Data‐driven design concept evaluation for full‐locked coil ropes

This article provides a detailed investigation of the design concept for full‐locked coil ropes, which are widely used in various applications in the construction sector. The current design concept in the ultimate limit state according to EN 1993‐1‐11 is explained by tracing its development through past standards. A database of breaking load tests is built up and used to evaluate the actual fill and spinning loss factors compared to the standard values. Afterwards, the current design equation and the revised equation of the third draft of prEN 1993‐1‐11 are statistically evaluated with the test‐based design procedure in EN 1990. The investigations reveal that the use of a fill factor is not required and should be omitted, while the spinning loss factor can be approximated by a constant value. The newly proposed equation in prEN 1993‐1‐11 shows improved accuracy in comparison with the current design equation of EN 1993‐1‐11 regarding the calculated partial factors. However, the determined partial factors are based exclusively on tensile tests under laboratory conditions. They could serve as a basis for the development of a new design concept in which project‐specific information such as local bending stresses can be considered individually. This would allow a more precise definition of the safe level of utilisation and thus lead to a more economical use. Further research is required to quantify these factors.


Introduction
Steel ropes are used as high-strength load-carrying elements in a wide range of applications: from cable-stayed and suspension bridges to wide-span supporting structures such as cable net roofs (Fig. 1).Especially for the latter, full-locked coil ropes (FLC) are often used.
Generally, FLC consist of an inner part of round wires and one or more outer layers of z-shaped wires (Fig. 2a).The wires of the different layers are arranged in a helix with different lay directions (Fig. 2b).According to EN 1993-1-11 [1], FLC are assigned to group B of tension members.
The calculation of the tension resistance of FLC in the ultimate limit state (ULS) is generally carried out according to the design rules of EN 1993-1-11 [1]: The first part of Eq. ( 1) refers to the characteristic value of the breaking strength F uk , while the second part relates to the characteristic value of the proof strength of the tension component F k .Within Eq. ( 1), a factor of 1.5 on the resistance side is used.The historic origin of this factor can be found in the German design code DIN 1073 [2] from 1974 for road bridges made of steel.The allowable stress factor for FLC for the load case 'H' (main loads) is given there as 0.42 [2].
With the revision of the subsequent standard DIN 18800-1 [3] in 1990, the semi-probabilistic safety concept was implemented, transferring the global safety factor of 1/0.42 = 2.38 into partial factors 1.35 to 1.5 on the load side.With a median value of 1.45, this leaves approximately 2.38/1.45= 1.65 = 1.5 • 1.1 on the resistance side [3].
The introduction of the EN 1993-1-1 [4] changed the partial factor γ M = 1.1 to γ M0 = 1.0 for the resistance of cross sections.This was implemented for the cable design as well, leading to formula Eq. ( 1) as it is used until now.Therefore, the partial factor γ R in Eq. ( 1) is set to 1.0.A reduction to γ R = 0.9 is possible if there are measures to minimise bending stresses at the anchorage [1].
Regarding the first part of Eq. ( 1), the F uk corresponds to the product of the minimum breaking force F min (MBF) This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.and a loss factor k e depending on the type of end termination.For the commonly used metal-and resin-filled sockets, k e is set to 1.0.The MBF F min is calculated according to EN 12385-2 [5]: In Eq. ( 2), d n corresponds to the nominal diameter of the rope and R r to the rope grade.The equation can be rewritten using the fill factor f and the spinning loss factor k instead of the minimum breaking force factor K leading to This reformulation is useful because both factors f and k, in contrast to K, have a direct physical meaning and can therefore be better evaluated.A detailed explanation of the design concept can be found in [6].
The second part of Eq. ( 1), the design against proof strength F k , does not become relevant for the usual prestretched FLC with the current factor of 1.5 against the calculated breaking strength F uk .Therefore, it is removed in the third draft for the revision of EN 1993-1-11 [7].However, in the light of a possible modification of the design against the calculated breaking strength, a verification of the serviceability limit state might become relevant instead.
The current research in the area of FLC primarily focuses on fatigue behaviour.It includes, among others, the determination of a failure criterion for the fatigue resistance [8] and the investigation of SÀ N-curves [9].The fatigue behaviour of FLC under bending stress is also investigated [10].The use of FLC for bridges is described in [11], but there is also no detailed explanation of the application of the design equation.With regard to the rope characteristics, Feyrer has established a database of reference values that can be used for the evaluation of tests [12].The revision of the Eurocodes is already in progress.Therefore, the third draft for the revision of EN 1993-1-11 [7] will be used as a comparison to the current version of the EN 1993-1-11 in this article.
From the perspective of structural engineering offices in construction practice, the design concept does not seem to represent reality well for modern cable structures.
Considering project-specific conditions of loading, movements, lateral pressure or relocation effects from saddles and clamps as well as imperfections may allow to define the partial factor more specific.
Therefore, a statistical evaluation of the design concept with regard to the required safety level according to EN 1990 is useful.Two main questions have to be answered: To answer these questions, a database of load breaking load tests of FLC is built up.Based on this database, the fill and spinning loss factors and their specifications in the standards are evaluated on the basis of measured data.Considering the results, the adapted equation of prEN 1993-1-11 is assessed together with the equation of EN 1993-1-11 using the test-based design procedure given in Annex D of EN 1990 [13].The resulting partial factors built on breaking load tests under laboratory conditions are compared as an indicator for the uncertainty of the different design equations.In the light of a modification of the design concept in the context of the revision of the Eurocodes, additional influence factors have to be investigated and taken into account.in the period between 2008 and 2021.Therefore, it will be assumed that the database is representative for ropes from European rope manufacturers over the last 15 years in the common diameter range.
Datasets had to be diminished if the data of the rope construction are insufficient or there are no valid breaking tests.Possible reasons are, for example, that some ropes were not tested to failure (test stopped when reaching the calculated MBF) or that the anchorage failed in the test.Tab. 1 shows how many values are available for the different parameters.It should be noted that in general for each parameter both calculated and measured values are required.
In addition, a database of wire breaking load tests is set up (see also Tab. 1).The database includes wire data provided by just three different manufacturers.Different diameters and wire heights as well as different strengths and cross-sectional shapes are investigated.
3 Assessment of the fill and spinning loss factors given in the standards

Evaluation of the fill factor
The axial design load capacity of FLC is strongly connected to the fill and spinning loss factors.Therefore, these reduction factors are analysed in the following on the basis of the available experimental wire and rope data.It is demonstrated that the fill factors given in the standards deviate strongly from the actual values and may be eliminated from the resistance equation, while the spinning loss factors as specified in the standards show good agreement with the mean measured values.
The fill factor f of a wire rope is defined as the ratio of the metallic cross-sectional area A met and the circumscribed area A u based on its diameter: Values for the fill factor are given in Tab.Furthermore, values for the fill factor can be calculated depending on the nominal rope diameter and the number of z-wire layers using the metallic cross-sectional areas given in EN 12385-10 [14].
Fill factors as ratio values cannot be measured directly.Therefore, the metallic cross-sectional area and the circular rope area are analysed separately.A major challenge is that both the determination of the actual metallic cross-sectional area and of the actual rope diameter are subject to large uncertainties.
Usually, there are no measured values for the metallic cross-sectional area of FLC; only measurements of the  wire diameters of round wires and the wire heights of zwires are given in wire test certificates.This results in the following two problems: Firstly, the cross-sectional area of z-wires cannot be determined exclusively by their height because of the individual z-wire geometries.Based on the nominal z-wire heights and nominal cross-sectional areas, it is assumed that the deviation of the cross-sectional area can be approximated using the squared wire heights h w,m 2 /h w,n 2 as a representative dimension, where the index m denotes the measured and the index n the nominal values (see Fig. 4).
Secondly, the exact wire geometries and the assemblies of the different wires inside the ropes are generally unknown.It is therefore impossible to determine the exact metallic cross-sectional area based only on measured wire data.However, the large wire dataset (see Tab. 1) contains major part of possible influencing factors.It is therefore assumed that the statistical distribution of the metallic cross-sectional area as the sum of any number of individual wires is consistent with the distribution of the individual wires.
The second component for calculating the fill factor, the circular area described by the rope diameter, is also difficult to measure precisely.According to EN 12385-1 [16], the rope diameter has to be measured at two points at least 1 m apart.At both points, two measurements must be taken at right angles to each other.The rope should be unloaded or loaded with a maximum of 5 % MBF.In practice, however, loads of up to 10 % MBF are also common.The use of a fill factor in rope dimensioning is considered inconclusive as only the load capacity and the stiffness of FLC are of interest for the designer of a cable structure.Afterwards, the required rope cross-section is determined by the manufacturer using these values, so that the actual metallic cross-sectional area is always known.The use of auxiliary variables such as the nominal rope diameter and the fill factor increases the uncertainty of the design concept because these variables do not accurately reflect the actual conditions of the rope.

Evaluation of the spinning loss factor
The spinning loss factor k describes the loss of axial load capacity caused by the stranding of the wires to a spiral rope (see Fig. 2b).It is defined as the ratio between the sum of the non-stranded wire breaking forces F e and the rope breaking force F: In DIN 18800-1 [15], the spinning loss factor was introduced as k = 0.92.In the subsequent revision [3] in 1990, it was increased to k = 0.95.According to EN 1993-1-11 [1], the spinning loss factor k = 0.92 is defined by the quotient of the nominal metallic area factor C and the breaking force factor K, given in Annex C of EN 1993-1-11.
For a total of 99 FLC tests, both the sum of the nonstranded wire breaking forces and the rope breaking forces are known.Using these measured breaking forces, the measured spinning loss factors are determined from Eq. ( 5) for FLC with one (FLC 1), two (FLC 2) and three or more z-wire layers (FLC 3) in Fig. 6.
The measured spinning loss factors vary over a wide range and no clear dependence on the nominal rope diameter or the number of z-wire layers can be recognised for the dataset.Other possible influencing factors, such as the lay length of the different wire layers, are usually not known to the structural engineers at the time of design and therefore not investigated herein.
The mean measured spinning loss factor is approximated by the constant value k = 0.92 given in EN 1993-1-11.For simplicity, the real spinning loss factors are therefore assumed to be normally distributed with a mean value of 0.92 in the following (see histogram in Fig. 6).

4
Evaluation of the adapted resistance equation in prEN 1993-1-11

Description of the resistance equation and the statistical evaluation
In the third draft for the revision of EN 1993-1-11 [7], an adapted equation for the calculation of the characteristic MBF is presented analogous to the results of the previous section.It can be formulated in the following way: where k is the spinning loss factor, k = 0.92, A w,i is the metallic cross-sectional area of the ith layer of wires and R w,i is the nominal wire grade of the ith layer of wires.
According to Section 3, the direct use of the wire breaking forces A w • R w allows for the dispensation of the fill factor, the rope diameter and the rope grade.
Eq. ( 6) is presently employed by rope manufacturers for the calculation of the MBF.However, for design purposes in accordance with EN 1993-1-11, Eq. ( 1) is mandated.The incorporated factor of 1.5 in Eq. ( 1) is derived for the MBF specified in Eq. ( 2), which incorporates highly uncertain variables, such as the fill factor or the rope grade.As a result, a distinct partial factor for Eq. ( 6) is investigated in this context.
The statistical evaluation of the breaking load tests is carried out on the basis of Annex D of EN 1990 [13], which gives guidance on the procedure of design assisted by testing.The evaluation of the resistance equations in this article follows the standard procedure for the statistical determination of resistance models described in Section D.8.As an evaluation exclusively with measured values of the basic variables is not possible due to a lack of measured data, mean values are used instead in this article.To account for the additional uncertainty, the coefficient of variation of the errors is adjusted using the coefficients of variation of the basic variables according to Bijlaard et al. [17].

Analysis of the statistical distribution of the basic variables
Before the test-based design procedure according to Annex D in EN 1990 can be carried out, it is necessary to determine the mean values and the coefficients of variation of all basic variables.As previously described, mean measured values are used for the evaluation, which are calculated from their deviation from the nominal values.
Reference values for the mean and the standard deviation of the ratio of actual and nominal values are given by Feyrer [18].They are shown in Fig. 7 for the metallic cross-sectional area A met,m /(π/4 • d n 2 • f), the rope tensile strength R m /R r and the spinning loss factor k m /k.It has to be noted that the values are based on 49 parallel lay wire ropes manufactured before 1992 [12].These can generally differ from FLC in terms of strengths and dimensions, but are used here due to the lack of comparative statistical data.The ratio of the spinning loss factor is calculated in reverse using all other given ratios; therefore, no standard deviation can be expressed.The exact definitions of all variables are given by Feyrer [12].
The reference values are compared with values calculated from the breaking load test database.Herein, the MBF F min is defined as per Eq. ( 3) according to EN 1993-1-11.In addition, the ratio values for the wire area A w,m /A w,n and wire strength R w,m /R w,n are shown as they are needed for the evaluation of Eq. ( 6).

Statistical comparison of the resistance equations
Based on the previous investigations on the fill and spinning loss factors in Section 3, the required partial factors γ M,lab for the resistance equations Eq. ( 3) and Eq. ( 6) are determined with a statistical evaluation according to Annex D, EN 1990 [13].It should be noted that the partial factor γ M,lab with the index 'lab' refers exclusively to the uncertainty under controlled test conditions as described above.For the general partial factor γ M , an additional partial factor γ M,imp must be considered: The partial factor γ M,imp must include all other effects that are not represented in the laboratory test, but are relevant for the practical use of FLC, such as bending effects at the anchorage, the influence of lateral pressure at clamps or saddles and imperfections of manufacture and handling of long cables compared to the small test samples.
As outlined in Section 4.1, the statistical evaluation presented herein relies on mean values with additional uncertainty considerations using the approach detailed by Bijlaard et al. [17].The mean resistance values are calculated as the product of the nominal resistance values and the mean value factors for all basic variables within each resistance equation sourced from the database depicted in Fig. 7.These factors delineate the disparity between nominal and mean measured values.The associated coefficients of variation for the basic variables are derived from the ratio of the provided standard deviations to the mean values.
Through the application of the statistical evaluation method in Section 8 of Annex D within EN 1990 [13], design values for resistance are determined directly.Subsequently, the partial factors are derived by dividing the nominal resistance by the design resistance.In comparison, 129 breaking load tests can be used for the assessment of Eq. ( 6) as the newly proposed equation of prEN 1993-1-11 based on the wire data.In this case, a required partial factor of γ M,lab = 1.14 results.
The statistical evaluation shows that the uncertainty in the resistance model may be significantly reduced by an adjusted resistance equation.Within the workshop design for the fabrication of a FLC, the wire layout and the wire grade for each layer are defined.A verification of the calculated MBF as per Eq. ( 6) is typically performed by the cable manufacturer as part of the workshop design.This can be defined as a regular process.The cable built-up can be considered to be known for cables used from stock, for cables being manufactured project specific and the recalculation of existing structures as far as the data are available.Subsequently, the use of Eq. ( 6) can be considered for all applications except for cases where the cable configuration remains unknown (recalculation of structures without existing detailed data).

Conclusion
In this article, a database of FLC was built up.This database was used to evaluate the current design equation in the ultimate limit state of EN 1993-1-11, particularly with regard to the fill and spinning loss factors.The adjusted resistance equation of the third draft of the revision of EN 1993-1-11 was compared to the equation of EN 1993-1-11 using the test-based design procedure of EN 1990, Annex D. The following conclusions were drawn: -An exact determination of the fill factor is difficult due to the high uncertainty of the measured rope diameter and not necessarily required for the dimensioning of FLC.
-Instead, the proposed equation of prEN 1993-1-11 agrees well with the minimum breaking force calculated using the sum of the wire breaking forces in combination with a constant spinning loss factor k = 0.92.-With this adapted resistance equation, the uncertainty for the calculated resistance can be significantly reduced compared to the current equation of EN 1993-1-11.
The results of this study provide valuable insights into the design concept of full-locked coil ropes and demonstrate the potential benefits of the revised equation of prEN1993-1-11 in combination with a reduction of the required partial factor γ M,lab .However, it has to be noted that all investigations are based on breaking load tests representing laboratory conditions.A design equation for construction application also must consider the effects of imperfections, deviation angles at the anchors as well as the effects of lateral pressure and local bending in the areas of clamps and saddles in a second partial factor γ M,imp .Detailed investigations of the various influencing factors are still required to determine the partial factor γ M,imp .A new design concept with a partial factor γ M = γ M,lab • γ M,imp should therefore offer the possibility of being able to take these influences into account in a projectspecific manner instead of using a constant partial factor in all cases.

1 .
Is it possible to reconstruct the design concept according to EN 1993-1-11 and the fill and spinning loss factors contained therein on the basis of measured rope data? 2. Can the uncertainties in the current design equation in EN 1993-1-11 be reduced by the design concept proposed in the third draft of prEN 1993-1-11?

Fig. 2 a
Fig. 2 a) Exemplary cross sections of FLC with different numbers of round and zshaped wire layers according to EN 1993-1-11 [1]; b) isometric illustration of a FLC showing the helically arranged wires 2.2 in EN 1993-1-11[1].For FLC1 (FLC with one z-wire layer), a fill factor f = 0.81 is stated.It is increased to f = 0.84 for FLC2 (two z-wire layers) and to f = 0.88 for FLC3 (three or more z-wire layers).The same values also result from the nominal metallic area factor C given in Annex C of EN 1993-1-11.These values were first introduced in the German national standard DIN 18800-1[15] in 1981 with a slight difference for FLC3 (f = 0.86) which was increased in the following revision of DIN 18800-1[3] in 1990.

Fig. 3
Fig. 3Overview of the database of breaking load tests for FLC of four different manufacturers A-D depending on the year of the breaking load test and the nominal rope diameter d n

For
the estimation of the actual fill factors, the measured rope diameter at 5 % MBF is used in accordance with EN 12385-1.The actual metallic cross-sectional areas are unknown for the database, so mean values are calculated using the distribution of the actual wire cross-sectional areas.The resulting fill factors are shown in Fig. 5 in comparison with the values of EN 1993-1-11.Due to the small number of only 15 mean fill factors, no conclusion can be drawn about the actual distribution of the fill factors, as can be seen from the histogram in Fig. 5.However, there are considerable deviations compared to the standard values in EN 1993-1-11.If the fill factor is to be used further in the future, more detailed investigations into the actual influencing factors are required.

Fig. 4 Fig. 5
Fig. 4Correlation between the nominal heights of the z-wires h w,n 2 and the nominal cross-sectional areas A w,n

Fig. 6
Fig. 6Comparison of the measured and standard values for the spinning loss factor as a function of the nominal rope diameter depending on the number of z-wire layers and the corresponding histogram of the spinning loss factors

Fig. 7
Fig. 7Mean values and standard deviations of the ratio of the actual and nominal values for parallel lay wire ropes given by Feyrer[12] and for FLC calculated from the database for both investigated resistance equations

2 Breaking load test database A
database of breaking load tests is built up using both project data from the structural engineering office sbp and data provided by four different European rope manufacturers.It includes a total of 199 FLC with nominal rope diameters d n from 20 mm to 155 mm and a nominal rope grade of 1570 N/mm 2 according to EN 12385-10 [14].Fig.3gives an overview over the breaking load tests

1
Overview of the usable number of measured and nominal values as well as the combination