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Fatigue failure analysis of steel wire rope sling based on share-splitting slip theory
Engineering Failure Analysis 105 (2019) 1189–1200
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Fatigue failure analysis of steel wire rope sling based on sharesplitting slip theory
T
Songling Xue, Ruili Shen , Menglong Shao, Wei Chen, Rusong Miao ⁎
School of Civil Engineering, Southwest Jiao tong University, Chengdu 610031, China
ARTICLE INFO
ABSTRACT
Keywords: Share-splitting slip Double index fatigue Damage evolution Calculation model Experimental analysis
At present, Section integrity hypothesis is mostly used to analyze the mechanical properties and identify the damage of the cable. Based on the actual situation and considering the slip effect between the wires of the cable, two simplified calculation models of the cable are proposed based on the theory of share-splitting slip, and the model parameters are obtained through experiments. Finally, on the background of Lijiang Railway Suspension Bridge in Yunnan Province, the fatigue test of the suspension cable is carried out by using the self-developed testing device which can consider both tension and bending fatigue, and the broken steel wire is further analyzed by means of electron microscopy. The results show that two simplified calculation models of wire rope sling are proposed based on the theory of share-splitting slip, which are close to the results of the original model, but greatly improve the calculation efficiency. Comparing the results of three models, it is found that the three models can reflect the development of the slip and fatigue damage of the wire rope sling, but they have some errors with the test. The fracture morphology of the steel wire is shown by electron microscopy photographs.
1. Introduction With the increasing span of bridge, the status of cable-supported bridges in long-span bridge has been continuously improved. Cables and slings are part of the core components of cable-supported bridges. Usually, they can be divided into parallel wire rope and steel wire rope. With the increase of service life, the wire rope may be damaged in Fig. 1, and the parallel wire rope may also be damaged [1,2]. Steel wire rope has been studied extensively in mechanical engineering, mining engineering and other fields, but there is little research in bridge field. Sling is one of the core force transmission components of the suspension bridge. It plays an important role by transmitting the load of the bridge deck to the main cable. Over the past 30 years, suspension bridge has developed rapidly, and the application of slings in bridge engineering has been increased. The theoretical design life of the sling is 30 years. However, the actual project shows that the sling is damaged to be different when it is far below the design service life, which makes the bridge must replace the cable in advance. As there were many examples of sudden damage of bridges caused by unscheduled cable replacement, it has become a necessary research topic to analyze the damage of cables. If the sling is hinged at the anchorage end, the sling can rotate freely without slipping. However, the actual constraint situation of slings at the anchorage end was between hinged and fixed restraint. Many scholars have studied the theory of layered slip of parallel steel wire cable [3–7]. The layered slip of parallel steel wire cable occurs within the distance from the end of the anchorage. Parallel steel wire cable can simplify the stacked beam model (Fig. 2) and other calculation models [8–11]. Because the structure of the wire rope sling is more complex than that of parallel steel wire cable, the failure mode of overlapping beam model cannot be calculated. In this paper, based on damage dynamics ⁎
Corresponding author. E-mail addresses: [email protected], [email protected] (R. Shen).
https://doi.org/10.1016/j.engfailanal.2019.07.055 Received 29 April 2019; Received in revised form 26 July 2019; Accepted 28 July 2019 Available online 31 July 2019 1350-6307/ © 2019 Published by Elsevier Ltd.
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Fig. 1. Steel wire rope failed in service.
Fig. 2. Computational model of stacked beams.
and slip theory, a simplified calculation model of steel wire rope sling was presented and its failure mode was analyzed. Some scholars have also done research on wire ropes. Most of the early models were based on linear elasticity assumption, and semi-continuous method and discrete method were used. In recent years, with the development of computer technology and finite element theory, many scholars have used the finite element method to simulate the steel wire rope. Based on the geometrical characteristics of the wire strands and the corresponding boundary conditions, a simplified finite element model of the wire strands was established by Jiang [12]. Since only 1/6 of the winding strand axis model is needed, the calculation efficiency of the model is high. However, complex boundary constraints have not been widely used. Imrak established the model of double helix wire rope by computer aided design (CAD) software, and simulated the performance of the strand under axial tension load by finite element method [13]. With the help of Abaqus/Explicit software, simulation of multi-layer steel wire rope strands with “1 + 6 + 12 + 18” structure under tension and bending support was carried out [14]. Judge used a 3D finite element model to study the axial load-strain curve and the failure load of multi-layer steel wire strand [15]. Kastratovic used two different methods (uniform axial force and uniform axial strain) to simulate the wire rope strand. It was found that the loading mode had a direct impact on the load distribution between wires of the wire rope strand [16]. In addition to the above research on the axial mechanical properties of wire rope, Yu studied the mechanical properties and the stress state of wire rope strands under transverse loads [17]. Juan used the finite element method to analyze the static performance and cross-section stress-strain distribution of steel wire ropes with asymmetric damage [18]. Fontanari used finite element simulation and test methods to study the elastoplastic properties of steel wire strands and the performance at different temperatures [19]. 1190
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Fig. 3. Friction coefficient test.
However, fatigue damage is a dynamic process, and it is unreasonable to evaluate fatigue damage by static method. In this paper, based on damage dynamics and slip theory, a simplified calculation model of steel wire rope sling is presented and its failure mode is analyzed. The basic structure of the article is as follows: The relevant background knowledge was introduced in section 1. Data of wire rope material was obtained through experiments, which provide a basis for determining the parameters of the numerical model based on damage dynamics in section 2. A share-splitting slip model based on damage dynamics was established and the fatigue failure time was calculated in section 3. In order to verify the validity of the numerical model, the bending fatigue test of the steel wire rope was carried out by the designed bracket based on the actual project in section 4. Finally, a summary of the full text was given. 2. Theoretical calculation model 2.1. Testing of friction coefficient between steel wires Friction is a complex problem, which is affected by many factors, such as contact material, indicating roughness, temperature and relative velocity. So far, there is no definite conclusion about the friction mechanism [20]. Under different conditions, the friction coefficient between steel wire and steel wire is different, and the accuracy of the friction coefficient is related to the correctness of theoretical calculation [21]. Many scholars have done a lot of research on the friction coefficient between steel wires, which range from 0.1 to 0.35 [22,23]. Previous team tests have shown that the friction coefficient of steel wires is about 0.3. Static friction coefficient is generally higher than dynamic friction coefficient. In this paper, a simple method of measuring friction coefficient was adopted. In order to simplify the calculation, the difference of static and dynamic friction coefficient was not considered. The measurement scheme of friction coefficient was shown in Fig. 3. Take out the same steel wire from the original steel wire rope sling and put it into the device shown in Fig. 3. The bottom plate was placed on the horizontal plane, and the angle between the steel wire and the bottom plate was adjusted and recorded when the steel wire slips. Ten different groups of steel wires were taken from the sling in this test, and the average value of each group was obtained three times. The result was shown in Fig. 4. The friction coefficient can be simplified as follows:
µ = tan
=
h L
(1)
The average value of Fig. 4 shows that the friction coefficient is 0.2473 in this analysis. Different friction coefficients may lead to differences. The influence of friction coefficients will be discussed in the next paper. 0.32
friction coefficient
0.30
0.28
0.26
0.24
0.22
0.20 0
2
4
6
8
test times
Fig. 4. Test results of friction coefficient. 1191
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2.2. Slip mechanism of sling rope Bending fatigue failure of cables has attracted the attention of many scholars and a lot of research has been done. The results show that the bending stress at the anchorage end is the primary factor causing cable bending fatigue. However, at present, the calculation of cable mechanical properties, cable force identification and damage identification are all based on the assumption that the cable section is a whole section. Numerous engineering practices show that bending stress exists near the anchorage section of the cable. The phenomenon of layered slip can be found in parallel steel wire, but for steel wire rope sling, because the structure of the steel wire rope sling is complex, the layered slip theory cannot be applied. In this paper, based on the test and engineering practice, a theoretical model for calculating the slip of steel wire rope sling is proposed to calculate the fatigue failure of the steel wire rope sling. Taking the most basic steel wire rope as an example, the theoretical stress calculation formula of the slip of the steel wire rope sling is deduced. Slip may occur in every steel wire of the wire rope sling, but in bridge engineering, slip occurs between side strand and side strand per share, and between side strand and core strand. According to structural mechanics, there is a corresponding relationship between bending moment and shear stress as follows:
Q (x ) = =
dM (x ) dx
4 QS = bIZ 3
(2)
exp
x
T EI
(3)
Q(x): cross-sectional shear force; M(x):section bending moment; ∆φ:angle of anchorage section; T:cable force of sling; E:elastic modulus; I:cross-sectional moment of inertia; τ:section shear stress. Before the slip occurs, the section can be known as a section and its stress can be calculated according to Eq. (3). When the slip occurs, the moment of inertia of the whole section in the integral formula (3) is replaced by the moment of inertia of each section. The shear stress distribution in both cases is shown in Fig. 5. When the maximum shear stress exceeds the ultimate friction force, the steel wire rope sling can slip, and the shear stress distribution pattern of the section becomes shown on the right side of Fig. 5. 2.3. Definition of rope strand damage mode Wire rope sling is composed of many steel wires, so it is very important to determine the damage model of steel wire for structural analysis [23,24]. In this paper, fatigue is analyzed not based on traditional miner criterion, but from the point of view of damage mechanics [25]. The damage degradation equation of the material is established, and the time and position of wire breakage can be calculated directly with the cyclic loading and high cycle fatigue in Abaqus. Because the numerical analysis is based on a simplified calculation model and takes each share as an entity, the performance of each share needs to be tested. Because the performance of each strand of wire rope is different, the whole mechanical parameters of each strand of wire rope were obtained by experiment. The strands of the wire rope are numbered, the core strand is No. 7, and the side strands are from 1 to 6. Each share (Fig. 6) was placed on the universal material experiment machine (Fig. 7). The stress-strain curve was shown in Fig. 8. According to damage mechanics, the damage degree (ω) is defined as the ratio of the bearing area (AD) after damage to that before damage (A).
=
AD A A = A A
(4)
Among them, A: the area of damage, ω = 0 means no damage, ω = 1means material fracture, and0 < ω < 1 means the state of damage process.The effective stress of the damaged material increases with the decrease of the bearing area caused by the damage, so the effective stress of the damaged material is as follows: ij
=
ij
(5)
1
In 1971, Lemaitre proposed the equivalent strain hypothesis, which enables indirect damage to be measured. The hypothesis was
Fig. 5. Segmental model of sling. 1192
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Fig. 6. The number of the share.
Fig. 7. Universal material experiment machine.
2000
stress(MPa)
1500
1 2 3 4 5 6 7
1000
500
0
0.0
0.5
1.0
1.5
2.0
2.5
strain(%) Fig. 8. The stress-strain curve.
that the deformation of materials containing damage can be represented only by effective stress, and the constitutive relation of the damage model can be expressed by the constitutive relation of non-destructive mode. The one-dimensional linear elastic relationship of damaged materials is as follows: e
=
ij
(1
)E
E = (1
)E
=
(6)
E
(7)
E is the modulus of elasticity of materials and E is the modulus of elasticity after damage. Steel wire rope used in this paper were high strength steel wires. The Ramberg-Osgood model considering strain hardening was 1193
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selected and the damage variable was added on the basis of the original constitutive model. The damage constitutive relationship is considered as follows: m
m
=
=
k
(1
(8)
)k
Among them, k and m are hardening coefficients. The damage is represented by replacing the hardening coefficient k with the hardening coefficient k after damage.
k = (1 is:
(9)
)k
In order to study the damage degradation process of materials, it is considered that the cumulative plastic strain rate of steel wire
p=
2 3
1
2
ij ij
(10)
ij :
strain tensor. The relationship between damage variable ωand cumulative plastic strain p was obtained by combining the dynamic equation of damage evolution with the damage constitutive eq. (8). 2s0 + m
=
c
p
2s0 + m
pR
m m
2s0 + m
pD
2s 0 + m
pD
m
(11)
m
s0: material parameters, m: hardening coefficient, pR: fracture strain and critical damage degree: ωc. c
=
k 2 (1 + ) + 3(1 2Es0 3
2
2 )
m eq
×
2s0 + m m m pR 2s0 + m
2s0 + m
pD
m
(12)
Based on the above theory and material test, the results of parameters are shown in Table 1. 3. Numerical analysis 3.1. Load acquisition and engineering background The main cable span of Jinsha jiang Suspension Bridge is 132 + 660 + 132 m. The main girder is a three-span continuous beam with a span composition of 110 + 660 + 98 m. The main girder is provided with vertical and transverse supports at the end of the girder and at the main tower. Jinsha jiang Railway Suspension Bridge is the first railway suspension bridge in China. The live load of railway suspension bridge accounts for a higher proportion of the total load. Its stress amplitude is larger than that of the highway bridge, and the suspension cable is prone to fatigue. For railway suspension bridge, the suspension cables are subjected to larger loads. The fatigue strength of the suspension cables may be the controlling factor in design and use. At the same time, the stiffening girder of suspension bridge will produce longitudinal and transverse displacement along the bridge under the action of vehicle operation, which will lead to the difference of longitudinal displacement between the upper and lower suspension points of the suspension cable anchored on the main girder. Therefore, it will lead to the bending of the sling and other phenomena, resulting in secondary stress and so on. In order to avoid fatigue failure of sling, it is necessary to calculate and analyze the response of sling under fatigue load in detail. Through the self-developed bridge structure static and dynamic non-linear analysis system BNLAS, the spatial analysis model is established (Fig. 9), and the most disadvantageous load is analyzed as showed in Table 0.2. In order to indicate the most dangerous sling location of the whole bridge, the number of sling and its internal force diagram can be seen in Fig. 10. Table 1 The results of parameters. Number
Elastic modulus (MPa)
Poisson ratio
Enhancement parameter(k)
Hardening parameter (m)
Failure plastic strain (pR)
Damage strain threshold
Damage critical coefficient(ωc)
1 2 3 4 5 6 7
1.13E+05 1.14E+05 1.17E+05 1.12E+05 1.21E+05 1.12E+05 1.14E+05
0.3 0.3 0.3 0.3 0.3 0.3 0.3
0.9745 0.9656 0.9728 0.9756 0.9765 0.9876 0.9954
0.16 0.16 0.16 0.16 0.16 0.16 0.18
0.36 0.36 0.36 0.36 0.36 0.36 0.33
0.023 0.023 0.023 0.023 0.023 0.023 0.027
0.25 0.25 0.25 0.25 0.25 0.25 0.24
1194
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Fig. 9. The spatial analysis model.
1200
Internal force (kN)
1000 800 The max axial force of sling The min axial force of sling The amplitude of axial force
600 400 200 0 0
10
20
30
40
50
60
The number of sling Fig. 10. The axial force of sling.
Fig. 11. three finite element models of steel wire rope sling.
3.2. Finite element model analysis Many engineering practices show that the short suspension cable of the suspension bridge is the most vulnerable place to damage. Due to the inconsistency of the top and bottom ends of the sling, the sling is not subject to simple tensile fatigue failure, but to a combined failure mode of tension and bending. According to the investigation of sling diseases over the past 30 years, it is found that the sling often suffers from corrosion fatigue damage due to the influence of accumulated water and other corrosive media (This paper does not consider the impact of corrosion). The specifications of the sling studied in this paper are 6 × 55SWS + IWR. Because of the complexity of wire rope sling, only the most dangerous section model is given in this paper. In this paper, three finite element models are given based on the proposed theory of split slip as showed in Fig. 11. Fig. 11 (a) is a wire rope model which is consistent with the actual situation. Its advantage is that it can simulate the failure mode of the wire rope sling most approximately, but it's modeling is very complex, the calculation consumes CPU and memory, and the calculation efficiency is very low. According to the simplified calculation model proposed by the theory of split slip, the model was simplified in a unit of one strand and the constitutive model and damage evolution equation of each strand through material testing machine were analyzed. The calculation speed of this model is much faster than that of Fig. 11(a), but its calculation efficiency is also relatively low in batch operation. Based on the idea of Fig. 11 (b), a calculation model of Fig. 11 (c) is proposed by combining 1195
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Fig. 12. User defined element in the model.
ABAQUS, Python and Fortran. On the basis of Fig. 11 (b), each solid model is equivalent to a spatial beam model. The cross section of the spatial beam is the same as that of the solid model. The material constitutive model and the damage evolution equation are the same as those of Fig. 11 (b). The beam element here is Timoshenko-beam, and the characteristics of the beam are similar to those of beam 189 in Ansys. The direct contact between each strand is represented by a user defined element (Fig. 12). The characteristics of the custom element are as follows: when the shear stress τ exceeds τmax, the user defined element is killed, otherwise the user defined element is activated Table 2. max
=µ×
(13)
c max
μ: the friction coefficient. σcmax: the maximum contact stress calculated in Fig. 13. (b). Fig. 13 (a)(b) numerical analysis results are as follows: for Fig. 13 (a) (b) model, Fixed constraints at one end of the model and the freedom of UX and UY is constrained at the other end. Because ABAQUS does not have the function of directly exerting cable force, the cooling method is used to exert 970KN cable force. (The solid model obtains the cable force by summing the stresses in the whole section). The UZ direction displacement is applied to one end to make the cable force to 1150 KN. In order to analyze the state, the analysis sub-step is set to 100 steps. The 2 million times loaded with direct loop function. In each load step, the internal force of the custom element should be extracted. In the three models presented in this paper, the phenomenon of share slip occurred near the fixed end. Slip displacement occurred at the contact between side strand and core strand at anchorage end. The length of the slip is shown in Fig. 14. Because the mesh size of Fig. 13 (c) is 2 mm, the slip length is a multiple of 2 mm. The results of three numerical models show that due to the restraint of the anchorage end of the steel wire rope sling, the shear stress at the end of the steel wire rope sling gradually increases beyond the ultimate shear stress, which results in the slip of the steel wire rope sling. Under multiple cyclic loads, the combined stresses of tension and bending exceed the damage threshold, which leads to damage degradation of materials and ultimately leads to fracture of steel wires. Model A can accurately simulate wire breakage and slip, but the calculation amount is relatively large (it takes about 300 h to calculate once in 20 core 40 thread server) and can only calculate very short length of the sling. Model B and model C, as showed in Fig. 13, can reflect the slip of sling. Model B results are close to model C. However, model C shows a step-by-step increase in slip length due to the element size is 2 mm 2 mm, which is smaller than other models but can improve accuracy by encrypting element seed. Model A and B can reflect broken wires, but because the broken wires in Abaqus are simulated by element failure, the fracture morphology cannot be revealed. About 300,000 times later, the wire rope was damaged. With the accumulation of damage, the speed of damage accelerated. As shown in Fig. 15, the results calculated by the three models were similar. Fracture morphology will be captured by electron microscopy in the experimental analysis. Model C can reflect the damage situation, but not the local fracture pattern, because each strand of the sling is replaced by a beam element, but its broken wire area can be deduced according to Eq. 4 and its calculation speed is fast (it takes about 1 h to calculate once in 20-core 40-thread server). 4. Experimental study 4.1. Introduction of experimental The acceleration sensor was installed to measure the acceleration at the corresponding place. The acoustic sensor shown was used to monitor the sound emitted when the wire breaks. The cable force sensor and reader were used to measure the cable force. Ultrasonic flaw detector was used to monitor the damage of pin shaft. The dynamic displacement of the model is measured by using the laser dynamic displacement sensor. Table 2 The most disadvantageous load. Sling number
Maximum internal Force/KN
Minimum internal force/KN
Internal force amplitude /KN
16
1150
970
180
1196
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Fig. 13. The results of the model.
10
Slip value(mm)
8
6
model a model b model c
4
2
0 0
20
40
60
80
100
Substep
Damage degree(%)
Fig. 14. The results of slip length.
model a model b model c
Cycle times(ten thousand ) Fig. 15. The results of slip damage degree.
The team has invented a patent for a cable tension and bending fatigue test device. Details can be found in the patent. The steel wire rope sling was tensioned to the designed cable force and stationed for at least 24 h to test the cable force state. After the cable force meets the test requirements, subsequent tests were carried out. The designed loading head was connected with the MTS actuator, and the loading head was installed in the middle position of the sling. The steel wire rope sling was fixed on the bracket through the cable clamp and steel anchor box, and was tensioned to the initial cable force by the jack (Fig. 16). The MTS loading system was used to adjust displacement amplitude slightly to meet the design requirements of the internal force amplitude. At this 1197
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Fig. 16. Model assembly diagram.
Fig. 17. Test process diagram.
time, steel wire rope sling is in the state of both tension and bending, and the double index function is successfully realized. Loading at 2HZ frequency, 100,000 times per load, stop to check the status of sling, record the status of wire breakage and sling wear. 4.2. Experimental analysis In the process of loading, it can be found by the cable force sensor reader that the cable force varies according to the designed stress amplitude, and the rotation angle of pin hinge rotates according to the designed angle. The fatigue test successfully realizes the double index fatigue of tension and bending, which has important research significance. Test process diagram is shown in Fig. 17. The phenomena in the experiment are as follows: (1) At the initial 300,000 times, everything was normal. During about 300,000 to 400,000 times, the wire rope at the clamp was slightly worn out. (2) During loading, black powder was produced at the joint of pin shaft and steel anchor box. Because the surface is powdery and uneven, the light irradiated on the powders produces diffuse reflection. So the color of the powder is black. (3) The phenomenon of share-splitting slip of steel wire has appeared. (4) There are two kinds of damage to the sling: the outermost layer of each strand contact begins to break; the steel wire at the sling clamp is worn obviously. 1198
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12
The maximum error is 11.4%
Damage degree(%)
10
8
6
4
model a model b model c experimental data
2
0 0
20
40
60
80
100
Cycle times(ten thousand ) Fig. 18. Damage degree contrast diagram.
The slip length is measured by varying the spraying pigments on the wire rope. The maximum slip length in the test is about 8 mm, which is smaller than the numerical calculation of 11 mm. The anchorage end is considered as a fixed constraint in numerical calculation. However, the bending stress caused by pin connection is not as large as that of numerical analysis in test and practical engineering. Similarly, the change of damage degree is not so great as that of numerical calculation. The comparison is shown in Fig. 18. Fig. 18 shows that no wire breakage occurs during 300,000 fatigue loads, either in finite element calculation or in test slings. When the number of fatigue loading reaches 400,000 times, wire breakage occurs, which proves that the three calculation models proposed in this paper can predict the occurrence of wire breakage. In the experiment, the damage degree calculated from broken wire is obviously less than that calculated by finite element method, because the bending stress of the test is not as large as that calculated by finite element method. After the test, the sling was removed and the sling was analyzed in detail by electronic method. The sling was shown in Fig. 19 under the electron microscope. It is known that the breakage of sling is caused by tension and bending
Fig. 19. Wire drawing under electron microscope. 1199
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fatigue, and there are pits caused by wear at the contact position of wire rope. In practice, the sling may break in advance due to the corrosion factor, which should be noticed by the designer. 5. Conclusion In view of the fact that the loads on the suspension cables of railway suspension bridge are much larger than those of highway bridge. Fatigue tests considering the coupling of tension and bending have not yet been reported. This paper designs a biaxial fatigue test which can consider both bending and tension of the wire rope. On the basis of practical engineering analysis, the most disadvantageous suspension position and load are calculated by BNLAS. According to the actual situation of the project, the theory of Share-splitting Slip is put forward and the three-middle numerical model is given. The material parameters are obtained by the material test. The experimental and finite element results show that the numerical model presented in this paper can reflect the failure process of the steel wire rope sling. Finally, the fracture of steel wire rope sling was photographed by the electron microscope. (1) In practical engineering, the bending stress and shear stress of the steel wire rope sling are produced due to the restraint action at the anchorage end, which results in slippage of the steel wire rope sling. (2) The simplified model presented in this paper can reflect slip and damage. (3) How to choose reasonable anchorage end restraint mode to reduce the error of testing and numerical analysis still needs to be studied. Acknowledgements This paper was financially supported by the National Natural Science Foundation of China (Grant No. 51178396/E080505). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.G. Metcalfe, R. Costanzi, Fatigue cracking of dragline boom support strands, Eng. Fail. Anal. 99 (2019) 46–68. D. Zhang, C. Feng, K. Chen, D. Wang, X. Ni, Effect of broken wire on bending fatigue characteristics of wire ropes, Int. J. Fatigue 103 (2017) 456–465. M. Giglio, A. Manes, Life prediction of a wire rope subjected to axial and bending loads, Eng. Fail. Anal. 12 (4) (2005) 549–568. J.F. Peng, M.H. Zhu, Z.B. Cai, J.H. Liu, K.C. Zuo, C. Song, W.J. Wang, On the damage mechanisms of bending fretting fatigue, Tribol. Int. 76 (2014) 133–141. Y.-L. Xu, C.-D. Zhang, S. Zhan, B.F. Spencer, Multi-level damage identification of a bridge structure: a combined numerical and experimental investigation, Eng. Struct. 156 (2018) 53–67. C.-D. Zhang, Y.-L. Xu, Multi-level damage identification with response reconstruction, Mech. Syst. Signal Process. 95 (2017) 42–57. H. Mouradi, A. El Barkany, A. El Biyaali, Steel wire ropes failure analysis: experimental study, Eng. Fail. Anal. 91 (2018) 234–242. M. Bonneric, V. Aubin, D. Durville, Finite element simulation of a steel cable - rubber composite under bending loading: influence of rubber penetration on the stress distribution in wires, Int. J. Solids Struct. 160 (2019) 158–167. Y. Yu, X. Wang, Z. Chen, A simplified finite element model for structural cable bending mechanism, Int. J. Mech. Sci. 113 (2016) 196–210. D. Wang, D. Zhang, S. Wang, S. Ge, Finite element analysis of hoisting rope and fretting wear evolution and fatigue life estimation of steel wires, Eng. Fail. Anal. 27 (2013) 173–193. K.S. Spak, G.S. Agnes, D.J. Inman, Modeling vibration response and damping of cables and cabled structures, J. Sound Vib. 336 (2015) 240–256. W. Jiang, J. Henshall, J. Walton, A concise finite element model for three-layered straight wire rope strand, Int. J. Mech. Sci. 42 (1) (2000) 63–86. C. Imrak, C. Erdönmez, On the problem of wire rope model generation with axial loading, Mathematical Comput. Appl. 15 (2) (2010) 259–268. E. Stanova, G. Fedorko, M. Fabian, S. Kmet, Computer modelling of wire strands and ropes part ii: finite element-based applications, Adv. Eng. Softw. 42 (6) (2011) 322–331. R. Judge, Z. Yang, S. Jones, G. Beattie, Full 3d finite element modelling of spiral strand cables, Constr. Build. Mater. 35 (2012) 452–459. G. Kastratovi, N. Vidanovi, V. Baki, B. Rašuo, On finite element analysis of sling wire rope subjected to axial loading, Ocean Eng. 88 (2014) 480–487. Y. Yu, Z. Chen, H. Liu, X. Wang, Finite element study of behavior and interface force conditions of seven-wire strand under axial and lateral loading, Constr. Build. Mater. 66 (2014) 10–18. J.F. Beltrán, E. De Vico, Assessment of static rope behavior with asymmetric damage distribution, Eng. Struct. 86 (2015) 84–98. V. Fontanari, M. Benedetti, B.D. Monelli, F. Degasperi, Fire behavior of steel wire ropes: experimental investigation and numerical analysis, Eng. Struct. 84 (2015) 340–349. D. Miler, M. Hoić, Z. Domitran, D. Žeželj, Prediction of friction coefficient in dry-lubricated polyoxymethylene spur gear pairs, Mech. Mach. Theory 138 (2019) 205–222. G.W. Rodgers, R. Herve, G.A. MacRae, J. Chanchi Golondrino, J.G. Chase, Dynamic friction coefficient and performance of asymmetric friction connections, Structures 14 (2018) 416–423. A.Y. Alali, A. Al-Khabbaz, S. Michael, M.V. Swain, Frictional coefficient during flossing of teeth, Dent. Mater. 34 (2018) 1727–1734. J.A.V. Bazán, A.T. Beck, J. Flórez-López, Random fatigue of plane frames via lumped damage mechanics, Eng. Struct. 182 (2019) 301–315. M. Bonneric, V. Aubin, D. Durville, Finite element simulation of a steel cable - rubber composite under bending loading: influence of rubber penetration on the stress distribution in wires, Int. J. Solids Struct. 160 (2019) 158–167. J.L. Martinez, J.C.R. Cyrino, M.A. Vaz, Continuum damage mechanics applied to numerical analysis of ship collisions, Mar. Struct. 56 (2017) 206–236.
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
Fatigue failure analysis of steel wire rope sling based on sharesplitting slip theory
T
Songling Xue, Ruili Shen , Menglong Shao, Wei Chen, Rusong Miao ⁎
School of Civil Engineering, Southwest Jiao tong University, Chengdu 610031, China
ARTICLE INFO
ABSTRACT
Keywords: Share-splitting slip Double index fatigue Damage evolution Calculation model Experimental analysis
At present, Section integrity hypothesis is mostly used to analyze the mechanical properties and identify the damage of the cable. Based on the actual situation and considering the slip effect between the wires of the cable, two simplified calculation models of the cable are proposed based on the theory of share-splitting slip, and the model parameters are obtained through experiments. Finally, on the background of Lijiang Railway Suspension Bridge in Yunnan Province, the fatigue test of the suspension cable is carried out by using the self-developed testing device which can consider both tension and bending fatigue, and the broken steel wire is further analyzed by means of electron microscopy. The results show that two simplified calculation models of wire rope sling are proposed based on the theory of share-splitting slip, which are close to the results of the original model, but greatly improve the calculation efficiency. Comparing the results of three models, it is found that the three models can reflect the development of the slip and fatigue damage of the wire rope sling, but they have some errors with the test. The fracture morphology of the steel wire is shown by electron microscopy photographs.
1. Introduction With the increasing span of bridge, the status of cable-supported bridges in long-span bridge has been continuously improved. Cables and slings are part of the core components of cable-supported bridges. Usually, they can be divided into parallel wire rope and steel wire rope. With the increase of service life, the wire rope may be damaged in Fig. 1, and the parallel wire rope may also be damaged [1,2]. Steel wire rope has been studied extensively in mechanical engineering, mining engineering and other fields, but there is little research in bridge field. Sling is one of the core force transmission components of the suspension bridge. It plays an important role by transmitting the load of the bridge deck to the main cable. Over the past 30 years, suspension bridge has developed rapidly, and the application of slings in bridge engineering has been increased. The theoretical design life of the sling is 30 years. However, the actual project shows that the sling is damaged to be different when it is far below the design service life, which makes the bridge must replace the cable in advance. As there were many examples of sudden damage of bridges caused by unscheduled cable replacement, it has become a necessary research topic to analyze the damage of cables. If the sling is hinged at the anchorage end, the sling can rotate freely without slipping. However, the actual constraint situation of slings at the anchorage end was between hinged and fixed restraint. Many scholars have studied the theory of layered slip of parallel steel wire cable [3–7]. The layered slip of parallel steel wire cable occurs within the distance from the end of the anchorage. Parallel steel wire cable can simplify the stacked beam model (Fig. 2) and other calculation models [8–11]. Because the structure of the wire rope sling is more complex than that of parallel steel wire cable, the failure mode of overlapping beam model cannot be calculated. In this paper, based on damage dynamics ⁎
Corresponding author. E-mail addresses: [email protected], [email protected] (R. Shen).
https://doi.org/10.1016/j.engfailanal.2019.07.055 Received 29 April 2019; Received in revised form 26 July 2019; Accepted 28 July 2019 Available online 31 July 2019 1350-6307/ © 2019 Published by Elsevier Ltd.
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Fig. 1. Steel wire rope failed in service.
Fig. 2. Computational model of stacked beams.
and slip theory, a simplified calculation model of steel wire rope sling was presented and its failure mode was analyzed. Some scholars have also done research on wire ropes. Most of the early models were based on linear elasticity assumption, and semi-continuous method and discrete method were used. In recent years, with the development of computer technology and finite element theory, many scholars have used the finite element method to simulate the steel wire rope. Based on the geometrical characteristics of the wire strands and the corresponding boundary conditions, a simplified finite element model of the wire strands was established by Jiang [12]. Since only 1/6 of the winding strand axis model is needed, the calculation efficiency of the model is high. However, complex boundary constraints have not been widely used. Imrak established the model of double helix wire rope by computer aided design (CAD) software, and simulated the performance of the strand under axial tension load by finite element method [13]. With the help of Abaqus/Explicit software, simulation of multi-layer steel wire rope strands with “1 + 6 + 12 + 18” structure under tension and bending support was carried out [14]. Judge used a 3D finite element model to study the axial load-strain curve and the failure load of multi-layer steel wire strand [15]. Kastratovic used two different methods (uniform axial force and uniform axial strain) to simulate the wire rope strand. It was found that the loading mode had a direct impact on the load distribution between wires of the wire rope strand [16]. In addition to the above research on the axial mechanical properties of wire rope, Yu studied the mechanical properties and the stress state of wire rope strands under transverse loads [17]. Juan used the finite element method to analyze the static performance and cross-section stress-strain distribution of steel wire ropes with asymmetric damage [18]. Fontanari used finite element simulation and test methods to study the elastoplastic properties of steel wire strands and the performance at different temperatures [19]. 1190
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Fig. 3. Friction coefficient test.
However, fatigue damage is a dynamic process, and it is unreasonable to evaluate fatigue damage by static method. In this paper, based on damage dynamics and slip theory, a simplified calculation model of steel wire rope sling is presented and its failure mode is analyzed. The basic structure of the article is as follows: The relevant background knowledge was introduced in section 1. Data of wire rope material was obtained through experiments, which provide a basis for determining the parameters of the numerical model based on damage dynamics in section 2. A share-splitting slip model based on damage dynamics was established and the fatigue failure time was calculated in section 3. In order to verify the validity of the numerical model, the bending fatigue test of the steel wire rope was carried out by the designed bracket based on the actual project in section 4. Finally, a summary of the full text was given. 2. Theoretical calculation model 2.1. Testing of friction coefficient between steel wires Friction is a complex problem, which is affected by many factors, such as contact material, indicating roughness, temperature and relative velocity. So far, there is no definite conclusion about the friction mechanism [20]. Under different conditions, the friction coefficient between steel wire and steel wire is different, and the accuracy of the friction coefficient is related to the correctness of theoretical calculation [21]. Many scholars have done a lot of research on the friction coefficient between steel wires, which range from 0.1 to 0.35 [22,23]. Previous team tests have shown that the friction coefficient of steel wires is about 0.3. Static friction coefficient is generally higher than dynamic friction coefficient. In this paper, a simple method of measuring friction coefficient was adopted. In order to simplify the calculation, the difference of static and dynamic friction coefficient was not considered. The measurement scheme of friction coefficient was shown in Fig. 3. Take out the same steel wire from the original steel wire rope sling and put it into the device shown in Fig. 3. The bottom plate was placed on the horizontal plane, and the angle between the steel wire and the bottom plate was adjusted and recorded when the steel wire slips. Ten different groups of steel wires were taken from the sling in this test, and the average value of each group was obtained three times. The result was shown in Fig. 4. The friction coefficient can be simplified as follows:
µ = tan
=
h L
(1)
The average value of Fig. 4 shows that the friction coefficient is 0.2473 in this analysis. Different friction coefficients may lead to differences. The influence of friction coefficients will be discussed in the next paper. 0.32
friction coefficient
0.30
0.28
0.26
0.24
0.22
0.20 0
2
4
6
8
test times
Fig. 4. Test results of friction coefficient. 1191
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2.2. Slip mechanism of sling rope Bending fatigue failure of cables has attracted the attention of many scholars and a lot of research has been done. The results show that the bending stress at the anchorage end is the primary factor causing cable bending fatigue. However, at present, the calculation of cable mechanical properties, cable force identification and damage identification are all based on the assumption that the cable section is a whole section. Numerous engineering practices show that bending stress exists near the anchorage section of the cable. The phenomenon of layered slip can be found in parallel steel wire, but for steel wire rope sling, because the structure of the steel wire rope sling is complex, the layered slip theory cannot be applied. In this paper, based on the test and engineering practice, a theoretical model for calculating the slip of steel wire rope sling is proposed to calculate the fatigue failure of the steel wire rope sling. Taking the most basic steel wire rope as an example, the theoretical stress calculation formula of the slip of the steel wire rope sling is deduced. Slip may occur in every steel wire of the wire rope sling, but in bridge engineering, slip occurs between side strand and side strand per share, and between side strand and core strand. According to structural mechanics, there is a corresponding relationship between bending moment and shear stress as follows:
Q (x ) = =
dM (x ) dx
4 QS = bIZ 3
(2)
exp
x
T EI
(3)
Q(x): cross-sectional shear force; M(x):section bending moment; ∆φ:angle of anchorage section; T:cable force of sling; E:elastic modulus; I:cross-sectional moment of inertia; τ:section shear stress. Before the slip occurs, the section can be known as a section and its stress can be calculated according to Eq. (3). When the slip occurs, the moment of inertia of the whole section in the integral formula (3) is replaced by the moment of inertia of each section. The shear stress distribution in both cases is shown in Fig. 5. When the maximum shear stress exceeds the ultimate friction force, the steel wire rope sling can slip, and the shear stress distribution pattern of the section becomes shown on the right side of Fig. 5. 2.3. Definition of rope strand damage mode Wire rope sling is composed of many steel wires, so it is very important to determine the damage model of steel wire for structural analysis [23,24]. In this paper, fatigue is analyzed not based on traditional miner criterion, but from the point of view of damage mechanics [25]. The damage degradation equation of the material is established, and the time and position of wire breakage can be calculated directly with the cyclic loading and high cycle fatigue in Abaqus. Because the numerical analysis is based on a simplified calculation model and takes each share as an entity, the performance of each share needs to be tested. Because the performance of each strand of wire rope is different, the whole mechanical parameters of each strand of wire rope were obtained by experiment. The strands of the wire rope are numbered, the core strand is No. 7, and the side strands are from 1 to 6. Each share (Fig. 6) was placed on the universal material experiment machine (Fig. 7). The stress-strain curve was shown in Fig. 8. According to damage mechanics, the damage degree (ω) is defined as the ratio of the bearing area (AD) after damage to that before damage (A).
=
AD A A = A A
(4)
Among them, A: the area of damage, ω = 0 means no damage, ω = 1means material fracture, and0 < ω < 1 means the state of damage process.The effective stress of the damaged material increases with the decrease of the bearing area caused by the damage, so the effective stress of the damaged material is as follows: ij
=
ij
(5)
1
In 1971, Lemaitre proposed the equivalent strain hypothesis, which enables indirect damage to be measured. The hypothesis was
Fig. 5. Segmental model of sling. 1192
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Fig. 6. The number of the share.
Fig. 7. Universal material experiment machine.
2000
stress(MPa)
1500
1 2 3 4 5 6 7
1000
500
0
0.0
0.5
1.0
1.5
2.0
2.5
strain(%) Fig. 8. The stress-strain curve.
that the deformation of materials containing damage can be represented only by effective stress, and the constitutive relation of the damage model can be expressed by the constitutive relation of non-destructive mode. The one-dimensional linear elastic relationship of damaged materials is as follows: e
=
ij
(1
)E
E = (1
)E
=
(6)
E
(7)
E is the modulus of elasticity of materials and E is the modulus of elasticity after damage. Steel wire rope used in this paper were high strength steel wires. The Ramberg-Osgood model considering strain hardening was 1193
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selected and the damage variable was added on the basis of the original constitutive model. The damage constitutive relationship is considered as follows: m
m
=
=
k
(1
(8)
)k
Among them, k and m are hardening coefficients. The damage is represented by replacing the hardening coefficient k with the hardening coefficient k after damage.
k = (1 is:
(9)
)k
In order to study the damage degradation process of materials, it is considered that the cumulative plastic strain rate of steel wire
p=
2 3
1
2
ij ij
(10)
ij :
strain tensor. The relationship between damage variable ωand cumulative plastic strain p was obtained by combining the dynamic equation of damage evolution with the damage constitutive eq. (8). 2s0 + m
=
c
p
2s0 + m
pR
m m
2s0 + m
pD
2s 0 + m
pD
m
(11)
m
s0: material parameters, m: hardening coefficient, pR: fracture strain and critical damage degree: ωc. c
=
k 2 (1 + ) + 3(1 2Es0 3
2
2 )
m eq
×
2s0 + m m m pR 2s0 + m
2s0 + m
pD
m
(12)
Based on the above theory and material test, the results of parameters are shown in Table 1. 3. Numerical analysis 3.1. Load acquisition and engineering background The main cable span of Jinsha jiang Suspension Bridge is 132 + 660 + 132 m. The main girder is a three-span continuous beam with a span composition of 110 + 660 + 98 m. The main girder is provided with vertical and transverse supports at the end of the girder and at the main tower. Jinsha jiang Railway Suspension Bridge is the first railway suspension bridge in China. The live load of railway suspension bridge accounts for a higher proportion of the total load. Its stress amplitude is larger than that of the highway bridge, and the suspension cable is prone to fatigue. For railway suspension bridge, the suspension cables are subjected to larger loads. The fatigue strength of the suspension cables may be the controlling factor in design and use. At the same time, the stiffening girder of suspension bridge will produce longitudinal and transverse displacement along the bridge under the action of vehicle operation, which will lead to the difference of longitudinal displacement between the upper and lower suspension points of the suspension cable anchored on the main girder. Therefore, it will lead to the bending of the sling and other phenomena, resulting in secondary stress and so on. In order to avoid fatigue failure of sling, it is necessary to calculate and analyze the response of sling under fatigue load in detail. Through the self-developed bridge structure static and dynamic non-linear analysis system BNLAS, the spatial analysis model is established (Fig. 9), and the most disadvantageous load is analyzed as showed in Table 0.2. In order to indicate the most dangerous sling location of the whole bridge, the number of sling and its internal force diagram can be seen in Fig. 10. Table 1 The results of parameters. Number
Elastic modulus (MPa)
Poisson ratio
Enhancement parameter(k)
Hardening parameter (m)
Failure plastic strain (pR)
Damage strain threshold
Damage critical coefficient(ωc)
1 2 3 4 5 6 7
1.13E+05 1.14E+05 1.17E+05 1.12E+05 1.21E+05 1.12E+05 1.14E+05
0.3 0.3 0.3 0.3 0.3 0.3 0.3
0.9745 0.9656 0.9728 0.9756 0.9765 0.9876 0.9954
0.16 0.16 0.16 0.16 0.16 0.16 0.18
0.36 0.36 0.36 0.36 0.36 0.36 0.33
0.023 0.023 0.023 0.023 0.023 0.023 0.027
0.25 0.25 0.25 0.25 0.25 0.25 0.24
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Fig. 9. The spatial analysis model.
1200
Internal force (kN)
1000 800 The max axial force of sling The min axial force of sling The amplitude of axial force
600 400 200 0 0
10
20
30
40
50
60
The number of sling Fig. 10. The axial force of sling.
Fig. 11. three finite element models of steel wire rope sling.
3.2. Finite element model analysis Many engineering practices show that the short suspension cable of the suspension bridge is the most vulnerable place to damage. Due to the inconsistency of the top and bottom ends of the sling, the sling is not subject to simple tensile fatigue failure, but to a combined failure mode of tension and bending. According to the investigation of sling diseases over the past 30 years, it is found that the sling often suffers from corrosion fatigue damage due to the influence of accumulated water and other corrosive media (This paper does not consider the impact of corrosion). The specifications of the sling studied in this paper are 6 × 55SWS + IWR. Because of the complexity of wire rope sling, only the most dangerous section model is given in this paper. In this paper, three finite element models are given based on the proposed theory of split slip as showed in Fig. 11. Fig. 11 (a) is a wire rope model which is consistent with the actual situation. Its advantage is that it can simulate the failure mode of the wire rope sling most approximately, but it's modeling is very complex, the calculation consumes CPU and memory, and the calculation efficiency is very low. According to the simplified calculation model proposed by the theory of split slip, the model was simplified in a unit of one strand and the constitutive model and damage evolution equation of each strand through material testing machine were analyzed. The calculation speed of this model is much faster than that of Fig. 11(a), but its calculation efficiency is also relatively low in batch operation. Based on the idea of Fig. 11 (b), a calculation model of Fig. 11 (c) is proposed by combining 1195
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Fig. 12. User defined element in the model.
ABAQUS, Python and Fortran. On the basis of Fig. 11 (b), each solid model is equivalent to a spatial beam model. The cross section of the spatial beam is the same as that of the solid model. The material constitutive model and the damage evolution equation are the same as those of Fig. 11 (b). The beam element here is Timoshenko-beam, and the characteristics of the beam are similar to those of beam 189 in Ansys. The direct contact between each strand is represented by a user defined element (Fig. 12). The characteristics of the custom element are as follows: when the shear stress τ exceeds τmax, the user defined element is killed, otherwise the user defined element is activated Table 2. max
=µ×
(13)
c max
μ: the friction coefficient. σcmax: the maximum contact stress calculated in Fig. 13. (b). Fig. 13 (a)(b) numerical analysis results are as follows: for Fig. 13 (a) (b) model, Fixed constraints at one end of the model and the freedom of UX and UY is constrained at the other end. Because ABAQUS does not have the function of directly exerting cable force, the cooling method is used to exert 970KN cable force. (The solid model obtains the cable force by summing the stresses in the whole section). The UZ direction displacement is applied to one end to make the cable force to 1150 KN. In order to analyze the state, the analysis sub-step is set to 100 steps. The 2 million times loaded with direct loop function. In each load step, the internal force of the custom element should be extracted. In the three models presented in this paper, the phenomenon of share slip occurred near the fixed end. Slip displacement occurred at the contact between side strand and core strand at anchorage end. The length of the slip is shown in Fig. 14. Because the mesh size of Fig. 13 (c) is 2 mm, the slip length is a multiple of 2 mm. The results of three numerical models show that due to the restraint of the anchorage end of the steel wire rope sling, the shear stress at the end of the steel wire rope sling gradually increases beyond the ultimate shear stress, which results in the slip of the steel wire rope sling. Under multiple cyclic loads, the combined stresses of tension and bending exceed the damage threshold, which leads to damage degradation of materials and ultimately leads to fracture of steel wires. Model A can accurately simulate wire breakage and slip, but the calculation amount is relatively large (it takes about 300 h to calculate once in 20 core 40 thread server) and can only calculate very short length of the sling. Model B and model C, as showed in Fig. 13, can reflect the slip of sling. Model B results are close to model C. However, model C shows a step-by-step increase in slip length due to the element size is 2 mm 2 mm, which is smaller than other models but can improve accuracy by encrypting element seed. Model A and B can reflect broken wires, but because the broken wires in Abaqus are simulated by element failure, the fracture morphology cannot be revealed. About 300,000 times later, the wire rope was damaged. With the accumulation of damage, the speed of damage accelerated. As shown in Fig. 15, the results calculated by the three models were similar. Fracture morphology will be captured by electron microscopy in the experimental analysis. Model C can reflect the damage situation, but not the local fracture pattern, because each strand of the sling is replaced by a beam element, but its broken wire area can be deduced according to Eq. 4 and its calculation speed is fast (it takes about 1 h to calculate once in 20-core 40-thread server). 4. Experimental study 4.1. Introduction of experimental The acceleration sensor was installed to measure the acceleration at the corresponding place. The acoustic sensor shown was used to monitor the sound emitted when the wire breaks. The cable force sensor and reader were used to measure the cable force. Ultrasonic flaw detector was used to monitor the damage of pin shaft. The dynamic displacement of the model is measured by using the laser dynamic displacement sensor. Table 2 The most disadvantageous load. Sling number
Maximum internal Force/KN
Minimum internal force/KN
Internal force amplitude /KN
16
1150
970
180
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Fig. 13. The results of the model.
10
Slip value(mm)
8
6
model a model b model c
4
2
0 0
20
40
60
80
100
Substep
Damage degree(%)
Fig. 14. The results of slip length.
model a model b model c
Cycle times(ten thousand ) Fig. 15. The results of slip damage degree.
The team has invented a patent for a cable tension and bending fatigue test device. Details can be found in the patent. The steel wire rope sling was tensioned to the designed cable force and stationed for at least 24 h to test the cable force state. After the cable force meets the test requirements, subsequent tests were carried out. The designed loading head was connected with the MTS actuator, and the loading head was installed in the middle position of the sling. The steel wire rope sling was fixed on the bracket through the cable clamp and steel anchor box, and was tensioned to the initial cable force by the jack (Fig. 16). The MTS loading system was used to adjust displacement amplitude slightly to meet the design requirements of the internal force amplitude. At this 1197
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Fig. 16. Model assembly diagram.
Fig. 17. Test process diagram.
time, steel wire rope sling is in the state of both tension and bending, and the double index function is successfully realized. Loading at 2HZ frequency, 100,000 times per load, stop to check the status of sling, record the status of wire breakage and sling wear. 4.2. Experimental analysis In the process of loading, it can be found by the cable force sensor reader that the cable force varies according to the designed stress amplitude, and the rotation angle of pin hinge rotates according to the designed angle. The fatigue test successfully realizes the double index fatigue of tension and bending, which has important research significance. Test process diagram is shown in Fig. 17. The phenomena in the experiment are as follows: (1) At the initial 300,000 times, everything was normal. During about 300,000 to 400,000 times, the wire rope at the clamp was slightly worn out. (2) During loading, black powder was produced at the joint of pin shaft and steel anchor box. Because the surface is powdery and uneven, the light irradiated on the powders produces diffuse reflection. So the color of the powder is black. (3) The phenomenon of share-splitting slip of steel wire has appeared. (4) There are two kinds of damage to the sling: the outermost layer of each strand contact begins to break; the steel wire at the sling clamp is worn obviously. 1198
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12
The maximum error is 11.4%
Damage degree(%)
10
8
6
4
model a model b model c experimental data
2
0 0
20
40
60
80
100
Cycle times(ten thousand ) Fig. 18. Damage degree contrast diagram.
The slip length is measured by varying the spraying pigments on the wire rope. The maximum slip length in the test is about 8 mm, which is smaller than the numerical calculation of 11 mm. The anchorage end is considered as a fixed constraint in numerical calculation. However, the bending stress caused by pin connection is not as large as that of numerical analysis in test and practical engineering. Similarly, the change of damage degree is not so great as that of numerical calculation. The comparison is shown in Fig. 18. Fig. 18 shows that no wire breakage occurs during 300,000 fatigue loads, either in finite element calculation or in test slings. When the number of fatigue loading reaches 400,000 times, wire breakage occurs, which proves that the three calculation models proposed in this paper can predict the occurrence of wire breakage. In the experiment, the damage degree calculated from broken wire is obviously less than that calculated by finite element method, because the bending stress of the test is not as large as that calculated by finite element method. After the test, the sling was removed and the sling was analyzed in detail by electronic method. The sling was shown in Fig. 19 under the electron microscope. It is known that the breakage of sling is caused by tension and bending
Fig. 19. Wire drawing under electron microscope. 1199
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fatigue, and there are pits caused by wear at the contact position of wire rope. In practice, the sling may break in advance due to the corrosion factor, which should be noticed by the designer. 5. Conclusion In view of the fact that the loads on the suspension cables of railway suspension bridge are much larger than those of highway bridge. Fatigue tests considering the coupling of tension and bending have not yet been reported. This paper designs a biaxial fatigue test which can consider both bending and tension of the wire rope. On the basis of practical engineering analysis, the most disadvantageous suspension position and load are calculated by BNLAS. According to the actual situation of the project, the theory of Share-splitting Slip is put forward and the three-middle numerical model is given. The material parameters are obtained by the material test. The experimental and finite element results show that the numerical model presented in this paper can reflect the failure process of the steel wire rope sling. Finally, the fracture of steel wire rope sling was photographed by the electron microscope. (1) In practical engineering, the bending stress and shear stress of the steel wire rope sling are produced due to the restraint action at the anchorage end, which results in slippage of the steel wire rope sling. (2) The simplified model presented in this paper can reflect slip and damage. (3) How to choose reasonable anchorage end restraint mode to reduce the error of testing and numerical analysis still needs to be studied. Acknowledgements This paper was financially supported by the National Natural Science Foundation of China (Grant No. 51178396/E080505). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.G. Metcalfe, R. Costanzi, Fatigue cracking of dragline boom support strands, Eng. Fail. Anal. 99 (2019) 46–68. D. Zhang, C. Feng, K. Chen, D. Wang, X. Ni, Effect of broken wire on bending fatigue characteristics of wire ropes, Int. J. Fatigue 103 (2017) 456–465. M. Giglio, A. Manes, Life prediction of a wire rope subjected to axial and bending loads, Eng. Fail. Anal. 12 (4) (2005) 549–568. J.F. Peng, M.H. Zhu, Z.B. Cai, J.H. Liu, K.C. Zuo, C. Song, W.J. Wang, On the damage mechanisms of bending fretting fatigue, Tribol. Int. 76 (2014) 133–141. Y.-L. Xu, C.-D. Zhang, S. Zhan, B.F. Spencer, Multi-level damage identification of a bridge structure: a combined numerical and experimental investigation, Eng. Struct. 156 (2018) 53–67. C.-D. Zhang, Y.-L. Xu, Multi-level damage identification with response reconstruction, Mech. Syst. Signal Process. 95 (2017) 42–57. H. Mouradi, A. El Barkany, A. El Biyaali, Steel wire ropes failure analysis: experimental study, Eng. Fail. Anal. 91 (2018) 234–242. M. Bonneric, V. Aubin, D. Durville, Finite element simulation of a steel cable - rubber composite under bending loading: influence of rubber penetration on the stress distribution in wires, Int. J. Solids Struct. 160 (2019) 158–167. Y. Yu, X. Wang, Z. Chen, A simplified finite element model for structural cable bending mechanism, Int. J. Mech. Sci. 113 (2016) 196–210. D. Wang, D. Zhang, S. Wang, S. Ge, Finite element analysis of hoisting rope and fretting wear evolution and fatigue life estimation of steel wires, Eng. Fail. Anal. 27 (2013) 173–193. K.S. Spak, G.S. Agnes, D.J. Inman, Modeling vibration response and damping of cables and cabled structures, J. Sound Vib. 336 (2015) 240–256. W. Jiang, J. Henshall, J. Walton, A concise finite element model for three-layered straight wire rope strand, Int. J. Mech. Sci. 42 (1) (2000) 63–86. C. Imrak, C. Erdönmez, On the problem of wire rope model generation with axial loading, Mathematical Comput. Appl. 15 (2) (2010) 259–268. E. Stanova, G. Fedorko, M. Fabian, S. Kmet, Computer modelling of wire strands and ropes part ii: finite element-based applications, Adv. Eng. Softw. 42 (6) (2011) 322–331. R. Judge, Z. Yang, S. Jones, G. Beattie, Full 3d finite element modelling of spiral strand cables, Constr. Build. Mater. 35 (2012) 452–459. G. Kastratovi, N. Vidanovi, V. Baki, B. Rašuo, On finite element analysis of sling wire rope subjected to axial loading, Ocean Eng. 88 (2014) 480–487. Y. Yu, Z. Chen, H. Liu, X. Wang, Finite element study of behavior and interface force conditions of seven-wire strand under axial and lateral loading, Constr. Build. Mater. 66 (2014) 10–18. J.F. Beltrán, E. De Vico, Assessment of static rope behavior with asymmetric damage distribution, Eng. Struct. 86 (2015) 84–98. V. Fontanari, M. Benedetti, B.D. Monelli, F. Degasperi, Fire behavior of steel wire ropes: experimental investigation and numerical analysis, Eng. Struct. 84 (2015) 340–349. D. Miler, M. Hoić, Z. Domitran, D. Žeželj, Prediction of friction coefficient in dry-lubricated polyoxymethylene spur gear pairs, Mech. Mach. Theory 138 (2019) 205–222. G.W. Rodgers, R. Herve, G.A. MacRae, J. Chanchi Golondrino, J.G. Chase, Dynamic friction coefficient and performance of asymmetric friction connections, Structures 14 (2018) 416–423. A.Y. Alali, A. Al-Khabbaz, S. Michael, M.V. Swain, Frictional coefficient during flossing of teeth, Dent. Mater. 34 (2018) 1727–1734. J.A.V. Bazán, A.T. Beck, J. Flórez-López, Random fatigue of plane frames via lumped damage mechanics, Eng. Struct. 182 (2019) 301–315. M. Bonneric, V. Aubin, D. Durville, Finite element simulation of a steel cable - rubber composite under bending loading: influence of rubber penetration on the stress distribution in wires, Int. J. Solids Struct. 160 (2019) 158–167. J.L. Martinez, J.C.R. Cyrino, M.A. Vaz, Continuum damage mechanics applied to numerical analysis of ship collisions, Mar. Struct. 56 (2017) 206–236.
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