Study on Stability of Equal-Leg Angle-Steel Members in Transmission Towers at Uniform Elevated Temperature

Abstract

Equal-leg angle-steel members are widely used as main load-bearing and bracing members in transmission towers. Under elevated-temperature environments such as mountain fires and forest fires, the elastic modulus and strength of steel degrade significantly, which may reduce the overall stability capacity of compression members and even lead to instability failure. To investigate the stability performance of equal-leg angle-steel members made of Q420 steel at uniform elevated temperatures, a shell-element finite-element model was established in Abaqus by considering temperature-dependent material properties of steel. Parametric analyses were carried out under constant compressional loading and uniform heating for the cases of pinned–pinned, fixed–fixed and eccentric–pinned conditions. The effects of slenderness ratio, stability load ratio, section dimension, and initial imperfection amplitude on the critical temperature were systematically analyzed. The results show that the critical temperature decreases significantly with increasing stability load ratio. The eccentric–pinned condition leads to a higher critical temperature than the other two conditions. The initial geometric imperfection will reduce the fire resistance of members. Based on the critical temperature method, design curves of the critical temperature for three boundary conditions were developed using the finite-element results, which are demonstrated to be more accurate than the existing Chinese and European codes.

1. Introduction

Transmission towers are key components of high-voltage power transmission systems. Equal-leg angle-steel members are widely used as main load-bearing and bracing members due to their high efficiency and ease of fabrication. With the expansion of transmission networks into mountainous and forested regions, these structures are increasingly exposed to wildfire-induced high-temperature environments. Elevated temperatures lead to significant degradation in the elastic modulus and yield strength of steel, reducing the load-bearing capacity of compression members.

The high-temperature mechanical properties of structural steel form the fundamental basis for fire-resistant performance analysis of steel structures. In China, extensive research has been conducted on the mechanical degradation behavior of ordinary carbon steel, high-strength steel, and fire-resistant steel during heating and post-fire cooling processes. Wang et al. [1] reviewed the research and engineering application of fire-resistant steel for high-rise buildings. Lou et al. [2,3] investigated the strength and stiffness degradation laws of high-strength fire-resistant steel under different elevated-temperature conditions through high-temperature and post-fire mechanical tests. Feng et al. [4] analyzed the influence of different elevated-temperature material models on the collapse simulation results of steel frames from the perspective of structural fire response. These studies indicate that the degradation of elastic modulus and yield strength with increasing temperature is a key factor governing the high-temperature load-bearing capacity and stability of steel members. For ordinary structural steels and high-strength steels, Wei et al. [5] studied the full-range true stress–strain relationship of Q355 and Q420 steels, providing a basis for material model input in the plastic stage of finite-element analysis. Hu et al. [6] investigated the cyclic-loading constitutive model of post-fire high-strength Q690 steel, and Li et al. [7] studied the post-fire mechanical properties of Q690 steel. Wang et al. [8] measured and analyzed welding residual stresses in stiffened plates made of Q345 and Q420 high-strength steels. Overall, existing studies provide useful data support for establishing elevated-temperature material models, but specialized research on angle-steel materials commonly used in transmission towers under fire-induced heating paths remains limited. Lu et al. [9] conducted systematic experimental studies on the post-fire mechanical properties of hot-rolled and cold-formed steels. Qiang et al. [10,11] investigated the post-fire mechanical degradation behavior of high-strength steels such as S460, S690, and S960. Gunalan and Mahendran [12] studied the post-fire mechanical properties of cold-formed steels. Zhang et al. [13] and Wang et al. [14] investigated the post-fire mechanical properties of Q345 and Q460 steels, respectively. The above studies demonstrate that yield strength and elastic modulus are highly sensitive to temperature, and that different steel grades, cooling paths, and strain definitions can significantly influence the fitting of constitutive models.

With regard to the stability of steel angle members, domestic studies have mainly focused on the load-bearing performance of main members in transmission towers, angle-steel tower columns, and strengthened angle-steel members under ambient temperature conditions. Li Jinghui et al. [15] proposed a parallel strengthening method using externally attached channel steel for the main members of aged angle-steel transmission towers, and discussed the improvement law of bearing capacity through experimental investigation and finite-element analysis. Relevant studies have indicated that when open-section angle members are subjected to compression, they tend to exhibit flexural buckling, torsional buckling, and flexural–torsional-coupled buckling. Their stable load-bearing capacity is significantly affected by factors such as the slenderness ratio, end restraints, and connection details. In recent years, researchers have also begun to focus on the stability of large-section angle-steel members and connected angle-steel members. Chen et al. [16] investigated the stability behavior of Q345 large-section angle-steel columns under axial compression through experimental and numerical methods, and pointed out that the end connection form, leg width-to-thickness ratio, and member slenderness ratio have significant effects on the ultimate stability capacity. It can be seen that the stability of open-section angle members is highly sensitive to parameters, and changes in joint configuration and end boundary conditions may alter their flexural–torsional-coupled buckling path. It should be noted that most of the above studies on steel angle members have focused on ambient-temperature conditions or strengthened bearing-capacity problems, and have not yet been systematically extended to the calculation of critical temperature under elevated-temperature conditions. For transmission tower structures, angle members are often connected to the overall structural system through end plates, bolts, and gusset plates. Their actual load-transfer boundaries differ significantly from those of typical I-section steel columns used in buildings. Therefore, it is necessary to establish an elevated-temperature stability analysis framework based on the structural characteristics of transmission tower members.

Research on the overall stability of steel axially compressed members at elevated temperatures mainly focuses on experimental investigation, numerical simulation, and simplified design methods. Taking welded I-section axially compressed members made of high-performance fire-resistant steel as the research object, Wu [17] systematically carried out overall stability tests at ambient and elevated temperatures, finite-element validation, and fitting studies based on the critical temperature assessment method. A design approach considering the effects of slenderness ratio and load ratio was proposed. This study provides an important reference for the technical route of “finite element database-critical temperature fitting-proposed design formula”. Internationally, a relatively mature methodological framework has been established in the field of fire analysis of steel columns. Liew and Chen [18,19] adopted fiber-element and hybrid-element methods to analyze the response of steel frames subjected to combined blast and fire actions, providing a theoretical and numerical basis for the analysis of steel structures at elevated temperatures. Franssen and Vila Real [20] systematically summarized the fire-resistant design methods for steel structures based on the European code system, while Eurocode 3 Part 1–2 also provides general rules for the structural fire design of steel structures [21]. These studies generally indicate that temperature-dependent material models, initial geometric imperfections, axial load level, and boundary restraints are key parameters governing the stability behavior of steel members at elevated temperatures. For the ambient-temperature design of transmission towers, the Chinese code DL/T 5154-2012 [22] specifies the fundamental requirements for tower and pole structures, including provisions relevant to the design and stability of angle-steel members. However, it does not provide a critical-temperature assessment method or specific fire-design provisions for these members, which further highlights the need for the present elevated-temperature study.

Compared with H-shaped and box sections, equal-leg angle steel is an open thin-walled section with geometric asymmetry and is highly sensitive to bending–torsional coupling effects and boundary conditions. However, existing research mainly focuses on building steel structures, while studies on the high-temperature stability behavior of transmission tower angle-steel members remain limited. Therefore, it is necessary to investigate the critical temperature behavior of such members and develop a reliable numerical evaluation method.

Based on the above review, the current research and design methods for the elevated-temperature stability of equal-leg angle-steel members used in transmission towers still have the following limitations:

(1)

Existing studies on steel structures at elevated temperatures mainly focus on H-shaped, box-section, and steel tubular members used in buildings. Experimental data and high-temperature stability databases for equal-leg angle steel axially compressed members in transmission towers remain insufficient. Most existing studies on the stability of angle-steel members have been conducted under ambient-temperature conditions, while the flexural–torsional-coupled buckling, eccentric compression effect, and influence of end boundary conditions of open-section angle members under elevated temperatures have not been sufficiently investigated [15,16,17,18,19,20,21].

(2)

The current Code for Fire Safety of Steel Structures in Buildings GB 51249-2017 mainly provides critical temperature calculation tables or design recommendations for common building steel members. However, it lacks directly applicable design parameters for transmission tower angle-steel members with special cross-sectional forms and complex boundary conditions [23].

This study investigates the high-temperature stability behavior of equal-leg angle-steel compression members in transmission towers. A finite-element model was developed in Abaqus considering temperature-dependent material properties, geometric imperfections, boundary conditions, and axial loading. The structural response under uniform heating was analyzed by extracting axial displacement, lateral displacement, and instability characteristics for determining the critical temperature. Five typical cross-sections considering three boundary/loading conditions occurring frequently in transmission towers were examined over the prescribed ranges of slenderness ratio, stability load ratio, and initial imperfection amplitude.

2. Finite-Element Model and Parameter Design

2.1. Basic Procedure of Finite-Element Analysis

The commercial software ABAQUS (version 2023) was used for the finite-element analysis. The analysis consisted of three sequential stages: eigenvalue buckling analysis, ambient-temperature static analysis, and temperature-deformation analysis. First, an eigenvalue buckling analysis was performed to obtain the first global buckling mode, which was scaled to the prescribed amplitude and introduced into the model as the initial geometric imperfection. Second, a constant compression load was applied in the static analysis until the target stability load ratio was reached. Third, while maintaining the axial load unchanged, a temperature-deformation analysis was performed by applying a time-dependent uniform temperature at a heating rate of 5 °C/min. Finally, the critical temperature was determined from the displacement response and overall instability characteristics. The complete analysis procedure is illustrated in Figure 1.

2.2. Finite-Element Model

The equal-leg angle-steel members were modeled using the SR4 shell element in ABAQUS. Compared with the three-dimensional solid element, the shell element can effectively simulate the overall flexural and torsional deformation of thin-walled open-section members while maintaining relatively high computational efficiency, making it suitable for the parametric analysis conducted in this study. As shown in Figure 2, the member was arranged along the global 1-axis, with the 3-axis corresponding to the leg-width direction. The member length and shell thickness are denoted by L and t, respectively. End plates were arranged at both ends of the member. Either of the end sections was tied to an end plate, which was connected to the reference points along the centroidal axis of the member, as shown in Figure 2. The boundary conditions and axial load can then be applied through the two reference points. A uniform mesh size of 8 mm was adopted for all parametric models, which is sufficient to obtain convergent solutions, as will be demonstrated in a mesh-sensitivity analysis later. Geometric nonlinearity was considered to account for large displacements and out-of-plane deformation during heating.

2.3. Temperature-Dependent Material Properties

The material used in this study was Q420 steel. The temperature-dependent mechanical and thermal properties were calculated in accordance with the Code for Fire Safety of Steel Structures in Buildings (GB 51249-2017) [23]. The elastic response was defined by the temperature-dependent elastic modulus and a constant Poisson’s ratio of 0.30. A rate-independent isotropic elastic–plastic model was used for the inelastic response. The engineering stress–strain relationship at each temperature was represented using the Ramberg–Osgood model. The engineering stress and strain were converted to true stress and true plastic strain before being entered into Abaqus. The corresponding temperature-dependent elastic modulus and yield strength are listed in

Table 1. Temperature-dependent mechanical properties of Q420 steel.

Temperature (°C)	Elastic Modulus (MPa)	Poisson’s Ratio	Yield Strength (MPa)
20	206,000	0.30	420.000
100	202,044.8	0.30	420.000
200	195,576.4	0.30	420.000
300	186,512.4	0.30	420.000
400	172,834	0.30	384.080
500	149,803.2	0.30	297.024
600	103,000	0.30	190.176
700	44,145.8	0.30	94.752
800	20,600	0.30	42.000

, and the temperature-dependent thermal properties are listed in

Table 2. Temperature-dependent thermal properties of Q420 steel.

Temperature (°C)	Thermal Conductivity (W/(m·°C))	Expansion Coefficient (°C−1)	Specific Heat (J/(kg·°C))
20	53.334	1.216 × 10−5	439.80
100	50.670	1.280 × 10−5	487.62
200	47.340	1.360 × 10−5	529.76
300	44.010	1.440 × 10−5	564.74
400	40.680	1.520 × 10−5	605.88
500	37.350	1.600 × 10−5	666.50
600	34.020	1.680 × 10−5	759.92
700	30.690	1.760 × 10−5	1008.16
800	27.360	0	803.26

, where the intermediate value can be obtained by linear interpolation between adjacent temperature levels.

2.4. Heating Regime and Temperature Amplitude

Following the elevated-temperature loading scheme adopted in [17], a spatially uniform heating rate of 5 °C/min was used. In the present finite-element model, this rate defines the time–temperature amplitude for a prescribed temperature field and is not intended to reproduce a specific wildfire heating curve. Since the analysis is quasi-static and the constitutive model is rate-independent, only the temperature is the governing loading variable, while the elapsed time is used to trace the displacement–temperature response. As can be seen from

Table 3. Temperature amplitude applied in the elevated-temperature stability analysis.

Time (s)	Temperature (°C)	Description
0	20	Initial ambient temperature
960	100	Heating rate: 5 °C/min
2160	200	Continued heating
3360	300	Continued heating
4560	400	Continued heating
5760	500	Continued heating
6960	600	Continued heating
8160	700	Continued heating
9360	800	Analysis termination temperature

, the temperature increased from 20 °C to 800 °C over 9360 s: 0 s corresponded to 20 °C and 960 s to 100 °C, and each subsequent increase of 100 °C required 1200 s.

2.5. Parameter Design

Q420 equal-leg angle-steel members, which are widely used in transmission towers, were selected to investigate elevated-temperature overall stability. The parametric variables were section size, slenderness ratio, boundary/loading condition, stability load ratio, and initial geometric imperfection. The complete parameter levels are summarized in

Table 4. Physical member lengths and load-ratio levels used in the finite-element models.

Cross-Section	Slenderness Ratio λ	Load-Ratio Levels R
L80×7	60~160 (Δλ = 20)	0.20~0.80 (ΔR = 0.05)
L100×8	60~160 (Δλ = 20)	0.20~0.80 (ΔR = 0.05)
L125×10	60~160 (Δλ = 20)	0.20~0.80 (ΔR = 0.05)
L140×12	60~160 (Δλ = 20)	0.20~0.80 (ΔR = 0.05)
L160×14	60~160 (Δλ = 20)	0.20~0.80 (ΔR = 0.05)

, where the physical length L can be determined by the slenderness ratio combined with the effective-length factor $\mu =1.0$ for pinned–pinned and eccentric–pinned members and $\mu =0.5$ for fixed–fixed members. The main matrix comprised 1170 parameter combinations with the standard initial imperfection amplitude of L/1000. A further 160 pinned–pinned cases were analyzed for various initial imperfection amplitudes, including L/2000, L/1500, L/1000, and L/500.

2.6. Boundary Conditions

In this paper, the following three boundary conditions are considered for finite-element analysis, since they are the most common in transmission towers:

(1)

Pinned–pinned condition

At one reference point of the member, U1, U2, U3, and UR1 are constrained, while at the other reference point, U2, U3, and UR1 are constrained, and an axial concentrated load N is applied in the U1 direction (as shown in Figure 2, U1, U2, U3, and UR1, UR2, and UR3 are the displacements and rotations along/about the corresponding axes).

(2)

Fixed–fixed condition

At one reference point of the member, U1, U2, U3, UR1, UR2, and UR3 are fully constrained, while at the other reference point, U2, U3, UR1, UR2, and UR3 are constrained and an axial concentrated load N is applied in this direction.

(3)

Eccentric–pinned condition

The restraints enforced at the reference points are the same as those for pinned–pinned members. In most cases in transmission towers, the axial load N is applied eccentrically at the midpoint of one leg of the angle-steel section (i.e., point C in Figure 2).

The stability load ratio $R$ of the member is used to represent the level of axial load $N$ applied to the member relative to its stability capacity, which is dependent on the applied boundary condition and loading considered above. For pinned–pinned and fixed–fixed members, the stability load ratio can be calculated by the following equation:

$$R={\displaystyle \frac{N}{{\phi }_{y}Af}}$$

(1)

where the overall stability coefficient ${\phi }_y$ of the member can be determined by the Standard for Design of Steel Structures (GB 50017-2017) [24], $A$ is the cross-sectional area, and $f$ the design strength of steel. On the other hand, since the stability capacity is controlled by flexural buckling about the weak axis of angle steels, the stability load ratio for eccentric–pinned members can be calculated according to [24] as:

$$R={\displaystyle \frac{1}{f}}\left[{\displaystyle \frac{N}{{\phi }_{y}A}}+{\displaystyle \frac{M_x}{W_x}}+{\displaystyle \frac{M_y}{W_y\left(1-\frac{0.8N}{N_{Ey}^{\prime }}\right)}}\right]$$

(2)

where the bending moments $M_x=N{e}_x$, $M_y=0$ ($e_x$ has been defined in Figure 2), $W_x$ and $W_y$ are the elastic section modules about the x- and y-axes. In the above equation, the Euler critical load about the weak axis is calculated as:

$$N_{Ey}^{\prime }={\displaystyle \frac{{\pi }^2{E}_{s}A}{1.1{\lambda }_y^2}}$$

(3)

where $E_s$ is the elastic modulus of steel, and ${\lambda }_y$ is the member slenderness ratio about the weak axis.

3. Finite-Element Results and Discussions

3.1. Validation of the Finite-Element Model

Since there are few experimental data reported in the literature on the stability behavior of angle-steel members against high temperature, the I-section members are selected for validation of the finite-element model applied in this paper. The shell element model established in the previous section is used herein to analyze two pinned–pinned I-section members considering the imperfection amplitude of L/1000 under axial compression, i.e., H180×70 × 10 × 10 and H270×200 × 10 × 10. The numerical results obtained are compared with the experimental results reported in Ref. [17] for the buckling mode and the axial displacement–temperature curve in Figure 3 and Figure 4, respectively. As can be seen, the calculated buckling mode of bending about the weak axis agrees well with the observed deformation of I-beams after heating. Moreover, the critical temperature calculated by the present finite-element model is 760.5 °C for H180×70 × 10 × 10 and 759.0 °C for H270×200 × 10 × 10, which is very close to the experimental values of 721.3 °C and 750.4 °C for the two cross-sections reported in [17]. It can thus be concluded that the present finite-element model is able to predict the buckling mode, deformation–temperature relations, and ultimately the critical temperature of angle-steel members under elevated-temperature conditions with sufficient accuracy.

To eliminate the mesh discretization error of the shell element model meshed by a uniform size of 8 mm, a mesh-sensitivity analysis was conducted for pinned-ended angle steels of cross-sections L80×7, L100×8, L125×10, L140×12, and L160×14. The slenderness ratio and stability load ratio were fixed at $\lambda =80$ and $R=0.30$, respectively. Four nominal mesh sizes (4, 8, 12, and 16 mm) were considered for each member, and the computed critical temperature is listed in

Table 5. Critical temperature of angle-steel members computed by various mesh sizes.

Section	Critical Temperature, Td (°C)
	16 mm	12 mm	8 mm	4 mm
L80×7	643.360	647.554	647.925	648.386
L100×8	680.413	680.413	680.413	680.413
L125×10	700.438	700.438	700.438	700.438
L140×12	677.543	679.308	679.308	679.308
L160×14	675.300	679.558	679.308	679.308

. As can be seen clearly, the mesh of uniform size 8 mm adopted in this paper leads to a convergent solution for each cross-section considered, and thus such a shell element model is sufficient to simulate the stability behavior of angle-steel members under elevated-temperature conditions with the mesh discretization error eliminated.

As shown in Table 4, the increments $\Delta R=0.05$ and $\Delta \lambda =20$ are selected in the parametric analysis to study the influence of the stability load ratio and that of the slenderness ratio of angle-steel members. To verify the validation of such increments, the pinned–pinned member made of L100×8 was additionally simulated considering various stability load ratios and slenderness ratios of finer increments $\Delta R=0.01$ and $\Delta \lambda =5$, respectively. The computed critical temperature against these two ratios for the selected increments and the finer ones is shown in Figure 5, and, as can be seen, the selected increments of the stability load ratio and the slenderness ratio are sufficient to reveal their influence on the instability behavior of angle-steel members under uniform heating.

3.2. Fire-Induced Buckling Modes

The equal-leg angle is a mono-symmetric open section with relatively low flexural stiffness (about the weak axis) and torsional stiffness. As its temperature-dependent stiffness and strength decrease, the member may develop global flexural–torsional buckling deformation. Figure 6 shows the representative buckling modes occurring at the critical temperature for the pinned–pinned, fixed–fixed, and eccentric–pinned conditions, respectively. The pinned–pinned member primarily exhibits global flexural buckling, with the largest lateral displacement accompanied by slight cross-sectional torsion at the midspan. The fixed–fixed member exhibits a smoother overall deformation curve, leading to greater stability resistance provided by the stronger end restraints. The eccentric–pinned member develops asymmetric bending from the initial heating stage, and then the additional moment and second-order effects produce an instability path distinct from that of the pinned–pinned and fixed–fixed members.

3.3. Displacement–Temperature Responses

Figure 7 shows the axial displacement–temperature responses at the loading end of angle-steel members under the representative case of λ = 100, R = 0.20, and initial imperfection amplitude L/1000. The end axial displacement initially increases smoothly and then reaches a peak point as the temperature-dependent stability resistance degenerates abruptly at the critical temperature. As can be seen, the fixed–fixed members undergo larger deformation than the pinned–pinned and eccentric–pinned ones; this is because such a strong restraint will lead to a higher stress level with the same stability load ratio, and the stress level plays an important role.

Figure 8 shows the transverse displacement–temperature responses at the midspan of angle-steel members of cross-section L100×8, for the representative case of λ = 100, R = 0.20, and initial imperfection L/1000. It can be observed that the buckling mode incorporates bending about the weak axis accompanied by some amount of torsion. After reaching the critical temperature, the transverse displacements of the eccentric–pinned member increase much more abruptly than those of pinned–pinned and fixed–fixed members, since the second-order effect for the eccentric–pinned case is the largest. As can be seen from the deformation–temperature responses in Figure 6 and Figure 7, the axial displacement always exhibits a peak accompanied by large transverse deformation, therefore the temperature at which this peak occurs is selected as the critical value beyond which fire-induced instability of the angle-steel member occurs.

3.4. Influence of Stability Load Ratio, Slenderness Ratio and Section Dimension

Applying the standard initial imperfection e = L/1000, the computed critical temperature for angle-steel members under the three boundary conditions, i.e., the pinned–pinned, fixed–fixed and eccentric–pinned ones, is summarized in

Table 6. Computed critical temperature for pinned–pinned angle-steel members.

λ	R	Td (°C) L80×7	Td (°C) L100×8	Td (°C) L125×10	Td (°C) L140×12	Td (°C) L160×14
60	0.65	436.06	530.89	530.24	530.02	527.64
	0.50	528.00	590.90	590.25	590.05	588.61
	0.40	579.97	630.91	629.70	630.06	628.63
	0.30	631.94	669.41	667.21	669.54	668.65
	0.20	679.91	700.42	700.43	699.31	697.58
100	0.65	516.01	573.77	572.56	572.51	571.1
	0.5	583.97	624.4	620.42	623.29	621.63
	0.4	623.94	652.41	653.42	655.3	653.56
	0.3	659.92	684.41	684.43	683.31	681.57
	0.2	695.9	700.42	700.43	699.31	701.58
140	0.65	555.98	584.4	584.41	587.28	606.53
	0.5	607.95	628.4	628.42	627.29	645.55
	0.4	639.93	656.41	656.42	655.3	669.57
	0.3	667.91	684.41	684.43	684.55	697.58
	0.2	699.89	700.42	700.43	699.31	701.58

,

Table 7. Computed critical temperature for fixed–fixed angle-steel members.

λ	R	Td (°C) L80×7	Td (°C) L100×8	Td (°C) L125×10	Td (°C) L140×12	Td (°C) L160×14
60	0.65	583.20	573.02	554.20	557.62	560.01
	0.50	631.00	623.99	609.78	612.66	617.54
	0.40	663.92	657.41	646.47	648.67	655.50
	0.30	691.90	689.41	681.42	683.08	690.58
	0.20	699.89	700.42	700.42	699.31	697.58
100	0.65	597.33	588.40	562.91	567.27	573.52
	0.50	639.93	633.41	617.42	620.52	628.05
	0.40	663.91	661.41	648.42	652.53	661.56
	0.30	691.90	688.41	680.43	680.54	689.58
	0.20	698.74	700.42	700.43	699.28	701.58
140	0.65	591.96	584.40	568.41	572.51	573.52
	0.50	631.93	632.41	620.42	620.52	625.54
	0.40	662.75	660.41	648.42	651.27	657.56
	0.30	686.74	688.41	680.43	683.28	687.06
	0.20	697.03	698.82	700.43	701.34	699.07

and

Table 8. Computed critical temperature for eccentric–pinned angle-steel members.

λ	R	Td (°C) L80×7	Td (°C) L100×8	Td (°C) L125×10	Td (°C) L140×12	Td (°C) L160×14
60	0.65	639.93	648.41	651.20	655.30	656.64
	0.50	675.91	680.41	683.21	687.31	684.66
	0.40	695.90	700.42	699.21	700.55	700.67
	0.30	699.90	700.42	700.43	700.55	701.58
	0.20	699.90	700.42	699.21	698.08	701.58
100	0.65	643.93	676.41	674.92	676.31	676.65
	0.50	675.91	700.42	699.49	699.31	700.67
	0.40	695.90	700.42	700.43	700.55	697.58
	0.30	699.89	700.42	700.43	700.55	701.58
	0.20	699.89	700.42	700.43	700.55	701.58
140	0.65	635.93	658.91	657.70	656.54	660.65
	0.50	663.92	684.41	684.43	683.31	685.66
	0.40	683.90	700.42	700.43	699.31	700.67
	0.30	699.89	700.42	700.43	699.31	700.67
	0.20	699.89	700.42	699.21	700.55	700.67

, respectively. The statistical results of the critical temperature of the pinned–pinned, fixed–fixed, and eccentric–pinned members are given in

Table 9. Factor-level statistics of critical temperature for pinned–pinned members.

R	Mean Value (°C)	Difference from Reference (°C)	Relative Change	Minimum Value (°C)	Maximum Value (°C)	Standard Deviation (°C)	CV (%)
0.2	699.08	0.00	0.00%	679.91	701.58	4.10	0.6
0.3	677.27	−21.81	−3.12%	631.94	697.58	13.62	2.0
0.4	644.49	−54.59	−7.81%	579.97	669.57	19.18	3.0
0.5	611.61	−87.47	−12.51%	528.00	645.55	25.36	4.1
0.6	577.65	−121.43	−17.37%	472.04	620.63	32.55	5.6
0.7	539.36	−159.72	−22.85%	397.69	593.61	41.80	7.7
0.8	501.18	−197.90	−28.31%	330.70	560.59	46.92	9.4

,

Table 10. Factor-level statistics of critical temperature for fixed–fixed members.

R	Mean Value (°C)	Difference from Reference (°C)	Relative Change	Minimum Value (°C)	Maximum Value (°C)	Standard Deviation (°C)	CV (%)
0.2	699.73	0.00	0.00%	697.03	701.58	1.26	0.2
0.3	686.41	−13.32	−1.90%	680.42	695.89	4.63	0.7
0.4	656.23	−43.51	−6.22%	646.47	668.16	6.19	0.9
0.5	625.20	−74.53	−10.65%	609.78	639.93	8.07	1.3
0.6	593.85	−105.89	−15.13%	573.12	611.95	11.04	1.9
0.7	556.54	−143.20	−20.46%	535.18	580.22	13.50	2.4
0.8	518.90	−180.83	−25.84%	495.87	546.63	15.90	3.1

and

Table 11. Factor-level statistics of critical temperature for eccentric–pinned members.

R	Mean Value (°C)	Difference from Reference (°C)	Relative Change	Minimum Value (°C)	Maximum Value (°C)	Standard Deviation (°C)	CV (%)
0.2	700.32	0.00	0.00%	698.08	701.58	0.70	0.1
0.3	700.40	0.08	0.01%	699.31	701.58	0.54	0.1
0.4	697.83	−2.49	−0.35%	679.90	700.67	5.13	0.7
0.5	685.99	−14.33	−2.05%	659.92	700.67	10.78	1.6
0.6	667.74	−32.58	−4.65%	635.93	700.42	13.65	2.0
0.7	650.29	−50.03	−7.14%	615.94	700.42	16.75	2.6
0.8	632.17	−68.15	−9.73%	599.95	694.32	19.05	3.0

, respectively. Also, a comparison of the critical temperature for members of various section dimensions can be clearly shown in Figure 9, Figure 10 and Figure 11 for the three boundary conditions, respectively. As can be seen, the overall influence of the section dimension on the critical temperature of angle-steel members under the three boundary conditions is insignificant, compared with that of the stability load ratio and the slenderness ratio. Only when the slenderness ratio is small and the member is pinned–pinned does the section dimension affect the critical temperature to some degree; in this situation, a smaller section dimension leads to a lower critical temperature. From Figure 9, Figure 10 and Figure 11, the following observations about the influence of the stability load ratio and the slenderness ratio for the three boundary conditions considered can be obtained:

(1)

The stability load ratio has a significant influence on the critical temperature of angle-steel members under the three boundary conditions considered. As the stability load ratio increases, the stress level of the member becomes larger, and thus its critical temperature decreases.

(2)

A higher slenderness ratio will lead to a higher critical temperature. This is because, for members of fixed stability load ratio, higher slenderness means a lower stability load, and thus the stress level of the member is lower, leading to stronger fire resistance of the member.

3.5. Influence of Initial Geometric Imperfection

The influence of the initial geometric imperfection is considered in the parametric analysis. Since the amount of data is too large, the case of pinned–pinned members with slenderness λ = 100 is selected as an example. Figure 12 compares the computed critical temperature of pinned–pinned angle-steel members of cross-sections L80×7 and L125×10 for various initial imperfection amplitudes e = L/500, L/1000, L/1500, and L/2000, from which the following observations can be obtained:

(1)

The larger the initial imperfection, the lower the critical temperature. The initial imperfection renders the axial compressive force at the column top to generate a bending moment in the cross-section due to the second-order effect induced by lateral deformation, thereby increasing the stress level of the member. A larger initial imperfection results in greater sectional stress. The increased sectional stress reduces the fire resistance of the member, thus leading to a decrease in the critical temperature.

(2)

When the initial imperfection is relatively large, its influence on the critical temperature becomes more significant. When the initial imperfections are L/1500 and L/2000, the critical temperatures show only a small difference compared with that for the standard initial imperfection L/1000. However, when the initial imperfection is L/500, the critical temperature differs significantly from that for L/1000.

(3)

The influence of initial imperfections varies for members with different stability load ratios. Members with a higher stability load ratio are more significantly affected by initial imperfections. This is because a larger axial force at the column top generates greater additional stress in the member cross-section under the same initial imperfection, resulting in increased sectional stress. Consequently, the fire resistance of the member is reduced, leading to a decrease in the critical temperature.

3.6. Influence of Boundary Conditions

In order to study the influence of the three boundary conditions considered, the critical temperature of angle-steel members of cross-sections L80×7 and L125×10 with the standard initial imperfection e = L/1000 is compared in Figure 13, from which the following observations can be concluded:

(1)

The critical temperature of pinned–pinned and fixed–fixed members is close. Only when the member is of small section dimension and of small slenderness, see Figure 12a, will the pinned–pinned condition lead to a lower critical temperature than the fixed–fixed one. This means stronger end restraints of the member will increase its fire resistance, and the fixed–fixed conditions provide a stronger restraint for the larger deformation of the member under higher temperature, as can be seen clearly from Figure 8.

(2)

The critical temperature of eccentric–pinned members is apparently higher than that of members under pinned–pinned and fixed–fixed conditions. Such a difference becomes more significant when the stability load ratio is larger. This is because the second-order effect of eccentric–pinned members always renders a much lower stability load, and the initial stress level will be higher under the same stability load ratio, consequently leading to stronger fire resistance of the member.

In this paper, the focus is placed on uniform elevated-temperature conditions, which are frequently encountered in main members at the bottom or horizontal bracing members when the transmission tower is subjected to a large scale of mountain fires or forest fires. It should be noted that the temperature may vary along the length of some members in some situations; in this regard, the buckling behavior of the member becomes much more complicated than that of uniformly heated members. In order to provide at least a rough assessment of the stability behavior of angle-steel members under non-uniform heating, the pinned–pinned members of slenderness λ = 100 and stability load ratio R = 0.5 were additionally simulated under non-uniform heating conditions, where a linear distribution of temperature was considered along the member.

Table 12. Computed critical temperature for pinned–pinned angle-steel members under non-uniform heating.

Cross-Section	λ	R	Temperature Gradient ${\mathit{T}}_{\mathit{m}\mathit{i}\mathit{n}}/{\mathit{T}}_{\mathit{m}\mathit{a}\mathit{x}}$
			1.0 (Uniform)	0.8	0.6	0.4	0.2
L80×7	100	0.5	583.97	621.56	643.51	677.20	690.12
L125×10	100	0.5	620.42	662.32	693.20	710.55	731.43

gives the critical temperature measured as the applied maximum temperature for various temperature gradients represented by $T_{min}/T_{max}=0.8, 0.6, 0.4, 0.2$. As can be seen clearly, non-uniform heating increases the critical temperature of angle-steel members significantly, and such an influence becomes larger as the temperature gradient increases. However, the influence of non-uniform heating needs much deeper investigation, which will be reserved for our future studies.

4. The Critical-Temperature Design Method

The critical temperature assessment method is a simplified fire-resistant design method based on the degradation characteristics of the high-temperature load-bearing capacity of members. Its basic principle is to take the temperature corresponding to the occurrence of overall instability of a member under a constant axial load as the assessment index, thereby characterizing the stability capacity of the member under fire circumstances. In this manner, the complex nonlinear finite-element analysis results at elevated temperatures can be transformed into practical engineering design parameters, providing a unified basis for evaluating the elevated-temperature stability of angle-steel members.

4.1. Design Formula for the Critical Temperature

Based on the finite-element analysis of the critical temperature of angle-steel members subjected to compression, the following formula is adopted in this study to describe the relationship between the critical temperature $T_d$ and the stability load ratio $R$:

$$T_d={\gamma }_1\mathrm{l}\mathrm{n}\left[{\displaystyle \frac{1}{{\gamma }_2{R}^{{\gamma }_3}}}-1\right]+{\gamma }_4$$

(4)

where the unit of $T_d$ is °C, R is the stability load ratio, and ${\gamma }_1,{\gamma }_2,{\gamma }_3,{\gamma }_4$ are fitting parameters. Using MATLAB (version 2024b), these parameters were obtained through curve fitting of the data points from the finite-element analysis for the pinned–pinned, fixed–fixed, and eccentric–pinned angle-steel members, as listed in

Table 13. Fitted parameters of the design formula of the critical temperature.

Parameters	Pinned–Pinned	Fixed–Fixed	Eccentric–Pinned
γ1	212	134	68
γ2	0.99983	0.99982	0.99986
γ3	0.00144	0.00144	0.00144
γ4	1958	1516	1119

. Since the formula is highly sensitive to the parameters ${\gamma }_2$ and ${\gamma }_3$, sufficient decimal places should be retained in order to ensure adequate accuracy. In order to verify the validation of the proposed design formula, sample results were obtained by simulating angle-steel members made of two other cross-sections L63×6 ad L90×8 using the finite-element model, considering the same design parameters listed in Table 4. A comparison between the finite-element results for fitting and those for the sample and the proposed curve in this study for the three boundary conditions is presented in Figure 14, and the proposed curve is sufficient to predict the instability behavior of angle-steel members under uniform heating in a conservative manner. In this Figure, the curves calculated according to the Chinese code (GB 51249-2017) [23] and the European code (EN 1993-1-2) [21] are plotted for comparison. It can be seen clearly that the proposed curve herein is more accurate for the pinned–pinned, fixed–fixed, and eccentric–pinned angle-steel members considered. Specifically, the existing codes [21,23] may lead to unsafe results for pinned–pinned members and thus the fire resistance of angle-steel members may be overestimated, and may lead to conservative results for fixed–fixed and eccentric–pinned members (especially the latter ones) and thus the economy of the designed section may be sacrificed.

Moreover, a comparison between the critical temperatures obtained from the finite-element analyses and those calculated using the proposed curve is presented in Figure 15. The mean value of the ratio of the numerical data to the proposed curve is 1.025, and the corresponding standard deviation is 0.045. The results indicate that the design curve provides a better prediction of the critical temperatures for angle-steel members under compression, which are widely encountered in transmission towers.

4.2. Steps of Design Procedure

Based on the above results, the following procedure is recommended for calculating the critical temperature of angle-steel members under compression used in transmission towers:

(1)

Determine the section dimensions, calculation length, boundary conditions, and material strength parameters of the angle-steel member.

(2)

Calculate the stability load ratio $R$ using Equation (1) for pinned–pinned and fixed–fixed members and using Equation (2) for eccentric–pinned members.

(3)

Calculate the critical temperature using the design curve Equation (4) by adopting the fitting parameters listed in Table 9 for the pinned–pinned, fixed–fixed, and eccentric–pinned members.

(4)

Compare the predicted critical temperature with the design or actual fire temperature. When the design or actual temperature is lower than the predicted value, the angle-steel member can be considered to satisfy the overall stability requirement under elevated-temperature conditions.

5. Conclusions

Taking equal-leg angle-steel members made of Q420 steel, widely used in transmission towers, as the research object, this study conducted a finite-element analysis considering the temperature-dependent material properties of steel. A parametric analysis was conducted to investigate the influence of the stability load ratio, slenderness ratio, section dimension, initial geometric imperfection, and boundary condition on the fire-resistance behavior of angle-steel members. Based on the finite-element results, a critical-temperature design method for assessing fire-resistance behavior appropriate to angle-steel members was proposed. The main conclusions are as follows:

(1)

As the stability load ratio increases, the stress level of the member becomes larger and thus leads to a lower critical temperature. A higher slenderness ratio will lead to a higher critical temperature, since higher slenderness means a lower stress level of the member. The influence of the section dimension of the member is negligible.

(2)

The critical temperature of pinned–pinned and fixed–fixed members is very close, while that of eccentric–pinned members is apparently higher; this is because the second-order effect of eccentric–pinned members always renders a higher stress level.

(3)

The larger the initial imperfection, the lower the critical temperature. The initial imperfection causes the axial compressive force to generate a bending moment in the cross-section due to the second-order effect, thereby increasing the stress level of the member and decreasing the critical temperature. Moreover, members with a higher stability load ratio are more significantly affected by initial imperfections.

(4)

Based on the critical temperature method, design curves for the critical temperature were developed using results obtained from finite-element analysis. The comparative results demonstrate that the design curves provide a more accurate prediction of the overall stability performance of angle-steel members subjected to uniform elevated temperature than those of the existing codes [21,23].

It should be noted that the proposed design method is limited to uniform elevated-temperature conditions, which can be frequently encountered in main members at the bottom or horizontal bracing members of fired transmission towers. In the future, the buckling behavior of angle-steel members under non-uniform heating will be studied.