Study on the Fire Resistance of Axially Restrained H-Shaped Steel Beams Under Real Fire

Abstract

The fire resistance performance of steel beams is of utmost importance to the fire safety of building structures and personnel evacuation. To address the deviation in the assumption of uniform temperature distribution in traditional studies, this study conducted multiple simulation tests. It was found that when the size of the vent was reduced by 50%, the difference in the heating rate in the early stage of the fire was 30% to 50%. Increasing the load ratio from 0.2 to 0.8 can significantly reduce the critical temperature of the steel beam by 15% to 20%, and the corresponding critical temperature is reduced from 670 °C to 565 °C. Based on parametric analysis, a simplified evaluation model of critical temperature for Q460 high-strength steel beams is proposed. The calculation error of the model is less than 5%, which provides a theoretical basis for the whole process of fire protection design. The research achievements break through the limitations of traditional methods and offer innovative approaches to predicting the fire resistance performance of steel beams and their optimized design.

1. Introduction

Scientific research should closely align with the practical demands of societal development. Among these, the study of building fires is a key field in addressing current safety challenges. Steel has been widely used in building frames and spatial structures due to its characteristics of lightweight and high strength. However, research shows that when the temperature exceeds 200 °C, the yield strength and elastic modulus of steel begin to decline [1,2,3]. Once the temperature of the steel exceeds 600 °C, most of the strength and stiffness will be lost. For high-strength steel, its performance will decrease rapidly after the temperature exceeds 400 °C. The steel beam is the core load-bearing component in a steel structure, and its performance degradation in fire will directly lead to the collapse of the whole building structure [4], thereby causing huge economic losses and serious social impacts. The major building fires in history and the collapse accidents they caused (Figure 1) are clear evidence. Therefore, an in-depth exploration of the fire-resistant performance of steel beams in building fires is of profound significance for enhancing the level of building fire protection design and ensuring public safety.

Experimental studies on the fire-resistant performance of restrained steel beams were initially initiated by Cardington [5]. Subsequently, the research initiated by Cardington was advanced by Liu et al. [6]. Through fire resistance tests on S275 restrained steel beams, the specific effects of parameters such as axial restraint, load ratio, and beam–column connection types on the fire-resistant performance of restrained steel beams were thoroughly investigated. Wang et al. [7] further broadened the research field. They carried out fire resistance tests on restrained steel beams with various joint forms, compared and analyzed the effects of these five joint forms, and summarized a variety of connection failure modes. Li et al. [8,9] have carried out a series of studies on Q235 restrained steel beams, including a fire resistance test, theoretical derivation of large deformation, and analysis of fire resistance performance parameters, and comprehensively studied the fire resistance of Q235 restrained steel beams in the heating and cooling stages. Changes in temperature field distribution, deformation, axial restraint force, and beam-end moments of H-shaped steel beams during fires were systematically summarized by Li Xiaodong et al. [10,11] through a series of full-scale tests, including those on H-shaped steel beams, columns, and frames. Sun Dongming et al. [12,13] analyzed the whole process of restrained steel beams during heating and cooling through experiments and finite elements, revealing the complete law of mid-span deflection of steel beams changing with temperature, and the specific influence of axial support stiffness and axial compression ratio on the failure mode and shape of steel beams. Liu et al. [14] conducted an in-depth study on the judgment method of the catenary effect of restrained steel beams and found that when the axial restraint stiffness reached a certain level, the catenary effect occurred almost simultaneously, and with the increase in load ratio, the catenary effect will appear in advance. After acquiring substantial fire response test data, numerous scholars began to utilize finite element simulations to conduct in-depth studies on restrained steel beams. Yin et al. [15,16] obtained the calculation methods of the axial force, mid-span bending moment, and beam-end bending moment of restrained steel beams through finite element simulation and theoretical derivation. Kodur et al. [17,18] utilized finite element models, analyzed refractory performance parameters of steel beam–steel column frames considering creep effect and studied parameter impacts on the fire resistance of restrained beams, including heating rate, slenderness ratio, restraint position, and high-temp creep. Heidarpour et al. conducted theoretical calculations of the flexural curve equations, axial forces, and bending moments of restrained steel beams and derived the corresponding calculation methods. Xi et al. [19] also performed theoretical analysis to derive the flexural curve equation for restrained steel beams under large deformation.

In summary, the research on the fire resistance of steel beams has been quite comprehensive, with various countries formulating corresponding design codes for fire-resistant restrained steel beams based on this research, and studies on restrained steel beams with different cross-sectional forms are gradually being deepened. However, previous studies on the fire resistance of steel beams have mostly been conducted by directly using ISO 834 (standard heating curve) [20], but in actual fire, the temperature change on the surface of the steel beams is more complicated.

Standard fire curves assume uniform temperature throughout the fire-exposed space, while in real fires, temperature distribution within the space is non-uniform both vertically and at different positions of the same height. To investigate the fire resistance of steel beams in real fires, constructing more realistic heating curves is required. A real fire test is constrained by cost and safety factors; thus, fire simulation technology plays a crucial role in practical applications. With the development of fire science and computer technology, theoretical models such as zone models and field models that represent the development of and variation in fire fields can be implemented through computational software, providing efficient computational tools for the study of real fire temperature fields. Quintiere [21] first proposed the basic equations of the zone model, and subsequently, a series of zone models based on this theory, such as RFIRES, ASET-B, and BRANZFIRE, were successively developed [22,23]. In addition, the CFAST software [24], introduced by the National Institute of Standards and Technology (NIST) in the US in 1993, remains one of the most commonly used zone simulation software platforms to date. Fire test data from numerous scholars indicate that the simulation results of CFAST software are in good agreement with experimental values [25,26], verifying its accuracy and reliability. But the zone model cannot accurately calculate the horizontal diffusion of temperature and hot smoke and can only describe temperature changes from a macroscopic perspective. For simulating a real temperature field, a more detailed model is required.

With the continuous advancement of computer technology and the increasing refinement of fluid dynamics models, the fire research field has developed field model software with computational fluid dynamics (CFD) as its core foundation. The field model method divides the building space into a large number of tiny control volume units and assumes that the physical parameters inside each unit are consistent. Then, according to the principle of material conservation and energy conservation, the partial differential equations describing the physical state changes of each unit are derived for calculation. Currently, field models are the most widely applied fire models. Among them, the fire dynamics simulator (FDS) developed by NIST, the most common in fire simulations, is widely used in fields like shipping, transportation, and tunnels and more extensively in building fires. The main functions of FDS include the following: (1) Building a fire scene, FDS can be used to build a real fire environment and set the type of fire source for fire simulation; (2) parameter analysis research, exploring the impact of various parameter changes on the fire field, studies the variables with the greatest impact. Therefore, fire simulation software is being widely used in fire research at present, and its accuracy and reliability have been fully verified. FDS still has some shortcomings: when simulating large-scale fire scenarios, the degree of mesh refinement directly affects the calculation accuracy. A mesh size that is too large will lead to inaccurate calculation, and a mesh size that is too small will lead to a significant increase in calculation time, which puts forward higher requirements for hardware resources. FDS has obvious shortcomings in structural response analysis, and Abaqus needs to be introduced to simulate the mechanical response of subsequent thermal–mechanical coupling analysis. Considering the relatively fine temperature field involved in the steel beams in this research, FDS is selected to simulate the fire heating curve, and the simulation results are transferred to Abaqus for subsequent fire resistance performance research.

There are still obvious limitations in the existing research. The current results are generally based on the ISO834 standard heating curve, ignoring the significant spatial inhomogeneity of the surface temperature of steel beams in real fires. Moreover, the existing research mainly focuses on the complete development stage of fire, and the high-temperature area in the fire growth stage (before flashover) will also cause the early performance degradation of steel beams. At present, this part of the content has not been fully discussed.

Based on the above research, this paper focuses on the fire resistance analysis of Q460 high-strength steel beams in real fire scenarios. FDS was used to simulate the whole process of fire in closed space, including the growth, flashover, and attenuation stages, and the quasi-real fire heating curve of a steel beam with uneven spatial distribution was obtained. The critical temperature of the steel beam under different working conditions was obtained by using the quasi-real fire temperature rise curve to study the mechanical properties, and the existing formula for calculating the critical temperature of the steel beam was modified. The purpose is to solve the limitation of the ISO834 uniform temperature rise hypothesis in traditional fire resistance research, and to provide a more accurate theoretical basis for fire protection design in the whole fire process.

2. Study on the Development Law of Temperature Field Under Different Fire Conditions

The complexity of temperature distribution in fire scenarios directly affects the fire resistance of steel beams. Traditional research methods are mostly based on the ISO834 standard heating curve, which assumes that the temperature of the whole heating space is evenly distributed, which is significantly different from the temperature gradient existing in real fires. Therefore, this chapter uses FDS fire simulation software to systematically study the influence of different parameters on the temperature field distribution in the confined space for uniform fire conditions and local fire conditions. By comparing and analyzing the temperature–time curves under different working conditions, the dynamic change law of the temperature gradient in the process of fire development is revealed, and the quasi-real fire temperature–time curve is proposed, which provides a more accurate temperature data basis for the subsequent research on the fire resistance of steel beams.

2.1. Uniform Fire Conditions

2.1.1. Model Parameters

To investigate real temperature field variations in confined spaces under uniform fire conditions, a 5.84 m × 6 m × 3 m closed-room model was constructed in FDS by defining room dimensions as specified in Reference [27]. After calculating the mesh size, considering factors such as computational efficiency and computational accuracy, the hexahedral element with a mesh of 8 cm × 8 cm × 7.5 cm was finally selected, with a total of 189,000. The doors and windows in the fire scene are designed to be open to ensure full combustion. The HRRPUA (heat release rate per unit area) is determined by the European standard; i.e., in the room, the HRRPUA linearly increases from 0 kW/m² to a certain value over 10 s and then remains constant. The floors and walls are modeled using the OBST (obstacles) command. The steel column is set as an obstacle with thermal conductivity, and its density and thermal radiation emission coefficient are 7850 kg/m³ and 0.7 (Figure 2).

The HRRPUA is 250 kW/m² for 34.4 m² of ground, and the HRR (total heat release rate) is 8.6 MW for the room. The comparison between the model in this paper and the HRR model in the literature is shown in Figure 3. Among them, HRR’ is the change curve of the literature. During the whole fire development process, the HRR change trend is consistent, indicating that the model parameter setting is reasonable and accurate. After the verification of the model, the subsequent uniform fire simulation is carried out.

In the uniform fire simulation for steel beams, multiple parameters are set to differentiate the influences of the heat release rate, vent height, and vent size by considering their effects, as listed in

Table 1. Model parameter settings under uniformly distributed fire conditions.

Model Number	HRRPUA/kw/m2	H/mm	S/m2
BE.1	250	500	2
BE.2	250	1500	2
BE.3	250	1000	2
BE.4	250	1000	4
BE.5	350	500	2
BE.6	350	1500	2
BE.7	350	1000	2
BE.8	350	1000	4

Note: HRRPUA denotes the heat release rate per unit area of the fire source, categorized into two scenarios of 250 kW/m² and 350 kW/m²; H represents the distance from the bottom of the vent to the ground; S represents the area of the vent.

. According to different parameters, the simulation under uniform fire conditions is carried out to obtain different temperature field distributions.

2.1.2. Study on the Early Heating Law of Uniform Fire Conditions

Under the uniformly distributed fire condition, the influence of hot flue gas and thermal radiation on each part of the steel beam is relatively uniform, taking BE.1 as an example (Figure 4). In subsequent research on the fire—resistance of steel beams, local fire conditions are emphasized, and uniform fire conditions are no longer considered, as all parts of the steel beam, whether the top, bottom, or sides, are subjected to similar heat effects, without causing the temperature of one part to be significantly higher than the others.

2.2. Local Fire Conditions

2.2.1. Model Parameters

To deeply investigate the impacts of different parameters on the temperature field distribution in confined spaces under local fire conditions, this paper conducts reasonable simplifications based on the uniform fire model. The size of the model room is 6 m × 4 m × 3 m, and the mesh uses a hexahedron unit of 8 cm × 8 cm × 6 cm, with a total of 187,500 units. A window is set as the vent, and the fire source is a 1 m × 1 m fire source with the HRRPUA of 1 MW/m² to simulate the ignition of a sofa in the room (Figure 5). The 1 m × 1 m fire source with the HRRPUA of 2 MW/m² is used to simulate the ignition of a liquid fire source in the room. The comparison of HRR changes between the two ignition sources is shown in Figure 6.

In the local fire simulation for steel beams, the influences of fire source types, vent height, and vent size are considered, with multiple parameters set to distinguish the effects of factors such as fire source types and vents, as shown in

Table 2. Model parameter settings under local fire conditions.

Model Number	HRRPUA/kw/m2	H/mm	S/m2
BL.1	1000	500	2
BL.2	1000	1500	2
BL.3	1000	1000	2
BL.4	1000	1000	4
BL.5	2000	500	2
BL.6	2000	1500	2
BL.7	2000	1000	2
BL.8	2000	1000	4

Note: HRRPUA represents the heat release rate per unit area of the fire source, which is divided into two scenarios of 1 MW/m² and 2 MW/m²; H denotes the distance from the bottom of the vent to the ground; S signifies the area of the vent.

. Simulations under local fire scenarios with various parameters are conducted to obtain different temperature field distributions.

2.2.2. Study on the Law of Temperature Rise in the Early Stage of Local Fire Conditions

The temperature rise curves of all the working conditions obtained by simulation calculation are shown in Figure 7. The comparisons (Figure 7a–c) reveal that, regardless of vent height, the steel beam exhibits significant temperature differences along its length, with higher temperatures at the fire-source end gradually decreasing with increasing distance and generally similar overall temperature distribution patterns. The reason is that each part of the steel beam is affected by hot flue gas similarly; greater temperature differences primarily result from the influence of thermal radiation from the fire source, with locations closer to the fire source experiencing stronger thermal radiation and higher measured temperatures.

The comparisons of Figure 7a,e,h show that, when the fire source has a higher heat release rate, room temperature is governed by ventilation conditions: better ventilation results in higher steel beam temperatures and more pronounced non-uniform temperature distributions across the steel beam’s surface transversely. This sufficiently demonstrates that, in the early stage of a local fire, a significant temperature gradient exists along the height direction of the steel beam surface, and the consideration of non-uniform temperature distribution along the length direction is necessary when analyzing the fire resistance performance of the steel beam.

2.3. Quasi-Real Fire Temperature–Time Curve

Currently, most research on structural fire—resistance performance is based on standard fire temperature—rise curves, such as furnace heating tests and finite—element numerical simulations, and the fire—protection community constantly emphasizes the limitations of standard fire curves. This curve is considered to represent the temperature rise of room components after flashover when the fire enters the fully developed stage. At this time, the overall temperature in the room tends to be consistent, regardless of the temperature gradient along the length direction.

The fire development process can be roughly divided into three stages [28]: fire growth stage, flashover, complete fire development stage, fire attenuation, and cooling stage. Most previous research efforts have focused on the fully developed fire stage, during which high temperatures and heat fluxes exist, and load-bearing structures are most susceptible to damage. The influence of the fire growth stage on the performance of steel columns and steel beams is rarely mentioned in the current fire resistance research; however, the results from the aforementioned simulations indicate that, during the early fire stage, influenced by the rise of hot flue gas, the high temperatures experienced by the upper ends of some support columns are sufficient to cause column failure, and similarly for steel beams. Taking the flashover occurrence time in compartment fires as the junction point, and using the CL.1 fire scenario as an example, the pre-fire heating curve simulated in FDS is connected with the standard heating curve of fire, and the quasi-real fire heating curve considering the whole process of fire development and the temperature gradient along the length direction is obtained (Figure 8); this can be used for the subsequent research on the fire resistance of steel columns and steel beams.

When there is a significant difference between the pseudo-real temperature rise curve and the ISO834 standard curve at the critical stage t = 300 s, the simulated temperature is 28.7% higher than the ISO 834 curve, indicating that the pseudo-real curve captures the local high temperature earlier, while the ISO 834 assumes uniform temperature rise, underestimating the early thermal shock. When t = 600 s, the temperature difference is the largest, and the simulated temperature is 42.3% higher than that of ISO 834, indicating that the pseudo-real curve can accurately reflect the sudden temperature rise during the sudden change in fire, while ISO 834 is seriously underestimated due to the neglect of the spatial gradient. When t = 1200 s, the temperature difference converges to 9.5%. The reason is that the quasi-real curve is consistent with the standard curve in the stable stage, and the cumulative effect of the early non-uniformity is retained. The above results show that the quasi-real fire temperature rise curve can more accurately reflect the non-uniform thermal effect of the high-temperature area on the steel beam in the early stage of the fire and can provide a reliable basis for the subsequent fire resistance analysis.

3. Analysis of Fire Resistance of Steel Columns Based on Quasi-Real Fire Heating Curve

The non-uniform distribution of steel beam surface temperature in real fire has a significant influence on its fire resistance, while the traditional research based on the ISO834 standard curve ignores this key factor. In order to break through this limitation, based on the quasi-real fire temperature–time curve proposed in Section 2, this chapter systematically studies the fire resistance of Q460 high-strength steel beams under axial constraints through thermal–mechanical coupling finite element simulation. By establishing a refined finite element model verified by experiments, the influence mechanism of load form, load ratio, and maximum temperature in the early stage of fire on the critical temperature of steel beams is deeply analyzed, and the existing critical temperature calculation formula is modified to provide an accurate theoretical basis for the fire protection design of steel beams in real fire scenarios.

3.1. Establishment of Finite Element Model of H-Section Steel Beam

3.1.1. High-Temperature Material Properties of Q460 Steel

(1) Thermal expansion coefficient

To accurately simulate the response of Q460 steel beams in fires, this study also decided to adopt the thermal expansion coefficients obtained by Xing et al. [29] through experiments, as shown in

Table 3. Thermal expansion coefficients of Q460 steel.

Temperature	Thermal Expansion Coefficient of Q460 Steel	Temperature	Thermal Expansion Coefficient of Q460 Steel
°C	×10−5 m/(m·°C)	°C	×10−5 m/(m·°C)
25	1.42	750	0
200	1.44	800	0
300	1.45	850	0
400	1.47	900	0
500	1.48	950	1.96
600	1.49	1000	1.96
700	1.50

.

(2) Thermal conductivity

To accurately simulate the Q460 steel beam’s response in fire, this study also decided to adopt the thermal conductivity coefficients obtained by Xing et al. [29] via tests, as shown in

Table 4. Thermal conductivity coefficients of Q460 steel.

Temperature	Thermal Conductivity of Q460 Steel	Temperature	Thermal Conductivity of Q460 Steel
°C	W/(m·°C)	°C	W/(m·°C)
25	50.971	600	36.820
100	49.125	700	34.359
200	46.664	800	31.898
300	44.203	900	29.437
400	41.742	1000	29.437
500	39.281

.

(3) Specific heat capacity

To obtain accurate simulation results, this paper also adopts the data obtained from experiments by Xing et al. [29], as shown in

Table 5. Specific heat capacity of Q460 steel.

Temperature	Specific Heat Capacity of Q460 Steel	Temperature	Specific Heat Capacity of Q460 Steel
°C	J/(kg·°C)	°C	J/(kg·°C)
25	462.40	600	733.57
100	495.55	700	895.14
200	529.10	800	840.68
300	559.68	900	628.41
400	597.27	1000	595.00
500	651.90

.

(4) Density

The density of Q460 steel is also assigned a constant value, with ρ_s = 7850 kg/m³.

3.1.2. High-Temperature Mechanical Properties of Steel

For this paper, the yield strength and elastic modulus of the steel are taken as 460 MPa and 200 GPa, as measured by Ding Yong et al. [30] through room temperature tensile tests. The mechanical properties of steel in fires gradually decrease as temperature rises, and in China’s Technical Code for Fire Protection of Building Steel Structures (GB 51249-2017) [31], the steel properties are only 50% of those at room temperature when the temperature reaches 600 °C. However, the domestic scholars Wang et al. discovered in their study on the high-temperature mechanical properties of Q460 steel that the reduction factors in the code are somewhat conservative; thus, to obtain more precise simulation results, this paper adopts the reduction factors for yield strength and elastic modulus at different temperatures from Wang et al.’s [32] research, as shown in

Table 6. Reduction coefficients for elastic modulus and yield strength of Q460 steel at high temperatures.

Temperature	Elastic Modulus Reduction Factor	Yield Strength Reduction Factor	Temperature	Elastic Modulus Reduction Factor	Yield Strength Reduction Factor
°C	ET/E	fy.T/fy	°C	ET/E	fy.T/fy
20	1.000	1.000	500	0.836	0.855
100	0.983	0.879	550	0.809	0.744
200	0.960	1.072	600	0.764	0.730
300	0.928	1.143	700	0.636	0.362
400	0.862	1.058	800	0.480	0.177
450	0.862	1.058

.

The Poisson ratio of Q460 steel in this paper is assigned as v = 0.3. To simplify calculations, the stress–strain model used for simulating Q460 steel in this paper also adopts a bilinear model, ignoring the strain-hardening segment of Q460 steel at high temperatures, as shown in Figure 9.

3.1.3. Steel Beam Model Establishment

(1) Model Overview

The thermal–mechanical coupling method is used to simulate the heating process and stress change process of the steel beam in the quasi-real fire. To verify the accuracy of the model, the steel beam model builds an H-shaped cross-section steel beam according to the specimen used in the literature [30]. The cross-section specification is H150 × 100, the length is 3000 mm, the element type is a four-node thermal–mechanical coupling shell element of S4RT, and the mesh size is taken as 1/10 of the width of the steel beam model. The material properties refer to the measured data obtained by Xing et al. [29], and the specific data refer to 3.1.2.

(2) Boundary conditions

To simulate the restraining effect of the steel column on the steel beam, axial and rotational restraints are set, respectively, at both ends of the beam (as shown in Figure 10), with magnitudes taking the test data values of 1956 N/mm and 6.8 × 10⁸ N·mm/rad in the verification literature. The area within 200 mm to the left and right of the mid-span of the steel beam is coupled to the point RP-7.

(3) Initial defects

In Abaqus, the eigenvalue buckling analysis of the steel beam is first performed through the buckle analysis step to identify its potential buckling mode. Next, a buckling mode that is closest to the initial deformation of the steel beam in the experiment is selected. This buckling mode is imported into the finite element model for high-temperature analysis to make the steel beam model more refined.

3.1.4. Model Verification

In Reference [30], the calculation results of the finite element model of the high-temperature steel beam are compared with the measured results of the test specimens in the four dimensions of steel beam surface temperature, displacement–temperature curve, axial force–temperature curve, and failure deformation diagram. The results are in good agreement, which proves the reliability of the finite element model of the constrained Q460 steel beam for thermal–mechanical coupling analysis. Among them, the difference between the surface temperature of the steel beam calculated by the model and the measured value of the test is within 3%; the difference between the change law of the mid-span displacement of the steel beam with temperature and the measured value of the test is less than 4%.

In order to verify the accuracy of the finite element model, the temperature–mid-span deflection curve and failure mode obtained from the finite element model results are compared with the experimental data in the reference paper.

The application of the concentrated load on the Q460 steel beam is kept consistent with the experiment. The concentrated load is applied to the coupling point RP-7 in the mid-span area (Figure 10). After the steel beam fails, the temperature–mid-span deflection displacement curve is obtained, and the failure mode is observed (Figure 10b). The loss of stability of the specimen is regarded as the failure condition of the specimen. As specified in GB/T 9978-1999 Test Methods for Fire Resistance of Building Components [33], when the maximum deflection of the steel beam exceeds L/20, it is considered to have lost stability, and the temperature of the steel beam at this time is the critical temperature.

In the process of thermal–mechanical coupling analysis, a static analysis step is first created to apply the predetermined load, which is kept stable in the subsequent analysis steps. Then, the time–temperature curves of the steel beams obtained from FDS simulations are set as air temperature rise curves and introduced into the model, being heated through the two mechanisms of surface thermal radiation and surface thermal convection. In this process, the thermal radiation coefficient of the model surface is specified as 0.5, and the heat transfer coefficient of surface thermal convection is set at 25 W/(m²·°C).

In Figure 10a, the comparison between the temperature–displacement curve from the finite element simulation and that from the experimental results reveals that the critical temperature obtained by the simulation is slightly lower than the experimental value, with an error within 5% and a good agreement. In Figure 10b, the comparison of the failure modes shows that the failure locations and deformation magnitudes are largely consistent, indicating that the finite element model can effectively simulate the failure modes of steel beams. In summary, the finite element simulation method used in this paper can better simulate the fire resistance response of Q460 steel beams.

3.2. The Whole Process of Behavior Analysis and Failure Mode of the Steel Beam Under Fire

3.2.1. Behavior Analysis and Failure Modes Under Fire

To obtain the fire resistance of Q460 steel beams under real fires, the validated finite element model with a steel beam cross-section specification of H200 × 150 × 8 × 12 is adopted for further research. The length is set to 3000 mm, the initial defect value is L/1000, and the quasi-real fire temperature rise curve under various working conditions obtained by FDS is used as one of the parameters, with two loading modes—concentrated and uniform loads—applied at load ratios ranging from 0.2 to 0.8 for fire resistance investigation (Figure 11 and Figure 12).

The results from the simulations (Figure 13) reveal that the failure mode of the simulated specimens is global buckling, with the failure locations of steel beams under quasi-real fire temperature–time curves—for both concentrated loads (Figure 13a) and uniform loads (Figure 13b)—approaching the high-temperature regions in the early-stage temperature field, which differ significantly from those under ISO834 temperature–time curves, primarily due to uneven temperature distribution causing higher temperatures at one beam end during the early fire stage, leading to damage there and yielding before other sections in the fire.

3.2.2. Comparison of Fire Resistance Temperature of Steel Beam

The longitudinal non-uniform temperature field is applied to the model in Abaqus, and the fire resistance simulation of steel beams under different loading modes and load ratios is carried out. Comparisons of the simulated results with the fire resistance results directly using the ISO834 heating curve reveal that higher failure temperatures are observed when lateral temperature non-uniformity of the steel beams is considered compared to those directly using the ISO834 heating curve, with temperature differences exceeding 100 °C under some fire conditions. Taking BL.1 as an example (Figure 14), comparisons between the simulation results and fire resistance results directly using the ISO834 heating curve reveal that, regardless of whether concentrated or uniformly distributed loads are applied, the failure temperatures accounting for lateral temperature non-uniformity in steel beams exhibit increases compared to those from direct use of the ISO834 heating curve, with the increase magnitude correlated with the load ratio. Under the same load ratio conditions, there is no significant difference in the critical temperature of the steel beam using the two loading methods of concentrated load and uniform load.

From the comparison of partial simulation results (Figure 15), one can observe that, under the same load ratio, the critical temperatures of steel beams under different fire scenarios exhibit differences:

(1) In local fire scenarios, the influences of concentrated and uniform loading modes on the critical temperature of steel beams are not significant (Figure 15a), with the difference from the critical temperature under direct application of the ISO834 heating curve approximating 100 °C.

(2) In local fire scenarios, the critical temperatures of steel beams when vent heights are higher or lower are slightly lower than those when vents are at the middle position (Figure 15b); however, the differences in critical temperature are insignificant; the reason lies in the fact that changes in vent positions affect the temperature field in the early stage of local fires; yet the transverse temperature variations of steel beams under the three scenarios are minimal, leading to insignificant differences in their critical temperatures.

(3) In local fire scenarios, when vent areas are larger, the critical temperatures are relatively close to those under direct application of the ISO834 heating curve, as under this scenario, the transverse temperature differences in steel beams remain minimal, their overall temperature tends to uniformity, and the effect is similar to that produced by direct ISO834 heating, thus leading to similar final critical temperatures.

(4) It can be found from Figure 15d that under the local fire condition, when the vent condition is constant, even if the heat release rate of the fire source increases, the surface temperature of the steel beam under BL.1 and BL.5 conditions is not much different due to insufficient ventilation; so the final critical temperature of the steel beam is also close.

3.2.3. Failure Temperature Parameter Analysis of Steel Beam

Through parametric analysis, the influences of load ratio, loading patterns, and transverse maximum temperature in the early fire stage on the critical temperature of steel beams were investigated. Taking BL.1-U as an example, under the same fire scenario, applying different load ratios, the simulation results reveal that as the load ratio increases, the critical temperature of steel columns decreases significantly (Figure 16): when the load ratio rises from 0.2 to 0.8, the critical temperature drops from 670 °C to 565 °C, with a difference of 105 °C and a decrease in amplitude of 15%. These observations reveal that the load ratio is a critical parameter influencing the critical temperature of steel columns in fires; thus, it requires substantial consideration in practical fire protection design.

Taking BL.1-C as an example (Figure 17), as the load ratio increases, the difference between the critical temperatures of the steel beams using a quasi-real fire heating curve and those under the direct ISO834 application increases progressively. When the load ratio reaches 0.8, the difference between the two reaches 227 °C. At higher load ratios, the non-uniform transverse temperature field of steel beams should be given substantial consideration.

4. Correction of Critical Temperature Calculation Formula of Steel Beam

4.1. Formula Correction

For the calculation of the critical temperatures of Q460 restrained steel beams, Ding Yong and other scholars [30] proposed a simplified calculation method (Equation (2)), which is derived from the yield reduction factor calculation formula for Q460 steel at high temperatures provided by Wang et al. [32] (Equation (1)).

$${\eta }_T=4.32e^{-{\displaystyle \frac{T}{880}}}-1.6$$

(1)

$$T_{cr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})$$

(2)

Note: where ${\eta }_T$ is the yield strength reduction factor of steel, calculated as formula (3); Tcr is the critical temperature of the steel beam.

The yield strength reduction factor of steel ${\eta }_T$ is used to integrate the interaction of temperature reduction, section characteristics, and load. The mathematical expression is derived as follows:

$${\eta }_T={\displaystyle \frac{K_M{M}_X}{{\phi }_{b,T}{W}_X{f}_y}}$$

(3)

Note: where $K_M$ is the bending moment coefficient, calculated as Formula (4); $M_X$ is the mid-span bending moment of the steel beam; $W_X$ is the section modulus of the steel beam; ${\phi }_{b,T}$ is the overall stability coefficient of the steel beam at temperature T, which is calculated according to Equation (5); $f_y$ is the yield strength of Q460 steel.

Ding [30] found that $K_M$ is related to the rotational restraint stiffness ratio, the axial restraint stiffness ratio, the load ratio and the span-depth ratio of the steel beam. The $K_M$ calculation formula obtained by data fitting analysis is as follows.

$$K_M=0.2675\times (1+0.9979R)\times {({\displaystyle \frac{l}{h}})}^{0.1643}\times {({\displaystyle \frac{K_r}{K_{r,b}}})}^{-0.1899}\times {({\displaystyle \frac{K_a}{K_{a,b}}})}^{0.0545}$$

(4)

Note: where $l/h$ is the span-to-height ratio of the steel beam; R is the load ratio; $K_r/K_{r,b}$ is the rotational restraint stiffness ratio of the steel beam; $K_a/K_{a,b}$ is the axial restraint stiffness ratio of the steel beam.

${\phi }_{b,T}$ refers to the study by Wang et al. [32] on the reduction in the elastic modulus of steel at high temperature.

$${\phi }_{b,T}={\beta }_b{\displaystyle \frac{{\pi }^2{E}_{T}Ah}{2f_{y,T}{\lambda }_y^2{W}_x}}\sqrt{1+{({\displaystyle \frac{{\lambda }_y{t}_f}{4.4h}})}^2}$$

(5)

Note: ${\beta }_b$ is the equivalent bending moment coefficient of the steel beam; $E_T$ is the elastic modulus of steel at temperature T; A is the cross-sectional area of the steel beam; $h$ is the height of the steel beam section; ${\lambda }_y$ is the slenderness ratio of the steel beam; $t_f$ is the thickness of the steel beam flange.

Comparing the simulation results with those calculated via formulas reveals that failure temperatures obtained from actual fire heating curves are relatively higher. In this paper, ∆T is defined as the difference between the critical temperature obtained by considering non-uniform temperature distribution in the early fire stage under the same load ratio and that calculated by Equation (2). Assuming a certain relationship between the critical temperature difference (∆T) and the longitudinal maximum temperature (T_max) in the early fire stage, a correlation between ∆T and T_max can be derived via regression analysis of quadratic fractions, as shown in Equation (6).

∆T = −3.61 × 10⁻⁵T_max² − 0.106 T_max + 345

(6a)

∆T = −1.12 × 10⁻⁴T_max² − 0.052 T_max + 247

(6b)

∆T = −5.41 × 10⁻⁵T_max² − 0.228 T_max + 345

(6c)

∆T = −1.47 × 10⁻³T_max² + 1.348 T_max−38

(6d)

∆T = −1.08 × 10⁻³T_max² + 0.588 T_max + 308

(6e)

∆T = −6.03 × 10⁻⁴T_max² + 0.301 T_max + 308

(6f)

∆T = 2.29 × 10⁻³T_max² + 1.59 T_max + 12

(6g)

Finally, a formula for predicting the critical temperature (Tpr) of steel beams can be determined, as shown in Equation (7).

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-3.61\times {10}^{-5}{T_{max}}^2-0.106 T_{max}+345 {\mathrm{P}}_{\mathrm{N}} = 0.2$$

(7a)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-1.12\times {10}^{-4}{T_{max}}^2-0.052 T_{max}+247 {\mathrm{P}}_{\mathrm{N}} = 0.3$$

(7b)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-5.41\times {10}^{-5}{T_{max}}^2-0.228 T_{max}+345 {\mathrm{P}}_{\mathrm{N}} = 0.4$$

(7c)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-1.47\times {10}^{-3}{T_{max}}^2+1.348 T_{max}-38 {\mathrm{P}}_{\mathrm{N}} = 0.5$$

(7d)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-1.08\times {10}^{-3}{T_{max}}^2+0.588 T_{max}+308 {\mathrm{P}}_{\mathrm{N}} = 0.6$$

(7e)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})-6.03\times {10}^{-4}{T_{max}}^2+0.301 T_{max}+308 {\mathrm{P}}_{\mathrm{N}} = 0.7$$

(7f)

$$T_{pr}=-880\times \ln ({\displaystyle \frac{{\eta }_T+1.6}{4.32}})+2.29\times {10}^{-3}{T_{max}}^2+1.59 T_{max}+12 {\mathrm{P}}_{\mathrm{N}} = 0.8$$

(7g)

4.2. Verification of Simplified Calculation Formula

Figure 18 presents the comparison between finite element analysis results under various scenarios and critical temperature results obtained from simplified calculation formulas for restrained Q460 steel beams. Through comparison, the critical temperature values of steel beams obtained from the simplified calculation formula are clearly seen to be highly consistent with the finite element analysis results, with errors maintained at a low level below 5%. Therefore, the simplified calculation method for the critical temperature of restrained Q460 steel beams proposed in this paper can be regarded as an effective reference for calculating the critical temperature in practical engineering projects, providing practical suggestions for relevant design work.

4.3. Limitations of the Formula

The simplified formula of critical temperature proposed in this paper is based on the study of a Q460 high-strength steel H-beam under specific fire and boundary conditions. The error of the calculation results is less than 5%, but the applicability is limited. The formula derivation depends on the high-temperature performance parameters of Q460 steel, and the applicability of other steel grades needs to be verified. The research is only for the H-beam section, and its temperature distribution and buckling behavior are specific, and the applicability to other cross-section forms needs to be studied. The formula is based on a specific axial and rotational constraint stiffness ratio setting, and the applicability may be limited when the actual boundary conditions change. In complex cases, it is recommended to supplement the correction coefficient or use numerical simulation to verify.

5. Conclusions

This paper uses the validated finite element model to conduct a parametric analysis of the fire resistance performance of Q460 steel beams, analyzing the impacts of loading patterns, longitudinal maximum temperature in the early fire stage, and load ratio on the critical temperature of steel beams. A theoretical analysis of the real fire response of Q460 steel beams during the heating stage is conducted, and the simplified calculation formula for the critical temperature is revised based on finite element analysis results. The following main conclusions are drawn:

(1) Under consideration of longitudinal temperature non-uniformity in the early fire stage, the critical temperatures of steel beams in all scenarios exhibit a significant increase compared to those under direct ISO 834 heating. The average increase was 100~120 °C. When the vent area is expanded to 4 m2, the transverse temperature difference of the steel beam is reduced to less than 50 °C, and the difference between the critical temperature and the ISO 834 working condition is reduced to less than 5%. The height of the ventilation openings has a minimal effect on the critical temperature of steel beams, as the distribution patterns and temperature differences of non-uniform temperature fields in steel beams during the early fire stage show little variation under various scenarios, thus exerting similar influences on the critical temperature of steel beams.

(2) Under the same scenario, for every 0.1 increase in load ratio, the critical temperature decreases by about 25 °C. When the load ratio increases from 0.2 to 0.8, the critical temperature decreases from 670 °C to 565 °C, with a decrease of 15.7%. Under the condition of a fixed load ratio, the failure temperature decreases by 8~12 °C for every 100 °C increase in the early maximum temperature. When the load ratio is fixed, as the maximum temperature in the early fire stage increases, the failure temperature of steel beams exhibits a moderate decrease, and the difference from the critical temperature under ISO 834 heating conditions gradually diminishes.

(3) The critical temperature of steel beams is modified, and the simplified calculation formula of critical temperature under different load ratios is proposed. Using the simplified calculation formula for steel beams, the critical temperature was calculated and compared with finite element analysis results; it was found that the calculation results were in good agreement with the finite element analysis results, and the difference was within 5%. Therefore, the simplified calculation method of the critical temperature of the constrained Q460 steel beam proposed in this paper can be used in practical engineering projects as an effective reference for calculating the critical temperature, and it can provide practical suggestions for related design work.