Experimental study and analysis on the collapse behavior of an interior column in a steel structure under local fire

Abstract

This article describes the experimental and analytical results of tests on two medium-scale steel frames to investigate the collapse behavior of a heated interior column in steel-framed structures and the fire-induced collapse mechanism of the structures. Because the heated column in the structure was under realistic inelastic end constraints (inelastic effects from the surrounding cool structures) as opposed to an idealized column in a furnace, the tests were designed to investigate the effect of inelastic end constraints and different load ratios. The interior column subjected to fire in the first story was connected with a force-measuring bearing, which was developed to measure the axial force in the heated column. This article presents the observations and analysis results of structural fire behavior, including the temperature distribution of the column, the development of deflection and axial forces, and the column failure modes. The results show that the effect of realistic inelastic end constraint is not significant for the column behavior during the pre-buckling stage, compared with the elastic constraint, but it is significant during the post-buckling stage. An analytical model was adopted to analyze the behavior of the column with inelastic end constraint under fire. Comparison of the theoretical and experimental results shows that the simplified analytical model can be used to predict the collapse behavior of the interior column in steel-framed structures under fire.

Keywords

collapse mechanism experiment fire resistance inelastic constraint steel frame

Introduction

Fire may lead to failure of structural components or even progressive collapse of building structures. The ultimate aim of structural fire engineering design is to prevent the collapse of steel structures caused by fires, it means that structures should be treated integrally in structural fire safety design (Bailey et al., 1999), and the interaction of the fire-affected structure with the cooler areas surrounding it should be taken into consideration. Moreover, the collapse limit state should be evaluated in structural fire safety design, because some local failure does not represent a structural collapse, as the full-scale fire tests at Cardington have shown (Wang, 2000), some of the beams experienced elevated temperatures greater than 1000°C and large deformations; however, the structure did not collapse.

Local fire generally caused local failure of structures, and the failure of individual members could initialize partially or global building collapse if alternative load-carrying path does not exist. Such as the interior column, its stiffness and load capacity decreased sharply with the increase in temperature (Alam et al., 2013), and nonlinear numerical analysis results of multi-story frames under fire showed that the column failure generally led to global collapse seriously than beam failure. So that the columns play a crucial role in terms of the global collapse of buildings subjected to fire. Analytical model of a column with constraint springs was usually used to analyze the collapse behavior of column under fire (Li et al., 2009; Poh and Bennetts, 1995); however, the constraint springs were often modeled as elastic and spring properties, and they were not precisely developed to represent real structures. Experiments and analysis on restrained steel columns subjected to fire were performed by Tan et al. (2007), Li et al. (2009), Choe et al. (2011), and Shepherd and Burgess (2011), and the constraint frames were elastic during the tests and analysis. Although extensive studies have been conducted to investigate the performance of restrained steel columns under fire, only a few studies focused on the inelastic boundary conditions. Considering inelastic and dynamic effects from the surrounding structures, Sun et al. (2012) investigated the progressive collapse performance of frame structures generated by the local failure of components under fire conditions with Vulcan software; the results showed that different loading ratio and beam sections can generate different local and global collapse mechanisms. However, the probable initial collapse mechanisms of frame structures fall broadly into four categories: beam failure, interior column failure, exterior column failure, and interactive failure. Certainly, one of these preliminary failure mechanisms may induce other mechanisms in subsequent events, which lead to progressive global building collapse. Quiel and Garlock (2008) analyzed the interactive failure of perimeter column and beam and proposed a mechanical model to analyze the interaction between the perimeter columns and beams, which can be used as reference for the performance-based design.

This article studies the collapse behavior of an interior column in steel-framed structures. Considering the realistic inelastic effect of surrounding components, multi-layer frame specimens were adopted in the tests. On the other hand, the hydraulic control load was applied to investigate the effect of different load levels. Major experimental results including thermocouple temperature measurements and displacement and axial force data of the heated column can be provided to serve as a database with which to check and validate numerical models. Finally, a simplified analytical model was adopted to analyze the collapse behavior of the column with inelastic end constraint under fire.

Test program

Specimens’ details

As mentioned in the “Introduction” section, the purpose of the tests was to study the initial collapse behavior of an interior column in a steel-framed structure. The heated column was under realistic nonlinear end constraints as opposed to an idealized column in a furnace or an idealized column with elastic end constraints. Therefore, the testing steel frame, as shown in Figure 1(a), was designed to be a three-story and four-bay frame with fixed bases, and the interior column in the first story was heated. All the columns were H section (H100 × 100 × 6 × 8) and were constructed from Q235 steel (yield strength: 235 MPa, average tensile strength: 375 MPa), and all the beams (I section H150 × 75 × 5 × 7) were made of Q235 steel, which is widely used in construction in China. The beam-to-column connections in the test frames were designed to transfer both moment and shear forces and were constructed with a 10-mm thick extended end plate, bolted with six M20 Grade G8.8 bolts, as shown in Figure 1(b).

Figure 1.

Dimensions of the specimen: (a) dimensions of the steel frame and (b) details of the connection.

The frame height, measured from the base to the topmost surface of the beam in the third story, was 5000 mm, and the center-to-center span length was 1700 mm. In order to measure the axial force of the heated column, the bottom of the interior column was connected with a bearing underground. The bearing was developed to measure the axial force, as shown in Figure 2(b), which consisted of a short steel beam, three thick-walled tubes, and a steel pad. The short steel beam was connected with the heated column by high strength bolts, and three thick-walled tubes were welded below the bottom of the short steel beam. The horizontal degree of freedom of the bearing was constrained through steel plate around the short beam, and the rotational degrees of freedom were also constrained, as shown in Figure 2(a). So that it represents the boundary condition of fixed end, but cannot resist tensile force. To measure the compression force, four strain gauges were pasted on each thick-walled tube. The arrangement and bridge connection of strain gauges were shown in Figure 2(c). Based on the bending strain (ε = ε1 − ε2 + ε3 − ε4) and the relationship between radial force and bending moment, the compression force of the tube can be calculated. Therefore, the axial compression force in the heated column was equal to the total vertical force of the three thick-walled tubes. In fact, the moment at the end of the heated column could be further considered by measuring the horizontal force of the constrained steel plate. However, the horizontal force was not measured during the tests because axial force variation can properly reflect the damage of the heated column and structure.

Figure 2.

Force measuring bearing: (a) boundary condition model, (b) A-A VIEW, and (c) strain gauges arrangement and bridge connection.

Test setup

The fire tests were carried out in the fire laboratory in Shandong Jianzhu University in China. The fire testing furnace was a rectangular box with dimensions of 2900 × 4200 × 3600 mm, as shown in Figure 3(a). Eight gas burners were symmetrically connected to both side walls of the test furnace and one exhaust was connected to the bottom of the furnace. There were three holes located in the upper cover, the front wall, and back wall of the furnace, so that the heated column and beam inside the furnace can be connected to the cooler structure outside the furnace. Asbestos was used to fill the holes after the installation of the whole frame specimen, and thermal expansion and large displacements were allowed. The furnace temperatures were recorded using three N-type thermocouples for each test.

Figure 3.

Side elevation of the furnace setup and the frame: (a) furnace setup and (b) frame setup.

The tests specimens were four-bay and three-story steel frames consisting of steel beams and columns. To investigate the effect of different load ratios, two specimens were prepared and they were subjected to different external loads. In order to ensure overall out-of-plane stability, two additional beams were arranged on both sides of the specimen. The both supplementary beams were connected together with high strength bolts and can move freely in the specimen plane. Moreover, the frame was restrained from lateral movement at the beam–column joints and was free to move in the plane direction of the frame as shown in Figure 3(b).

Steel columns are critical members to prevent collapse of the frame and should be protected under fire. In order to study the initial failure mechanism of the interior column, the interior column was unprotected in the tests, and the additional beams and the beam-to-column connection were fire protected by an alumino-silicate refractory fiber blanket as shown in Figure 4(a).

Figure 4.

Position of fire protection and loading device: (a) position of fire protection and (b) loading device.

The beams were subjected to external loads in a frame structure, and the external loads applied to the beams were converted to concentrated loads in the tests. Therefore, the vertical concentrated load was applied at the top of the interior column by hydraulic jacks (Figure 4(b)), and the load was checked continuously to maintain a constant value during the tests. To consider the influence of different axial compression ratios of the heated columns, both the test specimens were subjected to different external loads with 50 and 100 kN, respectively.

Instrumentation

In order to measure the temperature distribution in the structure, thermocouples were mounted on the columns, beams, and joint components. Three K-type thermocouples were installed around each steel column and beam profile at Sections 1-1 to 6-6 to measure the temperature distribution of the column and beam. Sections 1-1, 2-2, and 3-3 locate at mid-height and near the ends of the heated column, respectively, Sections 4-4 locates at the position of beam-to-column connection, and Sections 5-5 and 6-6 locate at the mid-span of the beams. The locations of the various sections and the locations of thermocouples for each section are shown in Figure 5(a), and the measured average temperature results are used in the analysis and discussions to be followed in this article.

Figure 5.

Arrangement of measurement devices on the specimen. (a) temperature measurement; (b) displacement measurement in detail.

The heated column was far away from the furnace wall, and the conventional displacement transducers were not suitable for the measurement of displacement because of the high temperature. In these tests, the displacements were measured outside the furnace by connecting molybdenum wires to the heated components inside the furnace, including the horizontal displacements at the mid-height of the heated column and the vertical displacements at mid-span of the beams and the heated column. The displacement measurement positions are shown in Figure 5(b), two displacement measuring points were placed at mid-height to measure the in-plane and out-of-plane horizontal displacements of the heated column, respectively, and other two displacement measuring points were placed at the positions of beam-to-column connections to measure the vertical displacements of the columns. Although molybdenum wires have the characteristics of high temperature resistance and low thermal expansion, their thermal expansion may affect the data; therefore, an additional molybdenum wire with the same length was used to measure the thermal expansion during the tests, and the displacement data were calibrated at high temperature to compensate for the expansion effect.

The bearing connected to the bottom of the heated column was fire protected with alumino-silicate refractory fiber blankets and located under the ground to avoid the effect of elevated temperature. The stains at the thick-walled tube were recorded through wires connected to the strain gauges under the ground, and the temperature compensation was realized by the full-bridge connection method. Thermocouple data during the tests show that the temperature of the bearing was close to room temperature within 30 min and increased slowly subsequently. Therefore, even the conventional strain gauges were also valid in the tests.

Test results and analysis

Temperatures

For test 1, six burner nozzles were used to heat the test frame and two thermocouples were used to monitor the furnace temperature. There were non-uniform temperature distributions in the furnace and the cross section of the heated column; however, only a small temperature gradient existed between the flange and web plate of the heated column, so that the average temperature–time curves for the furnace and the test frame were used in the following numerical analysis, as shown in Figure 6. The temperature in the furnace was greater than the beams and columns at the beginning. Temperature of column in fire reached 700°C at about 22 min. Temperature of the bearing was below 60°C during the test, which indicated that the protection of bearing worked well and could keep the high temperature resistance strain gauges at operating temperature well below the maximum temperature (250°C).

Figure 6.

Curves of temperature and time for test 1.

In test 2, all the eight combustion nozzles were used to replicate the standard fire temperature. Figure 7 shows the average temperature results. Temperature of column in the ground floor rose quickly without fire protection. About 10 min later, the temperature of the column reached 700°C. It was about 23 min later that the temperature of the beam reaches 700°C, and the highest temperature was lower than that of the heated column. Since the joint was protected from furnace heating, its temperature rose slowly compared to the heated column. At around 31 min, the hydraulic jacks were unable to apply the constant load on the frame, and the nozzle was turned off and the furnace was shut down. Temperature of furnace and steel column reduced rapidly, but temperature of middle story column and joint continued to rise at low speed and drop after a few minutes.

Figure 7.

Curves of temperature and time for test 2.

Deformation

Restrained thermal expansion can result in thermally induced stresses and deformation in the frame. This section describes the observed displacement and failure model of the tested specimens during the tests. The vertical displacements at the top of the heated columns plotted as a function of the average column temperature are shown in Figure 8, while the same horizontal displacements are plotted in Figure 9. Results indicate that the vertical displacements increased almost linearly until they reached the maximum value. In case of test 1, the displacement increased linearly until 600°C. Beyond 600°C, the vertical displacement became nonlinear, and the displacement of column reached a maximum value of 8.5 mm when the average temperature reached 670°C. As the temperature of the column rose above 670°C, the displacement decreased gradually. This is attributed to thermal degradation of material properties at high temperatures, although the furnace was still in the heating phase at this time. As the furnace temperature reached its maximum value, the vertical displacement remained almost unchanged, and the furnace was switched off.

Figure 8.

Vertical displacement of the heated column.

Figure 9.

Lateral displacement of the heated column.

For test 2, the load levels are different from test 1. The vertical displacement of column reached a maximum value of 11 mm when the average temperature reached 640°C. As the temperature of the column rose a little above 670°C, the vertical displacement dropped quickly, which indicates that the column buckled at 670°C. As temperature of the column continues to rise, the vertical displacement continues to decrease because of the performance degradation of steel. When buckling occurs, the column undergoes a lateral deflection in minor axis direction to a larger value (about 140 mm) in a short time, as shown in Figure 9. Because of the constraints of beams on both sides of the column, the increase of lateral displacement in major direction was very limited and it was less than 20 mm. Therefore, the main failure of the steel column is the buckling failure in the minor axis direction, and the column undergoes vertical displacement and a large out-of-plane lateral displacement after buckling. During the test, vertical displacement of the heated column increased rapidly as the temperature increased beyond 800°C, and it was hardly for the jack load to be maintained, which declared that the upper structures were failure, and the furnace was shut off.

Figure 10 shows the deformation of the frame in test 2 at two critical instants in time (pre-buckling instant and maximum furnace temperature instant). Since the actual values of the displacement were small compared to the dimensions of the frame, the displacements shown in Figure 10 have been magnified 10 times to improve visibility. Results indicate that the displacement of the test frame was almost symmetric. As the furnace temperature reached its maximum value, the vertical displacement reached 35 mm quickly and it was hardly for the jack load to be maintained, implying that the frame was undergoing failure. The frame structure and heating manner are similar to that of test 1. However, due to the differences of the load levels, the measured vertical and lateral displacements of the heated column during the post-buckled stage show a completely different response.

Figure 10.

Deformation of test frame at two different instants: (a) pre-buckling instant and (b) maximum furnace temperature instant.

Failure model analysis

After the furnace temperature reached its ambient value, the test frame was removed from the furnace. Figure 11 shows the deformation of a photograph of the heated columns. Visual observations clearly indicate the formation of a plastic hinge at the mid-span of the column. Since the beam–column connection was protected from heating and was restrained from lateral movement, plastic hinges also formed at the ends of the heated column. Moreover, the flanges suffer local buckling failure at the plastic hinge zone as shown in Figure 12. Therefore, three plastic hinges occurred after the buckling of the heated column and the collapse mode formed. As a result, the initial failure of a column in the frame structure is impacted by the surrounding components. For interior columns, the influences of the axial constraint are exerted by all the coupling beams at the upper floors.

Figure 11.

Test and numerical results of the heated columns: (a) Test 1 and (b) Test 2.

Figure 12.

Plastic hinge and local buckling modes of the heated column.

According to the test results, after first buckling of the heated column, the column undergoes lateral deflection to accommodate the thermal expansion until a new equilibrium position is found. Since the column bending moment caused by the lateral deflection is increasing, the column compression load reduces to maintain stability. If the column failure is defined as the point when the column compression load returns to its original value at begin, the heated column in test 2 fails at about 720°C according to the test results (Figure 13). After failure of the column, superstructure beams came into yield situation due to the external load.

Figure 13.

Axial force variance with temperature.

The consequences of the tests with different load levels are listed in Table 1. The results show that the column buckling temperature and failure temperature decrease with the increase of load levels. The beam rotation can be used to analysis the behavior of upper story beams. The maximum vertical displacements of the two specimens during the heating stage are about 7 and 40 mm, respectively, and then the maximum beam end rotations are about 1/250 and 1/40. However, the maximum elastic rotation is about 1/180 based on the research of Lee et al. (2009). Therefore, the upper beams in the two specimens are in the elastic stage and elastic–plastic stage, respectively. For the case of P = 100 kN, the structure yield since the upper beams could not resist the external load after the failure of the interior column. For the case of P = 50 kN, though the column buckling and failure temperatures increase, the upper beams are in elastic stage after failure of the column due to the decrease of the external load.

Table 1.

Results of the frames with different load levels.

Load levels		$T_{b}$	$T_{f}$	Upper story beams
50 kN	Test 1	670	900	Elastic
	FEA	645	655	Elastic
	Simplified	665	725	Elastic
100 kN	Test 2	640	720	Elastic–plastic
	FEA	620	620	Elastic–plastic
	Simplified	630	630	Elastic–plastic

FEA: finite element analysis; $T_{b}$ : column buckling temperature (°C); $T_{f}$ : column failure temperature (°C).

The results show that the different loading level can generate different failure temperatures of the heated column and different collapse mechanisms of the structure (local collapse or progressive collapse). Note that the process of deformation was a static process since hydraulic jack was used in these tests, which was in a state of unloading in the process of buckling stage and vertical displacement increased quickly, and the hydraulic should be kept on the initial load level. Nevertheless, dynamic response actually existed in the structure in the process of failure (Sun et al., 2012). Therefore, the dynamic response should be taken into account when fine models are adopted to analyze the structural collapse behavior under fire.

Finite element analysis

The ABAQUS finite element (FE) program is used to simulate the tests for a comparative analysis of the test results. In order to simulate the local buckling behavior in the structure more precisely, shell element is adopted to simulate the heated components in the furnace, and beam element is adopted for others. In the tests, the connecting bolts on the joint with fire protection still connected well after large deformation taking place on the structure; although several bolts yielded during test 1, the model of bolts is not build directly in the FE model, and rigid connection is assumed to simulate the connection between beams and columns. The results of numerical simulation for all the beams upside the heated column show that the rigidity of restraint at the end of the column at ambient temperature is 12.8 kN/mm, which is close to the test result of 11 kN/mm. Therefore, the rigid connection hypothesis for all the joints in the FE model is satisfactory and in agreement with the actual model. The stress–strain relationship of steel at elevated temperatures utilized in the analysis is based on the model in EC3 (European Committee for Standardization (CEN), 2001); the degradation of material properties with temperature is shown in Figure 14. The out-of-plane horizontal displacement of the beam-to-column joints was constrained in the FE model, and the columns were simulated with the boundary condition of fixed ends because the supports were anchored by rigid beams. Initial imperfection of the interior column is simulated by applying a load of 0.1 kN/m, which is equivalent to a shape that followed a sine wave and a magnitude at mid-height equal to 1/1000 times the length of the interior column. It is noted that the temperature of the flange and web of the heated column are very similar, and there is very litter temperature gradient in the section according to the test results; therefore, average temperature field was used in the FE analysis model.

Figure 14.

Material properties with temperature: (a) elastic modulus and (b) yield strength.

Some numerical simulation analysis showed that the constrained heated column would appear snap-through phenomenon in the process of structural local buckling until the structure obtained a new equilibrium. The snap-through behavior of columns has been studied for particular cases by many authors (Chandra et al., 2007; Franssen, 2000). Due to the discontinuity of the snap-through behavior, Franssen (2000) proposed an analysis method to switch the incremental analysis parameter from temperature to displacement control during the snap-through. Thus, this simplified method is applied to the numerical simulation in this article. In this displacement-control method, the temperature field was kept invariant in the unstable buckling process. When a new equilibrium position was found, conventional static analysis method was adopted again to analyze subsequent temperature load applied until the structural collapse.

Figures 8 and 9 show the relationships between vertical displacement and lateral displacement of the interior column and temperatures, respectively. The numerical consequences approximately matched up with the test results. The simulation results show that the process of buckling is obvious, but there was no obvious jumping buckling process in the tests, this is maybe the displacement decreased sharply when structural buckling, and the jack was in a state of unloading until it came to an equilibrium point to loading again. The ultimate failure modes of interior column in test 1 are compared with numerical simulation in Figures 11 and 12, which indicated the coincidence. In order to compare the numerical and experimental results, the numerical results are also listed in Table 1. In the case of test 1, column buckling temperatures of the numerical and experimental results were 645°C and 670°C, respectively. In the case of test 2, they were 620°C and 640°C, respectively. The numerical analysis predicts lower column buckling temperature and failure temperature than the test consequences. This is mainly because this analysis assumed a uniform column temperature distribution and rigid connection for the joints. In addition, the test results show that a gap was formed between the end plate and the column at the joint in the furnace and several bolts yielded, a fine FE model (Al-Jabri et al., 2004; Yang and Tan, 2012) considering the bolts is necessary if the local connection behavior is also needed to analysis.

Simplified analysis

Interior columns in these frames have equally restrained stiffness but not loading levels. They have similar growth rates of displacements at the top of columns and axial forces during the pre-buckling stage, but not during the post-buckling stage. With increasing loading level, the failure temperature of the column decreases and the upper beams yield gradually. According to the test results of the maximum beam end rotations, the upper beams in the two specimens are in the elastic stage and elastic–plastic stage, respectively. Therefore, the heated column was under nonlinear end constraints as opposed to an idealized column under elastic constraints.

Based on the test and numerical results, a modified analytical model of a column with constraint springs was used to analyze the collapse behavior of the heated column, as shown in Figure 15. The constraint springs are modeled with the nonlinear spring properties to take into account inelastic effects of the surrounding structures, and the column is modeled with three plastic hinges. In order to consider nonlinear spring, a simplified tri-linear model developed by Lee et al. (2009) is used for the analysis of load-bearing capacity of the upper story beams. This model consists of a tri-linear curve up to the limiting rotation and involves seven parameters: three beam action parameters (P_p, K_E, and u_p) and four catenary action parameters (K₁, K₂, u₁, and u_limit). In the case of the three-story frame in the fire tests, K₁/K_E = 0.0386, K₂/K_E = 0.098, u₁ = 167 mm, u_limit = 266 mm, as shown in Figure 16.

Figure 15.

Collapse analysis model of steel frames under fire.

Figure 16.

Model of beams.

The behavior of a restrained and axially loaded column at elevated temperatures can be divided into the three stages, pre-buckling stage, buckling stage, and post-buckling stage. Figure 17 shows the simplified behavior of pre-buckling stage and post-buckling stage.

Figure 17.

Simplified behavior of the restrained column under fire: (a) before heating, (b) pre-buckling, (c) post-buckling, and (d) post-buckling deformation.

During the pre-buckling stage

The free thermal expansion is $α (T - 20) l$ . The reduction of the free thermal expansion due to the external axial force P can be expressed as $P / (K_{1} + K_{c})$ , and the increase in the compression force due to the thermal expansion causes an extra reduction of the free thermal expansion given by $α (T - 20) l / (1 + K_{c} / K_{1})$ . Therefore, the total axial deflection of the model can be expressed as

\begin{matrix} u (T) = α (T - 20) l - P / (K_{1} + K_{c}) \\ - α (T - 20) l / (1 + K_{c} / K_{1}) \end{matrix}

(1)

where P is the external load, $K_{1}$ is the initial stiffness of the nonlinear spring, $K_{c} = k_{ET} EA / l$ is the axial stiffness of the column, and $k_{ET}$ is the reduction factor of elastic modulus with temperature. The column buckles when $P + K_{1} u (T) = R_{cr}$ (where $R_{cr}$ is the elastic buckling load).

During the post-buckling stage

According to the failure mechanism of the heated column in the tests, the fully plastic mechanism includes three plastic hinges at the column’s ends and mid-height as shown in Figure 17(d). Therefore, the equilibrium equation can be expressed as

P - Ku (T) = 2 M_{p} k_{yT} / Δ

(2)

where $k_{yT}$ is the reduction factor of yield strength with temperature and $Δ$ is the lateral displacements at column mid-span. The relationship between the axial and lateral mid-span deflections is approximately

Δ = \sqrt{{(\frac{l}{2})}^{2} - {(\frac{l - u - u_{0}}{2})}^{2}} \approx \sqrt{\frac{l (u + u_{0})}{2}}

(3)

where $u_{0}$ is the axial displacement before buckling. Therefore, the equation of axial deflection during the plastic post-buckling stage can be expressed as

\begin{matrix} u (T)^{3} + (u_{0} - \frac{2 P}{K}) u (T)^{2} + (\frac{P^{2}}{K^{2}} - \frac{2 P u_{0}}{K}) u (T) \\ + \frac{P^{2} u_{0}}{K^{2}} - \frac{8 M_{p}^{2} k_{yT}^{2}}{K^{2} l} = 0 \end{matrix}

(4)

Note that spring stiffness K is not a fixed value here. In the process of solving the cubic equation, the spring stiffness changes with the vertical displacement. If the axial force $P - Ku (T)$ in the column returns to its original value $P_{0}$ before the fire, where $P_{0} = P / (1 + K_{1} l / EA)$ , the column axial displacement can be expressed as

u (T) = [P - P / (1 + K_{1} l / EA)] / K

(5)

The column buckling temperature can be obtained through equation (1), and the column failure temperature can be obtained through equations (4) and (5). As for the upper structure, the state can be analyzed according to the column axial deflection.

Figure 18 shows the column vertical displacement plotted as function of temperature compared with the test results, pre-buckling curve (0-1) is obtained through equation (1), and post-buckling curve (2-3-4-5) is obtained through equation (4). Note that there is a sudden displacement change at the point 4 because stiffness of the spring has shifted from K_E to K₁. If the spring remains elastic, the post-buckling stage will travel along curve 2-3-6. Therefore, the results considering the nonlinear spring are more in line with the test results. The results show that the column buckling temperature is 630°C (point 1), and the column failure temperature is 630°C (point 3) because the intersection of equations (4) and (5) is located at the unstable stage (point a) and the structure snaps-through from point 1 to point 3. In the case of test 1, shown in Figure 19, the buckling temperature and failure temperature are 665°C and 725°C, respectively. The post-buckling curve indicates that the spring remains elastic even after the collapse of the column. The predicted results of the tests are listed in Table 1. The comparison of the theoretical and experimental results reveals that the analytical model is generally able to conservatively estimate the buckling temperature, failure temperature, and the upper structure state after column buckling.

Figure 18.

Column vertical displacement with temperature (compared with test result).

Figure 19.

Column vertical displacement with temperature (different load levels).

Summary and conclusion

Based on the study on initial collapse mechanism of a steel column in a frame structure with realistic inelastic end constraints under fire, the following conclusions or recommendations can be made:

Flexural buckling failure of interior columns is a kind of initial collapse mode in the frame structure, and the buckling failure mode is influenced mainly by the constraints of beams above the interior column and the load level. The column buckling temperature and failure temperature decrease with the increase of load levels.

The external load can be redistributed after the heated column buckled and be gradually supported by the surrounding structures. The upper beams maybe yield gradually. Therefore, the heated column in a structure is under nonlinear end constraints as opposed to an idealized column under elastic constraints.

The response of the heated column in steel-framed structures with realistic nonlinear end constraints is almost the same with that of column with elastic end constraints during the pre-buckling stage but is completely different during the post-buckling stage. The column failure temperature decreases and the vertical displacement gets larger; the structure is more in danger of collapsing.

The simplified analysis model considers column with lumped plastic hinge and nonlinear spring constraints. Based on the simplified analytical method, it is convenient to determine the column buckling temperature, failure temperature, and the upper structure state after column buckling for the tests. However, realistic evaluation of the constraint spring in this model is important for accurate analysis of structure under fire. In realistic structures, the constraint spring properties should be obtained through evaluation of the three-dimensional beams and slabs.

Footnotes

Acknowledgements

The authors would like to thank Dr Liu Chunyang and Wang Yuzhuo for their support in the tests. The authors would like to thank the reviewers for their thoughtful comments.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by the Natural Science Foundation (51278516) and Beijing Science and Technology Foundation (KM201210005025).

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