Stability Analysis of Transmission Towers in Mining-Affected Zones

Abstract

Transmission towers located above mined-out areas may experience collapse or instability due to mining-induced ground subsidence and deformation, which poses significant risks to the safe operation of power transmission lines. To clearly evaluate the deformation resistance and failure threshold of transmission towers under mining-induced ground deformation, this article examines a typical 220 kV self-supporting transmission tower located in a mining area of Northern Shaanxi Province through a detailed three-dimensional finite element model constructed and simulated using ANSYS 2022. The mechanical response and failure process of the tower structure were systematically simulated under five typical deformation conditions: tilt, horizontal compression, horizontal tension, tilt–compression, and tilt–tension. The results indicate that under individual deformation conditions, the critical deformation values of the tower are 35 mm/m for tilt, 10 mm/m for horizontal compression, and 8 mm/m for horizontal tension, demonstrating that the structure is most sensitive to horizontal tensile deformation. Under combined deformation conditions, the critical deformation values for the combined tilt–compression and tilt–tension conditions exhibited a marked reduction, reaching 8 mm/m and 6 mm/m. Compared to individual deformation conditions, transmission towers demonstrate a significantly higher susceptibility to structural failure under combined deformation conditions. The displacement at the tower head and the tower tilt angle exhibit a linear positive correlation with the values of ground surface deformation. Under individual deformation conditions, the tilt of the tower was approximately 0.903 times the tilt deformation value and 0.089 times the values of horizontal compression and tension deformation, indicating that tilt deformation exerts a more pronounced influence on the inclination of the tower. Under combined deformation conditions, the tilt of the tower reached approximately 0.981 times that of the tilt–compression deformation value and 0.829 times that of the tilt–tension deformation value. Compared to the tower tilt induced individually by horizontal compression or tension deformation, the tilt under combined deformation conditions demonstrated a significantly greater value. Under mining-induced ground deformation, a redistribution of support reactions occurs, exhibiting either nonlinear or linear increasing trends depending on the type of deformation. The findings of this article provide a theoretical basis and data support for disaster prevention and control, safety evaluation, and structural design of transmission lines in mining areas.

1. Introduction

With the large-scale extraction of coal resources and the continuous expansion of national power infrastructure, the routing of high-voltage transmission lines through coal mining-induced subsidence zones has become an unavoidable reality [1,2]. High-voltage transmission lines represent a specific type of structure that functions as a continuously coupled spatial system, comprising conductors, towers, foundations, and connecting elements [3,4]. Owing to the discrepancy in deformation mechanisms between transmission towers and the ground surface, the conventional mining-induced damage criteria established for buildings are not directly applicable for assessing the structural failure severity of such infrastructure [5,6]. Surface movement and deformation induced by coal mining will impose additional stresses and deformations on transmission towers [7], which may eventually lead to structural failure if these exceed allowable limits [8]. Hence, a comprehensive study on the relationship between transmission tower deformation and surface movement is essential for establishing the critical threshold values of ground deformation permissible for tower stability. Such findings are crucial for guiding the route selection of new power lines and ensuring the structural integrity of towers in mining-affected areas.

A considerable body of research has been devoted to analyzing how mining-induced ground movement and deformation affect the structural integrity of transmission towers. Zhao, B. [9,10,11] developed a mechanical model of the big board foundation through theoretical analysis and numerical simulation. By integrating the probability integral method, he derived the coordinated deformation behavior between surface movement and the foundation, thereby providing a novel methodology for predicting the extent of tower damage. Zhao, Y. [12,13] established a predictive model for estimating the displacement and deformation of high-voltage transmission towers under mining-induced conditions. The study revealed that the tilt deformation of the tower body is determined by both ground tilt and tower height, while the horizontal deformation of the foundation depends on the ground horizontal strain, the base width of the tower, and the properties of the foundation and soil. Guo, W. [14,15] conducted a case study of high-voltage transmission towers affected by mining activities. Using numerical simulation and the probability integral method, he examined the correlation between ground tilt/horizontal strain and the distribution of structural internal forces and deformation characteristics of the towers. This research provides a critical theoretical basis for the safety assessment of transmission towers in mining-affected areas. Yang, X. [16,17] developed an improved Analytic Hierarchy Process model by selecting nine key influencing factors as evaluation indicators. A comparison between the evaluation results and field investigation data on the site stability along high-voltage transmission lines demonstrated that the enhanced model can accurately reflect the actual stability characteristics of the goaf. Qi, W. [18] and Wang, H. [19] applied a fuzzy comprehensive evaluation method to assess specific indicators across multiple hierarchical levels. The results, determined by the maximum membership principle, indicated a “relatively stable” condition, suggesting that no additional safety reinforcement measures are required for the tower structure. Xu, S. [20,21], G, Vladimir. [22], and Zhang, S. [23] conducted an in-depth investigation through theoretical analysis, examining aspects such as subsoil deformation, tower foundation characteristics, simulation accuracy, structural deformation of towers, and the influence of cable suspension points. Their research elucidates the patterns and characteristics of surface deformation in goaf areas and proposes corresponding mitigation strategies. Yu, H. [24], Wu, J. [25], and Kang, B. [26] conducted an analytical study on a typical transmission tower under various foundation deformation conditions. They clearly determined the corresponding tilt rates of the tower and deformation rates of the foundation for each condition, providing critical reference data for the deformation-resistant capacity design and safety monitoring of tower foundations in goaf areas. Kouchaki, M. [27] and Sun, M. [28] employed machine learning and neural network algorithms, utilizing accuracy as the criterion to select the most effective method. By leveraging acceleration responses of a large transmission tower under various structural states, they evaluated the selected approach. The results demonstrate the feasibility of applying artificial neural networks for the qualitative assessment of structural health in transmission towers. Jia, J. [29] and Liu, J. [30] developed a finite element model of a transmission tower-line system using the finite element software ANSYS. The study comprehensively investigated the effects of various types of horizontal ground deformation and their directions of action on the mechanical behavior of the tower structure. The internal force distribution and evolution patterns of the structural members were analyzed, and the failure mechanisms of typical tower components were elucidated, thereby providing a theoretical basis for the safety assessment of transmission tower structures in mining-affected areas. Through a comparative analysis of six different working conditions, Shi, G. [31] concluded that uniform settlement imposes a significantly lower impact on transmission tower structures than uniform settlement. Based on this finding, he proposed the adoption of an integrated foundation design to enhance the stability of the tower. Yuan, G. [32] employed the finite element analysis method to systematically examine the influence of ground deformation on the internal force distribution and deformation characteristics of a typical transmission tower. The study revealed the correlation between structural failure modes and the load-bearing capacity as well as displacement of individual members. Zhou, Y. [33] developed a finite element model of a transmission tower-line system to calculate the structural internal forces and stresses in truss members. The model was validated against experimental data, leading to the formulation of a reinforcement strategy. Zhou, L. [34] investigated the performance of high-voltage transmission towers in resisting mining-induced deformation within loess hilly areas and proposed targeted protective mining techniques accordingly.

Although the aforementioned scholars have made considerable progress in studying the structural response of transmission towers in mining-affected zones, the critical surface deformation values leading to tower failure have not yet been definitively established. Building upon this foundation, the present study employs ANSYS numerical simulation software to construct a three-dimensional finite element model of a typical transmission tower. Five typical deformation conditions were selected and simulated. The article systematically examines the effects of mining-induced ground movement and deformation on transmission structures, determines critical deformation thresholds leading to failure under varying conditions, uncovers the underlying deformation response mechanisms and failure modes, and clarifies the characteristic behavior of towers subjected to surface deformation. Thereby, it establishes a theoretical framework for the design and safety evaluation of towers in mining-affected zones.

2. Finite Element Modeling and Failure Criteria for Transmission Towers

2.1. Establishment of the Model

This article focuses on a typical 220 kV single-circuit self-supporting transmission tower located above a mining area of Northern Shaanxi Province. The tower, with a total height of 44.84 m and a nominal height of 38.88 m, features a base width of 8 m in both the transverse and longitudinal directions. The main material of the tower adopts Q345 equal angle steel, and the diagonal material and auxiliary material adopt Q235 equal angle steel. The maximum section of the angle steel member used is L160 × 14, the minimum section is L40 × 3, and all sections are L-L-shaped. In the global finite element model, both the main and diagonal material were simulated using the BEAM188 3-D beam element in ANSYS, which incorporates seven degrees of freedom per node. The tower structure is primarily composed of steel angles, gusset plates, and high-strength bolts.

The steel material was modeled using an ideal elastoplastic constitutive model, with only self-weight considered. The material parameters for both Q345 and Q235 steel were uniformly defined as follows: density ρ = 7.85 × 10³ kg/m³, elastic modulus E = 2.06 × 10¹¹ Pa, and Poisson’s ratio μ = 0.3. Based on the aforementioned modeling criteria, the tower model comprised a total of 15,579 nodes and 1353 elements following the meshing process. To accurately characterize the large deformations that steel components may undergo under foundation-induced deformation, a multilinear stress–strain constitutive model was employed to simulate the kinematic hardening effect, with the large deformation option activated to account for geometric nonlinearity. The detailed structure of the finite element model of the transmission tower is presented in Figure 1.

2.2. Scheme of Numerical Simulation

Studies indicate that [35], following coal seam extraction, transmission towers overlying the goaf are susceptible to two predominant failure modes [36]. The first occurs at the margin of the goaf, where uniform subsidence causes one side of the foundation to sink significantly, inducing sliding toward the basin center and resulting in structural deformation of the tower. The second is observed in the central zone of the goaf, where non-uniform foundation settlement leads to partial subsidence accompanied by bilateral inward displacement, thereby distorting the tower body. From the perspective of mining-induced subsidence, the aforementioned damage can be attributed to ground tilt deformation, horizontal compression, tension deformation, as well as combined deformation effects.

Therefore, the above two forms of damage can be attributed to an individual deformation condition and a composite deformation condition, a total of 5 typical deformation conditions. The individual deformation conditions include condition 1: tilt deformation (i), condition 2: compression deformation (ε_c), and condition 3: tension deformation (ε_t). The combined deformation conditions include condition 4: tilt–compression deformation (i + ε_c), and condition 5: tilt–tension deformation (i + ε_t), as shown in Figure 2.

Based on this framework, this article will simulate the damage response of the transmission tower under five deformation conditions. The simulation methodology for each condition is described below, with detailed schemes for individual and combined deformation conditions given in

Table 1. Simulation scheme under individual deformation conditions.

Scheme	Condition 1			Condition 2			Condition 3
	i (mm/m)	B (m)	UY (mm)	εc (mm/m)	B (m)	UZ (mm)	εt (mm/m)	B (m)	UZ (mm)
1	3	8	−24	−2	8	−16	2	8	16
2	6	8	−48	−4	8	−32	4	8	32
3	10	8	−80	−6	8	−48	6	8	48
4	15	8	−120	−8	8	−64	8	8	64
5	20	8	−160	−10	8	−80	10	8	80
6	25	8	−200	−12	8	−96	12	8	96
7	30	8	−240	−14	8	−112	14	8	112
8	35	8	−280	−16	8	−128
9	40	8	−320
10	50	8	−400
11	60	8	−480

and

Table 2. Simulation scheme under combined deformation conditions.

Scheme	Condition 4				Condition 5
	i + εc (mm/m)	B (m)	UY (mm)	UZ (mm)	i + εc (mm/m)	B (m)	UY (mm)	UZ (mm)
1	2	8	−16	−16	2	8	−16	−16
2	4	8	−32	−32	4	8	−32	−32
3	6	8	−48	−48	6	8	−48	−48
4	8	8	−64	−64	8	8	−64	−64
5	10	8	−80	−80	10	8	−80	−80
6	12	8	−96	−96	12	8	−96	−96
7	14	8	−112	−112

, respectively.

Condition 1: fixed bearing C, D, bearing A, B applied vertical displacement (UY < 0), constrained UX, UZ, as shown in Figure 2a,

Condition 2: fixed bearing C, D, bearing A, B applied lateral inward displacement (UZ < 0), constraint UX, UY, as shown in Figure 2b.

Condition 3: fixed bearing C, D, bearing A, B applied lateral outward displacement (UZ > 0), constraint UX, UY, as shown in Figure 2c.

Condition 4: fixed bearing C and D, bearing A and B simultaneously applied vertical displacement (UY < 0) and lateral inward displacement (UZ < 0), as shown in Figure 2d.

Condition 5: fixed bearing C and D, bearing A and B simultaneously applied vertical displacement (UY < 0), and lateral outward displacement (UZ > 0), as shown in Figure 2e.

2.3. Failure Criterion of Tower

The transmission tower structure, fabricated from steel angles, is highly statically indeterminate. Members primarily under tension tend to exhibit strength-based failure, while those subjected to compression are susceptible to buckling instability. Under normal service conditions, the stresses in the structural members of the tower remain within the allowable material stress limits. Compressive members that have exceeded the allowable material stress retain residual post-buckling load-carrying capacity and continue to contribute to the global stiffness of the tower structure. When a tensile member undergoes stress beyond the material’s yield strength, it experiences strength failure, resulting in a loss of sectional load-bearing capacity and ceasing to contribute to the global structural stiffness of the tower. The tension resistance at both ends deteriorates rapidly, which may trigger progressive collapse and lead to global instability of the structure.

Therefore, the exceedance of the allowable stress in tensile members is defined as the failure criterion for the transmission tower [37]. The allowable stresses for Q235 and Q345 steel grades are specified as 235 MPa and 345 MPa, respectively [38]. Once the applied stress exceeds the allowable stress, the structural steel of the transmission tower undergoes failure. In the ANSYS simulation, it is assumed that the foundations remain intact, and ground movement-induced deformations are directly transferred through the four supports (A, B, C, D). Additionally, failure of the nodes will not precede that of the structural members.

3. Numerical Model Calculation Results and Analysis

3.1. Characteristics of Stress Distribution in Structural Members Under Five Deformation Conditions

Based on the simulation schemes detailed in Table 1 and Table 2, the damage response of the transmission tower under five distinct ground deformation conditions was simulated. Complete sets of nodal and elemental solutions were obtained for each condition. Due to space constraints, only selected contour cloud plots of axial stress distribution in the tower under specific ground deformation values are presented. In the figures, “MX” denotes the location of maximum tensile stress, while “MN” indicates the location of maximum compressive stress. Selected extreme values, including the maximum compressive and tensile stresses, extracted from the simulation results under various ground deformation conditions, are summarized in the table.

3.1.1. Stress Distribution Characteristics in Structural Members Under Individual Deformation Conditions

Figure 3 presents contour plots of stress distribution in the transmission tower under varying ground tilt deformation conditions, while

Table 3. Maximum stresses in structural members under tilt deformation values.

i (mm/m)	Member with Maximum Compressive Stress			Member with Maximum Tensile Stress
	Number	Type	σc (MPa)	Number	Type	σt (MPa)
3	6	Q235	26.167	951	Q235	15.884
6	6	Q235	51.198	951	Q235	35.639
10	6	Q235	87.882	8	Q235	68.998
15	6	Q235	124.295	8	Q235	105.127
20	6	Q235	167.036	8	Q235	147.729
25	6	Q235	198.101	8	Q235	178.035
30	6	Q235	230.323	8	Q235	209.172
35	6	Q235	261.113	8	Q235	240.365
40	6	Q235	282.431	8	Q235	260.846
50	6	Q235	324.701	8	Q235	302.136
60	6	Q235	345.861	8	Q235	322.753

summarizes the maximum stresses experienced by the structural members corresponding to different values of ground tilt. As illustrated in Figure 3, under a specific tilt deformation value, the maximum compressive stress in the structural members of the transmission tower occurs at the diagonal members near the fixed support C of the tower leg, while the maximum tensile stress is observed at the diagonal members adjacent to the movable support A of the tower leg. As the values of ground tilt increases, both the compressive stress in the members under compression and the tensile stress in the members under tension exhibit a corresponding increase.

As can be seen from Figure 3 and Table 3, when the ground tilt value reaches 35 mm/m, the tensile stress in the Q235 steel members of the transmission tower reaches 240.365 MPa, exceeding the allowable stress of 235 MPa for this material grade. Hence, a ground tilt deformation value of 35 mm/m can be established as the critical value for this transmission tower.

Figure 4 shows the fitted curves of the relationship between the maximum compressive and tensile stresses in the structural members and the tilt values under various deformation conditions, with the corresponding fitting equations provided in Equations (1) and (2).

$${\sigma }_c=-0.06602i^2+9.8379i-4.82119,R^2=0.99965$$

(1)

$${\sigma }_t=-0.06194i^2+9.4334i-17.56845,R^2=0.99919$$

(2)

As illustrated in Figure 4 and Equations (1) and (2), both the maximum compressive and tensile stresses in the structural members exhibit a quadratic nonlinear relationship with the ground tilt deformation values. These stresses increase monotonically in a quadratic manner as the tilt value rises. When the tilt value is less than 35 mm/m, the increasing trend of both the maximum compressive and tensile stresses is pronounced. Beyond the threshold of 35 mm/m, however, the parabolic growth rate decelerates.

Figure 5 presents contour plots of the stress distribution within the transmission tower under various ground compression deformation conditions, while

Table 4. Maximum stresses in structural members under compression deformation values.

ɛc (mm/m)	Member with Maximum Compressive Stress			Member with Maximum Tensile Stress
	Number	Type	σc (MPa)	Number	Type	σt (MPa)
2	33	Q235	54.485	8	Q235	42.753
4	309	Q235	103.507	8	Q235	94.757
6	309	Q235	154.813	8	Q235	146.461
8	490	Q235	227.958	8	Q235	184.864
10	489	Q235	300.631	323	Q235	242.231
12	480	Q235	335.852	323	Q235	306.833
14	489	Q235	401.372	323	Q235	396.963
16	480	Q235	431.985	323	Q235	453.655

summarizes the maximum stresses experienced by the structural members corresponding to different values of ground compression. As observed from Figure 5, when the horizontal compression deformation is less than 8 mm/m, the structural members most vulnerable to failure are the cross-diagonal members located immediately above the supports of the tower legs. As the value of ground compression increases beyond 8 mm/m, the failure region progressively transitions upward from the tower legs to the main body of the tower structure. As the ground compression deformation value increases, both the tensile and compressive stresses in the structural members exhibit a corresponding increase.

As shown in Figure 5 and Table 4, when the horizontal compression deformation reaches 10 mm/m, the stress in the Q235 steel tension members of the transmission tower reaches 242.231 MPa, exceeding the allowable stress of 235 MPa for the material. Therefore, the horizontal compression deformation value, denoted as 10 mm/m, can be identified as the critical ground compression deformation value for this transmission tower.

Figure 6 presents the fitted curves illustrating the relationship between the maximum compressive and tensile stresses in the structural members and the value of compression deformation under various conditions, with the corresponding fitting equations provided in Equations (3) and (4).

$${\sigma }_c=-0.54877{\epsilon }_c^2+37.43503{\epsilon }_c-22.1157,R^2=0.99921$$

(3)

$${\sigma }_t=0.68632{\epsilon }_c^2+17.07371{\epsilon }_c+8.14709,R^2=0.99922$$

(4)

As indicated in Figure 6 and Equation (3), the maximum compressive stress in the structural members exhibits a quadratically nonlinear monotonic increase with increasing ground compression deformation. When the deformation is below 10 mm/m, the rate of increase is particularly pronounced; beyond this threshold, the parabolic growth rate attenuates. Furthermore, as demonstrated in Figure 6 and Equation (4), the maximum tensile stress also increases in a quadratically nonlinear monotonic manner as the compression deformation rises. Notably, once the deformation exceeds 8 mm/m, the parabolic growth accelerates markedly, leading to a substantial increase in tension stress within the structural members.

Figure 7 presents contour plots of the stress distribution in the transmission tower under various ground tension deformation conditions, while

Table 5. Maximum stresses in structural members under tension deformation values.

ɛt (mm/m)	Member with Maximum Compressive Stress			Member with Maximum Tensile Stress
	Number	Type	σc (MPa)	Number	Type	σt (MPa)
2	8	Q235	48.533	309	Q235	36.081
4	8	Q235	133.174	309	Q235	113.482
6	329	Q235	230.865	350	Q235	151.987
8	330	Q235	295.539	304	Q235	251.768
10	330	Q235	323.018	304	Q235	294.965
12	330	Q235	334.152	304	Q235	338.835
14	385	Q235	338.642	354	Q235	347.589

summarizes the maximum stresses experienced by the structural members corresponding to different values of ground tension deformation. As observed in Figure 7, when the horizontal tension deformation is less than 4 mm/m, the most vulnerable structural components are the cross-diagonal members located adjacent to the supports of the tower legs. As the deformation exceeds 4 mm/m, the failure region progressively transitions from the tower legs toward the main body of the tower structure. As the ground tension deformation value increases, both the tensile and compressive stresses in the structural members exhibit a corresponding increase.

As indicated in Figure 7 and Table 5, when the horizontal tensile deformation reaches 8 mm/m, the tensile stress in the Q235 steel members reaches 251.768 MPa, exceeding the allowable stress of 235 MPa for the structural material. Therefore, the horizontal tensile deformation value of 8 mm/m can be regarded as the critical ground tensile deformation value for this transmission tower structure.

Figure 8 presents the fitted curves illustrating the relationship between the maximum compressive and tensile stresses in the structural members and the value of tension deformation under various conditions, with the corresponding fitting equations provided in Equations (5) and (6).

$${\sigma }_c=-2.3465{\epsilon }_t^2+62.67893{\epsilon }_t-75.55,R^2=0.99942$$

(5)

$${\sigma }_t=-1.4465{\epsilon }_t^2+49.68682{\epsilon }_t-57.98,R^2=0.99954$$

(6)

As demonstrated in Figure 8 and Equation (5), the maximum compressive stress in the structural members exhibits a quadratically nonlinear monotonic increase with increasing ground tensile deformation. When the deformation remains below 8 mm/m, the rate of increase is particularly pronounced; beyond this threshold, however, the parabolic growth rate attenuates. Furthermore, according to Figure 8 and Equation (6), the maximum tensile stress also demonstrates a quadratically nonlinear monotonic increase in response to greater tensile deformation, with the parabolic growth rate decelerating after the deformation exceeds 10 mm/m.

Under individual deformation conditions, the critical deformation values for the tower were determined as 35 mm/m for ground tilt, 10 mm/m for horizontal compression, and 8 mm/m for horizontal tension. The results demonstrate that the tower exhibits the highest resistance to ground tilt deformation, while being most sensitive to horizontal tension deformation, with members prone to failure under such conditions. Under ground tilt deformation, the most vulnerable members of the tower are located at the fixed supports of the tower legs. In contrast, under horizontal compression and tension deformation, failure initiates in the cross-diagonal members immediately above the supports of the tower legs. With increasing deformation, the critical failure region progressively propagates toward the main body of the tower. Throughout this process, both the tensile and compressive stresses in the members increase correspondingly. Both the maximum compressive and tensile stresses in the structural members exhibit a quadratically nonlinear increase, with the growth rate gradually decelerating as the deformation value rises.

3.1.2. Stress Distribution Characteristics in Structural Members Under Combined Deformation Conditions

Figure 9 presents contour plots of stress distribution in the transmission tower under varying combined tilt–compression deformation conditions, while

Table 6. Maximum stresses in structural members under tilt–compression deformation values.

i + εc (mm/m)	Member with Maximum Compressive Stress			Member with Maximum Tensile Stress
	Number	Type	σc (MPa)	Number	Type	σt (MPa)
2	6	Q235	83.694	951	Q235	63.404
4	6	Q235	157.882	951	Q235	124.334
6	6	Q235	231.903	951	Q235	185.259
8	6	Q235	304.126	951	Q235	245.985
10	6	Q235	363.762	951	Q235	293.793
12	6	Q235	420.651	951	Q235	340.310
14	6	Q235	462.679	951	Q235	374.592

summarizes the maximum stresses in the structural members corresponding to different values of combined tilt–compression deformation. As observed in Figure 9, the maximum tensile stress in the transmission tower occurs at the cross-diagonal members above fixed support D, while the maximum compressive stress is located at the cross-diagonal members above fixed support C. With increasing combined tilt–compression deformation, both the tensile and compressive stresses in the structural members exhibit a corresponding increase.

As indicated in Figure 9 and Table 6, when the combined tilt–compression deformation reaches 8 mm/m, the tensile stress in the Q235 steel members of the transmission tower reaches 245.985 MPa, exceeding the allowable stress of 235 MPa for the material. Hence, the combined tilt–compression deformation value of 8 mm/m can be identified as the critical ground deformation value under this combined condition for the tower structure.

Figure 10 shows the fitted curves of the relationship between the maximum stresses in the structural members and the values of combined tilt–compression deformation, with the corresponding fitting equations provided in Equations (7) and (8).

$${\sigma }_c=-0.80843{(i\cdot {\epsilon }_c)}^2+44.97683(i\cdot {\epsilon }_c)-5.89843,R^2=0.99958$$

(7)

$${\sigma }_t=-0.68785{(i\cdot {\epsilon }_c)}^2+37.32792(i\cdot {\epsilon }_c)-11.07014,R^2=0.99949$$

(8)

As derived from Figure 10 and Equations (7) and (8), both the maximum compressive and tensile stresses in the structural members exhibit a quadratically nonlinear relationship with the combined tilt–compression deformation values. As the deformation value increases, these stresses demonstrate a quadratically nonlinear monotonic increase. However, when the combined tilt–compression deformation exceeds 10 mm/m, the growth rate decelerates noticeably.

Figure 11 presents contour plots of the stress distribution in the transmission tower under varying combined tilt–tension deformation conditions, while

Table 7. Maximum stresses in structural members under tilt–tension deformation values.

i + εt (mm/m)	Member with Maximum Compressive Stress			Member with Maximum Tensile Stress
	Number	Type	σc (MPa)	Number	Type	σt (MPa)
2	963	Q235	80.914	8	Q235	94.907
4	963	Q235	140.268	8	Q235	169.503.
6	963	Q235	199.515	8	Q235	244.165
8	963	Q235	258.609	8	Q235	318.841
10	963	Q235	317.392	8	Q235	393.353
12	963	Q235	375.448	8	Q235	467.244

summarizes the maximum stresses in the structural members corresponding to different values of combined tilt–tension deformation. As shown in Figure 11, the maximum tensile stress in the transmission tower occurs in the cross-diagonal members above movable support A, while the maximum compressive stress is located in the cross-diagonal members above movable support B. With increasing combined tilt–tension deformation values, both the tensile and compressive stresses in the structural members show a corresponding increase.

As indicated in Figure 11 and Table 7, when the combined tilt–tension deformation reaches 6 mm/m, the tensile stress in the Q235 steel members reaches 244.165 MPa, exceeding the allowable stress of 235 MPa for the material. Hence, the combined tilt–tension deformation value of 6 mm/m can be identified as the critical ground deformation value under this combined condition for the transmission tower structure.

Figure 12 shows the fitted curves of the relationship between the maximum stresses in the structural members and the values of combined tilt–tension deformation with the corresponding fitting equations provided in Equations (9) and (10).

$${\sigma }_c=29.4734(i\cdot {\epsilon }_t)+22.37753,R^2=0.99998$$

(9)

$${\sigma }_t=37.2553(i\cdot {\epsilon }_t)+20.54442,R^2=0.99996$$

(10)

As derived from Figure 12 and Equations (9) and (10), both the maximum compressive and tensile stresses in the structural members exhibit a linear positive correlation with the combined tilt–tension deformation values. These stresses demonstrate a linear monotonic increase with rising deformation values.

Under combined deformation conditions, the critical deformation value for combined tilt–tension was observed to be 6 mm/m, which is lower than that of combined tilt–compression (8 mm/m), indicating that the former is more likely to induce failure in the structural members. In both conditions, failure initiated in the cross-diagonal members located immediately above the supports. Under combined tilt–compression deformation, the maximum compressive and tensile stresses in the members exhibit a quadratically nonlinear increase, with a decelerating growth rate as deformation progresses. In contrast, under combined tilt–tension deformation, these stresses demonstrate a distinct linear monotonic increasing trend. Compared to individual deformation conditions, the critical deformation values under combined conditions are significantly lower, indicating higher stress concentrations in the structural members and greater susceptibility to failure of the tower.

3.2. Characteristics of Displacement Distribution in Tower Under Five Deformation Conditions

Based on the simulation schemes detailed in Table 1 and Table 2, due to space constraints, only selected contour cloud plots of displacement distribution in the tower under specific ground deformation values are presented. In the figures, “MX” denotes the location of maximum displacement, while “MN” indicates the location of minimum displacement. The center of the tower head (elevation: 40.8 m; Node: 155) and the center of the cross-arm (elevation: 40.2 m; Node: 370) were selected as monitoring points. The displacements at these points were extracted to calculate the tilt angle (q) of the tower, and the results are summarized in the table. Based on the displacements of the monitoring points and the tilt angle of the tower compiled in the table, fitted curves illustrating the relationships between the ground deformation values and the displacement components in the x-, y-, and z-directions (Sx, Sy, Sz), as well as the tower tilt angle (q), were plotted. Additionally, the fitting expression describing the relationship between the tower tilt angle (q) and the ground deformation values is provided.

3.2.1. Displacement Distribution Characteristics in Tower Under Individual Deformation Conditions

Figure 13 presents contour plots of the displacement distribution in the transmission tower under various ground tilt deformation conditions.

Table 8. Displacement of monitoring points and tower tilt angle under tilt deformation.

i (mm/m)	Sx (mm)		Sy (mm)		Sz (mm)		S (mm)	q (‰)
	N.155	N.370	N.155	N.370	N.155	N.370
3	0.0138	0.0135	−13.59	−16.91	121.91	120.09	121.91	2.72
6	0.0269	0.0263	−24.48	−31.11	243.81	240.17	243.81	5.24
10	0.0443	0.0439	−39.58	−50.62	400.64	400.26	400.64	8.93
15	0.0663	0.0652	−59.37	−75.91	609.51	600.33	609.51	13.59
20	0.0922	0.0865	−80.19	−102.19	812.64	800.35	812.64	18.12
25	0.1152	0.1125	−102.04	−129.48	1015.19	999.76	1015.19	22.64
30	0.1381	0.1348	−124.89	−157.75	1217.41	1198.83	1217.41	27.15
35	0.1598	0.1588	−148.76	−186.98	1419.08	1397.34	1419.08	31.65
40	0.1842	0.1775	−173.63	−217.22	1620.18	1595.26	1620.18	36.13
50	0.2321	0.2318	−226.35	−280.56	2021.99	1990.64	2021.99	45.09
60	0.2762	0.2751	−268.55	−330.71	2423.39	2387.13	2423.39	54.05

summarizes the displacements of the monitoring points and the tilt angle of the tower under these ground tilt deformation conditions. Figure 14 shows the fitted curves of the relationships between the displacement components (Sx, Sy, Sz), the tower tilt angle (q), and the ground tilt values. As illustrated in Figure 13, under a specific ground tilt deformation value, the minimum displacement of the transmission tower occurs at fixed support D, while the maximum displacement is observed at the top of the tower. Furthermore, as the ground tilt value increases, the displacement progressively increases from the base to the top of the tower, with a corresponding rise in displacement across all structural components.

Equation (11) provides the fitting equation for the relationship between the tower tilt angle (q) and the ground tilt deformation value.

$$q=0.90271i-0.00798,R^2=0.9999$$

(11)

As can be seen from Table 8 and Figure 14, the displacement in the z-direction (Sz) is significantly greater than those in the x- and y-directions (Sx and Sy), with Sx being the smallest. Furthermore, Sx, Sy, Sz, and the tower tilt angle (q) all exhibit positive linear correlations with the ground tilt values. As the ground tilt values increase, the displacements Sx, Sy, Sz, and the tower tilt angle (q) exhibit a linear monotonic increase, with the tower tilt angle (q) being approximately 0.903 times the ground tilt value.

Figure 15 presents contour plots of the displacement distribution in the transmission tower under various ground compression deformation conditions.

Table 9. Displacement of monitoring points and tower tilt angle under compression deformation.

εc (mm/m)	Sx (10−4 mm)		Sy (mm)		Sz (mm)		S (mm)	q (‰)
	N.155	N.370	N.155	N.370	N.155	N.370
2	−0.91	26.8	−2.3248	−2.3249	−7.99575	−7.99642	−7.99575	0.178
4	−9.76	41.3	−1.6018	−1.6019	−15.99617	−15.99683	−15.99617	0.357
6	−18.51	55.8	−0.8986	−0.8987	−23.99661	−23.99721	−23.99661	0.535
8	−27.32	70.4	−0.2155	−0.2154	−31.99705	−31.99762	−31.99705	0.714
10	−35.94	84.8	0.4474	0.4474	−39.99752	−39.99818	−39.99752	0.892
12	−44.63	99.4	1.0889	1.0889	−47.99799	−47.99863	−47.99799	1.070
14	−53.21	114.4	1.7066	1.7067	−55.99848	−55.99914	−55.99848	1.245
16	−61.32	128.8	2.2972	2.2973	−63.99895	−63.99972	−63.99895	1.427

summarizes the displacements of the monitoring points and the tilt angle of the tower under these ground compression deformation conditions. Figure 16 shows the fitted curves of the relationships between the displacement components (Sx, Sy, Sz), the tower tilt angle (q), and the ground compression values. As observed in Figure 15, under a specific compression deformation value, the maximum displacement of the transmission tower occurs at movable support B, while the minimum displacement is located at fixed support D. The differential horizontal movement of the transmission tower supports resulted in contraction of the tower legs, leading to progressive distortion of the upper sections. As the compression deformation increased, displacements throughout the tower structure exhibited a corresponding rise.

Equation (12) provides the fitting equation for the relationship between the tower tilt angle (q) and the ground compression deformation value.

$$q=0.08908{\epsilon }_c+0.00051,R^2=0.9999$$

(12)

As can be seen from Table 9 and Figure 16, the displacement in the z-direction (Sz) is significantly greater than those in the x- and y-directions (Sx and Sy), with Sx being the smallest and approaching nearly zero. Furthermore, Sx, Sy, Sz, and the tower tilt angle (q) all exhibit positive linear correlations with the ground compression values. As the ground compression values increase, the displacements Sx, Sy, Sz, and the tower tilt angle (q) exhibit a linear monotonic increase, with the tower tilt angle (q) being approximately 0.089 times the ground compression deformation value.

Figure 17 presents contour plots of the displacement distribution in the transmission tower under various ground tension deformation conditions.

Table 10. Displacement of monitoring points and tower tilt angle under tension deformation.

εt (mm/m)	Sx (10−4 mm)		Sy (mm)		Sz (mm)		S (mm)	q (‰)
	N.155	N.370	N.155	N.370	N.155	N.370
2	17.1	16.8	−3.8299	−3.8312	8.00505	8.00521	8.00505	0.178
4	26.4	25.5	−4.6119	−4.6122	16.00543	16.00474	16.00543	0.357
6	35.2	34.9	−5.4135	−5.4137	24.00578	24.00509	24.00578	0.535
8	45.2	44.2	−6.2339	−6.2341	32.00612	32.00542	32.00612	0.714
10	54.7	53.7	−7.0713	−7.0716	40.00642	40.00571	40.00642	0.892
12	63.8	62.9	−7.9184	−7.9186	48.00665	48.00595	48.00665	1.070
14	72.9	71.9	−8.7631	−8.7632	56.00681	56.00615	56.00681	1.249

summarizes the displacements of the monitoring points and the tilt angle of the tower under these ground tension deformation conditions. Figure 18 shows the fitted curves of the relationships between the displacement components (Sx, Sy, Sz), the tower tilt angle (q), and the ground tension values. As observed in Figure 17, under a specific tension deformation value, the maximum displacement of the transmission tower occurs at movable support B, while the minimum displacement is located at fixed support C. As the ground tension deformation values increase, the displacements in various parts of the transmission tower exhibit a corresponding increase.

Equation (13) provides the fitting equation for the relationship between the tower tilt angle (q) and the ground tension deformation value.

$$q=0.08921{\epsilon }_t-0.00014,R^2=0.9999$$

(13)

As can be seen from Table 10 and Figure 18, the displacement in the z-direction (Sz) is significantly greater than those in the x- and y-directions (Sx and Sy), with Sx being the smallest and approaching nearly zero. Furthermore, Sx, Sy, Sz, and the tower tilt angle (q) all exhibit positive linear correlations with the ground tension values. As the ground tension values increase, the displacements Sx, Sy, Sz, and the tower tilt angle (q) exhibit a linear monotonic increase, with the tower tilt angle (q) being approximately 0.089 times the ground tension deformation value.

Under individual deformation conditions, the displacement distribution characteristics of the tower are as follows. During ground tilt deformation, the minimum displacement occurs at fixed supports C and D, while the maximum displacement appears at the tower top. In contrast, under horizontal compression or tension deformation, the minimum displacement remains located at fixed supports C and D, whereas the maximum displacement is observed at movable supports A and B of the tower. The displacement components Sx, Sy, and Sz induced by ground tilt deformation are significantly greater than those induced by horizontal compression and tension deformation, while the displacements under horizontal compression and tension conditions exhibit relatively minor differences. Under these three deformation conditions, the displacement components (Sx, Sy, Sz) and the tower tilt angle (q) exhibit positive linear correlations with the ground deformation values. Specifically, under ground tilt deformation, the tower tilt angle is approximately 0.903 times the ground tilt value, whereas under horizontal compression or tension deformation, the tower tilt angle is only about 0.089 times the corresponding deformation value. This indicates that ground tilt deformation has a significantly higher impact on the tilt behavior of the tower compared to horizontal compression and tension deformation.

3.2.2. Displacement Distribution Characteristics in Tower Under Combined Deformation Conditions

Figure 19 presents contour plots of the displacement distribution in the transmission tower under various ground tilt–compression deformation conditions.

Table 11. Displacement of monitoring points and tower tilt angle under tilt–compression deformation.

i + εc (mm/m)	Sx (10−3 mm)		Sy (mm)		Sz (mm)		S (mm)	q (‰)
	N.155	N.370	N.155	N.370	N.155	N.370
2	10.41	8.29	−10.822	−13.212	89.1787	87.9145	89.1787	1.989
4	19.84	15.29	−18.732	−23.133	177.9141	175.4997	177.9141	3.968
6	29.27	22.31	−26.843	−33.434	266.3993	262.7785	266.3993	5.941
8	38.76	29.55	−35.131	−43.891	354.5725	349.7456	354.5725	7.908
10	48.67	37.61	−43.614	−54.531	442.4343	436.4017	442.4343	9.867
12	59.89	48.01	−52.245	−65.322	529.9849	522.7471	529.9849	11.819
14	71.21	59.79	−61.032	−76.251	617.2243	608.7817	617.2243	13.765

summarizes the displacements of the monitoring points and the tilt angle of the tower under these ground tilt–compression deformation conditions. Figure 20 shows the fitted curves of the relationships between the displacement components (Sx, Sy, Sz), the tower tilt angle (q), and the ground tilt–compression values. As observed in Figure 20, under a specific combined tilt–compression deformation value, the minimum displacement of the transmission tower is located at fixed supports C and D, while the maximum displacement occurs at the top of the tower. As the combined tilt–compression deformation values increase, displacements in all components of the transmission tower exhibit a corresponding increase.

Equation (14) provides the fitting equation for the relationship between the tower tilt angle (q) and the ground tilt–compression deformation value.

$$q=0.98136(i\cdot {\epsilon }_c)+0.043,R^2=0.9999$$

(14)

As can be seen from Table 11 and Figure 20, the displacement in the z-direction (Sz) is significantly greater than those in the x- and y-directions (Sx and Sy), with Sx being the smallest. Furthermore, Sx, Sy, Sz, and the tower tilt angle (q) all exhibit positive linear correlations with the ground tilt–compression values. As the ground tilt–compression values increase, the displacements Sx, Sy, Sz and the tower tilt angle (q) exhibit a linear monotonic increase, with the tower tilt angle (q) being approximately 0.981 times the ground tilt–compression deformation value.

Figure 21 presents contour plots of the displacement distribution in the transmission tower under various ground tilt–tension deformation conditions.

Table 12. Displacement of monitoring points and tower tilt angle under tilt–tension deformation.

i + εt (mm/m)	Sx (10−3 mm)		Sy (mm)		Sz (mm)		S (mm)	q (‰)
	N.155	N.370	N.155	N.370	N.155	N.370
2	8.67	11.27	−9.342	−11.524	73.4124	72.2235	73.4124	1.637
4	16.72	21.41	−15.745	−20.124	147.5148	145.2145	147.5148	3.289
6	24.77	31.67	−22.305	−28.972	221.2448	217.5816	221.2448	4.934
8	33.01	42.03	−29.085	−37.986	295.6324	290.7298	295.6324	6.593
10	41.37	52.51	−36.051	−47.193	370.3418	364.1904	370.3418	8.259
12	49.87	63.12	−43.207	−56.596	445.3736	437.9641	445.3736	9.933

summarizes the displacements of the monitoring points and the tilt angle of the tower under these ground tilt–tension deformation conditions. Figure 22 shows the fitted curves of the relationships between the displacement components (Sx, Sy, Sz), the tower tilt angle (q), and the ground tilt–tension values. As observed in Figure 21, under a specific combined tilt–tension deformation value, the minimum displacement of the transmission tower is located at fixed supports C and D, while the maximum displacement occurs at the top of the tower. As the combined tilt–tension deformation values increase, displacements in all components of the transmission tower exhibit a corresponding increase.

Equation (15) provides the fitting equation for the relationship between the tower tilt angle (q) and the ground tilt–tension deformation value.

$$q=0.82927(i\cdot {\epsilon }_t)-0.03073,R^2=0.9999$$

(15)

As can be seen from Table 12 and Figure 22, the displacement in the z-direction (Sz) is significantly greater than those in the x- and y-directions (Sx and Sy), with Sx being the smallest. Furthermore, Sx, Sy, Sz, and the tower tilt angle (q) all exhibit positive linear correlations with the ground tilt–tension values. As the ground tilt–tension values increase, the displacements Sx, Sy, Sz, and the tower tilt angle (q) exhibit a linear monotonic increase, with the tower tilt angle (q) being approximately 0.829 times the ground tilt–tension deformation value.

Under combined deformation conditions, the displacement distribution characteristics of the tower are summarized as follows. Under both combined tilt–compression and tilt–tension deformation, the minimum displacement occurs at fixed supports C and D, while the maximum displacement is observed at the tower top. Under both conditions, the displacement components (Sx, Sy, Sz) and the tower tilt angle (q) exhibit positive linear correlations with the ground deformation values. Specifically, under combined tilt–compression deformation, the tower tilt angle is approximately 0.981 times the combined tilt–compression deformation value, while under combined tilt–tension deformation, the tower tilt angle is about 0.829 times the combined tilt–tension deformation value. Compared to individual deformation conditions, where the tower tilt angle is approximately 0.089 times the values of horizontal compression or tension deformation, the tilt angle under combined deformation conditions exhibits a significantly higher value. This further demonstrates that ground tilt deformation plays a dominant role in influencing the tilt behavior of the transmission tower.

3.3. Characteristics of Support Reaction Distribution Under Five Deformation Conditions

To clearly investigate the variation of support reaction forces of the tower with ground deformation values, the support reactions under five deformation conditions were obtained following the simulation schemes outlined in Table 1 and Table 2. The support reactions under these five deformation conditions were compiled into a table, and curve fitting was conducted to characterize the relationship between the support reactions of the transmission tower and the ground deformation values.

3.3.1. Support Reaction Distribution Characteristics Under Individual Deformation Conditions

Table 13. Y-direction support reaction forces FY under different ground tilt values.

i (mm/m)	Support A (kN)	Support B (kN)	Support C (kN)	Support D (kN)
3	27.228	27.373	31.035	30.911
6	25.323	25.607	32.933	32.683
10	22.783	23.253	35.465	35.046
15	19.606	20.311	38.631	38.011
20	16.429	17.369	41.794	40.956
25	13.912	15.147	44.183	43.258
30	11.859	13.351	46.169	45.17
35	11.141	12.393	47.201	45.811
40	10.847	12.073	47.543	46.066
50	10.609	12.041	47.692	46.216
60	10.266	11.905	47.831	46.429

presents the Y-direction support reaction force FY under tilt deformation, while Figure 23 provides the relationship curve between the support reaction force FY and the tilt deformation values.

As can be seen from Table 13 and Figure 23, under different ground tilt values, the reaction forces at all four supports of the tower remain consistently positive. Among these, the reaction forces at the two fixed supports C and D exhibit a nonlinear increasing trend, with both demonstrating closely similar patterns of growth. When the tilt value reached 35 mm/m, the growth rate of the reaction forces at fixed supports C and D gradually stabilized.

Under a given ground tilt value, the reaction force at Support D is observed to be less than that at Support C. In contrast, the reaction forces at the two movable supports A and B exhibit a nonlinear decreasing trend, with both demonstrating closely similar patterns of reduction. When the tilt value reaches 35 mm/m, the reduction rate of the reaction forces at supports A and B gradually stabilized. Under a given ground tilt value, the reaction force at support A is observed to be less than that at support B.

Table 14. Z-direction support reaction forces FZ under different ground compression and tension values.

εc/εt (mm/m)	Support A		Support B		Support C		Support D
	FZ (kN)		FZ (kN)		FZ (kN)		FZ (kN)
2	−7.825	1.702	−7.823	1.684	7.824	−1.701	7.824	−1.686
4	−12.569	6.473	−12.562	6.441	12.567	−6.469	12.565	−6.445
6	−17.293	11.232	−17.283	11.182	17.289	−11.225	17.287	−11.188
8	−21.987	15.944	−21.976	15.871	21.982	−15.932	21.981	−15.883
10	−26.629	20.496	−26.619	20.389	26.623	−20.472	26.626	−20.412
12	−31.163	24.501	−31.154	24.343	31.155	−24.452	31.162	−24.392
14	−35.434	27.414	−35.429	27.195	35.425	−27.334	35.438	−27.275
16	−39.169		−39.168		39.161		39.177

presents the Z-direction support reaction force FZ under compression and tension deformation, while Figure 24 provides the relationship curve between the support reaction force FZ and the compression and tension deformation values.

As can be seen from Table 14 and Figure 24, under horizontal compression deformation, the reaction forces at movable supports A and B exhibit a linearly increasing trend in the negative direction, with both demonstrating closely similar patterns of change. Conversely, the reaction forces at fixed supports C and D show a linearly increasing trend in the positive direction, also exhibiting highly consistent variation patterns. Under a specific compression deformation value, the reaction forces at the two movable supports and those at the two fixed supports are observed to be numerically equal but opposite in direction. The variation pattern of support reaction forces under horizontal tension deformation closely resembles that observed under compression deformation. However, under equivalent deformation magnitudes, the support reaction forces induced by compression are significantly greater than those generated under tension conditions.

3.3.2. Support Reaction Distribution Characteristics Under Combined Deformation Conditions

Table 15. Y- and Z-direction support reaction forces FY and FZ under different ground tilt–compression.

i + εc (mm/m)	Support A (kN)		Support B (kN)		Support C (kN)		Support D (kN)
	FY	FZ	FY	FZ	FY	FZ	FY	FZ
2	27.802	−7.824	28.031	−7.822	30.331	7.816	30.384	7.828
4	26.468	−12.564	26.942	−12.551	31.505	12.547	31.631	12.568
6	25.131	−17.28	25.872	−17.259	32.662	17.253	32.879	17.289
8	23.793	−21.963	24.822	−21.935	33.801	21.926	34.131	21.973
10	22.453	−26.593	23.788	−26.559	34.923	26.544	35.383	26.608
12	21.111	−31.116	22.769	−31.077	36.031	31.055	36.636	31.139
14	19.768	−35.383	21.759	−35.339	37.131	35.309	37.889	35.413

presents the Y- and Z-direction support reaction force FY and FZ under tilt–compression deformation, while Figure 25 provides the relationship curve between the support reaction force FY and FZ and the tilt–compression deformation values.

As indicated in Table 15 and Figure 25, under combined tilt–compression deformation, the Y-direction reaction forces at all four supports of the tower remain positive. Among these, the Y-direction reaction forces at the fixed supports C and D exhibit a linear increasing trend, with both demonstrating closely similar patterns of growth. Conversely, the Y-direction reaction forces at the movable supports A and B show a linear decreasing trend, with both exhibiting closely similar patterns of reduction. Under a specified tilt–compression deformation value, the Y-direction reaction force at Support D exceeds that at Support C, while the Y-direction reaction force at Support B is greater than that at Support A. The Z-direction reaction forces at the fixed supports C and D exhibit a positive increasing trend, while those at the movable supports A and B show a negative increasing trend. Under a specified tilt–compression deformation value, the Z-direction reaction forces at movable supports A and B are observed to be numerically equal but opposite in direction to those at fixed supports C and D.

Table 16. Y- and Z-direction support reaction forces FY and FZ under different ground tilt–tension.

i + εt (mm/m)	Support A (kN)		Support B (kN)		Support C (kN)		Support D (kN)
	FY	FZ	FY	FZ	FY	FZ	FY	FZ
2	27.913	1.703	28.211	1.688	30.462	−1.707	30.269	−1.683
4	26.672	6.478	26.972	6.451	31.784	−6.486	31.419	−6.443
6	25.411	11.247	25.745	11.203	33.102	−11.258	32.666	−11.192
8	24.129	15.972	24.422	15.907	34.414	−15.983	33.781	−15.895
10	22.826	20.538	23.108	20.442	35.719	−20.543	34.995	−20.437
12	21.506	24.526	21.811	24.377	37.006	−24.507	36.225	−24.396

presents the Y and Z-direction support reaction force FY and FZ under tilt–tension deformation, while Figure 26 provides the relationship curve between the support reaction force FY and FZ and the tilt–tension deformation values.

As observed in Table 16 and Figure 26, similar to the behavior under combined tilt–compression deformation, the Y-direction reaction forces at all four supports of the tower remain positive under combined tilt–tension deformation conditions. Among these, the Y-direction reaction forces at the fixed supports C and D exhibit a linear increasing trend, with both demonstrating closely similar patterns of growth. Conversely, the Y-direction reaction forces at the movable supports A and B show a linear decreasing trend, with both demonstrating closely similar patterns of reduction. Under a specified combined tilt–tension deformation value, the Y-direction reaction force at Support D is lower than that at Support C, while the Y-direction reaction force at Support A is lower than that at Support B. The Z-direction reaction forces at fixed supports C and D exhibit a negative increasing trend, while those at movable supports A and B demonstrate a positive increasing trend. Under a specified combined tilt–tension deformation value, the Z-direction reaction forces at supports A and B are numerically equal but opposite in direction to those at supports C and D. Compared to the combined tilt–compression deformation, the Y-direction reaction forces under tilt–tension deformation are essentially equivalent to those under tilt–compression deformation, whereas the Z-direction reaction forces under tilt–tension deformation are significantly smaller than those under tilt–compression deformation.

4. Conclusions

This article develops a detailed three-dimensional finite element model of a typical self-supporting transmission tower using ANSYS software. The model is used to analyze the stress distribution in structural members, displacement response, and support reaction force characteristics under various deformation conditions. The five critical deformation values obtained in this study only involve results from five typical deformation conditions in the y- and z-directions, and these values may differ under deformation conditions in other directions. The results provide important theoretical support for stability evaluation and risk mitigation of transmission towers located above mined-out areas. The main conclusions are as follows:

(1) Under individual deformation conditions, the critical deformation values for tilt, horizontal compression, and horizontal tension are 35 mm/m, 10 mm/m, and 8 mm/m, respectively, indicating that the tower exhibits stronger resistance to tilt deformation and is most sensitive to horizontal tension. Under combined deformation conditions, the critical values for tilt–compression and tilt–tension are 8 mm/m and 6 mm/m, respectively, which are significantly lower than those under individual deformation conditions. Among these, the tilt–tension condition is the most dangerous condition, substantially reducing the structural safety of the tower and demonstrating that combined deformations are more likely to cause failure.

(2) The failure mechanisms of the transmission tower under different deformation conditions exhibit significant differences. Under individual deformation conditions, failure occurs in the diagonal members near the supports during tilt deformation, whereas under horizontal compression and tension deformation, failure initially appears in the diagonal members adjacent to the supports and gradually propagates toward the main body of the tower as the deformation increases. Under combined deformation conditions, failure consistently occurs in the diagonal members near the supports.

(3) The displacement of the tower head and tilt angle exhibit a linear positive correlation with the ground deformation values. Under individual deformation conditions, the tower tilt angle is approximately 0.903 times the tilt deformation value and 0.089 times the values of horizontal compression and tension deformation, indicating that tilt deformation has a particularly significant influence on the tilt behavior of the tower. Under combined deformation conditions, the tower tilt angle reaches about 0.981 times the value of tilt–compression deformation and 0.829 times the value of tilt–tension deformation. Compared to horizontal compression and tension deformation, the tilt angle under combined conditions shows a notable increase.

(4) Under mining-induced ground movement, redistribution of support reaction force occurs in the transmission tower. Under individual deformation conditions, during ground tilt deformation, the reaction forces at the fixed supports increase nonlinearly, while those at the movable supports decrease nonlinearly. In contrast, under horizontal compression and tension deformation, the reaction forces at all four supports increase linearly and significantly. Under combined deformation conditions, both the Y-direction and Z-direction reaction forces exhibit linear increasing trends during both tilt–compression and tilt–tension deformation.