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Chains are typically used for tension load transfer. They are very flexible and allow easy length adjustment by hooking at the links. Steel is the traditional material for chains. Recently, synthetic link chains made from ultra-strong polyethylene fibers, branded as Dyneema®, are commercially available. These chains offer a highly improved strength to weight ratio. So far, one type of such chains is available, and it has a Working Load Limit of 100 kN. 50 of such chains, containing 6 links were tested to fracture. The strength of each chain and the location of the failed link were documented during testing for later interpretation. Weibull statistics was applied in order to extrapolate towards the allowable load for very low failure risks (high reliability). Two approaches were used. One extrapolation was based on all results; the other was applied after recognition that the end links failed under a slight negative influence by the connection to the testing equipment. Thus, in fact two populations are mixed, the chains with failing end links and the chains with failing central links. So considering the population without the failing end links is more representative for pure chain behavior without clamping effects. The results from this latter consideration showed a higher Weibull exponent, thus a more realistic extrapolation behavior. Both methods indicate that the reliability at the working load limit of 100 kN is very good.

Carrying tension load is typically achieved with cables or chains. Cables offer the best strength to weight ratio, whereas chains offer the possibility to attach hooks in various chain links and thus adjust the working length quickly. Moreover, chains are highly flexible and thus easy to handle and store and they coil easy. This is especially relevant in heavy haulage by load securing of irregular shaped freight on ship decks, train wagons or road trailers. Steel is the traditional material for chains. Steel link chains are very heavy. Synthetic link chains made from ultra-high strength gel-spun polyethylene fibers are recently commercially available. The fibers used for the manufacturing of those chains are Dyneema® fibers. An introduction to these fibers is presented earlier [

^{th} century mathematician). The reason for this construction [

as well for direct lashing. Some more details can be found on the TYCAN® website [

A number of 50 TYCAN® chains was ordered from the manufacturer and tested as received. A tensile test bench with 900 kN load capacity at an independent external test house was used (Hijs Service Astea B.V., Geleen, The Netherlands). The connection to the loading device was provided by an oval steel pin with a long axis of 60 mm and a short axis of 30 mm.

As one out of the fifty, a typical test diagram is presented in

Weibull statistics is often used in structural engineering and material science and was therefore also adopted for the present study. A reason is that it does not comprise negative strength values that are of course impossible and could result from using a Gaussian distribution, so the representation of low strength values (at low failure probabilities e.g. obtained by extrapolation) obtained from a Weibull distribution is expected to be more realistic than obtained from the Gaussian distribution. The original Weibull equation [

where R is the reliability at load P, and F represents the failure probability. V accounts for effects of volume. For the present case it can be replaced by chain length L. N accounts for the number of chains, as is stated under expression 2 as well, and K for shape effects. Shape effects should be treated with mechanics, rather than statistics, so this facility should be skipped from the equation. Shape effects are anyhow not part of the present consideration. The addition of i to V, K and N refers to the population considered. P0 is a reference strength level, close to the average strength. The exponent m is the so called Weibull modulus. The higher this value, the lower is the strength variability. Ignoring K, Equation (1) can be rewritten for the present chains as:

where L is the effective span length of one chain and N is the number of chains. Accordingly, when a number Ni of chains with length L is considered, the total chain length considered Li is:

Equation (3) can be substituted in Equation (2), reducing the first part between the brackets to total chain length only. A Graphical representation of a Weibull distribution can be made by plotting the natural logarithm of minus the natural logarithm of R on the vertical axis and the natural logarithm of the measured load P on the horizontal axis. R is than calculated from the rank number n (from strongest to weakest) of the rest result and the number of tests N according to:

Test number | Breaking load [kN] | Broken link number | Rank no. n | Reliability R | Test number | Breaking load [kN] | Broken link number | Rank no. n | Reliability R |
---|---|---|---|---|---|---|---|---|---|

1 | 216.4 | 2 | 27 | 0.53 | 26 | 226.8 | 4 | 12 | 0.23 |

2 | 220.3 | 5 | 20 | 0.39 | 27 | 227.5 | 3 | 11 | 0.21 |

3 | 217.0 | 2 | 25 | 0.49 | 28 | 220.1 | 4 | 21 | 0.41 |

4 | 193.3 | 1 | 48 | 0.95 | 29 | 220.9 | 6 | 18 | 0.35 |

5 | 235.6 | 4 | 4 | 0.07 | 30 | 215.7 | 2 | 28 | 0.55 |

6 | 211.7 | 3 | 39 | 0.77 | 31 | 220.9 | 1 | 19 | 0.37 |

7 | 219.1 | 6 | 22 | 0.43 | 32 | 228.2 | 5 | 10 | 0.19 |

8 | 233.0 | 5 | 5 | 0.09 | 33 | 216.6 | 6 | 26 | 0.51 |

9 | 199.4 | 1 | 47 | 0.93 | 34 | 223.4 | 3 | 16 | 0.31 |

10 | 212.9 | 4 | 34 | 0.67 | 35 | 238.4 | 4 | 2 | 0.03 |

11 | 223.3 | 3 | 15 | 0.29 | 36 | 230.2 | 5 | 8 | 0.15 |

12 | 187.7 | 6 | 50 | 0.99 | 37 | 218.1 | 1 | 23 | 0.45 |

13 | 210.4 | 1 | 40 | 0.79 | 38 | 208.2 | 6 | 44 | 0.87 |

14 | 214.8 | 4 | 29 | 0.57 | 39 | 232.7 | 3 | 6 | 0.11 |

15 | 206.7 | 6 | 46 | 0.91 | 40 | 213.7 | 4 | 33 | 0.65 |

16 | 211.8 | 1 | 38 | 0.75 | 41 | 209.0 | 5 | 43 | 0.85 |

17 | 207.5 | 2 | 45 | 0.89 | 42 | 212.9 | 5 | 36 | 0.71 |

18 | 209.0 | 2 | 42 | 0.83 | 43 | 214.0 | 5 | 32 | 0.63 |

19 | 224.2 | 3 | 13 | 0.25 | 44 | 214.1 | 6 | 31 | 0.61 |

20 | 237.6 | 5 | 3 | 0.05 | 45 | 222.4 | 6 | 17 | 0.33 |

21 | 212.6 | 2 | 37 | 0.73 | 46 | 214.8 | 6 | 30 | 0.59 |

22 | 212.9 | 4 | 35 | 0.69 | 47 | 231.2 | 1 | 7 | 0.13 |

23 | 209.2 | 3 | 41 | 0.81 | 48 | 224.0 | 6 | 14 | 0.27 |

24 | 192.0 | 6 | 49 | 0.97 | 49 | 217.2 | 6 | 24 | 0.47 |

25 | 228.5 | 3 | 9 | 0.17 | 50 | 248.9 | 3 | 1 | 0.01 |

Equation (4) is identical to Weibull’s original work. It can be understood such that the “reliability space” is divided in N blocks with rank n. The R value is placed in the center of each n^{th}-block, adding −0.5 provides that placement in the middle. A graphical representation is made as a first consideration for all as measured data from

It can be seen in

Ni is the number of chains of 0.5 meter. Indeed, inserting for Ni = 1 Million as an example for P = 121 kN, the resulting reliability R = 0.38. So suggesting a considerable chance of failure of one from those million chains, occurring at P = 121 kN. Another way of interpretation, is that a million times a chain of 0.5 meter, implies a total chain length of 500 km. So also a 500 km long chain, loaded with P = 121 kN has a reliability of 0.38. Loading of such a 500 km total chain length with the certified WLL = 100 kN still implies a reliability R = 0.987.

Rank Number | Fracture load [kN] | R experiment | R Weibull | R normal |
---|---|---|---|---|

47 | 199.4 | 0.93 | 0.925 | 0.944 |

48 | 193.3 | 0.95 | 0.962 | 0.983 |

49 | 192.0 | 0.97 | 0.967 | 0.987 |

50 | 187.7 | 0.99 | 0.980 | 0.995 |

An alternative consideration can be made after studying

Removal of the first and last link fractures from the data set, so removal of a non-representative sub population can be explored. The consequence of that segmentation is, that chains of 0.4 meter are remaining, so L = 0.4. It may be argued that removing lower values from the original data set causes an artificial non-realistic improvement. However, the removed links are not chosen based on their lower fracture load, but based on their position (the lower fracture load is only used for recognizing the non-representative positions). Moreover, the removal effect itself is compensated by correcting for a lower representative chain length in the Weibull equation. Also in practice a 16mm pin is used for load transfer into steel hooks. An additional argument for the segmentation is that the unfavorable interface fracture implies that the remaining interface of the removed links will have a fracture force, larger than the measured fracture force that was limited by the broken outer link. So potential high fracture loads belonging to the sub population of the considered central chain part were omitted, and the consideration may even be conservative. Summarizing, removal of the outer link fractures from the data set is defendable for providing a description of intrinsic chain behavior without any external outside world influence like end connections. Performing a Weibull fit on the remaining data for the central part yields:

This new Equation (6) shows a higher Weibull exponent indeed as compared to Equation (5). The index i has been replaced by index j in order to indicate a different population. It may be explored by inserting some numbers and comparing the result with the result of Equation (5). A length of 500 km chain is equivalent to Nj = 1,250,000. For a load of P = 121 kN, the reliability is R = 0.95, so considerably larger than according to Equation (5) representing the mixed population. A reliability of R = 0.38 as obtained with Equation (5) for a load of P = 121 kN for the mixed population, requires a load of P = 134.5 kN on the population of the central links according to Equation (6). Indeed this is a more attractive value than according to the first consideration, but not largely different (134.5 kN is only about 10% larger than 121 kN). The reliability according to Equation (6) for 500 km total chain length at WLL = 100 kN, now is R = 0.9997, so very close to R = 1.

Link Number | Number of fractures | Average Load [kN] | Standard Deviation [kN] |
---|---|---|---|

1 | 7 | 212.2 | 12.9 |

2 | 6 | 213.0 | 4.0 |

3 | 9 | 225.8 | 11.6 |

4 | 8 | 221.9 | 10.5 |

5 | 8 | 223.2 | 10.5 |

6 | 12 | 212.0 | 11.6 |

for low risk extrapolations). The main part of this steeper trend is set by three data points only. So, this deviation may still be judged coincidental and a linear fit for extrapolation is still adopted as a reasonable procedure.

Weibull statistics is applied on a set of 50 tensile tests on novel light weight synthetic link chains made from UHMWPE fiber instead of traditional steel wire. During tensile testing, a slight adverse measuring effect of the connection to the tensile test equipment became apparent. The effect was not found at the end termination itself, but at the opposite site of the link-machine interface. It can be explained by the deformation of the end links due to the thickness of the required connection pin. Two Weibull distributions were created that are used for extrapolation, one by fitting the data sets to all 50 results and one by fitting to a dataset without the outer links attached to the test equipment. The latter showed a higher Weibull exponent. The data sets were used for extrapolation to low failure risks or to very long total chain lengths (so the total length of a number of chains). The result is that the reliability of the chains is high, if the Work Load Limit WLL = 100 kN is respected. The extrapolations of the central part represent the intrinsic chain behavior. Additional effects like connection effects are excluded. So effects as caused by end connections and possible hooks are excluded. Of course such effects are relevant in practice, but beyond the scope of the present study. The present study indicates that the intrinsic consistency of chain properties is highly sufficient for safe use.

Chain length [m] | Estimated failure probability according to the mixed population. Equation (5) | Estimated Failure probability for the central link population. Equation (6) |
---|---|---|

1 | 2.7 × 10^{−8} | 10^{−9} |

5 | 1.4 × 10^{−7} | 3 × 10^{−9} |

50 | 1.4 × 10^{−6} | 2.8 × 10^{−8} |

500 | 1.4 × 10^{−5} | 2.8 × 10^{−7} |

10,000 | 2.8 × 10^{−4} | 5.6 × 10^{−6} |

500,000 | 1.4 × 10^{−2} | 2.8 × 10^{−4} |

5,000,000 | 1.3 × 10^{−1} | 2.8 × 10^{−3} |

Obviously, the failure probabilities remain low, so the reliabilities at WLL (100 kN) remain high, even for very long total chain lengths (5000 km). It was mentioned before that this far extrapolation is not an accurate prediction, but rather a reasonable expectation. Moreover, all chains are loaded at WLL = 100 kN, before leaving the factory, so fractures at WLL would occur before chains become in use. On the other hands, chains with a slightly larger strength than 100 kN would become in use. It is beyond the scope of this paper to speculate on the consequences of these effects, because reliability at 100 kN is high anyhow, even for large total chain lengths. It can be observed in

Roel Marissen,Dietrich Wienke,René Homminga,Rigo Bosman,Kjell Magne Veka,Anna Huguet, (2016) Weibull Statistics Strength Investigation of Synthetic Link Chains Made from Ultra-Strong Polyethylene Fibers. Materials Sciences and Applications,07,238-246. doi: 10.4236/msa.2016.75024