Analysis and Predictive Evaluation of Mechanical Properties of Steel Anchor Box for High-Speed-Railway Cable-Stayed Bridge

Abstract

As the stress amplitude in the anchorage zone of cable-stayed bridges vary considerably, it is essential to study the fatigue-load-bearing capacity of the anchorage structures of cable-stayed-bridge girders. In this study, the mechanical properties and force-transmission mechanism of a built-in double-fixed steel anchor box, as well as the influence of the geometric design parameters of the main pressure plates and support plates on its stress performance, were studied. A PSO-BP built-in double-fixed steel anchor box mechanical-performance-prediction-and-evaluation system was established, with the geometric design parameters of the important plates of the main body of the anchor box as the input variables and the mechanical properties of the main stress-transmitting welds of the steel anchor box as the output-evaluation variables. The results were as follows: the cable force was mainly transmitted in the form of shear force through the welds between the support plate and the web of the main beam; the transmission ratio of the four main welds was generally maintained at about 23%, and the stress distribution of each plate was relatively uniform, with most of the stresses in the range of 10~50 MPa. The shear-stress-inhomogeneity coefficient of the transmission weld was sensitive to the changes in the thickness of the support plate and the pressure plate, and the transmission ratio of the main weld was sensitive to the changes in the thickness and length of the support plate. The PSO-BP-based mechanical-performance-prediction-and-evaluation system can be used to explore the intrinsic relationship between the designed cable force, important plate-geometry parameters, and the force performance of steel anchor boxes during the iterative process, and to generate more accurate prediction and evaluation values.

1. Introduction

A cable-stayed bridge’s main girder and main tower form a stable structure through the cable-stayed cables. The main girders and the loads they carry are transferred to the foundations via the cable-stayed towers. The reliability of the anchorage structure of a cable-stayed bridge is therefore key to the safety of the whole bridge [1,2].

The current Chinese code, “Code for the Design of Steel Structures for Railway Bridges,” TB 10091-2017, only provides details on the construction of the anchorage structure of the cable girder, such as the welds, but does not provide the corresponding design dimensions of the key plates of the built-in double-fixed steel anchor box, so designers generally adopt the empirical method or finite-element trial-calculation method for the structural design of the steel anchor box, which often requires special experiments to verify the reliability of the box. Lin et al. [3] used a special lever arm to perform a full-scale fatigue test on the steel anchor box of an external cable beam, and the test showed that the weld between the steel anchor box of the cable beam and the web of the beam performed well under the applied fatigue load. Moreover, the developed finite element calculation model effectively obtained the general trend of the weld distribution. Shi et al. [4] used a finite element analysis and a full-scale test model to study the fatigue performance of the anchor-plate cable-girder anchorage structure of a large-span-railroad cable-stayed bridge, and the test results showed that the new anchor plate without ribs had a fairly uniformly lower stress distribution near the butt weld compared with the traditional ribbed-tension-anchor plate, and the fatigue performance of this anchorage structure met the design requirements of the bridge within the design’s service life. Ren et al. [5] conducted a full-scale static-load-fatigue-model test on an anchor-plate cable-beam anchorage structure, and studied its structural characteristics, the stress-concentration degree, and the fatigue performance of key structural features. The research showed that the anchor-plate cable-beam anchorage structure had a simple structure, clear force, and convenient maintenance. When the structural parameters were reasonable, it had good fatigue resistance and met the design requirements. Li et al. [6] conducted a fatigue test on a full-scale test model of an externally anchored box-girder anchorage structure, and analyzed the fatigue performance of the structure under cyclic loading on the basis of a theoretical analysis and experimental research results, which showed that the test model was reasonable and that the structure had a large fatigue-resistance safety reserve. Li et al. [7] conducted static-load tests on and a comparative analysis of four common cable-stayed bridge-girder anchorage structures, investigating the force-transfer mechanism, stress distribution, and stress-concentration phenomena of the four configurations and also proposing measures to improve the stress concentration.

Due to the excellent self-learning ability of artificial neural networks [8,9], they are increasingly used in research on steel properties. Petr Opěla et al. [10] established a feed-forward multilayer perceptron (FF-MLP) thermal-flow-stress regression model with steel temperature, strain rate, and strain as the input variables. The results showed that the model can effectively predict the thermal-flow stress of steel. Alexander Churyumov et al. [11] used an artificial neural network (ANN) model to predict the flow stress of high-alloy corrosion-resistant steel during hot deformation. The results showed that the ANN can accurately predict the flow stress of steel in a large range of alloy-element concentrations. Mohamed A. Shaheen et al. [12] developed a method based on an ANN to predict the mechanical properties of high-strength steel (HSS) at high temperature with chemical composition and temperature as the input parameters. The results showed that the model has good predictive ability. Ajeet Singh Rajput et al. [13] used an artificial neural network to develop a method that can quantify and predict the material loss caused by wear. However, the use of artificial-neural-network-based models to predict the anchorage structure of cable beams requires further exploration and discussion.

As mentioned above, the relevant codes do not impose clear requirements on the design dimensions of the plates of cable-girder anchorage structures to ensure their structural safety, and few scholars apply artificial neural networks in their research on the mechanical performances of built-in double-fixed steel anchor boxes. In order to fill this gap, in this study, a 1:2-scale-model fatigue test was conducted on the steel anchor box of the New Yujiang Special Bridge, on the Nanning–Yulin Railway, to study its force characteristics, force-transfer mechanism, and fatigue life, as well as the influence of the design dimensions of the key plates on the structural force performance. This study also established a PSO-BP neural-network-based built-in double-fixed cable-girder steel anchor-box mechanical-performance-prediction-and-evaluation system. The prediction-and-evaluation system is intended to provide a reference for the design of the main plate parameters in code, so that designers can quickly grasp the mechanical properties of the built-in double-fixed steel anchor boxes with different plate-design dimensions in the early stage of design.

2. Mechanical-Performance Analysis of Built-In Double-Fixed Steel Anchor Cable-Girder Box

2.1. Engineering Background

The new Yujiang River Bridge is a double-tower, double-rope surface-steel–concrete hybrid girder-cable-stayed bridge with a mileage range of DK107 + 393.65 to DK108 + 006.35. The span arrangement is 36 m + 40 m + 64 m + 330 m + 64 m + 40 m + 36 m, and the cable-stayed bridge adopts a semi-floating system, with diamond-shaped concrete towers and a main girder with a single three-compartment-box section. An overall fan-shaped symmetrical cable-staying arrangement is used, with 128 cables, 1 every 8 m at the main girders and 1 every 2 m at the towers. The bridge adopts a built-in double-fixed cable-girder anchorage structure, the elevation of which is shown in Figure 1.

Finite element analysis of the steel anchor box of the diagonal cable in the middle of the span showed that the equivalent stress on most of the plates of the steel anchor box corresponding to M16 cable reaches its maximum under the joint action of constant-load and live-load forces. Therefore, the steel anchor box at the M16 cable was selected as the object of this paper. The angle between the M16 cable and the longitudinal bridge direction α = 32.745°, the specification is PES7-253, and the cross-sectional area is 9737 mm². The structure of the steel anchor box for the Yujiang Bridge is shown in Figure 2.

2.2. Computational Models

In this paper, finite element analysis of the steel anchor box girder section at the M16 diagonal cable was carried out by using the large spatial finite-element software, Abaqus [14]. The original bridge is symmetrical about the center of the bridge cross-section, and the half-section was selected for analysis. The boundary conditions were solidified at both ends, and the beam-center surface was symmetrically constrained to slide up and down. The plates were made of Q370qD steel with an elastic modulus E = 2.06 × 10⁵ MPa and a Poisson ratio μ = 0.3. The anchor plates were made of 3D solid units, and the rest of the plates were made of continuous shell units; the welding relationship between the plates was simulated via constraint binding, and the tight-pressure-fit relationship between the pressure plate and the anchor plates was simulated via nodal coupling. The finite element model of the beam section where the M16 cable is located is shown in Figure 3.

2.3. Force-Transmission Mechanism

In order to clarify the force-transmission mechanism of the built-in double-fixed steel anchor cable-girder box, local finite element analysis of the steel anchor box was carried out. Four welds connected to the main beam were selected and numbered, among which the inner-web and outer-web welds were numbered N1–F1, N2–F1, N1–F2, and N2–F2 (see Figure 4 for the numbering).

In the local calculation model of the steel anchor box, the shear-stress distribution of the four main welds, N1–F1, N2–F1, N1–F2, and N2–F2 along the direction of the diagonal cable was plotted, as shown in Figure 5, with the apex point “O” of support plates N1 and N2 as the origin and the direction of the diagonal cable, pointing to the bridge tower as the positive direction.

Analysis of Figure 5 shows that the two connection welds, N1–F1 and N2–F1, in the inner web of the main beam had essentially the same shear stress values and the same stress variations along the diagonal cables, and that the same pattern occurred for the two connection welds in the outer web of the main beam. The shear stresses in the weld between the inner web and the support plate were slightly higher than those in the weld between the outer web and the support plate. Further analysis showed that the four welds, N1–F1, N2–F1, N1–F2, and N2–F2, had a large variation in shear stress along the diagonal cable direction, and that when the distance was between 0 m and 0.31 m, the shear-stress value was small. At 0.31 m, the shear stress increased steeply and reached its maximum value in the range of 0.5 m to 0.6 m. This was due to the fact that the cable force was transferred from the pressure plate N7 to the support plates N1 and N2 in the form of diffusion. When the distance was >0.6 m, the shear stress gradually decreased, but the shear stress still occurred at the end of the weld of the N1 and N2 plates, indicating that the rest of the shear force was transferred to the lower pressure plate N5 and the stiffening rib N6 through the steel anchor-box-support plates N1 and N2.

In order to further investigate the force-transmission mechanism of the built-in double-fixed steel anchor cable-girder box and determine the proportion of the load carried by the four main transmission welds, F1–N1, F1–N2, F2–N1, and F2–N2, the shear stresses along the diagonal cable direction at the welds were extracted through finite-element-analysis calculations and integrated to obtain the cable force and the proportion of the load carried by the four welds. The calculation results are shown in

Table 1. The main weld-cable-force transmission and proportion of the built-in double-fixed steel anchor cable-girder box.

Welding-Seam Number	Weld Length	Transmitted Cable-Force	Designed Cable-Force Value	Proportion of Shared Cable Force
	(m)	(kN)	(kN)	(%)
N1–F1	1.71	1089.59	4600	23.68
N2–F1	1.71	1092.27		23.74
N1–F2	1.71	1040.64		22.62
N2–F2	1.71	1059.12		23.02

As can be seen in Table 1, the main welds of the built-in double-fixed steel anchor box transferred a total of 93.08% of the cable force, in which the weld between the support plate N2 and the inner web transferred the largest cable force, with a value of 1092.27 kN, accounting for 23.74% of the total. Due to the symmetry of the steel anchor-box structure, the four welds transferred almost the same proportion, which was generally around 23%.

2.4. Stress Distribution

Under the design load, the built-in double-fixed steel anchor cable-girder box was subjected to a relatively uniform stress, with most of the stresses on the plates distributed between 10 and 50 MPa. However, a certain degree of stress concentration was observed in the inner and outer webs of the main beam, the support plates N1 and N2, and the pressure plate N7. In particular, the stress-concentration points of the support plates N1 and N2 appeared at the contact point between the weld and the pressure plate N7, and the maximum equivalent stress of this part was 135.4 MPa. The main reason for this is that this part was directly subjected to the force exerted by the pressure plate N7, resulting in obvious pressure and stress concentration on this part. The stress in this area gradually decreased along the cable direction, indicating that the cable force transmitted by the pressure plate N7 gradually passed through both sides of the weld to the beam web. The equivalent stress-contour map of the support plate N1 is shown in Figure 6.

The stress concentration in the inner web of the main beam was found at the weld with N1 and N2, where the maximum equivalent stress was 40.45 MPa and gradually spread around due to the transfer of the cable force in the form of the shear force through the weld between the support plate and the inner web. The equivalent stress-contour map of the inner-web F1 is shown in Figure 7.

The stress concentration in the pressure plate N7 was relatively prominent, appearing at the connection point with the support plates N1 and N2; the maximum equivalent stress in this part was 66.15 MPa, which was due to the more concentrated vertical action of the cable force in this area. At the same time, the node was subject to strong restraint, which in turn led to significant stress concentration in this area. The equivalent stress-contour diagram for the pressure plate N7 is shown in Figure 8.

From the above analysis, it can be seen that the stress-transmission path of the built-in double-fixed steel anchor cable-girder box was clear, and that the ratio of the transmission of each weld was even and reasonable. However, the stress concentration still occurred in the main plates, although it was not serious. The stress-concentration area at the connection weld in the web of the main beam is sensitive to fatigue damage and should be focused on.

3. Experimental-Model-Design Scheme

With reference to the previous test-based research data and the test requirements described above, the test model of the Yujiang steel anchor box was designed. A large number of finite-element-model calculations were carried out during the design of the test model, and through continuous calculation and debugging, the test model was compared with the original bridge model to ensure that the test model could effectively simulate the stress state of the original bridge structure. At the same time, the auxiliary structure in the test model was locally strengthened to ensure that the test model would not be damaged in advance before the required test load was reached, and the test was carried out smoothly. The design process is shown in Figure 9, below.

The shear stresses on the inner-web force-transmission weld in the test model and the original bridge model were compared and analyzed, as shown in Figure 10 and Figure 11. It can be seen in the figure that the finite-element-calculation results of the original bridge shear stress on the welds of the support plates N1 and N2 and the inner web of the main beam were generally in agreement with the finite-element-calculation results of the test model. Based on the above comparison of the calculation results of test on the original bridge and the test model of the inner-web force-transmission weld, it can be seen that the test model can be used in the experimental study of the built-in double-fixed cable-girder steel anchor box on the Yujiang Bridge, and that it can better reflect the actual stress state of the steel anchor box.

4. Loading Scheme and Measuring-Point Arrangement

4.1. Loading Program

The test model used Q370qD. The mechanical properties of the steel plates with different thicknesses are shown in

Table 2. Material and mechanical properties of Q370qD.

Thickness (mm)	Yield Stress	Ultimate Tensile Stress	Elongation
	(Mpa)	(MPa)	(%)
16	556	644	21
20	459	564	23
30	441	548	27

. The fatigue test was loaded with a 2000-kN MTS servo actuator from MTS, USA. The strain at the measuring point was measured using a three-way strain flower, and the displacement of the main body of the steel anchor box was monitored using a 5G10X-series linear-displacement meter. The DH3816 and DH3820 static-data-acquisition systems and the DH5922D dynamic-data-acquisition system produced by China Donghua Testing Company were used to collect the test data. The fatigue-test-loading diagram is shown in Figure 12. The test was carried out in three stages: (1) the static-load-test stage; (2) the fatigue-test-verification stage; and (3) the fatigue-test-damage stage. On the fatigue test, the effect of the constant loads and live loads on the fatigue performance of the structure was also taken into consideration, with the upper limit of fatigue load P_max = 920 kN and the lower limit P_min = 640 kN. In the fatigue-test-damage stage, the upper fatigue-load limit P_max = 1050 kN and the lower limit P_min = 640 kN, and 1 million fatigue cycles were completed to verify the fatigue-safety reserve. The load grades and load values in static-loading stage of fatigue test is shown in

Table 3. Load grades and load values in static-loading stage of fatigue test.

Test Phase		Validation Phase	Destruction Phase
Number of fatigue loads/10,000 cycles		0~200	200~300
Static load test classification	1	0	0
	2	170	170
	3	320	320
	4	470	470
	8	620	620
	6	770	770
	7	920	920
	8 *	-	1050 *

Note: “*” represents the additional load rating and load value during the fatigue-damage phase.

.

4.2. Layout of Measuring Points

The main body of the steel anchor box, the weld of the inner and outer webs of the main beam and the stress concentration and fatigue-sensitive areas according to the finite element calculations were arranged. Finally, as many points as possible were arranged in the main plates according to the maximum data-collection capacity of the test-data collection equipment. However, due to the limited operating space of the test model, some of the important measurement points in the test model could not be placed, and the stress could only be calculated through the finite element analysis.

5. Test-Result Analysis

5.1. Static-Load-Test Results

The strain of most of the measuring points on the static-load test was linear with the load and the residual strain was very small, indicating that the test model was still in the elastic stage during the whole loading process. Under the static load of the upper fatigue limit, the stress-test value of each measuring point was low, and the stress distribution was uniform. The static-test values of the main plates in the built-in double-fixed cable-girder steel anchor box were analyzed as follows.

5.1.1. Inner-Web F1

Located on the inner side of the F1, the main tensile-stress direction of the measuring point on the outer side of the connection weld of the support plates N1 and N2 was about 45° in the weld direction, and the stress-distribution law was smaller at both ends and larger in the middle. Among the measuring points of the inner-web F1 on the main beam, the maximum tensile stress, which was 17.22 MPa, occurred at the weld between the support plate N2 and the inner-web F1. The maximum compressive stress, which was −18.12 MPa, occurred at the weld between the support plate N1 and the inner-web F1. The stress-test value of the inner web of the main beam F1 is shown in Figure 13.

5.1.2. Support Plate N1

The stress value of the measuring points at the weld between the outer side of the N1 plate and the beam web was large in the middle and small on both sides, and the compressive stress at most of the measuring points was larger than the tensile stress. The upper part of the supporting plate N1 was mainly subjected to compressive stress, while the lower part was mainly subjected to tension. The maximum tensile stress, which was 25.88 MPa, occurred at the measuring points N1–2; the maximum compressive stress, which was −60.12 MPa, occurred at the measuring points N1–8, located at the weld connected to the pressure plate N7. The test value of the support plate N1 is shown in Figure 14.

5.1.3. Pressure Plate N7

The stress distribution of the two rows of measuring points on the bottom of the pressure plate, which were mainly under pressure, was smaller in the middle and larger on both sides. The compressive stress at both ends of the weld connecting with the support plate was the largest, with a maximum value of −65.30 MPa. The measuring points on the top surface of the pressure plate were located on the top surface of the weld with the support plate, and were all under pressure, with a maximum value of −45.25 MPa. The test values of the measuring points of the pressure plate N7 are shown in Figure 15.

5.2. Fatigue Test Results

After first completing 2 million fatigue loads, an additional 1 million fatigue loads of 1.5 times the fatigue amplitude were applied, and there was still no significant change in the stress values at each test point, with no cracks observed on the surface of the structure.

Table 4. Equivalent stress-test values of some measuring points after cyclic fatigue loading.

Point Number	Measured Stresses
	0 m	0.50 m	1.00 m	1.50 m	2.00 m	2.50 m	3.00 m
F1–1	20.9	20.5	20.11	21.12	20.13	23.45	23.33
F1–2	21.97	21.96	22.8	22.24	22.44	23.51	24.01
F1–3	25.5	25.98	26.36	26.57	26.19	29.61	29.78
F2–1	18.49	18.58	19.66	18.7	18.3	21.19	21.30
F2–2	17.30	18.41	18.12	17.91	17.29	20.11	20.33
F2–3	22.20	22.3	22.74	23.26	22.61	26.12	26.73
N1–1	18.08	19.17	18.86	18.4	19.96	20.03	19.9
N1–2	41.95	44.05	44.57	41	43.5	44.01	46.45
N1–3	34.66	33.54	36.2	34.65	34.31	38.21	38.05
N2–1	18.13	19.08	18.81	18.18	18.12	19.12	19.92
N2–2	39.94	39.39	39.09	38.64	38.27	44.16	44.95
N2–3	34.77	34.01	34.71	34.91	36.64	39.79	40.96
N5–1	7.93	8.05	8.57	8.66	8.34	9.67	9.26
N5–2	11.65	11.75	12.69	12.06	11.91	13.77	14.09
N5–3	10.93	11.39	11.54	11.37	11.98	12.44	13.16

Note: m = million fatigue-load cycles.

shows the von Mises equivalent stress-test values at some of the test points after the cyclic fatigue loading. Figure 16 shows the von Mises equivalent stress-test values at some of the test points as a function of the number of cyclic loading cycles.

By analyzing the data from the fatigue-test-verification stage, it can be seen that after 2 million fatigue cycles, the von Mises equivalent stress values of most of the measuring points remained unchanged. After repeated observations, no cracks were found on the test model, and no significant fatigue damage occurred in the structure. This shows that the fatigue resistance of the built-in double-fixed steel anchor cable-girder box was good. At the fatigue-failure-test stage, the stress law of the model’s measuring point was essentially the same as that in the fatigue-verification stage after the model underwent an additional 1 million fatigue cycles. After the fatigue loading was completed, the main plates and welds in the test model were inspected using an ultrasonic flaw detector, and no abnormalities were found in the main stressed plates or welds. According to this analysis, the built-in double-fixed steel anchor cable-girder box had a sufficient fatigue-strength safety reserve.

6. Prediction and Evaluation System of Mechanical Properties of Steel anchor Box Based on PSO-BP Neural Network

The large number of plates and complex welds in the steel anchor boxes of cable girders and the large cable forces to which they are subjected to make the rational design of their structure essential. The analysis of the structure of the built-in double-fixed cable-girder steel anchor box showed that it was mainly through the pressure plate that the cable force was transmitted to the support plate, followed by the weld between the support plate and the web of the main beam in the form of shear force. Based on the experimentally validated finite-element-calculation model, the influence of the changes to the geometric parameters of the key plates on the mechanical properties of the main transmission welds was investigated, and a prediction-and-evaluation system, based on a PSO-BP neural network, for the mechanical properties of the built-in double-fixed steel anchor box of the cable-girder was established. The system used the geometrical parameters of the key plates and the designed cable force as the input variables, and the shear-stress-inhomogeneity coefficient, force-transfer ratio, maximum shear stress, and total force-transfer ratio of the four main force-transferring welds as the output-evaluation parameters, with a view to providing a reasonable and effective supplementary verification-and-evaluation method for the design stage of the built-in double-fixed cable-girder steel anchor box.

6.1. Influence of Key Design Parameters on the Force-Transmission Mechanism

Because the steel anchor-box structure mainly transmits the cable force through the weld between its support plate and the main girder web, the stress state and stress level of each key plate in the steel anchor box are mainly determined by the stress state of the corresponding weld. At the same time, compared with the plate, the stress-concentration problem at the weld position is more prominent, which is one of the main characteristics of steel anchor boxes [15]. In order to quantitatively study the force-transfer mechanism of the built-in double-fixed cable-girder steel anchor box, three parameters were introduced: the shear-stress-inhomogeneity coefficient μ of the weld, the transfer ratio p, and the total transfer ratio P.

$${\mu }_i=\frac{\max ({\tau }_1^i,{\tau }_2^i,\cdots {\tau }_n^i)}{{\displaystyle \sum _j^n{\tau }_j^i/n}}$$

(1)

$$p_i=\frac{t_i{\displaystyle {\int }_0^{l_i}{\tau }_j(l)dl}}{{\displaystyle \sum _{i=1}^4{t}_i{\displaystyle {\int }_0^{l_i}{\tau }_j(l)dl}}}\times 100(\%)$$

(2)

$$P={\displaystyle \sum _{i=1}^4{p}_i}\times 100(\%)$$

(3)

where i = 1, 2, 3, 4 denote the welds N1–F1, N1–F2, N2–F1, and N2–F2 respectively; τ₁, τ₂, …, τ_n indicate the shear stress of welds N1–F1, N2–F1, N1–F2, and N2–F2; n is the number of shear-stress-sampling points along the weld direction; p_i (i = 1, 2, 3, 4) indicates the ratio of the force transmission of the above four welds, respectively; P is the total force-transfer ratio of the four welds; and t_i and l_i indicate the plate thickness and the length of the supporting plate, respectively, corresponding to the four welds.

Based on the theoretical model validated by the tests, the influence of the key design parameters on the mechanical properties of the built-in double-fixed steel anchor cable-girder box was further investigated on the basis of a discussion of the force-transmission paths of the steel anchor box, with a view to further investigating the force-transmission mechanism of the steel anchor box. Due to the complexity of the structural system and the factors influencing the built-in double-fixed steel anchor cable-girder box, the key design parameters, i.e., the thickness of the plates and the length of the support plates of the main members, were selected as design variables for the study. In order to eliminate the interference of other factors, the single-variable research method was adopted in the research process, so as to more accurately explore the influence of the design parameters on the mechanical characteristics of the structure.

6.1.1. Plate-Thickness-Design Effect

The sensitivity of the force characteristics of the anchor box to the designed thickness of the plates was investigated by varying the thickness of the support plates N1 and N2 and the pressure plate N7, while keeping the other design parameters constant. The designed thickness of each plate was determined by considering the range of possible values for the actual structural design, as shown in

Table 5. Designed thickness value of each plate.

No.	Weld Name	Original Design Thickness (mm)	Thickness Values (mm)
1	Support plate (N1)	32	30, 32, 34
2	Support plate (N2)	32	30, 32, 34
3	Pressure plate (N7)	40	38, 40, 42, 44, 46

.

The results of the stress-inhomogeneity coefficient and the force-transmission ratio of the weld for the different design thicknesses of the support plates N1 and N2 and the pressure plate N7 are shown in Figure 17. The figure uses the left and right vertical axes to represent the shear-stress-inhomogeneity factor and the force-transfer ratio, with the hollow icon line indicating the welds’ shear-stress-inhomogeneity-factor results, referring to the left vertical axis, and the solid icon line indicating the force-transfer-ratio results, referring to the right vertical axis.

The results of the study showed the following: ① the shear-stress-inhomogeneity coefficient of the transmission weld decreased with the increase in the thickness of the pressure plate N7, and the force-transmission ratio of the weld did not change significantly with the increase in the thickness of the pressure plate N7; ② the shear-stress-inhomogeneity coefficient of the transmission weld decreased with the increase in the thickness of the support plates N1 and N2, and the force-transmission ratio of the weld increased with the increase in the thickness of the support plates N1 and N2; ③ the inner-web weld had a larger force transfer than the outer-web weld, but the shear-stress-inhomogeneity coefficient was smaller; ④ the variation in the thicknesses of the support plates N1 and N2 had a greater effect on the force characteristics than the thickness variation in the pressure plate N7.

6.1.2. The Influence of Plate-Length Design

The designed length of the weld seam between the built-in double-fixed steel anchor cable-girder box and the web of the main beam is one of the most important parameters affecting the structural force characteristics of the steel anchor box. By varying these parameters while keeping the other parameters constant, the sensitivity of the force characteristics of the steel anchor box and its force-transmission mechanism to the designed length of the plates, i.e., the weld length, was investigated, and the values of the plate length were taken, as shown in

Table 6. Designed length values of each plate.

No.	Weld Name	Original Design Length (mm)	Length Values (mm)
1	N1–F1, N1–F2	1710	1600, 1710, 1800, 1900, 2000
2	N2–F1, N2–F2	1710	1600, 1710, 1800, 1900, 2000

. The results of the force-transmission ratio and the total force-transmission ratio for the different design length of the support plates N1 and N2 are shown in Figure 18.

The results of the study showed the following: ① the force-transmission ratio of each main transmission weld in the built-in double-fixed steel anchor cable-girder box increased with the increasing weld length; ② the total force-transmission ratio of the built-in double-fixed steel anchor cable-girder box gradually increased with the increasing weld length; ③ the influence of the designed lengths of the plates on the mechanical properties of the steel anchor box was greater in the range of 1600–1800 mm than in the range of 1800–2000 mm.

6.2. Prediction-and-Evaluation System of Mechanical Properties of Steel Anchor Box Based on PSO-BP

It was clear from the results of the study on the influence of the key design parameters on the force-transfer mechanism that the problem of determining reasonable design parameters for the anchorage structure of the built-in double-fixed cable girder would be complex and would require a comprehensive consideration of factors such as the ultimate structural load capacity, structural stability, and fatigue performance before it could be finally determined. Therefore, a structural-design-parameter database and a mechanical-performance-prediction-and-evaluation system based on a PSO-BP neural network of the built-in double-fixed steel anchor cable-girder box were established.

6.2.1. Database Establishment

Based on the finite element model established through the experimental verification, a large number of calculation models with different design parameters were constructed to study the influence of the design load and key plate geometric parameters on the four main force-transmission welds, namely, N1–F1, N1–F2, N2–F1, and N2–F2. The design parameters considered in the simulation model included the thickness of the support plates N1 and N2 and the pressure plate N7, the lengths of the support plates N1 and N2, and the designed bearing capacity. In this study, a total of 375 cases were calculated.

Table 7. Input parameters for the calculation of the force-transmission state of the weld.

Parameters	Values (mm)
Support plates N1 and N2, thickness	30, 32, 34
Pressure plate N7, thickness	38, 40, 42, 44, 46
Support plates N1, N2, length	1600, 1710, 1800, 1900, 2000
Design load	4000, 4600, 5000, 6000, 7000

lists the variation range of each parameter.

6.2.2. PSO-BP Prediction Model

BP Neural Network

The BP neural-network model consists of three components: the input, hidden, and output layers. Furthermore, the model includes two processes: the forward propagation of the signal and the backward propagation of the error. The input signal is processed layer by layer from the input layer through the hidden layer and transferred to the output layer in the forward-propagation process, where the state of the neurons in each layer only affects the state of the neurons in the following layer. If the output result was not optimal, the network-connection weights and thresholds are changed through reverse propagation for repeated training to obtain the optimal result [16,17,18]. The BP neural network is illustrated in Figure 19.

Particle Swarm Optimization (PSO) Algorithms

The PSO algorithm originated in the study of birds’ predation behavior. Each particle in the algorithm corresponds to the role of a bird in a flock, defining the position of each particle as a possible solution to the optimization problem [19,20]. The PSO algorithm is initialized as a flock of random particles, each representing a potential solution to the optimization problem, and then it is iterated to find the optimal solution, updating individual positions by following individual extremes and global extremes in each iteration, with each position representing a potential solution to the problem. Assume that there is a population of N particles in a D-dimensional search space, where the position of the ith particle in the D-dimensional search space is represented as a D-dimensional vector, and the fitness value of each particle position can be calculated according to the fitness function.

When the individual and global extremes are found, the particle updates its velocity and position using Equations (4) and (5).

$$V_{id}^{k+1}=w{V}_{id}^k+c_1{r}_1(P_{id}^k-X_{id}^k)+c_2{r}_2(P_{id}^k-X_{id}^k)$$

(4)

$$X_{id}^{k+1}=X_{id}^k+V_{id}^{k+1}$$

(5)

where w is the inertia weight, k is the number of current iterations, V is the velocity of the particle, X is the position of the particle, and c₁ and c₂ are the learning factors, also known as acceleration factors.

PSO-BP-Prediction-Model Construction

The hidden nodes in a BP neural network are usually determined by repeated forward pass and backward propagation. By modifying or constructing the training method to change the number of hidden nodes, the corresponding initial weights and thresholds are also changed, thus affecting the convergence and learning efficiency of the network. In this study, in order to reduce the impact, the adjustment of the weights and thresholds was optimized by using a BP neural-network model based on the particle-swarm algorithm, thus speeding up the convergence of the network and improving its learning efficiency. The specific computational flow chart was shown in Figure 20.

In this paper, Windows 7 was used as the system platform and MATLAB 2016a was used as the processor. The length and thickness of the support plates N1 and N2, the thickness of the pressure plate N7, the designed load-bearing cable force were used as the input variables, and the number of input nodes was four. The shear-stress-inhomogeneity coefficient, the ratio of the force transfer, the maximum shear stress of the weld, and the ratio of four total force transfers were used as the output-evaluation variables, and the number of output nodes was 13. In order to minimize the training error, the number of hidden-layer nodes was determined using Equation (6). The final topology determined in this paper was 4-10-13.

$$L<\sqrt{a+b}+c$$

(6)

where L is the number of neurons in the hidden layer, a is the number of neurons in the output layer, b is the number of neurons in the input layer, and c is in the range of [0–10], where a constant is taken.

The cross-validation method was used in this research to divide 70% of the data into the training set, 15% into the validation set, and the remaining 15% into the test set, and the consistency of the data distribution was maintained in the data division. The statistical parameters of each data set are shown in

Table 8. Statistical parameters of input variables.

Statistical Parameters of Input Variables
Dataset	Sample Size	Statistical Parameters	X1 Thickness of N1, N2	X2 Length of N1, N2	X3 Thickness of N7	X4 Designed Load-Bearing Cable Force
Training set	263	Max	34	2000	46	7000
		Min	30	1600	38	4000
		Average	32.027	1803.064	42.027	5371.212
		Stdev.s	1.636	159.026	2.780	1068.271
Validation set	56	Max	34	2000	46	7000
		Min	30	1600	38	4000
		Average	32.158	1804.737	42.000	5344.737
		Stdev.s	1.667	142.748	3.013	1064.693
Test set	56	Max	34	2000	46	7000
		Min	30	1600	38	4000
		Average	31.632	1780.263	41.684	5418.421
		Stdev.s	1.636	137.930	3.103	1063.128

and

Table 9. Statistical parameters of output variables.

Statistical Parameters of Output Variables
Dataset	Sample Size	Statistical Parameters	μ1	p1	τmax1	μ2	p2	τmax2	μ3	p3	τmax3	μ4	p4	τmax4	P
Training set	263	Max	2.008	24.84%	60.002	2.286	23.16%	61.934	2.183	24.76%	62.298	2.103	24.11%	58.313	96.49%
		Min	1.781	23.01%	27.738	1.954	21.69%	28.974	1.852	22.63%	29.681	1.783	21.82%	26.709	89.42%
		Average	1.892	0.240	40.957	2.078	0.226	42.380	2.025	0.235	43.032	1.934	0.229	40.147	92.89%
		Stdev.s	0.065	0.005	8.555	0.090	0.004	8.807	0.088	0.004	8.947	0.081	0.003	8.452	1.42%
Validation set	56	Max	1.991	24.82%	58.313	2.283	23.16%	60.067	2.171	24.21%	61.040	2.089	23.26%	57.194	95.33%
		Min	1.782	23.04%	27.853	1.969	21.78%	29.103	1.861	22.66%	29.883	1.796	22.47%	26.873	90.65%
		Average	1.893	0.240	40.667	2.091	0.226	42.055	2.024	0.236	42.734	1.934	0.229	39.766	93.05%
		Stdev.s	0.058	0.006	8.886	0.097	0.004	9.200	0.093	0.004	9.218	0.085	0.002	8.843	1.51%
Test set	56	Max	2.005	24.84%	59.769	2.268	23.13%	61.638	2.183	24.20%	62.108	2.103	23.28%	57.905	95.35%
		Min	1.789	23.02%	29.799	1.957	21.70%	30.923	1.868	22.65%	30.760	1.803	22.37%	28.899	90.33%
		Average	1.885	0.238	42.492	2.081	0.225	43.975	2.024	0.234	44.577	1.945	0.228	41.670	92.54%
		Stdev.s	0.057	0.005	8.067	0.089	0.004	8.192	0.092	0.004	8.473	0.079	0.003	7.901	1.47%

. The activation function of the BP neural network from the input layer to the hidden layer was logsig, and the function of the hidden layer to the output layer was the purlin function. The training function was the default trainlm function, which is a method that combines Newton’s method and gradient descent, and can improve the training speed and accuracy when training neural networks; the learning function was the default learngdm function, which is a gradient descent with a momentum weight and bias-learning function, which can update the weights and biases of a neural network according to the gradient of the network’s performance, and can use the momentum factor to accelerate the convergence process. According to the PSO algorithm, the initial position and velocity of the particles were generated randomly within the allowed range. The population size was 40 and the inertia weight was set to 1. The learning factor used c₁ = c₂ = 2.5. The weights and thresholds of the trained PSO-BP network model are shown in Appendix A.

Prediction Results and Discussion

In this paper, two different evaluation indicators, namely, the mean squared error (MSE) and the Nash efficiency coefficient (NS), were used with the following mathematical expressions:

$$MSE=\frac{1}{n}{\displaystyle \sum _{i=1}^n{\left(x_i-p_i\right)}^2}$$

(7)

$$NS=1-\frac{{\displaystyle {\sum }_{i=1}^n{(x_i-p_i)}^2}}{{\displaystyle {\sum }_{i=1}^n{(x_i-\overline{x})}^2}}$$

(8)

where x_i is the true value, p_i is the predicted value, and x is the average of the true values. The calculation results are shown in

Table 10. PSO-BP algorithm’s performance-evaluation results.

	Training Set	Validation Set	Test Set
MSE	0.12119	0.12250	0.07065
NS	0.99970	0.99972	0.99984

Note: the closer the Nash efficiency coefficient (NS) is to 1, the higher the likelihood that the model has a high degree of confidence.

, below.

The prediction results of three cases were selected for comparison with the calculated results to demonstrate the workability of the model. The comparison results are shown in

Table 11. Comparison of calculated values and predicted results.

		Input Variable Array 1		Input Variable Array 2		Input Variable Array 3
		Calculated Value	Predicted Value	Calculated Value	Predicted Value	Calculated Value	Predicted Value
N1–F1	μ1	1.902	1.915	1.865	1.867	1.840	1.839
	p1	23.87%	23.88%	23.69%	23.65%	24.57%	24.60%
	τmax1	32.433	32.314	35.444	35.387	36.223	36.059
N1–F2	μ2	2.068	2.069	2.010	2.009	2.036	2.032
	p2	22.56%	22.60%	22.62%	22.60%	23.12%	23.11%
	τmax2	33.665	33.699	36.941	36.781	37.592	37.505
N2–F1	μ3	2.091	2.064	1.974	1.982	2.007	2.010
	p3	23.17%	23.17%	23.39%	23.45%	24.04%	24.09%
	τmax3	33.806	33.817	37.354	37.430	38.484	38.598
N2–F2	μ4	1.975	1.979	1.881	1.880	1.893	1.891
	p4	22.78%	22.72%	22.70%	22.74%	23.16%	23.14%
	τmax4	31.864	31.983	34.863	34.882	35.010	34.845
P		92.38%	92.40%	92.40%	92.44%	94.89%	94.90%

Note: input-variable array 1—X1 = 30, X2 = 1900, X3 = 38, X4 = 4000; input-variable array 2—X1 = 32, X2 = 1710, X3 = 40, X4 = 4600; input-variable array 3—X1 = 34, X2 = 1800, X3 = 42, X4 = 5000.

. The analysis of the results in Figure 21 and Table 11 showed that the correlation coefficients for the predicted mechanical properties of the main transmission welds in the three stages of training, testing, and prediction were all greater than 0.99, and that the MSEs were all less than 0.13, indicating that the model can be used to explore the intrinsic relationship between the design load, the plate-geometry parameters, and the mechanical properties of the main transmission welds in the process of iterative training, and generate more reasonable predictions of the mechanical properties of the main transmission welds.

7. Conclusions

In this paper, 1:2-scale fatigue tests were carried out on a built-in double-fixed steel anchor cable-girder box on a high-speed-railway cable-stayed bridge, and a finite element model based on the test verification was established to analyze the force characteristics, the force-transmission mechanism of the built-in double-fixed steel anchor cable-girder box, as well as the influence of different design parameters on its mechanical properties. Furthermore, a dataset was established, and a mechanical-property-prediction-and-evaluation system for the built-in double-fixed steel anchor cable-girder box was built based on a PSO-BP neural network. The conclusions are summarized as follows:

• By analyzing the mechanical properties of the main force-transmission welds in the original bridge model and the test model, it was shown that the test model effectively simulated the force state of the real bridge.
• Under fatigue loading, the stress distribution in the built-in double-fixed steel anchor cable-girder box was homogeneous, and that the cable force was mainly transmitted through the weld between the support plate and the web of the main beam, where the stress-concentration problem was more prominent. The weld can be considered as the main fatigue-sensitive feature of the built-in double-fixed steel anchor cable-girder box.
• The fatigue-test results showed that most of the stresses calculated through the finite element analysis were in good agreement with the experimental measurements. The stress measured in the whole structure remained stable after 2 million fatigue-load cycles. After 3 million fatigue-load cycles, no surface cracks were observed, indicating that the fatigue performance of the built-in double-fixed steel anchor cable-girder box was good and met the designed load-bearing-capacity requirements.
• The influence of the key design parameters on the force-transmission mechanism showed that variations in the thickness of the support plate had a greater effect on the shear-stress distribution in the steel anchor box than the variations in the thickness of the pressure plate, and that the designed length of the weld seam had a more significant effect on the force characteristics of the anchorage structure of the built-in double-fixed cable beam than the plate thickness.
• The PSO-BP neural-network-based predictive evaluation system for built-in double-fixed cable-beam steel anchor boxes can be used to explore the intrinsic relationship between the designed cable force, the plate-geometry parameters, and the mechanical properties of the main force-transfer welds during the iterative training process, and that it can generate reasonable predictions of the mechanical properties of the main force-transfer welds, providing a new and effective evaluation method for the design phase.