Each link in a fiber chain is made by repeatedly winding fiber fabric to form a belt shape with a constant inner diameter and fixing the fabric using separate stitches. Many such links are connected to realize the chain length, and the product is completed by adding a shackle at the end of the chain. A fiber chain can be used with a crane or winch for heavy-duty lifting. Fig. 1 shows examples of its application by Green Pin® and the geometry of a fiber chain link. The main design variables of a link are the fabric width, inner diameter length, and number of times that the fabric is repeatedly wound to form a link.

The capstan equation was used for the fundamental design of the fiber chain in consideration of the load gradient: where, T_load represents the tension applied on the line, T_hold represents the resulting force exerted at the opposite side of the cylinder, μ is the coefficient of friction between the line and cylinder material, and ϕ is the total angle swept by all the turns of the line. The capstan equation is also known as Eytelwein’s formula and relates the holding force of a flexible line to the loading force when the line is wound around a cylinder. Because of the interaction between frictional forces and tension forces, the tension on the line around the cylinder may differ from one side of the cylinder to the other. A small holding force applied on one side of the cylinder may exhibit a much larger loading force on the opposite side, depending on the wrapping count. This is how a capstan-type device operates. A fiber chain is configured using the wrapping count at a constant inner diameter, so a foundation design with a gradual load was developed using the capstan equation. Tthe layout of the equation is provided in Fig. 2.

The load according to the wrapping count can be calculated at the design breaking strength of 32 t based on the use of HMPE DM20 and assuming that the coefficient of friction is 0.1 and a wrapping count of 10, as provided in Table 1. When the wrapping count was set at its maximum of 10, the load applied to the outermost fabric of the fiber chain was calculated as 0.06 t, which is within the allowed range of the design standard.

The manufacturing process for the fiber chain involved braiding yarn to produce the fiber chain fabric and loading the braid onto a loom to weave the fabric in the form of webbing. The woven fabric was then wound to produce chain links. The webbed fabric took the form of a strip, and the fabric was woven according to the pick count. The pick count is the number of intersection points for the wrap and weft contained in 25.4 mm, which affects the shape stability of the fabric and tensile performance until failure. Thus, different pick counts were used to analyze the tensile strength characteristics.

The shape of the webbing fabric woven with a 30 mm width is shown in Fig. 3. The pick count values used in the fabric webbing were 10.7, 13.7, and 14.7, and a tensile strength test (ISO 2307- Fiber ropes, 2019) was performed on the woven fabric. The breaking strength results according to the pick count are provided in Table 2. As revealed by the breaking strength values, a smaller pick count resulted in a higher breaking strength for the woven fabric. Breaking strengths of 52.1 and 42.4 kN were observed for pick counts of 10.7 and 14.7, respectively. Because the fiber chain fabric was subjected to a tensile load in the weft direction, a lower pick count resulted in better tensile strength, and a larger pick count produced a larger fiber density, which improved the shape stability and increased the elongation. It can be seen that the shape stability and the breaking strength are parameters to consider in the design of the fiber chain.

The breaking strength is the primary performance factor to consider for a fiber chain. Because a fiber chain should be designed to satisfy a target strength, an experimental design method was applied to analyze the influences of the design factors on the breaking strength and derive the optimal combination of factors. During the manufacture of a fiber chain, the valid design variables include the pick count, wrapping count, and inner length. Therefore, a factorial experiment design with these design variables was conducted, and their correlations to the breaking strength were analyzed. The ranges of the design variables are provided in Table 3. The inner length is the minimum length to start wrapping the fabric. For each design variable, the variance range was set by employing one upper level and one lower level from the reference model.

The design of experiments method was used to minimize the number of experiments conducted by designing an apparatus on a statistical basis. An analysis of the results based on a statistical evaluation of the hypothesis made it possible to define the relationship between the input and output variables as a function and provide the maximum amount of information with minimum experimentation and cost. In this study, an orthogonal array with three factors was adopted to analyze the relationship between the input parameters (design variables) and an output parameter (breaking strength) through experiments.

The fiber chain specimens were fabricated, and tensile strength tests were conducted with various values for the design variables: the pick count, wrapping count, and inner length. Rope tensile-strength test equipment with 150 t capacity at DSR Corp. was used to apply loading conditions to the fiber chain. This test equipment and the experiment are shown in Fig. 4. ISO 2307 standards were used in the tests.

A synthetic fiber rope is subjected to pre-tension due to the uneven torsional stress and torsional strength of the inner lower rope. This pre-tension improves the shape stability of the synthetic fiber rope. The effect of pre-tension on the tensile strength of the fiber chain was analyzed prior to the complete experiment because a fiber chain shares many similarities in the material and manufacturing process with a synthetic fiber rope. Generally, the pre-tension of a synthetic fiber rope is within 60% of the breaking load. When more than 85% of the breaking load is applied, the rope core may undergo plastic deformation, which significantly decreases the elastic modulus despite the increased elongation. Because of this risk of a decreased elastic modulus, a load within 60% of the breaking load is applied. Therefore, the fiber chain was prepared with pre-tension equivalent to 50% of the breaking load, and the effect was analyzed.

Fiber chain samples with a width of 30 mm were produced and used in strength tests with and without pre-tension. As summarized in the test results in Table 4, the fiber chain without pre-tension showed a breaking strength of 239.8 kN and elongation of 41.5%, while the fiber chain with pre-tension demonstrated a breaking strength of 286 kN and elongation of 14.5%. The breaking strength was the average value of three tests, and the deviations from the mean in test results were 5.41 and 8.34, respectively. The pre-tension effect caused a 16% difference in the breaking strength and a 27% difference in the elongation.

The yarn of the fiber chain gained residual tensile strength in the spinning process with torsional stress produced in the twisting and weaving processes. The residual tensile stress and torsional stress were major factors affecting the structural elongation of the fiber chain. Because the imposition of pre-tension could partially or completely eliminate the residual tensile strength and torsional stress, the increased breaking strength under the pre-tension effect was postulated to be due to the increase in the elastic modulus of the fiber chain. In addition, the fabric structure of the fiber chain became denser as a result of the pre-tension, and the elongation rate decreased.

The results of the breaking strength experiments with various design variable values are presented using an orthogonal array in Table 5. The analysis of variance (ANOVA) results for the breaking strength in relation to the pick count, wrapping count, and inner length are shown in Table 6. The ANOVA results indicate the degree of freedom (DF), sum of the adjusted squared deviation (Adj SS), sum of the adjusted mean square (Adj MS), standard error of the regression (S), coefficient of the determination (R²), F-statistic, and P-value for each term in the model. The variance (the sum of the deviation squared) determines the magnitude of the change with respect to the measurement. The sum of the deviation squared divided by the degree of freedom is the sum of the mean squared (unbiased variance). The F-statistics refer to the ratio of variances, and the P-value is the expected value of variance. An evaluation of a variable was conducted using the F-statistic and P-value, while the P-value was used in this experiment.

The Adj SS values for the pick count, wrapping count, and inner length were 1548.83, 532.04, and 151.0, respectively, as shown in Table 6. The variations in the breaking strength due to the pick count and wrapping count were greater than those of the inner length. This occurred because the density of the warp and weft constituting the cross-sectional layer of the fiber chain were direct factors in the breaking strength, and the wrapping count affected the capstan effect. The ANOVA of the breaking strength showed that the F-value was 21.66 for the pick count and 7.44 for the wrapping count, and the expected variance P-value was less than or equal to 0.05 for both variables. Therefore, it was concluded that the pick count and wrapping count were the main factors influencing the breaking strength at a significance level of 5%.

Fig. 5 displays the main effects of the pick count, wrapping count, and inner length on the breaking strength. In the main effect plot, the dotted line in the middle represents the total average of the breaking strengths, and each dot corresponds to the average value at a specific level. The breaking strength appeared larger-the-better characteristics with respect to the changes in the design variables. The main effect plots for the changes in the levels of the design variables in Fig. 5 reveal that the breaking strength increased as the pick count decreased, the wrapping count increased, and the inner length increased.

The sensitivity of each design variable to the breaking strength was determined using the delta values (difference between the maximum and minimum values) of the objective function, which are summarized in Table 7. The variable with the highest sensitivity was determined to be the pick count with a delta value of 32.1, followed by the wrapping count and inner length. The analysis revealed that the level changes in the pick count resulted in a change of up to 11.8% for the breaking strength, while changes in the wrapping count and inner length caused 6.7% and 3.5% changes, respectively. A large strength variance was observed in the pick count range from −1 to 0 and in the wrapping count range from 0 to 1.

A contour plot represents the response curve for a combination of design variables, and the value of an objective function is constant on each contour line. The contour lines for the pick count and wrapping count with respect to the breaking strength are plotted in Fig. 6. The maximum breaking strength is observed at a wrapping count of 1 and a pick count of −1 in Fig. 5, and the figure displays the contour lines under the same conditions. Using the results of the contour plot analysis, the breaking strength can be estimated to be greater than 300 kN when the wrapping count is fixed at 1, the pick count is less than −0.65, and the inner length is greater than 0.38. The combination of design variables made it possible to identify the region of optimal conditions that satisfy the target strength. Through additional experimentation and analysis, the derivation of a optimal design-variable combination is expected.