Robustness of Corner-Supported Modular Steel Buildings with Core Walls

Abstract

This paper studies the dynamic response of corner-supported modular steel buildings with a core wall system, under progressive collapse scenarios, associated with corner module removals. Since using secondary systems such as concrete core in mid- to high-rise buildings is currently unavoidable, understanding their impact on load transfer between modules during collapse scenarios becomes essential. The designated four-, eight-, and twelve-story buildings were modelled using the macro-model-based finite element method in Abaqus. In addition, three different locations are considered for the concrete shear core within the building plan, leading to nine various case scenarios. Each vertical and horizontal inter-module connection was modelled by one axial and two shear springs with predefined nonlinear force-displacement behavior. The local and global buckling, which plays an essential role in the building’s stability, was considered to obtain accurate results. Finally, parametric studies on the building response were carried out, including the intra-module connection rigidity and inter-module connection stiffness. The results demonstrated that the core wall could maintain the robustness of a modular steel building through two mechanisms dependent on its location within the plan. In addition, preventing plastic hinges from forming in beams could be introduced as an anti-collapse mechanism in the corner module removal scenarios.

1. Introduction

Modular steel buildings (MSBs) are construction systems made of repeating pieces known as prefabricated or prefinished modules. This system has quickly gained popularity in many countries as it offers several benefits, including fast-track construction, off-site construction, cost-effectiveness, less site waste, and lower environmental impact [1,2,3]. The corner-supported modular building, as shown in Figure 1, is one type of MSB in which each module is tied at its corners by horizontal connections (HC) as well as vertical connections (VC), which are called inter-module connections (see Figure 1a) and intra-module connections (Figure 1b). The interconnection with corner posts is the sole component responsible for transferring loads from one module to another.

A typical structural layout for modular structures is comprised of stacked modules to carry gravity loads and a core system to mainly resist lateral loads [4]. This paper presents the impact of a core system on the collapse capacity of the MSB against corner module loss scenarios by adopting the results of nine different cases.

The inter-module connections act as the linking elements between modules for load distribution and efficient load transfer to the foundation. Since these connections play a key role in the stability of the entire building [5], most of the research studies in the literature have been focused on the structural performance or constructional challenges [6,7,8]. Lacy et al. [6,9] studied the stiffness of bolted connections and showed that rotational stiffness has a modest impact on global structural responses. It was concluded that simplified modelling of inter-module connections could accurately predict overall building response under gravity and lateral demands [10]. Peng et al. [11] developed a tenon inter-module connection using spring elements in Abaqus, which significantly lowered the computational expenses. Feng et al. [12] studied the seismic performance of four types of inter-module connection for modular box buildings, and the results showed that none satisfied Chinese code limitations such as inter-story drifts. Similarly, Peng et al. [13] showed that 12-story MSBs cannot meet the drift requirements against service wind loads.

Some notions have been put forward in recent research to overcome the former shortcomings, ranging from developing innovative inter-modular connections [14,15,16] as well as precast concrete core wall systems [17,18,19] to various module layouts [20,21]. Therefore, a secondary system, such as a core wall system, is essential in mid- to high-rise MSBs [22]. Chua et al. [18] and Yee [23] investigated the lateral performance of a high-rise MSB with a central concrete core. Bi et al. [19] also studied the lateral performance of a multi-story MSB with two concrete cores under wind loads, as a case study located in China.

From the MSB collapse perspective, several studies have been undertaken. Luo et al. (2019) studied steel-frame progressive collapse by assuming that all connections are rigid [24]. However, actual connection behaviors would also be necessary to understand the extent of damage in the aftermath of a blast or fire events. Alembagheri et al. [25,26] and Sharafi et al. [27] investigated the collapse capacity of rigid modular buildings in the macro model context, in which the connections were modelled by translational axial and shear nonlinear springs. In addition, their early work on the progressive collapse of a flexible six-story modular steel frame revealed that the global column buckling dominates the progressive collapse response of the building [28]. The behavior of semi-rigid joints is one of the issues that should be addressed in simplified numerical methods [29,30]. Thai et al. [31] performed a parametric study on the progressive collapse of a 12-story braced MSB. The results showed that the robustness of modular buildings is significantly increased by a bracing system, as it decreases the effective length of columns. Likewise, it was indicated that the MSB‘s ability to withstand collapse can be improved by up to 50% when wall panels are tied in each module [32].

As using alternative lateral-force-resisting systems in mid- to high-rise MSBs to meet drift requirements is unavoidable, a comprehensive study on their role in collapse scenarios is essential. In the current literature, this is mostly performed based on the progressive collapse of modular steel buildings without considering the contribution of the secondary systems such as shear wall (core) or bracing systems. This paper studies the dynamic response of corner-supported modular steel buildings incorporating a core wall system when subjected to progressive collapse scenarios, with a focus on recognizing the role of secondary systems. The primary aim of the current study is to provide insight into the impact of these systems on load redistribution and the structural robustness of the building during the loss of a corner module. For this purpose, nine modular building cases are considered, with four, eight, and twelve stories and three different locations on the plan for a core wall system. The inter-module connections are modelled using the macro-model-based finite element method. A guideline based on the alternative path method is used to study the collapse response of modular buildings, and the corner modules are considered a missing vertical-load-bearing component. Then, a parametric analysis is conducted to examine the effect of inter-module connection stiffness, intra-module joint rigidity, and plastic hinges on the MSBs’ robustness.

2. Gravity-Induced Progressive Collapse

All buildings contain critical components that, if lost under extreme conditions such as blast, might cause the entire structure to collapse. The alternative load path (ALP) method has been developed to address this issue. The ALP method is a threat-independent method, in which a critical component of a building is removed, and then the ability of the building to survive the loss scenario is investigated [33,34,35,36,37,38]. If the other components are capable of resisting the redistributed loads, the building is deemed to be robust; otherwise, it may result in a cascading failure and progressive collapse unless an essential element design or segmentation is introduced [39]. The probable ALPs due to corner module loss are shown in Figure 2.

3. Simulated Modular Building

A typical plan shown in Figure 3a is chosen to study the robustness of modular buildings with cores. There are 15 modules in three rows, with the middle module considered a corridor. In this study, the shear walls are in three zones: zone (1), zone (2), and zone (3). There are two types of modules: the first has a 6 m length, 3 m width, 3 m height, and a total mass of 20 tons; the other has a 3 m length, 3 m width, 3 m height, and a total mass of 10 tons. It should be noted that the mass of each module is calculated by adopting dead loads plus 25% of live loads specified by GSA guidelines [37]. The horizontal and vertical clearances between neighboring modules are 0.2 m, as depicted in Figure 3b. Three building heights are considered in this study.

All buildings were designed in accordance with the requirements of AS1170 and AS4100 [40]. The design resulted in SHS 100 × 100 × 9 for columns, 180 UB for floor beams, and 150 UB for ceiling beams. The steel material for columns was assumed as a tri-linear elastoplastic [41] model with an initial yielding stress of 350 MPa, and an ultimate strength of 490 MPa at a plastic strain of 0.02. For beams, however, steel is treated as an elastic material because plastic hinges are regarded at their ends. Furthermore, a density of 7800 kg/m³, Young’s modulus of 200 GPa, and Poisson’s ratio of 0.25 were considered for steel material. A thickness of 250 mm was adopted for the concrete core wall with elastic material behavior, and the concrete density, Young’s modulus, and Poisson’s ratio were, respectively, assumed to be 2400 kg/m³, 30 GPa, and 0.3. The finite element (FE) mesh discretization developed in the Abaqus program is shown in Figure 4. All beams and columns were modelled using two-node linear beam in space (B31) elements and the core wall was simulated using four-node doubly curved thin shell elements with reduced integration, hourglass control, and finite membrane strain (S4R). A tie constraint was adopted to link shell nodes to the adjacent beam nodes. A clamped restraint was adopted to fix column nodes to the ground.

This study had two types of connections: inter-module and intra-module connections. Adjacent modules were connected to each other via inter-module connections including 6M24 class 10.9 bolts, simplified with two horizontal connectors (HCx, HCz) and one vertical connector (VC), as depicted in Figure 4.

A Cartesian connector element, available as a built-in connection in Abaqus, was defined with the force-displacement relations proposed in [6], as shown in Figure 5. The Cartesian connection type connects two nodes where the response in three local connection directions is defined. It was assumed that vertical and horizontal connections were eliminated from the model after reaching their failure points as listed in

Table 1. Failure limits of inter-module connections.

Connection	Axial Failure Force	Shear Failure Force
Vertical (VC)	1700 KN	2000 KN
Horizontal (HC)	1200 KN	2000 KN

. Note that the rotational stiffness of inter-module connections was neglected based on previous research [25] because the load-bearing mechanism in MSBs is primarily governed by shear transfer.

Intra-module connections, referred to as beam-column connections within a module, were assumed to be rigid. However, concentrated plastic hinges were assigned to both ends of the beam element to consider the effects of local buckling. The hinges were modelled as rotational springs with a symmetric quadrilinear moment rotation relationship for positive and negative rotations. The capacity of these hinges was based on the theoretical moment-curvature (M-ɵ) relationships of the cross-sections using the ASCE41-17 provision [42].

The nonlinear dynamic analysis was performed using the implicit dynamic solver in Abaqus [43]. At first, an eigenvalue buckling analysis was performed, and buckling mode shapes were determined and imposed as an initial state to the model. This analysis that predicts the collapse mechanism was performed using the subspace iteration method to extract the buckling modes. This analysis provides buckling eigenvalues and buckling modes, which are used in the post-buckling analysis by determining the initial geometric imperfection. Next, the gravity loads, including modules’ dead and live loads, were applied gradually through the quasi-static step. The corner module was eventually removed from the model within 0.0001 s, and the dynamic response of the remaining model was monitored until it reached an equilibrium state.

Due to the limited availability of experimental tests specifically addressing the dynamic collapse of modular buildings, the validation of the numerical modelling procedure presented in this study relies on the utilization of previously published numerical models. To achieve this, we developed a steel modular building, which had been previously examined by Luo et al. [24], employing the methodology described in this paper. As shown in Figure 6, there is an acceptable agreement between the developed model that had been verified in our previous research [25] and the original model of Luo et al. [24].

4. Response under Module Loss Scenario

The dynamic responses of different cases, reported in

Table 2. Overview of collapse analysis cases and corresponding building configurations.

Case No.	Story No.	Building Frame Types
1	4	Bare frame (4 s)
2	8	Bare frame (8 s)
3	8	Bare frame + shear core at zone 1 (8 sw1)
4	8	Bare frame + shear core at zone 2 (8 sw2)
5	8	Bare frame + shear core at zone 3 (8 sw3)
6	12	Bare frame (12 s)
7	12	Bare frame + shear core at zone 1 (12 sw1)
8	12	Bare frame + shear core at zone 2 (12 sw2)
9	12	Bare frame + shear core at zone 3 (12 sw3)

, are presented and compared in this section, to evaluate the core wall’s role in the building’s robustness. To that end, the response history of corner roof displacements, the maximum stress of the shear wall, plastic hinges distribution, and internal forces in critical members in Figure 7 are reported. In addition, maximum dynamic to static response ratio (DSR = r_max,dyn/r_static), the dynamic increase factor (DIF = r_max,dyn/r_final), demand-capacity ratio (DCR), and some design implications are addressed in this section. Note that r_static is considered as a force at the end of the quasi-static gravity step, and r_max,dyn is the maximum element force after the corner modules’ removal.

4.1. Four-Story Structure

In the first case, the structure was robust after the corner module’s sudden removal because no column buckling occurred. The time history of lateral and vertical displacements of the roof corner, labelled A, is shown in Figure 8a. The maximum vertical and lateral displacements are about 18 cm and 7 cm, respectively.

In this case, Column#1 carried the highest load, about 560 kN, with a corresponding DSR = 3.25 and DIF = 1.15. The critical buckling load of columns was about 860 kN. In the corner module removal scenario, Column#1 has a key role in the stability of the entire building because, as shown in Figure 9, substantial load distribution occurs. The time history of axial load in beams shows that they are in compression, and Beam#3 has the highest value. Additionally, it was observed that, despite employing plastic hinges in the finite element model of the MSBs, no plastic hinges formed. This aligns with the findings of the previous study [28], which discussed that in the scenario of corner module removal in low-rise MSBs, the formation of plastic hinges can be disregarded as global column buckling is the dominant behavior.

After removing the corner module, no inter-module connection failure is observed (see Figure 10). As mentioned above, the failure values for inter-module connections are summarized in Table 1. The compressive force in VC#3 is similar to that in Column#1. HC#1 carries the highest shear force, corresponding to DCR = 160/2000 = 0.08, indicating that the redistributed load flows mainly in the X direction rather than Y.

4.2. Eight-Story Structure

In the second case, the eight-story MSB was inadequate to maintain its robustness and collapsed. The collapse sequence of case no. 2 is shown in Figure 11. In the previous case, it was shown that Column#1 is very prone to buckling because of substantial load redistribution. Accordingly, in case no. 2, the collapse was triggered by the buckling of Column#1 located close to the removed module and was accompanied by the buckling of Column#2 after a while. As depicted in Figure 12b, after a second of removing the corner modules, the maximum vertical and lateral displacements are about 49 cm and 20 cm, respectively. Then, due to successive buckling of columns in the front row, the building tends to overturn around the x-axis.

The initial plastic hinges, which play an essential role in the robustness of the buildings, are shown in Figure 12a. The hinges are formed in the ceiling beams of the modules right close to the removed one because the rotation of these beams resulting from unsymmetrical deflection is more pronounced than those above the removed modules. After removing the corner modules, the main part of the gravity load tolerated by the removed columns is transferred to Column#1. Moreover, the plastic hinge formation in the beams connected to the Column#1 increases the effective length factor, K, so that exacerbates the situation of the partially buckled column. The K-factors determining the effective length of columns in corner-supported steel modular buildings are investigated by Farajian et al. [44,45,46], with consideration for both sway and non-sway frames.

To explore the impact of the core wall on the collapse behavior of the MSB against corner module removal, the core wall system was installed in three zones (case no. 3–5), and their responses were compared with that of the bare frame (case no. 2).

It was shown that the core wall located at the center of the building (case no. 3) arrested the collapse of the building under the corner module loss scenario. The lateral displacements in cases no. 2 and 3 were approximately equal when the building was in a stable state. Despite the buckling of Column#1 and the significant vertical displacement of about 520 mm depicted in Figure 13a, the building maintains its robustness under dynamic loads. The time history of lateral and vertical displacements of the roof corner is shown in Figure 13b. The distribution of the first plastic hinges is the same as in case no. 2.

The time history of axial forces in several beams and columns in case no. 2 and case no. 3 is compared in Figure 14. In case no. 2, the DSR of Column#1 and Column#2 are 2.87 kN and 3.34 kN, respectively. The highest load redistribution in Column#2 for case no. 3 and case no. 2 is about 867 kN and 993 kN, respectively. In case no. 3, where the core wall was cantered, beams experienced more fluctuation in their axial load immediately after module removal. The axial forces in Beam#1 and Beam#2 were the opposite of each other. As Beam#2 was linked to HC#1, it had to bear tensile load, while Beam#1 suffered compressive load. In case no. 3, after the buckling of Column#1 occurred, twice as many loads were redistributed in these beams compared to case no. 2. This showed that the concrete core led to better load sharing through the beams in the vicinity of removed modules. Thus, the less axial load is redistributed to Column#2, as shown in Figure 14a. The maximum redistributed load in Column#2 is reduced by 12% compared with case no. 2. In both cases no. 2 and no. 3, Column#1 carried a load of about 1100 kN, which was higher than the buckling capacity.

Regarding inter-module connections, as shown in Figure 15, when Column#2 buckles in case no. 2, the load-bearing capacity of HC#2 tends to zero. Despite the high deformation of the building, no inter-module connection failure was observed because the dynamic redistribution in inter-module connections, especially HC#1, which was considered the most critical connection, was much lower than failure values. In both cases, The HC#1 maximum shear force was about 300 kN with a DCR of 300/2000 = 0.15. The highest force redistribution through the inter-module connections was the axial force in VC#3, which was approximately equal to the buckling load in Column#1.

In the following, the outcomes of cases no. 4 and 5 are compared with case no. 3. The results of case no. 4 and case no. 5 in terms of displacements, axial forces, and inter-module connection forces are very close to each other. However, the axial force in Column#2 of case no. 5, which was attached to shear walls, was reduced by about 50% compared with case no. 4. Contrary to case no. 3, when the core wall was directly connected to the removal zone through inter-module connections, the shear wall reduced horizontal displacements, as observed in Figure 16b. The lateral Z displacement in case no. 4 is about 50% less than in case no. 3.

Figure 17 shows that the time history of axial forces in the columns is virtually the same; however, the redistributed load for Column#2 in case no. 4 is a bit higher than in case no. 3. Although the shear core did not reduce the redistributed load in Column#2, the vertical displacement that the building undergoes is a little less than in case no. 3. This is because of decreasing the horizontal displacements provided by the stiff core wall, connected directly to the modules above the missing modules. In fact, as illustrated, the horizontal displacements can accelerate the column’s buckling, and restraining them helps the building stay robust.

It can be seen from Figure 18 that the tensile envelope contours mostly appeared on the walls that are parallel to the X-direction. The maximum tensile stress is about 2.7 MPa, which is lower than the ultimate tensile strength of concrete. The majority of the envelope contour is located in the walls of the first four stories of the MSB. Moreover, the bracing effect on these walls can be seen. Therefore, walls with less thickness or fewer braces should be considered for upper levels. On the other hand, the maximum compressive stress is just limited to the vicinity of columns.

4.3. Twelve-Story Structure

In the sixth case, the 12-story MSB, considered as a high-rise building, the structure collapses two seconds after removing the corner module. The collapse sequence of this case is shown in Figure 19. Analogous to the eight-story MSB, the collapse initiates with the buckling of Columns#1 and #2 but continues with the buckling of the rest of the columns at the first and second story simultaneously. As seen in Figure 19, 0.5 s after removing the corner modules, the maximum vertical and lateral displacements are about 500 mm and 110 mm, respectively. Then, due to further column buckling at the front row, the building tends to overturn about the x-axis.

As shown in Figure 20a, the first set of plastic hinges formed in 0.6 s after the sudden removal of corner modules at the first elevation. Unlike the eight-story MSB, no hinges are observed at the first story immediately after module removal. In addition, the hinges are extended to the center modules at the first elevation, demonstrating that more beams participated in load bearing.

In the following, the effect of the shear wall (case no. 7–9) on the collapse behavior of a 12-story MSB is studied. It is shown that the core wall system located at the center of the building (case no. 7) does not prevent the building from collapsing. The time history of lateral and vertical displacements of the roof corner is shown in Figure 21b. The maximum lateral displacements at the beginning are about zero, eventually reaching about 500 mm. Similar to case no. 6 (bare frame), the slope of the vertical displacement is constant because columns at the first and second floor buckle successively. The distribution of the first set of plastic hinges is the same as in case no. 6.

The time history of axial forces in several beams and columns in case no. 6 and case no. 7 is compared in Figure 22. Like the previous cases, a part of the loads is redistributed by an additional load path, owing to the core wall, in Beam#1 and #2. In both cases, the redistributed load in Column#2 and Column#1 is more than the critical buckling load, which is about 860 kN, as shown in Figure 22a. In case no. 6, Column#1 tolerates the highest load of about 1051 kN with the corresponding DSR = 1071/465 = 2.3.

As shown in Figure 23, there is still a high level of strength reserve capacity in the inter-module connections. VC#3 suffers the maximum compressive force with corresponding DCR = 1071/1700 = 0.63. In case no. 7, the shear force in HC#1 increases to 500 kN due to the buckling of a column close to Column#3 on the second floor.

As the results of case no. 8 and case no. 9 were approximately identical, case no. 8 is compared with case no. 7. The results, depicted in Figure 24, in terms of displacements, axial forces, and inter-module connection forces are very close. In these cases, neither alternative loading paths due to the shear core nor limiting the lateral displacements help the building’s robustness.

In terms of the concrete core’s stress pattern, like the 8-story MSB, most envelope contour is located at the first half-height of the 12-story MSB. The maximum compressive stress is localized in the vicinity of columns, and the maximum tensile stress is found in the floor and ceiling beams. The maximum tensile stress is about 3.44 MPa, near the ultimate tensile strength of concrete, and the maximum compressive stress is about 12 MPa, which is much lower than its capacity.

In summary, the final state of each case against the corner module loss scenario is presented in

Table 3. Summary of building’s stability results.

Cases	Stability State	Main Reasons
(1) 4 s	stable	No column buckling
(2) 8 s	collapsed	Buckling C#1 and C#2
(3) 8 sw1	stable	Less redistributed load in C#2
(4) 8 sw2	stable	Less lateral displacements
(5) 8 sw3	stable	C#2 constrained with concrete core
(6) 12 s	collapsed	Several columns buckling
(7) 12 sw1	collapsed	Several columns buckling
(8) 12 sw2	collapsed	Several columns buckling
(9) 12 sw3	collapsed	Several columns buckling

. In cases no. 6–9, because many columns start buckling at the same time, the core wall cannot safeguard the building’s stability via improving load sharing or reducing lateral displacements.

5. Parametric Investigation

A parametric study is conducted to explore the effects of the inter-module connection’s behavior and the rigidity of intra-module connections on the overall response of the building, and the results are presented. The former section indicated that the ultimate capacity of the initial design of inter-module connections is much higher than the corresponding demand. Therefore, the effects of the weakened connections in terms of stiffness and strength on the overall response of the building are studied. In addition, the effect of the rigidity of the beam-to-column connection on the collapse capacity of the building, which is yet to be identified, is investigated.

5.1. Inter-Module Connection: Stiffness and Strength Change

The horizontal and vertical springs are softened, as described in Figure 25 [25], to study the impact of stiffness and strength reduction on the robustness of the MSB. In this process, the force at each displacement is decreased, whereas the corresponding displacement remains constant. The effects of the HCs and VCs are studied separately.

The HCs’ and VCs’ springs are softened by 50% and 90%; then, the results are compared to the baseline models (case no. 2 and case no. 6). Starting with the eight-story MSB, no inter-module connection failure is observed when the horizontal or vertical springs are softened by 50%. Therefore, as depicted in Figure 26, the vertical displacements and redistributed loads in critical connections are approximately the same as in the base model. On the other hand, the situation for 90% softening is different because of the failure of several inter-module connections. In the model where the vertical connections are 90% softened, the collapse will happen sooner because of the failure of VC#3. As shown in Figure 27a, its value reaches zero 0.5 s after corner module removal. As shown in Figure 28a, the failure propagates to the upper stories and side modules. The early-stage failures that take 2 s are limited to the fourth row of modules.

When the horizontal connections are 90% softened, failure of the inter-module connections leads to a delay in the collapse of the whole building for 12 s. The failure sequence of the inter-module connections is shown in Figure 28. After the buckling of Column#1, bridging of the remaining load to Column#2 is postponed because of the failure of several HC connections, as shown in Figure 28b; then, two columns in the second story, shown in this figure, start to buckle. After 12 s, the collapse will happen because of the buckling of Column#2 and the numerous columns of the first and second stories. This shows that the building had a higher resistance capacity against buckling and collapse. In addition, unlike the past study where the modules were solid, the successive failure of inter-module connections is not started by losing HC#1.

To recap, softening horizontal connections can convey the redistributed loads from critical columns to other members. Moreover, identifying the first inter-module connection prone to failure in critical situations helps the designers prevent damage propagation to other modules.

In the 12-story MSB, like the 8-story model, no inter-module connection fails when they are softened by 50%. Consequently, as depicted in Figure 29, the vertical displacements and redistributed loads in critical connections also have the same values as the base model. In 90% HCs, softening the premature failure in inter-module connections is not limited to HCs of the first row. Several HCs of the first and second stories, connecting the first-row modules to the corridor modules, failed. Since the base columns must bear more loads in 12-story buildings than in 8-story building, more than just one column starts to buckle immediately after module loss. Thus, the failure of inter-module connections cannot change the situation as it does in the eight-story building. It should be noted that the first inter-module connection that begins to fail for both 90% HCs and 90% VCs is the same as an eight-story building.

5.2. Intra-Module Connection: Pin/Rigid, Plastic Hinges, and Anti-Collapse Remediation

To investigate the impact of plastic hinge formation on the robustness of MSB, the eight-story MSB (case no. 2) is envisaged precluding local buckling. Figure 30 shows that this assumption conservatively ensures the building’s robustness. It should be noted that this assumption for the building with fewer stories, such as in case no. 1 and the one in the former study [23] with less likelihood of plastic hinge formation, can substantially improve computational runtime.

To study the effect of fully pinned intra-module connections on the robustness of the eight-story building, all semi-rigid beam-column connections turn into pinned ones. In the pinned model, since there is less restraint at both ends of columns, the second- and third-story columns become easily distorted after a sudden removal of the corner modules. In Figure 31, at the time frame of 0.5 s after module removal, the first- and second-row columns in the second and third story drift oppositely. This results in the buckling of the base columns and the collapse of the whole building. The maximum horizontal displacement of the building has occurred at the level where a module is removed. In contrast, the maximum displacement in the semi-rigid model happened at the roof level.

In the pinned model, more kinetic energy dissipates due to more building deformation. In Figure 32, the redistributed load in VC#3 and Column#1 decreased considerably from 1042 kN to 710 kN. It can be concluded that enhancing the flexibility of the building is beneficial in terms of column buckling unless it leads to excessive relative displacements in columns of the lower stories.

Figure 33 compares the time history of vertical displacement of the pinned and semi-rigid twelve-story model. Like the eight-story model, the pinned model does not have enough redundancy, losing its robustness due to high deflection and successive buckling of columns.

6. Conclusions

This research was focused on studying the role of the concrete core wall in the robustness of MSBs against immediate corner module removal. The verified numerical finite element macro model of the building is simulated by incorporating material and geometric nonlinearities. The inter-module connections are modelled with nonlinear simplified springs, and the intra-module connections are assumed rigid. The dynamic responses, including lateral displacements, inter-module connection forces, and beams’ and columns’ internal forces, are reported for MSBs with different stories and shear core wall positions. In the end, parametric studies were undertaken, including stiffness softening of inter-module connections and intra-module rigidity. The main findings from this study are summarized as follows:

• Unlike low-rise traditional steel buildings, four-story MSBs are robust against corner module removal. No columns buckle at the base level, and no plastic hinges appear in structural members. In addition, in the corner module removal scenario, inter-module connections showed a significant safety factor in all cases, so no inter-module failure was observed.
• In 12-story MSBs, where many columns collapse simultaneously, the core wall has a minimal impact on robustness because there might not be enough time for load sharing. However, eight-story MSBs benefited from the core wall system and maintained their robustness. The core wall helps the robustness of MSBs in two ways: the first is enhancing load sharing when the core is located at the center (zone 1); the second is reducing lateral displacements, provided that the core wall is directly connected to the modules above the missing module (zone 2 or 3).
• Softening horizontal inter-module connections can worsen or improve the performance of the remaining modules against gravity-induced progressive collapse. In an eight-story modular building, 50% and 90% of softening in HCs accelerated and delayed the progressive collapse, respectively, by increasing flexibility. However, this was accompanied by many connection failures that led to the collapse of the whole building. Therefore, choosing an optimized inter-module connection stiffness is essential for the robustness of MSBs.
• Preventing plastic hinges from forming can be considered as an anti-collapse mechanism. It turned the unstable bare-frame eight-story MSB’s response against the corner module removal scenario into a robust one.
• In the corner module removal scenario, the early set of plastic hinges formed in the ceiling beams of the first elevation (in the X direction), which means structural members in this region have higher participation in carrying redistributed loads.

The progressive collapse of steel modular buildings with a shear core was simulated numerically using the finite element software Abaqus 2021. Load redistribution and progressive collapse responses of modular steel buildings (MSBs) with a typical layout under corner module loss scenario were investigated. Various modular building cases with three different heights and core wall locations were considered. Moreover, a parametric analysis was conducted to assess the impact of inter-module connection stiffness, intra-module connection rigidity, and plastic hinges on the robustness of the MSB under gravity loading.

The research can be expanded to include other types of layouts and configurations such as different module sizes, geometries, material properties, and collapse scenarios. Furthermore, the assumed instantaneous removal time, representing a blast scenario, can be adjusted to a longer duration, which is more representative of a fire scenario.