Dynamic Research on Steel Wire Rope Rigging Under Impact Bending Wave Load

Abstract

Wire rope joints are critical components requiring detailed mechanical analysis. This study investigates the stress/strain characteristics at the joint root under axial impact and combined tension-bending loads. A mathematical model was derived from the rope’s spatial structure, enabling the construction of 3D simulation and finite element models. Explicit dynamic analysis revealed distinct stress evolution patterns. Under axial impact, the joint root wires experience instantaneous peak stress causing core, inner, and outer wire yielding, though stress rapidly decreases and stabilizes. During stable loading, maximum stress (67% of impact peak) occurs on the joint root’s secondary outer wire. Under combined tension-bending, maximum stress dynamically shifts to the tension-side secondary outer wire at the joint root. Critically, both loading conditions identify the joint root’s secondary outer wire as the primary danger zone, with combined tension-bending producing a maximum local stress 1.04 times higher than axial impact. These findings highlight consistent failure locations and quantify relative stress magnitudes under complex loading.

1. Introduction

Steel wire rope rigging, as a kind of flexible and high-strength load-bearing equipment, is widely used in various fields of social life and industrial production. The analysis of mechanical properties in steel wire rope slings serves as a fundamental theoretical approach for evaluating their load-bearing capacity and operational safety. Considerable research efforts have been devoted to this field by domestic and international scholars in recent years. Utilizing the spiral symmetry and periodic characteristics of strands, Jiang [1,2,3,4] developed a representative sector finite element model within ANSYS. This model was subsequently employed to investigate the frictional contact interactions among the wires within two configurations: a 1 × 7 single-strand wire rope and a more complex straight strand rope consisting of three layers. Nawrocki [5] performed a comprehensive investigation into the mechanical response of single-strand wire ropes subjected to axial loading. Separately, Stanova et al. [6,7] utilized Catia V5 for 3D geometric modeling and Abaqus for finite element analysis to create a model of a 1 × 37 wire rope, which was then used to examine its behavior under analogous axial load conditions. In their studies [8,9], Ma constructed a finite element model to simulate a 6 × 7 + IWS wire rope, subsequently conducting stress analysis and comparing the mechanical performance of ropes manufactured using different laying techniques. Wang et al [10]. conducted finite element simulations in ABAQUS 6.14 on 6 × 19 + IWS and 1 + 6 + 12 structured hoisting wire ropes to investigate their stress distributions and fatigue parameters. Such wire ropes are widely utilized in engineering applications for lifting and towing operations via pulleys or winch drums. While Kmet et al. investigated the bending behavior of wire ropes on a cable saddles [11], the inherent complexity of this problem has hindered the establishment of a unified standard for both modeling methodologies and analytical theories. In a recent study, Cao et al. [12] introduced a novel mechanical modeling approach tailored for multi-strand wire ropes under bending conditions. This method holds considerable general significance from both theoretical and modeling perspectives, thereby facilitating advancements in the mechanical modeling of wire ropes. Zhang et al. [13] examined the 3D contact mechanics of wire ropes and friction pulleys. Their results demonstrated that the contact and wear positions predicted by the finite element model closely matched the experimental findings. Based on beam theory, the work of Guo et al. [14] examined the effects of cross-sectional type, nominal diameter, prestress level, and boundary constraints on the bending behavior of steel wire ropes. Zhang et al. [15] investigated the transmission of contact pressure within cables and subsequently computed the magnitude of this pressure in a parallel-wire cable configuration. Xia et al. [16,17] developed a complex special-shaped steel wire rope and proposed a new method for extracting the stress of the steel wire rope. The above research has laid the foundation for obtaining the stress information of steel wire rope rigging, but it is still limited to the static mechanics category mainly involving axial static tension and does not involve the connection parts of the rigging, so it cannot accurately describe the stress and strain conditions of individual wires in the dangerous areas when the rigging is subjected to impact and high-speed bending loads.

A certain special rigging uses 6-strand round-strand steel wire rope and cast-type joint structure. During service, wire ropes are subjected to complex multi-axial stress states. They must endure not only axial tensile impacts but also high-speed bending effects. These bending effects occur at the joints due to the propagation of bending waves through the rope structure. Therefore, wire breakage at the root of the joint is a typical failure phenomenon. This paper mainly combines the load characteristics of the rigging and decouples the loads it bears into axial impact and composite tension-bending loads for finite element analysis, providing a reference for the mechanical property analysis and design optimization of such rigging.

2. Mathematical Model

The special rigging wire rope is of the 6 × 31 WS-FC structure, that is, 31 wires of different diameters are layered and helically twisted to form strands, and then these 6 strands are helically twisted around a core to form the rope. Therefore, the rope is composed of a primary helix line and a secondary helix line in space, as shown in Figure 1.

The key to establishing the three-dimensional model of the steel wire rope is to establish the equation of a single helical line representing the center line of the wire strand and the equation of a second helical line representing the space of 30 side line steel wires wound around the central steel wire of the wire strand.

First, establish the right-angled coordinate system CSO_r on the cross-section of the steel wire rope, as shown in Figure 2a, with the coordinate origin O_r (fixed) located at the center of the cross-section of the steel wire rope, the coordinate axis z_r coinciding with the axis line of the steel wire rope and pointing upwards in the positive direction, and the coordinate plane x_rO_ry_r used to measure the helical angle θ_s of the wire strand. Then establish the local coordinate system CSO_w as shown in Figure 2b, with the coordinate origin O_w (movable) located on the central line of the wire strand, as shown in Figure 2a. P₁ is the projection of O_w on the x_rO_ry_r plane, the direction of the coordinate axis z_w is consistent with the tangent direction at point O_w on the central line of the wire strand, the positive direction of z_w is the direction of the helix ascent; the coordinate plane x_wO_wy_w is used to measure the helical angle θ_w of the steel wire, x_wO_wy_w is perpendicular to the z_w axis.

When z_r = 0, the center line of the rope strand intersects with the x_r axis, and at this time θ_s = 0. Then, in the entire rope coordinate system CSO_r, the vector equation of the center line of the rope strand is expressed by the following formula:

$$\overrightarrow{R}=x_s·\overrightarrow{i_r}+y_s·\overrightarrow{j_r}+z_s·\overrightarrow{k_r}$$

(1)

The parameter expressions on the three components (x_s, y_s, z_s) can be calculated using the following formula:

$$\left\{\begin{array}{c}{x}_s=r_s·\cos \left({\theta }_s\right)\\ y_s=r_s·\sin \left({\theta }_s\right)\\ z_s=r_s·{\theta }_s·\tan \left({\theta }_s\right)\end{array}\right.$$

(2)

In the formula, r_s represents the first helical radius of the rope strand centerline, and α_s represents the twist angle of the rope strand centerline.

Similarly, when z_w = 0, θ_w = 0. Then, in the rope strand coordinate system CSO_w, the vector equation of the steel wire centerline is expressed by the following formula:

$$\overrightarrow{Q}=x_w·\overrightarrow{i_w}+y_w·\overrightarrow{j_w}+z_w·\overrightarrow{k_w}$$

(3)

The parameter expressions on the three components (x_w, y_w, z_w) can be calculated using the following formula:

$$\left\{\begin{array}{c}{x}_w=r_w·\cos \left({\theta }_w\right)\\ y_w=r_w·\sin \left({\theta }_w\right)\\ z_w=r_w·{\theta }_w·\tan \left({\theta }_w\right)\end{array}\right.$$

(4)

In the formula, r_w represents the first helical radius of the steel wire centerline around the strand axis, and α_w represents the twist angle of the steel wire centerline.

The second helical model is composed of the vector R defined in the global rope coordinate system CSO_r and the vector q defined in the coordinate plane x_wO_wy_w of the ropestrand, as shown in Figure 3.

Suppose the position vector P defined in the entire rope coordinate system points to the end point of vector q (x_w1, y_w1, z_w1), and the point (x_w1, y_w1, z_w1) is also on the center line of the yarn of the rope strand, then the vector P can be expressed as:

$$\overrightarrow{P}=x_{rp}·\overrightarrow{i_r}+y_{rp}·\overrightarrow{j_r}+z_{rp}·\overrightarrow{k_r}$$

(5)

The vector q located on the coordinate plane x_wO_wy_w can be expressed in the rope strand coordinate system as:

$$\overrightarrow{q}=x_{w1}·\overrightarrow{i_w}+y_{w1}·\overrightarrow{j_w}$$

(6)

Since vector q lies on the coordinate plane, therefore z_w1 = 0. Here, the coordinate components (x_w1, y_w1) can be obtained from Equation (4). Also, since vector R points to the starting point of vector q, vector P can be obtained by vector summation, that is:

$$\overrightarrow{P}=\overrightarrow{R}+\overrightarrow{q}$$

(7)

Next, we will solve for the projection coordinates of the vector q defined in the rope fiber coordinate system CSO_w in the entire rope coordinate system CSO_r. Given: (1) The coordinate axis x_w is parallel to the line segment on the coordinate plane x_rO_ry_r that indicates the rotation angle of the rope fiber; (2) The coordinate axis z_w points in the tangential direction of the rope fiber center line, and it has a helical pitch angle (twist angle) αs; (3) z_w1 = 0. Then, the x_w1 component of the vector q in the entire rope coordinate system can be expressed as:

$$\left\{\begin{array}{c}{x}_{r1}=x_w·cos\left({\theta }_s\right)\\ y_{r1}=y_w·sin\left({\theta }_s\right)\\ z_{r1}=0\end{array}\right.$$

(8)

According to the theorem of orthogonal projection: For two intersecting lines that are perpendicular to each other, if one of the lines is parallel to a certain projection plane, then the projections of the two lines on that projection plane still reflect the right-angle relationship. Then the projection of the y_w1 component of vector q in the entire coordinate system can be expressed as:

$$\left\{\begin{array}{c}{x}_{r2}={-y}_w·sin\left({\alpha }_s\right)·sin\left({\theta }_s\right)\\ y_{r2}=y_w·sin\left({\alpha }_s\right)·cos\left({\theta }_s\right)\\ z_{r2}={-y}_w·cos\left({\alpha }_s\right)\end{array}\right.$$

(9)

Therefore, the vector q can be expressed in the global coordinate system CSO_r as:

$$\overrightarrow{q}={(x}_{r1}+x_{r2})·\overrightarrow{i_r}+{(y}_{r1}+y_{r2})·\overrightarrow{j_r}+{(z}_{r1}+z_{r2})·\overrightarrow{k_r}$$

(10)

Similarly, vector P can be expressed in the global coordinate system CSO_r as:

$$\overrightarrow{P}=\overrightarrow{R}+\overrightarrow{q}={(x_s+x}_{r1}+x_{r2})·\overrightarrow{i_r}+{(y_s+y}_{r1}+y_{r2})·\overrightarrow{j_r}+{(z_s+z}_{r1}+z_{r2})·\overrightarrow{k_r}$$

(11)

After substituting the parameters of the above formula with their respective expressions, the parametric equation of vector P in the integral coordinate system CSO_r can be obtained. This equation is the mathematical model of the second-order helical line of the wire centerline.

$$\begin{gathered} {x_{rp}=x_s+x}_{r1}+x_{r2}=r_s·\mathit{cos}\left({\theta }_s\right)+x_w·\mathit{cos}\left({\theta }_s\right)-y_w·\mathit{sin}\left({\alpha }_s\right)·\mathit{sin}\left({\theta }_s\right) \\ =r_s·\mathit{cos}\left({\theta }_s\right)+x_w·\mathit{cos}\left({\theta }_s\right)-r_{w1}·sin\left({\theta }_w\right)·\mathit{sin}\left({\alpha }_s\right)·\mathit{sin}\left({\theta }_s\right) \end{gathered}$$

(12)

$$\begin{gathered} {y_{rp}=y_s+y}_{r1}+y_{r2}=r_s·\mathit{sin}\left({\theta }_s\right)+x_w·\mathit{sin}\left({\theta }_s\right)+y_w·\mathit{sin}\left({\alpha }_s\right)·\mathit{cos}\left({\theta }_s\right) \\ =r_s·\mathit{sin}\left({\theta }_s\right)+x_w·\mathit{sin}\left({\theta }_s\right)+r_{w1}·sin({\theta }_w)·\mathit{sin}\left({\alpha }_s\right)·\mathit{cos}\left({\theta }_s\right) \end{gathered}$$

(13)

$$\begin{gathered} {z_{rp}=z_s+z}_{r1}+z_{r2}=r_s·{\theta }_s·\mathit{tan}\left({\alpha }_s\right)+0-y_w·\mathit{cos}\left({\alpha }_s\right) \\ =r_s·{\theta }_s·\mathit{tan}\left({\alpha }_s\right)-r_{w1}·\mathit{sin}\left({\alpha }_s\right)·\mathit{sin}\left({\theta }_w\right)·\mathit{cos}\left({\alpha }_s\right) \end{gathered}$$

(14)

3. Finite Element Model

3.1. Finite Element Model of the Steel Wire Rope

Using the mathematical model of the steel wire centerline obtained through the above derivation, programming was carried out in MATLAB R2019b to calculate the centerline coordinates of all the steel wires that constitute the steel wire rope. Based on the obtained coordinate points, the centerline of the sling steel wire was fitted and generated as shown in Figure 4.

The centerline, which was computed using MATLAB R2019b, was subsequently imported into the finite element software Abaqus to construct a geometric model of the single wire strand, as depicted in Figure 5.

The three-dimensional models of the strands were arranged in a helical pattern to generate all six strand assemblies. The core was geometrically represented as a cylindrical body, leading to the complete three-dimensional solid model of the wire rope, illustrated in Figure 6a. Based on the structure of the steel wire rope, the finite element mesh is divided using the sweeping method, and the element type is selected as C3D8R. The length of the steel wire rope is set to 1 m. Each steel wire and the rope core are distributed along the circumference of the cross-section. The overall distribution density of the mesh is controlled at 5 mm, and the minimum mesh size is controlled at 0.1% of the global size. The mesh model of the entire steel wire rope is shown in Figure 6b.

There are two types of contact between wires inside the wire rope: point contact and line contact. Due to the complex structure of the wire rope, the contact algorithm used in the finite element model must be accurate and efficient. ABAQUS 2020/Explicit offers two algorithms for modeling contact and interaction problems: the general contact algorithm and the contact pair algorithm. While contact pair algorithms can provide more specialized contact conditions, the general contact algorithm in ABAQUS 2020/Explicit utilizes sophisticated tracking algorithms to ensure that proper contact conditions are enforced efficiently [18]. This is achieved through the use of a global contact search that determines the nearest master surface facet for each slave node in a given contact pair. To minimize the computational expense of these searches, a bucket sorting algorithm is employed. Therefore, the contact type between wires is set to general contact, and hard contact is chosen as the normal behavior to prevent surface penetration.

3.2. Finite Element Models of Joints and Riggings

The detailed structure of the connection joint does not affect the analysis of the service performance of the rigging. Therefore, the connection part can be simplified to a cylindrical body with one end open, as shown in Figure 7a,b. The meshing method was adopted to divide the grid for the joint model. The overall grid density was 4 mm, and the local grid density of the rounded corner part was 3 mm. The unit type was C3D8R. As shown in Figure 7c.

By assembling the finite element models of each component, the finite element model of the sling is obtained, as shown in Figure 8. To simulate the connection between the joint and the wire rope end, the segment of the rope residing within the joint’s cavity (indicated by the bright yellow region in Figure 9 is rigidly coupled to the joint body. This connection fully restrains all six degrees of freedom of the joint.

4. Transient Dynamics Analysis

Considering the structure of the wire rope rigging, there are numerous contacts between the wires, and the contact surfaces constantly change during the loading process, featuring strong contact nonlinearity characteristics. Additionally, during the loading process, the rope itself will also undergo significant deformation, causing nonlinear responses of the structure. To enhance computational efficiency and mitigate convergence difficulties arising from strong nonlinearities in the model, an explicit dynamic algorithm was adopted for simulation. This method has been demonstrated to accurately predict the axial, torsional, and bending stiffness of wire ropes [19].

In this paper, the elastic modulus of the wire in the rigging is 200 GPa, and the Poisson’s ratio is 0.3. The elastic–plastic material parameters used in the simulation are obtained through splitting the steel wire tensile test. The tension, stress, time, and angle in the text are in the normalized unit P.U., where 1 P.U. tension = 0.1 times the minimum breaking force of the wire rope, 1 P.U. stress = the tensile strength of the inner steel wire, 1 P.U. time = twice the time it takes for the tension of the rigging to rise to the maximum value during the impact loading, and 1 P.U. angle = the relative maximum allowable bending angle of the joint to the wire rope design.

4.1. Analysis of the Force on the Rigging Under Axial Impact Load

Based on the established finite element model of the rigging, an axial impact load was applied. Given that the joint exhibits substantially higher macroscopic stiffness compared to the wire rope, all six degrees of freedom of the joint were fully constrained in the simulation. The nodes on the end cross-section of the wire rope were kinematically coupled to the sectional center point. All degrees of freedom of this reference point were restricted except for translation along the axial direction. An axial impact load of 4 P.U. with a loading duration of 10 P.U. was then applied at this center point, as illustrated in Figure 10.

Following the simulation, the mechanical response of the rigging under a 4 P.U. axial impact load (upon load stabilization) is illustrated in Figure 8 (cross-section of the wire rope) and Figure 11 (cross-section at the joint root). In Figure 11, labels A, B, C, and D denote the outer layer, the second outer layer, the inner layer, and the core wire, respectively.

It can be observed that under tensile loading, the maximum stress in the rigging occurs in the second outer layer of the wire rope at the joint root. The peak equivalent stress reaches 0.62P.U., which is 1.8 times greater than that of the outer wire. Stress levels in the inner and core wires are comparable, and the stress distribution across the strand cross-section exhibits an approximately fan-shaped pattern. This phenomenon can be attributed to the following factors: First, due to the larger helical radius of the outer wires compared to the inner wires, the outer wires have a longer effective length over the same rope segment. During tension, the outer wires experience less elongation, resulting in lower stress development. Second, the constraint imposed by the joint induces stress concentration near the contact region, causing the stress distribution to shift toward the joint side. As a result, the second outer layer bears higher stress than both the inner and core wires. Along the length of the rope, significant stress concentration is observed near the joint root due to end effects, while regions farther away show more uniform stress distribution, indicating effective load transfer characteristics of the wire rope.

Further analysis of the stress evolution at locations A, B, C, and D (as indicated in Figure 12) during axial impact loading is presented in Figure 13. It was observed that upon impact, the stress in all wires increased abruptly. Among them, the maximum instantaneous stress of the inner layer, the second outer layer, and the core wire reached 0.93 P.U., which was 1.5 times the maximum stress of the steel wires in the stable state. Moreover, the steel wires underwent yielding before reaching the stable state. With the continuous loading, the stress of the steel wires in the corresponding area decreased rapidly and stabilized. This indicates that the impact load is an important form of load causing damage to the rigging.

4.2. Analysis of the Forces Acting on the Rigging Under Tension Bending Loads

For the tension bending load of the rigging, the method of decoupling tension and bending loads is adopted. In this method, the joint end and the axial impact loading are the same. The joint and the steel wire rope inside the joint are subject to rigid constraints, and all nodes are constrained to the center point of the end face close to the steel wire rope direction (P1); the end face node of the steel wire rope is constrained to the center point of the end face (P2), as shown in Figure 13. The combined tension-bending loading is applied in two time steps. During the tension application stage, all constraints are applied at point P1, and an axial impact load of 4 P.U. is applied at point P2. During the bending application stage, the tension at point P2 remains constant, and a bending load is applied at point P1 (where the object rotates 1 P.U. around the Z-axis).

Figure 14 shows the stress distribution of the rigging under tension bending load. It can be seen that the maximum stress still exists on the outermost layer of steel wires, at 0.97 P.U. However, at the root of the joint, the stress distribution on the steel wire rope has significantly changed compared to when it only bears axial load. The stress distribution on the upper and lower sections of the steel wire is no longer uniform, and the stress on the side of the steel wire rope that is under tension during bending has significantly increased; in the length direction of the steel wire rope, as the distance from the joint increases, the maximum stress of the entire cross-section of the rope gradually decreases, and the stress distribution becomes more uniform. This is because a "fixed end" constraint similar to casting was adopted between the joint and the steel wire rope, which has extremely high stiffness. The positions of the steel wires on the cross-section of the joint area are relatively fixed. However, when the rigging bears a large tension bending load, the steel wires away from this cross-section can automatically adjust their positions to better bear the load, while the positions of the steel wires on the cross-section of the joint area are rigidly constrained and fixed, so there is inevitably a large force imbalance.

The curves showing the stress and strain of each layer of steel wires on the tensioned side of the joint root in Figure 14 (the steel wire at point P in the figure) as a function of the bending angle are shown in Figure 15.

It can be seen that as the bending angle of the joint of the steel wire rope increases, the stress of each layer of steel wires on the tension side of the joint root increases rapidly. When the bending angle exceeds 0.15 P.U., the stress gradually slows down as the bending angle increases. This is because at this time, the steel wires have undergone yielding. As the strain increases, the stress no longer increases linearly. The strain of each layer of steel wires changes in an opposite trend to the stress with the change of the bending angle: when the bending angle is not more than 0.15 P.U., the strain of each steel wire changes little. When the bending angle is greater than 0.15 P.U., the strain of the steel wire approximately continuously increases linearly. This is because when the bending angle is not more than 0.15 P.U., the steel wire has not yet yielded, and its deformation is mainly elastic deformation. The steel wire material relies on its own strength to resist the bending deformation; when the bending angle is greater than 0.15 P.U., the steel wire yields, and the plastic deformation increases rapidly. Among them, the plastic deformation of the outer layer steel wire is greater than that of the inner layer steel wire, because the outer layer steel wire is farther from the bending rotation center, and after bending, a larger deformation is required to coordinate.

5. Conclusions

This paper uses the method of dynamic simulation to study the mechanical response of a certain wire rope accessory under impact and bending wave loads. The following conclusions are drawn:

• Under axial impact load (4 P.U.), the steel wire at the root of the joint reaches the stress peak instantaneously under the impact action. Among them, the core, inner layer, and secondary outer layer steel wires have already yielded, and their local maximum stress reaches 0.93 P.U. However, the peak stress lasts for a very short time and then decreases and tends to stabilize. After the loading becomes stable, the maximum stress in the rope is found on the secondary outer layer steel wire at the root of the joint, with a maximum stress of 0.62 P.U., which is only 67% of the maximum peak stress. At this time, the stress is approximately fan-shaped distribution on the cross-section of the rope strands.
• Under the combined tension-bending load (tension: 4 P.U.; bending: 1 P.U.), the maximum stress of the accessory occurs on the secondary outer layer steel wire on the tensioned side of the joint root. The maximum stress is 0.97 P.U., and the stress distribution changes dynamically during the bending loading process. When the bending angle is less than 0.15 P.U., the stress of each steel wire in the tensioned side rope strand increases rapidly with the bending angle; when the bending angle exceeds 0.15 P.U., the stress increases more slowly with the bending angle. This is because at this time, the steel wires have yielded, and as the strain increases, the stress no longer increases linearly. The strain of each layer of steel wires changes little when the bending angle is not more than 0.15 P.U., and when the bending angle is greater than 0.15 P.U., the steel wires yield, and the strain increases approximately linearly with the bending angle. Among them, the plastic deformation of the outer layer steel wire is greater than that of the inner layer steel wire. Due to the tension-bending loading of the accessory, the plastic deformation of the steel wires in the local area occurs, which to some extent affects the service life of the accessory.
• Comparing the axial impact and the simulated bending wave tension-bending loading conditions, the dangerous areas of both are located on the secondary outer layer steel wire at the root of the joint. Under the axial impact load, the maximum peak stress is 0.93 P.U., and after the combined tension-bending load, the maximum stress increases to 0.97 P.U., an increase of 1.04 times.