1 Introduction

The progressive collapse of a building structure refers to the disproportionate or overall structural collapse caused by an initial local failure which propagates in the structural system [1]. The initial local failure may be induced by incidental actions, such as a fire, a vehicle impact, or a gas explosion. During an actual fire, the progressive collapse of the entire structure may occur under certain circumstances following the local failure of a structural component. Table 1 lists several disastrous incidents involving the fire-induced collapse of RC structures that have occurred in China in the last 25 years. Among them, the collapse of a textile factory in Xinyu claimed 70 lives; this collapse exemplifies how a fire-induced collapse of an RC building can result in the serious loss of property and human lives [2].

Table 1 Examples of fire-induced collapse incidents of RC structures in China during the last 25 years

2 Numerical analysis method

2.1 Analysis process

The analysis framework for simulating a fire-induced progressive collapse of super-tall RC buildings is shown in Figure 1. An FE model of the overall structural system was established first. The numerical analysis was divided into the following three steps. Step 1 involved gravity loads being applied to the structure at room temperature to generate the initial internal force in the structure. Step 2 involved the selected fire scenarios being applied to the structure to generate the temperature distribution field. Step 3 involved the simulated temperature fields of the components being assigned to the structure as boundary conditions to perform the fire-induced progressive collapse analysis.

Figure 1 Framework for simulating the fire-induced progressive collapse of super-tall RC buildings

2.2 Numerical model

An accurate and efficient numerical model that considers high temperature effects is critical for analyzing the fire-induced collapse of overall structures. In the current study, the models for unheated and heated structural components were introduced, respectively, as follows:

2.2.1 Unheated structural components

The unheated columns and beams were simulated using conventional fiber beam elements [20]. The slabs were also considered because of their significant effects on the vulnerability of RC structures to resist progressive collapse [21]. The unheated floor slabs, shear walls, and coupling beams were simulated using conventional multi-layer shell elements [22, 23], as shown in Figure 2. Previous studies [20, 22] indicated that these models had satisfactory accuracy and efficiency for predicting the dynamic collapse of RC structures.

Figure 2 Fiber beam and multi-layer shell elements

2.2.2 Heated structural components

The temperature fields of RC components subjected to fire vary significantly within their cross section and along their longitudinal axis. In this study, the conventional fiber beam elements and multi-layer shell elements were extended to cover the heated RC components. The temperature fields within the cross sections of the various components were computed by the transient heat transfer analysis module in MSC.Marc [24]. The cross sections were meshed with 4-node quadrilateral elements for planar heat transfer simulations. The conventionally used heat transfer coefficient of 40 W/(m²¡¤K) and the emissivity of 0.7 of the thermal boundary condition [25] were adopted in the current study. Different temperatures were assigned to the different fibers or layers to account for the non-uniform temperature distribution within the cross-sections of the beams, columns, walls, and slabs [26]. The non-uniform temperature field along the longitudinal axis of the components was considered by subdividing the component into many individual elements. The constitutive laws of the materials [27, 28], that account for high-temperature effects, were used to simulate the material behaviors subjected to the evaluated temperature. The load redistribution, induced by the failure of structural elements, played an important role for the propagation of the collapse through the structural systems. This feature was also considered by incorporating the elemental failure criteria and the element deactivation technique in the numerical model. A detailed introduction of the constitutive laws and element failure criteria is shown in Appendix A.

  The fiber beam elements and the multi-layer shell elements were implemented in the general purpose FE code of MSC.Marc [24]. For each iteration of the calculation, MSC.Marc provides the temperature of each fiber or layer and the deformation (i.e., the translation and rotation) of each node of the fiber beam elements and the multi-layer shell elements. Then, the internal forces and the tangential stiffness of the fiber beam element or multi-layer shell element were calculated by the developed model, according to the material constitutive laws. Next, such internal forces and tangential stiffnesses were sent back to MSC.Marc to implement the solution of the dynamic equations of the overall structure.      

  By comparing the fire experiment results of different types of typical components, the developed model was validated, namely, that it can accurately predict the behavior of RC components under high temperatures. Furthermore, the computational efficiency was high enough to satisfy the demands of the overall structural collapse analysis. The validations of the numerical model are shown in Appendix B.

2.3 Numerical method

In the simulation, the implicit static and the dynamic solvers available in the MSC.Marc [24] were employed, for the pre- and post-analysis of the triggered dynamic progressive collapse. The transition from the static solver to the dynamic solver was determined by iterative computation, i.e., Steps (1)-(4), as shown in Figure 3. If the maximum velocity of the elemental nodes in Step (1) trended to continuously increase after a structural member failed, the dynamic progressive collapse was considered to have been triggered, and the dynamic analysis occurred , i.e., Step (2). Otherwise, the dynamic collapse did not occur, and the analysis was restarted, where the structural state variables (the stress-strain and temperature fields) were reloaded immediately after the failure of the member, i.e., Step (3). Consequently, the static analysis was performed until the failure of the subsequent member occurred, i.e., Step (4). Note that a standard 5% Rayleigh damping ratio was used in the dynamic collapse simulation [29]. The total computational time for each case of the studied super-tall building was approximately 60 hours on a desktop computer with an Intel E5-2620 CPU and 64-GB of memory, which proved the computational efficiency, as stated in Section 2.2.2.

Figure 3 Analysis procedure

3 Structural and analysis details of the case study

3.1 General building information

The building case studied herein is a typical super-tall RC frame-core tube building. It was designed by Lu et al. [29] with a standard inner core-tube and outer frame layout in accordance with the Chinese design code [30]. It has the common structural parameters, including the total height, the story height, and the sectional sizes of the main components. Such a structural arrangement is commonly used in the super-tall structures with the height exceeding 100 m. The building has 42 stories and a 6.1-m high penthouse on the top, with a total height of 141.8 m. It is noteworthy that this building is also very similar to the typical super-tall RC frame-core tube building proposed by the Tall Buildings Initiative (TBI) project of the Pacific Earthquake Engineering Research Center (PEER) [31]. The three-dimensional model and planar layout of the structure are shown in Figure 4. The FE model of the building was established using the method proposed in Section 2.2, and contains 25,476 fiber beam elements and 17,352 multi-layer shell elements.

Figure 4 Structural arrangement and position of the fire-affected area (unit: mm)

3.2 Fire scenario

When preventing a fire-induced progressive collapse is considered as the fire safety goal for performance-based design, a fire scenario is required to be specified first. The extreme fire loads which can cause progressive collapse are much more severe than the ordinary fire loads. Significantly, the existing fire design codes [32, 33], guidelines [34], or progressive collapse design codes [35, 36] do not explicitly provide the extreme fire scenario for assessing a fire-induced progressive collapse. As such, the real-world failure cases of the Windsor Tower and the WTC are referred to herein as extreme fire scenarios. The Windsor Tower was a 97-m tall, 28-story, steel frame-RC core tube building. A severe collapse occurred after the structure experienced a 20-hour fire in 2005 [37]. The fire was initiated on the 21^st story and spread from the 11^th to the 28^th stories, at a speed of 6.5 min/story upward, and 30 min/story downward. A similar phenomenon was also observed in the fire-induced collapses of WTC-1, WTC-2 (which collapsed after a 1.5-hour fire exposure and the 78^th-84^th stories were fully exposed to the fire), and WTC-7 (which collapsed after a 10-hour fire exposure and more than 10 stories were exposed to the fire) [38]. Thus, it is evident that fire loads causing the progressive collapse of an overall structure yield a much larger affected area and take a much longer period of time than those considered in the conventional fire safety design. For example, in the Chinese fire design code [35], a building structure is considered to be fire safe when all structural members can withstand a standard ISO fire [39] longer than the regulated duration time (e.g., 3 hours for columns). However, although the performance-based design guideline [33] allows the building owner or engineers to specify the fire load for the design, the extreme fire scenarios leading to catastrophic progressive collapse can hardly be predicted by researchers or engineers.  

Referring to the actual fire-induced collapse accidents of tall buildings, and the collapse accidents listed in Table 1, this study adopted a simplified extreme fire scenario in which a 10-hour ISO standard heating process was applied to multiple stories of a building. It is noteworthy that such a fire load is much more severe than the ISO fire used for the experiments of the structural members and substructures (usually approximately 3 hours). The fire spreading process was ignored due to the rapid vertical flame spreading speed in tall buildings (e.g., the speed of the upward flame spreading in the Windsor Tower was 6.5 min for each story). Further, it was recognized that such a simplified extreme fire scenario cannot fully represent the complicated fire action of an extreme fire. Nevertheless, this simplification was considered acceptable for the present work given that this was the first trial to implement the fire-induced progressive collapse simulation of tall RC buildings. Although a simplified extreme fire scenario was employed, a total of 10 fire scenarios were considered in the simulation to cover the complicated nature of extreme fire scenarios.

3.3 Fire-affected areas on each story

In tall and super-tall buildings, fire compartment zones are often constructed on each story, according to the design code, so as to more effectively control the fire affected areas [40]. As such, it is unlikely that the entire story is fully affected by fire. To reflect this situation, three zones were selected as typical fire-affected areas, with the fire compartment arrangement in the building being considered (Figure 4). The zones included two edge zones (Zones I and II along the X and Y directions, respectively) and a corner zone (Zone III). In addition, the influences of the number of fire-affected stories on the progressive collapse resistance were also discussed. This resulted in a total of 10 cases being simulated, as listed in Table 2. Importantly, this table also includes the simulation results to be discussed in Section 4. The table indicates that Cases A, B and C had the same number of fire-affected stories (i.e., 10 stories between the 11^th and 20^th stories), but they differed in the fire-affected areas. In contrast, Cases B1 to B7 had the same fire-affected areas on each story, but the numbers of the fire-affected stories were different.

Table 2 Simulation cases

Note: Partial - Part of the floor slabs and beams in the fire-affect area failed. No columns failed;

     Severe - Most of the slabs, beams and columns in the fire-affect area failed.

3.4 Simulation of the temperature field

The thermal analysis module of MSC.Marc was used to predict the internal temperature field of the fire-affected structural components. The material thermal parameters required that the temperature field analysis be determined according to Eurocode 2 [33]. Subsequently, the calculated temperature field was transferred to the structural analysis module.

In the thermal analysis, four thermal boundary conditions for the different components were set as follows: (1) for the concrete shear walls at the edges of the fire-affected areas, their surfaces exposed to fire were assumed to be heated; (2) for the slabs and beams located at the top of the fire-affected areas, the bottom surfaces of the slabs and the bottom and side surfaces of the beams were heated; (3) for the slabs and beams located at the bottom of the fire-affected areas, the top surfaces of the slabs and beams were heated; and (4) for the components inside the fire-affected areas, all the column, beam and slab surfaces were heated.

4 Overall fire resistance¨C a summary of analysis outcomes

Fire resistance analyses were conducted for the 10 cases introduced in Section 3.3, with the analysis results shown in Table 2. For all cases, serious damage occurred after 10 hours of fire exposure. Among these cases, the damage modes can be primarily classified into two types: partial damage and severe damage. Partial damage involves the collapse to several slabs and beams in the fire-affected areas, without the failure of the columns. Subsequently, the collapse is restricted to a local region, while a progressive collapse does not occur. An example of partial damage is shown in Figure 5a. Severe damage involves a collapse of most of the slabs, beams and columns in the fire-affected areas; the damage results in a large-scale progressive collapse. However, due to the sufficient redundancy of the entire structure, the collapse does not spread to the unheated regions of the building, as shown in Figure 5b. The severe damage mode is very similar to the fire-induced collapse of the Windsor Tower [37], in which the outer frame, exposed to the fire, collapsed, while the inner concrete core tube survived.

More specifically, Cases A, B and C exhibited different damage modes; for example, progressive collapses occurred in Cases B and C, but not in Case A. This phenomenon suggests that different fire-affected areas yield different failure modes. Similarly, different numbers of fire-affected stories also result in different failure modes; for example, the progressive collapses occurred in Cases B5 to B7, which had a greater number of fire-affected stories, but this did not occur in Cases B1 to B4, which had less fire-affected stories. The mechanisms of the different failure modes are discussed in Section 6.

5 Detailed analysis of a typical fire-induced progressive collapse case (Case B)

5.1 Collapse process

Case B is discussed in detail in this section to illustrate the collapse process, the failure mechanisms, and the interactions of the different components in the building. To facilitate the discussions, the structural components are designated as ¡°Column/Beam story number-component number¡±. For example, ¡°Column 11-5¡± represents Column 5 (see planar layout in Figure 4) on the 11^th story (see elevation in Figure 6 where the key points and components are illustrated).

Figure 6 Positions of the key points and key components

The progressive collapse process of Case B is shown in Figure 7. As the temperature increases, the heated slabs are deformed vertically due to the combined effects of the temperature and the gravity loads. Simultaneously, due to thermal expansion, the slabs and beams push the peripheral columns outward. When t = 1.5 h, the peripheral columns deform laterally up to 5 cm (Figure 7a). When t = 3.3 h, the vertical deflection of the slabs reaches 50 cm. Meanwhile, the tensile strain of steel in the middle span of the slabs exceeds the ultimate strain and, in turn, ruptures. Despite this situation, such a local collapse of the slabs does not affect the stability of the entire structure (Figure 7b). When t = 6.5 h, most of the slabs in the fire-affected areas were found to collapse. Also the peripheral beams located on the lower stories of the fire-affect areas collapse due to the rupture of reinforcement at a large deflection and a high temperature. When t = 7.3 h, Column 11-5 collapses due to a large lateral deformation (Figure 7d). The detailed failure mechanism of Column 11-5 is discussed in Section 5.3. Two seconds after the collapse of Column 11-5, Column 11-6 also collapsed, which triggered a progressive collapse (Figure 7e). Ten seconds after, the fire-affected structural areas collapsed completely (Figure 7f). Due to the large redundancy of the tall building, such a local collapse does not cause the further collapse of the residual structure.

5.2 Analysis of the internal force in the collapsed areas

The simulation results presented above reveal that the progressive collapse was triggered by the failure of Columns 11-5 and 11-6. The horizontal displacement of Point D at the top of Column 21-5 (the roof, as shown in Figure 6) is shown in Figure 8, which clearly shows that the responses of the structure can be divided into the following three phases:

(1) Phase A: static loading phase under gravity load before heating;

(2) Phase B: quasi-static heating phase before progressive collapse; and

(3) Phase C: dynamic progressive collapse phase.

Figures 9a - 9c show the time histories of the vertical displacement, axial forces, and bending moments in Columns 11-5 and 11-6. Before the progressive collapse occurred (i.e., t ¡Ü 7.7 h), no obvious change was found in the vertical displacement of the columns. However, the axial forces of the columns changed significantly (up to 26.8%). The axial compressive forces increased firstly, and then decreased when approaching progressive collapse. In addition, the bending moments of the columns changed significantly and became reversed during the heating process due to the thermal expansion of the fire-affected floor slabs and beams.

5.3 Analysis of the collapse mechanism

The numerical simulation outlined in Section 4 showed that the failure of Column 11-5 triggered the progressive collapse of the entire fire-affected structural areas. The detailed collapse mechanisms are explained in Figure 10. During the initial phase of the heating, the fire-affected stories pushed the column laterally outward due to thermal expansion. Figure 10a shows the lateral deformation of the column when t = 3.3 h. Subsequently, the fire-affected slabs and beams on the 11^th through 13^th stories failed consecutively (Figures 7b and 7c). The axial forces from the failed beams and slabs vanished, which resulted in an inward movement of ¡®Point a¡¯ on the column, due to the recovery of elastic deformation. Meanwhile, ¡®Point b¡¯ on the column was continuously pushed outward by the thermal expansion of the slabs and beams on the 14^th to 20^th stories. Such deformation resulted in a significant increase of curvature at ¡®Point a¡¯ (Figure 11). Finally, Column 11-5 failed at ¡®Point a¡¯ when t = 7.7 h, due to extensive flexural deformation.

A previous study by Li et al. [19] indicated that the additional axial forces, due to restrained thermal expansion induced, in the main, the failure of columns in multi-story RC structures. However, when multiple stories were heated at the same time in a super-tall building, the columns deformed together outwardly, due to the thermal expansion of the fire-affected floor system, which released the axial thermal expansion of the columns. Furthermore, due to the relatively small axial load ratio in super-tall buildings (e.g., the maximum axial load ratio of the columns was approximately 0.3 in this case), compressive failure did not occur in the columns. In contrast, the lateral deformation of the columns resulted in additional bending moments that lead to the flexural failure of the columns. Such a failure mode was significantly different from that of low- to medium-rise RC structures.

5.4 Internal forces of surrounding components

The failure of the fire-affected region induced a change in the internal forces of the surrounding components. Figure 12 shows the changes of the internal forces in the surrounding components of Case B. Columns 21-5 and 21-6 are located just above the fire-affected stories. Figure 12 clearly shows that Columns 21-5 and 21-6 moved downward when the progressive collapse was initiated, and the axial forces changed from compression to tension. Such tensile forces were able to transfer the unbalanced gravity load to other vertical components (i.e., columns and shear walls) and prevented the floor system of the 21^st story from collapsing. This, in turn, prevented the subsequent collapse of the surrounding components.

Figure 12c shows the axial forces of the surrounding columns on the 15^th story. Before the fire-induced progressive collapse occurred, the axial force of Column 15-3 significantly decreased and the axial force in Column 15-7 increased. The axial forces in Column 15-4 remained relatively constant. The mechanisms of this phenomenon are explained in Figure 13.

Figure 13a shows the deformation mode of the fire-affected areas when t = 1 h, and Figures 13b and 13c demonstrate the relative values of the displacements and the internal forces of the typical points compared with those before the structure was subjected to fire, respectively. Because the axial compression was negative, the increase in the axial compressive force was also negative. Point P3 moved upward and Point N3 moved downward due to the thermal expansion of Columns 11-6 through to 20-6 (Figure 13b), which increased the axial compressive forces in these columns (i.e., Point M3 in Figure 13c). However, due to the lateral thermal expansion of the fire-affected floor system, Columns 11-7 to 20-7 and Columns 11-3 to 20-3 formed a Vierendeel truss system. Thus, the deformation of the Vierendeel truss system resulted in the downward movements of Points P2/N1 and the upward movements of Points P1/N2 (Figures 13a and 13b). This Vierendeel truss action also increased the axial compressive forces in Columns 11-7 through 20-7 (i.e., Point M2 in Figure 13c) and decreased the axial compressive forces in Columns 11-3 through 20-3 (i.e., Point M1 in Figure 13c). The above discussion clearly demonstrates the complicated nature of the fire-induced internal force redistribution of super-tall buildings. The internal forces were redistributed horizontally and vertically. For example, the horizontal redistribution of the internal force resulted in, approximately, a 20% increase in the axial force in Column 11-5 (Figures 9b and 12c). Meanwhile, the vertical redistribution of the internal force lead to a 100% increase in the axial force in Column 15-7 (Figure 12c).

Figure 12d shows the fire response of the shear wall, where the locations of Points E and F are shown in Figure 6. The compressive stress at Point E increased when the progressive collapse occurred, which redistributed the gravitational load from the failed columns. However, due to the sufficient redundancy of the core-tube, such a stress increase was insignificant. Therefore, the core-tube was a critical alternative load path for transferring the redistributed gravity load when the fire-affected columns failed.

6 Comparative analysis of different cases

6.1 Comparison of different fire-affected areas on each story (Cases A, B, C)

Cases A, B and C were compared in relation to the influences of different fire-affected areas on each story. Progressive collapses occurred in Cases B and C, but not in Case A. Figure 14 shows the lateral deformations of typical fire-affected columns in Cases A, B and C. Because the fire-affected floor systems in Cases B and C were much larger than those in Case A, Column 11-5 of Case B and Column 11-7 of Case C experienced significant lateral deformations that resulted in the flexural failure of the columns. By contrast, the lateral deformation of Column 15-1 in Case A was much smaller and, therefore, the collapse can be prevented.

6.2 Comparison of different numbers of fire-affected stories (Cases B1-B7)

Figure 15 shows the curvatures of ¡®Point a¡¯ in Column 11-5 (Figure 10) for Cases B1 through to B7. The progressive collapse occurred in Cases B5, B6 and B7, which is explained by the significant increase in the curvature shown in Figure 15, and it is similar to the phenomenon shown in Figure 11. In contrast, the changes in curvature were much smaller for those cases without a progressive collapse (i.e., Cases B1 to B4). The more stories exposed to fire the larger the lateral displacement of the outer columns (e.g., ¡®Point b¡¯ in Figure 10) was, resulting in a substantial increase in the curvature. Figure 16 also shows the axial force in Column 11-5 for Cases B1 through to B7. The axial forces of the collapsed cases (Cases B5 to B7) were slightly larger than those of the non-collapsed cases (Case B1 to B4), which was due to the P-D effect; the larger lateral displacements were developed in the collapsed cases. The significant development of the bending deformations and the smaller changes in the axial forces confirmed that the progressive collapse was not induced by the additional axial force, but the bending deformation was due to the thermal expansion.

7 Conclusion

In this work, the progressive collapse of a typical super-tall RC frame-core tube building model, exposed to extreme fires, was simulated using the proposed analysis framework and numerical model. The primary conclusions demonstrated the main findings of this study are given as follows. Firstly, when subjected to extreme fire loads, the progressive collapse may originate in the fire-affected areas. However, due to the large redundancy of the super-tall RC frame-core tube building, the progressive collapse of the residual structure outside of the fire-affected areas can be prevented via alternative load paths. Secondly, the progressive collapse of the super-tall building analyzed was triggered by the flexural failure of the peripheral columns, due to the outward push by the thermal expansion of the upper floors and the inward contraction of the lower floors. The more stories subjected to fire, the greater the possibility of progressive collapse. Finally, when multiple stories were subjected to fire, the internal forces in the components were redistributed in the horizontal and vertical directions through the Vierendeel truss mechanism, due to thermal expansion deformation. In summary, the present work reveals the mechanisms of the fire-induced progressive collapse of a typical RC super-tall building, and provides an effective analysis framework for further study of the fire safety of tall and super-tall RC buildings, in particular, in evaluating the collapse behavior of all potential regions of such a building under fire.

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Appendix A

1 Material constitutive laws

It is well known that the strains on concrete at high temperatures are made up of 4 components: stress strain (¦Å_¦Ò), thermal expansion strain (¦Å_th), thermal creep strain (¦Å_cr), and thermal transient strain (¦Å_tr) [27]. The behaviors of concrete subject to these four strain components are simulated using the models proposed by Guo and Shi [27, 28]. The stress-strain (¦Å_¦Ò) curve of concrete is shown in Figure A.1. Specifically, for concrete layers in multi-layer shell elements, the elasto-plastic-creep constitutive model provided by MSC.Marc [41] is adopted. The relationships between the equivalent stress and the stress strain (¦Å_¦Ò) at different temperatures are inputted into MSC.Marc by defining a series of stress-strain curves at different temperatures. The thermal expansion strain (¦Å_th) is simulated by defining a thermal expansion coefficient curve at different temperatures in MSC.Marc. The deformation of concrete due to the thermal creep strain (¦Å_cr) and thermal transient strain (¦Å_tr) are simulated by programming the user subroutine CRPLAW in MSC.Marc. For the concrete fibers in the fiber beam elements, the behavior of concrete is defined using the subroutine UBEAM, following the previous work of Li et al. [19].

Note: T₁, T₂ and T₃ are the temperatures of concrete; f^T_c, ¦Å^T_p, E^T_o and ¦Å^T_u are the peak compression strength and corresponding strain, initial elastic modulus and ultimate compression strain of concrete under the temperature T, respectively; f^T_t, ¦Å^T_tcr and ¦Å^T_tu are the tension strength, cracking strain and ultimate tension strain of concrete under the temperature T, respectively; a is the cracking factor, set to 0.5.

The constitutive model of steel is much simpler. The constitutive law of steel accounting for high temperature effects proposed by Guo and Shi [28] is adopted and implemented in MSC.Marc according to the work of Li et al. [19]. The stress-strain curve of steel is shown in Figure A.2.

Note: T₁, T₂ and T₃ are the temperatures of steel; f^T_y, f^T_u, E^T_s, ¦Å^T_y and ¦Å^T_su are the yield strength, ultimate strength, elastic modulus, yield strain and ultimate strain of steel under the temperature T, respectively

2 Elemental failure criteria

During the collapse process, the structural components either crack or crush to small fragments. This phenomenon is simulated with the elemental deactivation technology. When a specified elemental-failure criterion is reached, the element will be ¡°deactivated¡± and be removed from the FE model.

In this study, if the principal compressive strain in a layer/fiber exceeds the ultimate strain of concrete (i.e., the softening branch of the concrete approaching zero), or the principal tensile strain exceeds the rupture strain of steel, the stress and stiffness of this layer/fiber are deactivated. Thus, this layer/fiber no longer contributes to the computation of the entire structure. If all the layers of a shell element, or all the fibers in a fiber beam element are deactivated, the element is considered fully deactivated from the model [24]. The ultimate strains of steel and concrete at high temperatures suggested by Guo and Shi [28] are used in this work.

Appendix B

1 Validation using RC column test results

Lie et al. [25] performed a series of fire tests of full-scale RC columns. In this study, a test specimen with a sectional size of 305 ¡Á 305 mm is selected to validate the developed numerical model (Figure B.1). This column is fixed at both ends, and a constant axial load is applied on the column throughout the heating process. The temperature in the furnace is controlled according to the ASTM E119 standard heating curve [42]. The temperature fields measured in the test and predicted by MSC.Marc are shown in Figure B.1b. A good agreement is obtained. Some differences exist between the predicted and measured temperatures during the earlier stage of fire exposure (0 - 50 min). Such a difference is also found by Lie et al. [25] which is due to the migration of moisture. Note that at a later stage, good agreement between the calculated and measured temperatures is obtained, which is an important stage when predicting fire resistance.

The axial displacements measured in the test and predicted by MSC.Marc are shown in Figure B.1c. Because the predicted temperature is lower than the experimental values during the initial stage, the simulated deformation is also smaller. At a later stage, the predicted deformation agrees well with the test results. Therefore, the developed model can be used to analyze the fire-induced failure of columns. The analyses using 105% and 95% of the material strength are also conducted to demonstrate the effect of material uncertainty on the structural response of the RC column exposed to fire. It can be seen from Figure B.1c that the material strength certainly influences the deformation of the column after 40 min. At this time, a large proportion of the column section is under high temperature. The material degradation significantly increases the actual axial load ratio, which consequently influences the mechanical response of the column in fire.

2 Validation using RC beam test results

Lin et al. [43] performed a series of fire-resistance tests of RC beams. One of the beams is selected to validate the developed numerical model. The boundary conditions, geometric dimensions and reinforcement information are shown in Figure B.2a. The diameter and yield strength of the steel bar were 19 mm and 435.8 MPa, respectively. The cylinder compressive strength of the concrete was 29.46 MPa. In this test, the bottom and both sides of the beam were exposed to an ASTM E119 standard fire [42].

  The predicted temperature fields using MSC.Marc are shown in Figure B.2b, which is consistent with the test results. The predicted deformation is shown in Figure B.2c. During the early stage of heating, the simulated result is slightly smaller than the test results. However, the simulation at and beyond 30 min agrees well with the test results, which are significant for a collapse simulation. Note that no measured data are available for the test after 80 min of heating due to the excessively large deformation of the beam. However, the numerical simulation can trace the deformation of the components throughout the failure process.

3 Validation using RC slab test results

Lin and Wade [44] tested a series of two-way RC slabs. Two specimens (DH12 and D147) are simulated herein with identical sizes and loads (Figure B.3a). However, Specimen D147 had a smaller reinforcement ratio than Specimen DH12. A four-edge simple-supported boundary condition was applied to the specimen during the test. The slabs were heated at the bottom in the furnace. A constant distributed load was applied to the specimens throughout the heating process.

The multi-layer shell element model is used to simulate the fire temperature distribution. The simulated temperature distribution agrees well with the test results, as shown in Figure B.3b. Furthermore, the predicted deformation is consistent with the test (Figure B.3c). Note that some discrepancies exist between the test results and the predicted deformation of Specimen D147 when the deformation is greater than 186 mm. This is because such a large deformation is beyond the capabilities of the test instrument. However, the numerical simulation is capable of predicting the collapse behavior under large deformations. Therefore, the comparison of these two specimens further validates the accuracy of the developed model. The analysis also indicates that the 5% variation of the material strength has little effect on the mechanics response of the RC slabs. This is because the slabs exhibit the tensile membrane action under fire in which the two-way load transferring mechanism dominates the deformation of the slabs.