Abstract
Carbon Fiber Reinforced Polymer (CFRP) cables, due to their outstanding performance in terms of specific stiffness and strength, are usually found in civil construction applications and, more recently, in the Oil & Gas sector. However, experimental data and theoretical solutions for these cables are very limited. On the contrary, several theoretical and numerical approaches are available for isotropic cables (metallic wire ropes), some of them with severe simplifications, nonetheless showing good agreement with experimental data. In this study, experimental tensile results for 1×7 CRFP cables were compared to a simplified analytical model (assumed transversally isotropic) and to a 3D finite element model incorporating the experimental uncertainty in important input parameters: longitudinal elastic modulus, Poisson’s ratio, static friction coefficient and ultimate tensile strain. The average experimental breaking load of the cable was 190.25 kN (coefficient of variation of 1.74%) and the agreement with the numerical model predictions were good, with an averagevalue deviation of –1.15%, which is lower than the experimental variations. The simplified analytical model yielded a discrepancy above 10%, indicating that it needs further refinement although much less time consuming than the numerical model. These conclusions were corroborated by statistical analyses (i.e. Kruskal–Wallis and MannWhitney).
Keywords
Composite cables; finite element analysis; analytical model; design of experiments; uncertainty quantification
1 INTRODUCTION
Carbon Fiber Reinforced Polymer (CFRP) cables present high stiffnesstoweight and strengthtoweight ratios, damping capabilities and high resistance to environmental degradation ( Adanur et al., 2015 Adanur, S., Mosallam, A.S., Shinozuka, M., Gumusel, L., (2015). A comparative study on static and dynamic responses of FRP composite and steel suspension bridges. Journal of Reinforced Plastics and Composites 30(15):12651279. ), being natural candidates for harsh environments such as offshore applications. While the cost of composite materials is generally greater than that of traditional structural materials, their extended life leads to reduced longterm costs ( Fabbrocino et al., 2016 Fabbrocino, F., Modano, M., Farina, I., Carpentieri, G., Fraternali, F., (2016). Optimal prestress design of composite cablestayed bridges. Compos. Struct. 169:167172. ). Other benefits include low energy consumption during manufacturing, construction and execution processes ( Dhand et al., 2015 Dhand, V., Mittal, G., Rhee, K.Y., Park, S.J., Hui, D., (2015). A short review on basalt fiber reinforced polymer composites. Compos. Part BEng. 73:166180. ; Son et al, 2013 Son, B.J., Lee, S.Y., Ji, H.S., (2013). Longterm performance of a fiberreinforced polymer slab bridge superstructurefield load test and ratings. Compos. Part BEng. 45(1):644656. ; Wang and Wang, 2015 Wang, Z.Y., Wang, Q.Y., (2015). Fatigue strength of CFRP strengthened welded joints with corrugated steel plates. Compos. Part BEng. 72:3039. ).
CFRP cables are traditionally manufactured by pultrusion using epoxy resin, allowing the use of long fibers and high fiber volume fraction. According to Meier (2012) Meier, U., (2012). Carbon Fiber Reinforced Polymer Cables: Why? Why Not? What If? Arabian Journal for Science and Engineering 37:399411. , Meier et al. (1982) Meier, U., Müller, R., Puck, A., (1982). GFKBiegeträger Unter Quasistatischer und Schwingender Beanspruchung. Proceedings of Internationale Tagung Über Verstärkte Kunststoffe, 35:135. , Meier (1992) Meier, U., (1992). Carbon Fiber Reinforced Polymers: Modern Materials in Bridge Engineering. Structural Engineering International 2(1):712. and Rohleder et al. (2008) Rohleder Jr., W.J., Tang, B., Doe, T.A., Grace, N.F., Burgess, C.J., (2008). CFRP Strand Application on Penobscot Narrows Cable Stayed Bridge. Transportation Research Board Journal 2050:169176. , in order to encourage the use of composite cables in structural applications, it is necessary to fully study and understand their behavior, which could be translated as evaluating analytical solutions, constructing numerical models and executing experimental tests. Indeed, there is very limited experimental data in the literature for these cables and theoretical solutions still require further development.
On the contrary, several theoretical approaches have been developed for isotropic cables (metallic wire ropes), some of them with severe simplifications, nonetheless showing good agreement with experimental data. Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. extensively investigated isotropic cables developing analytical models based on beam theory. Unlike most of the previous analytical solutions, he treated the wires as rods, allowing bending and torsion stiffness analysis. Usabiaga and Pagalday (2008) Usabiaga, H., Pagalday, J.M., (2008). Analytical procedure for modelling recursively and wire by wire stranded ropes subjected to traction and torsion loads. International Journal of Solids and Structures 45(21):55035520. , also using beam theory, developed an analytical solution for isotropic cables submitted to tensile stress considering rod rotation, but neglected the Poisson’s effect, and verified a small difference with the results of Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. for Poisson’s ratio of 0.0 and 0.3.
On the numerical field, Erdonmez and Imrak (2011) Erdonmez, C., Imrak, C., (2011). A finite element model for independent wire rope core with double helical geometry subjected to axial loads. Sadhana 36(6):9951008. analyzed 6×7 Stranded isotropic cables using the FEM (finite element method). They investigated the minimum length to be modeled that could still give reliable results, saving computational time and allowing effective contact between wires. They concluded that wire contraction played a small role on the mechanical behavior. Jiang (2012) Jiang, W.G., (2012). A concise finite element model for pure bending analysis of simple wire strand. International Journal of Mechanical Sciences 54(1):6973. analyzed isotropic singlelayered strand cables in pure bending considering plastic strain and friction and also achieved good agreement with Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. analytical solution. Ghoreishi et al. (2007) Ghoreishi, S.R., Messager, T., Cartraud, P., Davies, P., (2007). Validity and limitations of linear analytical models for steel wire strands under axial loading, using a 3D FE model. International Journal of Mechanical Sciences 49(11):12511261. compared their FEM model results with experimental data for a 1×7 singlelayered isotropic cable under axial loading, varying helix angle, and obtained small variations. They also analyzed the same cable using nine different analytical models, including Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. , with different considerations and assumptions, and reported satisfactory estimations of the elastic stiffness constants for lay angles below 20°.
Composite cables have been scarcely used in the petroleum sector. Jackson et al. (2005) Jackson, D., Shepheard, B., Kebadze, E., Teles, R., Rossi, R., Gonçalves, R.C., (2005). CFRP Mooring lines for MODU applications. In: Offshore Technology Conference, Houston, United States, pp. 16. used a spiral Carbon Fiber Composite Cable (CFCC) and ran tensile and cyclic fatigue tests with satisfactory results. However, they reported issues with the manufacturing process and no analytical or numerical model was used. Sparks et al. (2003) Sparks, C., Zivanovic, I., Luyckx, J., Hudson, W., (2003). Carbon Fiber Composite Tendons for Deepwater Tension Leg Platforms. In: Offshore Technology Conference, Houston, United States, pp. 15. studied CFCC in the offshore industry, more specifically in TLP (tension leg platforms) used in ultradeep waters, and reported advantages as: good fatigue resistance, high level of flexibility, safety due to the nonpropagation of defects, and unlimited size for mooring.
Even if a stateoftheart numerical model was available, the predictions could differ from experimental results as they, unlike the latter, do not include statistical variations ( Grover et al., 2017 Grover, N., Sahoo, R., Singh, B.N., Maiti, D.K., (2017). Influence of Parametric Uncertainties on the Deflection Statistics of General Laminated Composite and Sandwich Plates. Composite Structures, doi: http://dx.doi.org/10.1016/j.compstruct.2017.03.036.
http://dx.doi.org/10.1016/j.compstruct....
). And since quality of composite materials is severely influenced by the quality control of processes, operating conditions and environmental effects, uncertainties in input parameters can propagate through different modelling scales and significantly influence other parameters and the final output due to risk accumulation ( Shaw et al., 2010 Shaw, A., Sriramula, S., Gosling, P.D., Chryssanthopoulos, M.K., (2010). A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macromechanical analysis. Composites Part B: Engineering 41(6):446453. ).
To allow associating experimental design to a virtual model simulation, the methodology of Design of Experiments (DEX) was applied in the analysis of tensile strength of a commercial 1×7 CFRP composite cable to account for uncertainties ( Fong et al., 2013 Fong, J.T., Cimini, C.A., Melo, J.D.D., Heckert, N.A., Filliben, J.J., (2013). Theoretical Failure Envelopes Of Open Hole Composite Laminates With A And BBasis Allowables Estimated From Smooth Specimen Properties. In: 19th International Conference On Composite Materials (ICCM19), Montréal, Canada, pp. 111. ). More specifically, the study focused on how carbon fiber composite cables responses in terms of tensile load and on the sensitivity analysis of the strength estimate to four main design parameters, ultimate tensile strain of the rod, longitudinal elastic modulus, Poisson’s ratio and the static friction coefficient between rods/wires. This work is organized as follows, (ii) Experiments shows how the rods and cables were tested for ultimate tensile strain, longitudinal elastic modulus, Poisson’s ratio and static friction, (ii) Simplified Analytical Model for Transversely Isotropic Material presents the equations used to describe the cable’s behavior, (iii) Finite Element Model exhibits the numerical method and its boundary conditions, (iv) Results shows and analyzes the experimental data and its correlation with the models, and (v) our conclusions.
2 EXPERIMENTS
Standard 1×7 commercial CFRP cable was selected for this study. This cable is composed of six outer wires (3.5 mm diameter) helically twisted around a core of the same diameter. Nominal diameter of the cable is 10.5 mm and pitch length is 152 mm. Fiber volume fraction of the rods was experimentally determined according to the ASTM D3171 as 68%.
Tensile tests of the cables were performed according to the Japanese Standard JSCEE 531 (1995) JSCEE 531 (1995). Test Method for Tensile Properties of Continuous Fiber Reinforcing Materials. Japan Society of Civil Engineers, Tokyo, Japan. using a 170ton horizontal testing machine ( Figure 1 a). The length of the anchor region at the cable ends, which was socketed with expansive cement, was 330 mm, and the central gauge region was 3360 mm long, as shown in Figure 1 b.
(a) CFRP cable attached to the tensile testing equipment, (b) detailing of the cable geometry and socketing scheme.
Strain data was not collected in this test because it was not possible to attach strain gauges to the core rod of the cable which was closely surrounded by six other rods. Instead, a similar test program was performed for single straight CFRP rods (the core rod) to calculate ultimate tensile strain (
From these measurements, it was possible to use the invariantbased method proposed by Tsai and Melo (2014) Tsai, S., Melo, J., (2014). An invariantbased theory of composites. Composites Science and Technology 100:237243. , which introduced the masterply concept (same tracenormalized unidirectional lamina stiffness properties). This is expected to yield good results for thermoset resins reinforced with high volume fraction of carbon fibers, which is adequate for the CFRP analyzed in this study.
The Static Friction Coefficient (
3 SIMPLIFIED ANALYTICAL MODEL FOR TRANSVERSELY ISOTROPIC MATERIAL
The analytical model used in this study is a simplified model based on Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. , adapted for a transversely isotropic material. Since the rods used in the CFRP cables were produced by pultrusion, where all the fibers are aligned in the same direction and homogeneously distributed within a circular crosssection, the transversely isotropic symmetry was chosen to describe them.
The 1×7 cable is the subelement of more complex wire ropes. Therefore, this model consists in the use of kinematics of a thin wire to get the equilibrium equations. The full based analytical model, with all the assumptions, limitations and original equations can be found in Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. . Figure 3 shows the loads acting on the cable (a) and on a single helical rod (b), and Equations (1)  (8) describe the static response of a 1×7 stranded cable, which, in this study, differs from the original Costello (1997) Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York. due to anisotropic material assumptions:
where
4 FINITE ELEMENT MODEL
The numerical finite element model was developed within the commercial finite element platform Abaqus^{TM} using a tetrahedral mesh with tennode linear brick elements and three degrees of freedom per node (C3D10 in Abaqus^{TM} mesh library) as seen in Figure 4 a. The 1×7 Stranded cable geometry was modeled using 10,533 nodes and 32,075 elements. The orthotropic material of each wire in the strand was oriented with its fiber axis set along the helix axis, as seen in Figure 4 b. Surfacetosurface contact type was defined between surfaces of adjacent wires of the strand, 12 pairs in total, and one of them is highlighted in Figure 4 c. Friction was taken into account in the contact based on the static friction coefficient. Boundary conditions considered builtin restrictions in both cable ends, but allowing torsion rotation along the longitudinal (z) axis at one end. Pure tensile stress loading was applied and maximum strain failure criterion was considered in all numerical simulations, using
3D numerical finite element model of the CFRP cable: (a) mesh with 8node tetrahedral elements, (b) description of material orientation (fiber axis) along the helix length and (c) highlighting of one of the 12 contact pairs.
5 RESULTS
Table 1 presents the tensile breaking loads of four 1×7 CFRP cable specimens (mean value of 190.25 kN). Table 1 also shows the results from single straight CFRP rods, with mean longitudinal elastic modulus, major Poisson’s ratio and ultimate tensile strain of 141.6 GPa, 0.28 and 1.64%, respectively. From the friction tests, the mean
CFRP rod elastic properties estimated using Tsai and Melo (2014) Tsai, S., Melo, J., (2014). An invariantbased theory of composites. Composites Science and Technology 100:237243. trace invariant method.
In order to insert probabilistic uncertainties into the simulations, both models incorporated variations in the main input properties. Uncertainty was represented by one standard deviation (
Considering the full population for each approach, the numerical model presented an average load break of 188.1 kN (
The smaller difference presented by the numerical model can be justified by its higher complexity, since it considers a 3D geometry, friction and nine elastic constants. Results from the analytical model indicates that it needs refinement, nonetheless, the numerical model is much more timeintensive than the analytical model.
As for the statistical analysis, a Kolmogorov–Smirnov test was applied in order to check the hypothesis of normality. Levene's test was also applied to check the hypothesis of homogeneity of variance. The pvalue from both tests was less than 0.05 ( Table 4 ), indicating that: (i) the data set is nonparametric and (ii) there is no equality of variances in at least one group. Therefore, only nonparametric tests were applied so forth.
Statistical analysis for differences between results and sensitivity of both models to individual parameters.
In order to check for statistically significant differences between results, a Kruskal–Wallis H test was applied. The pvalue was less than 0.05 indicating that there is significant difference between at least one model compared to the experimental results. To find out which group is different, a MannWhitney U test was used, and the results can be seen in Table 4 . This analysis indicates that there is no significant difference between experimental and numerical results (pvalue of 0.852). However, significant difference was found for the analytical model (pvalue 0.022). That is, the numerical result was similar to the experiment, but they both differed from the analytical model.
Lastly, in order to verify sensitivity of both models to individual parameters, another Kruskal–Wallis H test was applied. The numerical data was organized into three groups: (i) Runs 5, 14 and 23, (ii) Runs 11, 14 and 17, and (iii) Runs 13, 14 and 15; and the analytical data into two groups: (i) Runs 46, 1315 and 2224, and (ii) Runs 1012, 1315 and 1618. The results indicate that are no difference between the groups for both models (pvalue 1.00 for both analyses). That is, the individual parameters give similar contribution to the model’s results.
6 CONCLUSIONS
CFRP cables have a great potential to be use in high performance applications for different engineering fields. Experimental tensile testing was conducted in 1×7 CFRP cable specimens resulting in average breaking load of 190.25 kN (
Acknowledgments
Authors would like to acknowledge the support of FAPEMIG, CAPES, CNPq and Petrobras.
References
 Adanur, S., Mosallam, A.S., Shinozuka, M., Gumusel, L., (2015). A comparative study on static and dynamic responses of FRP composite and steel suspension bridges. Journal of Reinforced Plastics and Composites 30(15):12651279.
 Cai, H., Aref, A.J., (2015). On The Design And Optimization Of Hybrid Carbon Fiber Reinforced PolymerSteel Cable System For CableStayed Bridges. Composites: Part B 68:146152.
 Costello, G.A., (1997). Theory of Wire Rope. Mechanical Engineering Series, SpringerVerlag, New York.
 Dhand, V., Mittal, G., Rhee, K.Y., Park, S.J., Hui, D., (2015). A short review on basalt fiber reinforced polymer composites. Compos. Part BEng. 73:166180.
 Erdonmez, C., Imrak, C., (2011). A finite element model for independent wire rope core with double helical geometry subjected to axial loads. Sadhana 36(6):9951008.
 Fabbrocino, F., Modano, M., Farina, I., Carpentieri, G., Fraternali, F., (2016). Optimal prestress design of composite cablestayed bridges. Compos. Struct. 169:167172.
 Fong, J.T., Cimini, C.A., Melo, J.D.D., Heckert, N.A., Filliben, J.J., (2013). Theoretical Failure Envelopes Of Open Hole Composite Laminates With A And BBasis Allowables Estimated From Smooth Specimen Properties. In: 19th International Conference On Composite Materials (ICCM19), Montréal, Canada, pp. 111.
 Ghoreishi, S.R., Messager, T., Cartraud, P., Davies, P., (2007). Validity and limitations of linear analytical models for steel wire strands under axial loading, using a 3D FE model. International Journal of Mechanical Sciences 49(11):12511261.
 Grover, N., Sahoo, R., Singh, B.N., Maiti, D.K., (2017). Influence of Parametric Uncertainties on the Deflection Statistics of General Laminated Composite and Sandwich Plates. Composite Structures, doi: http://dx.doi.org/10.1016/j.compstruct.2017.03.036.
» http://dx.doi.org/10.1016/j.compstruct.2017.03.036  Jackson, D., Shepheard, B., Kebadze, E., Teles, R., Rossi, R., Gonçalves, R.C., (2005). CFRP Mooring lines for MODU applications. In: Offshore Technology Conference, Houston, United States, pp. 16.
 Jiang, W.G., (2012). A concise finite element model for pure bending analysis of simple wire strand. International Journal of Mechanical Sciences 54(1):6973.
 JSCEE 531 (1995). Test Method for Tensile Properties of Continuous Fiber Reinforcing Materials. Japan Society of Civil Engineers, Tokyo, Japan.
 Meier, U., (2012). Carbon Fiber Reinforced Polymer Cables: Why? Why Not? What If? Arabian Journal for Science and Engineering 37:399411.
 Meier, U., (1992). Carbon Fiber Reinforced Polymers: Modern Materials in Bridge Engineering. Structural Engineering International 2(1):712.
 Meier, U., Müller, R., Puck, A., (1982). GFKBiegeträger Unter Quasistatischer und Schwingender Beanspruchung. Proceedings of Internationale Tagung Über Verstärkte Kunststoffe, 35:135.
 Rohleder Jr., W.J., Tang, B., Doe, T.A., Grace, N.F., Burgess, C.J., (2008). CFRP Strand Application on Penobscot Narrows Cable Stayed Bridge. Transportation Research Board Journal 2050:169176.
 Schmidt, J.W., Bennitz, A., Täljsten, B., Pedersen, H., (2010). Development of Mechanical Anchor for CFRP Tendons Using Integrated Sleeve. Journal of Composites for Construction 14:397405.
 Shaw, A., Sriramula, S., Gosling, P.D., Chryssanthopoulos, M.K., (2010). A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macromechanical analysis. Composites Part B: Engineering 41(6):446453.
 Sparks, C., Zivanovic, I., Luyckx, J., Hudson, W., (2003). Carbon Fiber Composite Tendons for Deepwater Tension Leg Platforms. In: Offshore Technology Conference, Houston, United States, pp. 15.
 Son, B.J., Lee, S.Y., Ji, H.S., (2013). Longterm performance of a fiberreinforced polymer slab bridge superstructurefield load test and ratings. Compos. Part BEng. 45(1):644656.
 Tsai, S., Melo, J., (2014). An invariantbased theory of composites. Composites Science and Technology 100:237243.
 Usabiaga, H., Pagalday, J.M., (2008). Analytical procedure for modelling recursively and wire by wire stranded ropes subjected to traction and torsion loads. International Journal of Solids and Structures 45(21):55035520.
 Wang, X., Wu, Z., (2010). Integrated HighPerformance ThousandMetre Scale CableStayed Bridge With Hybrid FRP Cables. Composites: Part B 41:166175.
 Wang, Z.Y., Wang, Q.Y., (2015). Fatigue strength of CFRP strengthened welded joints with corrugated steel plates. Compos. Part BEng. 72:3039.
Publication Dates

Publication in this collection
04 June 2018 
Date of issue
2018
History

Received
29 June 2017 
Reviewed
17 Nov 2017 
Accepted
01 Dec 2018