables to ensure safety and reliability. Keywords: Weibull distribution; Length effect; Daniels’ effect; Asymptotic threshold length; Longitudinal composites 1. Introduction The application of cable-stayed bridges to cross wide rivers or sea bays has developed rapidly in the past 30 years, and they have become a widely-used type of long-span bridges, due to their superior self-balancing structural system, higher overall stiffness and better aerodynamic behavior in comparison to suspension bridges [1]. Currently, cable-stayed bridges have already reached the thousand-meter span, which was previously economical only for suspension bridges [2]. Moreover, even longer spans have been planned [3, 4]. A cable-stayed bridge is characterized by its stay cables, which are prime candidates for replacement by new materials due to its simplicity in terms of required mechanical properties. The major disadvantages of conventional steel cables in a super long-span cable-stayed bridge lie in the pronounced sag effect, which will reduce overall stiffness of the bridge. In addition, the durability deficiency induced by corrosion, which will greatly limit the initial advantages of a cable-stayed bridge with super long-span [5]. The CFRP is the most prospective among the materials for stay cables because of its good mechanical and chemical behaviors compared with the various types of FRP materials. Several previous studies have revealed that FRP as stay cables exhibit prominent advantages compared with steel cables, such as allowing longer spanning capacity, increasing safety level of a bridge, and avoiding of resonance between the bridge and cables [2, 3, 6, 7]. As a matter of fact, plastic deformation in

2

metals tends to reduce stress concentrations arising from defects, and these materials display much less strength scatter and size effect than brittle materials. In practice, the influence of size on the strength of metallic structures is rarely, if ever, considered, whereas it is a key consideration for ceramics [8]. The question of size effect exists for all composite materials. FRP have several characteristics that are typical of brittle materials. One of the most important similarities to ceramics is that FRP lack plasticity to reduce the influence of stress concentrations arising from defects. However, in the literatures on the design of FRP cables [2, 5, 9], the length effect on the tensile strength of the FRP wires was not considered, which may result in an unsafe result. This is a fatal question which needs to be solved before FRP materials are used as stay cables. Moreover, to evaluate the strength of the parallel FRP wire cables, the Daniels’ effect on the strength of parallel system should be further considered [10-12]. The stochastic behavior of the tensile strengths of long slender components (e.g., fibers, bundles and FRP wires) at different lengths has been observed, i.e., the strength decreases as the specimen length increases [13, 14]. Since the stay cables of long-span cable-stayed bridge are very long, such as the 577.082 m long cable of Sutong Yangtze River Bridge, it is impossible to obtain the true strength of such long structural components through experiments directly. The potential solution is to predict the strengths of practical structural components based on the test results of small-scale specimens. Models for the tensile strength of the long slender components that incorporate the length effects have been extensively investigated [13, 15-20]. The weakest-link model is

3

most frequently employed to establish the extreme value distribution of the tensile strength of the longitudinal components. In the weakest-link model, suppose that a longitudinal component with a length of L can be divided into n elements with uniform lengths of L0 ; thus, n = L L0 , and the strength of each element is an independently identically distributed (iid) random variable. The weakest-link theory assumes that failure of the entire component occurs when the weakest element fails, which implies that the tensile strength of the entire component is dependent on the strength of the weakest element only. Based on the weakest-link model, the Weibull distribution has been extensively employed as a mathematical model of the tensile strength for both the fibers and composites. The weakest-link Weibull cumulative distribution function (CDF) can be written as  L  σ β  FL (σ ; σ 0 , β ) = 1 − exp −    .  L0  σ 0  

(1)

The characteristic strength for length L is

σ ( L) = σ 0 (L L0 )

−1 β

.

(2)

where σ 0 is the characteristic strength, which is a reference value for length L0 , and

β is the shape parameter. Eq. (1) describes the length effect in terms of the number of elements ( n = L / L0 ). In

spite of its wide applications, the weakest-link hypothesis is difficult to verify by the experimental data [16, 21]. The empirical evidence demonstrates that the extrapolation from short specimens, based on the hypothesis of independence, does not lead to reliable predictions [22]. Extrapolation based on Eq. (1) is reliable only when a specimen length 4

of L in testing is larger than the threshold length of Lρ . However, there are few studies on determining the threshold length thus far. Some fibers are prone to length effects, i.e., the characteristic strength of a filament is larger than that predicted by Eq. (2) when the specimen length L is longer than L0 , whereas those with shorter lengths are weaker. To consider this feature, Watson and Smith [16] modified the power 1 β in Eq. (2) to α β , to obtain the following:

σ ( L) = σ 0 (L L0 )

−α β

(3)

where α is a scale parameter, and α ∈[0, 1] . Thus, the underlying Weibull distribution for the specimen with length L becomes   L α  σ  β  FL (σ ; σ 0 , β ,α ) = 1 − exp −      .   L0   σ 0  

If parameter α

(4)

is equal to zero, then the model degenerates as a standard

two-parameter Weibull distribution, and the model is free of length effects. If parameter

α is equal to 1, then the model degenerates according to Eq. (1). Eq. (4) provides a good fit for some of the experimental data on fiber strength. Phoenix et al. [23] suggested that α = 0.60 for Kevlar 49 fibers, and Watson and Smith [16] obtained α = 0.90 for carbon fibers and α = 0.58 for impregnated bundles of parallel carbon fibers. In contrast, data for super high-strength polyethylene fibers yielded a value for α that was close to zero [13]. However, this model conflicts with the empirical evidence in that the model is not asymptotic to the weakest-link model when

α does not have the value of 1 regardless of how long the length is. The extrapolation outside the range of the test data is questionable since the strengths of the components could be overestimated. 5

Until now, the threshold length of the longitudinal element has still been an intractable issue in actual engineering applications. To estimate the length effect and extrapolate the strength of longitudinal composites with actual length, an asymptotic weakest-link Weibull model is proposed based on plausible and physically acceptable assumptions in this paper. The size effects on the strengths of parallel CFRP wire cables are of main concern of this paper, and both the length effects on the strengths of long slender CFRP wires and the Daniels’ effects on the strengths of parallel CFRP wire cables are studied. The structure of this paper is as follows: the asymptotic weakest-link Weibull model with three parameters is proposed in Section 2, and the asymptotic threshold length can be determined based on the model parameters estimated from the experimental data with different lengths. In Section 3, the log-likelihood ratio test and the Bayesian information criterion (BIC) are adopted as the evaluation for model selection. The strength obtained by Bader and Priest [24] for single carbon fibers and impregnated bundles with different lengths, which were investigated exhaustively by various researchers [16-18], is employed to validate the proposed model in Section 4. The actual strength data of CFRP wires with various lengths are introduced in Section 5. Also, the length effect and asymptotic threshold length on the strength of the CFRP wires are analyzed by the proposed model. The corresponding results obtained from the random strength field model proposed by Vořechovský and Chudoba [25, 26] are illustrated and discussed in detail. Further, the strengths of parallel CFRP wires cable are illustrated by Monte Carlo simulation to consider the combined influences of length effects and Daniels’ effects in Section 6. Then, the implication of parameters for the proposed model and length effects

6

on longitudinal composites are discussed in detail in Section 7. Finally, Section 8 concludes the paper. 2. The Asymptotic Weakest-Link Weibull Model

As discussed above, the real spatial distribution of the strengths of longitudinal composites is difficult to know exactly; thus, the threshold length for the independence assumption of the strengths of neighboring elements cannot be determined at the current stage. To resolve this issue, we assume that parameter α in Eq. (4) is a function of the length L , and without loss of generality L0 is regarded as unit length. We further assume that α ( L) is asymptotic to 1 with an increase in the length L ; therefore, the model is asymptotic to the weakest-link model. Suppose that the characteristic strength is expressed as

σ ( L) = σ 0 L−α ( L ) β

(5)

and

 L

α ( L) = 1 − exp −  , λ > 0 .  λ

(6)

It can be seen that α ( L) ∈ [0, 1] . Thus, the CDF and probability density function (PDF) of the proposed asymptotic weakest-link Weibull model for a specimen with length L is as follows:  1−exp  − L    σ  β  FL (σ ; σ 0 , β , λ ) = 1 − exp − L  λ   ⋅      σ 0     L  1− exp  −    λ  

f L (σ ; σ 0 , β , λ ) = L

(7)

 1− exp  − L    σ  β  βσ β −1  λ  ⋅ ⋅ exp − L ⋅    . σ 0β  σ0   

(8)



The mean value E (σ ) and variance of V (σ ) of the strength at length L are respectively, 7

−

E (σ ) = σ 0 ⋅ L

2 0

1  L  1− exp  −    λ 

2  L  − 1− exp  −   β  λ 

V (σ ) = σ ⋅ L

 1 ⋅ Γ1 +   β

(9)

  2  1  ⋅ Γ1 +  − Γ 2 1 +   β    β

(10)

β 

where λ is the scale parameter, and λ > 0 , σ 0 is the characteristic strength with unit length and σ 0 > 0 . β is the shape parameter, and β > 0 . In fact, there are no independence assumptions adopted in Eq. (7). The unknown parameters λ , σ 0 and β can be estimated by all of the test data of the specimens with different lengths. Once the above parameters are obtained, the asymptotic threshold length Lρ can be determined from Eq. (6) using the following condition:

α ( Lρ ) ≅ 1 .

(11)

Then, the corresponding characteristic strength can be expressed as σ Lρ = σ 0 L−ρ1 β . It can be seen from Eqs. (6) and (11) that the scale parameter λ dominates the asymptotic threshold length Lρ . The asymptotic threshold length Lρ is defined as α ( Lρ ) = 0.99 in this study. Once

the asymptotic threshold length Lρ and characteristic strength σ Lρ are obtained, the strength of the longitudinal element with length L > Lρ can be predicted by Eq. (1) or (7). This means that the strength of longitudinal element for each segment of length L > Lρ becomes identical to the extremes of iid. As the length increases, the proposed

model is asymptotic to the weakest-link model with the asymptotic threshold length Lρ . When the length L is less than the unit length, it is assumed that the spatial fluctuations in strength become insignificant and negligible in macroscale, the random strength is regarded as a single random variable with length L = 1 . Thus Eq. (7)

8

degrades to the standard two-parameter Weibull distribution. It can be seen that the characteristic strength σ 0 in Eq. (7) is a reference value with length L = 1 . For a specimen with length 1 < L < Lρ , it is a transition zone for the random strength of longitudinal element. In the transition zone, the strengths are length-dependent and can be predicted directly by using Eq. (7) with the estimated parameters. The basic assumption of the proposed asymptotic weakest-link Weibull model is similar to the standpoint of the random strength field model that is proposed in [25, 26]. To approximate the length effect, two suggested formulas were given in [25] (see Eqs. (9) and (10) in [25]). The characteristic strengths for length L are rewritten as

 L L  σ ( L) = σ 0  + a   La La + L 

−1 β

.

(12)

or

L +L  σ ( L) = σ 0  a  La 

−1 β

.

(13)

where La is the autocorrelation length proposed in [25] and controls the spatial fluctuations of tensile strengths. For the random strength field model, the local random strength is phenomenologically modeled as a random field with a certain autocorrelation function La . There are three zones of the statistical size effect on the random strength field, single random variable as L / La → 0 , autocorrelated random field when

L / La ≅ 1 and the set of independent identically distributed random variables as L / La → ∞ . In fact, the characteristic strengths σ 0 used in Eqs. (12) and (13) are the reference values when the length L → 0 . Obviously, for the right asymptote as the weakest-link model, the parameter σ 0 is not the characteristic strength for

9

autocorrelation length La , which shows the characteristic strengths should be overestimated for L >> La . The results of random strength field will be illustrated and compared with that of the proposed asymptotic weakest-link Weibull model in following examples. The main purpose of this paper is to predict the strength of CFRP wire with actual length of stay cable in macroscale. Therefore, the asymptotic threshold length Lρ is critical to validate whether the independence hypothesis is approximately accepted and the prediction of the strength is reasonable. The detailed computation process will be illustrated in subsequent sections. Suppose that longitudinal elements with k lengths are performed to obtain the strengths of the specimens, which are represented by L1 , L2 ,L, Lk . ni specimens with length Li are employed and ni strengths of σ ij ( i = 1,2,L, k ; j = 1,2,L, ni ) are obtained. The maximum likelihood estimation (MLE) approach is employed to estimate the parameters in the proposed model, as follows: k

ni

i =1

j

l (σ 0 , β , λ ) = ∏∏ f L (σ ij ; Li )

(14)

or equivalently, to maximize the logarithm of the likelihood function, log l (σ 0 , β , λ ) , as follows:   L  log l (σ 0 , β , λ ) = log β ∑ ni + ∑ ni 1 − exp − i  log Li + (β − 1)∑∑ log σ ij  λ    L  1−exp  − i   λ  i

− β log σ 0 ∑ ni − ∑ L

σ  ∑  σ ij   0

β

(15)

where the summations are with respect to i and j over the appropriate ranges. The maximization can be performed by taking the partial derivatives of the 10

log-likelihood function with respect to each of the three parameters β

 L 

1− exp  − i   σ  β β ∂ log l = − ∑ ni + ∑∑ Li  λ   ij  =0 ∂σ 0 σ0  σ0  σ0

∂ log l 1 = ∂β β

∑ n + ∑∑ log σ i

ij

 L  1− exp  − i   λ  i

− log σ 0 ∑ ni − ∑∑ L  L 

(16) β

 σ ij  σ    log ij  = 0 (17)  σ0  σ0 

β

1− exp  − i  ∂ log l L L σ   L  L = −∑ ni 2i exp − i  log Li + ∑∑ Li  λ  2i  ij  exp − i  log Li = 0 .(18) ∂λ λ λ  σ0   λ  λ

The three nonlinear equations are numerically solved to obtain the estimates of the parameters σ 0 , β and λ in this paper. The inverse of the Hessian matrix of − log l (σ 0 , β , λ ) provides an estimate of the covariance matrix of the estimators. In particular, the square roots of the diagonal elements of the covariance matrix are the standard errors of σ 0 , β and λ . 3. Model Selection and Data Interpretation

Model selection is a critical and integral issue of data analysis that leads to valid inference. To illustrate the validation of the underlying models, the log-likelihood ratio test and BIC, which are based on asymptotic likelihood inferences are used as the methods of model selection. 3.1 Log-Likelihood ratio test

Watson and Smith [16] previously proposed to test whether the weakest-link hypotheses of the strengths for the single fibers and impregnated bundles were accepted by using the log-likelihood ratio test. Suppose that the standard two-parameter Weibull distribution is valid as a statistical model of the strength data for each specified length. For a total of k lengths, there are k standard two-parameter Weibull distributions with 2 k parameters, and the characteristic strength is unconstrained for each length. 11

However, for the other models, e.g., Eqs. (1), (4), (7), (12) and (13) the length L is regarded as a covariate of the characteristic strength σ (L ) . Consequently, the log-likelihood ratio test is used for testing whether a model is appropriate for the observed data compared with a more general distribution family. The log-likelihood test statistic is X L = 2[log l * − log l0 ] .

(19)

where log l * is the summation of the individual maximized log-likelihood of the standard two-parameter Weibull distribution for each of the k length subsamples, and log l0 is the maximized log-likelihood of the underlying distribution under the null hypothesis. The appropriate test is to reject the null hypothesis at the 1 − α level of significance if X L > χ v2,α , where χ v2,α denotes the upper α -point of the chi-square distribution with v degrees of freedom, and v = 2k − q , where q is the number of parameters of the underlying distribution under the null hypothesis. 3.2 Bayesian information criterion

To evaluate the underlying models, a simpler model selection procedure in terms of the Bayesian information criterion [27] is used to determine the best model. This criterion is based on the log-likelihood log l , the number of parameters in the distribution q , and the total number of observations N = ∑ ni . For each candidate model, the BIC is formally defined as BIC = −2 log l + q log( N ) .

(20)

The candidate distribution with the smallest BIC value is the model that fits the experimental data with the smallest error. It has been shown that under mild

12

assumptions, for sufficiently large N , the model selected by the BIC procedure approaches the true underlying distribution, if it exists [28]. In general, the larger the value of the parameter q is in a distribution, the smaller the negative log-likelihood in Eq. (18). Thus, the first term represents the gain by using a model with more parameters. However, the larger the value of q is, the larger the second term in Eq. (20), which represents a penalty by having more parameters in the distribution. Therefore, the BIC is a trade-off between the gain and penalty. 3.3 Graphic method for data interpretation

Additionally, graphical methods have been used to present and interpret the data. Therefore, Eq. (7) under the specified length of Li is transformed to a Weibull probability plot, as follows:   L  log( − log(1 − FLi )) = β log(σ i, j ) − β log(σ 0 ) + 1 − exp −  i  log Li .  λ  

(21)

The log( − log(1 − FLi )) versus log(σ i , j ) is linear for each length Li with a slope of

β . The EDF (Empirical distribution function) FL for each length Li is calculated by i

using the mean rank estimator, as follows: FˆLi =

j . ni + 1

(22)

4. Validation of the Model for Strength Prediction of Fibers and Bundles

The test data of the strengths of single carbon fibers and impregnated bundles of different lengths, which was obtained by Bader and Priest [24], reported explicitly in Smith [17] and previously analyzed by Watson and Smith [16], are utilized to validate the proposed model.

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4.1 Tensile Strength of Carbon Fibers

Bader and Priest [24] obtained the strengths for single carbon fibers through testing at several gauge lengths. In their experiment, fibers were selected at random from a 1000-filament high-strength carbon-fiber tow (Celion 1000); cut to specimens with gauge lengths of 1, 10, 20 and 50 mm mounted on standard “window” cards; and tested in tension on an Instron machine with a load cell at an extension rate of 0.1 mm/min. There are total 257 observations, 57 at 1 mm, 64 at 10 mm, 70 at 20 mm, and 66 at 50 mm. The average diameter of the fibers, measured by laser diffraction to an accuracy of ±2%, was 8 microns. More detailed information is available in [24]. Table 1 presents the estimated values and 95% confidence intervals of the parameters of the proposed asymptotic weakest-link Weibull model (Eq. (7)) using the tensile strength of the carbon fibers with four lengths. Once three parameters are estimated, Eq. (7) can be transformed to a standard two-parameter Weibull distribution with the specified length Li and the corresponding scale and shape parameters can be accepted by a χ 22 test and KS test at the 95% level of significance. The results of the log-likelihood ratio test and BIC for different models are shown in Table 2. Padgett et al. [18] proposed a variation model of the Weibull distribution with three parameters; this model was a linear model to analyze the strength data of the carbon fibers, and their results are shown in Table 2 for comparison. The results by MLE of random strength field models using Eqs. (12) and (13) are also listed in Table 2. The estimated value of parameter β in Eq. (7) is slightly larger than that of existing models. Watson and Smith [16] mentioned that if the log-likelihood ratio test rejects the

14

weakest-link hypothesis at the 95% level of significance, then the weakest-link hypothesis will be questionable as an assumption. In fact, log-likelihood ratio tests reject the Weibull model in Eq. (4), the linear model and the random strength field models using Eqs. (12) and (13) at the 95% level of significance. However, the log-likelihood ratio test does not reject the proposed model in Eq. (7) at the 95% level of significance. The results indicate that the proposed Weibull model gives the smallest BIC value and fits the test data of the strengths of the carbon fibers with the smallest error. Additionally, among the Weibull models with three parameters, Eq. (7) gives the largest value of the log-likelihood function. Fig. 1 shows the tensile strengths of the carbon fibers on a Weibull probability plot. The results obtained by Watson and Smith [16], Padgett et al. [18] and random strength field models using Eqs. (12) and (13) are also shown in Fig. 1. It can be seen from the Weibull probability plot that the proposed model shows slightly better results compared with existing models. The slope of the proposed model is little larger than that of existing models due to its estimated shape parameter is slightly larger. The results of linear model and random strength field models are extremely close. As shown in Fig. 1, the lower tails of the empirical data plots tend to curve downward from the Weibull lines for all of the lengths except for L = 20 mm . One possible explanation for this phenomenon is that in the tedious process of extracting single fibers from a tow, there is a risk of breaking off the weaker fibers, which results in tests of fibers that have been somewhat “proof tested” in preparation [18]. Another possible explanation is the

15

intermittent presence of a competing flaw population. The relationships of the characteristic strengths and lengths of carbon fibers as calculated by the proposed model are shown in Fig. 2. The results of Eq. (4), the linear model and the random strength field models are shown in Fig. 2 simultaneously, for comparison. The symbols in Fig. 2 are the characteristic strength and 95% confidence intervals estimated directly by the standard Weibull distribution based on the strength samples at each length. It can be seen from Fig. 2 that in a bi-logarithmic coordinate, the characteristic strength obtained from Eq. (4) is linear, the characteristic strengths obtained from linear model and random strength field models are linear and quite close except for the shorter length as L < 1 mm. However, the characteristic strengths obtained from Eq. (7) are nonlinear, and Eq. (7) provides a slightly better estimation for the characteristic strength at each testing length. When length L is shorter than unit length, the random strength of the proposed model is regarded as a single random variable, therefore the characteristic strength is constant. The extrapolations of the characteristic strengths of different models outside the range of the testing lengths are quite close, except the results obtained from Eq. (4). Fig. 3 shows that Lα ( L ) is asymptotic to L as L increases, which implies that α ( L) is asymptotic to 1 and the weakest-link effect gradually becomes dominant. The estimated value of the asymptotic threshold length is 24.36 mm for carbon fibers based on the four lengths of the experimental data. 4.2 Tensile Strength of Impregnated Bundles

Impregnated tows of parallel carbon fibers in an epoxy resin matrix of various lengths

16

are selected as another example. The tensile strength of the specimens obtained in the test Bader and Priest [24] is employed to validate the proposed model and further obtain the asymptotic threshold length. There are total 119 observations, 28 at 20 mm, 30 at 50 mm, 32 at 150 mm, and 29 at 300 mm. More detailed experimental parameters are available in Bader and Priest [24]. Table 3 presents the estimated values of the shape and scale parameters in Eq. (7) using the tensile strength data of impregnated bundles with four lengths. For Eq. (7) with the specified length Li , the corresponding scale and shape parameters of the standard two-parameter Weibull distribution can be accepted by the χ 22 test and KS test at the 95% level of significance. The results of the log-likelihood ratio test and BIC for different models are shown in Table 4. The log-likelihood ratio test provides fairly strong evidence against the weak-link hypothesis for the impregnated bundles, due to the log-likelihood test statistic X L = 28.22 , which is much greater than χ 62, 0.05 = 12.59 . The log-likelihood ratio test

also rejects the Weibull model as in Eq. (4), at the 95% level of significance. However, the log-likelihood ratio test does not reject the random strength field models and the proposed model at the 95% level of significance. The results of BIC show that the proposed asymptotic weakest-link Weibull model gives the lowest BIC value and fits the strength data of the impregnated bundles best. The tensile strengths of the impregnated bundles on a Weibull probability plot are shown in Fig. 4. The MLE results of Eq. (4) obtained by Watson and Smith [16] and the MLE results of random strength field models using Eqs. (12) and (13) are also shown in

17

Fig. 4. It can be seen that the proposed model fits the experimental data better. The relationships of the characteristic strengths and lengths of the impregnated bundles obtained from different models are shown in Fig. 5. The symbols in Fig. 5 are the characteristic strength and 95% confidence intervals, estimated directly by the standard two-parameter Weibull model at each length. The proposed model gives a more accurate prediction for the characteristic strength at each testing length as shown in Fig. 5. However, the extrapolations of the characteristic strengths of different models outside the range of the testing lengths show visible differences. Regardless of how long the impregnated bundle is, Eq. (4) is not asymptotic to the weakest-link model, which is inconsistent with the empirical evidence. It was shown in Fig. 4 that the estimated autocorrelation lengths La of random strength field model using Eqs. (12) and (13) are 61.49 mm and 55.94 mm, respectively. Fig. 6 shows that Lα ( L ) is asymptotic to L as L increases, which implies that α ( L) is asymptotic to 1, and the weakest-link effect

gradually becomes dominant. The estimated value of the asymptotic threshold length is 3892.89 mm for impregnated bundles. The estimated asymptotic threshold length of the proposed model is much larger than the estimated autocorrelation length of random strength field models, which causes the differences of characteristic strengths with longer length. It can be seen that the asymptotic threshold length of the impregnated bundles is much longer than that of the constituent fibers. Compared with Figs. 2 and 5, once the length exceeds 20 mm, the characteristic strengths of the impregnated bundles gradually become greater than that of the fibers. The length effect becomes relatively mild for the

18

impregnated bundles. 5. Strength Analysis of CFRP Wires

Unidirectional CFRP wires with the nominal diameter of 4mm are fabricated through pultrusion, which is a conventional process for manufacturing fiber/resin composite with a constant cross-section. The pultruded CFRP wire with a fiber volume content of approximate 66% is composed of 19 bunches of Toray T700SC-12K carbon fiber tows, which are impregnated in bisphenol-A type epoxy resin prior to entering a hearted die. The nominal values of the tensile strength, elastic module and elongation for the Toray T700SC-12K carbon fibers are 4.9 GPa, 230 GPa and 2.1% respectively. The failure load of the CFRP wires is measured by the load cell, and the specimens are all broken in free length. The details of the anchorage for the single CFRP wire and CFRP stay cable will be presented in another paper. There are a total of 42 observations of the tensile strength of the CFRP wires, 12 at 1 m, 10 at 4 m, 10 at 10 m and 10 at 20 m. The experimental setups are shown in Fig. 7. Table 5 shows the test data of the tensile strengths, the mean and the standard deviation of the CFRP wire at each length. The strength means of the CFRP wires decrease with an increase in the length. Photographs of representative specimens before and after testing are shown in Fig. 8. Table 6 presents the MLE of the shape and scale parameters of the proposed Weibull model in Eq. (7) for the tensile strengths of the CFRP wires with four lengths. For Eq. (7) with the specified length Li , the corresponding scale and shape parameters of the standard two-parameter Weibull distribution can be accepted by the χ 22 test and KS test at the 95% level of significance.

19

The results of the log-likelihood ratio test and BIC for different models are shown in Table 7. The log-likelihood ratio test does not reject the proposed model and random strength field model at the 95% level of significance for the log-likelihood ratio statistic. The results of log-likelihood function and BIC show that the random strength field model using Eq. (13) fits the strength data of CFRP wires slightly better. The tensile strengths of the CFRP wires on a Weibull probability plot are shown in Fig. 9. The MLE results of random strength field models are also shown in Fig. 9. The relationships of the characteristic strengths and lengths of the CFRP wires obtained from different models are given in Fig. 10, and the characteristic strength and 95% confidence intervals for the specimens at each testing length are also shown in Fig. 10. For CFRP wires, the estimated autocorrelation lengths of random strength field model using Eqs. (12) and (13) are 4.91 m and 5.14 m, respectively (Fig. 9). As the length of CFRP wire increases, the proposed model is asymptotic to the weakest-link model (see Fig. 11), and the asymptotic threshold length of the CFRP wires is Lρ = 160.70 m. It is similar with results of impregnated bundles that the estimated asymptotic threshold length of the CFRP wires is much larger than the estimated autocorrelation lengths of random strength field models using Eqs. (12) and (13). It can be seen in Fig. 10 that the estimated autocorrelation lengths La of random strength field models using Eqs. (12) and (13) locate the transition zone of the strength from a random variable with the shorter length to the extreme of iid (the weakest-link Weibull model) with the longer length. When the lengths of specimens exceed the autocorrelation length, the characteristic strength begins to decrease almost linearly as length increase with slope

20

− 1 / βˆ

in a bi-logarithmic coordinate. However, for the proposed asymptotic

weakest-link Weibull model, the characteristic strength begins to decrease linearly when the lengths exceed the asymptotic threshold length. Also, the characteristic strengths for L >> La are overestimated for random strength field models due to the parameter σ 0

with reference length L → 0 . In Fig. 10, the characteristic strength of the proposed model is lower approximately 0.177 GPa than that of the random strength field strength using Eq. (13) when the length of CFRP wire reaches 1000 m. The estimated characteristic strengths, mean strengths and strengths with 5% failure probability for different models are listed in Table 8 with length L = 1000 m. It can be seen from Fig. 10 and Table 8 that the random strength field models extrapolate higher strengths for CFRP wires with actual length which maybe deduce an unsafe conclusion. Therefore, the proposed model is used to estimate the tensile strengths of CFRP wires and cables with actual lengths below. As an example, the longest cable of the Sutong Yangtze River Bridge, which has a span of 1088 m, is 577.082 m. If the CFRP wires were used in manufacturing parallel CFRP wire stay cables, then the characteristic strength of the CFRP wire would be 2.07 GPa and the tensile strength with a 5% failure probability would be 1.83 GPa with length L = 577.082 m. By using the strength data on the CFRP wires with length L = 1 m,

the estimations of the characteristic strengths and tensile strengths with a 5% failure probability are 2.72 GPa and 2.41 GPa, respectively. The tensile strength with a 5% failure probability decreases about 24.1% which should be considered in the design of parallel CFRP wire stay cables.

21

6. Strength Analysis of Parallel CFRP Wire Cables

The parallel CFRP wire cable can be regards as a parallel system [12], see Fig. 12. To evaluate the strength of the parallel CFRP wire cables, the Daniels’ effects of parallel system should be further considered [10-12]. For the situation studied, the effect of length of length and parallel coupling can be treated separately, and they are independent and do not interact [26]. The model within this case consists of a single component of m elements in parallel with the applied load shared equally among surviving elements. Then if the strengths of the individual longitudinal elements with length L are denoted by σ L ,1, σ L , 2 , L , σ L , m and their ascending ordered values by

σ L ,(1) , σ L ,( 2) , L, σ L ,( m) , the strength of the system is given by m − r +1  Qm* = max σ L ,( r ) ⋅ . 1≤ r ≤ m m  

(23)

The distribution of Qm* was investigated by Daniels [10] under the assumption that the strengths of elements are iid random variables with known common distribution function FL . Daniels’ recursive formula is as follow m m  mx  Gm ( x) = P(Qm* ≤ x) = ∑ (−1) r +1  [ FL ( x)]r Gm−r   m−r r =1 r 

(24)

where G0 ( x) ≡ 1 and G1 ( x) = FL ( x) . As m becomes larger than 40 the recursive formula (Eq. (22)) becomes very demanding and then the only way to estimate the probability distribution is stochastic simulation. Exact and asymptotic (m → ∞) results were obtained by Daniels [10]. As m → ∞ the distribution function Gm ( x) converges to the normal distribution. In particular, Daniels obtained positive constants µ * and

σ * such that m1/ 2 (Qm* − µ * ) converges in distribution to a normal random variable

22

with zero mean and a standard deviation of σ * . In this paper, the Monte Carlo simulation is employed to compute the distribution of the strengths for parallel wire cables in terms of Eq. (21). Two parallel CFRP wire cables with length L = 4 m and number of wires m = 61 were fabricated for tensile strength tests. Before tensile strength tests, the two cables endured 2×10 6 pulsating fatigue cycles. The acoustic emission (AE) technology was used to detect the fatigue damage and broken process of CFRP wires during fatigue testing. No significant damage or wire broken was detected by AE monitoring during fatigue process. After 2×106 fatigue cycles, the actual ultimate tensile strengths of the two cables were obtained by tensile tests. The fatigue stress ranges and the actual ultimate tensile strengths of the two cables are listed in Table 9. The photographs of the experimental setups and fracture morphology for tensile strength tests are shown in Fig. 13. After the first few wires broke during the tensile process, the tensile force increased continuously and the tensile force achieved peak value before several CFRP wires broke simultaneously. The broken CFRP wires were checked to verify that they broke outside the anchorage after tensile tests. Based on the estimated parameters of CFRP wires in Table 6, the PDF and CDF of the strength for the tested cable are computed by Monte Carlo simulation and shown in Fig. 14. The result shows that the cable strength obeys normal distribution with a mean of 2.2936 GPa and a standard deviation of 0.0520 GPa. The actual strengths of the two cables are within the range of simulated strengths. The mean and standard deviation of the cable strength are less than those of CFRP wire with the same length.

23

There are total 313 steel wires in the longest cable of the Sutong Yangtze River Bridge with length L = 577.082 m. Suppose that the CFRP wires are used in the cable, PDF and CDF of cable strength are shown in Fig. 15. The cable strength obeys normal distribution with a mean of 1.7505 GPa, and a standard deviation of 0.0188 GPa. The CDFs of cable strength with different number of CFRP wires are shown in Fig. 16. With the numbers of CFRP wires increase, the cable strength gradually converts from the Weibull distribution to the asymptotically normal distribution as shown in Fig. 16. The reduction factor and COV of cable strengths are illustrated as a function of the number of wires in Fig. 17. The reduction factor is the ratio of the mean of cable strengths to that of wire strengths with L = 1 m. The cable strength decreases about 15% while the number of CFRP wires increases to 1000 with length L = 1 m. However, as the cable strength decreases, the COV of the cable strength decreases even faster simultaneously. Together the two effects are often referred to as the Daniels’ effect. For cable length L = 577 .082 m, the strength reduction amounts to about 35% while the number of

CFRP wires increases to 1000, which is the combined action of both Daniels’ effect and length effect. However, the COV curve of strength for cable length L = 577.082 m is almost identical with that for cable length L = 1 m as CFRP wires increase in cables. The CDFs of cable strength with different lengths and number of wires m = 313 are shown in Fig. 18, also the reduction factors of cable strengths are given in Fig. 19. The CDFs of cable strengths move to the left and the cable strengths decrease as the lengths increase. The impacts of length effect on cable strengths are significant, since strengths of CFRP wires decrease with the length. Apparently, the Daniels’ effect and length

24

effect on the strengths of CFRP cables should be taken into account in the design of CFRP cables; otherwise, the results could be unsafe and unreliable. 7. Discussion

As a matter of fact, this article is focus on the evaluation of tensile strengths for parallel CFRP wires stay cable with actual length. The results of different models which were proposed to evaluate the length effect on tensile strengths are compared and discussed in detail. If the adopted L0 is smaller than the threshold length, then Eq. (1) underestimates the strength, while Eq. (4) overestimates the strength of the longitudinal composites for extrapolation outside the range of the testing lengths. Obviously, extrapolations by using Eq. (1) based on long-specimen data would be more approximate. However, there have been few available methods to determine the threshold length until now. Castillo et al. [22] proposed an asymptotic function with three unknown parameters to determine the threshold length for the fatigue life estimation of longitudinal elements. The model needs more test lengths to improve the fitting precision of the three unknown parameters. Vořechovský and Chudoba [25, 26] proposed the random strength field model and defined the autocorrelation length to incorporate the length-scale. The two suggested formulas Eqs. (12) and (13), are compared with the proposed asymptotic weakest-link Weibull model. The assumptions are similar, nevertheless the obtained threshold lengths and strengths are different. Vořechovský and Chudoba suggested to use the mean size effect curve which is a curve in the bi-logarithmic plot of the size vs. mean bundle strength to determine the autocorrelation length La [25]. In this article, the

25

autocorrelation lengths of random strength field model as Eqs. (12) and (13) are estimated by MLE with strength data of different lengths. For the impregnated bundles and the CFRP wires, the estimated asymptotic threshold lengths of the proposed model are much longer than the estimated autocorrelation lengths of random strength field models, which cause the differences of characteristic strengths for the longer length. When the lengths of specimens exceed the autocorrelation length or the asymptotic threshold length, the characteristic strengths begin to decrease almost linearly as the length increase with slope − 1 / βˆ in a bi-logarithmic coordinate where the estimated shape parameters βˆ

of different models are almost similar. The estimated

autocorrelation lengths of random strength field model (see Figs. 5 and 10) seem to locate the transition zones where the strengths are not independent for each length. In addition, for the right asymptote as the weakest-link model, the parameter σ 0 is not the characteristic strength for autocorrelation length La which shows the characteristic strengths are overestimated for L >> La . Therefore, the results may be unsafe since the strengths are overestimated. Essentially, the broken fibers can maintain load carrying ability to some degree since the epoxy matrix transmits the loads between the fibers at the fiber breakages through shear. Strength models for unidirectional composites have significantly advanced in the past few decades [29]. Fiber packings, load sharing rules, dynamic stress concentrations and multiple fiber breaks are considered to predict the strength of unidirectional fiber-reinforced composites based on shear-lag model [30, 31] and finite element (FE) methods [32, 33]. However, for the practical issues of CFRP cables in this study, the

26

lengths of CFRP wires are over five hundred meters and about 228 K carbon fibers are in the CFRP wires, it should be noted that both Monte Carlo approaches based on shear-lag model and FE methods are generally limited under current computing power which may not be enough for such a realistic composite. To extrapolate the strengths of longitudinal composites outside the range of the testing lengths reasonably, especially for CFRP wires with actual length, the asymptotic weakest-link Weibull model is proposed. The longitudinal composites are regarded as integrated elements in the macroscale. If the threshold length can be determined, then the weakest-link model is appropriate and correct for the integrated element with longer length. The plausible and physically acceptable assumption for the proposed asymptotic weakest-link Weibull model is that two distant segments are asymptotically independent with an increase in length. In the proposed model, the parameter λ dominates the range of the transition zone and the asymptotic threshold length. As the parameter λ increases, the asymptotic threshold length Lρ increases simultaneously. Apparently, the parameters λ and β

dominate the characteristic strength of longitudinal

composites with length L . For parameters estimation of the proposed model, it is unnecessary to have testing lengths larger than the threshold length. Obviously, the proposed asymptotic weakest-link Weibull model is an effective method to evaluate the strength and the asymptotic threshold length for longitudinal composites. Harlow and Phoenix [34] noted that the Weibull shape parameter for a composite is two or three times that of the constituent fibers. The shape parameter of impregnated bundles is approximately three times that of a single fiber from the parameter

27

estimations of the proposed model. The shape parameter is even larger for CFRP wires. Additionally, the asymptotic threshold lengths of the carbon fibers, impregnated bundles and CFRP wires become much longer. The length effects become relatively mild while the number of fibers increases in the longitudinal composites. Nevertheless, the length effects on the tensile strength should be considered for the CFRP wires used in stay cable. Furthermore, Daniel’s effects should be considered to evaluate the tensile strength of parallel CFRP wire stay cable. 8. Conclusions

The theoretical basis for assessing the strength of parallel CFRP wire stay cables was studied considering the length effect and Daniels’ effect in this paper. Especially, an asymptotic weakest-link Weibull model was proposed to study the length effects on the tensile strength for the longitudinal composites with actual length. With the increase in length, the proposed Weibull model is asymptotic to the weakest-link model. From the statistical point of view, the strength data with different lengths are pooled together to improve the accuracy of parameters estimation. The parameter λ dominates the asymptotic threshold length of the longitudinal composites. The results of the log-likelihood ratio test and BIC justify that the asymptotic weakest-link Weibull model fits the strength data of the carbon fibers and impregnated bundles better than the existing strength models in the macroscale with respect to the length effects. The differences of random strength model and the proposed model are discussed in detail. The asymptotic weakest-link model provides a very useful tool to predict the strength distribution of the longitudinal composites when accounting for the

28

length effects, especially for very long longitudinal elements. The asymptotic threshold length is approximately 24.36 mm for carbon fibers and 3892.89 mm for impregnated bundles based on the strengths obtained by Bader and Priest [24]. The asymptotic threshold length is approximately 160.70 m for CFRP wires based on the strengths obtained by authors. The shape parameters and asymptotic threshold lengths of the CFRP composites are much larger than those of the constituent fiber, and the length effects become mild for longitudinal composites when the number of fibers increases. Although the length effects for the CFRP wires become mild, the results show that the tensile strength with a 5% failure probability decreases 24.1% when the length of the CFRP wires increases to 577.082 m. With the number of CFRP wires increase in the cable, the strength distribution of the cable gradually converts from the Weibull to the asymptotically normal. With the CFRP wire numbers increase, the mean strength of the cable decreases, and the COV of the strength decreases even faster. Based on the simulated results, the Daniels’ effect on the strength reduction of the cable amounts to about 15% while the number of CFRP wires increases to 1000 for the tested CFRP wires in this study. Therefore, the length effect and the Daniels’ effect on the tensile strength should be considered in the design of the CFRP stay cables. Acknowledgements

This study is financially supported by the NSFC project (Grant No. 51478039), Beijing Nova program (Grant No. Z151100000315053), the Program of International S&T Cooperation (Grant No. 2015DFG82080), the Fundamental Research Funds for the

29

Central Universities of China (Grant No. FRF-TP-15-001C1), and the Ningbo Science and Technology Project (No. 2015C110020). References [1] N.J. Gimsing, Cable supported bridge: concept and design, 2nd ed. Chichester(UK): Wiley; 1997. [2] X. Wang, Z. Wu, Integrated high-performance thousand-meter scale cable-stayed bridge with hybrid FRP cables, Compos. Part B 41 (2) (2010) 166-175. [3] U. Meier, Proposal for a carbon fibre reinforced composite bridge across the Strait of Gibraltar at its narrowest site, P. I. Mech. Eng. B-J. Eng. 201 (2) (1987) 73-78. [4] M. Nagai, Y. Fujino, H. Yamaguchi, E. Iwasaki. Feasibility of a 1400 m span steel cable-stayed bridge, J. Bridge Eng. 9 (5) (2004) 444-452. [5] X. Wang, Z. Wu, Evaluation of FRP and hybrid FRP cables for super long-span cable-stayed bridges, Compos. Struct. 92 (10) (2010) 2582-2590. [6] C. Kao, C. Kou, X. Xie, Static instability analysis of long-span cable-stayed bridges with carbon fiber composite cable under wind load, Tamkang J. Sci Eng. 9 (2) (2006): 89-95. [7] X. Zhang, L. Ying, Aerodynamic stability of cable-supported bridges using CFRP cables, J. Zhejiang Univ. – Sci. A. 8 (5) (2007) 693-698. [8] C. Zweben, Is there a size effect in composites? Compos. 25 (6) (1994): 451-454. [9] X. Wang, Z. Wu, G. Wu, H. Zhu, F. Zen. Enhancement of basalt FRP by hybridization for long-span cable-stayed bridge. Compos. Part B 44 (1) (2013) 184-192 [10] H.E. Daniels, The statistical theory of the strength of bundles of threads. I, Proceedings of the Royal Society of London. Series A, 1945, 183: 405-435. [11] R.L. Smith, The asymptotic distribution of the strength of a series-parallel system with equal load-sharing, Ann. Probab. 10 (1) (1982) 137-171. [12] M.H. Faber, S. Engelund, R. Rackwitz, Aspects of parallel wire cable reliability, Struct. Saf. 25 (2) (2003) 201-225. [13] P. Schwartz, A. Netravali, S. Sembach, Effects of strain rate and gauge length on the failure of ultra-high strength polyethylene fibers, Text. Res. J. 56 (8) (1986) 502-508. [14] L.S. Sutherland, R.A. Shenoi, S.M. Lewis, Size and scale effects in composites: I. Literature review, Compos. Sci. Technol. 59 (2) (1999) 209-220. [15] B.D. Coleman, Statistics and time dependence of mechanical breakdown in fibers, J. Appl. Phys. 29 (6) (1958) 968-983. [16] A.S. Watson, R.L. Smith, An examination of statistical theories for fibrous materials in the light of experimental data, J. Mater. Sci. 20 (9) (1985) 3260-3270. [17] R.L. Smith, Weibull regression models for reliability data, Reliab. Eng. Syst. Safe. 34 (1) (1991) 55-76. [18] W.J. Padgett, S.D. Durham, A.M. Mason, Weibull analysis of the strength of carbon fibers using linear and power law models for the length effect, J. Compos. Mater. 29 (14) (1995) 1873-1884. [19] L.C. Wolstenholme, A nonparametric test of the weakest-link principle, Technometrics. 37 (2) (1995) 169-175. [20] L.S. Sutherland, C.G. Soares, Review of probabilistic models of the strength of composite materials, Reliab. Eng. Syst. Safe. 56 (3) (1997) 183-196. [21] B.C. Arnold, E. Castillo, J.M. Sarabia, Modeling the Fatigue Life of Longitudinal Elements, Nav. Res. Log. 43 (6) (1996) 885-895. [22] E. Castillo, A. Fernández-Canteli, J.R. Ruiz-Tolosa, J.M. Sarabia, Statistical models for analysis of fatigue life of long elements, J. Eng. Mech. 116 (5) (1990) 1036-1049. [23] S.L. Phoenix, P. Schwartz, H.H. Robinson IV, Statistics for the strength and lifetime in creep-rupture of model carbon/epoxy composites, Compos. Sci. Technol. 32 (2) (1988) 81-120. [24] M.G. Bader, A.M. Priest, Statistical aspects of fibre and bundle strength in hybrid composites. In: Progress in Science and Engineering of Composites. Hayashi T., Kawata K., Umekawa S., eds., ICCM-IV, Tokyo, 1982, 1129-1136. [25] M. Vořechovský, R. Chudoba, Stochastic modeling of multi-filament yarns: II. Random properties over the length and size effect, Int. J. Solids. Struct. 43 (3-4) (2006) 435-458. [26] M. Vořechovský, Incorporation of statistical length scale into Weibull strength theory for composites, Compos. Struct. 92(9) (2010) 2027-2034. [27] G. Schwarz, Estimating the dimension of a model, Ann. Stat. 6 (2) (1978) 461-464. 30

[28] E.T. Lee, J.W. Wang, Statistical methods for survival data analysis, 3rd ed. Hoboken, New Jersey: John Wiley & Sons; 2003. [29] Y. Swolfs, I. Verpoest, L. Gorbatikh, A review of input data and modelling assumptions in longitudinal strength models for unidirectional fibre-reinforced composites, Compos. Struct. 150 (2016) 153-172. [30] T. Okabe, N. Takeda, Y. Kamoshida, M. Shimizu, W.A. Curtin, A 3D shear-lag model considering micro-damage and statistical strength prediction of unidirectional fiber-reinforced composites, Compos. Sci. Technol. 61 (2001) 1773-1787. [31] Elastoplastic shear-lag analysis of single-fiber composites and strength prediction of unidirectional multi-fiber composites, Compos. Part A 33 (2002) 1327-1335. [32] L. Mishnaevsky Jr, P. Brøndsted, Micromechanisms of damage in unidirectional fiber reinforced composites: 3D computational analysis, Compos. Sci. Technol. 69 (7-8) (2009) 1036-1044. [33] L. Mishnaevsky Jr, G. Dai, Hybrid carbon/glass fiber composites: Micromechanical analysis of structure-damage resistance relationships, Comput. Mater. Sci. 81 (2014) 630-640. [34] D.G. Harlow, S.L. Phoenix, Bounds on the probability of failure of composite materials, Int. J. Fracture 15 (4) (1979) 321-336.

31

Figure and Table Captions Table 1 Parameter estimations for carbon fibers Table 2 Results of different Weibull models (carbon fibers) Table 3 Parameter estimations for impregnated bundles Table 4 Results of different Weibull models (impregnated bundles) Table 5 Strength data of the CFRP wires (units: GPa) Table 6 Parameter estimations for CFRP wires Table 7 Results of different Weibull models (CFRP wires) Table 8 The estimated strengths of CFRP wires with length L=1000 m Table 9 Fatigue test conditions and actual tensile strengths of the two cables Fig. 1.

Tensile strengths of carbon fibers on a Weibull probability plot

Fig. 2.

The relationships of the characteristic strength and length for carbon fibers

Fig. 3.

The asymptotic threshold length for carbon fibers

Fig. 4.

Tensile strength of the impregnated bundles on a Weibull probability plot

Fig. 5.

The relationships of the characteristic strength and length for impregnated bundles

Fig. 6.

The asymptotic threshold length for impregnated bundles

Fig. 7.

Experimental setups for tensile testing of CFRP wires

Fig. 8.

Photographs of the specimens before and after testing

Fig. 9.

Tensile strength of CFRP wires on a Weibull probability plot

Fig. 10. The relationships of the characteristic strength and length for CFRP wires Fig. 11. Asymptotic threshold length of CFRP wires Fig. 12. Parallel system for the modelling of parallel CFRP wire cable Fig. 13. The photographs of the experimental setups and fracture morphology for tensile strength tests Fig. 14. The PDF and CDF of the cable strength with length L = 4 m and number of wires m = 61 Fig. 15. The PDF and CDF of the strength for the longest cable of the Sutong Yangtze River Bridge Fig. 16. The CDFs of cable strength with different number of CFRP wires Fig. 17. The reduction factor and COV of cable strengths Fig. 18. The CDFs of the cable strength with different lengths Fig. 19. Reduction factor of cable strength 32

Table 1

Parameter estimations for carbon fibers Parameter

λ

σ0

β

Estimated value

5.29

4.65

5.67

95% Confidence intervals

(4.03, 6.56)

(4.51, 4.79)

(5.33, 6.01)

33

Table 2

Results of different Weibull models (carbon fibers) Model

βˆ

log l

XL

χ v2,0.05

q

BIC

Standard two-parameter Weibull model for each length [16]

-

-220.1

-

-

8

484.59

Eq. (1) [16]

5.58

-229.1

18

χ 62, 0.05 = 12.59

2

469.30

Eq. (4) [16, 18]

5.31

-227.7

15.2

χ 52,0.05 = 11.07

3

472.05

Linear model [18]

5.30

-226.8

13.4

χ 52,0.05 = 11.07

3

470.25

Eq. (12)

5.31

-226.52

12.84

χ 52,0.05 = 11.07

3

469.69

Eq. (13)

5.30

-226.75

13.3

χ 52,0.05 = 11.07

3

470.15

Eq. (7)

5.67

-225.17

10.14

χ 52,0.05 = 11.07

3

466.99

34

Table 3

Parameter estimations for impregnated bundles Parameter

λ

σ0

β

Estimated value

845.33

2.90

17.05

95% Confidence intervals

(526.09, 1164.58)

(2.85, 2.95)

(14.70, 19.39)

35

Table 4

Results of different Weibull models (impregnated bundles) Model

βˆ

log l

XL

χ v2,0.05

q

BIC

Standard two-parameter Weibull model for each length [16]

-

35.8

-

-

8

-33.37

Eq. (1) [16]

18.66

21.7

28.22

χ 62,0.05 = 12.59

2

-33.84

Eq. (4) [16]

16.83

29.6

12.4

χ 52,0.05 = 11.07

3

-44.86

Eq. (12)

16.92 31.37

8.86

χ 52,0.05 = 11.07

3

-48.40

Eq. (13)

16.86 30.83

9.94

χ 52,0.05 = 11.07

3

-47.32

Eq. (7)

17.05 31.70

8.20

χ 52,0.05 = 11.07

3

-49.06

36

Table 5

Strength data of the CFRP wires (units: GPa) Length 1m 4m 10 m 20 m

Strength data 2.316, 2.490, 2.517, 2.574, 2.623, 2.708, 2.715, 2.723, 2.743, 2.783, 2.798, 2.833 2.437, 2.458, 2.492, 2.569, 2.572, 2.583, 2.608, 2.678, 2.696, 2.773 2.293, 2.446, 2.458, 2.508, 2.525, 2.569, 2.613, 2.638, 2.719, 2.752 2.286, 2.398, 2.43, 2.452, 2.491, 2.539, 2.551, 2.594, 2.653, 2.672

37

Mean

Standard deviation

2.652

0.153

2.587

0.107

2.552

0.137

2.507

0.120

Table 6

Parameter estimations for CFRP wires Parameter

λ

σ0

β

Estimated value

34.90

2.69

24.15

95% Confidence intervals

(10.17, 59.62)

(2.64, 2.74)

(18.66, 29.65)

38

Table 7

Results of different Weibull models (CFRP wires) Model

βˆ

log l

XL

χ v2,0.05

q

BIC

Standard two-parameter Weibull model for each length [16]

-

29.15

-

-

8

-28.40

Eq. (12)

24.53

28.31 1.68

χ 52,0.05 = 11.07

3

-45.41

Eq. (13)

24.54

28.58 1.14

χ 52,0.05 = 11.07

3

-45.45

Eq. (7)

24.15

27.97 2.36

χ 52,0.05 = 11.07

3

-44.73

39

Table 8

The estimated strengths of CFRP wires with length L = 1000 m Model Random strength field model using Eq. (12) Random strength field model using Eq. (13) The proposed model

Characteristic strength

Mean strength

Strength with 5% failure probability

2.178

2.130

1.930

2.198

2.150

1.947

2.021

1.976

1.788

40

Table 9

Fatigue test conditions and actual tensile strengths of the two cables Cable No. 1 2

Upper stress 800 MPa 850 MPa

Stress range 200 MPa 300 MPa

41

Tensile strength 2.3952 GPa 2.3221 GPa

Strength GPa 0.8

1

2

3

4

5

6

7 2.0

0.999

L = 1 mm L = 10 mm L = 20 mm L = 50 mm

0.99 0.96

1.5

MLE fits using Eq.(4) [16, 18] β = 5.31, α = 0.90, σ0 = 4.63

0.9

1.0

MLE fits using linear model [18] β = 5.30, γ = 0.55, σ0 = 4.97

0.5

MLE fits using Eq.(12) β = 5.31, La = 1.04, σ0 = 4.90

0.75

MLE fits using Eq.(13) β = 5.30, La = 0.55, σ0 = 5.55

0.0

MLE fits using Eq.(7) β = 5.67, λ = 5.29, σ0 = 4.65

0.5

-0.5

Probability

0.25 -1.5

-2.0 0.1 -2.5

0.05

-3.0

-3.5 0.02

-4.0

-4.5

0.01

-5.0

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-5.5 2.0

Log(Strength (GPa)) Fig. 1. Tensile strengths of carbon fibers on a Weibull probability plot

42

Log(-Log(1-F))

-1.0

6 Carbon fibers

Characteristic strength GPa

5 4 3 Eq.(4) [16, 18] Linear model [18] Random strength field model using Eq.(12) Random strength field model using Eq.(13) Eq.(7)

2

1

0

10

1

2

10

10

3

10

L mm Fig. 2. The relationships of the characteristic strength and length for carbon fibers

43

2

10

Asymptote

L

α(L)

α(L) = 1

Lρ

1

10

Model

0

10

0

1

10

10

L Fig. 3. The asymptotic threshold length for carbon fibers

44

2

10

Strength GPa 1.7 0.999

0.99 0.96 0.9

0.75

1.9

2.1

2.3

2.5

2.7

2.9

3.1

3.3 2.0

L = 20 mm L = 50 mm L = 150 mm L = 300 mm

1.5

MLE fits using Eq.(4) [16] β = 16.83, α = 0.58, σ0 = 3.25

1.0

MLE fits using Eq.(12) β = 16.92, La = 61.49, σ0 = 2.91 MLE fits using Eq.(13) β = 16.86, La = 55.94, σ0 = 2.96

0.5

MLE fits using Eq.(7) β = 17.05, λ = 845.33, σ0 = 2.90

0.0

0.5

Probability

-1.0 0.25 -1.5

-2.0 0.1 -2.5

0.05

-3.0

-3.5

0.02

-4.0

-4.5

0.01

0.5

0.6

0.7

0.8

0.9

1.0

1.1

-5.0 1.2

Log(Strength (GPa)) Fig. 4. Tensile strength of impregnated bundles on a Weibull probability plot 45

Log(-Log(1-F))

-0.5

4

Characteristic strength GPa

Impregnated bundles

3

2

Eq.(4) [16] Random strength field model using Eq.(12) Random strength field model using Eq.(13) Eq.(7)

1 0 10

1

10

2

10

3

10

4

10

L mm Fig. 5. The relationships of the characteristic strength and length for impregnated

bundles

46

5

10

Asymptote 4

10

α(L) = 1

3

10

L

α(L)

Lρ 2

10

1

10

Model 0

10

0

10

1

10

2

3

10

10

4

10

L Fig. 6. The asymptotic threshold length for impregnated bundles

47

5

10

Load cell

CFRP wire

Fig. 7. Experimental setups for tensile testing of CFRP wires

48

(a) Before testing

(b) After testing

Fig. 8. Photographs of the specimens before and after testing

49

Strength GPa 2.1

2.4

2.7

3

0.999

0.99

0.96

0.9

L=1m L=4m L = 10 m L = 20 m

2.0

1.5

MLE fits using Eq.(12) β = 24.53, La = 4.91, σ0 = 2.71 1.0

MLE fits using Eq.(13) β = 24.54, La = 5.14, σ0 = 2.73 MLE fits using Eq.(7) β = 24.15, λ = 34.90, σ0 = 2.69

0.5

0.75 0.0

0.5

-1.0 0.25 -1.5

-2.0 0.1 -2.5

0.05

-3.0

-3.5

0.02 0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

-4.0 1.10

Log(Stress (GPa)) Fig. 9. Tensile strength of CFRP wires on a Weibull probability plot

50

Log(-Log(1-F))

Probability

-0.5

3

Characteristic Strength GPa

CFRP wires

2.8 2.6

2.4 Random strength field model using Eq.(12) Random strength field model using Eq.(13) Eq.(7)

2.2

2 0

10

1

2

10

10

3

10

L m Fig. 10. The relationships of the characteristic strength and length for CFRP wires

51

3

10

Asymptote

2

10

L

ρ

L

α(L)

α(L) = 1

1

10

Model

0

10

0

10

1

2

10

10

L Fig. 11. Asymptotic threshold length of CFRP wires

52

3

10

1 2 3 m-1 m Cable with m CFRP wires

Fig. 12. Parallel system for the modelling of parallel CFRP wire cable

53

(a) Experimental setups

(b) Fracture morphology of parallel CFRP wire cable Fig. 13. The photographs of the experimental setups and fracture morphology for tensile

strength tests

54

10

1.0 Histogram Estimated PDF

8

0.8

6

m = 61 L = 4m

0.6

4

0.4

2

0.2

0 2.0

2.1

2.2

2.3

2.4

CDF

PDF

Estimated CDF

0.0 2.5

Strength GPa Fig. 14. The PDF and CDF of the cable strength with length L = 4 m and number of

wires m = 61

55

30

1.0 Histogram Estimated PDF

24

0.8

18

m = 313 L = 577.082m

0.6

12

0.4

6

0.2

0 1.65

1.70

1.75

1.80

CDF

PDF

Estimated CDF

0.0 1.85

Strength GPa Fig. 15. The PDF and CDF of the strength for the longest cable of the Sutong Yangtze

River Bridge

56

1.0 Normal CDF

CDF

0.8

Weibull CDF

m = 1000

0.6

m = 500 m = 100

0.4

m = 50

0.2 m = 10

0.0 2.1

2.2

2.3

m=1

2.4

2.5

2.6

L = 1m

2.7

2.8

2.9

Strength GPa (a) L = 1m

1.0 Normal CDF

CDF

0.8

Weibull CDF

m = 1000

0.6

m = 500 m = 100

0.4

m = 50

0.2

m = 10

m=1

L = 577.082m

0.0 1.6

1.7

1.8

1.9

2.0

2.1

2.2

2.3

Strength GPa (b) L = 577.082m Fig. 16. The CDFs of cable strength with different number of CFRP wires

57

Reduction factor

0.95 0.90 0.85 0.80

L = 577.082m L = 1m

0.75 0.70

COV of cable strength

0.65 0.05 L = 577.082m L = 1m

0.04 0.03 0.02 0.01 0.00 101

102

Number of CFRP wires in Cable Fig. 17. The reduction factor and COV of cable strengths

58

103

1.0 m = 313

CDF

0.8 0.6 L =1000m

0.4 L =500m

0.2

L =50m L =100m

0.0 1.6

1.7

1.8

L =1m

L =10m

1.9

2.0

2.1

2.2

2.3

2.4

Strength GPa Fig. 18. The CDFs of the cable strength with different lengths

59

0.90

Reduction factor

m = 313

0.85 0.80 0.75 0.70 0.65 Reduction factor of cable strength

0.60 100

101

102

Length m Fig. 19. Reduction factor of cable strength

60

103

61