Nonlinear Pushover Analysis of Retrofitted Transmission Towers Under Static Wind Loads

Abstract

This study numerically investigated retrofitted transmission towers subjected to static wind loads using nonlinear pushover analysis, emphasizing enhanced structural resilience and safety. Transmission towers are critical infrastructure that must withstand extreme wind conditions. However, aging structures and increasing load demands necessitate effective retrofitting strategies. The nonlinear pushover analysis employs advanced finite element modeling to simulate the nonlinear inelastic behavior of towers under incremental static wind loads until failure. Six retrofitting methods are presented and compared to identify the most effective retrofitting approach for the considered tower. The findings reveal that retrofitting significantly improves the capacity, ductility, and stiffness of transmission towers under static wind loads, delaying buckling and failure. The proposed retrofit method enhances tower capacity by at least 10% compared to the non-retrofitted configuration.

1. Introduction

Many steel lattice towers have been in operation for over 20 years, and their current load-carrying capacity is insufficient to meet today’s demands. This is primarily due to changes in design codes relevant to wind loads and the increased need for electrical power. For this reason, these towers are often subjected to loads that exceed their original design limits, leading to failures in several countries. To avoid further collapses, it is essential to improve the capacity of these towers to support higher load requirements. Upgrading existing towers is preferred rather than building new ones, mainly for environmental reasons. Due to the high costs associated with power interruptions, any upgrades must be carried out while the towers remain operational.

There are two typical retrofit methods of transmission towers: reinforcement of diaphragm bracing systems (horizontal members and diagonal members) and reinforcement of chord members (tower legs). These typical members are presented in Figure 1. The first method is employed to provide additional buckling resistance to the panels by reducing the slenderness ratios of long compression members. Albermani et al. [1] enhanced the strength of steel structures by adding diaphragm bracings, which significantly increased the buckling load of diagonal members in both bending and torsion, improving it by a factor of 2.56 to 3.89. Without the strengthening, the sub-assemblages exhibited brittle behavior, whereas the retrofitted structures demonstrated more ductile behavior with reduced deformation in the diagonal bracings [2]. The introduction of diaphragms altered the buckling mode of the sub-assemblage members, shifting from flexural–torsional buckling to flexural buckling, which occurred at a higher load. This indicated that the diagonal members were effective in providing lateral support to the main legs. Qiang and Jian [3] proposed a retrofitting technique on diagonal bracings to enhance the load-carrying capacity of existing transmission towers based on a thorough exploration of their failure mechanism. A pair of double-panel sub-assemblages of a transmission tower with and without retrofitted diagonal members were tested to investigate the failure modes and retrofitting effects. Experimental results revealed that the ultimate lateral and vertical loads that could be applied on the retrofitted sub-assemblage were significantly enhanced (by 80% and 77%, respectively) compared with the original one, highlighting the efficiency of the proposed retrofitting technique.

The second method focuses on enhancing the capacity of existing tower legs by retrofitting them with additional angle members, commonly referred to as “reinforcing members”, which are attached parallel to the existing legs from the base upward. Various tower member cross-sections are typically used, such as angle sections for bracing members, angle and C-shaped sections for horizontal members, and angle and circular pipe sections for tower legs. Numerous studies have proposed retrofit methods to improve the capacity of tower legs [4,5,6,7,8,9,10]. For example, Balagopal et al. [4] found that, among five retrofitting patterns, the cleat angle and double cross-plate connections significantly enhanced compression strength compared to other connection types. The cleat angle connection was also more cost-effective. Additionally, the choice of strengthening pattern influences the failure mode of the main angle member. Li et al. [5] introduced a new steel angle reinforcement method for prefabricated composite structures. This method offers several advantages over traditional steel angle reinforcements, including easier installation, no need for welding or drilling, and other benefits. Their results indicated that the bearing capacity of the reinforced steel angles increased by 39% to 174%, highlighting the significant impact of this non-destructive reinforcement method. Recently, Liang et al. [9] proposed an innovative clamp-type (IC) method that does not require drilling into the core leg members of the tower, bringing considerable convenience in strengthening members at the construction site. It was found that the bearing capacity of the compound members with the IC interconnector was approximately 41–62% higher than that of the single angle member. Similar to Liang et al. [9], Liu et al. [10] recommended an innovative clamp reinforcing technique for the main leg and diagonal members of operational transmission towers. This technique employed star-shaped plates with stiffeners, angle steel clamps, connecting plates, and bolts to connect the reinforcing members to the original members of the transmission tower without drilling or welding. It was demonstrated by performing sub-assemblage tests that the ultimate bearing capacity of the reinforced structure can be enhanced by 21.96% compared with the original structure while maintaining a strong synergy between the reinforcing and original members. Guo et al. [11] investigated the mechanical behavior of in-service angle steel members strengthened by welding through an experimental and numerical study. The test results indicated that the strengthened members had a higher bearing capacity and better ductility than the nonstrengthened ones. Square tube and cross-shaped strengthened sections, and a strengthening member with the same thickness as the base member, were recommended. Gorripotu and Raghavan [12] assessed the efficiency of the retrofitting method using bi-angled cruciform sections under preload conditions through numerical investigation. The analysis results demonstrated that the proposed retrofitting can enhance the tower capacity by up to 70% compared to a non-retrofitted one.

To the best of the authors’ knowledge, all existing studies have proposed retrofit methods to reinforce the capacity of transmission towers by investigating the effects of these methods on individual tower members or tower panels under concentric or eccentric compressive loadings, rather than on the entire tower structure under wind loading. However, this approach may not accurately reflect the actual behavior of the retrofitted tower under wind loading, as the interaction between individual tower members and its impact on the overall tower behavior has not been considered. Additionally, no study has been conducted to investigate the effects of retrofitting on the nonlinear inelastic response of entire transmission towers under wind loading. The effects of retrofitting methods on the tower under various wind directions have also not been investigated. Therefore, it is crucial to conduct a study to investigate the effects of different retrofitting methods on the nonlinear behavior of transmission towers subjected to various wind directions and to propose a suitable retrofit method.

On the other hand, in practical design, fragility analysis is a critical tool in assessing the vulnerability of transmission towers to wind-induced failures. This analysis helps in understanding the probability of structural failure under varying wind conditions, thereby informing design improvements and mitigation strategies. Numerous works have been conducted on the fragility analysis of transmission towers under extreme winds [13,14,15,16,17], in which several fragility analysis methods have been proposed to evaluate the collapse of the tower [18,19,20]. Based on the fragility analysis results, the critical wind speed for the tower under wind loading can be determined for practical design purposes. However, to the best of the authors’ knowledge, fragility analyses have only been conducted for non-retrofitted towers and not retrofitted ones under wind loading.

As stated above, research gaps have been outlined for the investigation of retrofitted transmission towers under wind loading. This work was conducted to address certain research gaps, with the main contribution as follows:

(a)

Investigating the effects of six retrofit methods on the capacity of whole tower structures subjected to static wind loading through nonlinear pushover analysis.

(b)

Investigating the effects of different retrofit methods on the tower capacity under different wind directions.

(c)

Comparing the capacity, stiffness, and ductility between non-retrofitted and retrofitted towers to demonstrate the efficiency of the retrofit method, and suggesting the most suitable retrofit method for the considered tower.

In summary, the general framework for this study is presented in Figure 2.

2. Static Wind Loading Acting on Transmission Towers

In this study, the static wind loading on the lattice transmission tower subjected to the yaw wind angle only was determined according to the IEC 60826:2017 standard [21]. In this standard, the wind loadings include wind loads on supports (tower body), conductors, ground wires, and insulator strings. Since wind loads on insulators strings are small, they were neglected in this work. Only wind loads on supports, conductors, and ground wires were taken into consideration.

For the wind loads on supports, two methods—“wind on panels” and “wind on all tower members”—are presented in the IEC 60826:2017 standard. In the current work, the “wind on panels” method was used to calculate the static wind load on the tower body. To compute the influence of the wind on the lattice tower itself, the whole tower was divided into different panels. For a lattice tower with a square/rectangular cross-section, the wind loading A_t in the direction of the wind, applied at the center of gravity of this panel, made up of various support members, is calculated as follows:

$$A_t=q_0\left(1+0.2{\sin }^{2}2\theta \right)\left(S_{t1}{C}_{xt1}{\cos }^2\theta +S_{t2}{C}_{xt2}{\sin }^2\theta \right)G_t$$

(1)

where

• q₀ is the dynamic reference wind pressure (N/m²), given by

$$q_0={\displaystyle \frac{1}{2}}\tau \mu {(K_R{V}_{RB})}^2$$

(2)

where μ is the air mass per unit volume equal to 1225 kg/m³ at a temperature of 15 °C and an atmospheric pressure of 101.3 kPa at sea level; τ is the air density correction factor taken in the IEC 60826:2017 standard; K_R is the roughness factor in the IEC 60826:2017 standard; V_RB is the reference wind speed for terrain category B; θ is the angle of incidence of the wind direction with the perpendicular to face 1 of the panel in a horizontal plane (Figure 3); S_t1 is the total surface area projected normally on face 1 of the panel (m²); S_t2 is the total surface area projected normally on face 2 of the panel (m²); C_xt1 and C_xt2 are the drag coefficients peculiar to faces 1 and 2 for wind perpendicular to each face given in the IEC 60826:2017 standard for lattice supports made of flat-sided or rounded members, respectively; and G_t is the combined wind factor for the supports given in the IEC 60826:2017 standard for different terrain categories.

For the wind loads on conductors and ground wires, the load due to the impact of the wind pressure on a wind span L, applied at the support and blowing at an angle Ω with the conductors, is determined as follows:

$$A_c=q_0\times C_{xc}\times G_c\times G_L\times d\times L\times {\sin }^2\Omega$$

(3)

where C_xc is the drag coefficient of the conductor taken as equal to 1 for the generally considered stranded conductors and wind speeds; Gc is the combined wind factor for the supports given in the IEC 60826:2017 standard for different terrain categories; G_L is the span factor given in the IEC 60826:2017 standard; d is the diameter of the conductor or ground wire (m); L is the wind span of the support, equal to half the sum of the lengths of adjacent spans of the support; and Ω is the angle between the wind direction and the conductor or ground wire, given in Figure 3.

3. Finite Element Modeling of Non-Retrofitted Towers Using Nonlinear Pushover Analysis

3.1. Nonlinear Pushover Analysis

The nonlinear static pushover analysis (NSP) is firstly employed to estimate the capacity of structures subjected to seismic forces. In this approach, a constant loading profile is applied while incrementally increasing the load to generate load–displacement curves. These curves are utilized to identify the yield and maximum ultimate load of the structure. Recently, the NSP method was used for the estimation of load–displacement responses of transmission towers under wind loads [18,22,23,24]. When applying the NSP method to transmission towers under wind loads, the resulting load–displacement or speed–displacement curves are referred to as capacity curves, from which the tower’s yield and maximum capacities can be determined. In this context, the applied load represents the total wind load, equivalent to the base shear force in NSP analysis. The general capacity curve is presented in Figure 4. In this study, the NSP method was implemented using the Riks analysis [25,26] available in ABAQUS software (version 6.14) [27]. Transmission towers are subject to complex loading conditions, including wind forces, which can induce nonlinear responses such as large deformations and buckling. Traditional linear analysis may not adequately capture these behaviors. The Riks method is well-suited to solving nonlinear problems, especially when post-buckling behavior or large deformations are involved. The nonlinear pushover analysis aims to simulate the tower’s response under incremental loads, in this case, wind loading. The Riks method is particularly effective in tracking equilibrium paths for structures that experience instability or load reversal, which is critical when modeling the wind-induced loads on a tower, especially as they grow in magnitude. Transmission towers may exhibit snap-through or snap-back behaviors, where the structure suddenly changes its configuration under extreme wind loading. The Riks method is designed to handle such instabilities by controlling the path of the solution even when convergence issues arise, ensuring that the analysis can proceed smoothly despite these challenges. Additionally, the Riks method is capable of identifying both ultimate strength and failure mechanisms. This is crucial in wind loading analysis, where the goal is to understand not only the elastic response of the tower but also the behavior at higher load levels when plastic deformations or other failure modes occur. For this reason, the Riks method was successfully applied for the nonlinear pushover analysis of transmission tower in previous works [28,29,30].

3.2. Finite Element Modeling Procedure

In this study, nonlinear pushover analysis was conducted on both non-retrofitted and retrofitted towers using ABAQUS software (version 6.14) [27]. Figure 5 shows a non-retrofitted transmission tower with a height of 79.3 m. The surrounding terrain is assumed to be type B, as defined in IEC 60826:2017 [21]. The reference wind speed is taken as 30.5 m/s, which corresponds to the 100-year wind speed in Gangneung, Korea. The wind span is assumed to be 410 m. The section dimensions and material properties of tower members are presented in

Table 1. Section dimensions and material properties of tower members.

Tower Members	Section Dimensions (mm)	Yield Strength (MPa)	Ultimate Strength (MPa)
Main leg 1	140 × 14	410	540
Main leg 2	120 × 10	410	540
Main cage	100 × 8	410	540
Main cross-arms	90 × 8	250	400
Horizontal braced members	80 × 7	250	400
Other members	50 × 5	250	400

, while material properties of the conductors and ground wires are provided in

Table 2. Properties of conductors and ground wires.

Parameters	Unit	Conductor (2xLGJ-400/35)	Ground Wire (JLB20A-150)
Cross-section area	mm2	425.24	148.07
Diameter	mm	26.82	15.75
Density	kg/m	1.349	0.9894
Maximum tension	kN	78.96	37.70
Breaking tension	%	25	100

.

All tower members are simulated using two-node linear beam elements (B31), allowing for linear elastic transverse shear deformation [18,30,31,32,33]. The cross-section of the tower members is L-shaped, matching the section of the beam elements. The element size of tower members is selected as 10, as suggested in [31]. The material constitutive model is assumed to be elastic–perfectly plastic, as suggested by previous works [18,23,31,32]. The nonlinear regime is defined by the yield strength (f_y) at zero plastic strain and the ultimate strength (f_u) at plastic strain (ε_u) equal to 20%. The detailed stress–strain relationship of steel material is presented in Figure 6.

Wind loads on the transmission towers are divided into two categories: wind loads acting directly on the tower, and wind loads transmitted through conductors and wires. Initially, only the tower’s self-weight is applied to the model, with other types of loading not considered. Static wind loads are then applied to the tower as concentrated forces at specific nodes. The static wind loads acting on each panel are calculated according to the IEC standard [21], and these panel wind loads are converted into nodal wind loads for the tower. Additionally, static wind loads transmitted through conductors and wires are applied to the tower. The final loading on the tower consists of a combination of its self-weight and static wind loads [18,22,32].

Figure 7 illustrates the flowchart for the nonlinear pushover analysis of the transmission tower, which is performed in three phases. The first phase involves a static analysis of the tower under its self-weight. The second phase is an eigenvalue buckling analysis, used to identify the appropriate eigenmode that serves as the initial imperfection for the nonlinear pushover analysis. These initial imperfections, typically arising from the construction process, are incorporated into the nonlinear finite element model based on the lowest buckling eigenmode [23,24,28,33]. This eigenmode is then applied to the tower’s initial geometry, scaled by an imperfection factor of H/1000, where H is the tower height [34,35]. The third phase is the nonlinear static pushover analysis. In this phase, the tower is loaded proportionally according to the force distribution described earlier. The static analysis incorporates geometric nonlinearity to account for large deformations using the Nlgeom option available in the Riks analysis step. The boundary conditions are assumed to be fixed at the tower legs, while tower members are considered to be rigidly connected [18,30,31,32,33]. The loading and boundary conditions for the tower are shown in Figure 8.

3.3. Finite Element Modeling Results

By using the procedure presented in Figure 7, nonlinear pushover analysis of the tower under static yawed wind is implemented. Figure 9 illustrates the failure modes and load–displacement responses of the tower. Based on the locations of the failed members identified in this analysis, six retrofit methods will be proposed and investigated in the next section.

4. Finite Element Modeling of Retrofitted Towers Using Nonlinear Pushover Analysis

This section presents the effects of six retrofit methods on the capacity of non-retrofitted transmission towers under yawed wind using nonlinear pushover analysis. It is noted that all geometric dimensions and material properties of the considered tower members are identical with those of non-retrofitted towers. A suggested retrofit method is to provide additional members on the compressive leg members of the tower. The additional members will be placed as shown in Figure 10.

The general approach is to provide additional members on the yielding members of non-retrofitted towers at two stages, as shown in Figure 9: (a) stage 1—initial yielding members, and (b) stage 2—the ultimate stage. Six retrofit methods are presented in Figure 11. It is noted that the modeling procedure of the retrofitted towers under wind loading is similar to that of the non-retrofitted tower.

5. Results and Discussion

By performing nonlinear pushover analyses of six retrofitted towers under various wind yaw angles, a comparison was made of the load–displacement curves for the non-retrofitted and retrofitted towers, as presented in Figure 12. From this figure, it can be observed that retrofit methods 4, 5, and 6 significantly enhance both the base shear capacity and stiffness of the towers for all considered wind directions. In contrast, the remaining methods (1, 2, and 3) only notably improve the base shear capacity and stiffness for all wind directions except when θ = 180°. This is because methods 1 to 3 only add additional members to the main members located on one side of the tower. When the members on this side are in compression, methods 1 to 3 significantly improve the capacity of the tower. However, when the wind direction shifts to the opposite side, these members will be subjected to tension, rendering methods 1 to 3 insufficient in this case.

Figure 13 presents a comparison of the load–displacement curves of retrofitted towers under different wind directions for the six retrofit methods. In this work, five yaw angles—0° (transverse direction), 90° (longitudinal direction), 45°, 135°, and 180°—are taken into consideration. It can be observed that at a wind direction of 45°, the tower capacity is the lowest compared to the other considered wind directions for almost all retrofit methods, except method 1. This suggests that a wind direction of 45° is the most unfavorable direction for almost all retrofit methods, except method 1. On the other hand, the tower capacity is highest at a wind direction of 90° for all the retrofit methods considered.

Additionally, the tower capacities at θ = 180° obtained from methods 1, 2, 3, 5, and 6 are lower than those at θ = 0°. This is because the length of retrofitting members in the compressive area of the towers at θ = 180° is shorter than that at θ = 0°. The tower capacities at θ = 45° and θ = 135° are identical, as they have the same length of retrofitting members in both compression and tension.

Figure 14 presents the failure modes of the non-retrofitted tower and three retrofitted towers (methods 1 to 3) at the ultimate stage for the transverse wind direction (θ = 0°). There is a significant change in the failure member locations for different retrofit methods. This occurs because when some of the main members in the compressive area of the non-retrofitted tower are strengthened, their buckling capacities are significantly increased, which in turn leads to failure in some members of the tension area. Therefore, the ductility of the towers using these retrofit methods is much higher than that of the non-retrofitted tower, as shown in Figure 12a.

On the other hand, Figure 15 presents the stress contours obtained from the non-retrofitted and retrofitted towers using Method 5. It can be observed that the maximum stresses in both cases exceed the yield strength of the steel material in all tower members (Table 1), indicating that these towers failed due to yielding rather than buckling. A similar observation can be made for the towers retrofitted using the other retrofitting methods.

Table 3. Comparison of maximum base shear forces from six retrofitted methods.

Retrofitted Method	Maximum Base Shear Force (kN)					Difference (%)
	θ = 0°	θ = 45°	θ = 90°	θ = 135°	θ = 180°	θ = 0°	θ = 45°	θ = 90°	θ = 135°	θ = 180°
No retrofit	642.8	394.7	727.4	394.7	642.8
Method 1	699.9	441.5	806.7	410.1	655.3	8.9	11.9	10.9	3.9	2.0
Method 2	740.5	459.2	843.3	476.2	646.2	15.2	16.4	15.5	20.7	0.5
Method 3	808.3	468.3	914.0	503.7	649.0	25.8	18.6	25.6	27.6	1.0
Method 4	698.0	436.1	802.0	436.1	698.0	8.6	10.5	10.2	10.5	8.6
Method 5	744.6	488.7	866.9	500.8	705.9	15.8	23.8	19.2	26.9	9.8
Method 6	914.2	489.1	971.4	489.3	708.8	42.2	23.9	33.5	24.0	10.3

presents our comparison of the maximum base shear forces of retrofitted towers under different wind directions for six retrofit methods. Method 6 achieves the highest base shear force across most wind directions, with substantial percentage differences compared to the non-retrofitted case. However, at wind directions of 45°, 135°, and 180°, there are only slightly improvements in tower capacity between Methods 5 and 6. The lengths of additional members in the compressive area of the tower from Method 6 (30.2 m) are significantly longer than those for Method 5 (14 m). Moreover, installing very long members on-site poses significant challenges. As a result, Method 5 is considered more suitable than Method 6 in this case.

On the other hand, it can be observed from this table that at a wind direction of 180°, Methods 1–3 are insufficient to improve the tower capacity. That is because with this wind direction, adding members are provided in the tensile area but not the compressive area of the tower. Since wind directions in nature are random, additional members should be provided in both the compressive and tensile areas of non-retrofitted towers to significantly enhance their capacity. Based on all the above observations, it is suggested that Method 5 is the most suitable retrofit method for the considered tower subjected to yawed wind, as this method not only significantly increases the tower’s capacity but also its stiffness and ductility, regardless of the wind direction, highlighting the efficiency of the proposed method.

6. Conclusions

This work highlights the critical importance of retrofitting strategies in enhancing the structural performance of transmission towers under static wind loads. Through nonlinear pushover analysis using advanced finite element modeling, this study systematically evaluated six retrofit methods, demonstrating their varying impacts on the structural behavior of the towers. The analysis results confirm that retrofitting effectively delays the onset of buckling and failure, thereby enhancing the overall capacity and ductility of the considered tower. Among the evaluated methods, the proposed retrofitting approach under Method 5 achieves the most notable improvement, increasing the tower capacity approximately from 10% (θ = 180°) to 24% (θ = 45°) compared to the non-retrofitted configuration. Nonetheless, all retrofit methods significantly improve tower stiffness in almost considered cases. These results underscore the value of targeted retrofitting solutions in extending the lifespan and reliability of transmission towers.

Although this study investigated the effects of retrofitting on the nonlinear response of the entire tower under wind loading, several limitations remain. Firstly, the modeling procedure did not include damage criteria to assess the progressive collapse response of transmission towers. Additionally, geometric imperfections in tower members were not considered. Furthermore, fragility analyses of retrofitted transmission towers were not conducted. Future studies will address these limitations by incorporating damage parameters, geometric imperfections of tower members, and joint eccentricity in the modeling of retrofitted towers. Additionally, fragility analyses of retrofitted transmission towers under wind, seismic, or combined wind and seismic loadings will be performed to ensure the proposed retrofit method is applicable to practical design.