Allowable Strain Value for Contact Wires Taking into Account Probability of Failure

Abstract

The allowable strain value for all types of contact wire, including high-strength contact wires, has been set to 500×10⁻⁶ based on the fatigue characteristics of a basic hard-drawn copper. However, as train speeds increase, the strain value of contact wires may increase to more than 500×10⁻⁶ in the future. Therefore, in this paper, we propose a method for setting allowable strain values for each contact wire, taking into account the probability of failure. This probability is consistent with the margins of the conventional allowable strain value of 500×10⁻⁶. In addition, using this method, we propose allowable strain values for four types of high-strength contact wires.

1. Introduction

When a pantograph slides over a contact wire, bending strain is generated in the contact wire. This bending strain tends to increase as the train speed increases. Depending on the magnitude of the strain, the contact wire may break due to fatigue. To suppress the fatigue breakage of the contact wire, the allowable strain value for contact wires has been set at 500 × 10⁻⁶, and railway operators confirm that the strain of contact wires is below the allowable value.

This allowable strain value was set by adding a certain margin to the fatigue properties of hard-drawn copper wire under no tension conditions [1]. So far, this allowable strain value has been applied to all types of copper and copper alloy contact wire. However, if the contact wire strain increases in the future as a result of further increases in train speeds, railway operators may not be able to keep contact wire strain below the allowable value, even if they improve overhead contact line equipment.

It is now well known that there is a roughly positive correlation between the mechanical strength and fatigue resistance of metallic materials. Therefore, it is possible to increase the allowable strain value from 500 × 10⁻⁶ for high-strength copper alloy contact wires used in the simple catenary systems that are intended for operating speeds of over 300 km/h (hereafter referred to as “high-speed simple catenary system”) [2]. However, as described below, the conventional methods used to set allowable strain values have some problems, such as not quantifying margins. Therefore, in order to set different allowable strain values for each type of contact wire, it is necessary to quantify the basis for margins.

In this study, we propose a method for evaluating fatigue properties that can be applied to any type of contact wire by setting the statistical probability of failure equivalent to the margin in the conventional allowable strain value. In addition, using this method, we propose allowable strain values for four types of high-strength contact wires. We also investigate a method for calculating the allowable strain value of contact wires when the mean tensile stress is different.

2. Conventional method for setting allowable strain values

The conventional method for setting allowable strain values [1] is described below. First, bending fatigue tests under no tension on a wire made of the same material as the hard-drawn copper contact wire with a diameter of 2 mm are carried out. Then, based on the fatigue test results, an S-N curve for the wire (Fig. 1) is created. The bending stress amplitude of 120 MPa at 10⁷ cycles on the S-N curve is defined as the fatigue limit. This is because, if number of cycles of 10⁷ is regarded as the fatigue life of the contact wire, it is thought that a contact wire will need replacing due to wear before the wire fails due to fatigue.

Fig. 1　S-N curve of non-tension hard-drawn copper wire

Note: Figure 1 is redrawn based on reference [1].

Since the mean tensile stress is applied to a contact wire by the catenary tension in a real installation, the fatigue limit of the contact wire decreases from one that is not under tension. Therefore, the fatigue limit under non-tension is corrected to the fatigue limit under certain tension using the fatigue limit diagram called the Smith diagram shown in Fig. 2. In this diagram, when a stress equal to the tensile strength of a metal material (350 MPa in the case of hard-drawn copper) is applied, the material will fail immediately. So, the repeated stress at this point is 0 (point A in Fig. 2). There is an empirical law whereby the repeated stress as the fatigue limit decreases linearly with increasing mean tensile stress. And as mentioned above, the fatigue limit of hard-drawn copper with non-tension is 120 MPa, so the fatigue limit corresponding to the mean tensile stress forms a triangle ABC in Fig. 2. Then, in order to set a safer allowable strain value, the practical minimum fatigue limit of the contact wire (i.e., the fatigue limit of the contact wire that has been worn to its limit) is calculated. The mean tensile stress of the GT110 hard-drawn copper contact wire with a tension of 9.8 kN at the wear limit (residual diameter of 7.5 mm at the time) is 145 MPa. The repeated stress as the practical minimum fatigue limit is calculated to be 70 MPa at the 145 MPa position on the horizontal axis in Fig. 2. Furthermore, a margin for uncertainties (such as deformation of the overhead contact line height over time and surface deterioration of the contact wire [1]) is added to this value to set the allowable stress 60 MPa. Finally, the value converted to strain using Young's modulus of hard-drawn copper, 120 GPa, is the conventional allowable strain value of 500 × 10⁻⁶.

Fig. 2　Smith line of hard-drawn copper wire

There are two problems with the above method of setting the allowable strain value. The first problem is that the quantitative basis for the margin between 70 MPa and 60 MPa has not been presented. From past experience, it is true that fatigue failure can be prevented by keeping the contact wire strain in actual overhead equipment at 500 × 10⁻⁶ or less, so the above margin is considered to be sufficient for safety. However, there is a possibility that this margin is excessive.

Another problem is that the conventional allowable strain value is set based on the S-N curve obtained from the fatigue test results of hard-drawn copper wire under non-tension. The S-N curve should be prepared by matching the size of a test specimen and repeated bending stress as closely as possible to actual conditions of use. It is therefore desirable to set an allowable strain value based on the S-N curve obtained from fatigue test results of real contact wires under tension.

As mentioned above, the margin for setting the conventional allowable strain value is intended to include the effects of deterioration over time. However, this margin does not reflect the quantitative deterioration of the contact wire. It is thought to be a value that gives a sufficient margin in the fatigue characteristics of a new contact wire to compensate for deterioration. Therefore, to quantify how much safety the conventional allowable strain value margin gives for a new contact wire, we decided to evaluate it using the statistical probability of failure.

3. Margin of the conventional allowable strain value

3.1 Test method

In this chapter, we carry out fatigue tests on real contact wires of hard-drawn copper under tension. We also obtain a P-S-N curve (S-N curve for P% probability of failure) from the fatigue test results and clarify a probability of failure that is consistent with the conventional margin. In the following, a graph in which the vertical axis (stress amplitude) of the S-N curve is converted to strain amplitude is also treated as an S-N curve.

The tests were carried out using the “wire/metal fittings vibration testing machine” shown in Fig. 3. This machine is capable of applying tension to a real contact wire, and also has a mechanism to repeatedly generate any bending strain by vibrating the vibration roller. The fatigue tests were carried out until the contact wire broke, and the number of repetitions until the break was measured. The vibration waveform was a sine wave, the vibration frequency was 5 Hz, and the tension was 9.8 kN, which is the standard tension at the time of installation.

Fig. 3　Wire/metal fittings vibration testing machine

The contact wire used in the test was a hard-drawn copper contact wire, GT110. Table 1 shows the allowable stress of GT110 (tensile strength divided by a safety factor of 2.2) and the mean tensile stress corresponding to each residual diameter. The hard-drawn copper contact wire was machined to a residual diameter of 7.1 mm to meet the mean tensile stress condition (156.2 MPa) corresponding to the wear limit at a tension of 9.8 kN. Figure 4 shows a cross-section of the hard-drawn copper contact wire after machining.

Table 1　Mean tensile stress of hard-drawn copper contact wire, GT110

Residual diameter (mm)	Cross-sectional area (mm2)	Tensile strength (MPa)	Allowable stress (MPa)	Tension (kN)	Mean tensile stress (MPa)
12.34	111.1	344	156.4	9.8	88.2
7.3	65.2				150.4
7.2	64.0				153.2
7.1	62.7				156.2
7.0	61.5				159.3
6.9	60.3				162.6

Fig. 4　Cross section of the hard-drawn copper contact wire, GT110

The fatigue limit was set to a strain amplitude corresponding to 10⁷ cycles, similar to the method used to set the conventional allowable strain value. Other test procedures and the construction of the S-N curve were carried out in accordance with the 14S-N test method in reference [3].

3.2 Test results

The results of the fatigue test are shown in Table 2. And the S-N curve of GT110 created from the test results is shown in Fig. 5. Based on the data from the slope in Table 2, the slope data of the regression line (approximation line corresponding to a 50% probability of failure) of the S-N curve in Fig. 5 was set. Furthermore, the estimated standard deviation of the strain amplitude, $\widehat{\sigma }$, was calculated to be 93 × 10⁻⁶ using the data from the slope.

Table 2　Test results of the hard-drawn copper contact wire, GT110

Slope data			Horizontal data
No.	Strain amplitude	Number of cycles	No.	Strain amplitude	Number of cycles
1	1930×10−6	2.11×105	1	530×10−6	>1.00×107＊
2	1910×10−6	1.86×105	2	630×10−6	>1.00×107＊
3	1580×10−6	5.79×105	3	730×10−6	>1.00×107＊
4	1580×10−6	5.50×105	4	800×10−6	>1.00×107＊
5	1240×10−6	8.95×105	5	900×10−6	9.00×106
6	1230×10−6	1.51×106	6	810×10−6	8.84×106
7	880×10−6	5.37×106
8	880×10−6	7.53×106

^＊> represents ‘Not broken’.

Fig. 5　S-N curve of the hard-drawn copper contact wire, GT110

A staircase method was used to estimate the strain amplitude corresponding to 10⁷ cycles, which is the fatigue limit [3]. The horizontal data in Table 2 shows the test results using the staircase method. In addition, using the horizontal data, the mean strain amplitude corresponding to a 50% probability of failure, ${\widehat{S}}_{{10}^7}$, was calculated to be 762 × 10⁻⁶.

From the above $\widehat{\sigma }$ and ${\widehat{S}}_{{10}^7}$, the failure probability factor for any strain amplitude can be calculated as follows:

  

$$k=\frac{{\widehat{S}}_{{10}^7}-S}{\widehat{\sigma }}$$（1）

where k is the failure probability factor, and S is any strain amplitude.

Substituting the conventional allowable strain value of 500 × 10⁻⁶ for S in (1), k is calculated to be 2.81. According to reference [3], this value corresponds to a probability of failure of approximately 0.3%. This means that the margin included in the conventional allowable strain value of 500 × 10⁻⁶ is equal to the strain between 50% and 0.3% probability of failure at 10⁷ cycles.

This margin roughly coincides with the general indicator of data variability, the 3σ interval (an interval that contains data with a 99.7% probability when the distribution of measurement data follows a normal distribution). This means that the above margin is equivalent to the minimum strain based on the general statistical indicator, and is therefore of an appropriate quantity.

3.3 How to set the allowable strain value

There are two approaches to setting the allowable strain value of a contact wire. The first approach is to use a width of the margin of hard-drawn copper contact wire as it is. The margin of hard-drawn copper contact wire is defined by the following formula: (strain amplitude with 50% probability of failure) - (strain amplitude with 0.3% probability of failure) = (762 − 500) × 10⁻⁶ = 262 × 10⁻⁶. This approach defines the allowable strain value as the value obtained by subtracting 262 × 10⁻⁶ from the 50% probability of failure strain amplitude of the target contact wire. The second approach defines the allowable strain value as the strain amplitude with 0.3% probability of failure of the S-N curve of the target contact wire.

Now, the variation in fatigue properties depends on the material and processing of the contact wire. Therefore, applying the margin of hard-drawn copper contact wire as it is, may lead to a dangerous evaluation in some cases. For this reason, the second approach, which allows the variation in fatigue characteristics to be evaluated for each type of contact wire, is considered more appropriate.

From the above, this paper proposes a method for evaluating the fatigue properties of contact wires as follows: Obtain the S-N curve of a contact wire under mean tensile stress conditions equivalent to the wear limit, and evaluate the strain amplitude with a 0.3% probability of failure at 10⁷ cycles.

4. Allowable strain values for high-strength contact wires

This chapter describes the results of setting the allowable strain values of four types of high-strength contact wires using the method proposed in the previous chapter. These contact wires are indium-containing hard-drawn copper contact wire, SNN170 (cross-sectional area 170 mm²), precipitation-hardened copper alloy contact wire, PHC110 (cross-sectional area 110 mm²) and PHC130 (cross-sectional area 130 mm²), and cobalt-phosphorus copper alloy contact wire, CPS130 (cross-sectional area 130 mm²). They are used in high-speed simple catenary systems, etc.

First, fatigue tests were carried out on these contact wires, and the S-N curves were obtained. Table 3 shows the allowable stress, tension, mean tensile stress, and residual diameter conditions for each contact wire. Tension was set to the tension when each contact wire was used in a high-speed simple catenary system. The residual diameter condition was set to the value when the mean tensile stress was equivalent to the wear limit, as described above. The other test methods were the same as in the previous chapter.

Table 3　Allowable stress of high-strength contact wires and test conditions

Contact wire type	Allowable stress (MPa)	Test condition
		Tension (kN)	Residual diameter (mm)	Mean tensile stress (MPa)
SNN170	196.5	22.54	10.4	194
PHC110	241.4	19.60	8.7	240
PHC130	241.4	24.50	10.2	240
CPS130	241.4	22.54	9.5	240

The S-N curves of each contact wire obtained from the fatigue test results are shown in Fig. 6 to Fig. 9. Table 4 shows the standard deviations obtained from each S-N curve and the strain amplitudes corresponding to failure probabilities of 50% and 0.3%. From Table 4, it can be seen that the allowable strain value of all high-strength contact wires is larger than the conventional allowable value of 500 × 10⁻⁶.

Fig. 6　S-N curve of SNN170

Fig. 7　S-N curve of PHC110

Fig. 8　S-N curve of PHC130

Fig. 9　S-N curve of CPS130

Table 4　Standard deviation of S-N curve, strain amplitude with 50% and 0.3% probability of failure

Contact wire type	Mean tensile stress＊ (MPa)	Standard deviation $\widehat{\sigma }$×10−6	Strain amplitude with 50% probability of failure $\widehat{S}$N×10−6	Strain amplitude with 0.3% probability of failure ×10−6 ($\widehat{S}$N−2.75$\widehat{\sigma }$) (i.e. allowable strain value)
SNN170	194	63	1107	932
PHC110	240	74	1263	1061
PHC130	240	85	1016	781
CPS130	240	99	1247	975

^＊These data are at worn limit condition.

5. Allowable strain for different mean tensile stresses

5.1 Method of calculating allowable strain when mean tensile stress is different

In the previous chapter, we showed how to obtain the minimum allowable strain value in practical use by setting the mean tensile stress condition of the contact wire equivalent to the wear limit. However, in reality, contact wires are rarely used to their wear limit for safety reasons, and are often replaced with some margin before they reach their wear limit. As the residual diameter increases, the mean tensile stress decreases, so the allowable strain should also increase. In this chapter, we consider how to calculate the allowable strain of a contact wire when the mean tensile stress conditions are different.

Reference [4] shows that fatigue test results with different mean stresses fall within a single variation band when organized by (maximum stress × stress amplitude)^1/2. It has also been reported that the equation proposed in reference [4] is applicable to hard-drawn copper contact wires [5]. Therefore, we convert the strain amplitudes in the fatigue test results of the high-strength contact wires mentioned above to strain amplitudes when the mean tensile stress is different using the above equation. Hereafter, we will add ' (dash) to the symbol after changing the mean tensile stress. The following formula is obtained:

  

$${\left\{\left({\sigma }_m+{\sigma }_a\right){\sigma }_a\right\}}^{1/2}={\sigma }_e={\left\{\left({\sigma }_m\text{'}+{\sigma }_a\text{'}\right){\sigma }_a\text{'}\right\}}^{1/2}$$（2）

  

$$\mathrm{\therefore }{\sigma }_a\text{'}=\frac{-{\sigma }_m\text{'}+\sqrt{{\sigma }_m{\text{'}}^2+4{\sigma }_e{}^2}}{2}$$（3）

  

$${\sigma }_m\text{'}=T\text{'}/A\text{'}$$（4）

where σ_m: mean tensile stress;

σ_a: stress amplitude,

σ_e: (maximum stress × stress amplitude)^1/2, that is,

{(σ_m+ σ_a)σ_a}^1/2,

T: tension; and,

A: cross-sectional area of contact wire.

Furthermore, by calculating σ_a′ using (2) to (4), a strain amplitude after mean tensile stress conversion can be calculated using the following equation:

  

$${\epsilon }_a\text{'}={\sigma }_a\text{'}/E$$（5）

where ε_a: strain amplitude; and

E: Young's modulus of contact wire.

(Note that in section 5.2 and 5.3, E = 117.6 GPa.)

5.2 When residual diameter is different

First, we consider the case where the tension T of contact wire is constant and the residual diameter (i.e., A) is different. As an example, we use the test results of CPS130 in Fig. 9 and assume that the residual diameter is set before the mean tensile stress conversion to 9.5 mm (wear limit) and after the conversion to 12.0 mm. Figure 10 shows a comparison of the S-N curves before and after the conversion. To avoid complication, only the dots of the slope data are shown in Fig. 10. This figure shows that the larger the residual diameter, the larger the strain amplitude for the same number of cycles. Furthermore, comparing the allowable strain values showed that the allowable strain value increased by approximately 120 × 10⁻⁶ due to this conversion. Figure 11 shows the strain amplitude corresponding to a 0.3% probability of failure (i.e., allowable strain value) when the residual diameter is changed from 9.5 mm to 13.38 mm (new wire). From Fig. 11, it is possible to set the allowable strain value corresponding to the residual diameter of the replacement standard set by each company.

Fig. 10　Comparison before and after residual diameter conversion

Fig. 11　Relation between residual diameter and strain amplitude with 0.3% probability of failure

5.3 When tension is different

Next, we consider the case where the residual diameter of the contact wire is constant and the tension T is different. As in the previous section, the test results of CPS130 were used, and the tension was set to 22.54 kN before the conversion of the mean tensile stress, and to 14.7 kN after the conversion. Figure 12 shows a comparison of the S-N curves before and after conversion. This figure shows that the lower the tension, the larger the strain amplitude for the same number of cycles, and the difference is approximately 200 × 10⁻⁶. Figure 13 shows the allowable strain value when the tension is changed from 9.8 kN to 22.54 kN. Using Fig. 13, it is possible to set the allowable strain value corresponding to the tension of the contact wire.

Fig. 12　Comparison before and after tension conversion

Fig. 13　Relation between tension and strain amplitude with 0.3% probability of failure

6. Conclusions

We quantified the margin of the conventional allowable strain value of contact wires using the probability of failure and proposed a new method for evaluating the fatigue properties of contact wires. We also set allowable strain values for four types of high-strength contact wires using the above method. Furthermore, we investigated a method for calculating the allowable strain value of contact wires when the mean tensile stress is different. The main results are as follows:

・The fatigue test results of GT110 contact wires under tension showed that the conventional allowable strain value 500 × 10⁻⁶ corresponds to the strain amplitude with a probability of failure of about 0.3% at 10⁷ load cycles.

・In the P-S-N curve of a real contact wire under mean tensile stress equivalent to the wear limit, we propose to set the strain amplitude with a 0.3% probability of failure at 10⁷ load cycles as the allowable strain value.

・Using the above method, we propose allowable strain values for four types of high-strength contact wires (Table 4).

・We organize the fatigue test results with different mean tensile stresses by (maximum stress × stress amplitude)^1/2, and present a method for calculating the allowable strain value of a contact wire at any mean tensile stress.

References

• [1]  Railway Technical Research Institute, Manual and explanation for conventional railway operation speed improvement test [Translated from Japanese.], Railway Research-Culture Promotion Foundation, pp. 203-209, 1993 (in Japanese).
• [2]  Tsunemoto, M., Shimizu, M., Irikura, K., Morimoto, H., Yamashita, C., and Ikeda, K., “Development of Simple Catenary System for Operation over 300 km/h,” Quarterly Report of RTRI, Vol. 61, No. 1, pp. 47-53, 2020.
• [3]  Japan Society of Mechanical Engineers, Statistical Fatigue Testing Methods (Revised Edition) [Translated from Japanese.], Japan Society of Mechanical Engineers, 1994 (in Japanese).
• [4]  K. N. Smith, P. Watson, T. H. Topper, “A Stress-Strain Function for the Fatigue of Metals,” Journal of Materials, Vol. 5, No. 4, 1970.
• [5]  Sugahara, A., Yamashita, C., “Evaluation of Factors Which Affects Fatigue Life of Hard-drawn Copper Contact Wire,” RTRI Report, Vol. 20, No. 9, pp. 17-22, 2006 (in Japanese).

Authors

	Takuya OHARA Assistant Senior Researcher, Current Collection Maintenance Laboratory, Power Supply Technology Division Research Areas: Fatigue of Overhead Contact Line, Vibration Engineering
	Chikara YAMASHITA, Ph.D. Eng. Senior Chief Researcher, Head of Current Collection Maintenance Laboratory, Power Supply Technology Division Research Areas: Tribology under Current Flowing Condition, Fatigue of Overhead Contact Line