Numerical investigation on moment redistribution of continuous reinforced concrete beams under local fire conditions

Abstract

This article presents a numerical investigation on the mechanical behaviours, such as the fire resistance, the moment redistribution and the failure mode, of continuous reinforced concrete beams with two spans and three spans under the standard fire of ISO-834. Firstly, a 3D finite element model was established and validated against the fire test beam. Secondly, the three stages associated with the fire time of fire behaviour for the continuous reinforced concrete beams were divided and explained. An index of the moment redistribution amplitude was modified and used to evaluate the fire performance of continuous reinforced concrete beams. A series of parametric analyses for continuous reinforced concrete beams with two spans were conducted in order to investigate the influence of some parameters such as the load ratio, the load position, the support condition and the sectional size. Finally, the distributions of the vertical deflection and the plastic hinge within beam spans and the failure modes for continuous reinforced concrete beams with three spans under local fire conditions were discussed emphatically.

Keywords

Continuous beam fire resistance moment redistribution plastic hinge failure mode local fire

1. Introduction

Reinforced concrete (RC) structural systems are commonly used in high-rise buildings due to a series of advantages such as good weather resistance and durability. Fire incidents occasionally occur during their design lifetimes. Therefore, they have to achieve sufficient fire resistance and excellent fire performance to ensure life safety and property protection under fire. Moreover, the provision of fire resistance for structural members is a significant aspect of structural design. At present, the fire resistance of the continuous beams is generally established using prescriptive approaches, which are based on either the standard fire resistance tests or the empirical calculation methods (Dwaikat and Kodur, 2008).

From the perspective of research subjects, initially, focus is mainly given to the fire resistance of simply supported beams under fire condition (Bratina et al., 2003; Choi and Shin, 2011; Ding et al., 2018; Dwaikat and Kodur, 2009; Gao et al., 2016; Kodur and Dwaikat, 2007, 2008; Limin et al., 2015; Shi et al., 2002; Zha, 2003). However, many practical RC beams are designed as multi-span continuous ones, which have more complicated fire performance than simply supported ones, mainly due to the different redistribution amplitudes of internal forces (including shear and axial fore, especially bending moment) as a result of thermal restraint. That is a mechanical restraint from continuous boundaries against uneven thermal expansion, including axial expansion and bending deflection, resulting from of non-uniform thermal distribution within a concrete section of beam members due to thermal inertia of concrete. A larger number of fire tests on continuous beams have been conducted by some scholars (Desai, 1998; Gustaferro and Lin, 1986; Riva and Franssen, 2008; Talamona et al., 2002; Xu et al., 2015). These test results indicate that the failure modes of all tested continuous beams under fire are flexural failure mode at the positions of the middle support or the point load, and the moment redistribution within continuous beams occurs in the whole fire process. Later on, analyses on the effect of the axial restraints on the fire performance of restrained beams (Dwaikat and Kodur, 2008, 2009) and the rotational restraints on that of continuous beams by some researchers have been conducted (Guo and Shi, 2011; Xu et al., 2015). A series of papers (Capua and Mari, 2007; Lamont et al., 2004, 2007; Sanad et al., 2000; Terro, 1998; Usmani et al., 2001) have presented both the theoretical descriptions and the finite element (FE) analysis on fire exposed structure and its structural beam in-depth, in which a combination of thermal actions and end restraints is considered. However, there are limited research studies on the process of moment redistribution and plastic hinge formation, which are difficultly achieved with the help of present test methods, for continuous beams with multi-spans under local fire conditions.

The aim of this study is, therefore, to employ the numerical method to achieve a fire performance analysis on continuous RC beams with multi-spans under local fire conditions. For this purpose, a validated three-dimensional FE model was established for investigating the structural behaviour of a continuous RC beam during the whole period of fire exposure. Then the analytical result of moment redistribution for a continuous beam under fire conditions was discussed. Finally, the influences of parameters, such as load ratio, load position, support condition, sectional size and local fire condition, on the fire resistance and moment redistribution for continuous RC beams under fire were conducted.

2. Numerical modelling

2.1. Numerical models

The thermo-mechanical analysis is performed in two separate steps: the first step is to conduct thermal analysis, and in the second step, the 3D model continues with mechanical analysis. The FE model is carried out using ABAQUS/Standard 6.10 (2010).

The following assumptions are made in numerical modelling:

The thermal and mechanical analyses are independent and uncoupled;

the temperature of the furnace gas is uniform within a single furnace chamber; and

the mechanical restraint supports are not in the fire.

2.1.1. Thermal analysis

In this study, the density of concrete is taken as a constant value of 2300 kg/m³ (Gao et al., 2013). The conductivity of concrete given by Lie (1994) and the specific heat of concrete recognised in EN 1992-1-1 CEN (2004) are adopted. The density of reinforcement is taken as 7850 kg/m³, and the conductivity of reinforcement given by Lie (1994) and the specific heat of reinforcement proposed by Li et al. (1991) are adopted.

3D elements are used in the thermal analysis so that the results can be conveniently imported to the subsequent mechanical analysis. The 8-node heat transfer brick element DC3D8 is used for the concrete, and the 2-node heat transfer link element DC1D2 is used for the reinforcement. The structured meshing technique, which generates structured meshes using simple predefined mesh topologies in ABAQUS/CAE and transforms the mesh of a regularly shaped region onto the geometry of the meshed region, is adopted in mesh generation. The tie constraint is applied on nodes to bond the concrete with reinforcement. The radiative emissivity and integrated heat transfer coefficient on the exposed surface are set as 0.5 and 25 W/(m²°C), respectively (CEN, 2002). The top surface of the beam is exposed to ambient temperature, and the heat transfer coefficient is taken as 9 W/(m²°C) (Xu et al., 2015). The interaction, including surface radiation and surface film condition, is created in thermal analysis, in which the radiative emissivity, integrated heat transfer coefficient and fire temperature amplitude are edited. The initial temperature of the model is defined as 20°C. The node temperatures at each increment time are recorded, which will then be used in the mechanical analysis.

2.1.2. Mechanical analysis

A non-linear damaged plasticity model is used to simulate a concrete material (Han et al., 2012). The yielding surface and the plastic behaviour have been predefined in ABAQUS programme using the equivalent uniaxial stress–strain relationship of concrete. The non-linear stress–strain property of concrete under uniaxial compression at different temperatures is as follows (Ding and Yu, 2006)

y = {\begin{cases} \frac{k_{1} x + (m_{1} - 1) x^{2}}{1 + (k_{1} - 2) x + m_{1} x^{2}} x \leq 1 \\ \frac{x}{α_{1} {(x - 1)}^{2} + x} x > 1 \end{cases}

(1)

where y (= σ/f_c^T) and x (= ε/ε_c^T) are the normalised stress and strain ratios of concrete to the uniaxial compressive concrete at high temperatures, respectively; σ and ε are the stress and strain of concrete; f_c^T is the uniaxial compressive strength of concrete at high temperature; and ε_c^T is the strain corresponding with the uniaxial compressive peak stress of concrete at high temperature. Parameters k₁, m₁ and α₁ are adopted to describe the ascending and descending phase of the stress–strain relationship, respectively, which have been taken as k₁ = 9.1f_cu^−4/9, m₁ = 1.6 (k₁ – 1)² and α₁ = 2.5f_cu³ × 10⁻⁵, where f_cu is the compressive strength of concrete cubes.

The tensile stress–strain relationship of concrete under uniaxial tension at different temperatures is adopted as follows (Ding and Yu, 2006)

y = {\begin{cases} \frac{k_{2} x + (m_{2} - 1) x^{2}}{1 + (k_{2} - 2) x + m_{2} x^{2}} x \leq 1 \\ \frac{x}{α_{2} {(x - 1)}^{2} + x} x > 1 \end{cases}

(2)

where y (= σ/f_t^T) and x (= ε/ε_t^T) are the normalised stress and strain ratios of concrete to the uniaxial tensile concrete at high temperature, respectively; σ and ε are the stress and strain of concrete; f_t^T is the uniaxial tensile strength of concrete at high temperature; and ε_t^T is the uniaxial tensile peak strain of concrete at high temperature. Parameters k₂, m₂ and α₂ are adopted to describe the ascending and descending phase of the stress–strain relationship, respectively, which have been taken as k₂ = 1.306, m₂ = 5 (k₂ – 1)^2/3 = 0.15 and α₂ = 0.8. Given the beneficial effect of tensile reinforcement on increasing stress of concrete in FE analysis, the value of α₂ is taken as 0.8. Details of the model under high temperatures are presented (Ding and Yu, 2006).

To avoid the local stress concentration at supports and loading positions, cushion blocks are created. The cushion blocks are connected to the continuous beam with tie constraints. The steel reinforcements are embedded in the concrete so that no slippage between them occurs. However, different element types should be defined in mechanical analysis. The 8-node brick element C3D8 is used for concrete and cushion blocks, with three translation degrees of freedom at each node. The 2-node truss element T3D2, which could only support axial force and plastic deformation, is used to model the steel reinforcements and stirrups. Figure 1 shows the thermal and mechanical analysis model for continuous beams.

Figure 1.

Finite element model of continuous reinforced concrete beams: (a) whole numerical model and (b) reinforcement cage elements.

The thermal boundary conditions are applied in the mechanical analysis as follows: in the predefined field management, the ambient temperature of 20°C is directly specified in the initial step; the temperature field is reset to initial in the followed step in which the beam is applied constant load only; and the nodes temperature distribution from results or output database file in thermal analysis is used as an input in heating phase. Same mesh and same node number have been used to facilitate the transfer of results from the thermal analysis to the mechanical analysis. Implicit analysis is adopted in solving the model.

2.2. Model validation

In this section, the above numerical model is validated against the fire test of beam TCB1-1 reported by Guo and Shi (2011) in terms of thermal analysis and mechanical analysis, respectively. The geometric and material properties of the tested beam are given in Figure 2. The beam is tested under the designed fire condition, and hence, the beam is also analysed by exposing its three sides to the designed fire curves, which are shown in Figure 3.

Figure 2.

Properties of the tested beam TCB1-1 from Guo and Shi (2011).

Figure 3.

Curves of the designed fire and ISO-834 standard fire (ISO, 1999).

Comparison of the temperatures at measured point location and the maximum deflections at loading position obtained from experiment and numerical model are shown in Figure 4, respectively. The calculated results agreed well with experiment. The reaction forces as functions of furnace temperature for tested specimen obtained from both numerical model and experiment have been shown in Figure 5. Good agreements have been achieved in the first stage of initial development. However, deviations have occurred in the second stage of stable development, especially for the reaction force at the middle support. In this stage, the temperature gradient within the middle support cross-section was undervalued because the beam segment at the middle support plate was exposed to high temperatures in the numerical simulation, which is different from the experimental condition where it was exposed to ambient temperatures. This caused differences of the reaction force at middle support between the calculated and the measured curves. Moreover, deformation of continuous beams at elevated temperatures is considerable, and the influence of geometric non-linearity on the mechanical performance, such as the reaction force, could not be taken into account in the numerical simulation. Moreover, differences have been existed in the third stage of plastic hinge formation, which was captured in the numerical model and difficultly achieved through present test method. Overall, the proposed numerical model could predict the mechanical behaviours, such as the vertical deflection, the formation of plastic hinges and the failure mode, of continuous beams subjected to elevated temperatures.

Figure 4.

Comparison between the measured and calculated results in the span of the continuous beam TCB1-1: (a) temperature distribution and (b) maximum deflection.

Figure 5.

Comparison between the measured and calculated reaction forces of the continuous beam TCB1-1.

3. Discussion on moment redistribution

It should be illustrated that the discussion on moment redistribution is based on the numerical analysis results, and the experimental results are used to validate against the numerical models only.

3.1. Process of moment redistribution

Due to restraints to thermal bowing during the whole fire process, redistributions of internal forces such as moments and reaction forces will occur within continuous RC beams, which are statically indeterminate structures. The reaction forces at supports vary with furnace temperature seen in Figure 5. The maximum sagging moment in the span (M₀ = βlR_A, where β is the load position ratio defined as the distance between the concentrated load and the end support to the length of span) is proportional to the reaction forces R_A. Similarly, the hogging moment at middle support (M_B = R_B (1 − β)l − M₀) is also proportional to the reaction forces R_B. Therefore, the trend of moment redistribution is the same as the reaction forces. Stress contours in the loading section and the middle support sections at the ends of three moment redistribution stages from the FE model are shown in Figure 6, where colourful highlights indicate the compressive areas and darker shades indicate the tensile areas of critical cross sections, respectively. Combining Figures 5 and 6, there are three stages in the process of moment redistribution within continuous beams during fire heating.

Stage of initial development: OA

Figure 6.

Stress contours in the loading section and middle support sections at the end of three moment redistribution stages: (a) in the loading section and (b) in the middle support section. (‘T’: tension zone; ‘C’: compression zone).

At the initial stage (before 500°C), the large temperature gradient within the cross section leads to non-uniform expansion in the concrete. Flexural deformation, which bows toward the high temperature zone, results in the increase of reaction force at the middle support (R_B) and the decrease at the end supports (R_A). Compared to the initial fire time corresponding to Point-O, when the top area of the concrete section is in compression and the bottom area is in tension, the compressive area is significantly reduced at the loading section of the continuous RC beam corresponding to Point-A. On the contrary, that is increased at the middle support section. It indicates that the sagging moment in the loading section (M₀) reduces and the hogging moment at the middle support (M_B) increases over fire time.

2. Stage of stable development: AB

The temperature gradient reduces significantly when the specimen is heated between 500°C and 900°C, which leads to the decrease of the reaction force at the middle support (R_B) and the increase of reaction force at the end supports (R_A). Simultaneously, the flexural rigidity and the moment capacity reduce significantly due to the rapid deterioration of material properties, which result in the increase of reaction force at the middle support (R_B) and the decrease of reaction force at the end supports (R_A). Under the two opposite effects, the development of reaction forces and sectional moments maintain relative stability in this temperature interval.

3. Stage of plastic hinge formation: BC

With the continuous increase of furnace temperature, the moment resistance in the loading section (M_0u), where the tension zone is at higher temperature, is reduced more quickly than that at the middle support (M_Bu), where the tension zone is at lower temperature. Therefore, the first plastic hinge is formed in the loading section early, which is identical to the normal stress contour of the concrete section, as seen in Figure 6(a). When the end of heating, the sagging moment in the loading section slightly reduces and the hogging moment at middle support slightly increases, which is identical to the normal stress contour of the concrete section, as seen in Figure 6. Using the above numerical model, the formation of plastic hinge within continuous RC beams during fire heating is proposed originally in this study, which is difficultly achieved through present test methods.

3.2. Amplitude of moment redistribution

The ratio between the reactions at the end support when the first plastic hinge is formed and that before heating is taken as an index of the redistribution amplitude of the internal force of the continuous beam after heating (Guo and Shi, 2011). However, from the perspective of geometric composition, not until the second plastic hinge is formed does the two-span continuous beam fail. It is, therefore, some more reasonable to define an index of the moment redistribution amplitude for continuous beams with two spans as the ratio of the moment when the second plastic hinge is formed at the critical cross-section to that before heating. The critical cross-section of continuous beams refers to the positions of the middle support and the point load along their longitudinal axis, where the corresponding hogging or sagging bending moment is the highest under point load at ambient temperature. Given the process of plastic hinge formation and moment redistribution, the index of the moment redistribution amplitude could be adopted to quantitatively calculate and qualitatively compare the fire resistance of continuous beams with different failure modes. Specifically, when the second plastic hinge is formed at the loading section in span, the smaller index represents the adequate moment redistribution within continuous beams. On the contrary, when the second plastic hinge is formed at the middle support section, the bigger index represents the adequate moment redistribution within continuous beams (as shown in Table 1).

Table 1.

Numbers and calculated results of analysed continuous beams.

Number	Load position (β)	Load level (m)	Support condition	Local fire condition	Sectional Size	Fire resistance (t_f)	M _0,20	M _0u	M _B,20	M _Bu	M_0u/M_0,20	M_Bu/M_B,20
Number	/	/				min	kN·m	kN·m	kN·m	kN·m	/	/
TCB1-1	1/3	0.25	Continuous	Two spans in fire	Smaller	71	2.09	1.55	(1.73)^f	(3.31)	0.74	(1.91)
TCB1-1m^a	1/3	0.5	Continuous	Two spans in fire	Smaller	49	4.16	3.56	(3.52)	(5.29)	0.86	(1.50)
TCB1-1β^b	2/3	0.25	Continuous	Two spans in fire	Smaller	95	(1.28)	(0.76)	2.08	3.77	(0.59)	1.81
TCB1-1S^c	1/3	0.25	Simply	Two spans in fire	Smaller	56	2.67	2.67	/	/	1.00	/
TCB1-1L^d	1/3	0.25	Continuous	Two spans in fire	Larger	/	161.01	134.45	(140.81)	/^g	0.84	/
TCB1-1F^e1	1/2	0.25	Continuous	Side span in fire	Smaller	79	(2.04)	(1.28)	1.91	3.93	(0.63)	2.06
TCB1-1F2	1/2	0.25	Continuous	Middle span in fire	Smaller	172	1.91	0.23	(5.04)	(5.84)	0.12	(1.16)
TCB1-1F3	1/2	0.25	Continuous	Side and middle span in fire	Smaller	84	5.04	5.99	(2.04)	(1.96)	1.19	(0.96)
TCB1-1F4	1/2	0.25	Continuous	Three spans in fire	Smaller	63	(2.04)	(2.11)	5.04	4.92	(1.03)	0.98

^am: load ratio.

^bβ: load position ratio.

^cS: simply supported.

^dL: large sectional size.

^eF: fire condition.

^f( ): ultimate failure mode.

^g/: not available.

4. Parametric analysis

Five specimens with two equal spans are designed to investigate the effects of the load ratio, the load position ratio and the support condition on mechanical behaviours of continuous beams, and four specimens with three equal spans are designed to investigate the effect of the local fire conditions. All analysed beams are exposed to standard fire of ISO-834 (ISO, 1999), and supports are in an area at room temperature. The geometric dimension and arrangement of steel reinforcements of all analysed beams are identified with TCB1-1 tested by Guo and Shi (2011). Numbers and calculated results of analysed continuous beams are listed in Table 1.

What needs illustration is that two different heating curves were applied on continuous beams successively, including the designed fire and the standard fire of ISO-834. The designed fire was used to validate the thermal analysis, which is in accordance with the prior experiment, and the standard fire of ISO-834 was used to all subsequent parametric analysis.

4.1. Load ratio

The load ratio (m) is defined as the load applied on the beam to the ultimate capacity at ambient temperature

m = \frac{P_{f}}{P_{u}}

(3)

where P_f is the actual concentrated load applied on the beam; P_u is the expected ultimate capacity of the beam at ambient temperature and calculated by the numerical model.

Two load ratios of 0.25 and 0.5 are considered. The failure of the analysed beam is assumed when the plastic hinges generate successively in the span and at the middle support of continuous beams with two spans, and a mechanism of one degree of freedom is formed. The moment redistribution within continuous beams has existed throughout the whole heating process (seen from Figure 7).

Figure 7.

Evolution of bending moments in span and at middle support sections and comparisons against their ultimate limits under ISO-834 for analysed secondary beams: (a) TCB1-1, (b) TCB1-1m and (c) TCB1-1β.

In comparison with the continuous beam of TCB1-1 (shown from Figure 7(a)–(b)), with a 100% increase of load ratio and the same load position and heating curve, the duration time of the first stage for beam TCB1-1m is the same due to the same temperature gradient within the cross-section. However, that of the second stage is shortened by 41.5% (53 min to 31 min) due to the significant increase of bending moment of critical cross-sections; the fire resistance is reduced by 31.0% (71 min to 49 min), and the moment redistribution amplitude is decreased by 21.5% (1.91 to 1.50). It is mainly because the duration time of the second stage is cut down that the fire resistance of continuous beams reduces with the increase of load ratio.

The fire resistance of the beam is calculated by the moment strength criteria based on complete plastic hinge mechanism. Figure 7 shows the calculated fire resistance for the analysed secondary beams, where M₀ and M_B are the bending moments at the loading section and middle support section, respectively and M_0u and M_Bu are the ultimate limits calculated from the 500°C isotherm method suggested in EN 1992-1-1 (CEN, 2004).

4.2. Load position ratio

The load position ratio (β) is defined as the distance between the concentrated load and the end support to the length of the span. Two load position ratios of 1/3 and 2/3 are considered in parametric analysis on continuous beams. In comparison with the continuous beam of TCB1-1 (shown in Figure 7(a) and (c)), the duration time of both the first stage and second stage for beam TCB1-1β are the same basically. However, at the third stage of plastic hinge formation, the formation between the first plastic hinge (formed in the middle support) and the second plastic hinge (formed in the mid-span) experiences a longer fire time.

The fire resistance is increased by 33.8%, with a 100% increase of the load position ratio. It is therefore concluded that increasing load position ratio can yield an increase on fire resistance for continuous beams. It may be because that the reduction of bending moment in span (M₀), as a result of external load closer to the middle support, can effectively delay the occurrence of the plastic failure in span in which the tension zone is exposed to high temperatures and then contribute to the improvement of fire resistance. That is to say, with the increase of load position ratio, the failure mode of continuous beams transfers favourably from ‘mid-span failed first’ to ‘mid-support failed first’. It is therefore concluded that the fire resistance of continuous beams increases significantly when the load position is closer to the support where the rotational restraint is stronger. It is mainly due to the favourable transformation of failure modes from ‘mid-span failed first’ to ‘mid-support failed first’.

4.3. Support condition

The support condition is a key parameter which influences the mechanical behaviours and fire resistance of RC beams. In the subsequent analysis, the boundary conditions of the middle support will be focused on. Both continuous and simply supported boundary conditions will be applied to the middle support, while the other parameters are kept the same as mentioned above. In comparison with the two-span simply supported beam, which is statically determinate structures, only one rotational restraint at middle support is exceeded for the continuous beam with the same span, which is indeterminate structures.

The indexes of moment redistribution for all simply supported fired beams are 1.00, which indicates that no moment redistribution exists within simply supported beams during fire exposure. However, the moment redistribution exists within continuous beams throughout the whole heating process. The comparison between TCB1-1S and TCB1-1 shows that the fire resistance is reduced by 21.1% (from 71 min to 56 min). It is therefore indicated that the fire resistance of the continuous beams is much higher than that of the simply supported beams due to the occurrence of moment redistribution during a fire as a result of thermal restraint.

4.4. Sectional size

Analyses and comparisons of the mechanical behaviour of continuous RC beams with larger and smaller sectional size subjected to the standard fire of ISO-834 are conducted. Properties of the continuous beam with a smaller sectional size, of which the expected ultimate bearing capacity at ambient temperature is 40 kN, are shown in Figure 2. Properties of the continuous beam with a larger sectional size, of which the expected ultimate bearing capacity at ambient temperature is 520 kN, are shown in Figure 8. The characteristic strengths of stirrups and reinforcements are taken as 300 MPa and 400 MPa, respectively.

Figure 8.

Cross section of analysed continuous girder beams.

The plastic hinge at loading section, of which the tensile zone is in high temperatures, is formed firstly, and that at middle support is not formed during the whole fire time of 200 min for continuous with larger sectional size, because of the slowdown in hogging moment resistance at middle support, of which the tensile zone is in low temperatures.

In comparison with the continuous beam with a smaller sectional size of TCB1-1 (shown from Figures 9 and 7(a)), with the same load position ratio of 1/3, load ratio of 0.25 and heating curve of ISO-834, the duration time of the first stage is enlarged twice for beam TCB1-1L, of which the larger temperature gradient exists in cross-section due to thermal inertia of concrete. The duration time of the second stage is increased by 81.1% (from 53 min to 96 min) for beam TCB1-1L (with a larger sectional size) because the degradation ratio of sagging moment resistance at the loading section, which is the slopes of evolution curves at the second stage, is much slower than TCB1-1 (with a smaller sectional size). It is therefore concluded that the fire resistance of continuous beams with a larger sectional size is much higher than that with smaller ones due to the combined effects of the evolution of temperature gradient at the first stage and the degradation ratio of sagging moment resistance at the second stage.

Figure 9.

Evolution of bending moments in span and middle support sections and comparisons against their ultimate limits under ISO-834 for analysed girder beams TCB1-1L.

4.5. Local fire condition

Fully developed fires can occur in compartments that share a continuous load bearing element with adjacent compartments. In other words, it can be the case if the fire area is localised within the compartment. So, even if the continuous beams are statically indeterminate, a proper analysis of the structural behaviour must take into account the spatial variability of local fire scenarios.

Defining the thermal exposure from real-world fires is rather difficult. Changing gas phase temperatures and emissivity through varying heat fluxes are difficult to resolve accurately in time and space. Hence, the investigation on structural performances under real-world fires often has to fall back to the standard fires. Moreover, the real-world fires could have been approximately normalised to the standard fires. How to relate the structural performance of building elements in standard fires to their performance in real-world fires has been approximately resolved by the researcher Harmathy (1980, 1981).

Therefore, the investigation on the structural performance of continuous beams with multiple supports under local standard fires has theoretical significance.

Figure 10 shows four local fire conditions for continuous beams with three equal spans on which the concentrated load of the same position (at the middle point) and value is acted. The failure of continuous beams with three spans occurs when a mechanism of three degrees of freedom is formed or when a local component is disabled to continue to sustain the external load. Figure 11 shows the evolutions of bending moments at mid-span and middle support sections and comparisons against their ultimate limits under local fire conditions.

Figure 10.

Distributions of vertical deflections and plastic hinges for analysed beams under different local fire conditions: (a) TCB1-1F1 under side span in fire only, (b) TCB1-1F2 under middle span in fire only, (c) TCB1-1F3 under side span and middle span in fire and (d) TCB1-1F4 under three spans in fire.

Figure 11.

Evolutions of bending moments in span and at middle support sections and comparisons against their ultimate limits for analysed beams under different local fire conditions: (a) TCB1-1F1, (b) TCB1-1F2, (c) TCB1-1F3 and (d) TCB1-1F4.

4.5.1. Side span in fire only

The vertical deflections in three spans exhibit downwards under the initial load at ambient temperatures. When the beam TCB1-1F1 is under local fire in one side span, the maximum deflection of the heated side span CD increases with fire time, that of the adjacent middle span BC under ambient condition turns a negative into a positive because of the rotation of middle support C as a result of the flexural rigidity degradation of the heated side span, and that of another side span AB under ambient condition increases slightly.

As the rotational angles of the two adjacent beam ends at the support B are in the same direction during fire time, the hogging moment at the middle support B (M_B) decreases and that at the middle support C (M_C) increases, which leads to the first plastic hinge is formed early at middle support C. A mechanism of two degrees of freedom is formed after the second plastic hinge is formed at the mid-section of the heated side span CD due to disability to continue to carry on the external load. The moment redistribution amplitude of beam span is 0.63, and the fire resistance time of continuous beam TCB1-1F1 is 79 min.

4.5.2. Middle span in fire only

Only under the initially symmetrical loads do the downward deflections in three spans distribute equally at ambient temperatures. When the beam TCB1-1F2 is under local fire in the middle span only, the vertical deflections are also symmetrical about the middle span. The maximum deflection of middle span under fire temperatures increases with fire time, and that of side spans under ambient temperatures turns a negative into a positive because of the rotation of middle supports as a result of the flexural rigidity degradation of the middle span.

Because the flexural rigidity of two side spans (AB and CD) at ambient temperatures is much bigger than middle span BC under fire temperatures, the sections at the middle supports (B and C) are rotated during heating. Therefore, the negative moment of middle supports decreases, and the positive moment of the middle span increases with fire time. A mechanism of three degrees of freedom is formed after one plastic hinge appears at the heated middle span initially and two hinges are formed at middle supports later. The moment redistribution amplitude of middle supports is 1.16, and the fire resistance time of continuous beam TCB1-1F2 is 172 min.

4.5.3. Side span and middle span in fire

When the beam TCB1-1F3 is under local fire both in side span and middle span, the maximum deflection of two heated spans increases with fire time, and that of the other one under ambient condition turns a negative into a positive because of the rotation of middle support B as a result of the flexural rigidity degradation of the heated middle span.

As the beam TCB1-1F3 is heated in one side span and middle span, only, angular rotation of the section at the middle support B is certain to occur. Then the bending moment (M_B) has to be reduced, and the moment at the mid-span is increased correspondingly. Moreover, because of the reduction in moment bearing capacity of the middle span, where the tensile zone is exposed to fire, the first plastic hinge appears earlier at the mid-span section of the heated span BC and the second in the span CD. Finally, the third plastic hinge is formed at the middle support C due to moment redistribution within beams under local fire conditions. The moment redistribution amplitude of the middle support section is 0.96, and the fire resistance of continuous beam TCB1-1F3 is 84 min.

4.5.4. Three spans in fire

When the beam TCB1-1F4 is uniformly heated in all three spans, the vertical deflections are symmetrical about the middle span. Because of the flexural stiffness degradation of all heated spans, the maximum deflections of all three heated spans increase with fire time.

Moreover, the ultimate bending moments in the span sections, where the tension zone is exposed to high temperature, are reduced more quickly than those at the middle supports. Therefore, the plastic hinges are formed early in the heated three spans and distribute symmetrically both under symmetrical loading and local fire conditions. A mechanism of three degrees of freedom is formed after one plastic hinge appears at the heated middle span initially and two hinges are formed at the heated side spans later. The moment redistribution amplitude of beam spans is 1.03, and the fire resistance time of continuous beam TCB1-1F4 is 63 min.

Based on the above analyses, the following conclusion can be drawn: ‘middle span in fire only’ is the most favourable local fire condition. However, ‘three spans in fire’ is the most unfavourable one, which should be considered in a fire-resistant design for multi-span continuous beams.

5. Conclusion

This study presents a numerical investigation on the mechanical behaviours, especially on the fire resistance, the moment redistribution and the failure mode, of multi-span continuous RC beams subjected to local fire conditions using FE simulations. Parametric studies are conducted in order to investigate the influence of some parameters on the fire behaviours of continuous beams, such as the load ratio, the load position, the support condition, the sectional size and the local fire conditions. Based on the results of this study, the following conclusions can be drawn:

The process of moment redistribution of continuous beams under fire heating can be divided into three stages: initial development, stable development and plastic hinge formation.

An index of the moment redistribution amplitude is modified and used to evaluate the fire performance of continuous beams with different failure modes.

When the load position is closer to the support in which there exists much stronger rotationally restraint, the failure mode transforms favourably from ‘mid-span failed first’ to ‘mid-support failed first’.

Middle span in fire only’ is the most favourable local fire condition; however, ‘three spans in fire’ is the most unfavourable one, which should be considered in the fire-resistant design for multi-span continuous beams.

Footnotes

Authors’ Note

Zhe Li performed the FE analysis and data processing. Zhe Li and Fa-xing Ding conceived and designed the study. Shanshan Cheng reviewed and edited the manuscript.

Declaration of conflicting interest

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

Funding

This research work was financially supported by the Natural Science Foundation of Hunan Province, China (Grant no. 2018JJ3202), the Science Research Program from the Provincial Education Department of Hunan Province, China (Grant no. 17C0681), the Science Fund for Distinguished Young Scholars of Hunan Province, China (Grant no. 2019JJ20029), and the Science Research Program from Hunan University of Science and Engineering (Grant no. 16XKY070). This work was partially supported by the Construct Program of Applied Characteristic discipline in Hunan University of Science and Engineering.

Data availability statement

The datasets used or analysed during the current study are available from the corresponding author on reasonable request.

ORCID iD

Fa-xing Ding

Appendix

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