Load identification and fatigue evaluation via wind-induced attitude decoupling of railway catenary

2.1 FE model description

In FEA with ABAQUS/CAE commercial software, a local coordinate system is created at the registration point, where the x-axis is along the lateral direction, y-axis is along the vertical direction, and z-axis is along the railway line. Only half of the suspension span (20 m) is modeled centering about the catenary support pole because of its geometric and loading symmetry. Symmetric constraint is applied at the two ends of catenary suspension, in which the displacement along z-axis and rotation about x-axis and y-axis are set to zero. To make a general solution between the environmental wind load, nodal displacement, and nodal force, all catenary components are modeled with BEAM elements. Mechanical parameters of catenary components are listed in Table 1.

Before FE modeling of the overall railway catenary system, the initial shape of the suspension is the first step to ascertain. The catenary system tested on-site has no hanging invasions because of the high bridge, and no ice coating because of the dry climate. The dead weight of suspension is only considered to determine its initial shape. In railway catenary suspension, the pre-sag of contact wire is compensated by the lift forces from the messenger wire aided with droppers. In FEA, the initial configuration of catenary suspension greatly affects the accuracy of its numerical model. The solution for catenary equilibrium state is to find the initial suspension shape according to the specific design parameter [45]. The separation method is proposed here by decomposing the catenary suspension into the messenger wire and dropper system, as well as the contact wire and dropper system, to determine the pre-sag of the messenger wire and the dropper lengths, as shown in Figure 5.

Figure 5

Equilibrium state calculation by separate model method.

In mechanical equilibrium of catenary suspension, the catenary wire is segmented by droppers. Each segment is supported by the locator or a dropper, which can be regarded as a small segment of free suspended cable, as the dashed hanging line in Figure 5. The support forces on locators and dragging forces on droppers can therefore be calculated by adding the 50% supported weight of each suspension segment at both sides, when all connections between droppers and catenary wire are in level position. Therefore, dragging forces from droppers in catenary wire-dropper system are calculated in equation (1) as follows:

where T _S is the dragging force from the first and last dropper at two ends, T _M is the dragging force from droppers in the middle, L _S is the distance between locator and its adjacent dropper (L _S = 5,000 mm), L _M is the distance between droppers at middle (L _M = 6,000 mm), and g _c is the weight of catenary wire per unit length (g _c = 13.271 N/m). Accordingly, the dragging forces from all droppers are T _S = 73.0 N, T _M = 79.6 N.

In messenger wire-dropper system, the shape of single messenger wire is numerically simulated with the gravity force acting evenly along the whole span, and the concentrating dragging forces from all droppers at connection locations underneath. In FEA, the pretension force of catenary wire and messenger wire are denoted by T _CW (25 kN) and T _MW (25 kN), respectively. The weight of messenger wire per unit length is denoted by g _m (13.145 N/m). The messenger wire is meshed into 400 BEAM elements, pre-stress of which is defined as 166.7 MPa to represent the 25 kN-pretension in wire section area both of 120 mm². After simulation, nodal displacement along the whole span in 40 m is depicted in Figure 6.

Figure 6

Nodal displacement of messenger wire after shape finding.

The FE model of messenger wire is defined by creating nodes with same coordinates as its simulated configuration in shape finding. Catenary wire is modeled with meshed nodes in level. Beam elements are modeled between nodes in pretension stress of 166.7 MPa for both messenger wire and catenary wire. Droppers are created as connection between catenary wire and messenger wire. After shape finding, gravity no longer acts on both wires to maintain the suspension equilibrium state as in shape finding.

Apart from catenary suspension, support structures including the locator, cantilever beam, insulator, and support pole are modeled with simplified beam elements in half span (20 m) at two sides considering its geometrical symmetry, as shown in Figure 7. The nodes at the span ends are set in symmetric constraint on the translation freedom along z-axis, the rotation freedom about x-axis and y-axis. The spring element is modeled between the positioning tube and cantilever beam in stiffness of 7.67 N/mm, which is calibrated in experiment.

Figure 7

Catenary components and positioning spring modeling.

2.2 Wind load identification

Cross wind is the prevailing wind along railway line in Xinjiang, according to the on-site wind test results at different heights and locations. The wind load F _W is considered as acting in a 2D plane which is perpendicular to the direction along railway line. In this plane, the wind load can be decomposed into lateral and vertical forces (respectively denoted as F _WL and F _WV), acting on suspension wires, cantilever beams, and support structures. For each wind load, there is a 2D displacement at registration point because of structural deformation under environmental wind load and the load transmitted from catenary suspension. Therefore, the practical registration displacement under coupled wind load can be decoupled into 2D displacements under individual wind loading, as shown in Figure 8. Theoretical formulation between suspension attitude and wind load follows this decoupling rule as stated.

Figure 8

Wind loading decoupling rule on suspension attitude.

In lateral wind loading, the wind-induced displacement at registration point is denoted as X _WL in lateral direction and Y _WL in vertical direction, respectively. Mechanical relation between lateral wind load and 2D displacement is described in equation (2) as follows:

where f _WL() and g _WL() are the mathematical function that relates lateral force with lateral and vertical registration displacement, respectively, which are acquired by data fitting from FEA simulations in pure lateral wind loading.

In vertical wind loading, resulting displacement at registration point is denoted by X _WV in lateral direction and Y _WV in vertical direction, respectively. Mechanical relation between vertical load and 2D displacement is described in equation (3) as follows:

where f _WV() and g _WV() are the mechanical function that respectively relates vertical force with lateral and vertical registration displacement, acquired by data fitting from FEA simulations in pure vertical wind loading.

In coupled loading, the environmental wind load is decomposed into two perpendicular wind loads. Accordingly, the resulting registration displacement can be calculated as the blend of displacement induced by lateral and vertical loading, respectively. Therefore, the registration displacement under environmental wind load F _W is given in equation (4) as follows:

where X _W and Y _W are the resulting lateral and vertical displacement at registration point under coupled wind loading, respectively.

Load identification is the solution of lateral wind load F _WL and vertical wind load F _WV from the system of two element equation as (4). In FEA simulation, environmental wind load is defined with 12 load cases in the range of the historical maximum wind speed, 64 m/s, to obtain adequate fitting data from both directions. As the wind load magnitude differs from catenary components because of their different windage area, predefined wind load is described in the wind pressure other than the wind force, so that the load unit is unified on all catenary components. In calculation, wind pressure and wind force are given in equations (5) and (6), respectively, as follows:

where Q _W is the wind pressure, ρ is the air density (1.2 kg/m³ in normal temperature and pressure), and v _W is the wind speed in equation (5). C _w is the aerodynamic shape factor of catenary components (1.1 for suspension wires, 0.9 for tubular support structure [46]), and A _w is the projected area of the component in equation (6).

In different load cases, calculation results of nodal displacement at registration point are listed with wind pressure, wind load, and wind speed in Table 2.

Mathematical function is formulated by fitting the registration displacement and wind pressure in two directions as shown in Figure 9 and equation (7). The correlations between lateral pressure and 2D registration displacement, vertical pressure and vertical registration displacement follow a linear relationship. while there is a second-order polynomial relation between vertical pressure and lateral registration offset.

where C _XWL is the fitting coefficient between lateral offset and lateral wind pressure (C _XWL = 1.2613 × 10⁻² mm/Pa), C _XWV1 and C _XWV2 are the coefficients between lateral offset and vertical wind pressure (C _XWV1 = –2.0755 × 10⁻⁶ mm/Pa², C _XWV2 = 4.5273 × 10⁻³ mm/Pa), C _YWL is the coefficient between vertical offset and lateral wind pressure (C _YWL = 1.3669 × 10⁻³ mm/Pa), and C _YWV is the coefficient between vertical offset and vertical wind pressure (C _YWV = 6.9964 × 10⁻² mm/Pa).

Figure 9

Fitting curve between 2D nodal displacement and wind pressure.

To explain, the mechanical model of the locator can be simplified as a beam model with one hinged end O, one free end with vertical wind load F _WV, and the spring attached point where a resistance force F _SPR works in balance, as shown in Figure 10. When the locator rotates under vertical loading, the circle trajectory of the registration point gives an inflection which is defined as “Extreme” and the reference location to describe the lateral offset, when dX _WV/dF _WV = 0, that is F _WV = – C _XWV2/(2C _XWV1) = 1090.65 Pa. If the vertical load is smaller or greater than that critical value, the lateral offset will not surpass the “Extreme” regardless of the elastic deformation of the locator, such as locations and depicted both in Figures 9 and 10.

Figure 10

Mechanical diagram of the locator.

Therefore, the movement at registration point is composed of displacements induced by lateral and vertical wind load, which are given as:

where X _W and Y _W are the lateral and vertical resultant registration displacement, respectively.

The two groups of wind load solution can be analyzed through a parabolic equation acquired from equation (8) in equation (9) as below.

where the constant A = C _XWV1·C _YWL, B = C _XWV2·C _YWL – C _XWL·C _YWV, and C = C _XWL· Y _W – C _YWL· X _W .

As the lateral and vertical wind load varies within ± 2.5 kPa in scattering interval of 100 Pa, in a same description of the wind speed ranging within 63.3 m/s, the resultant registration offset X _W in lateral and Y _W in vertical are calculated after equation (8) and plotted in Figure 11(a) and (b). The maximum and minimum offset at registration point are, respectively, 55.82 and −34.00 mm in lateral direction and 178.33 mm and −178.33 mm in vertical direction. The components A and B in equation (9) remain constant, while component C changes with different resultant registration offset X _W and Y _W. Figure 11(c) gives the 3D plot of component C in a range from −2.208 to 2.173 mm²/Pa, which is only related to vertical wind load F _WV.

Figure 11

3D plot under wind loading limits. (a) Plot of lateral offset X _W; (b) plot of vertical offset Y _W; (c) plot of equation component C .

In the range of variable equation component C and constant component A , B as in equation (9), the parabolic curve is shown in Figure 12, where the intersections between the curve and the axis y = 0 indicate the double roots. Apparently, the root on the left side varies in a reasonable range of ±2.5 kPa, while the rightward root reveals an impossible solution as identified load.

Figure 12

Root selection of vertical wind load identification.

Accordingly, the wind load identification can be re-formulated in equation (10) by abandoning the impossible root at the right side.

Validation undergoes in numerical simulation. To cover possible loading scenarios, 10 load cases of coupled wind loading are randomly generated in the range of practical scenarios. The lateral and vertical wind loads vary from −2,500 Pa to 2,500 Pa. Validation load cases are illustrated in Table 3.

In each case, lateral and vertical displacements at registration point are read and substituted into the load identification model. The identified wind loads (denoted as “ID”) are compared with the given loads (denoted as “GV”). Relative deviation is calculated by |“ID” − “GV”|/“GV” × 100% and listed in Table 4.

Comparison reveals that the deviation (“ID” − “GV”) between identified and given loads varies within −3 to 4 Pa. The relative deviation is within 0.59 and 3.39% for vertical and lateral wind load, respectively. Comparison suggests good agreement between the proposed method and numerical simulation.

2.3 Structural stress calculation

On the beam-modeled catenary support structure, the nodal forces at connections are defined in two directions in the local coordinate system other than the global system. In the local system, if the nodal force is perpendicular to the beam which is embraced by the connection component, it is defined as the axial nodal force. If along the beam, the nodal force is described as the shear force. From the previous section, the wind loads in the global coordinate system are identified by the registration displacement which can be measured via visual detection. The nodal forces at each connection can thereby be given in relation with wind loads as below.

where R _S ⁽ⁱ⁾ and R _A ⁽ⁱ⁾ are the shear and axial reaction force at component connection i# (i = 1–9), the numbering and name of which are shown in Figure 13. H _WL and H _WV are the functions between nodal shear force and lateral wind load, vertical wind load, respectively. J _WL and J _WV are the functions between nodal axial force and lateral wind load, vertical wind load, respectively. Above functions suggest a linear relationship between wind loads and nodal forces.

Figure 13

Connection components numbering and precise FE modeling.

Therefore, nodal axial and shear force calculation can be reformulated in equation (12) as below in terms of coefficient matrix.

where C _{S_WL} ⁽ⁱ⁾ and C _{S_WV} ⁽ⁱ⁾ are the transfer coefficient from lateral wind pressure, vertical wind pressure to nodal shear force for component i#, respectively. C _{A_WL} ⁽ⁱ⁾ and C _{A_WV} ⁽ⁱ⁾ are the transfer coefficient from lateral wind pressure and vertical wind pressure to nodal axial force for component i#, respectively.

Reading nodal forces at the nine connection nodes in the same load cases as set in wind load identification, the mechanical relation between nodal forces and global wind pressure can be acquired in Table 5, where the four coefficients are accordingly listed as four columns. Connection components’ numbering and detailed FE model are shown in Figure 13.

Table 5

Correlation between nodal force and wind pressure (Unit: N/Pa)

Component ID	Nodal shear force R S (i)		Nodal axial force R A (i)
	Lateral wind load C S_WL	Vertical wind load C S_WV	Lateral wind load C A_WL	Vertical wind load C A_WV
	0.1324	0.4636	0.4942	−0.1242
	0.0000	0.4849	0.5117	0.0000
	0.0852	1.3123	0.2783	2.6196
	−0.0852	0.1764	0.7451	−2.7321
	0.1865	−0.1465	0.1639	−0.0735
	0.1865	−0.1465	0.1639	−0.0735
	−0.0988	−0.5500	0.4856	−0.8638
	0.0261	0.8638	−0.0988	−1.0300
	0.0179	−0.4797	−0.5114	−0.0168

For each connection component, the structural stress is computed in individual model with precise mesh. Intentions include the strength check and fatigue damage assessment of the initial component structure, as well as the component after structural optimization under the same given nodal forces. Therefore, further calculation from nodal forces at connections to the structural stress of the full-size component can be given in equation (13) as below.

where C _{xx_S} ⁽ⁱ⁾, C _{yy_S} ⁽ⁱ⁾, C _{zz_S} ⁽ⁱ⁾, C _{xy_S} ⁽ⁱ⁾, C _{yz_S} ⁽ⁱ⁾, and C _{xz_S} ⁽ⁱ⁾ are the transfer coefficient from nodal shear force to the 6-component stress. C _{xx_A} ⁽ⁱ⁾, C _{yy_A} ⁽ⁱ⁾, C _{zz_A} ⁽ⁱ⁾, C _{xy_A} ⁽ⁱ⁾, C _{yz_A} ⁽ⁱ⁾, and C _{xz_A} ⁽ⁱ⁾ are the transfer coefficient from nodal axial force to the 6-component stress. The structural stress is only the nodal force-induced stress and excludes the structural pre-stress because of bolt pre-tightening.

The FE modeling of connection components can be categorized into four kinds, the contact clamp (component ), messenger clamp (component ), support base (components and ), and bushing ears (components –). The meshed FE models, their boundary conditions, and loading locations are illustrated in Figure 14.

Figure 14

Detailed FE model of connection components. (a) Contact clamp; (b) messenger clamp; (c) support base; (d) bushing ears.

In the mesh model, linear hexahedral and quadratic tetrahedral elements are adopted to fit the complex components configuration. The mesh size is set to 2 mm to enhance simulation accuracy. Material properties of the connection components are as follows: elastic modulus E = 206 GPa, Poisson’s ratio υ = 0.3, density ρ = 7.85 g/cm³. For each connection component, the mesh properties, boundary constraint, and loading areas are defined as follows:

1. Contact clamp: FE model consists of 225,360 elements and 252,763 nodes. The hollow cylinder at the top is connected with the locator, the inner surface is therefore set in radial constraint and the bottom is vertically constrained. Axial and shear loading are both applied on the clamp area at bottom. The pretension is set to 12.5 kN on the two bolts to simulate pre-tightening as shown in Figure 14(a).

2. Messenger clamp: The model has 822,306 elements and 306,486 nodes. The bottom clamp links the top cantilever beam through two bolts, the inner face and bolt holes are set in radial constraint. Nodal axial and shear forces are applied on the inner face of top clamps which embraces the messenger wire. The four tightening bolts are in 18.3 kN-pretension as shown in Figure 14(b).

3. Support base: The mesh is comprised of 580,891 elements and 443,666 nodes. The left hoop embraces the support pole through double long bolts in pretension of 18.3 kN. The inner face of the hoop is set in radial constraint and vertical translation restriction. The nodal axial force is applied on the connecting bolt, which is in pretension of 18.3 kN as well. Nodal shear force is applied on its neighboring level planes as shown in Figure 14(c).

4. Bushing ears: There are 257,210 elements and 373,719 nodes in FE model. The hinged hoop attaches the cantilever beam through two bolts in pretension of 12.5 kN. The inner hoop face is set in radial constraint and the translation restriction along the embraced beam. Nodal axial and shear force are applied on the bolt connecting to the other beam. The pretension of the bolt is set in 18.3 kN as shown in Figure 14(d).

To cover all possible nodal forces under the lateral and vertical wind load range in ±2.5 kPa, wind load scattering is generated in an interval of 100 Pa to combine 2D wind loads, and nodal axial and shear forces are calculated according to equation (12) on those scattering wind loads. After that, the nodal force range for each connection component is confirmed based on the minimum and maximum value on above scatterings, as listed in Table 6. However in stress calculation, the acting nodal forces on connection components are not the exact calculated nodal force range. All nodal force calculations are rounded up to the closest outer bound in unit of 50 N, as the nodal force input in stress calculation simulation, which is denoted as “FE range” in Table 6. For each kind of component, the nodal force range in FE simulation starts from the minimum and ends at the maximum extreme, because what the structural stress correlates with the nodal axial and shear force for each component kind is the same even at different connections.

Table 6

Nodal force range and FE load magnitude definition (Unit: N)

Component type	Component ID	Nodal shear force R S (i)		Nodal axial force R A (i)
		Calculation range	FE range	Calculation range	FE range
Contact clamp		±1490.0	±1,500	±1546.0	±1,550
Messenger clamp		±609.7	±610	±817.4	±850
Support base		±1569.4	±1,600	±3051.7	±3,100
		±324.0		±1343.3
Bushing ears		±454.1	±1,100	±340.2	±1,700
		±454.1		±340.2
		±625.7		±1696.3
		±1061.5		±1205.0
		±620.2		±799.0

For each component type, nodal force range in FE simulation is segmented into seven uniformly spaced scatters. In pure axial loading and shear loading, six-component stress of parent metal and bolts at stress concentration locations are read and fitted with nodal forces in distinct magnitude. A linear relationship is observed between stress and nodal forces, and the transfer coefficients are listed in Table 7 for nodal axial force and shear force, respectively. For connection components, parent metal and bolts are individually analyzed because of their different material properties. The initial stress of connection components is also given in Table 7 because of bolts pre-tightening. The contour plot of von Mises stress of four component categories is illustrated in Figure 15.

Figure 15

Stress concentration contour plot of the four connection components. (a) Bushing ears under axial force; (b) bushing ears under shear force; (c) support base under axial force; (d) support base under shear force; (e) messenger clamp under axial force; (f) messenger clamp under shear force; (g) contact clamp under axial force; (h) contact clamp under shear force.

As revealed above, stress concentrating region mainly occurs around the bolts and installing hole, no matter whether it is of axial or shear loading. This is, on one hand, because of the bolt pre-tightening and, on the other hand, the axial and shear forces transmission via bolt connections. Taking account into the higher stress and weaker material properties of the parentmetal on components, bolts are no longer considered in the following analysis.

In fatigue stress calculation, the standard von Mises stress is calculated as equation (14).

The standard calculation is generally a poor choice for fatigue evaluation. The range acquired from the strictly positive history of the von Mises stress does not include any potentially negative part of the stress cycle. The signed von Mises stress (SVM), which can take the negative values, is thus given by equation (15) as the stress calculation rule.

where I ₁ ⁽ⁱ⁾ is the first stress invariant, I ₁ ⁽ⁱ⁾ = σ _x ⁽ⁱ⁾ + σ _y ⁽ⁱ⁾ + σ _z ⁽ⁱ⁾.

3.1 Visual detection model

Wind test of catenary suspension was carried out at one end of Sangequan Grand Bridge in Nanjiang railway line, where frequent pantograph operation faults were reported because of vertical wind-induced uplift to catenary wire.

To get the best viewpoint and illumination conditions, area-array CCD cameras were installed between cantilever mounting bases on catenary support pole. The camera is DS-2CD5A24FWD-IZH, manufactured by HIKVISION. Image sampling rate is set to 24 image frames per second, which surely covers low frequency characteristic requirement of wind-induced vibration of catenary suspension (mostly 1.2–1.8 Hz) [47]. Infrared light source was set along with the camera during nighttime to avoid the risk of mistaking visible lighting source as the train operation signal light. At the registration point, reflective coating is used to enhance the visibility of dynamic detecting target, the motion of which is observed by the camera in terms of image pixel offset. The mileage plate serves as the static target to represent the wind-induced camera vibration. On one hand, image of the plate demonstrates a signboard with numbers on it at daytime, and a white light-reflecting plate at nighttime. On the other hand, the plate is regarded as a static point even in the strongest wind because of large rigidity of the steel portal structure, as shown in Figure 16.

Figure 16

Wind-induced attitude recognition scheme.

The imaging model between actual and image pixel offset of observing target is given as equation (16).

where {ΔX, ΔY} and {Δx, Δy} denote the 2D actual and pixel offset of the target, respectively. L _R and L _P denote the actual and pixel size of the target, respectively. K _C represents the self-calibration coefficient, which values 2.7 mm/pixel at the registration point.

To enhance detection efficiency, image processing is performed in a subregion where the registration point movement does not exceed the transverse and vertical boundary even under the strongest wind. A rectangle target template is predefined to compare with the matching area transversely and vertically by pixels, as shown in Figure 17. The matching difference between target template and matching area is formulated as equation (17).

where G _OBJ(i,j) and G _IMG(i,j) denote the gray-scale value of pixel point at coordinate (i,j) in target template and image matching area, respectively. x _LIM and y _LIM denote the pixel length and width of the target template. X _LIM and Y _LIM denote the pixel length and width of image processing sub-region. Parameters k and l represent the pixel coordinate of the present matching area in vertical and transverse direction, which are calculated as in equation (18).

Figure 17

Target template matching and 3D contour plot of matching difference [47].

By calculating the matching difference between matching area and target template of registration point, the 3D contour map is illustrated in Figure 17. The matching difference peak reaches the minimum when the target template coincides with the matching area, which indicates the pixel location of identified registration point target.

Pixel coordinates of the identified target are denoted as (k ₀, l ₀). The modified target pixel coordinates are then corrected with target template size considered, as in equation (19).

In detection, the gray feature of pixels in target area varies because of illumination changes from day to night. Therefore, if the target template remains constant as the initial template, it will fail to match the actual target in the present image. To better fit in complex illumination, the target template is updated by the recognized target at each processing image frame.

In addition, a threshold for matching difference to tell whether the matching area is the target is defined as T. The threshold T is determined by the maximum of all minimum matching differences after processing all image frames over 24-h period. The optimal threshold T, which values 5.0, is substituted into all image frames to recognize the target and calculate the deviation between recognized target location and actual position. Calculation indicates that the maximum deviation is within only 2.7 mm in engineering dimension, which proves that the optimal threshold contributes to the best recognition results both in accuracy and efficiency.

Strong wind gives rise to camera vibration on the support pole and brings about the false displacement of dynamic target detected in the camera. To eliminate the detection error, a static target is set via the mileage plate on the steel portal structure in the background. The mileage plate remains stationary in the most unfavorable wind environment because of the large rigidity of steel portal structure. Moreover, imaging of the plate presents a numbered signboard during daytime and a white rectangle plate during nighttime, which can be easily captured by the camera. Through detection, the transverse and vertical false displacements of static target are denoted by Δx _{STA_i} and Δy _{STA_i} , respectively. The transverse and vertical displacements of dynamic target, the registration point, are denoted by Δx _{DYN_i} and Δy _{DYN_i} , respectively. Accordingly, the absolute displacements of dynamic target referred to the background are given in equation (20).

where ΔX _{MOV_i} and ΔY _{MOV_i} indicate the absolute transverse and vertical displacement of registration point in image i# (unit in mm), respectively.

The detection precision of presented method depends on the pixel size and resolution of the camera. In this work, the resolution of the detecting camera is 1,920 × 1,080, the target area of registration point is 53 × 31 pixels, describing the real size of 50 mm × 29 mm. Therefore, the minimum motion that can be detected by the visual detection system is 50 mm/53 pixels = 0.94 mm for each pixel. Considering the relative deviation 1.8% of the visual detection stated in ref. [47], the recognition capacity and deviation determines the minimum motion, that is, 0.94 mm × (1 + 1.8%) = 1.02 mm.

3.2 Results and discussion

In wind test of one catenary span at Sangequan Grand Bridge in Nanjiang railway line, the environmental wind speed was tested by anemometer, which was set 50 m away from the railway bridge perpendicularly.

To get a more clear view of the wind changes, the environmental wind speed data were averaged per 10 min. The experimental data, which includes the wind speed and 2D suspension displacement, are selected lasting 24 h from 5:00 on September 24, 2017. The time-history curve of the transverse and vertical displacement to the 10-min mean wind speed is shown in Figure 18.

Figure 18

Transverse and vertical absolute displacement to wind speed curve.

From the experiment, it is observed that the wind speed climbs up to 41 m/s from 16 m/s, and then drops to 14 m/s. The magnitude of wind-induced attitude of registration point shows a similar trend to that in wind speed. The maximum vertical target displacement reaches 86.7 mm and −21.3 mm, which is 6.7 times greater than that transversely (12.9 and −17.7 mm). It is explained that the vertical target movement is affected by the vertical spring at the registration root, while the transverse target movement is mainly determined by the rigidity and elasticity of the registration tube.

4.1 Load spectra development

The load spectra used in catenary design for service life and damage tolerance assessment are based on a predicted average usage. It is important for catenary safety and maintenance planning reasons to acquire the real load spectra because the inspection intervals are based on the predicted spectra. In this work, the procedure in use to handle the tracking of real service loads is making use of the recorded wind-induced registration attitudes and the theoretical model to identify the loads indirectly. Peak values of the nodal forces and SVM stress at 9 connection components are listed in Table 9. Time-history of the greatest in each component type is given according to the recorded displacement of catenary registration point in Figure 19.

Table 9

Peak values of nodal forces and structural stress at connection components

Component type	Component ID	Nodal shear force (N)		Nodal axial force (N)		SVM stress (MPa)
		Max	Min	Max	Min	Max	Min
Contact clamp		452.57	−1028.11	1843.80	−973.82	281.88	29.68
Messenger clamp		511.46	−817.373	609.68	−150.85	278.45	104.49
Support base		1569.41	−384.15	3051.70	−862.67	263.93	114.05
		324.00	−118.81	1343.28	−4264.80	153.73	114.23
Bushing ears		200.92	−454.07	166.34	−340.15	158.76	113.79
		200.92	−454.07	166.34	−340.15	158.76	113.79
		216.88	−625.74	642.10	−1696.26	201.71	116.26
		1061.46	−260.21	329.83	−1205.01	262.99	134.64
		155.10	−620.16	799.03	−511.99	161.77	110.84

Note: “SVM” is short for “Signed Von Mises.”

Figure 19

Time-history of nodal forces and SVM stress of connection components. (a) Nodal forces at contact clamp; (b) SVM at contact clamp; (c) nodal forces at messenger clamp; (d) SVM at messenger clamp; (e) nodal forces at support base (bottom); (f) SVM at support base (bottom); (g) nodal forces at bushing ears (top frame anchor); (h) SVM at bushing ears (top frame anchor).

To calculate the total fatigue damage, time domain-based methods are often used by applying deterministic methods, such as the rain-flow counting algorithm. Rain-flow counting is easier to use but requires higher computational or experimental effort, and the approach that is most widely used in different fields and standard codes, as well as part of the framework of classic fatigue theory. Its outstanding feature is that a series of closed stress-strain hysteresis loops are determined according to the nonlinear relationship between the measured material stress and strain. The actual measured load history is simplified into several load cycles, and the stress load spectra are compiled for the subsequent estimation of fatigue life.

In rain-flow counting, the load cycles counted from the recorded real service load data are grouped according to the limit of mean and amplitude of each load cycle when 2D load spectra are concerned. If all load cycles are grouped into n grades, the interval between adjacent grades is calculated as equation (24):

where G _M and G _A are the interval between adjacent grade of mean and amplitude load cycles, respectively. S _{M_MAX} and S _{M_MIN} are the maximum and minimum of mean load among all load cycles, respectively. S _{A_MAX} and S _{A_MIN} are the maximum and minimum of amplitude load among all load cycles, respectively.

Thus, the boundaries for the ith grade load spectra are calculated in equation (25) as follows:

where S _M(i)LOW and S _M(i)UP are the lower and upper limits of the ith grade load spectra for mean load cycles, respectively. S _A(i)LOW and S _A(i)UP are the lower and upper limits of the ith grade load spectra for amplitude load cycles, respectively.

The rain-flow counting will statistically count the load cycles depending on which grade of the load spectra they belong to, both in mean and amplitude load cycles. To find the best group number in balance with calculation efficiency and precision, 2D fatigue damage which considers both mean and amplitude stress is computed in distinct groups from 2, 4, 8, 16, 32, 64, 128, 256, to 512 grades. Taking into account the most unfavorable circumstance, 1D load spectra are not involved because the mean stress remains positive for all components and therefore contributes to the equivalent cyclic amplitude stress and fatigue damage. Resulting damage of the experimental test is transformed into base-10 logarithm to fit the plot range in challenge of damage differences, respectively, for each component kind with maximum stress at selected locations, as seen in Figure 20.

Figure 20

Fatigue damage convergence with load spectra group number.

Results reveal that from 32 or more groups, fatigue damage remains unchanged for contact clamp, messenger clamp, and bushing ears. However for support base, the convergence starts from the 64 groups. It is recommended that 64-grade load spectra are adopted in wind-induced fatigue assessment for railway catenary system. In the following sections, all evaluations go in 64-grade to meet calculation precision and efficiency.

After stress transformation and spectra developing, 2D SVM stress spectra are plotted in 64 grades respectively for contact clamp, messenger clamp, bushing ears, and support base, as shown in Figure 21.

Figure 21

2D load spectra of SVM stress in 64 grades. (a) Contact clamp; (b) messenger clamp; (c) support base at bottom; (d) bushing ears at top frame anchor.

Above spectra indicates that the stress amplitude climbs up to 115, 87, 75, and 64 MPa, respectively, for contact clamp, messenger clamp, support base, and bushing ears. However, load cycles concentrate in the amplitude stress range of 0–15 MPa, which accounts for 63.9% of the total cycle counts in mean stress range of 129–187 MPa for the contact clamp, and 71.12% of the total counts in mean stress range of 147–201 MPa for the messenger clamp. For support base and bushing ears, the spectra concentration deviates to a lower mean stress region of 122–156 MPa and 136–166 MPa, and contributes 65.8 and 78.0% of the total cycle counts in stress amplitude range of 1–10 MPa, respectively. Spectra distribution difference can be explained with the initial stress because of different pre-tightening of connecting bolts.

4.2 Fatigue damage evaluation

In calculation of fatigue damage from the 64-grade SVM stress spectra in Figure 21, the sum of damage in the 24 h wind test for four connection components is listed in Table 10. The results are given in terms of 50 and 99.9% SR, 1D and 2D load spectra.

It is seen that the contact clamp suffers the largest damage, followed by the messenger clamp, support base, and bushing ears. It is also revealed that the 2D damage is more than two times of 1D damage, because the positive mean SVM stress contributes to the amplitude of transferred cyclic stress.

According to the statistics of the fatigue damage of the four connection components, the contour map can be obtained as shown in Figure 22.

Figure 22

64-grade fatigue damage spectra of connection components. (a) Contact clamp in 50% SR; (b) contact clamp in 99.99% SR; (c) messenger clamp in 50% SR; (d) messenger clamp in 99.99% SR; (e) support base in 50% SR; (f) support base in 99.99% SR; (g) bushing ears in 50% SR; (h) bushing ears in 99.99% SR.

Unlike the VM stress spectra, amplitude stress from 0 to 30 MPa of the contact clamp, which accounts for 93.7% of all load cycles, contributes only 1.7% of the total fatigue damage. However, amplitude stress from 30 to 115 MPa gives 98.3% contribution of the total damage. Amplitude stress ranges of 55–87 MPa, 47–75 MPa, and 44–64 MPa account for 61, 49, and 68% of the total damage, respectively, for messenger clamp, bottom support base, and bushing ears in 99.9% SR.

To extend the fatigue damage in the test to the designed life of catenary components, meteorological data of wind days (>20.4 m/s) from 2015 to 2020 are statistically collected from wind monitoring station at Sangequan Grand Bridge, in divisions from the 8th wind grade (v _w8 = 19.0 m/s) to the 17th (v _w14 = 43.8 m/s). To make it equivalent in wind load magnitude, the maximum wind speed tested in field experiment is defined as v _wts = 41 m/s, statistically counted in wind monitoring station are denoted as v _w8 = 19.0 m/s, v _w9 = 22.6 m/s, v _w10 = 26.5 m/s, v _w11 = 30.6 m/s, v _w12 = 34.8 m/s, v _w13 = 39.2 m/s, v _w14 = 43.8 m/s, v _w15 = 46.8 m/s, v _w16 = 53.5 m/s, v _w17 = 58.7 m/s, respectively, for the 8th, 9th, to the 17th-grade wind. Wind days for each wind grade are described as N ₈, N ₉, N ₁₀, N ₁₁, N ₁₂, N ₁₃, N ₁₅, N ₁₆, and N ₁₇. Taking the quadratic power relation between wind pressure and wind speed, the equivalent annual wind days is given in equation (26) as below.

Wind day counts categorized by the wind grade from 2015 to 2020 are given in terms of column bar plot as shown in Figure 23. Equivalent wind day calculations based on equation (26) are 178.8, 201.3, 196.2, 205.8, 175.3, and 171.7 days, respectively, from 2015 to 2020. Thus, equivalent wind days in damage extension to 1 year is the average of wind days in the concerning 6 years, which makes 188.2 days per year.

Figure 23

Statistical data of wind days (>20.4 m/s) in the past 6 years (“G” – is short for Grade).

In extension of the fatigue damage from the 24 h experimental calculation to the designed life cycle, the equivalent damage of all components is computed by the multiplying of experimental damage, equivalent annual wind days, and the designed life years. Therefore, the equivalent fatigue damage and estimated life are given in Table 11 for four types of connection components under 50 and 99.9% SR, 1D and 2D stress spectra evaluation.

The rating list of equivalent fatigue damage in descending sequence is the contact clamp (1.07), bottom support base (0.458), bushing ears (0.348), and messenger clamp (0.231) in 99.9% SR. Among them, life estimation of the contact clamp is 9.3 years, which is smaller than the designed life cycle. The estimated life of other components can meet the requirement of their design, but the fatigue evaluation is performed in pure wind vibration. It is thus recommended that the maintenance period of connection components be shortened according to the fatigue behavior because of both pantograph-catenary impacts and wind-induced vibration.