1 Introduction

Following the catastrophic incidents at Ronan Point in 1968, the Murrah Federal Building in 1995, and the World Trade Center in 2001, the academic community has displayed an unparalleled level of concern regarding the phenomenon of progressive collapse. This has resulted in the proliferation of a plethora of experimental and numerical studies, as documented in references [1,2,3,4,5,6,7,8,9]. According to reference [10], progressive collapse refers to a state in which the failure of one or more structural constituents initiates a cascading failure of contiguous members and the eventual collapse of the entire structure. As per the American Society of Civil Engineers’ definition [11], progressive collapse refers to the failure of structural members, commencing with the impairment of a single constituent, ultimately leading to the eventual collapse of the entire structure or a significant portion thereof. For mitigation of progressive collapse, the Alternative Load Path Method (APM) or the strength of critical members which are essential in maintaining structural stability is prescribed in regulations of the General Services Administration [12], Unified Facilities Criteria (UFC) [13, 14], and other relevant codes and standards. In APM, the structural integrity is assessed by removing critical load-bearing members to determine whether the structure can bridge between removed members or not.

Most previous studies to reduce the risk of progressive collapse on reinforced concrete (RC) structures have focused on the use of methods such as RC shear walls [15,16,17,18], steel braces [19, 20], steel cables [21], and FRP technique [22]. Fang and Linzell [16] carried out an investigation into the capability of a 13-story reinforced concrete structure, featuring a pair of concrete shear walls situated in the central area of the floor plan, to withstand progressive collapse. They investigated the building’s response to column removal in different locations before, during, and after the column removal scenario. Their results provide insights for the improvement of new and existing RC building structures with a core-wall system against sudden column removal. Three precast concrete specimens with and without infilled frames were subjected to middle column removal scenarios [15]. The results of their study revealed that the pull-out of anchorage bars at the location of the exterior beam–column joint played a crucial role in precipitating the collapse of the precast concrete frame. Additionally, adding the infill wall could increase the load-bearing capacity.

In reference [19], Mohamed explored the effectiveness of implementing the Unified Facilities Criteria 4-023-23 [13] as a safeguard against the progressive collapse of corner floor panels in instances where their dimensions exceed the damage threshold. Through the utilization of the APM, an analysis and design of an RC building were undertaken for the purposes of examination. Yu et al. [20] conducted a numerical investigation into the RC frames which were strengthened with steel braces under various column removal scenarios. The finite element (FE) models produced outcomes that demonstrated the optimal placement of braces has to be on the uppermost floor. Additionally, the fragility analysis of the buildings’ frames through incremental dynamic analysis was performed. Six small-scale T-concrete beams with different widths of flanges and equipped with steel cables were subjected to monotonic loading to simulate a column removal scenario by Qiu et al. [21]. The findings derived from the experimental testing established that the primary components of the beam’s resistance mechanism were the flexural and compressive arch actions, catenary action, and cable tension.

A novel strengthening approach for reinforced concrete (RC) buildings, involving the installation of near-surface mounted steel bars and fiber-reinforced polymer (FRP) wrapping, was proposed by Elsanadedy et al. [22] to mitigate the consequences of column removal scenarios. The retrofitted specimen exhibited a remarkable enhancement in its peak load and energy dissipation capacity, reaching approximately 16.9 and 12.4 times greater values than the unmodified precast specimen. In addition, the authors developed a sophisticated three-dimensional (3D) finite element (FE) model that accounted for rate-dependent material nonlinearity and bonding behavior at the interface between the FRP and concrete. Examining high-performance fiber-reinforced cementitious composites in beam–column joints enhances RC frame buildings’ resistance to progressive collapse, improving stiffness, load-bearing capacity, and deformability. Reduced damages, even with fewer reinforcements, demonstrate their effectiveness [23]. Zhang et al. [24] optimized FRP retrofitting for RC frame structures, identifying schemes that enhance progressive collapse resistance without compromising seismic performance. Analytical methods for strength calculation under the catenary mechanism are proposed based on parametric analyses. In another study, X-tension reinforcement and steel trusses were applied to enhance progressive collapse resistance in an L-shaped RC frame. Finite element analysis revealed improved structural integrity and risk reduction, with steel trusses proving more effective in load transfer after component failure [25].

According to the available literature, a few studies have examined the impact of removing columns at various locations in a building’s plan that has been reinforced with steel shear walls and steel belt strips (SBS) around the frame. This paper introduces four distinct approaches to improve the capacity of irregular reinforced concrete (RC) buildings to resist progressive collapse in the event of unexpected column removal. These proposed methods include: reinforcing the RC slab with high-performance fiber-reinforced cementitious composites (HPFRCC), reinforcing the RC slab with carbon fiber-reinforced polymers (CFRP), employing steel plate shear walls (SPSW), and adopting a novel steel belt strip (SBS) approach that is introduced for the first time in this study. The primary goal of this paper is to utilize model analysis to assess the likelihood of progressive collapse in irregular reinforced concrete (RC) buildings, both with and without the implementation of the four mitigation approaches introduced in this study. To this end, a nonlinear dynamic analysis was initially conducted on an irregular RC building designed in accordance with the ACI 318-14 [26] code for RC frames and subjected to 10 independent column failure scenarios in the first-floor structural response using the model’s analysis software (ABAQUS [27]). Second, the strengthened structure with the proposed methods was reanalyzed to ensure the effectiveness of the strengthening strategies. Subsequently, a pushdown analysis was conducted to determine the potential for progressive collapse. The subsequent sections define the proposed approaches for mitigating progressive collapse and present the outcomes of the strengthening methods.

The progressive collapse of irregular reinforced concrete (RC) buildings poses a significant threat to life safety and property damage. This paper addresses the challenge of mitigating progressive collapse in irregular RC buildings by evaluating four promising approaches: reinforcing the RC slab with high-performance fiber-reinforced cementitious composites (HPFRCC), reinforcing the RC slab with carbon fiber-reinforced polymers (CFRP), employing steel plate shear walls (SPSW), and adopting a novel steel belt strip (SBS) approach. The primary objectives of this study are to:

Assess the likelihood of progressive collapse in irregular RC buildings using nonlinear dynamic analysis and pushdown analysis.

Evaluate the effectiveness of four mitigation approaches in preventing progressive collapse: HPFRCC reinforcement, CFRP reinforcement, SPSW, and a novel SBS approach.

Compare the performance of the four mitigation approaches and identify the most effective strategies for enhancing the progressive collapse resistance of irregular RC buildings.

2 Numerical Modeling

2.1 Structural Features of the Building

A six-story irregular reinforced concrete (RC) building was selected to examine the potential for progressive collapse. A longitudinal span of 6 m and floor height of 3.5 m were chosen. An RC slab system with a thickness of 200 mm was utilized for the ceiling. Figure 1 illustrates the plan and 3D model of the structure. Tables 1 and 2 provide details on the materials properties and design loads of the building, respectively. To simulate concrete in the models, a brittle cracking model was employed, which considers tension stiffening and shear retention. ABAQUS [27] offers the functionality to precisely define the brittle failure of a material within this model. In other words, when any of the local direct cracking strain components at a material point reach the input value for failure strain, the material point is considered to have failed, and all stress components are set to zero. In accordance with the GSA [28] guidelines, distinct gravity load combinations are applied to floor regions that are remote from the eliminated columns in nonlinear dynamic and static analyses. The gravity load combinations for both types of analyses, nonlinear dynamic analysis and nonlinear static analysis, are shown in Fig. 2. The corresponding mathematical formulations for the load combinations are presented in Eqs. (1) and (2), respectively.

$${\text{Comb}}\,\,1 = 1.2 \;{\text{DL}} + 0.5\;{\text{LL}}$$
(1)
$${\text{Comb}}\;2 = 2 \times \left( {1.2\;{\text{DL}} + 0.5\;{\text{LL}}} \right) + \left( {1.2\;{\text{DL}} + 0.5\;{\text{LL}}} \right)$$
(2)

where DL and LL are the abbreviations of dead load and live load, respectively.

Fig. 1
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Plan and 3D model of case-study building

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Table 1 Material properties
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Table 2 Design loads of the building
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Fig. 2
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a Gravity loads combinations in dynamic and static analyses as per GSA [28], b loading model in dynamic analysis

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Figure 3 displays the specifications of the beam’s size and its reinforcement details. To simplify the explanation of the two types of column designs for all floors, Table 3 shows the dimension and reinforcement details of these two types of columns.

Fig. 3
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Longitudinal reinforcement details for beams

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Table 3 Reinforcement details of column sections
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2.2 Description of Numerical Modeling

The current study primarily focused on numerical modeling of HPFRCC panels to evaluate their behavior under accidental column removal. The primary objective was to assess the effectiveness of incorporating HPFRCC panels beneath a reinforced concrete slab and examine their contribution to the overall structural response. It is noteworthy that the compressive strength of the concrete employed in structural elements is 25 MPa. The numerical models examined in this study involved the utilization of a CFRP sheet with two layers arranged in the x and y directions atop the slab. The CFRP layer employed in the study had a thickness of 1 mm and a width of 50 mm, with an interlayer spacing of 150 mm. Additionally, the placement of SPSWs on the first floor was also explored, and the findings are presented in Fig. 4. Material properties are depicted in Tables 4, 5, 6, and 7. The thickness of the steel plate shear wall is 8 mm, and the mechanical properties of the steel used therein are specified in Table 8. The study contributes to the ongoing endeavors to bolster the resilience and safety of structures under extreme loading scenarios.

Table 4 Characteristics of normal concrete
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Table 5 Characteristics of HPFRCC
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Table 6 Properties of CFRP sheets and epoxy
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Table 7 Properties of epoxy-bonded CFRP sheets
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Fig. 4
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Placement of SPSWs in the plan

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Table 8 Mechanical characteristics of steel material
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The method of SBS is proposed for the first time in the present paper. SBS encompasses vertical low-yield stress steel plate infills that are firmly linked to the adjacent beams and columns and are primarily situated beneath the beams at the uppermost story. The thickness of the SBS plate is 8 mm. Yield stress and ultimate stress are 93 and 272 MPa, respectively [29]. Also, \(W14\times 22\) and \(W5\times 16\) are used for column and beam, respectively. The yield stress and ultimate stress of boundary elements of SBS are 250 and 400 MPa, respectively. Accordingly, the SBS schema is shown in Fig. 5.

Fig. 5
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The SBS schema

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2.3 Material Constitutive Models

The investigation centers on examining structural elements composed of diverse materials, encompassing normal concrete, HPFRCC concrete, steel rebars, and steel plate. The behavior of each material under various loading conditions is meticulously scrutinized, leading to a comprehensive comprehension of their responses to external forces. This research contributes to the development of resilient design methodologies and aids in guiding the selection of optimal materials for enhanced structural performance in construction engineering. Overall, it offers valuable insights for constructing safer and more efficient structures.

2.3.1 Normal Concrete

In this study, various concrete models were evaluated for their applicability in ABAQUS [27] finite element analysis software. The three concrete models, namely the brittle cracking, damaged plasticity, and smeared cracking models, were examined. Although the smeared cracking model has its limitations, the damaged plasticity model is considered to be quite intricate and presents difficulties in terms of calibration. Additionally, it does not allow for the deletion of damaged elements, leading to numerical uncertainty. The brittle cracking model, on the other hand, is more user-friendly, and its calibration is straightforward. However, it assumes linear elastic material behavior in compression, limiting its reliability to instances where tensile failure dominates the concrete behavior. In this study, the user subroutine VUSDFLD was used to incorporate the nonlinear compressive behavior of concrete into the brittle cracking model. The concrete elasticity modulus (Ec) was defined as a function of strain (εc) using the CEB-FIP Model Code [30] to obtain the σc–εc relation that describes the concrete’s uniaxial compression behavior.

$$\frac{{\sigma_{c} }}{{f_{{{\text{cm}}}} }} = - \frac{{k\eta - \eta^{2} }}{{1 + \left( {k - 2} \right)\eta }}\; \ \ \ \ \ \ \ \varepsilon_{c} < \varepsilon_{{\text{c,lim}}}$$
(3)

where εc, lim denotes the strain at concrete crushing in compression, \(\eta ={\varepsilon }_{{\text{c}}}/{\varepsilon }_{{\text{c}}1}\), εc1 represents the strain at the maximum compressive stress fcm, \(k={E}_{ci}/{E}_{c1}\) indicates the number of plasticities, Ec1 denotes the secant modulus gained by the connection of the diagram origin to the curve peak, that is, (εc1, fcm), and Eci represents the initial modulus of elasticity. Equation (4) gives the Ec–εc relation considering that \({E}_{{\text{c}}1}={f}_{{\text{cm}}}/{\varepsilon }_{c1}\) and \({E}_{{\text{c}}}={\sigma }_{{\text{c}}}/{\varepsilon }_{{\text{c}}}\).

$$E_{{\text{c}}} = \left( { - \frac{k - \eta }{{1 + \left( {k - 2} \right)\eta }}} \right)E_{{{\text{c}}1}} \; \ \ \ \ \ \ \ \varepsilon_{{\text{c}}} < \varepsilon_{{\text{c,lim}}}$$
(4)

where Ec represents the secant modulus achieved by the connection of the diagram origin to a point on the \({\sigma }_{c}-{\varepsilon }_{c}\) curve. Equations (3) and (4) yield the \({\sigma }_{c}-{\varepsilon }_{c}\) and Ec–εc relations, schematic views of which are indicated in Fig. 6 (see also Table 4).

Fig. 6
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Relationship between the \({\sigma }_{{\text{c}}}-{\varepsilon }_{{\text{c}}}\) and Ec–εc based on Eqs. (3) and (4)

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2.3.2 HPFRCC Concrete

Concrete cracking is a major problem in reinforced concrete (RC) structures, which can lead to structural deterioration. The maximum axial strain of concrete during fracture is different and depends on the type of concrete, fiber content, fiber type, and type of aggregate. Normal concrete has a maximum axial strain of around 0.003. High-performance concrete has a maximum axial strain of around 0.004. Fiber-reinforced concrete has a maximum axial strain of around 0.006. This is because the fibers in fiber-reinforced concrete help to bridge cracks and prevent them from propagating. The higher the fiber content, the more resistant the concrete is to cracking and the higher the maximum axial strain. The type of fiber also affects the maximum axial strain. For example, steel fibers are more effective at bridging cracks than polypropylene fibers. The type of aggregate also affects the maximum axial strain. For example, lightweight concrete has a lower maximum axial strain than normal weight concrete. Researchers have shown that adding fiber reinforcement to concrete can improve its mechanical properties and mitigate cracking [31, 32]. Among various fiber-reinforced materials, HPFRCC has received particular attention due to its high ductility, durability, and strain-hardening behavior under uniaxial tensile stress [33, 34]. Figure 7 illustrates the typical stress–strain behavior of HPFRCC in tension. The tension and compression envelope curves of HPFRCC are shown in Fig. 8, which include important parameters such as first cracking stress (σto), strain corresponding to initial cracking (εto), tensile strain capacity (εtu), compressive strength (σcp), strain corresponding to maximum compressive strength (εcp), ultimate compressive strength (σcr), maximum tensile strength (σtp), strain corresponding to maximum tensile strength (εtp), and ultimate strain in compression (εcu) [35] (see Table 5). The unique properties of HPFRCC make it a promising material for enhancing both the performance and durability of reinforced concrete structures.

Fig. 7
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Stress–strain curves for tensile strength of both normal concrete and HPFRCC

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Fig. 8
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Tensile and compressive stress–strain envelope curves of HPFRCC

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2.3.3 Steel Material

A bilinear model with strain hardening is adopted by the constitutive relations of reinforcement and structural steel in ABAQUS/CAE [27]. The materials’ large deformations were not taken into account. It was assumed that the material’s actual strain and stress were identical to the nominal strain and stress. Equation (5) shows the strain–stress relations of the reinforcement and structural steel.

$$\sigma_{{\text{s}}} = \left\{ {\begin{array}{*{20}l} {E_{{\text{s}}} \varepsilon_{{\text{s}}} } \hfill & {0 \le \varepsilon_{{\text{s}}} \le \varepsilon_{{\text{y}}} } \hfill \\ {f_{{\text{y}}} + 0.01E_{{\text{s}}} \left( {\varepsilon_{{\text{s}}} - \varepsilon_{{\text{y}}} } \right)} \hfill & {\varepsilon_{{\text{y}}} < \varepsilon_{{\text{s}}} } \hfill \\ \end{array} } \right.$$
(5)

where σs represents the steel stress, εs denotes the steel strain, εy indicates the steel yield strain, Es represents the steel elastic modulus (200 GPa), and fy denotes the steel yield stress.

2.3.4 CFRP Behavior

Up to a rupture failure, fibers show a linear elastic behavior. It is possible to model CFRP as a lamina linear elastic element. The characteristics of the epoxy material and CFRP sheets used in the present work are indicated in Table 6.

Mallick [36] presents Eqs. (6)–(11), which are employed for evaluating the mechanical characteristics of the combined CFRP sheet and adhesion. Table 7 summarizes the characteristics of the combined adhesive material and FRP sheets.

$$E_{1} = E_{f} V_{{\text{f}}} + E_{{\text{a}}} \left( {1 - V_{{\text{f}}} } \right)$$
(6)
$$E_{2} = \frac{{E_{f} E_{{\text{a}}} }}{{V_{{\text{f}}} E_{{\text{a}}} + E_{f} \left( {1 - V_{{\text{f}}} } \right)}}$$
(7)
$$G_{12} = G_{13} = \frac{{G_{f} G_{{\text{a}}} }}{{G_{{\text{a}}} V_{{\text{f}}} + G_{f} \left( {1 - V_{{\text{f}}} } \right)}}$$
(8)
$$G_{23} = \frac{{E_{2} }}{{2\left( {1 + \nu_{23} } \right)}}$$
(9)
$$\nu_{23} = \nu_{{\text{f}}} V_{{\text{f}}} + \nu_{{\text{a}}} \left( {1 - V_{{\text{f}}} } \right)$$
(10)
$$\sigma_{{{\text{co}}}} = \sigma_{u} V_{{\text{f}}} + \sigma_{u} \left( {\frac{{\left( {1 - V_{{\text{f}}} } \right)E_{{\text{a}}} }}{{E_{f} }}} \right)$$
(11)

where G12, G13: Plane shear moduli, E1: Elastic modulus in the longitudinal direction, E2: Elastic modulus in the transverse direction, G23: Normal to plane shear modulus, σco: Ultimate tensile strength, σu: Maximum strength, Vf: Volume fraction of CFRP, Ea: Adhesive material’s elastic modulus, Ga: Shear modulus of the adhesive material, νa: Poisson’s ratio of the adhesive material, Ef: Elastic modulus of CFRP, Gf: CFRP shear modulus, and νf: Poisson’s ratio of CFRP.

2.4 Element Selection

The specimens studied in this research consisted of three primary constituents: structural steel, reinforcement cage, and concrete. For concrete modeling, eight-node linear brick elements (C3D8R) were employed, and the integration was reduced to minimize overly rigid components and improve hourglass control, which suppresses spurious deformation modes within the model mesh. Additionally, the model was designed to prevent excessive distortion by monitoring the elements during the analysis. Shell elements (S4R) were employed to represent SBS and SPSW sections, which are commonly used in metallic structures. In adherence with the approach proposed by Jamkhaneh et al. [37], the primary and transverse reinforcements were simulated using two-node linear 3D truss elements (T3D2), while four-node shell elements (S4R) were used to model the CFRP. The dimensions of column and beam elements were set at 50 mm, and for the roof slab, it was taken as 100 mm. The dimensions for steel shear wall elements were established at 100 mm, and for polymer sheets, they were considered to be 50 mm.

2.5 Modeling of Interfaces

The interaction between concrete and CFRP surfaces was modeled using a tied contact model, which effectively merges the CFRP and concrete elements into a single entity. This simplified approach ensures that the two materials move and deform in unison, making it computationally efficient for uncomplicated scenarios. Additionally, the concrete surface–SBS interface and the boundary element–SPSW contact were treated as tied contacts, further streamlining the analysis. In contrast, the reinforcement–concrete bond was modeled using an embedded contact model, a more sophisticated approach that captures the intricate bond characteristics between the reinforcement and the concrete substrate. This enhanced model incorporates frictional forces, enabling a more realistic representation of the bond behavior and its influence on the overall structural performance.

3 Verification of Models

3.1 Validation of HPFRCC Concrete

In the current investigation, an analytical framework was employed to replicate the performance of HPFRCC material. Subsequently, the deflection curve obtained from the numerical analysis was juxtaposed against the empirical findings. The experimental work involved casting prism specimens with dimensions of 80 × 75 × 400 mm and testing them under a four-point bending load to determine flexural parameters such as flexural strength and mid-span beam deflection. Hooked-end steel fibers were added to the mixture to reinforce the concrete, and the compressive strength and flexural strength at the 28-day specimen were found to be 35.6 MPa and 4.68 MPa, respectively. The maximal deformation at the mid-span of the beam was ascertained to be 1.43 mm.

The numerical simulation was fine-tuned via a comparison of its deflection curve against the empirical findings, whereby the maximal flexural strength attained from the model was 4.32 MPa at a deformation magnitude of 0.655 mm. This value closely matched the experimentally measured flexural strength of 4.25 MPa at the same deflection. The deflection curve obtained from the numerical model had two peak points at the deflection of 0.702 and 1.072 mm. The initial slope of the test specimen and numerical model was 14.5 and 15.7 MPa/mm, respectively. These remarkable agreements between the model’s predictions and experimental results demonstrate the model’s high level of accuracy in simulating the behavior of HPFRCC (Fig. 9).

Fig. 9
figure 9

Comparison of analytical and experimental results of the HPFRCC deflection test

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3.2 Validation of Strengthening a Slab with CFRP Sheet

Fiber-reinforced polymer (FRP) is a material commonly used to strengthen and repair concrete structures, and many studies have been conducted on reinforcing concrete structural elements using FRP [38,39,40,41,42]. One method for resisting the progressive collapse of irregular reinforced concrete buildings is to strengthen slabs with Carbon Fiber-Reinforced Polymers (CFRPs). An analytical model of CFRP was calibrated using experimental work conducted by Agbossou et al. [43]. The specimen was a slab with dimensions of 1.25 m × 1.25 m × 0.1 m, composed of concrete possessing a compressive resistance of 30 MPa. The superior and inferior steel reinforcement had diameters of 9 mm and 7 mm, correspondingly, with a tensile strain of 500 MPa and a concrete encasement of 25 mm. The slab was supported on four parallel 1.2-m lengths, upheld by simple supports at the perimeter. The CFRP exhibited a modulus of elasticity of 80 GPa and a tensile breaking stress of 950 MPa. The CFRP’s dimensions comprised a thickness of 1 mm and a width of 50 mm, while the spacing between successive strata was 150 mm.

Figure 10a depicts a meticulously meshed model, encompassing the assigned boundary conditions, while Fig. 10b presents a comparative analysis of the numerical and experimental outcomes for mid-span bending of a slab fortified with CFRP sheets. The test results revealed an initial stiffness of 37.21 kN/mm and a failure load of 139.5 kN. The numerical simulation closely mirrored these values, predicting an initial stiffness of 42.51 kN/mm and a failure load of 134.2 kN. These numerical predictions deviate from the experimental results by approximately 14% and 4%, respectively, indicating a high degree of accuracy. Figure 10c illustrates a correlation between the tensile degradation of concrete in the experimental sample and the numerical simulation, with the punching disintegration of the concrete slab exhibiting an analogous pattern in both the test specimen and the computational model. These findings collectively validate the effectiveness of the numerical model in predicting the flexural behavior of CFRP-reinforced slabs.

Fig. 10
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Results of comparing numerical and experimental curves for bending the middle of the strengthened slab with CFRP sheets; a a meshed model, b load–deflection curves, and c distribution of tensile damage in test specimen and numerical model

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3.3 Validation of SPSW

SPSWs are constructed using steel plate fillers that interconnect with the adjacent beams and columns and are generally installed within one or multiple intermediate spaces. SPSW is known for its high initial stiffness and ductile behavior under loading, which allows it to resist significant lateral loads and deformations while maintaining its strength and stability. Another advantage of SPSW is its ability to dissipate a significant amount of energy during loading, which can help to reduce damage to the structure and prevent progressive collapse. As a result of these properties and design requirements, numerous studies have been conducted on SPSW behavior and design, exploring various aspects such as seismic performance, structural detailing, and optimization [44,45,46,47,48].

The third validation model was selected from a study by Chen and Jhang [29] on an SPSW consisting of an infill plate with low yield stress. In this study, they used an infill plate with yield stress and ultimate stress of 93 MPa and 275 MPa. The performance of five test specimens was evaluated under cyclic loading. Sample No.1 of their research has been selected for the second stage of validation. The dimension and thickness of the square-shaped infill plate were 1250 mm and 8 mm, respectively. The beam and column sections were \(H244\times 175\times 7\times 11\) and \(H250\times 250\times 9\times 14\), respectively. The tabulated information in Table 8 denotes the mechanical characteristics of the steel materials.

The investigation entails a comparison of the empirical and numerical outcomes of a loaded configuration employing a loading procedure depicted in Fig. 11. Figure 12 presents a comparative analysis of the force–displacement curves obtained from both experimental and numerical simulations. The results indicate that the experimental specimen exhibits a remarkably higher lateral load-carrying capacity, reaching a maximum of 1190 kN. This value surpasses the numerical model’s prediction by approximately 2.5%, highlighting the experimental specimen’s enhanced resistance to lateral forces. Furthermore, the comparison reveals a slight discrepancy (8%) in the stiffness behavior during the initial loading cycle.

Fig. 11
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Numerical model: a cyclic loading protocol and b boundary conditions

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Fig. 12
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Comparison of the results of test and FE model: a horizontal load–drift angle curves and b stiffness–cycle number relationships

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4 Numerical Methods

4.1 Nonlinear Dynamic Analysis of the Structure Without Strengthening Methods

Ten column failure scenarios were tested on the first floor of the building, and load combinations were performed using 120% dead load and 50% live loads. The load was applied linearly over five seconds, followed by a constant load for two seconds to account for dynamic effects. The relevant column was suddenly removed in the seventh second, and the response of the structure was examined. It should be noted that axial loads are not evenly distributed in irregular buildings, so load maximization was done in a way that minimized dynamic excitation in columns during the first five seconds. Figure 13 shows the location of the column failure. The name of the models is a form of S1-C-4-B; this means that scenario S1 is considered for the analysis and the column at the junction of axes 4 and B is removed.

Fig. 13
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Location of the removed columns

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Figure 14 illustrates the axial load distribution in columns after different column removal scenarios. The findings evince a gradual augmentation in the axial load of columns subsequent to the removal of the target column, with the utmost magnitude transpiring approximately at the ninth second. Columns 2C and 3C typically experienced the highest axial loads in most scenarios. Table 9 provides a summary of the results.

Fig. 14
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The axial load of columns in 10 column failure scenarios

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Table 9 Maximum axial load and increasing percentage of axial force at columns in different column removal scenarios, where percentage is shown in parentheses (Units: kN, %)
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Table 9 presents the maximum axial load of columns in the first story for each column removal scenario, along with the peak axial load before any columns were removed (named “Intact”). Also, in Table 9, the increasing percentage of axial force is shown in parentheses, and the maximum percentage is presented in bold. The results show that column 3C had the highest maximum axial load in all scenarios. However, the maximum percentage increase in force did not occur in this column during the column removal scenarios. In scenario S4, when column 2B was removed, the highest percentage increase in force was observed in column 2A (289.6%), which had the highest value among all the different column removal situations.

Moreover, it was observed that removing columns located around the building’s plan (S2, S3, and S4) and at the intersection of the corner axes caused an additional force to be exerted on the adjacent columns. Furthermore, columns with fewer beams around them and less integrity in the plan had a higher percentage of tolerated force. For instance, in scenario S1, there were two columns around column 1A, including 2A and 1B, and among them, column 2A had an increase of 81.6% compared to 43.6% for column 1B. This was due to the integrity of the beam around the columns.

Additionally, removing column 2B in scenario S4 caused the maximum vertical load at the base of the building with a value of 704 kN, which was 1.39 times the intact model. These results are crucial in comprehending the response of the structures in column removal scenarios and can serve as a foundation for the construction of forthcoming structures.

Figure 15 presents the vertical displacement at a specific point located just above the removed column in each scenario. The maximum vertical displacement was observed at point 2A under the S2 scenario, with a deflection of 500 mm. The displacement values in other scenarios varied between 200 and 400 mm, indicating the sensitivity of the structure to column removal. It is worth noting that in the concrete brittle cracking model implemented in the ABAQUS software [27], element deletion occurs when the stress and strain of the elements surpass a certain amount. In the 3D models used in this study, the maximum element deletion was observed for the column removal scenarios of S2, S4, and S9, indicating the criticality of these scenarios in terms of structural safety. Thus, it is imperative to consider such scenarios in the structural design process to guarantee the resilience of the building against unanticipated contingencies.

Fig. 15
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Vertical displacement of the structure at the location of the removed column over time

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According to the data presented in Fig. 16, the analysis of the numerical models indicates that the greatest magnitude of horizontal displacement occurred at the proximal point to the removed column. This outcome was consistently observed across all scenarios examined, where the highest values of horizontal displacement were registered at the uppermost point of the columns neighboring the removed column. As an illustration, when column 2B was released, the point situated at the summit of columns 2A and 2C exhibited the largest horizontal displacement of 1 mm and 0.6 mm, respectively.

Fig. 16
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The top side of columns displacement in x direction among the 10 scenarios

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Figure 17 depicts the three-dimensional deformed models of S2, S4, and S9, each subjected to various column removal scenarios. As per the visual representation, the removal of columns 2A, 2B, and 3C resulted in a disconnection between the column above and the reinforced concrete slab, rendering the columns above incapable of withstanding tensile stresses. This, in turn, led to the removal of structural elements.

Fig. 17
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3D models of S2, S4, and S9

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4.2 Assessment of the HPFRCCs Method

As specified in Sect. 3.1, the investigation involved the placement of HPFRCC panels, with a thickness of 300 mm, beneath the reinforced concrete (RC) slab. This approach was utilized to assess its impact on the structural response under scenarios involving accidental column removal. The results indicated that the incorporation of HPFRCC in S2 resulted in a remarkable reduction of vertical displacement, up to 99.89%, in comparison with the primary structure. However, no substantial alteration was observed in S4 and S9. It is noteworthy that the HPFRCC material exhibits behavior akin to regular concrete in compression, whereas in tensile mode, its performance surpasses that of conventional concrete. Figure 18 delineates the vertical displacement underneath the extracted column 2A, along with the columns’ axial load and the HPFRCC stresses, thereby portraying a comprehensive and detailed visualization of the structure’s response to the real conditions. It is discernible from Fig. 18 that the usage of HPFRCC enhances the load-carrying capacity of columns while simultaneously reducing the vertical displacement under the removed column.

Fig. 18
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a Vertical displacement under removed column 2A, b columns axial load, and c HPFRCC stresses

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4.3 Assessment of the Carbon Fiber-Reinforced Polymers (CFRPs) Method

The placement of a CFRP sheet on the slab involved using one layer in the x direction and one layer in the y direction. In contrast to the HPFRCC method, where the axial load of columns increased significantly (by over 100 kN in this study) following column removal, the use of CFRP sheets did not result in a substantial reduction in vertical displacement under the removed column. However, in S2, the incorporation of CFRP sheets into the RC slab reduced vertical displacement by up to 95% (as shown in Fig. 19). Furthermore, a 9% reduction in the primary building was observed in S4 and S9 scenarios. In comparison with the HPFRCC approach, the CFRP method did not result in a considerable increase in the axial load of columns in the primary model following column removal. Figure 20 displays the axial load of columns in S2 and the vertical displacement under columns in S2 and S4. According to Fig. 20a, column 3C experienced a maximum axial load of approximately 109.5 kN, which is approximately 11% lower than that of the primary structure. The maximum reduction in axial load was observed in column 1A, which decreased by 35% compared to the S2 scenario in the primary structure.

Fig. 19
figure 19

Reducing vertical displacement under removed column (S2) strengthened with FRP system

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Fig. 20
figure 20

a Columns axial load in S2 and b vertical displacement under columns in S2 and S4

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4.4 Assessment of the SPSW Method

The behavior of SPSW under the effect of column removal was examined in S2, S4, and S9 scenarios, and the deformation of SPSW is analyzed using Fig. 21. In S2, despite the presence of a large amount of axial load, the SPSW deformation was not significantly affected by the removal of the column. In contrast, S4 and S9 scenarios demonstrated more substantial SPSW deformations compared to S2, indicating that the resistance of the SPSW to deformations decreased with the removal of the column. This result could be attributed to the redistribution of the load following the removal of the column, causing the SPSW to undergo more deformation in S4 and S9 scenarios. Therefore, it can be concluded that the behavior of SPSW under accidental column removal is highly dependent on the load redistribution pattern in the structure.

Fig. 21
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SPSW deformation in S2, S4, and S9

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Upon removal of a column in a building that employs SPSW, the weight that was formerly borne by the extracted column is apportioned among the adjacent columns through a process of load redistribution. This redistribution of load results in a significant increase in axial load in the adjacent columns due to the interaction and pressure between the SPSW and the concrete elements. As a consequence, this increased load may lead to cracking and other forms of damage in the adjacent concrete members, which can compromise the structural integrity of the building.

Accordingly, it is advised against deploying SPSW in neighboring columns as a means of reinforcing the load-carrying capability of the structure in the event of inadvertent column elimination. Instead, it is advised to consider alternative methods that can help to distribute the load more evenly among the remaining columns without significantly increasing the axial load in any particular column. This may include techniques such as the use of HPFRCC or CFRP, which have been shown to improve the load-bearing capacity and reduce vertical displacement under removed columns without significantly increasing the axial load in adjacent columns.

4.5 Assessment of the SBS Method

The application of Steel-Belted SBS in S2, S4, and S9 has been found to result in a significant reduction of 97%, 89%, and 25.9%, respectively, in terms of vertical displacement under the removed column. This underscores the efficacy of SBS as a feasible alternative for augmenting the structural resilience of buildings that encounter inadvertent column detachment scenarios. Figure 22 clearly demonstrates the impact of SBS on the structural response against progressive collapse in S2 and adjacent column axial load in S2. The results indicate that the use of SBS significantly improves the structural resistance against progressive collapse in S2, as evidenced by the considerable reduction in vertical displacement under the removed column. Additionally, SBS appears to have a negligible effect on the axial load in the adjacent columns in S2, which suggests that it does not cause any significant increase in the load-bearing capacity of these columns.

Fig. 22
figure 22

a Vertical displacement under removed column in S2, S4, and S9, b effects of SBS on structural resistance against progressive collapse in S2, and c comparison of the distribution of load between the primary model and models with SBS method

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It is noteworthy that the use of SBS in S4 and S9 scenarios results in a relatively lower reduction in vertical displacement under the removed column compared to that observed in S2. Nevertheless, SBS still remains an effective method for improving the structural response of buildings subjected to accidental column removal scenarios, especially when implemented in combination with other methods such as HPFRCC or CFRP. The effectiveness of SBS in reducing vertical displacement under the removed column in S2, S4, and S9 highlights its potential as a viable option for improving the structural safety and resilience of buildings subjected to accidental column removal scenarios.

Figure 22c demonstrates that the implementation of SBS has an insignificant impact on the axial load of the column in different column detachment scenarios. It is noteworthy that the use of SBS resulted in a mere 8.5% reduction in the axial load of column 1A in S2. However, despite the limited effect on column axial load, the use of SBS can still play a crucial role in reducing the vertical displacement under the removed column, which is a key factor in mitigating progressive collapse.

5 Nonlinear Static Analysis Method (Pushdown Analysis)

Pushdown analysis is a crucial tool for assessing the resistance of structures against progressive collapse. The assessment takes into account the force exerted at the extremity of the removed column under the continued influence of gravitational loads on the spans located above the removed column. In this study, the loading factor is calculated as the ratio of the ultimate applied load at each analysis step to the full gravity load using Eq. (2) as described in Kim and Lee [49].

Figure 23 presents the results of the pushdown analysis for S2, S9, and S4 scenarios. Table 10 provides a comparison of the loading factors for each method and scenario. The results indicate that S2 (column 2A) is the most vulnerable scenario to progressive collapse with a loading factor of 0.77. However, the loading factor can be significantly improved up to 2.14 times by using the SBS method. This manifests the effectiveness of SBS in fortifying the resistance of constructions against progressive collapse. In Fig. 23, S2 means that the column at the junction of axes 2 and A is removed. S2-HPFRCC/CFRP/SBS/SPSW means that the model S2 is retrofitted by several methods.

Fig. 23
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Pushdown analysis results for a S2, b S4, and c S9 scenarios

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Table 10 Loading factor changes in different scenarios and methods
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6 Conclusions

The present study introduces a pioneering approach to reinforce the progressive collapse resistance of an asymmetrical RC building. Four mitigation methods were evaluated to improve the resistance of an irregular RC building designed based on ACI 318-14 code to progressive collapse, including HPFRCCs, CFRPs, SPSW, and the newly proposed SBS. The findings are presented in terms of displacement at the point of the removed column and the deformed shape of the structures. In light of the analyses conducted, several inferences can be drawn.

  1. 1.

    After evaluating 10 different column removal scenarios on the first floor of the primary building, three scenarios (S2, S4, and S9) were selected for further analysis to assess the effectiveness of the proposed methods using nonlinear dynamic analysis. The decision to choose these scenarios was based on several factors, including the maximum vertical displacement observed under the removed column, the maximum loads generated in adjacent members after column removal, and the maximum damage caused to structural members after column removal.

  2. 2.

    Results of the nonlinear dynamic analysis revealed that RC slab with HPFRCC can reduce vertical displacement under the removed column up to 99.89% in S2 proportional to the primary building; however, in S4 and S9, no significant change was observed in vertical displacement under the removed column. Also, it was observed that using HPFRCC led to an increase in axial load in adjacent removed columns up to 1.81 times proportional to the primary building.

  3. 3.

    The incorporation of HPFRCC panels under the reinforced concrete slab reduced vertical displacement by up to 99.89% in scenario S2, while no significant impact was observed in S4 and S9.

  4. 4.

    The CFRP method effectively reduced vertical displacement by up to 95% in scenario S2 and 9% in S4 and S9. It also prevented a substantial increase in axial load in the primary model.

  5. 5.

    SPSW is not effective in reducing vertical displacement under column removal. It causes a significant increase in axial load in adjacent columns. Better alternatives include HPFRCC and CFRP, which can evenly distribute the load and improve load-bearing capacity without causing damage to adjacent columns.

  6. 6.

    SBS effectively reduces vertical displacement under column removal by up to 97% in S2, 89% in S4, and 25.9% in S9. It also has a negligible impact on axial load, making it a safe and effective strengthening method.

  7. 7.

    The pushdown analysis results demonstrate that SBS effectively mitigates progressive collapse by increasing the loading factor up to 2.14 times in S2, 1.92 times in S4, and 1.81 times in S9. This highlights the effectiveness of SBS in fortifying the resistance of constructions against progressive collapse.

  8. 8.

    The research outcomes presented in this study hold substantial potential for practical applications in engineering. They offer valuable insights for the safer design and construction of irregular reinforced concrete buildings, retrofitting existing structures to enhance their resistance to progressive collapse, informing the development of building codes and standards, facilitating risk assessment and evaluation, stimulating innovation in materials and construction technology, and improving engineering education and training. By implementing the mitigation approaches and methodologies outlined in this research, engineers can contribute to the creation of more resilient and secure built environments, ultimately enhancing public safety and the longevity of critical infrastructure.