# Determination of the Bending Properties of Wire Rope Used in Cable Barrier Systems

## Abstract

**:**

## 1. Introduction

## 2. The 3 × 7 19-mm Wire Rope

#### 2.1. Characteristics of the Wire Rope

^{2}. The material density is equal to 7948 kg/m

^{3}[35].

#### 2.2. Geometry of the Wire Rope

_{w}. However, it has to be noted that the actual cross-section of the wire is elliptical in a plane normal to the rope, see, e.g., [3,13,24]. Herein, differences resulting from this simplification should be negligible and a similar assumption can be found in the paper [35]. In the first step, the geometry of a single strand was determined in a three-dimensional x’y’z’ coordinate system, where the z’-axis coincides with the centroidal wire (Figure 4c). The analyzed wire rope consists of three such strands, so, in order to determine the final geometry, a new xyz coordinate system was assumed, in which the z-axis is in the longitudinal direction of the wire rope. In this coordinate system, the angle between each strand is 120 degrees. To determine the final coordinates, the points x’, y’, and z’ have to be transformed from the x’y’z’ systems of each strand into the xyz global system, as shown in the cross-section in Figure 4d. In this figure, for better clarity, the strands are separated from each other.

_{1}is the distance from the origin of the xyz global system to the center of the inner wire of the strand (initial radius of the helix of the inner wire of the strand); R

_{2}is the distance from the center of the inner wire to the center of the outer wire in the strand (initial radius of the helix of the outside wire in the strand); φ

_{1}denotes the initial rotation angle of the strand; φ

_{2}stands for the initial rotation angle of the outer wire in the strand; and P

_{1}and P

_{2}are the pitch of the strand and the pitch of the single wire, respectively. The symbols are also shown in Figure 4c,d. The equations were used to construct the geometry of a straight section of the wire rope (see, e.g., the 3D model in Figure 1a).

## 3. Moment–Curvature Relationship

_{x}acting over the cross-section is equal to the bending moment M:

^{4}determined experimentally.

## 4. Experimental Bending Testing

#### 4.1. Test Specimens

#### 4.2. Test Stand

_{f}− u

_{i}. This measurement was used to determine wire rope curvatures, assuming that, between the loading points, the curvature was described by the equation of the circle. Because of the irregular geometry of the wire rope, the transducer had to be laid on the specimen via a stabilizing pipe, which prevented the transducer’s head from sliding on the rope (Figure 5b).

#### 4.3. Test Results

## 5. Numerical Bending Testing

#### 5.1. Numerical Model

#### 5.1.1. Beam Model

#### 5.1.2. Solid Model

#### 5.1.3. Parts of the Universal Testing Machine

#### 5.2. Numerical Test Results

#### 5.2.1. Comparison with Experimental Results

_{T}is the displacement of the traverse.

_{T}of approximately 70 mm; this is ~60% of the analyzed range (Figure 16a). Then, the slope of the curve changes. The maximum force obtained from the beam model is 489 N, whilst from the solid model, it is 440 N. The displacements at the midspan of the wire rope increase linearly, similar to the TC1 configuration, and coincide well with experimental outcomes (Figure 16b). The maximum displacement for the beam and solid model is 14.8 and 15.0 mm, respectively. For the TC2 configuration, the solid model shows better agreement with the experimental measurements, and the beam model appears to be slightly more rigid.

#### 5.2.2. Analysis of Numerical Results

_{UTC_WR}has more of an impact on the results than the friction coefficient between the wires FC

_{WR_WR}. These preliminary tests revealed that if the FC

_{UTC_WR}is reduced from 0.1 to 0.02 without changing FC

_{WR_WR}, the final force F

_{UTC}from the displacement-force curve can decrease by up to 6%. However, when only increasing FC

_{WR_WR}from 0.1 to 0.3, the F

_{UTC}force remains almost the same. If both coefficients, FC

_{UTC_WR}and FC

_{WR_WR}, are increased from 0.1 to 0.3, the F

_{UTC}force can increase by 20%. This indicates that the FC

_{UTC_WR}coefficient has a greater effect on the results than FC

_{WR_WR}. Finally, in the numerical models, one coefficient of friction equal to 0.1 was assumed (see Section 5.1). To thoroughly investigate the impact of friction, a detailed contact stress analysis was carried out. Figure 20 illustrates the normal contact stresses between the two strands in the neighborhood of the middle of the wire rope (see location in Figure 20) obtained from the solid model, with the TC2 configuration. Large stresses occur locally at points of contact between the wires. Higher values are observed between strands (e.g., 11.5 and 8.0 MPa) than between the wires within the single strand (e.g., 6.1 and 2.2 MPa). However, for most surfaces, the contact stresses equal zero. This results from the geometry of the wire rope; most surfaces are not in contact with other parts. A similar view of the contact stress distribution is shown in [17]. It is worth noting that the contact stresses between the wires are smaller than between the wires and UTM’s parts. For instance, the maximum contact compressive stress between the wire rope and loading pin is 65.4 and 153.9 MPa for TC1 (T = 240 s, D

_{T}= 80 mm) and TC2 (T = 360 s, D

_{T}= 120 mm) configurations, respectively. This also confirms that the friction coefficient between the wires and the UTM affects the results more than the friction coefficient between the wires.

## 6. Results and Discussion

#### 6.1. Moment–Curvature Relationship for the Wire Rope

_{c}and y

_{c}are the center coordinates and ρ is the radius. Since the coordinates of three points (two loading pins and the displacement at the mid-span) were known, the equation of the circle could be determined.

_{i}deflection in Figure 7). The sought curvature κ is the reciprocal of the radius ρ. In this way, the moment–curvature relationships were determined for all experimental tests, as shown in Figure 22.

^{6}Nmm

^{2}. Assuming that the modulus of elasticity of the non-prestretched wire rope is 79.9 GPa [35], the moment of inertia of the cross-sectional area of the wire rope I is equal to 172 mm

^{4}.

#### 6.2. Discussion

^{−1}was determined (Figure 23). The obtained curvature corresponds to bending the wire rope into a circle with a radius of 25 cm. This relationship is comparable to that proposed by Reid et al. [20,35]; however, it should be noted that Reid’s relationship covers a higher range of curvatures, up to 0.05 mm

^{−1}. The curves, compared in Figure 23, differ in the initial range of curvatures. The proposed curve is characterized by a smaller slope at the origin point and consequently, the obtained flexural rigidity of 13.7 × 10

^{6}Nmm

^{2}is lower than the value of 32.3 × 10

^{6}Nmm

^{2}given in [35]. Considering the cross-section of the wire rope as either 21 independent wires or as a single solid cross-section, the moment of inertia is 83.5 and 2660 mm

^{4}, respectively. The experimentally determined moment of inertia I = 172 mm

^{4}indicates that the wire rope works more as a set of 21 separated wires than as one solid section. A similar observation can be found in [35]. The reason for this is that the wrap of the rope, contact surfaces, and friction forces between the wires do not provide sufficient cross-sectional bonding to allow it to work as a single solid structure.

## 7. Conclusions

- Developing equations for a wire rope geometry;
- Developing two advanced nonlinear FE models of wire rope utilizing beam and solid finite elements;
- Conducting an analysis of 19 experimental tests and four numerical simulations of four-point bending. The simulations were validated against the experimental results;
- Detailed analysis of numerical results including both cross-sectional and contact stress analyses;
- Determination of the nonlinear elastic moment–curvature relation for the wire rope.

- The responses of non-prestretched and prestretched wire rope in the range of curvatures up to 0.004 mm
^{−1}(i.e., the radius of curvature equals 25 cm) are similar and one moment–curvature relation is assumed for both prestretched and non-prestretched ropes; - In the analyzed range of curvatures, the wire rope worked in the elastic range. Plastic strains in wires appeared solely locally under the loading pins. This suggests that in real-life accidents, the wire rope may work in the elastic range as well; however locally, in the vicinity of a point where a vehicle impacts the barrier, plastic strain may emerge;
- The considered wire rope works more as a set of 21 separated wires than as one single solid section;
- The interwire friction coefficient does not substantially affect the results. This is due to the wire rope geometry as most of the wire surfaces are not in contact with other wires.

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Wire rope models: (

**a**) Detailed 3D model (for short section); (

**b**) simplified (for long rope section, here used in a cable barrier model).

**Figure 4.**Geometry of the wire rope: (

**a**) Simple 1 + 6 strand; (

**b**) 3 × 7 wire rope structure; (

**c**) single strand cross-section; (

**d**) cross-section of a wire rope consisting of three strands.

**Figure 8.**WR1: Load vs. displacement curve from a Universal Testing Machine (UTM) (

**a**), and displacements from a transducer (

**b**).

**Figure 10.**WR3: Load vs. displacement curve from a UTM (

**a**), and displacements from a transducer (

**b**).

**Figure 11.**WR4: Load vs. displacement curve from a UTM (

**a**), and displacements from a transducer (

**b**).

**Figure 13.**General view of wire rope made of beam finite elements (FEs) (visualization of the cross-section of the beam FEs is turned on).

**Figure 15.**Comparison of numerical and experimental results for the TC1 configuration (A = 80 cm, B = 40 cm): Load vs. displacement of the traverse curve (

**a**), and relative displacements at the midspan (

**b**).

**Figure 16.**Comparison of numerical and experimental results for the TC2 configuration (A = 57 cm, B = 17 cm): Load vs. displacement of the traverse curve (

**a**), and displacements from the transducer (

**b**).

**Figure 20.**Contact compressive stress distribution. Note: Maximum of the fringe level was set to 0.5 MPa.

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**MDPI and ACS Style**

Bruski, D.
Determination of the Bending Properties of Wire Rope Used in Cable Barrier Systems. *Materials* **2020**, *13*, 3842.
https://doi.org/10.3390/ma13173842

**AMA Style**

Bruski D.
Determination of the Bending Properties of Wire Rope Used in Cable Barrier Systems. *Materials*. 2020; 13(17):3842.
https://doi.org/10.3390/ma13173842

**Chicago/Turabian Style**

Bruski, Dawid.
2020. "Determination of the Bending Properties of Wire Rope Used in Cable Barrier Systems" *Materials* 13, no. 17: 3842.
https://doi.org/10.3390/ma13173842