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Article

Effects of Floor System on Progressive Collapse Behavior of RC Frame Sub-Assemblages

by
Said Elkholy
1,2,
Ahmad Shehada
1 and
Bilal El-Ariss
1,*
1
Civil and Environmental Engineering Department, United Arab Emirates University, Al Ain 15551, United Arab Emirates
2
Department of Civil Engineering, Fayoum University, Fayoum 63514, Egypt
*
Author to whom correspondence should be addressed.
Buildings 2022, 12(6), 737; https://doi.org/10.3390/buildings12060737
Submission received: 27 April 2022 / Revised: 21 May 2022 / Accepted: 26 May 2022 / Published: 30 May 2022
(This article belongs to the Section Building Structures)
Browse Figures
Versions Notes

Abstract

:
The ability to predict the resistance of reinforced concrete (RC) structures to progressive collapse as a result of an interior column removal has become a need in structural design. In general, three resistance mechanisms characterize the structure resistance to progressive collapse, flexural action, compressive arch action, and tension catenary action. The objective of this study is to investigate the effects of floor system configurations on the progressive collapse-resistance of RC frame sub-assemblages and the amount of energy dissipated in each resistance mechanism. This investigation employs a fiber element-based modeling technique to present findings into the effects of beam size and reinforcement details on the progressive collapse-resistance and energy dissipation of RC beam-column sub-assemblages with four equal spans. Three different span lengths of 5, 6, and 7 m were considered. A total of 38 floor system designs for gravity loads were performed in accordance with the ACI 318-14 design code. The modeling technique employed in this study was validated and utilized by the authors in previously published works. The study shows that beam size and the presence of slab are critical as they significantly affect the energy dissipation and progressive collapse-resistance and failure pattern of the sub-assemblage frames. Moreover, the presence of a slab was found to increase the energy dissipation by around 28%.

1. Introduction

Progressive collapse is defined as the loss of a relatively small load-carrying member, such as a column, as a result of intentional damage made or due to a natural or accidental load, which leads to the disproportionate failure of the building. Currently, design regulations and standards such as US Design Guidelines (DoD 2016 [1] and GSA 2016 [2]) recommend that building structures use different load paths to avoid non-proportional failure and propose few mitigation procedures as well. Guidelines and regulations were reviewed, detailing and damages due to various design loads were examined, and a few recommendations for buildings were offered to ensure that buildings perform robustly when challenged by unexpected loading [3,4,5,6].
To investigate the behavior and resistance of a building to progressive collapse, a load-carrying vertical member should be removed to simulate the building progressive collapse. Previous experimental studies [7,8,9] have mostly concentrated on beam–column sub-assemblages and have largely ignored the effect of slabs on the frame progressive collapse resisting performance. RC slabs, on the other hand, play a vital role in dispersing loads and spanning initial local failures. This is due to the fact that slabs are cast monolithically with beams and serve as horizontal members for transferring the imbalanced gravity loads caused by the column first local failure. Under a corner-column failure scenario, Qian and Li [10] performed static tests on the progressive collapse of beam-slab sub-assemblages and beams. By comparison, the slab contribution to progressive collapse-resistance was evaluated. The slab contribution, according to their test results, can greatly lower the risk of collapse. Similarly, Qian and Li [11] conducted dynamic collapse experiments on beam–slab sub-assemblages in the corner sections of RC frames, comparing the results of their static testing [10] to the influence of slabs during the dynamic testing.
Other investigations [12,13,14,15,16] have primarily focused on secondary capacity mechanisms established in beams, such as compression arch action (CAA) and tensile catenary action (TCA). Test data [17,18,19] showed that at the small deformations, CAA improved the peak value of the vertical load on beams designed according to ACI 318–05 [20], while a TCA was activated at the large deformations by the fracture of bottom reinforcement bars close to the middle joint [17,19] or the crushing of concrete [18]. However, according to a static test conducted by Qian et al. [21] on progressive collapse on specimens with a removed penultimate column, CAA and TCA did not fully develop due to the horizontal displacement of the specimen edge column. The catenary arch action and the tension catenary action may be numerically approximated using fiber plastic hinges at critical sections [22], fiber-based beam members [15], or an analytical model [21,23,24].
Quasi-static testing [8,14,25,26,27] and pushdown analyses [13,26,28,29] were used in several studies to examine the resistance of RC substructure to progressive collapse under several column-elimination scenarios. Unlike substructures that rely merely on the CAA and TCA [30,31,32], the existence of slabs not only upgraded the substructure resistance to progressive collapse through compression and tension membrane action, CMA, and TMA [10,33,34,35,36,37] but also made the structural progressive collapse-resistance reliant on the loading scheme, i.e., uniformly distributed (CL) or concentrated load (CL). Although CL can be used to highlight the key load-transfer mechanisms, the results revealed that UDL is a better representative loading scheme for investigating the progressive collapse-resistance of beam-slab sub-assemblages [38,39]. El-Ariss and Elkholy [40] and Elkholy and El-Ariss [41,42,43,44] presented a modeling procedure and strengthening arrangement for RC continuous members to withstand progressive failure as a result of a missing internal column. The method features external cables attached to the beam at constraint and holding locations without being stressed. The presented method improved the strength and energy dissipation of the beams. Shehada et al. [45] and Elkholy et al. [46] investigated the resistance of RC sub-assemblages to progressive collapse due to the exclusion of an interior column using the fiber element technique. The numerical findings showed good agreement with the results from 10 tests in the literature.
Elsanadedy and Abadel [47] proposed a simple high-fidelity FE model to assess the risk level of RC ordinary moment frames with discontinuous sagging beam reinforcement over beam-column connections and subjected to interior column removal. The axial load on columns, the percentage of continuous sagging beam reinforcement over the beam-column joints, and the number of spans were parameters investigated in their study. Iribarren et al. [48] suggested a multilayered beam element to evaluate the dynamic response of structures under sudden column loss. The effects of reinforcement ratio and material parameters were investigated. Substantial variances in the progressive collapse patterns were highlighted as the considered parametric changed. Elkholy et al. [49] confirmed that the beam height had a tangible effect on the sub-assemblage resistance to progressive collapse and demonstrated that varying the concrete and steel quantities had a significant impact on the ductility and the cost of the frame structure.
The studies given in this literature search revealed that the investigations of the effect of member dimensions, slab thickness, and reinforcements while examining the structural progressive collapse of structures are rare. As a result, it is necessary to evaluate and emphasize the importance of the design of the floor system configuration (member dimensions, slab thickness, and reinforcements) to the structure behavior, strength, and ability to dissipate energy to resist significantly large deformations in response to a missing interior column. The impact of the design on the structure ability to initiate supplemental load-carrying capacity via arch and catenary action mechanisms is of great importance.

2. Framework of the Study

Typically, after the structural system has been designed to meet applicable design codes, progressive collapse-resistance is usually examined. In this study, varying design options were considered for the structure, and each design option was numerically examined to evaluate the structure progressive collapse-resistance. The framework of the study is as follows:
  • The minimum slab thickness was calculated according to the ACI 318-14 [50] design code considering the relative rigidity between the slab and beams.
  • The slab thickness was used to accurately calculate the dead load of the floor system.
  • spSlab/spBeam software [51] was used to design the floor system in accordance with ACI 318-14 design code requirements.
  • The numerical model using SeismoStruct software [52] was employed to perform nonlinear push-down analyses and examine the progressive collapse-resistance of the different designs of the structure under a middle-column removal scenario. The validity of the numerical model was demonstrated in previously published work by the authors [53].
  • The analysis and interpretation of the numerical results were presented.
  • The conclusions and future recommendations were drawn and proposed.

3. RC Frame Sub-Assemblages with Varying Design Options

The choice of a given design configuration may indeed affect the structural failure pattern. Therefore, it is important to examine the influence of varying design options on the structure resistance to progressive collapse due to the loss of the interior column.

3.1. Sub-Assemblage Configurations and Material Properties

Low- to moderate-rise reinforced concrete structures consist of beams, columns, and beam-column joints. Accordingly, practical bay span length ranges in general from 5 to 7 m. In this study, the bay span in one direction was kept constant at 5 m, while it varied in the other direction (5, 6, and 7 m) to study the effects of varying the bay span length on the structural resistance to progressive collapse as a result of column removal. Therefore, the structure shown in Figure 1 with six floors and an equal number of bays in both directions was considered in this study. The structure has four equal bays with a constant span of 5 m in one direction and four equal bays in the orthogonal direction with different span length of 5, 6, and 7 m. Software spSlab/spBeam [51] was used to design the sub-assemblage beams for gravity loads and to check the slab thicknesses and reinforcements of the sub-assemblages in accordance with the ACI 318-14 [50] design code requirements for varying design options and the same material properties. In addition to the self-weights of slabs, beams, and columns, gravity loads comprised a superimposed dead load of 2 kN/m2, a wall load of 7.2 kN/m, and a live load of 1.92 kN/m2, for residential building as per the ASCE 7-22 standard [54]. The column size and reinforcement were kept the same in all floor systems to exclude the column effects on the total performance of the floor system under the removed column scenario. The equivalent frame of one of the floors as shown in Figure 1 was considered as the sub-assemblage frame, which was numerically modelled to examine the effects of varying design options on its progressive collapse-resistance, as discussed in the next sections.
To examine the effects of the slab reinforcement in the direction of the beams on the sub-assemblage progressive collapse-resistance and energy dissipation due to the removal of the middle column, the sub-assemblages were numerically modeled in Section 4 twice by considering the beams as T-beams with an effective flange width (slab) and as rectangular beams (R) without a slab, as shown in Figure 1. It is worth noting that in the two configurations, T-beams and R-beams, all member dimensions and reinforcement detailing were kept the same with the exception of the effective flange width removal in the rectangular beam configuration. Slab reinforcement in the direction of the beams was considered in the T-beam models.
Varying design options with various sectional dimensions (i.e., beam depth, width, slab thickness, and corresponding reinforcement ratios) and various spans were considered in this study.
The sub-assemblage slab thicknesses, beam dimensions, and column size considered in this study are shown in Table 1. The sub-assemblage concrete and steel material properties used are shown in Table 2.

3.2. Sub-Assemblage Varying Design Options

Considering the varying design options of Table 1, the software spSlab/spBeam [51] was used to analyze and design the sub-assemblage beams for gravity loads and to check the corresponding varying slab thicknesses and reinforcements of the floor system in accordance with the ACI 318-14 [50] code design requirements. The software spSlab/spBeam follows the ACI 318-14 design code, which adopts the rectangular stress distribution as a material model for concrete and elasto-plastic behavior as a material model for steel bars. The moment capacities and demands of the beams and slab column strips were calculated. The longitudinal sagging and hogging reinforcement of the beams and slab column strips were computed. Beam web reinforcements were estimated as well. Samples of the spSlab/spBeam [51] outcomes are displayed in Figure 2.
The sub-assemblage details of all the varying design options considered, using spSlab/spBeam [51], are tabulated and displayed in Table 3. The locations of sections A, B, C, and D listed in the table are shown on the frame sub-assemblage in Figure 3.

4. Numerical Modeling for Progressive Collapse Analysis of Designed Sub-Assemblages

Using the fiber element-based software SeismoStruct [52] and employing the authors’ previously calibrated and utilized modelling technique [53], the RC frame sub-assemblages designed in this study and described in Table 3 and Figure 3 were numerically modelled to investigate the impact of varying design options of the floor system on the structure progressive collapse behavior and amount of energy dissipated. The authors’ modeling technique employed in this study was extensively validated by the authors in a previously published work [53], in which a comprehensive database of experimental results related to the progressive collapse of RC frame sub-assemblages was collated and utilized for the comparison, tuning, and verification of the model results. Based on the authors’ previously validated Seismostruct model [53], the Chang–Mander nonlinear concrete model and the Menegotto–Pinto steel model [52] were adopted and used in this study. The RC sub-assemblage members were divided into a number of segments according to their physical cross-section dimensions and reinforcement ratios in their hogging and sagging moment regions. Each segment was modeled using a single inelastic force-based plastic-hinge frame element (infrmFBPH) element [52]. The member section was divided into at least 150 fibers, and a plastic hinge length of 50% of the element length was implemented. The analysis was carried out by applying displacement-controlled nonlinear static push-down at the location of the failed column to simulate the removal of the column and to predict the progressive collapse of the structure. The load is applied in increments, and the stresses and strains calculated for each element are compared with some material performance criteria such as concrete cracking and crushing, steel yielding, and fracture to determine whether or not the element has failed. The applied vertical load at the location of the removed column is equal to the summation of all vertical support reactions of the sub-assemblage due to the applied displacement at the failed column location. The complete information on the fiber-element-based numerical model parameters and framework can be found in [53]. The authors have also used the fiber element-based software SeismoStruct in previous studies [40,41,42,43,44,45,46,49] on the simulation of the progressive collapse of RC structures.
To examine the effects of the slab on the sub-assemblage progressive collapse-resistance and energy dissipation due to the removal of the middle column, two configurations of the designed sub-assemblages were numerically modeled by considering the beams as T-beams with an effective flange width (slab) of 1250 mm and as rectangular beams (R) without a slab, as shown in Figure 1. It is worth noting that in the two configurations, all member dimensions and reinforcement detailing were kept the same, with the exception of the effective flange width removal in the rectangular beam configuration. Slab reinforcement in the direction of the beams was considered in the T-beam models.

5. Numerical Results and Discussions

5.1. Influence of Varying Design Options on the Load-Displacement Curves

Figure 4 shows the relationships of the applied load versus vertical displacement at the removed column location for the different RC sub-assemblage design configurations. The figure clearly demonstrates that varying structural design options have a direct influence on the obtained collapse patterns and, consequently, on the progressive collapse-resistance of the sub-assemblages, with T-beams or R-beams.
In addition, Figure 4 reveals that increasing the depth of the beam improved the first peak load (also see Figure 5), indicating that the compressive arch action capacity was enhanced followed by a softening in the arch action. The improvement in the first peak load was even more pronounced as the beam width increased. In addition, increasing the beam width resulted in a beneficial effect on the sub-assemblage deformation capacity manifested in larger vertical displacements when reaching the first peak load and the transition load (also see Figure 5). This behavior can be attributed to the large vertical component of the compressive force developed in the compressive arch action at relatively small deformations (notably less than the beam depth) and large beam height and width. However, sub-assemblages with higher beam depths did not display a substantial catenary action after the softening of the arch action, and the structures failed at a load less than the first peak load. This may be accredited to the corresponding small reinforcement ratios leading to the premature rupture of the reinforcement bars as the deflection and rotations of the beams increased. Hence, the compressive arch action substantially increased the resistance to progressive collapse as the beam height increased and the reinforcement ratios decreased.
Sub-assemblages with small beam depths demonstrated less arch action resistance proceeded by remarkably shallow softening behavior and followed by more noticeable catenary action than sub-assemblages with larger beam depths. Increasing the beam width, on the other hand, further improved the catenary action resistance but did not enhance the deformation capacity of the sub-assemblage significantly. This is credited to the smaller vertical component of the arch action compressive force and to the increase in the design hogging and sagging reinforcement ratios. As the vertical deflection increased, the arch action compressive force quickly flattened and changed into pulling tensile force, causing the sub-assemblage to regain its strength as deformations increased (notably more than the beam depth) due to the hogging reinforcement. The sub-assemblages with low beam depths failed when they reached their ultimate catenary capacity, which were higher than their first peak load.
Figure 4 also discloses that for the same span length, the varying design options did not have a significant effect on the ductility of the designed sub-assemblages that failed. It can be concluded from this remark that design options of large sections can be considered as designs that provide an alternate load path in which the sub-assemblage bridges across the failed column, whereas the design options of the small sections can be considered as designs that provide tie forces that can transfer the load from the failed column to the undamaged neighboring elements.
It is worth noting that the behavior patterns of the two sub-assemblage configurations, with T- beams and with R-beams, were almost identical.

5.2. Influence of Varying Design Options on the Capacity and Dissipated Energy

The behavior pattern of the T- or R-beam configuration under column removal scenario is governed by the typical behavior shown in Figure 5. This typical behavior shows that upon the removal of a column, the load on the removed column is resisted by the sub-assemblage flexure action followed by the arch action. Once instability is reached, the load is resisted by catenary action. Five key points in this typical behavior can be recognized in general, as depicted in Figure 5: (1) the yielding load (Py) under beam flexure/arch action; (2) the first peak load (Pf) under the beam flexure representing the compressive arch action capacity; (3) the transition load (Pt), representing the capacity at the end of the arch action softening; (4) the rupture load (Pr), representing the load at which reinforcement bar ruptures; and (5) the failure load (Pfa), representing the catenary action ultimate capacity.
The T- and R-beam configurations in this study have the same model; however, the model of T-beam configuration has an extra component that is the slab-effective flange width with reinforcement in the beam direction. As a result, the investigation revealed that sub-assemblages with T-beams have more load capacity and can dissipate more energy, as shown in Figure 6 and Figure 7 below, due to the inclusion of the slab reinforcement in the beam direction and due to increasing the lever arm between the tension and compression forces in the beam cross section.

5.2.1. Influence of Varying Design Options on the Capacity

The focus in this section is the flexure, transition, and catenary capacities. Figure 6 shows the effects of the varying design options on the progressive collapse capacities of the designed sub-assemblages.
Despite few discrepancies, the figure demonstrates that, overall, the sub-assemblage flexure capacity (Pf) and the transition capacity (Pt) increased steadily as the beam dimensions increased, regardless of the considered span length. The figure also shows that the magnitudes of flexure capacity (Pf) are higher than the magnitudes of failure capacity/catenary (Pfa) for sub-assemblages with higher beam depths and widths, indicating the sub-assemblages failed at lower loads than the first peak loads. Such design options can be considered as designs that provide a load path in which the sub-assemblage bridges across the failed column.
On the contrary, the flexure capacity (Pf) values are lower than the failure capacity (Pfa) values for sub-assemblages with small beam depths and widths, indicating that the sub-assemblages failed at higher loads than the first peak loads. However, the transition capacities (Pt) and failure capacities (Pfa) did not have a consistent and comparable pattern. For some sub-assemblages, Pt is higher than Pfa, and for other sub-assemblages they are lower. This could due to the fracture of the reinforcement bars according to the size and number of bars produced by a specific design option. Such design options can be considered as designs that provide tie forces that transfer the load from the failed column to the undamaged neighboring elements.
From Figure 6, the same trend is observed for both the T-beam and R-beam configurations. However, the capacities of sub-assemblages with T-beams were more enhanced than those with R-beams as the span increased. This enhancement was due to the increase in the effective flange/slab thickness and the provision of more reinforcement within the regions of hogging moments and in the direction of the beam. Increasing the slab thickness requires larger slab reinforcement according to the slab requirement of minimum reinforcement ratio; such an increase in the slab thickness boosted the progressive collapse-resistance of the designed sub-assemblages with T-beams.
It should be mentioned that missing bars in some charts in Figure 6 are due to the removal of sections not meeting the design requirements of the design code.

5.2.2. Influence of Varying Design Options on the Dissipated Energy

The energy dissipated by the structure is the area under the load-displacement curve as portrayed in Figure 5. Using the load-displacement curves in Figure 4, the energy dissipated by the designed sub-assemblages are shown in Figure 7.
The energy dissipation capacity of the designed sub-assemblages depends on various parameters such as the shape of the beams, the size of the beam cross-sections and corresponding reinforcement ratios, and the span of the beams, as clearly demonstrated in Figure 7.
The figure shows that the T-beam configuration demonstrated a higher energy dissipation capacity than the R-beam configuration as a result of the presence of the effective flange/slab. This indicated that additional resistance was provided by the slab and, therefore, the sub-assemblages experienced and enhanced strength and performed better than sub-assemblages with R-beam. Alternatively, as the size of the beam increased, the energy dissipation of the sub-assemblages fluctuated and did not show a clear pattern for both of the configurations (the T- and R-beams). This is attributed to the varying reinforcement ratio associated with each beam size, leading to the variation and fluctuation of the energy dissipated by the flexure action (FA) mechanism, the compression arch action/softening (CAA) mechanism, and the tension catenary action (TCA) mechanism.
Alternatively, Figure 6 shows the percentages of the energy dissipated by each of the above mechanisms (FA, CAA, and TCA). It can be seen from the figure that, in general, TCA was the most efficient and had the highest energy dissipation percentage when the beam depth was smallest at the shortest span. This is because of the provision of tensile axial/tie forces in the beams as a result of the small beam size and high reinforcement ratio that was needed when the member was originally designed according to the code to resist shear and flexure and later was subjected to large displacements. These tie forces in the beams transferred the load of the failed column to the adjacent members and ensured more ductility. Conversely, as the beam depth became larger, the dissipated energy percentage of FA and CAA increased, providing an alternate load path bridging over the missing column. As the span increased, higher energy was dissipated, indicating that the slab had a greater impact on the progressive collapse behavior than the R-beam configuration.
An analysis of such numerical results is challenging due to the contribution and effects of many parameters, such as span length, beam size, slab thickness, and sagging and hogging reinforcement ratios. Some of the parameters are coupled together, which makes the performance pattern vary and differs from one design option to another. Part of the variation and difference is due to the demand-to-capacity ratio between the varying design options/models. As the demand-to-capacity ratio changes from model to model, different behavior patterns appear among the models.
It should be mentioned that the missing bars in some charts in Figure 7 are due to the removal of sections not meeting the design requirements of the code.

5.2.3. Impact of Slab on Energy Dissipation

To underline the impact of the flange width/slab presence on the performance of the designed sub-assemblages, the difference between the total dissipated energy of the T- and R-beam configurations and the total dissipated energy of the R-beam ratio, (ET − ER)/ER%, is plotted versus the beam size in Figure 8. The graph shows a positive difference that highlights the importance of modeling the beams as T-beams when investigating the progressive collapse dissipated energy of the designed sub-assemblages. Based on the designed configurations, sub-assemblages with T-beams showed that the slab effects on dissipating energy ranged between 7% and 64%, with an average of 28.3% based on the beam size and span, indicating that the resistance of the R-beam to progressive collapse could be conservative. The effect of the slab was more pronounced in dissipating energy when the beam span increased.

5.2.4. Performance Index

Performance is generally synonymous with strength; yet, in beams subjected to progressive collapse, deflection; rotation; and, as a result, energy, are just as significant as the loading. As a result, one of the most essential features of the progressive collapse performance of the structures is their ability to dissipate energy effectively. To optimize the performance, maximum energy should be dissipated with minimum deformations. The performance index has been previously proposed in the literature to mitigate the progressive collapse of the existing structures [43].
As a result, the performance index is defined in this study as the highest relative energy ratio to the maximum middle displacement ratio of the beams to provide a quantitative measurement. The higher the performance index value, the bigger the dissipated energy and the smaller the displacement in the system under progressive collapse. The performance index is then used as an indicator of the optimum system configuration that dissipates most of the energy with the least total displacement.
In Figure 9, the performance index is plotted versus the beam span-to-depth ratio (L/a). The figure reveals that by increasing the width of the beam, the performance index increased for both T- and R-beams. As the beam span-to-depth ratio (L/a) increased, the performance index varied between 1.07 and 0.68, 1.27 and 0.71, and 1.17 and 0.77 for spans of 5 m, 6 m, and 7 m, respectively.
The largest performance index value, 1.27, indicated that the R-beam configuration with a span length (L) of 6 m and the largest beam size b = 400 mm and a = 700 mm (L/a = 8.57, Figure 9) was the optimum configuration among all the considered options as it dissipated the largest energy associated with the least displacement. As for the T-beams, the optimum design option for resisting the progressive collapse following the column elimination was the configuration with a performance index value of 1.12, Figure 9. This system configuration was also with a beam span (L) of 6 m and the largest beam size b = 400 mm and a = 700 mm (L/a = 8.57, Figure 9). The trend of the performance index is verified by the findings shown in Figure 4 and Figure 7.
Hence, it can be concluded that the highest performance index value is a good gauge to identify the system configuration that dissipates most of the energy with the least total displacement. In this study, the highest performance index value was obtained by decreasing the beam span-to-depth ratio (L/a) and increasing the beam size of the structure configuration.

6. Conclusions

The study reported the numerical results of the effects of floor system configurations on the progressive collapse-resistance of the RC frame sub-assemblages and the amount of energy dissipated as a result of a column removal scenario. The performance pattern of sub-assemblages varied and differed from one design option to another. This could be due to the change in the demand-to-capacity ratio from one design option to another.
Increasing the beam depth improved the flexure resistance and the total energy dissipated by the sub-assemblages/system, with T- or R-beams, mainly due to the beam compression arch action. Conversely, decreasing the beam depth did not improve the flexure resistance; however, it maximized the dissipated energy. This is primarily due to the addition of compression reinforcement, which improved the effectiveness of the tension catenary action. On the other hand, as the beam size increased the total energy dissipation capacity of the systems fluctuated and did not show a clear pattern, for both the T- and R-beams. This could be attributed to the varying reinforcement ratio associated with each beam size leading to a change in the energy dissipated by the flexure action, compression arch action/softening, and tension catenary action. In addition, for the same span length, the varying design options did not have a significant effect on the ductility of the designed sub-assemblages that failed.
In this study, the T- and R-beam systems had the same model; however, the model of the T-beam system had an extra component, which was the slab effective flange width with reinforcement in the beam direction. As a result, the investigation revealed that, depending on the design configuration, the T-beam systems had a greater load capacity and dissipated between 7 and 64%, with an average of 28%, and more energy, depending on the design options, when compared to R-beam systems. Thus, the resistance of the R-beam systems to progressive collapse due to internal column elimination could be considered conservative.
A performance index (PI) was proposed and defined as the highest relative energy ratio to the maximum middle displacement ratio of the beams to provide a quantitative measurement of the system resistance to progressive collapse due to internal column removal. The higher the PI value, the bigger the dissipated energy and the smaller the displacement in the system under progressive collapse. In this study, the largest performance index value was 1.27 and 1.12 for the R-beam and T-beam systems, respectively, indicating the design options of a length of 6 m, and the largest beam sizes of 400 mm by 700 mm were the optimum design options for the R- and T-beam systems to resist the progress collapse as they dissipated the largest amount of energy associated with the least displacement at the location of the removed column.
It can also be concluded from the findings that the design options of the large sections can be considered as designs that provide an alternate load path in which the sub-assemblage bridges across the failed column, whereas the design options of small sections can be considered as designs that provide tie forces in the beams that can transfer the load from the failed column to the undamaged neighboring elements.

Author Contributions

Conceptualization, B.E.-A. and S.E.; methodology, B.E.-A., S.E., and A.S.; software, S.E. and A.S.; validation, B.E.-A., S.E., and A.S.; formal analysis, B.E.-A. and S.E.; investigation, B.E.-A., S.E., and A.S.; resources, S.E. and A.S.; data curation, B.E.-A., S.E., and A.S.; writing—original draft preparation, B.E.-A.; writing—review and editing, B.E.-A. and S.E.; visualization, B.E.-A., S.E., and A.S.; supervision, B.E.-A. and S.E.; and project administration, B.E.-A. and S.E. All authors have read and agreed to the published version of the manuscript.

Funding

No external funding was received in this research.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

On reasonable request, the authors will provide the data that support the findings of this study.

Acknowledgments

The authors acknowledge SEiSMOSOFT for providing the SeismoStruct software with a free academic license.

Conflicts of Interest

There are no conflicts of interest declared by the authors.

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Buildings 12 00737 g001 550
Figure 1. The structural framing of the building and the sub-assemblage configuration.
Figure 1. The structural framing of the building and the sub-assemblage configuration.
Buildings 12 00737 g001
Buildings 12 00737 g002 550
Figure 2. Samples of the spSlab/spBeam analysis and its design outcomes.
Figure 2. Samples of the spSlab/spBeam analysis and its design outcomes.
Buildings 12 00737 g002
Buildings 12 00737 g003 550
Figure 3. The sub-assemblage detailing (only half of the structure is shown due to symmetry).
Figure 3. The sub-assemblage detailing (only half of the structure is shown due to symmetry).
Buildings 12 00737 g003
Buildings 12 00737 g004a 550Buildings 12 00737 g004b 550Buildings 12 00737 g004c 550
Figure 4. Load-displacement relations of the different sub-assemblage designs.
Figure 4. Load-displacement relations of the different sub-assemblage designs.
Buildings 12 00737 g004aBuildings 12 00737 g004bBuildings 12 00737 g004c
Buildings 12 00737 g005 550
Figure 5. The compressive arch action, catenary action, and energy under the curve.
Figure 5. The compressive arch action, catenary action, and energy under the curve.
Buildings 12 00737 g005
Buildings 12 00737 g006 550
Figure 6. Load capacities of the different sub-assemblage designs.
Figure 6. Load capacities of the different sub-assemblage designs.
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Buildings 12 00737 g007a 550Buildings 12 00737 g007b 550
Figure 7. The energy dissipated by the sub-assemblage varying design options and spans.
Figure 7. The energy dissipated by the sub-assemblage varying design options and spans.
Buildings 12 00737 g007aBuildings 12 00737 g007b
Buildings 12 00737 g008 550
Figure 8. The impact of slab presence on energy dissipation.
Figure 8. The impact of slab presence on energy dissipation.
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Buildings 12 00737 g009a 550Buildings 12 00737 g009b 550
Figure 9. The performance index.
Figure 9. The performance index.
Buildings 12 00737 g009aBuildings 12 00737 g009b
Table
Table 1. The varying design options of the sub-assemblages.
Table 1. The varying design options of the sub-assemblages.
Beam Dimensions, Varying Design Options (mm)Beam Span (m)Thickness of Slab (mm)All Columns
Width (b)Height (a)Width (b)Height (a)Width (b)Height (a)Dimensions and Detailing (mm)
1503002503004003005, 6, and 7varies from 125 to 205400 × 400
8#32
150400250400400400
150500250500400500
150600250600400600
150700250700400700
Table
Table 2. The concrete and steel material properties used in the considered frame sub-assemblages.
Table 2. The concrete and steel material properties used in the considered frame sub-assemblages.
MaterialMaterial Properties
Steel reinforcementElastic modulus (MPa)200,000
Yield strength (MPa)420
Fracture strain (%)6.5–7.0
ConcreteCompressive strength (MPa)35
Mean tensile strength (MPa)2.2
Modulus of elasticity (MPa)27,806
Table
Table 3. The varying design options, and the details of the designed RC sub-assemblages.
Table 3. The varying design options, and the details of the designed RC sub-assemblages.
Span (m)Sub-Assemblage Beam Varying Design Options bxa (mm)Slab
Thickness,
h (mm)		Sub-Assemblage Beam Reinforcement of the Varying Design Options
Bottom/Sagging ReinforcementTop/Hogging Reinforcement
Section
A *		Section
B *		Section
C *		Section
D *		Section
A *		Section
B *		Section
C *		Section
D *
5150 × 3001453 #132#132#132#13-3 #19-3 #16
150 × 4001253 #132#132#132#13-4 #16-4 #16
150 × 5001253 #132 #132 #132 #13-4 #16-4 #16
150 × 6001252 #132 #132 #132 #13-5 #13-5 #13
150 × 7001252 #192 #132 #132 #13 - 5 #10 - 5 #10
250 × 3001404 #133 #133 #133 #13-5 #16-4 #16
250 × 4001253 #133 #133 #133 #13-4 #16-4 #13
250 × 5001254 #132 #132 #132 #13-5 #13-5 #13
250 × 6001253 #132 #132 #132 #13-6 #13-6 #13
250 × 7001252 #195 #105 #105 #104 #134 #134 #134 #13
400 × 3001354 #133 #133 #133 #13-5 #16-5 #16
400 × 4001254 #133 #133 #133 #13-5 #13-5 #13
400 × 5001254 #133 #133 #133 #13-5 #13-5 #13
400 × 6001253 #133 #133 #133 #13-6 #13-6 #13
400 × 7001253 #135 #105 #105 #10 - 4 #13 - 4 #13
6150 × 300section did not meet the design requirements of the ACI 318-14 code
150 × 4001703 #132 #132 #132 #13 - 5 #13 - 5 #13
150 × 5001504 #133 #133 #133 #13 - 5 #16 - 5 #13
150 × 6001504 #133 #133 #133 #13 - 6 #13 - 4 #13
150 × 7001503 #133 #133 #133 #13 - 2 #19 - 4 #13
250 × 300section did not meet the design requirements of the ACI 318-14 code
250 × 4001605 #134 #134 #134 #13 - 6 #16 - 5 #16
250 × 5001504 #134 #134 #134 #13 - 5 #16 - 5 #13
250 × 6001504 #133 #133 #133 #13 - 6 #13 - 4 #13
250 × 7001504 #133 #133 #133 #13 - 5 #13 - 2 #19
400 × 300section did not meet the design requirements of the ACI 318-14 code
400 × 4001505 #134 #134 #134 #13 - 4 #19 - 5 #13
400 × 5001505 #134 #134 #134 #13 - 5 #16 - 5 #13
400 × 6001505 #133 #133 #133 #13 - 5 #16 - 5 #13
400 × 7001505 #133 #133 #133 #13 - 5 #16 - 2 #19
7150 × 300section did not meet the design requirements of the ACI 318-14 code
150 × 400section did not meet the design requirements of the ACI 318-14 code
150 × 5001904 #133 #133 #133 #13 - 4 #19 - 6 #13
150 × 6001705 #133 #133 #133 #13 - 4 #19 - 6 #13
150 × 7001704 #132 #132 #132 #13 - 5 #16 - 5 #13
250 × 300section did not meet the design requirements of the ACI 318-14 code
250 × 4002003 #133 #133 #133 #13 - 6 #13 - 4 #13
250 × 5001806 #134 #134 #134 #13 - 5 #19 - 5 #16
250 × 6001705 #134 #134 #134 #13 - 4 #19 - 4 #19
250 × 7001705 #134 #134 #134 #13 - 5 #16 - 5 #13
400 × 300section did not meet the design requirements of the ACI 318-14 code
400 × 4001905 #134 #134 #134 #13 - 6 #16 - 5 #16
400 × 5001706 #135 #135 #135 #13 - 5 #19 - 5 #16
400 × 6001705 #165 #135 #135 #13 - 4 #19 - 5 #16
400 × 7001705 #164 #134 #134 #13 - 4 #19 - 5 #16
* Sections A, B, C, and D are shown in Figure 3.
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Elkholy, S.; Shehada, A.; El-Ariss, B. Effects of Floor System on Progressive Collapse Behavior of RC Frame Sub-Assemblages. Buildings 2022, 12, 737. https://doi.org/10.3390/buildings12060737

AMA Style

Elkholy S, Shehada A, El-Ariss B. Effects of Floor System on Progressive Collapse Behavior of RC Frame Sub-Assemblages. Buildings. 2022; 12(6):737. https://doi.org/10.3390/buildings12060737

Chicago/Turabian Style

Elkholy, Said, Ahmad Shehada, and Bilal El-Ariss. 2022. "Effects of Floor System on Progressive Collapse Behavior of RC Frame Sub-Assemblages" Buildings 12, no. 6: 737. https://doi.org/10.3390/buildings12060737

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