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Article

Progressive Collapse Resistance Assessment of a Multi-Column Frame Tube Structure with an Assembled Truss Beam Composite Floor under Different Column Removal Conditions

by
Rongguo Zhao
1,2,*,
Guangfei Chen
1,3,
Zaihua Zhang
4,5,* and
Wei Luo
1,3
1
Key Laboratory of Dynamics and Reliability of Engineering Structures of College of Hunan Province, Xiangtan University, Xiangtan 411105, China
2
School of Mechanical Engineering and Mechanics, Xiangtan University, Xiangtan 411105, China
3
College of Civil Engineering, Xiangtan University, Xiangtan 411105, China
4
Civil Engineering College, Hunan City University, Yiyang 413000, China
5
Hunan Engineering Research Center of Development and Application of Ceramsite Concrete Technology, Hunan City University, Yiyang 413000, China
*
Authors to whom correspondence should be addressed.
Buildings 2024, 14(1), 111; https://doi.org/10.3390/buildings14010111
Submission received: 23 November 2023 / Revised: 15 December 2023 / Accepted: 30 December 2023 / Published: 31 December 2023
(This article belongs to the Section Building Structures)
Browse Figures
Versions Notes

Abstract

:
To estimate the progressive collapse resistance capacity of a multi-column frame tube structure with an assembled truss beam composite floor (ATBCF), pushdown analysis and nonlinear dynamic analysis are conducted for such a structure using the alternate load path (ALP) method. The bearing capacities of the remaining structures under three different work conditions, which are the side middle column removal, the edge middle column removal, and the corner column removal, are individually studied, and the collapse mechanism of the remaining structures is analyzed based on the aspects of the internal force redistribution and the failure mode of the second defense line. Simultaneously, the influence of the column failure time on the dynamic response of the remaining structure and the dynamic amplification coefficient is discussed. The results indicate that the residual bearing capacity of the remaining structure following the bottom corner column removal is higher than that of the one following the side or edge middle column removal, while the latter has a stronger plastic deformation capacity. When the ALP method is adopted to operate the progressive collapse analysis, it is reasonable to take the column failure time as 0.1 times the period of the first-order vertical vibration mode of the remaining structure, and it is suitable to set the dynamic amplification coefficient as 2.0, which is the ratio of the maximum dynamic displacement to the static displacement of the remaining structure under the transient loading condition.

1. Introduction

Since the collapse of the Ronan Point apartments in the United Kingdom in 1968, the progressive collapse resistance of building structures has received wide attention in the world for the first time. The progressive collapse of a building structure refers to the structure’s local failure caused by accidental loads, which are transmitted among the components and trigger a chain reaction, ultimately leading to the collapse of the whole structure or causing a large-scale collapse of the structure that is disproportionate to the initial local failure [1].
To analyze the progressive collapse resistance performance and mechanism, a lot of research works were carried out on building structures, among which reinforced concrete (RC) structures were the most common building structures [2]. In terms of the standardized design, the UK takes the lead in formulating regulations that require buildings with five or more stories to consider the effects of accidental loads, and dividing the regulations into three levels, which are the Building Regulations [3], the Approved Document [4], and the British Standard [5]. To enhance the progressive collapse resistance of building structures, the subsequent requirements propose the need to improve the capacity of key components to resist accidental loads and increase the connection strength of structural components. In the design of high-rise buildings in the United States, the current design codes, the GSA [6] and DoD [7], specify the alternate load path (ALP) method that takes the possibility of progressive collapse into account. However, the Chinese code [8] stipulates the requirements for the removal component method and the structural design faced to the progressive collapse resistance, and when the structural safety level is of level 1 or 2, the requirements for conceptual design should be satisfied.
In the field of theoretical research, Starossek [9] established the types of progressive collapses of structures and classified them into the mixed collapse, the unstable collapse, the pancake collapse, the zipper collapse, the domino collapse, and the cross-sectional collapse. Xiong et al. [10] studied the progressive collapse performance and mechanism of RC structures and derived the resistance calculation formulas for the beam mechanism, catenary mechanism, and composite mechanism by using analytical methods. Alshaikh et al. [11] summarized the previous research works on the progressive collapse of RC structures and concluded that the mechanism of structural progressive collapse resistance includes three stages, namely the bending action stage (or the Vierendeel stage), the compressive arch action stage, and the catenary action stage, respectively. In addition, among the numerous analysis methods, the ALP method is usually utilized in experiments or tests to analyze the progressive collapse resistance of RC structures [12].
In a numerical analysis and experimental studies, Zhou et al. [13,14] adopted the removal component method in the experiment to evaluate the progressive collapse resistance of a frame structure under the conditions of removing the columns in different parts and compared the experimental results with the numerically simulated ones obtained by using the finite element method (FEM). The results indicated that the progressive collapse risk of removing the middle column is lower than that of removing the corner column. Based on summarizing the previous research works on the progressive collapse resistance of the remaining structure with one column failure, Qian et al. [15] conducted an experimental and numerical simulation study on the progressive collapse resistance of an RC structure with both the corner column and the adjacent column failure. It was shown that the mechanical response of the RC beam–slab–column structure under this failure condition was like that of the cantilever slab, and the insufficient horizontal constraints between the beam and slab resulted in the inability to form an effective arch compression mechanism, catenary mechanism, or membrane mechanism to resist progressive collapse. Zhou et al. [16] conducted static load tests on half-scale RC and two half-scale PC substructures bearing the bending moment and simulated the mechanical behaviors of the remaining structure by removing the middle column using the FEM, and they evaluated the remaining structure’s properties for resisting the progressive collapse, such as the load transfer mechanism, deformation capacity, steel strain, crack distribution, and failure mode. Using an equivalent linear finite element updating technique based on sensitivity analysis, Moaveni et al. [17] carried out a study on the progressive damage identification for a masonry-infilled RC frame that was tested on a shake table and identified the mode parameters of the damage structure by applying the deterministic–stochastic subspace identification method. The results suggested that the proposed method could effectively identify the location and degree of the damage observed in the tests.
The dynamic response of building structures, especially assembly building structures, under the dynamic loads has been becoming an important topic in the field of building structure research. The effect of the dynamic load on the building structure is usually related to many uncertainties and random factors, such as the frequency, intensity, and duration of the dynamic load. Cao [18] operated a dynamic analysis for the five different precast concrete structures by using the PSD-based procedure and verified the effectiveness of such an iterative procedure. In recent years, with the development of reinforcement and strengthening technologies for existing aged building structures, the externally attached substructure technique has been receiving extensive attention. Cao et al. [19] developed a novel precast steel-reinforced concrete and ultra-high performance concrete composite braced-frame that was applied to externally attached seismic retrofitting for existing aged buildings, proposed a CSM-based displacement-oriented design strategy, and the structural capacity of the braced-frame and the effectiveness of the design strategy were verified via a case study. Following that, Cao et al. [20] developed an externally attached BFRP-bar reinforced precast concrete sub-frame for upgrading the seismic bearing capacity, and the effectiveness of such a sub-frame for seismic upgrading was verified through the application cases of three various concrete frame constructions.
Although there is considerable understanding of the collapse performance and mechanism of structures, most of these research works are focused on the field of frame structure. The large-span ATBCF is a new type of spatial floor structure, and the multi-column frame tube is its combining structure to resist lateral loads [21]. Although this structural system has the advantages of a large-span structure and assembled structure, such a structure needs to pay more attention to the overall resistance performance of the floor against progressive collapse. However, for the progressive collapse performance of the assembled multi-column frame tube structures or the assembled composite floors, there are few theoretical and experimental studies. In this paper, a numerical model of a multi-column frame tube structure with an ATBCF under different column removal conditions is conducted by adopting the ALP method, the progressive collapse performance of the remaining structure is simulated using MIDAS/Gen v8.75 software, and the mechanical mechanism is analyzed. The effect of the column failure time on the dynamic response of the remaining structure is studied, the vertical displacements of the remaining structures are individually solved using the linear static calculation method (LSCM), the nonlinear static calculation method (NSCM), and the nonlinear dynamic calculation method (NDCM), and the calculated results are compared and verified by detecting the dynamic amplification factor.

2. Structure Model and Mechanical Analysis

2.1. Assembled Structural System

The multi-column frame tube structure with an ATBCF is a new type of fully assembled structure. For such a structure, the outer multi-column frame tube is utilized as its basic lateral load resistance system, and the interior of each single tube structure is connected to form a spatial structural system by using a fully assembled open web truss beam composite floor. The floor is assembled from four modular board units, with a plan size of 15.6 m × 15.6 m. The fully dry assembly connection of the floor system is achieved using the specially designed assembly connection components between the plate units, and between the assembled floors and the frame columns [22]. The composition of a multi-column frame tube structure with an ATBCF is shown in Figure 1.
To investigate the progressive collapse resistance performance of this type of assembled structure, combined with the removal component method (also known as the ALP method), a progressive collapse resistance analysis on a multi-column frame tube structure with a ATBCF under different column failure conditions is conducted using the pushdown approach and the transient loading technique considering the initial state. Firstly, the residual bearing capacity, internal force redistribution mechanism, and progressive collapse mechanism of the multi-column frame tube structure with an ATBCF under different column removal conditions for this structural system are studied using the nonlinear static analysis method. Secondly, the dynamic responses of the remaining structure at various column failure times are studied using the nonlinear dynamic analysis method, and the influence of the column failure time on the dynamic response of the remaining structure is investigated. Finally, the load dynamic amplification factor, which should be considered in the static analysis of the remaining structure, is discussed.

2.2. Model and Loading Conditions

In the multi-column frame tube structure with an ATBCF, as shown in Figure 1, a typical multi-layer frame tube structure is selected as the research object. According to Chinese standards [23,24], the ultimate bearing capacity design for such a structure is designed. The structure consists of 6 floors, with a multi-column frame possessing 4 spans in both the transverse and longitudinal directions, a column spacing of 3.9 m, and a floor height of 3.6 m. The structural elevation and layout plan are individually shown in Figure 2 and Figure 3, and the isometric layout of the truss beams used for the modular plate components is shown in Figure 4. The cross-section forms and dimensions of the components of the structure, such as the truss beams, frame beams, and columns, are shown in Table 1. To investigate the progressive collapse resistance of the structure, the dead loads on the floor and roof are individually taken as 3.2 kN/m2 and 4.0 kN/m2, and the live loads are taken as 2.5 kN/m2 and 0.5 kN/m2, respectively. The dead load on the exterior wall is taken as 3.0 kN/m, and E355C steel is applied in this work. The main chemical composition of E335C steel is shown in Table 2. The mechanical property parameters of E335C steel at ambient temperature are shown in Table 3. C30 concrete is utilized to manufacture the concrete panels. The bulk density of C30 concrete is 2300 kg/m3, the tensile modulus of elasticity of C30 concrete is 30 GPa, and the compressive strength of C30 concrete is 34.5 MPa.
For the multi-column frame tube structure with an ATBCF, a finite element model is built using MIDAS/Gen software, as shown in Figure 5. In the mechanical analysis of such a structure, the tensile effect of the floor slab is ignored, and the components of the structure are simulated using the beam elements. When the pushdown analysis of the structure is conducted, the P–M–M plastic hinges are assigned to both ends of the frame column, the M3 hinges are selected to both ends of the frame beam, and the P hinges are adopted at both ends of the chord or web member of the truss beam. According to FEMA 356, the parameters of the plastic hinges are determined. The fixed connection between the column and the foundation of each column is modeled by assigning the displacement constrains in the lateral, longitudinal, and vertical directions for the lower end of the bottom column. In the numerical simulations of the pushdown analysis and nonlinear dynamic analysis of the remaining structures with the bottom side middle column, edge one, or corner column removal, the loads are acting on the corresponding nodes. When the nonlinear dynamic analysis of the structure is operated, the damping of the remaining structure is presumed as the Rayleigh damping, with a damping ratio of 0.02.
Due to the presence of an assembly slab seam in the assembled floors, as shown in Figure 3, the progressive collapse resistance performance of the structure may vary in different directions. Therefore, in this study, the side middle column of the bottom frame that is perpendicular to the assembly slab seam, the edge middle column of the bottom frame that is parallel to the assembly slab seam, and the corner column at the bottom frame, as shown in Figure 3, are individually selected as the removal objects for the progressive collapse resistance analysis of the structure. The three types of column removal conditions are sequentially named work condition 1, work condition 2, and work condition 3, respectively.

2.3. Analysis Methodology

2.3.1. Pushdown Analysis of the Remaining Structure

The pushdown analysis, which belongs to the nonlinear static analysis, is one of the commonly adopted methods in the progressive collapse analysis of structures [25]. With this method, a gradually increasing vertical load is acting on the remaining structure, until the structure collapse is generated, the relationship curve between the vertical load and the displacement of the structural controlling point is obtained, and the residual bearing capacity of the remaining structure is quantitatively evaluated. During the loading process, the internal force change and plastic development of the remaining structure at different stages can be detected, and the internal force redistribution mechanism and failure modes of the second defense line during the process of structural progressive collapse can be analyzed as well.
Under the three types of work conditions, the pushdown analysis works are individually operated for the finite element model of the remaining structure using MIDAS/Gen software. To obtain the ultimate state of the progressive collapse of the remaining structure, the applied vertical load should be large enough. By using the trail calculation method, the target value of the vertical load can be achieved. In the nonlinear analysis, the stiffness of the remaining structure gradually decreases with the increasing vertical load, which will result in the deterioration of the remaining structure. When the stiffness coefficient of the remaining structure decreases to a critical value, the remaining structure will no longer bear the external load, and ultimately, will be followed by the collapse of the remaining structure. Furthermore, according to the Chinese Code CSCE 392 [26], when the nonlinear static analysis method and the nonlinear dynamic one are individually adopted to investigate the static behaviors and the dynamic responses of building structures, for the case of the plastic rotation angle θp,e of the horizontal components of the remaining structure beyond the allowable one [θp,e], it should be considered that the building structure does not satisfy the design requirements for the progressive collapse resistance. For steel beams without adopting any weakening or strengthening measure to the flange, the allowable plastic rotation angle [θp,e] is taken as 0.0213–0.00012H/h. Here, H and h individually denote the cross-sectional height of the steel beam and a unit displacement with a dimension centimeter. Therefore, in this work, the termination conditions of the pushdown analysis of the multi-column frame tube structure with an ATBCF under different column removal conditions are as follows: (1) under a certain incremental step, the ratio of the structural current stiffness to initial one is of zero; and (2) the plastic rotation angle of a certain horizontal component reaches the allowable plastic rotation angle [θp,e].

2.3.2. Nonlinear Dynamic Analysis of the Remaining Structure

In this paper, the transient loading method with an equivalent load considering the initial state is adopted. After obtaining the internal force of the component to be removed under the condition of a specific load combination, the component is removed and the internal force at the failed end is converted into the equivalent node load P0, which is applied to the corresponding node, making the remaining structure statically equivalent to the original structure, that is, the initial deformation and internal force of the structure are considered. To obtain the dynamic response of the remaining structure with column failure, based on this situation, a dynamic load Pt is applied to the corresponding nodes to conduct the dynamic time history analysis. The loading curve of the remaining structure is shown in Figure 6. In the loading curve, when the time is t0, the load P(t0) is P0, with the direction opposite to the node force, and t0 represents the failure time of the component. In this work, according to the requirements in the Chinese Code CSCE 392, a vertical load combination with 1.0 times the dead load and 0.5 times the live load acting on the multi-column frame tube structure with an ATBCF under the condition of column removal are adopted for the nonlinear dynamic analysis.
Although the dynamic effects of the remaining structure can be accurately reflected using the dynamic analysis method, a lot of computational time is required for both the theoretical and numerical calculations. Therefore, in the process of the actual structure design, the progressive collapse resistance performance of the remaining structure is preliminarily studied using static analysis methods. In the static analysis, to take the dynamic effect caused by the transient failure of the structural component bearing loads into account, a dynamic amplification coefficient αd is introduced according to the Chinese Code CSCE 392. When the progressive collapse resistance of the remaining structure is calculated using the linear static analysis method, the dynamic amplification coefficient is determined as 2.0, and when that is solved using the nonlinear static analysis method, the one is taken as 1.35 for the steel structure. The applicable range of the load amplification is determined as the span between the columns connected to the removed column, and the layer is located above that of the removed column. In addition, the determination of the value of the dynamic amplification coefficient in the Chinese Code CSCE 392 is more focused on traditional frame structures, while for the multi-column frame tube structure with an ATBCF, further study is needed.

3. Results

3.1. Pushdown Processing Analysis

3.1.1. Pushdown Processing of the Remaining Structure

Under the three types of different removal column work conditions, the curves of the vertical load coefficient αd versus the vertical displacement δ of the removal column’s upper node (hereinafter referred to as the “failure point”) are shown in Figure 7. The endpoints of the curves corresponding to work condition 1 and work condition 2 are determined by the collapse criterion, while for the curve corresponding to work condition 3, at the endpoint of the curve, the ratio of the current structural stiffness to the initial one is equal to zero, which can be seen in Figure 7. As the load gradually increases, the working state of the structure changes from elastic deformation to elastoplastic one, which reflects the curve changing from linear to nonlinear. Such a change is marked by the first occurrence of a break point in the curve. The load coefficient at the first break point in the curve of the load coefficient versus the displacement is defined as αd,y. The maximum load coefficient, which is called as the ultimate one, is named as αd,max, and the corresponding displacement, which is called the ultimate displacement, is noted as δmax. The results are listed in Table 4.
Comparing the slopes of the linear segments of the curves shown in Figure 7, it can be seen that the stiffness of the remaining structure under work condition 3 is larger than that under work conditions 1 and 2. For work condition 3, when the load curve enters the nonlinear deformation stage, the bearing capacity of the remaining structure quickly reaches its ultimate value and the remaining structure reflects a similar “brittle” property, with the value of αd,max only increasing by 0.71% compared to the value of αd,y. For work conditions 1 and 2, in the nonlinear deformation stage, the remaining structure undergoes a long period of deformation development, followed by a certain degree of recovery in stiffness, and the load coefficient increases with the increasing displacement and the remaining structure exhibits a similar “plastic” property. Compared to the value of αd,y, the values of αd,max individually increase by 23.81% and 23.85%, and the ultimate displacements are also greater than that under work condition 3. Simultaneously, by comparing the curves under work conditions 1 and 2, it can be found that the shapes of the two curves are similar, and the ultimate load coefficient of work condition 2 is slightly larger than that of work condition 1.
It can be seen from the above analysis that under the condition of the corner column removal at the bottom floor, the stiffness of the remaining structure in the elastic stage is greater than that of the one under the condition of the middle column removal at the bottom floor, which indicates that the residual bearing capacity of the remaining structure with the corner column removal is higher than that of the one with the middle column removal. However, the plastic deformation capacity of the remaining structure with the middle column removal is superior to that of the one with the corner column removal, which can be seen in Figure 7.

3.1.2. Progressive Collapse Mechanism of the Remaining Structure

With the removal of the middle column or the corner column of the multi-column frame tube structure with an ATBCF, the vertical load is transmitted to the remaining columns via the frame beams and the truss beams. Due to the higher stiffness of the truss beam, the load transmitted by the truss beam is greater than that of the frame beam. Under work condition 1 with side middle column removal and work condition 2 with edge middle column removal, the numerical simulations are individually conducted for the finite element models corresponding to the remaining structures using MIDAS/Gen software. When the iteration steps are 33 and 34, the deformation diagrams of the remaining structures are as shown in Figure 8a,b.
Combined with the load coefficient and displacement curve under work conditions 1 or 2 (as shown in Figure 7), it can be found that during the linear loading stage, the overall remaining structure is in an elastic serving state. When the load closes on the first break point, the remaining structure has almost no plastic hinges or only a small number of plastic hinges. At the first break point, a lot of plastic hinges suddenly appear in the remaining structure, and the plastic hinges firstly appear on the web members of the truss beams near the removal column end at the bottom floor and rapidly develop toward the upper floors, as shown in Figure 8a,b. The web members of the truss beam yield and quickly fail, resulting in a rapid decrease in structural stiffness and an increase in structural deformation. The internal force redistribution occurs in the remaining structure, the bending moment at the end of the frame beam at the failure point rapidly increases, and the frame beam is in an elastoplastic working state. Due to the tie effect of the upper and lower chords of the truss beam and the bending resistance performance of the frame beam, the remaining structure can still bear greater loads. Through the internal force redistribution, the structural stiffness has been restored to a certain extent, corresponding to the second break point and following with the second upward stage in the load coefficient versus the displacement curve, as shown in Figure 7. The plastic rotation angle at the end of the frame beam increases with the increasing external load until it approaches the ultimate value [θp,e] and the remaining structure no longer satisfies with the requirements for resisting the progressive collapse. Currently, the loading processes are terminated, the corresponding iteration steps for work conditions 1 and 2 are 149 and 80, and the ultimate plastic hinge distributions and deformation diagrams of the remaining structures are shown in Figure 8c and Figure 8d, respectively. Moreover, the yield performance of the nodes near the end of the removal column can be explained using the modified MCC model [27], which reasonably describes the bi-modulus extension and the hardening property during the forming process of plastic hinges in the steel truss beams, and the P-wave and S-wave generated at transient time for column removal cause the dynamic damage to the components of the remaining structure and accelerate the plastic deformation of the steel truss beams, as shown in Figure 7.
Under work condition 1 with side middle column removal, the curves of the bending moment My and plastic rotation angle θp,e of the frame beam end at the failure point versus the load coefficient αd are as shown in Figure 9. It can be seen from Figure 9 that the changes in the bending moment and plastic rotation angle at the beam end are synchronous. When the bending moment My reaches its yield value Mp = 90.00 kN·m, the plastic rotation angle at the beam end begins to emerge. When the bending moment My increases to 101.56 kN·m, the plastic rotation angle θp,e at the beam end is of 0.1897 rad, which is larger than the allowable one [θp,e] = 0.0189 rad. It can be seen from the above analysis that under the condition of the side middle column failure, the structure possesses a good second defense line to resist progressive collapse. Therefore, the frame beam can effectively improve the safety storage of the remaining structure against the progressive collapse.
Under work condition 3 with corner column removal, the numerical calculation is operated on the finite element model of the remaining structure using MIDAS/Gen software. When the iteration step is 149, the deformation diagram of the structure is as shown in Figure 10a. It can be seen from Figure 10a that the progressive collapse process of the multi-column frame tube structure with an ATBCF under work condition 3 is different from that under work condition 1 or 2. When the corner column of the structure fails, as the vertical load gradually increases, the plastic hinges firstly emerge on the web members of the truss beams near the column end at the bottom floor, as shown in Figure 10a. Distinguishing work conditions 1 and 2, the upper and lower chords of the structural truss beam at the corner cannot form effective ties like that at the mid span. With the failure of the web members, the stiffness of the truss beam rapidly decreases, and the proportion of the load endured by the frame beam greatly increases. The load redistribution between the truss beam and the frame beam causes the rapid yielding and failure of the frame beam, resulting in the severe degradation of the stiffness of the overall remaining structure, which corresponds to the break point of the load coefficient versus the displacement curve under work condition 3 in Figure 7 and ultimately results in progressive collapse in the region at the corner columns. At this moment, the corresponding iteration step for work condition 3 is 171 and the ultimate plastic hinge distribution and deformation diagram of the remaining structure is as shown in Figure 10b. Thus, it can be found from the above analysis that in the actual design of the multi-column frame tube structure with an ATBCF, the frame beam, as the second defense line against progressive collapse, should possess sufficient safety storage to ensure that the structure still possesses the capacity of progressive collapse resistance under the condition of corner column removal.

3.2. Nonlinear Dynamic Analysis

3.2.1. Failure Time of the Column

When an engineered structure is subjected to an unexpected load, such as an explosion load or an impact load, resulting in the failure of its components, the failure time of the structural components is generally very short [28]. It is shown that the failure time of the load-bearing component at the bottom floor has a significant impact on the dynamic response of the remaining structure. The shorter the failure time is, the stronger the dynamic response [29]. For the value of failure time t0, the DoD 2013 specification [30] stipulates that the failure time of the column shall not exceed 0.1 times the period of the first-order vertical vibration mode Tv1 of the remaining structure, while the Chinese Code CSCE 392 stipulates that the failure time of the removed column shall not be greater than 0.1 times the primary vibration period T1 of the remaining structure. To obtain a reasonable value for the failure time t0, and to investigate the dynamic response of the remaining structure under the action of a slope load, in the following study, the multi-column frame tube structure with an ATBCF under work condition 1 is selected as a research object. Based on the eigenvalue analysis (or mode analysis) of the frame structure with the side middle column removal, the period of the primary vibration mode T1 and the period of the first-order vertical vibration mode Tv1 of the remaining structure are 1.327 s and 0.2717 s, respectively. Thus, the failure time t0 is individually taken as 0.001 s, 0.02717 s (0.1Tv1), 0.1 s, 0.1327 s (0.1T1), and 0.5 s, and the vertical displacement versus time curves of the remaining structure at various failure times are as shown in Figure 11. Extracting the maximum displacement of each vertical displacement time history curve, the relationship curve between the maximum displacement and the failure time can be obtained, as shown in Figure 12.
It can be seen from Figure 11 and Figure 12 that the shorter the failure time t0 of the column, the greater the dynamic response of the remaining structure. When the failure times are 0.001 s and 0.02717 s (0.1Tv1), the vertical displacement time history curves of the remaining structures are very similar, with the corresponding maximum vertical displacement being 14.53 mm and 14.22 mm, respectively. The difference between the both values is only 0.31 mm, with a decrease of only 2.2%. When the failure times are 0.1 s and 0.1327 s (0.1T1), the maximum vertical displacements of the remaining structure are individually 11.72 mm and 11.32 mm, which are 19.3% and 22.1% lower than the result at the failure time of 0.001 s. When the failure time is 0.5 s, the maximum vertical displacement of the remaining structure is 10.52 mm, which is a decrease of 27.6% compared to the value at the failure time of 0.001 s. For this case, the dynamic effect of the remaining structure has significantly decreased. Therefore, it is reasonable to take a failure time for the structural components not greater than 0.1 times the period of the first-order vertical vibration mode (0.1Tv1) of the remaining structure. In this situation, the calculation results can fully reflect the maximum dynamic response of the remaining structure.

3.2.2. Dynamical Magnification Factor

For work condition 1 with the side middle column failure, work condition 2 with the edge middle column failure, and work condition 3 with the corner column failure, the failure times of the columns are taken as 0.1 times the periods of the first-order vertical vibration modes of the corresponding remaining structures. A nonlinear dynamic analysis is conducted on the remaining structures to obtain the displacement time history curves at the structural components’ failure points under the three work conditions, as shown in Figure 13. It can be seen from Figure 13 that under the three types of different working conditions, the maximum displacements at the failure points of the remaining structures are 14.22 mm, 13.27 mm, and 8.20 mm, respectively. Then, according to the Chinese Code CSCE 392, the displacements at the failure points of the remaining structures are individually solved using the linear static calculation method (LSCM) and the nonlinear static calculation method (NSCM). The vertical displacements at the failure points of the multi-column frame tube structure with an ATBCF under the three working conditions obtained using the linear static calculation method, the nonlinear static calculation method, and the nonlinear dynamic calculation method (NDCM) are listed in Table 5.
It can be seen from Table 5 that under the three working conditions, the vertical displacements obtained using the LSCM are greater than those obtained using the NDCM, and the calculation results obtained using the LSCM and NDCM are relatively close. Compared with the vertical displacement obtained using the NDCM, the vertical displacements obtained using the LSCM are individually only larger by 2.5%, 2.9%, and 7.2%, which indicates that when adopting the LSCM, if the dynamic amplification coefficient is selected as 2.0, then the calculation results can accurately reflect the dynamic effects caused by the transient failure of the structural load-bearing components.
In addition, when an arbitrary load is acting on the remaining structure, an analytical solution for the dynamic displacement of the remaining structure can be obtained using the Duhamel integral equation, that is:
δ ( t ) = 1 m ω ∫ 0 t P ( τ ) sin [ ω ( t − τ ) ] d τ
where δ is the dynamic displacement, m is the mass of the remaining structure, ω is the natural circular frequency of the remaining structure, P is the external load, t is the time, and τ is the moment of load action.
When the load acting on the remaining structure is a slope load, as shown in Figure 6, the function of the slope load can be represented as:
P ( t ) = { P 0 ( t / t 0 ) ,   ( 0 ≤ t ≤ t 0 ) P 0 ,                       ( t ≥ t 0 )
where P0 is the maximum load, t0 is the rise time of the slope load reaching the maximum value, and t is the time. Submitting Equation (2) into Equation (1), the dynamical displacement of the remaining structure is calculated as:
δ ( t ) = { δ st 1 t 0 [ t − sin ( ω t ) ω ] ,                                                                                     ( 0 ≤ t ≤ t 0 ) δ st { 1 − 1 ω t 0 [ sin ( ω t ) − sin ( ω ( t − t 0 ) ) ] } ,         ( t ≥ t 0 )
where δst = P0/(mω2), which denotes the static displacement of the remaining structure under the action of the static load P0.
The dynamic amplification factor β is defined as the ratio of the maximum dynamic displacement to the static displacement, that is:
β = [ δ ( t ) ] max / δ st
It can be seen from Equation (3) that the dynamic amplification factor is between 1 and 2. When the rise time of the slope load is very long, the dynamic amplification factor approaches 1, while when the rise time of the slope load is very short, the dynamic amplification factor is close to 2. Especially, when the rise time is equal to zero, the slope load is converted into a suddenly applied load. By using the limitation solution of Equation (3) for letting the rise time t0 tending to zero, or by directly using the Duhamel integral with respect to the suddenly applied load, the dynamic displacement of the remaining structure can be derived as:
δ ( t ) = P 0 m ω 2 [ 1 − cos ( ω t ) ] = δ st [ 1 − cos ( ω t ) ]
According to the Equation (4) and Equation (5), when the load acting on the remaining structure is a suddenly applied load, the dynamic amplification factor is calculated as:
β = [ δ ( t ) ] max / δ st = 2
In fact, for the multi-column frame tube structure with an ATBCF, it is necessary to take time to dismantle the column from the main structure, so when an equivalent load is applied to the node corresponding to the end point of the removal column, the load acting on the node of the remaining structure is undoubtedly a slope load, as shown in Figure 6, and the rise time of the slope load is related to the failure time of the column. Therefore, no matter how short the failure time of the removal column is, the dynamic amplification factor of the remaining structure must be less than 2. In the above calculation and analysis, when the LSCM is adopted to calculate the displacement of the remaining structure, the dynamic amplification factor is set as 2, which inevitably leads to the vertical displacement calculated using the LSCM being slightly larger than that calculated using the NDCM. Thus, the effectiveness of the calculated vertical displacements of the remaining structure using the LSCM and NDCM is verified and validated.
However, under the three working conditions, the vertical displacements obtained using the NSCM are smaller than those obtained using the NDCM. Compared with the vertical displacements obtained using the NDCM, the vertical displacements obtained using the NSCM are individually smaller by 17.4%, 18.8%, and 21.3%, which indicates that when utilizing the NSCM, if the dynamic amplification coefficient is taken as 1.35, then the calculation results of the vertical displacements at the failure points of the remaining structure may be smaller than the actual ones.
The vertical displacements of the failure points of the remaining structures with the side middle column removal, the edge middle column removal, and the corner column removal are recalculated using the NSCM. Currently, the dynamic amplification coefficients are determined as 1.92, 1.92, and 1.84, and the corresponding vertical displacements at the failure points of the remaining structures are calculated as 14.23 mm, 13.31 mm, and 8.22 mm, respectively. At this point, the calculation results obtained using the NSCM are very close to those obtained using the NDCM, with only the differences of 0.01 mm, 0.04 mm, and 0.01 mm between the calculated results. Therefore, when the NSCM is adopted to solve the displacement at the failure point of the remaining structure of the multi-column frame tube structure with an ATBCF, a range of dynamic amplification factors from 1.80 to 1.95 is recommended.

4. Discussion

The plastic deformation capacity of the remaining structure with the side column removal or with the edge middle column removal is significantly higher than that of the remaining structure with the corner column removal. Its mechanical mechanism can be explained by the fact that the upper and lower chords of the mid-span truss beam can generate effective ties, while the upper and lower chords of the truss beam at the end almost cannot militate the effective ties. However, for the remaining structure dismantling the side column that is perpendicular to the assembly slab seam, its plastic deformation capacity approximately equals that of the remaining structure dismantling the edge column that is parallel to the assembly slab seam, while the bearing capacity of the second defense line is slightly smaller than the latter’s bearing capacity of the second defense line, which suggests that when such a multi-column frame tube structure with an ATBCF is designed, more sufficient attention should be paid to the treatment of the assembly slab seam. The reliable connections between the concrete panel and steel truss keel of the composite floor, and those between the assembly floor units, can help improve the residual bearing capacity of the remaining structure. Of course, the mechanical mechanism formed by the difference in the load-bearing capacity of the second defense line of the remaining structure under the two working conditions of removing the side column and removing the edge one is a problem worthy of in-depth research.
In this work, the progressive collapse performances of the remaining structures with column removal are simulated using the finite element method (FEM). Besides the FEM, there are other approaches for the structural analysis of damage caused to the structural components such as rods, beams, plates, and shells from macro to nano, including meshfree methods, characteristics orthogonal polynomials (COPs), and assumed mode methods. In terms of the meshfree methods for rods or wires at a nano scale, the motion equations for the magnetically affected–damaged rod-like nanostructures were derived, and the extracted nonlocal-integral governing equations were solved for natural frequencies using the FE-meshfree method [31]. Then, for the bimaterially defective nanowires, the nonlocal-surface energy-based motion equations were derived using the DNM and INM, and the results predicted using the meshfree method based on the DNM were verified by comparing them with those obtained using the Galerkin-based admissible mode method [32]. In terms of the COPs method for the plates, the dynamic analysis of cracked thin rectangular plates subjected to a moving mass was investigated, the governing equation was solved using the eigenfunction expansion method, and the results indicated that the moving mass had a greater impact than the moving load on the dynamic responses of cracked plates [33]. Following that, the dynamical analysis of such a class of plates bearing a moving non-stationary random load was conducted, and an inclusive parametric study was performed to investigate the influences of the inclined crack angles and the crack lengths on the non-dimensional functions of the squared mean values at the middle point of the undamped and damped cracked plates [34]. During the process of the column removal, the internal force redistribution of the upper and lower chords and web members in the truss beam occurs, causing the plastic deformation of the nodes near the removal column’s end and forming plastic hinges, which is similar to the failure mode of the rod with fixed ends and ultimately results in the progressive collapse of the multi-column frame tube structure with an ATBCF. Therefore, how to identify the failure mode of the components and the meso- and micro-deformation mechanism of the associated truss members is a worthwhile research topic.
In addition, the calculation and analysis of the progressive collapse resistance of the multi-column frame tube structure with an ATBCF under the conditions of different column removals indicate that when applying the LSCM for calculating the vertical displacements of the remaining structures, it is reasonable to take 2.0 as the recommended dynamic amplification factor according to the Chinese Code CSCE 392, while when adopting the NSCM for calculation, if the dynamic amplification coefficient is taken as 1.35, the calculation results may be smaller than the actual values. So, when the nonlinear static calculation method is adopted to solve the failure point displacement of the remaining structure of the multi-column frame tube structure with an ATBCF or other similar assembly frame structures, how to determine an appropriate dynamic amplification factor is also a question worth further research.

5. Conclusions

To assess the progressive collapse resistance capacity of a multi-column frame tube structure with an ATBCF under different column removal conditions, the progressive collapse resistance performance and mechanism of the remaining structure are studied using the ALP method and the effect of the column failure time on the dynamic response of the remaining structure is investigated. The conclusions of this study are as follows:
(1) The stiffness and residual bearing capacity of the remaining structure with the corner column removal are larger than those with the middle column removal, while the ultimate displacement of the former is smaller than that of the latter; therefore, the remaining structure with the middle column removal possesses a stronger plastic deformation capacity.
(2) For the remaining structure with the side or edge middle column failure, the frame beam elevates the safety storage, which results in the remaining structure having an excellent second defense line against progressive collapse. However, for the remaining structure with the corner column failure, due to the lack of reliable ties of the upper and lower chords of the truss beam at the corner, the safety storge of the frame beam above the failure point needs to be enhanced to ensure the remaining structure’s capacity to resist progressive collapse under the condition of corner column failure.
(3) The dynamic analysis of the structure with an ATBCF indicates that the shorter the column failure time, the greater the dynamic response of the remaining structure is. When the ALP method is adopted to perform the progressive collapse analysis of the remaining structure, it is reasonable to take the column failure time as 0.1 times the period of the first-order vertical vibration mode of the remaining structure, and it is suitable to set the dynamic amplification coefficient as 2.0.
The numerical simulations of the progressive collapse resistance for a multi-column frame tube structure with an ATBCF are carried out using the finite element software MIDAS/Gen, the residual bearing capacity and the plastic deformation capacity of the remaining structures are analyzed and compared, and the vertical displacements of the remaining structures under three work conditions are calculated using the LSCM, NSCM and NDCM. The dynamic deformation of the remaining structure is solved using the Duhamel integral equation with the slope load, the dynamic amplification factor is discussed, and the numerical calculation results obtained using the LSCM and NDCM are verified and validated. The test verification of the pushdown analysis of the remaining structure will be the work to be conducted in the future. Furthermore, all the obtained results in this work have been extracted for a six-story multi-column frame tube structure with an ATBCF with four spans in each direction, and as for the influence of structural parameters such as the span and story number on the progressive collapse resistance of the remaining structures, it will be a topic that will be further studied in the future.

Author Contributions

Conceptualization, R.Z. and Z.Z.; methodology, R.Z. and G.C.; software, Z.Z. and G.C.; validation, R.Z., Z.Z. and G.C.; formal analysis, Z.Z.; investigation, R.Z. and Z.Z.; resources, W.L.; data curation, Z.Z. and G.C.; writing—original draft preparation, R.Z., G.C. and Z.Z.; writing—review and editing, R.Z. and G.C.; visualization, W.L.; supervision, R.Z.; project administration, Z.Z. and G.C.; funding acquisition, R.Z. and G.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the Scientific Research Project of Education Department of Hunan Province (19A095) and the Hunan Natural Science Foundation (2018JJ2020, 2020JJ5018).

Data Availability Statement

All data and models generated or used during this study appear in the submitted article.

Acknowledgments

This research work was supported by the Key Laboratory of Dynamics and Reliability of Engineering Structures of the College of Hunan Province, and by the Hunan Engineering Research Center of Development and Application of Ceramsite Concrete Technology.

Conflicts of Interest

The authors have no relevant financial or non-financial interests to disclose. The authors have no competing interests to declare that are relevant to the content of this article.

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Figure 1. Composition of a multi-column frame tube structure with an ATBCF.
Figure 1. Composition of a multi-column frame tube structure with an ATBCF.
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Figure 2. Structural elevation of the multi-column frame tube structure with an ATBCF.
Figure 2. Structural elevation of the multi-column frame tube structure with an ATBCF.
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Figure 3. Layout plan of the multi-column frame tube structure with an ATBCF.
Figure 3. Layout plan of the multi-column frame tube structure with an ATBCF.
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Figure 4. Isometric layout of the truss beams used for the modular plate components.
Figure 4. Isometric layout of the truss beams used for the modular plate components.
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Figure 5. Finite element model of the multi-column frame tube structure with an ATBCF.
Figure 5. Finite element model of the multi-column frame tube structure with an ATBCF.
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Figure 6. The loading curve of the remaining structure.
Figure 6. The loading curve of the remaining structure.
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Figure 7. The curves of the loading coefficient versus the displacement under the various work conditions.
Figure 7. The curves of the loading coefficient versus the displacement under the various work conditions.
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Figure 8. Plastic hinge distribution of the truss beams under work conditions 1 and 2: (a) step 33 for work condition 1; (b) step 34 for work condition 2; (c) ultimate step 149 for work condition 1; and (d) ultimate step 80 for work condition 2.
Figure 8. Plastic hinge distribution of the truss beams under work conditions 1 and 2: (a) step 33 for work condition 1; (b) step 34 for work condition 2; (c) ultimate step 149 for work condition 1; and (d) ultimate step 80 for work condition 2.
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Figure 9. Curves of the bending moment and plastic rotation versus the loading coefficient.
Figure 9. Curves of the bending moment and plastic rotation versus the loading coefficient.
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Figure 10. Plastic hinge distribution of the truss beams under work condition 3: (a) step 149 for work condition 3; and (b) ultimate step 171 for work condition 3.
Figure 10. Plastic hinge distribution of the truss beams under work condition 3: (a) step 149 for work condition 3; and (b) ultimate step 171 for work condition 3.
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Figure 11. Vertical displacement time history curves at various failure times.
Figure 11. Vertical displacement time history curves at various failure times.
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Figure 12. Relationship curve between the maximum displacement and failure time.
Figure 12. Relationship curve between the maximum displacement and failure time.
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Figure 13. Vertical displacement time history curve at the failure points of the remaining structure under the various work conditions: (a) work condition 1; (b) work condition 2; and (c) work condition 3.
Figure 13. Vertical displacement time history curve at the failure points of the remaining structure under the various work conditions: (a) work condition 1; (b) work condition 2; and (c) work condition 3.
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Table 1. The physical parameters of the rock specimens.
Table 1. The physical parameters of the rock specimens.
CompositionComponentSpecification of Cross-Section
FloorChords of the main beamsCold-formed channel steel 200 × 100 × 8
The web members of the main beamsHot-rolled steel equal angle 50 × 4
Chords of edge beams and end beamsCold-formed channel steel 100 × 100 × 8
Web members of edge beams and end beamsCold-formed channel steel 80 × 60 × 4
Vertical ventral membersCold-formed channel steel 100 × 100 × 8
Single cylinder frameMiddle columnH400 × 400 × 10 × 20
Side columnH300 × 300 × 8 × 14
Corner columnH200 × 200 × 8 × 8
Frame beamH200 × 150 × 6 × 8
Table 2. The main chemical composition of E335C steel (wt.%).
Table 2. The main chemical composition of E335C steel (wt.%).
ElementFeCMnSiPSNbVTiCrNiMo
Content/%96.6980.2001.7000.5000.0350.0350.0700.1500.2000.3000.0120.100
Table 3. The mechanical property parameters of E335C steel at ambient temperature.
Table 3. The mechanical property parameters of E335C steel at ambient temperature.
ParameterYoung’s Modulus
E (GPa)		Yield Strength
σy (MPa)		Tensile Strength
σb (MPa)		Impact Toughness
αk (J/cm2)		Elongation
δ (%)		Hardness
H (HB)
Value2103455106720140
Table 4. Load coefficient and vertical displacement under the various work conditions.
Table 4. Load coefficient and vertical displacement under the various work conditions.
Work Conditionαd,yαd,maxδmax (mm)
13.2474.02055.821
23.3384.13455.324
38.4578.51737.425
Table 5. Vertical displacements gained by using the three calculation methods under the various work conditions.
Table 5. Vertical displacements gained by using the three calculation methods under the various work conditions.
Work
Condition		Displacement by Using LSCM
δs (mm)		Displacement by Using NSCM
δns (mm)		Displacement by Using NDCM
δnd (mm)
114.5811.7414.22
213.6610.7713.27
38.796.458.20
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MDPI and ACS Style

Zhao, R.; Chen, G.; Zhang, Z.; Luo, W. Progressive Collapse Resistance Assessment of a Multi-Column Frame Tube Structure with an Assembled Truss Beam Composite Floor under Different Column Removal Conditions. Buildings 2024, 14, 111. https://doi.org/10.3390/buildings14010111

AMA Style

Zhao R, Chen G, Zhang Z, Luo W. Progressive Collapse Resistance Assessment of a Multi-Column Frame Tube Structure with an Assembled Truss Beam Composite Floor under Different Column Removal Conditions. Buildings. 2024; 14(1):111. https://doi.org/10.3390/buildings14010111

Chicago/Turabian Style

Zhao, Rongguo, Guangfei Chen, Zaihua Zhang, and Wei Luo. 2024. "Progressive Collapse Resistance Assessment of a Multi-Column Frame Tube Structure with an Assembled Truss Beam Composite Floor under Different Column Removal Conditions" Buildings 14, no. 1: 111. https://doi.org/10.3390/buildings14010111

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