Numerical and Theoretical Studies on Axial Compression Performance of Modular Steel Tubular Columns Grouped with Shear-Key Connectors

Abstract

Shear-keyed inter-modular connections (IMCs) are integral components of high-rise modular steel structures (MSSs), providing robust interconnectivity to support grouped tubular columns across modules, thereby introducing column discontinuities and distinctive structural behavior. This study conducted a comprehensive numerical assessment and theoretical analysis of the axial compression behavior of grouped tubular columns based on a validated finite element model (FEM), which captured the member-to-structural level behavior of steel hollow section (SHS) columns and accommodated geometric imperfections. An FEM was initially developed and validated using 28 axial compression tests documented in the literature, comprising 15 tests on cold-formed and 13 on hot-rolled steel hollow section (SHS) columns. The primary parameters explored in tests included material properties (stainless/carbon), processing methods (cold-formed/hot-rolled), cross-section sizes (D/B), cross-sectional or member slenderness ratios (D/t_c, B/t_c, or L_c/r), and the number of columns (1, 7, and 11). A comprehensive parametric numerical study involving 103 grouped tubular column FEMs then investigated the influence of initial imperfection, shear-key height (L_t), thickness (t_t), steel tube length (D), width (B), thickness (t_c), and height (L_c) alongside the effects of space between tube and key, and the gap between tubes. The results indicated that the load-shortening behavior of the grouped columns consists of linear elastic, inelastic, and recession stages. The failure modes observed primarily displayed an S-shaped pair of inward and outward local buckling on the outer sides and double S-shaped local buckling on the interior sides. The buckling arose near the shear key or at 1/4 or 1/2 of the column height. None of the considered models experienced global buckling. Increasing t_t, L_t, t_c, D, or B enhances strength and stiffness, while L_c or L_c/r linearly affects stiffness and ductility. The columns’ nominal axial strength was reduced because of the shear keys, which decreased compression yielding and caused localized elastic buckling. Subsequently, the theoretical analysis revealed that the design codes do not capture this behavior, and thus, their capacity estimate yields inaccurate findings. This discrepancy renders existing code prediction equations, including those from Indian (IS800), New Zealand (NZS400), European (EC3:1-1), Canadian (CSA S16), American (AISC360-16), and Chinese (GB50017) standards, as well as the model proposed by Li et al., non-conservative. To assure conservative results, the paper recommended modification of existing standards and proposed prediction equations based on a fourth-order differential equation that describes the actual behavior of modular steel columns grouped with shear keys. The proposed design approach accurately predicted the axial compression capacity of modular steel-grouped columns, proving conservative yet effective. This provides valuable data that could transform design and construction techniques for MSSs, extending to various column and IMC forms through adaptable design parameters. This enhancement in structural performance and safety significantly contributes to the advancement of modular construction practices.

1. Introduction

Modular steel structures (MSSs) are gaining popularity in urban regions, where construction is complex due to space and ecological requirements [1]. Many experts consider MSSs as a game-changing technology for the future of the global construction sector [2], owing to its integrated nature wherein the entire building is composed of factory-based, ready-made room-sized volumetric modules that are transported and assembled onsite [3,4]. The importance of MSSs has increased significantly due to their cost- and time-effectiveness [2], superior quality [5], improved safety [6], and reduced environmental impacts [7]. Figure 1 illustrates how MSSs differ from traditional steel structures (TSSs), particularly in terms of the arrangement and discontinuity of columns and various structural elements [8]. Steel modules are increasingly preferred over modules made of timber, precast, or composite materials because of their superior strength, ductility, robustness, rigidity, durability, lightness, and ease of operation [9]. Modules can be categorized as corner- or continuous-supported, depending on their load-bearing components. Continuous-supported modules have light steel walls continuously supported at 300 to 600 mm to withstand gravity loads up to three stories [10]. Columns at corners of corner-supported modules resist loads, whereas walls function as partitioning members, benefiting from a clear load transfer path and space flexibility [11,12,13]. Thus, they are widely used in engineering projects due to their ability to extend to high-rise structures with an efficient lateral stabilization system [2,14]. Steel hollow section (SHS) columns, known for their excellent torsional, compressive, and bending resistance, are employed to manage loads at corners [15,16,17]. Therefore, a thorough study of the axial behavior of SHS columns is necessary for future development of MSSs. MSSs combine discrete modular units whose structural characteristics differ from TSSs. Their structural behavior is primarily determined by the modules and their deformation coordination [18]. Unlike TSSs, the structural integrity of MSSs hinges on the effectiveness of inter-modular connections (IMCs), which link modules both vertically and horizontally [19]. Consequently, to attain structural cohesion between SHS tubes in MSSs, various types of IMCs, such as welded [20], bolted [19], and prestressed [21,22] IMCs, are employed at the corners of modules. Nonetheless, technical challenges associated with the unique structural limitations of MSSs, including issues of instability, robustness, and the intricate nature of interior connection fastening, rank as some of the most critical concerns. Addressing these issues is vital for assuring the safety and quality of MSSs [23,24]. A variety of joint configurations, especially those between SHS columns, have been suggested to tackle these issues. This has led to the accumulation of several systematic studies, which have been consolidated and discussed in recently published review articles focused on IMCs [4,19,25,26,27,28,29,30,31].

MSS technology has emerged as an influential force in current buildings, providing unprecedented efficiency and flexibility to fulfill modern design demands. However, assuring the safety and structural integrity of MSS components, especially steel columns, is critical. Imperfections cause the catastrophic compression failure of steel columns without warning, and this fact applies to all columns along with other structures that are imperfection-sensitive; thus, imperfections in column loads and the impact of member–system interactions are critical considerations, as highlighted in recent studies [32,33]. In [32], the authors emphasize accurately identifying compressive column loads in steel buildings. The study highlights their function in supporting repairs and determining frame safety in the context of uncertainties such as dead loads. Similarly, another study [33] underlines the need to evaluate member–system interaction effects in structural behavior evaluation, demonstrating that isolated analysis of structural members might result in performance differences when compared to analyzing the entire structural system. Building on these findings, this study aims to develop MSS technology by thoroughly investigating the axial behavior of SHS columns, focusing on shear-keyed IMCs. The present research attempts to improve the integrity, stability, and overall performance of MSS constructions by employing unique joint configurations and advanced analytical methodologies such as numerical analysis, theoretical modeling, and experimental validation. It intends to reduce the risk of compression failures in steel columns by resolving imperfections and incorporating insights from current studies, enabling safer, more efficient, and more sustainable MSS construction techniques.

Figure 1. Comparison of design features between TSSs and MSSs [34].

Figure 1. Comparison of design features between TSSs and MSSs [34].

2. Literature Review

A comprehensive summary of the literature review concerning the mechanical behavior of shear-keyed IMCs and SHS members in TSSs and MSSs is presented in Table 1. The literature underscores the robustness and efficiency of column-to-column IMCs as a method for joining modules that vertically interconnect SHS columns at corners. Various forms of shear keys, including solid or hollow box-shaped, threaded-shaped, socket-shaped, and cruciform-shaped, are commonly employed to ensure proper module mounting and eliminate discreteness [31]. The MSS projects in Figure 2a,b and Figure 3a–f also utilized shear-keyed IMCs between SHS columns in corner-supported MSSs, affirming the practicality of shear keys in real projects. Recent studies have delved into shear-keyed IMCs and shear-keyed tubes, including experimental research by Hajimohammadi et al. [35]. Their experiments revealed that increasing the loading angle (from 0° to 45°) reduces the ultimate capacity of shear keys, rendering BS-3580 [36], ASME-B1.1 [37], and ISO/TR-16224 [38] standards inapplicable. Additionally, Chen et al. [39,40] demonstrated the excellent seismic capacity of shear-keyed IMCs, albeit with observed instances of column tearing. Other researchers, such as Khan et al. [41,42,43], Bowron [44], and Pang et al. [45], have noted that shear-keyed IMCs provide semi-rigid connectivity and shear resistance to SHS columns, although high stresses near shear-key zones have been observed. Studies by Dai et al. [46,47] found grouted shear-keyed IMCs contribute rigidly to load resistance. On the other hand, research by Deng et al. [48], Zhang et al. [49], and Ma et al. [50] explored bolted and welded shear-keyed IMCs, highlighting the facilitation of horizontal connectivity and shear capacity. However, the lack of welding within the modules led to rotational movements around the columns. Nadeem et al. [51] introduced an IMC featuring a self-locking shear key, which they observed to offer enhanced initial resistance to both lateral and slip forces [35].

Recent studies have also investigated pre- and post-tensioned MSSs with shear-keyed IMCs. Liew et al. [52,53] and Chen et al. [21] found that shear-keyed IMCs effectively transmit lateral forces, while Sanches et al. [54,55] noted their resistance to lateral force via friction, with thickness as a governing factor. Lacey et al. [56,57] observed that the shear-slip resistance of shear-keyed columns could be enhanced through sandblasting or increasing the contact area. Despite the considerable focus on lateral behavior, the literature review reveals a lack of understanding regarding shear-keyed column structural behavior under pure axial compression. Existing studies have often overlooked buckling resistances and joint rotations, assuming firm welding of shear keys to columns, leading to inadequately conservative designs. Therefore, research into the compression behavior of columns equipped with shear keys is essential, considering their widespread application in MSS projects, as illustrated in Figure 2 and Figure 3.

Modular steel structures that incorporate SHS using shear keys are known for their enhanced structural performance [4]. The compression behavior of various SHS configurations has been thoroughly examined within the context of TSSs. Research findings have indicated that the EC3 effective width equation and the Class 3 slenderness limit for stainless steel SHS columns, both stub and long, tend to be conservative [58]. Similarly, evaluations of square/rectangular hollow cold-formed stub columns against standards like CSA S16-19 [59], AISC360-16 [60], and AISI S100-16 [61] also suggest conservative design approaches [62]. It has been discovered that slenderness limits for mild steel columns across different classes in EC-3 [63] and ANSI/AISC 360-16 [60] are deemed non-conservative for certain steel grades [64]. The direct strength method’s application to cold-formed boxes has been criticized for its non-conservative nature [65]. Investigations into the compressive response of SHS columns under various temperature conditions reveal significant findings, alongside reports marking AISC360 [60], EC3 [63], and GB50017 [66] as conservative for stainless steel [67,68,69,70]. In contrast, evaluations of high-strength steel (HSS) columns indicate instances of both unsafe capacity and overly conservative classification [71,72]. Studies also show that reducing the diameter-to-thickness ratio (D/t_c) can lead to increased capacity, with observations of local buckling in both stub and long columns [73,74]. Additionally, it has been verified that traditional standards often overestimate the compressive strengths of hot-rolled SHS multi-column walls due to a lack of consideration for the unique aspects of MSSs [15,16,17]. Extensive research spans the compressive behavior of stub or long columns made from stainless, mild, hot-rolled, cold-formed, and HSS at varying temperatures in TSSs. Still, their assumptions and conclusions were exclusive to TSSs because the determination of ultimate resistance was predicated on the assumption of column continuity at both or one end without any IMCs, while group columns and neighboring column effects were lacking. Moreover, Xu et al. [75] and Sharafi et al. [76] found that the combined action of neighboring columns and beams has a beneficial effect on the mechanical performance of MSSs [75]. Because of the interfacial behaviors, interior MSSs would have stronger stiffnesses and strengths, load, and varying failure behavior than with only corner connection [77]. Moreover, Ma et al. [50] and Xu et al. [78] observed that the fully fastened group members had a greater capacity than the independent members. Li et al. [79] introduced a Diameter-Adjustable Mandrel for enhancing metal tube bending processes, demonstrating that parameters such as the amount of support blocks and effective diameter improve tube forming quality. Wang et al. [80] offered a Bo-LSTM-based tool for accurately forecasting the cross-sectional profiles of metal tube bending segments. This method outperforms others by utilizing Bayesian optimization for hyper-parameter selection. Yang et al. [81] created a unique displacement-amplified mild steel bar joint damper that consumes three times more energy than standard dampers. Liang et al. [82] reported that a steel–aluminum composite sandwich construction slider reduces mass by 18.9% and energy consumption by 6.1% compared to a traditional steel slider, enabling greener manufacturing while preserving performance. According to Wei et al. [83], concrete-filled steel tubular columns with ultra-high performance concrete increase initial stiffness by 13.7% and hysteretic energy dissipation by 41.2% when a steel plate is substituted. Chen et al. [84] analyzed orthotropic steel deck rib-to-deck double-sided welded joints for fatigue resistance using welding residual stress, which dramatically impacts crack behavior, increasing the effective stress intensity factor and decreasing fatigue life.

The studies effectively addressed various aspects, such as the bending, tensile, shear, fatigue, and seismic behavior of individual shear-keyed IMCs, whether integrated into beams or modular frames. However, within the realm of TSSs and MSSs, the research predominantly focused on the compressive behavior of single tubular columns. Despite this extensive investigation, the literature has thus far overlooked the impact of IMCs, neighboring columns, and the boundary conditions at the ends of modular steel grouped tubular columns under axial compression loads, neglecting the critical examination of axial compression behavior and the buckling and stability analysis in modular grouped configurations (including double, triple, and tetra configurations) linked by shear-keyed IMCs at their ends. Therefore, further studies are warranted to assess the axial compression behavior and stability response of modular tubular columns grouped with shear-key connectors within MSSs while considering the influence of horizontal IMCs in both directions. This is because studies have shown that addressing the mechanical performance of grouped structural components is crucial for understanding the behavior of MSSs. In the context of MSSs, integrating modules often involves using SHS as grouped tubular columns connected by IMCs in vertical and horizontal directions, which can introduce discontinuities and gaps at each floor level in both vertical and horizontal directions. Furthermore, modular columns, such as shear-keyed grouped tubular columns in MSSs, may demonstrate increased slenderness and potential deficiencies in rotational rigidity at column ends caused by insufficient connection stiffness. This highlights the importance of performing stability analyses on MSS columns in order to evaluate their buckling performance and determine critical design aspects. However, existing studies listed in Table 1 offer limited insight into these columns’ behavior, especially in grouped columns when shear keys are inserted at both ends, leading to variations in effective lengths, critical loads, boundary conditions, and ultimate resistance [85]. The suitability of conventional design standards for such shear-keyed grouped columns is also questioned due to their inability to consider the contribution of IMCs in buckling and compressive resistance, underscoring the necessity to analyze the conservatism of traditional steel standards in the context of MSSs. Moreover, column designs developed for conventional or modular buildings that do not account for shear key connectors vertically and the horizontal IMC/connecting plate effect laterally on columns may not be suitable for shear-keyed modular grouped column design directly, emphasizing the importance of investigating their axial compression behavior, particularly for application in high-rise MSSs [2,14]. The existing theoretical approaches fail to consider the varying moments of inertia and rigidities at the ends and mid-height of modular grouped columns. These columns are reinforced with shear-keyed IMCs of varying thicknesses and heights at the ends and are hollow at mid-height with no IMC. This results in different effective length factors, buckling loads, and compressive resistances of shear-keyed grouped tubular columns compared to both individual or standard grouped hollow columns. Consequently, there is a significant need for a comprehensive study to develop theoretical design equations and propose design recommendations for traditional design standards. These should be based on the findings obtained from a detailed parametric study that accounts for the varying rigidities of shear-keyed grouped columns, aiming to offer conservative and accurate designs [86,87].

Consequently, this study addresses the compressive behavior of modular grouped columns, focusing on shear-keyed IMCs, beginning with the development of a nonlinear finite element model (FEM). The model was validated through 28 axial compression tests on cold-rolled and hot-rolled SHS columns drawn from previous studies. This investigation encompasses various factors, including initial imperfections, dimensions of steel tubes (length, width, thickness, and height), dimensions of shear keys (height and thickness), and the influence of tube–key space and tube–tube gap. This study also develops new theoretical buckling models and updated prediction equations aligned with standards such as the Indian (IS800) [88,89], New Zealand (NZS400) [90], European (EC3:1-1) [63], Canadian (CSA S16) [59], American (AISC360-16) [60], and Chinese (GB50017) [66] standards, to assess the ultimate compressive resistances and propose the accurate and conservative design method for modular steel grouped columns. Furthermore, this study extensively explores the compression behavior of modular grouped tubular columns, considering both neighboring column effects, vertical shear-keyed IMCs inserted in columns, and horizontal IMCs connecting columns in both horizontal directions.

Table 1. Summaries of the literature reviews on SHS members and shear-keyed IMC mechanical behavior related to TSSs and MSSs.

Ref.	Authors	Year	Experiments	Topic	Findings
[35]	Hajimohammadi et al.	2022	Yes	IMC	According to the study, VectorBloc’s registration-pin shear-keyed IMC often fails due to thread stripping. It shows thicker lifting plates or coarser threads boost lifting capacity, while loading angles from 0°–45° decrease it.
[39,40]	Chen et al.	2017	Yes	IMC	The tests on the innovative bolted shear-keyed IMC studied static and seismic performance, showing how weld quality, stiffeners, and floor/ceiling beam/column stiffness affect connection performance. Diagonal stiffeners improve lateral bearing and bending stiffness.
[41,42,43]	Khan et al.	2020/2021	No	IMC	The numerical study revealed that corner, middle, and interior shear-keyed IMCs exhibited semi-rigid connectivity, adequate stiffness, lateral capacity, seismic performance, and ductility. However, it also identified high stresses near shear-key zones, resulting in strong-column and weak-beam responses.
[45]	Pang et al.	2016	No	IMC	Installed modules can prevent the inspection of shear-keyed IMCs. Additionally, if the columns are not cast in concrete or waterproofed, corrosion can become a problem.
[46,47]	Dai et al.	2020/2021	Yes	IMC	Grouted shear-keyed IMCs contribute to axial and bending load resistances, acting as semi-rigid connections in non-sway/braced MSSs.
[48]	Deng et al.	2017	No	IMC	Welded shear-keyed IMCs affect ultimate load and end shortening, with the shear-key length being the most critical factor in compressive loads.
[49]	Zhang et al.	2021	Yes	IMC	Bolted shear-keyed IMCs improve MSS strength and stiffness and produce stable flag-shaped hysteretic responses with good self-centering in earthquake scenarios.
[50]	Ma et al.	2021	Yes	IMC	Bolted shear-keyed IMC accomplishes the identical rotation of twin beams; however, the rotational movement of the upper columns is greater than that of the lower columns.
[51]	Nadeem et al.	2021	No	IMC	Self-locking shear-keyed IMCs demonstrated improved initial resistance to lateral and slip stresses, meeting the EC3 and AISC standards for semi-rigid connections and special moment frames in modular construction.
[52,53]	Liew et al.	2018/2019	No	IMC	Pretensioned shear-keyed IMCs also offer effective transmission of lateral forces with one bar.
[21]	Chen et al.	2017	Yes	IMC	After lateral loading, the pre-tensioned shear-keyed IMC frame exhibited self-centering deformation restoration and strength deterioration, mainly due to the loss of modular column bonding or concrete slippage.
[54,55]	Sanches et al.	2018/2019	Yes	IMC	The thickness of the post-tensioned shear-keyed IMC determines its frictional resistance to lateral forces.
[56,57]	Lacey et al.	2019	Yes	IMC	Varying bolt preload and the faying surface slip factor control the slip stress in post-tensioned shear-keyed IMCs. Additionally, preload and sandblasting enhance resistance to slipping and increase load capacity.
[58]	Theofanous et al.	2009	Yes	Column	Stainless steel columns under compression have demonstrated conservative Class 3 slenderness limits and effective width equations.
[62]	Tayyebi et al.	2021	No	Column	Post-production galvanizing reduces residual stress in directly-formed SHS/RHS. Moreover, the effective width and direct strength methods proved conservative according to standards AISC 360-16 [60], CSA S16-19 [59], and AISI S100-16 [61].
[64]	Liu et al.	2022	Yes	Column	According to stub column compression tests on press-braked Q355 and Q460 columns, EC3 [63], ANSI/AISC 360-16 [60], and the direct strength technique have unconservative slenderness limits for classifications between Class 1–3 (Non-slender) and Class 4 (Slender) sections.
[65]	Rahnavard et al.	2021	No	Column	The compression investigation revealed that the Effective Width and Direct Strength methods were not conservative for cold-formed, built-up sections with connecting plates and a single row of fasteners.
[70]	Liu et al.	2003	Yes	Column	A reliability analysis recommends that fixed-end cold-formed stainless steel SHS columns be designed according to the Australian/New Zealand Standard, which proves marginally more reliable than the American/European requirements.
[67,68,69]	Yan et al.	2021/2022	Yes	Column	Low-temperature compression studies on stainless steel stub tubular columns demonstrate that while strength increases, ductility decreases. Additionally, the prediction formulas from AISC360 [60], EC3 [63], and GB50017 [66] codes are found to be conservative.
[91]	Huang et al.	2021	No	Column	Existing design requirements can predict strengths, but the Direct Strength technique and the European Code are the most accurate and conservative for modeling the compression behavior of cold-formed stainless steel columns at extreme temperatures (24–960 °C).
[71]	Li et al.	2022	Yes	Column	Imperfections and residual stresses had a lesser impact on the compression of 800 MPa HSS welded box-section columns. GB50017-2017 [66] overestimated the local buckling load, while AISC360-16 [60] overestimated, and both GB50017-2017 [66] and EC3 [63] underestimated the ultimate load.
[72]	Wang et al.	2017	Yes	Column	HSS sections met ductility criteria during compression testing; however, higher-strength materials may not achieve satisfactory ultimate-to-yield strain ratios. Furthermore, the current Class 3 EC3 [63] slenderness limitations for internal elements under compression and the Class 4 effective width formula were called into question.
[73]	Huan et al.	2013	Yes	Column	The compression investigation found that steel-bar stiffeners delay local buckling, increase load-bearing capacity, and reduce ductility in square, thin-walled CFST columns. Additionally, while higher cross-section area ratios decrease deformation capacity, using steel bars as stiffeners and spot welding can reinforce columns cost-effectively.
[92]	Guo et al.	2007	No	Column	This compression study examines the effects of depth-to-thickness ratios on stub composite columns and proposes a novel equation for steel area computation and buckling bearing capacity. It was found that concrete-filled tubes bear loads more effectively than hollow steel tubes.
[74]	Key et al.	1998	Yes	Column	Compressive tests on cold-formed SHS columns indicated higher yield strength, reduced ductility, and outer tensile and inner compressive residual stresses. While these tests verified AISI’s criteria for slenderness limit, post-ultimate ductility, and unloading behavior, they did not confirm the predicted ultimate loads.
[15,16,17]	Khan et al.	2022	Yes	Walls	Global and local buckling, particularly in mid-column, was observed in MSS compression studies on planar and C-shaped SHS walls, where sidewalls restrained corner columns. Regarding safety and accuracy in predicting ultimate resistance, GB50017 [66] was the safest, EC3:1-1 [63] was the least secure, and AISC360 [60] was the most accurate.
[75]	Xu et al.	2020	Yes	Beams/IMC	The mechanical behavior of laminated unequal channel beams in MSSs was examined, revealing that interfacial connections dramatically affected flexural failure modes and significantly increased loading capacities and stiffness.
[76]	Sharafi et al.	2018	Yes	IMC	Dynamic analysis under intense loads revealed that integral interlocking connections among modules simplify building processes, reduce force requirements, and enhance the integrity and stability of multi-story MSSs.
[77]	Choi et al.	2016	No	IMC	Grouped components and connection behavior vary in lateral stiffness and strength, which results in different load-carrying mechanisms in 3- and 5-story MSSs compared to TSSs. The assumption that components are entirely composite and that unit-module connections are fixed can lead to overestimations.
[50]	Ma et al.	2021	Yes	Beams/IMC	Bending experiments demonstrated that fully bolted shear-keyed IMCs enhance stability and seismic resistance by integrating both top and bottom unit beams and left and right columns.
[78]	Xu et al.	2022	Yes	Beams/IMC	The study tested laminated channel beams in MSSs under lateral loads, finding that larger ceiling beams and bolt connections enhanced bending performance. Additionally, interfacial sliding altered the load distribution and failure modes.
[79]	Li et al.	2023	Yes	Tubes	This study introduces a novel concept of a Diameter-Adjustable Mandrel designed to enhance metal tube bending processes. Accommodating tubes within a specific diameter range improves forming quality and reduces manufacturing complexity.
[80]	Wang et al.	2023	No	Tubes	This paper introduces a novel Bo-LSTM-based approach that effectively forecasts the cross-sectional characteristics of metal tube bending segments. Incorporating Bayesian optimization for hyper-parameter selection, this method surpasses previous approaches in accuracy and efficiency.
[81]	Yang et al.	2023	No	Joints/dampers	This research introduces a unique displacement-amplified mild steel bar joint damper to enhance energy dissipation during small earthquakes. This damper efficiently absorbs and dissipates energy by leveraging the lever principle to amplify node displacements.
[82]	Liang et al.	2023	No	Joints	This research presents a steel–aluminum composite sandwich structure to minimize energy consumption in power presses. By adopting a lightweight design for the slider, this structure achieves an 18.9% reduction in mass and a 6.1% decrease in energy consumption compared to traditional steel sliders.
[83]	Wei et al.	2023	Yes	Columns/Composite	This study utilizes pseudo-dynamic testing to evaluate the seismic performance of concrete-filled steel tubular composite columns reinforced with ultra-high-performance concrete plates. The results indicate that ground motion characteristics significantly influence the seismic response of these structures.
[84]	Chen et al.	2023	No	Slider	A slider featuring a steel–aluminum composite bionic sandwich structure has been developed, achieving an 18.6% reduction in mass and a 6.1% increase in energy efficiency.
[86]	Chen et al.	2019	No	Column/IMC	The research found that shear keys and gusset plates effectively address internal tying problems. However, to adequately account for buckling behavior, it is essential to consider factors such as rotational capacity, shear-key length, and IMC stiffness in horizontal and vertical directions.
[87]	Khan et al.	2023	Yes	Frame/IMC	The study examines the impact of beam-to-column connection stiffness on sway modular interior frames, presenting buckling load models that demonstrate increased accuracy. It shows that considering the stiffness of the IMCs results in more precise buckling load predictions than those assuming pinned IMCs.

Table 1. Summaries of the literature reviews on SHS members and shear-keyed IMC mechanical behavior related to TSSs and MSSs.

Ref.	Authors	Year	Experiments	Topic	Findings
[35]	Hajimohammadi et al.	2022	Yes	IMC	According to the study, VectorBloc’s registration-pin shear-keyed IMC often fails due to thread stripping. It shows thicker lifting plates or coarser threads boost lifting capacity, while loading angles from 0°–45° decrease it.
[39,40]	Chen et al.	2017	Yes	IMC	The tests on the innovative bolted shear-keyed IMC studied static and seismic performance, showing how weld quality, stiffeners, and floor/ceiling beam/column stiffness affect connection performance. Diagonal stiffeners improve lateral bearing and bending stiffness.
[41,42,43]	Khan et al.	2020/2021	No	IMC	The numerical study revealed that corner, middle, and interior shear-keyed IMCs exhibited semi-rigid connectivity, adequate stiffness, lateral capacity, seismic performance, and ductility. However, it also identified high stresses near shear-key zones, resulting in strong-column and weak-beam responses.
[45]	Pang et al.	2016	No	IMC	Installed modules can prevent the inspection of shear-keyed IMCs. Additionally, if the columns are not cast in concrete or waterproofed, corrosion can become a problem.
[46,47]	Dai et al.	2020/2021	Yes	IMC	Grouted shear-keyed IMCs contribute to axial and bending load resistances, acting as semi-rigid connections in non-sway/braced MSSs.
[48]	Deng et al.	2017	No	IMC	Welded shear-keyed IMCs affect ultimate load and end shortening, with the shear-key length being the most critical factor in compressive loads.
[49]	Zhang et al.	2021	Yes	IMC	Bolted shear-keyed IMCs improve MSS strength and stiffness and produce stable flag-shaped hysteretic responses with good self-centering in earthquake scenarios.
[50]	Ma et al.	2021	Yes	IMC	Bolted shear-keyed IMC accomplishes the identical rotation of twin beams; however, the rotational movement of the upper columns is greater than that of the lower columns.
[51]	Nadeem et al.	2021	No	IMC	Self-locking shear-keyed IMCs demonstrated improved initial resistance to lateral and slip stresses, meeting the EC3 and AISC standards for semi-rigid connections and special moment frames in modular construction.
[52,53]	Liew et al.	2018/2019	No	IMC	Pretensioned shear-keyed IMCs also offer effective transmission of lateral forces with one bar.
[21]	Chen et al.	2017	Yes	IMC	After lateral loading, the pre-tensioned shear-keyed IMC frame exhibited self-centering deformation restoration and strength deterioration, mainly due to the loss of modular column bonding or concrete slippage.
[54,55]	Sanches et al.	2018/2019	Yes	IMC	The thickness of the post-tensioned shear-keyed IMC determines its frictional resistance to lateral forces.
[56,57]	Lacey et al.	2019	Yes	IMC	Varying bolt preload and the faying surface slip factor control the slip stress in post-tensioned shear-keyed IMCs. Additionally, preload and sandblasting enhance resistance to slipping and increase load capacity.
[58]	Theofanous et al.	2009	Yes	Column	Stainless steel columns under compression have demonstrated conservative Class 3 slenderness limits and effective width equations.
[62]	Tayyebi et al.	2021	No	Column	Post-production galvanizing reduces residual stress in directly-formed SHS/RHS. Moreover, the effective width and direct strength methods proved conservative according to standards AISC 360-16 [60], CSA S16-19 [59], and AISI S100-16 [61].
[64]	Liu et al.	2022	Yes	Column	According to stub column compression tests on press-braked Q355 and Q460 columns, EC3 [63], ANSI/AISC 360-16 [60], and the direct strength technique have unconservative slenderness limits for classifications between Class 1–3 (Non-slender) and Class 4 (Slender) sections.
[65]	Rahnavard et al.	2021	No	Column	The compression investigation revealed that the Effective Width and Direct Strength methods were not conservative for cold-formed, built-up sections with connecting plates and a single row of fasteners.
[70]	Liu et al.	2003	Yes	Column	A reliability analysis recommends that fixed-end cold-formed stainless steel SHS columns be designed according to the Australian/New Zealand Standard, which proves marginally more reliable than the American/European requirements.
[67,68,69]	Yan et al.	2021/2022	Yes	Column	Low-temperature compression studies on stainless steel stub tubular columns demonstrate that while strength increases, ductility decreases. Additionally, the prediction formulas from AISC360 [60], EC3 [63], and GB50017 [66] codes are found to be conservative.
[91]	Huang et al.	2021	No	Column	Existing design requirements can predict strengths, but the Direct Strength technique and the European Code are the most accurate and conservative for modeling the compression behavior of cold-formed stainless steel columns at extreme temperatures (24–960 °C).
[71]	Li et al.	2022	Yes	Column	Imperfections and residual stresses had a lesser impact on the compression of 800 MPa HSS welded box-section columns. GB50017-2017 [66] overestimated the local buckling load, while AISC360-16 [60] overestimated, and both GB50017-2017 [66] and EC3 [63] underestimated the ultimate load.
[72]	Wang et al.	2017	Yes	Column	HSS sections met ductility criteria during compression testing; however, higher-strength materials may not achieve satisfactory ultimate-to-yield strain ratios. Furthermore, the current Class 3 EC3 [63] slenderness limitations for internal elements under compression and the Class 4 effective width formula were called into question.
[73]	Huan et al.	2013	Yes	Column	The compression investigation found that steel-bar stiffeners delay local buckling, increase load-bearing capacity, and reduce ductility in square, thin-walled CFST columns. Additionally, while higher cross-section area ratios decrease deformation capacity, using steel bars as stiffeners and spot welding can reinforce columns cost-effectively.
[92]	Guo et al.	2007	No	Column	This compression study examines the effects of depth-to-thickness ratios on stub composite columns and proposes a novel equation for steel area computation and buckling bearing capacity. It was found that concrete-filled tubes bear loads more effectively than hollow steel tubes.
[74]	Key et al.	1998	Yes	Column	Compressive tests on cold-formed SHS columns indicated higher yield strength, reduced ductility, and outer tensile and inner compressive residual stresses. While these tests verified AISI’s criteria for slenderness limit, post-ultimate ductility, and unloading behavior, they did not confirm the predicted ultimate loads.
[15,16,17]	Khan et al.	2022	Yes	Walls	Global and local buckling, particularly in mid-column, was observed in MSS compression studies on planar and C-shaped SHS walls, where sidewalls restrained corner columns. Regarding safety and accuracy in predicting ultimate resistance, GB50017 [66] was the safest, EC3:1-1 [63] was the least secure, and AISC360 [60] was the most accurate.
[75]	Xu et al.	2020	Yes	Beams/IMC	The mechanical behavior of laminated unequal channel beams in MSSs was examined, revealing that interfacial connections dramatically affected flexural failure modes and significantly increased loading capacities and stiffness.
[76]	Sharafi et al.	2018	Yes	IMC	Dynamic analysis under intense loads revealed that integral interlocking connections among modules simplify building processes, reduce force requirements, and enhance the integrity and stability of multi-story MSSs.
[77]	Choi et al.	2016	No	IMC	Grouped components and connection behavior vary in lateral stiffness and strength, which results in different load-carrying mechanisms in 3- and 5-story MSSs compared to TSSs. The assumption that components are entirely composite and that unit-module connections are fixed can lead to overestimations.
[50]	Ma et al.	2021	Yes	Beams/IMC	Bending experiments demonstrated that fully bolted shear-keyed IMCs enhance stability and seismic resistance by integrating both top and bottom unit beams and left and right columns.
[78]	Xu et al.	2022	Yes	Beams/IMC	The study tested laminated channel beams in MSSs under lateral loads, finding that larger ceiling beams and bolt connections enhanced bending performance. Additionally, interfacial sliding altered the load distribution and failure modes.
[79]	Li et al.	2023	Yes	Tubes	This study introduces a novel concept of a Diameter-Adjustable Mandrel designed to enhance metal tube bending processes. Accommodating tubes within a specific diameter range improves forming quality and reduces manufacturing complexity.
[80]	Wang et al.	2023	No	Tubes	This paper introduces a novel Bo-LSTM-based approach that effectively forecasts the cross-sectional characteristics of metal tube bending segments. Incorporating Bayesian optimization for hyper-parameter selection, this method surpasses previous approaches in accuracy and efficiency.
[81]	Yang et al.	2023	No	Joints/dampers	This research introduces a unique displacement-amplified mild steel bar joint damper to enhance energy dissipation during small earthquakes. This damper efficiently absorbs and dissipates energy by leveraging the lever principle to amplify node displacements.
[82]	Liang et al.	2023	No	Joints	This research presents a steel–aluminum composite sandwich structure to minimize energy consumption in power presses. By adopting a lightweight design for the slider, this structure achieves an 18.9% reduction in mass and a 6.1% decrease in energy consumption compared to traditional steel sliders.
[83]	Wei et al.	2023	Yes	Columns/Composite	This study utilizes pseudo-dynamic testing to evaluate the seismic performance of concrete-filled steel tubular composite columns reinforced with ultra-high-performance concrete plates. The results indicate that ground motion characteristics significantly influence the seismic response of these structures.
[84]	Chen et al.	2023	No	Slider	A slider featuring a steel–aluminum composite bionic sandwich structure has been developed, achieving an 18.6% reduction in mass and a 6.1% increase in energy efficiency.
[86]	Chen et al.	2019	No	Column/IMC	The research found that shear keys and gusset plates effectively address internal tying problems. However, to adequately account for buckling behavior, it is essential to consider factors such as rotational capacity, shear-key length, and IMC stiffness in horizontal and vertical directions.
[87]	Khan et al.	2023	Yes	Frame/IMC	The study examines the impact of beam-to-column connection stiffness on sway modular interior frames, presenting buckling load models that demonstrate increased accuracy. It shows that considering the stiffness of the IMCs results in more precise buckling load predictions than those assuming pinned IMCs.

Figure 2. Engineering background of MSSs with SHS columns grouped with shear-keyed IMCs [93]. Note: (i) Ma et al. [50], (ii) Zhang et al. [49].

Figure 2. Engineering background of MSSs with SHS columns grouped with shear-keyed IMCs [93]. Note: (i) Ma et al. [50], (ii) Zhang et al. [49].

Figure 3. Description of Tianjin Ziya office building, China, with shear-keyed IMCs [39,40,94].

Figure 3. Description of Tianjin Ziya office building, China, with shear-keyed IMCs [39,40,94].

3. Development of Modular Steel Tubular Columns Grouped with Shear-Keyed IMCs

3.1. Design of Columns

The configuration of grouped columns featuring shear-keyed IMCs, showcased in Figure 1 and Figure 2, draws on the authors’ expertise in designing a five-story, corner-supported MSS office building which complies with the Chinese steel design standard GB 50017-2017 [66]. The real-life application of shear-keyed grouped tubular columns was observed in constructing the five-story corner-supported MSS Haoshi office building in Tangshan, China, as shown in Figure 2a. The authors included similar columns in their Tianjin Ziya office building design, a five-story MSS project in the Jinghai District of Tianjin, China, as depicted in Figure 3a–f [39,40]. The Haoshi office building was built using 68 modular units, each measuring 13.8 × 3.6 × 3.5 and 14.4 × 3.6 × 3.5 m. It can be seen that Tianjin Ziya office building used two blocks consisting of 314 modules. These modules had dimensions of 8.5 × 3.0 × 3.0 and 6.7 × 3.0 × 3.0 m. In addition, each of these hybrid forms of MSSs included two steel-braced moment-resisting frames that served as stairs to limit lateral sway and IMC rotation, hence avoiding flexure buckling and enhancing the buckling strength of columns [86]. In addition, these case studies employed tubular columns with a reduced cross-sectional and member slenderness (D/t_c and L_c/r_c) ratio to improve their stability, prevent flexural buckling, and guarantee a 50-year design life of the MSS against 8-degree seismic forces. In addition to being braced with steel frames, these MSSs also incorporate columns with fixed-end boundary conditions to ensure column safety and prevent further deterioration of the IMC against lateral deflection and instability caused by the P-Δ effect [86]. The primary objective was to conduct thorough parametric and theoretical investigations. Consequently, the focus was predominantly on non-slender hot-rolled column cross-sections. Additionally, the selection of fixed-end boundary conditions was based on the actual building design, as depicted in Figure 3f, corroborated by references [86]. Following relevant research in the literature, column subassembly techniques were employed to determine tube height, with the inflection point set by designing the column height as half the actual height, as depicted in Figure 2b [30]. This research employed box-shaped, grouped shear keys, which were welded to both the upper and lower plates [31]. To mimic real-world scenarios, shear keys were placed within SHS tubes, and connecting plates were left unwelded to allow for rotation. A 1 to 2 mm space was maintained between the shear key and tube to accommodate construction tolerances. Additionally, the spacing between double, triple, and tetra columns was left at 24 mm, as illustrated in Figure 4a–c.

3.2. Column Geometry

The geometric specifications of the tubular columns grouped with shear-keyed IMCs are outlined in Figure 4a–c. Given the study’s focus on assessing the effectiveness of shear keys, various parameters were designed. The standard length (L_c) for tubes was set at 1.5 m, with variations extending down to 1.0 m. Additionally, the cross-section dimensions (D, B, and t_c) of the tube’s length, width, and thickness varied from 200 × 200 × 8 to 200 × 200 × 5, 200 × 200 × 7, 200 × 200 × 9, 200 × 200 × 10, 150 × 150 × 10, 180 × 180 × 10, 220 × 220 × 10, 250 × 250 × 10, 160 × 80 × 8, 200 × 120 × 8, 220 × 140 × 8, and 250 × 180 × 8 mm. Notably, the size of the connecting plate remained consistent and aligned with the dimensions of the actual project’s connection gusset plate: 524 × 484 × 20 mm in length, width, and thickness, respectively.

4. Experimental Studies on SHS Column Compression Behaviors

4.1. Testing Details

Theofanous and Gardner [58,95] conducted a series of 12 axial compression tests on stainless steel stub columns and flexural tests on 12-long square, rectangular, and elliptical SHS tubes. Flat-ended stub columns were compressed using 300-ton Amsler hydraulic testing machine parallel plates, while long tubes were tested on an Instron apparatus with a 600 kN jack with pinned boundaries at Imperial College London’s Structures Laboratory. Their aim was to assess the conservativeness of various parameters such as the EC-3 Class 3 classification limit, effective width formula, and column buckling curves. The study incorporated varying parameters, including tube length (D), width (B), height (L_c), longest diameter (a), shortest diameter (b), and thickness (t_c). All specimens were cold-formed and seam-welded, as outlined in

Table 2. Design details and findings of compressive loading on varying configurations of tubular columns.

Sp. #	D (mm)	B (mm)	tc (mm)		Lc (mm)	Col. (#)	${\mathit{f}}_{\mathit{y},\mathit{w}}$ (MPa)		${\mathit{f}}_{\mathit{u},\mathit{w}}$ (MPa)		${\mathit{E}}_{\mathit{s},\mathit{w}}$ (GPa)		${\mathit{f}}_{\mathit{y},\mathit{C}}$ (MPa)	${\mathit{f}}_{\mathit{u},\mathit{C}}$ (MPa)		${\mathit{E}}_{\mathit{s},\mathit{C}}$ (GPa)	$\mathit{\nu }$	$\mathit{\epsilon }$ (%)	${\mathit{P}}_{\mathit{u},\mathit{T}\mathit{e}\mathit{s}\mathit{t}}$ (kN)	${\mathit{P}}_{\mathit{u},\mathit{F}\mathit{E}}$ (kN)	$\frac{{\mathit{P}}_{\mathit{u},\mathit{T}\mathit{e}\mathit{s}\mathit{t}}}{{\mathit{P}}_{\mathit{u},\mathit{F}\mathit{E}}}$	Col. Failure
S1	60	60	3		240	1S	755		839		209		885	1026		212	0.3	22	615	631	0.97	LB
S2	80	80	4		332	1S	679		773		199		731	959		210	0.3	22	919	920	1.00	LB
S3	80	40	4		238	1R	734		817		199		831	962		213	0.3	22	710	704	1.01	LB
S4	100	100	4		400	1S	586		761		198		811	917		206	0.3	22	1030	1059	0.97	LB
S5	60	60	3		2000	1S	755		839		209		885	1026		212	0.3	22	162	179	0.91	GB + LB
S6	60	60	3		1600	1S	755		839		209		885	1026		212	0.3	22	232	224	1.03	GB + LB
S7	60	60	3		1200	1S	755		839		209		885	1026		212	0.3	22	327	362	0.90	GB + LB
S8	60	60	3		800	1S	755		839		209		885	1026		212	0.3	22	447	471	0.95	GB + LB
S9	80	80	4		1200	1S	679		773		199		731	959		210	0.3	22	672	673	1.00	GB + LB
S10	80	80	4		2000	1S	679		773		199		731	959		210	0.3	22	362	381	0.95	GB + LB
S11	80	40	4		1600	1R	734		817		199		831	962		213	0.3	22	160	167	0.96	GB + LB
S12	80	40	4		1200	1R	734		817		199		831	962		213	0.3	22	237	247	0.96	GB + LB
S13	80	40	4		800	1R	734		817		199		831	962		213	0.3	22	367	360	1.02	GB + LB
Sp. #	ac (mm)	bc (mm)		tc (mm)		Lc (mm)		Col. (#)		$f_y$ (MPa)		$f_u$ (MPa)			$E_s$ (GPa)		$\nu$	$\epsilon$ (%)	$P_{u,Test}$ (kN)	$P_{u,FE}$ (kN)	$\frac{P_{u,Test}}{P_{u,FE}}$	Col. Failure
S14	121	76		2		242		1E		193		380			676		0.3	22	234	225	1.04	LB
S15	121	76		3		242		1E		194		420			578		0.3	22	444	443	1.00	LB
Sp. #	D (mm)	B (mm)		tc (mm)		Lc (mm)		Col. (#)		$f_y$ (MPa)		$f_u$ (MPa)			$E_s$ (GPa)		$\mu or \nu$	$\epsilon$ (%)	$P_{u,Test}$ (kN)	$P_{u,FE}$ (kN)	$\frac{P_{u,Test}}{P_{u,FE}}$	Col. Failure
S16	80	80		3		2815		5S		441		521			206		0.3	26	1287	1254	1.03	GB + LB
S17	80	80		5		2815		5S		403		480			206		0.3	26	1829	1735	1.05	GB + LB
S18	100	80		3		2815		5R		425		506			206		0.3	26	1495	1407	1.06	GB + LB
S19	140	80		4		2815		5R		391		522			206		0.3	26	2222	2101	1.06	GB + LB
S20	140	80		6		2815		5R		359		509			206		0.3	26	2812	2704	1.04	GB + LB
S21	160	80		5		2815		5R		403		480			206		0.3	26	3027	2767	1.09	GB + LB
S22	200	80		10		2815		5R		365		500			206		0.3	26	4805	5105	0.94	GB + LB
S23	100	80		3		2815		11R		425		506			206		0.3	26	3208	3154	1.02	GB + LB
S24	160	80		5		2815		11R		403		480			206		0.3	26	6373	6028	1.06	GB + LB
Sp. #	D = B (mm)	tc/tt (mm)		d = b (mm)		Lc (m)		Lt (mm)		fy (MPa)		fu (MPa)			ES (GPa)		$\mu or \nu$	$\epsilon$ (%)	$P_{u,Test}$ (kN)	$P_{u,FE}$ (kN)	$\frac{P_{u,Test}}{P_{u,FE}}$	Col. Failure
S25	200	8/-		-		1.5		-		380		434			206		0.3	23	2043	2030	1.01	SLB
S26	200	8/10		180		1.5		150		380		434			206		0.3	23	1862	1854	1.00	SLB
S27	200	8/25		180		1.5		250		380		434			206		0.3	23	2104	2099	1.00	SLB
S28	200	8/25		180		1.5		100		380		434			206		0.3	23	2068	2031	1.02	SLB

#: Specimen number; D: diameter or length of the column (mm); B: width of the column (mm); t_c: thickness of the column’s wall (mm); L_c: height of the column (mm); Col. (#): number and shape of columns in the grouped configuration; $f_{y,w}$/$f_{y,c}$: yield strength of the tube’s flat wall and corner regions (MPa); $f_{u,w}$/$f_{u,c}$: ultimate strength of the tube’s flat wall and corner regions (MPa); $E_{s,C}$/$E_{s,w}$: elastic modulus of the flat wall and corner regions of the tube (GPa); $\mu /\nu /\epsilon$: friction coefficient/Poisson’s ratio/plastic strain; $P_{u,Test}$/$P_{u,FE}$: ultimate resistance determined through experimental tests and FEA (kN); a_c/b_c: longest and shortest diameters for elliptical tubes (mm); f_y/f_u/E_S: general material properties including yield strength, ultimate strength, and elastic modulus (MPa for strengths and GPa for modulus); S/R/E: square, rectangular, and elliptical tube columns; LB/GB/SLB: local buckling/global buckling/sinusoidal (S)-shaped local buckling.

. Four different cross-sectional sizes (D × B × t_c), namely 100 × 100 × 4, 80 × 80 × 4, 60 × 60 × 3, and 80 × 40 × 4 mm, were designed for the square (S) and rectangular (R) columns, while elliptical (E) stub columns were characterized by two different diameters (a × b × t_c), specifically 121 × 76 × 2 and 121 × 76 × 3 mm. The height of the columns was determined as four times the square tube’s cross-sectional length and two times that of the rectangular tube or the largest diameter of elliptical tubes, aimed at avoiding the global buckling of stubs. Furthermore, tests conducted on long columns utilized cross-sections (D × B × t_c) of 60 × 60 × 3, 80 × 80 × 4, and 80 × 40 × 4. For the 80 × 40 × 4 specimens, both minor and major axis buckling were considered. Heights were selected as 800, 1200, 1600, and 2000 mm, resulting in non-dimensional slendernesses ranging from 0.57 to 2.02. Table 2 thoroughly lists the columns chosen for the study, including detailed cross-sectional information, material behaviors, and capacities. These columns are marked with identifiers S1 to S15.

Khan et al. [15,16] and Hou et al. [96] conducted seven full-scale tests on planar walls and two on C-shaped walls to examine the compression behavior of multi-column tubular sectioned walls. Wall compression tests were conducted at Tianjin University’s Structural Engineering Laboratory using a 1500-ton compression testing machine. The top and bottom rigid support beams were employed, supported by steel grooves on ceiling and floor beams, to establish pin-ended boundaries, as depicted in Figure 5. The study aimed to compare the compressive behavior of planar walls with that of C-shaped walls, analyzing the impact of the number of columns and module geometry. Each specimen was 3000 mm tall and 2560 mm long. In planar walls, five columns with dimensions of 80 × 80 × 3, 80 × 80 × 5, 100 × 80 × 3, 140 × 80 × 4, 140 × 80 × 6, 160 × 80 × 5, and 200 × 80 × 10 mm were arranged at 600 mm intervals. Conversely, C-shaped walls featured five columns in the front panel and three columns on each sidewall, with dimensions of 100 × 80 × 3 and 160 × 80 × 5 mm, also spaced at 600 mm intervals. The cross-sectional width (B) of the columns remained constant at 80 mm, while the length (D) and thickness (t_c) ranged from 80 to 200 mm and 3 to 10 mm, respectively. Ceiling beams of L80 × 80 × 7 mm, angle supports measuring L70 × 70 × 6 mm, and floor beams of C140 × 80 × 8 mm were welded to the columns. A 35 mm solid cushion block, matching the column cross-section, was welded to the ceiling beam, and a 10 mm block, 20 mm longer, was welded to the floor beam beneath the column. When $D\le$ 100 mm, a 10 mm thick stiffener was welded under each column in the channel floor beam. For $D$ > 100 specimens, two stiffeners were welded. The floor channel beams and stringers were sized at C140 × 80 × 8 mm, with lengths of 2560 mm for planar walls and 1360 mm for C-shaped walls. Table 2 lists the multi-column walls chosen for the investigation, including cross-sectional data, material behaviors, and capacities (S16~S24).

In their research, Khan et al. [97] conducted comprehensive tests on individual steel modular columns fitted with shear keys to evaluate their effectiveness. The specimens were mounted with steel bolts on Tianjin University’s Structural Engineering Laboratory’s 500-ton compression machine. After installing the bottom connecting plate, the column was inserted. The machine’s upper beam was lowered to accommodate the tube’s upper end after attaching the upper shear key. The research aimed to compare the compressive behavior of a standard column (S25) with shear-keyed columns (S26–S28), as listed in Table 2. These columns shared similar dimensions, including a height (L_c) of 1.5 m and a tube size of 200 × 200 × 8 mm (D × B × t_c). The shear keys had dimensions of 180 × 180 mm (d × b) with thicknesses (t_t) ranging from 10 to 25 mm and heights (L_t) from 100 to 250 mm. The connecting plate measured 524 mm in length, 490 mm in width, and 20 mm in thickness.

4.2. Testing Outcomes

The compression test outcomes on stubs and flexural test results on long square, rectangular, and elliptical stainless steel columns showed localized inward/outward buckling in stubs, while longer columns exhibited both global and localized buckling, as detailed in Refs. [58,95]. Further, axial compression tests on planar and C-shaped tubular walls revealed global buckling in the middle/lower sections of columns with smaller/larger cross-sections. Local inward/outward buckling was observed at various lower sections of the columns, absent in larger ones. Sidewall reinforcements were reported to mitigate local buckling in corner columns, as documented by Khan et al. [15,16] and Hou et al. [96]. Additionally, Khan et al. [97] found that compressive tests on individual modular shear-keyed columns displayed a pair of elastic and plastic local inward and outward buckling, forming an S-shaped pattern on their sides. This study incorporates the results from cited works solely to validate the developed FEM, with detailed reporting in subsequent sections.

5. Establishment of a Nonlinear Finite Element Model

The failure modes and load-shortening curves derived from the referenced tests have been instrumental in developing a robust FEM for modular steel, shear-keyed, grouped tubular columns. This model will facilitate the analysis of parametric effects.

5.1. General

A finite element analysis (FEA) was performed using ABAQUS 6.13 [98], a commercially available FE software. ABAQUS/CAE 6.13 was utilized for modeling. Alternately, a linear elastic eigenvalue buckling analysis was performed using the subspace iteration method to determine the buckling modes. The Riks method, a variant of the classical arc-length method, was adopted in the nonlinear analysis to determine the load-shortening and failure mechanism.

5.2. Finite Element Model

The refined FEM accurately replicates the specific cross-sectional geometries, heights, boundary conditions, and loading patterns of each specimen tested for validation. Illustrations of the FEM for both a cold-formed stainless-steel stub and a long column are provided in Figure 6. In addition, the FEM for hot-rolled tubular column walls captures the nuances of various planar and C-shaped wall configurations, as shown in Figure 7. The FEM for hot-rolled individual modular columns, covering both standard and shear-keyed tubular designs, is depicted in Figure 6. These models encompass a range of shapes, including SHS columns, angle columns, floor channel and angle ceiling beams, and connecting plates with holes. They also include the floor chassis and shear-keyed IMCs welded onto connecting plates, enhancing the accuracy of simulations concerning the ultimate strength and failure modes of shear-keyed grouped columns. All models adhere to the design dimensions and material properties listed in Table 2.

5.3. Boundary, Loading, Interactions, and Geometric Imperfections

In line with the experimental protocols, the FEM for stub column simulations constrained all degrees of freedom at the ends of the column cross-sections, allowing for vertical translation to mimic displacement loading and enable vertical shortening through kinematic coupling. Constraints were strategically placed on reference points at the top and bottom cross-sectional ends, where the loading and boundary conditions were applied. For the FEMs addressing flexural buckling in longer tubes, similar constraints were imposed, except for a rotational degree of freedom around the buckling axis, which was left unrestrained to emulate pin-ended boundary conditions. The incorporation of geometric imperfections was based on analyzing buckling shapes observed during tests and comparing load-shortening behaviors as reported in Refs. [58,95]. Initial steps included eigenmode analysis to identify multiple buckling modes, followed by selecting the buckling mode most closely aligned with the observed test failure mode for applying geometric imperfections. Specifically, local imperfections were introduced to stub columns, while both local and global imperfections, the latter through eccentricity, were used to long columns prone to flexural buckling, with imperfection magnitudes set at t/100 for local and L/1500 for global deviations.

For the simulation of hot-rolled tubular walls, movement at the top and bottom was restricted in all directions, with allowances made for vertical movement and rotation at the bottom to simulate column shortening. Displacement-controlled loading was executed through kinematically coupled center-reference points on top and bottom cushion plates, with beams, columns, and angles welded to cushion blocks. Modular floors were simulated with welded floors and stringer beams, employing a “tie constraint” for surface-to-surface contact, which prevented relative movement. This approach was consistent with the application of geometric imperfections, adopting a standard magnitude of L/600 for the FEM. Additionally, for individual modular steel columns, constraints were applied to the top and bottom plates in all directions, with an exception made for vertical movement at the bottom, facilitating shortening under displacement loading applied through coupling constraints at the cross-sections’ midpoint. The interaction between the connecting plates, column, and between the steel tube’s inner face and the shear key’s outer face was defined by “surface-to-surface hard contact” for normal behavior and “finite sliding” for tangential behavior, with a friction coefficient of 0.3. The variability in failure modes, such as 1st for S25 and S26, 3rd for S27, and 4th for S28, was attributed to the buckling position, with local amplitude adjustments of 5/16t_c or L_c/600 yielding results closely aligning with experimental observations.

5.4. Element Types and Mesh Sizes

Table 3. Finite element model details for validated column and wall models in MSSs.

FEM	Constraint	Interaction	Imperfection	FE Used	Elements of FEM (#)	No. of FEM (#)	Mesh/Nodes of the Column (#)	Column Mesh Size (mm)
S1	Fixed	S-NC	tc/100 (L)	S4R	5700	5776	5700/5776	3 × 3 × 3
S2	Fixed	S-NC	tc/100 (L)	S4R	8004	8096	8004/8096	4 × 4 × 4
S3	Fixed	S-NC	tc/100 (L)	S4R	3906	3968	3906/3968	4 × 4 × 4
S4	Fixed	S-NC	tc/100 (L)	S4R	10,712	10,816	10,712/10	4 × 4 × 4
S5	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	53,280	53,360	53,280/53,360	3 × 3 × 3
S6	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	42,720	42,800	42,720/42,800	3 × 3 × 3
S7	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	32,000	32,080	32,000/32,080	3 × 3 × 3
S8	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	21,280	21,360	21,280/21,360	4 × 4 × 4
S9	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	26,400	26,488	26,400/26,488	4 × 4 × 4
S10	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	44,000	44,088	44,000/44,088	4 × 4 × 4
S11	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	24,000	24,060	24,000/24,060	4 × 4 × 4
S12	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	18,000	18,060	18,000/18,060	4 × 4 × 4
S13	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	12,000	12,060	12,000/12,060	4 × 4 × 4
S14	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	19,360	19,520	19,360/19,520	2 × 2 × 2
S15	Pinned	S-NC	tc/100 (L) and Lc/1500 (G)	S4R	8560	8667	8560/8667	3 × 3 × 3
S16	Pinned	S-ST	Lc/600 (L)	C3D8R	49,629	10,3016	1904/3876	Max: 30 × 30 × 3 Min: 30 × 3 × 3
S17	Pinned	S-ST	Lc/600 (L)	C3D8R	61,389	11,6126	4256/6498	Max: 30 × 30 × 5 Min: 30 × 5 × 5
S18	Pinned	S-ST	Lc/600 (L)	C3D8R	35,840	74,902	2576/5244	Max: 30 × 30 × 3 Min: 30 × 3 × 3
S19	Pinned	S-ST	Lc/600 (L)	C3D8R	55,074	11,3768	2576/5244	Max: 30 × 30 × 4 Min: 30 × 4 × 4
S20	Pinned	S-ST	Lc/600 (L)	C3D8R	55074	11,3768	2576/5244	Max: 30 × 30 × 6 Min: 30 × 6 × 6
S21	Pinned	S-ST	Lc/600 (L)	C3D8R	55,733	11,5048	2800/5700	Max: 30 × 30 × 5 Min: 30 × 5 × 5
S22	Pinned	S-ST	Lc/600 (L)	C3D8R	74,404	12,6731	4032/7125	Max: 30 × 30 × 5 Min: 30 × 5 × 5
S23	Pinned	S-ST	Lc/600 (L)	C3D8R	53,100	11,2558	2576/5244	Max: 30 × 30 × 3 Min: 30 × 3 × 3
S24	Pinned	S-ST	Lc/600 (L)	C3D8R	57,148	12,0082	2576/5244	Max: 30 × 30 × 5 Min: 30 × 5 × 5
S25	Fixed	S-SF	5/16tc or Lc/600 (L)	C3D8R	2802	5840	1920/2904	Max: 25 × 25 × 8 Min: 25 × 8 × 8
S26	Fixed	S-SF	5/16tc or Lc/600 (L)	C3D8R	3408	7076	1920/3904	Max: 25 × 25 × 8 Min: 25 × 8 × 8
S27	Fixed	S-SF	5/16tc or Lc/600 (L)	C3D8R	3458	7168	1920/3904	Max: 25 × 25 × 8 Min: 25 × 8 × 8
S28	Fixed	S-SF	5/16tc or Lc/600 (L)	C3D8R	3170	6592	1920/3904	Max: 25 × 25 × 8 Min: 25 × 8 × 8

# represents specimen number (No.), S4R refers to a four-node double-curved shell element, while C3D8R represents an eight-node linear brick element with reduced integration and hourglass control. S-NC, S-ST, and S-SF are surface-to-node coupling, surface-to-surface tie, and surface-to-surface friction interactions involving columns and other attached members. Additionally, t_c and L_c denote the thickness and length of the column, while L and G indicate local and global imperfections, respectively.

presents the FE models, such as S1~S28, used in the validation study, the types of FE elements used, the total number of elements and nodes in each FE model, the number of elements and nodes columns, and the mesh size of the columns. Thin-walled cold-formed stainless steel tube sections were discretized using shell elements, explicitly employing the four-node double-curved shell element (S4R), as illustrated in Figure 6. The element sizes were set to match the material’s thickness (t_c) across both corners and flat surfaces to ensure precise results. For root radii, the model divided them at twice the material thickness away from the edges, based on the assumption that their shapes closely resemble circular arcs. For connecting plates with perforations and other deformable solid components, the model applied advanced hexagonal sweep mesh controls and hexagonally structured mesh controls, respectively. These utilized an eight-node linear brick element with reduced integration and hourglass control (C3D8R), as showcased in Figure 7. While mesh size affects the calculated strengths, optimal mesh dimensions for various structural components such as SHS columns, corner angle columns, stiffening plates, and connection plates, including PFC floor beams and cushion blocks, were established as 30 × 10 × t, 30 × 30 × t, 30 × 8 × t, and 7 × 7 × t, respectively. Figure 6 presents a mesh model of shear-keyed tubular columns employing C3D8R for the shear keys, tubes, and connecting plates, indicating that mesh sizes ranging from a maximum of 25 × t mm to a minimum of t × t mm yielded results that closely align with experimental findings.

5.5. Material Simulation

In the material simulation for cold-formed sections, the strength enhancement resulting from the cold-forming process, particularly notable in the corner regions of stainless-steel cross-sections, was modeled to extend 2t from the curved corner regions into the adjacent flat areas. This approach attributed increased tensile properties to the corners. At the same time, the adjacent flat sections were modeled with compressive properties, maintaining consistency with the property values listed for the flat regions in Table 2, in accordance with methodologies described in Refs. [58,95,99,100,101]. A conventional elastic-plastic model employing kinematic hardening based on the von Mises yield criterion was utilized to simulate hot-rolled tubular walls and shear-keyed tubular columns. This model overlooked the enhancements in corner strength and the intricacies of root radii, also disregarding the effects of bending, welding, and residual stresses from temperature variations as seen in similar FEM approaches cited in [69]. The material properties, both elastic and plastic, required for this simulation criterion are detailed in Table 2 for both cold-formed and hot-rolled columns. As specified in Section C.6 of EN 1993-1-5:2006 (E) [102], engineering stress–strain measurements are transformed into true stress–strain figures to ensure precise simulation. A Poisson’s ratio of 0.3 was selected for these simulations.

5.6. Validations

Figure 8a–v presents a comparison between finite element (FE) simulations and experimental load-shortening curves, including the analysis of over- and underestimation ratios. This comparison reveals that the FE models closely mirror the actual shortening behavior, exhibiting only minor discrepancies in stiffness and post-peak recession. These differences can be attributed to factors such as using soft support conditions, adopting bi-linear stress–strain models for steel, simplifications in modeling, and the limited representation of geometric imperfections. The inconsistencies between experimental data and FE curves for sample S28 are apparent. The load-shortening curves diverge due to FEM simplifications like bi-linear stress–strain models for steel and a limited representation of geometric imperfections, which are common in computational analyses but can significantly affect behavior during plastic or failure stages. These simplifications can cause earlier capacity decreases than predicted by FE models in shear-keyed tubular columns when the actual experimental setup inherently includes minor imperfections that drastically affect performance during the plastic or failure stage. FE models and test findings match from elastic to ultimate compressive strength. However, during the post-ultimate phase, where local buckling occurs, the experimental specimens bent slightly and lost capacity abruptly. The FE model predicted overall buckling, but localized deformations and their severe impact on load capacity, notably under bending and shear, may have been overlooked. Support conditions also cause these inconsistencies. Unintentional soft support situations during testing allow specimens to translate or rotate slightly between tubes and shear keys, causing column capacity to decline abruptly after attaining the ultimate load. However, FE models, with ideally stable supports, do not fully recreate this translational or rotational flexibility, resulting in more extended recession capacity than experimental results. According to Table 2, the ratios derived from test-to-FE analysis for 28 tests average out to 1.0 with a coefficient of variation (Cov) of 0.05, indicating negligible errors in predicting the ultimate load (P_u). Ratios above 1.0 suggest underestimations by the FE models (accounting for 50% of the cases), while ratios below 1.0 signal overestimations of the actual responses (representing 46% of the cases). Figure 9a–n showcases the deformation patterns and von Mises stress distribution from FE analysis alongside experimental observations, underscoring the models’ capability to accurately forecast the failure mechanisms in cold-formed tubes, hot-rolled multi-column structures, and shear-keyed tubular columns. The observed failure phenomena include local inward (IB) and outward buckling (OB) at load application points or mid-column, global buckling (GB), stiffener bending, extrusion of channel beams, angle weld fractures, the restraining effects of corner columns, and S-shaped local buckling patterns. These validation efforts affirm the FEM’s efficiency in predicting axial compression responses accurately at both the individual member and overall structural levels in TSSs and MSSs.

6. Parametric Studies on Modular Steel Columns Grouped with Shear-keyed IMCs

6.1. Parametric Study

In Section 6, the numerical investigations into the behavior of modular steel tubular columns grouped with shear-keyed IMCs within MSSs were enhanced through a detailed parametric study using 103 validated FEMs. This study assessed the impact of various parameters, including initial imperfections, shear-key dimensions (height L_t and thickness t_t), and the dimensions of steel tubes (length D, width B, thickness t_c, and height L_c), on their ultimate strength behavior. The models were systematically divided into following groups to explore a wide range of variables: t_t variations (15, 20, and 25 mm with L_t at 100, 150, and 250 mm) represented by F1~F9, L_t adjustments (75, 150, 250, 300 mm with t_t at 15, 20, 25, and 35 mm) denoted by F10~F25, t_c modifications (5, 7, 8, and 9 mm with L_c at 1 and 1.5 m, and L_t at 100, 150, and 250 mm) with D/B of 150/150 and 200/200, denoted by F26~F49, changes in D/B ratio (150/150, 180/180, 200/200, 250/250, 160/80, 200/120, 220/140, and 250/180 mm with L_t at 100, 150, and 250 mm) represented by F50~F73, altering space between tubes and keys (0, 4, and 6 mm with L_t 120, 270, 400 mm) represented by F74~F82, modifying space between neighboring tubes (0, 24, 36, and 50 mm with L_t 180, 230, and 330 mm) represented by F83~F94, and a comprehensive array of tubes quantity values (2, 3, and 4 with L_t 180, 230, and 330 mm) represented by F95~F103 [69]. The research focused on the correlation between ultimate compressive load capacity (P_u), axial shortening (Δ_u), initial stiffness (K_e), and the ductility index (DI) in the context of shear-keyed grouped tubular columns, aiming to elucidate their load-resistance capabilities and ductile attributes comprehensively. The P_u, Δ_u, K_e, and DI of shear-keyed grouped columns, as determined by the standard tube parameter, are compared via rise or fall (+/−). The load-shortening curves derived from the modular steel grouped tubular columns were instrumental in determining P_u and Δ_u, while K_e and DI were calculated using the following equation:

$$K_e=\raisebox{1ex}{$P_{45\%}$}\!\left/ \!\raisebox{-1ex}{${\mathsf{\Delta }}_{45\%}$}\right., DI=\raisebox{1ex}{${\mathsf{\Delta }}_{85\%}$}\!\left/ \!\raisebox{-1ex}{${\mathsf{\Delta }}_u$}\right.$$

(1)

where $P_{45\mathrm{\%}}$, ${\mathsf{\Delta }}_{45\%}$, and ${\mathsf{\Delta }}_{85\%}$ represent the load at 45% of P_u, axial shortening at 45% of the ultimate load, and shortening at 85% of the ultimate load, respectively. This approach facilitates a deeper understanding of the pre- and post-ultimate ductility of the columns. The specific design details for each column configuration, along with the calculated values of P_u, Δ_u, K_e, and DI, are thoroughly documented in Section 7, aligning with the methodologies referenced in the literature [103].

6.2. Column Design

The FEM illustrated in Figure 10a–c includes connecting plates, steel tubes, and grouped shear keys as its main elements, all of which are modeled using C3D8R. This mesh configuration is a standard approach for simulating modular IMCs and structural components. Building on previous research (S25~S28), the design incorporates partitioned tubes and shear keys at the corners. It employs the hot-rolled section specifications from a prototype project, adopting a mesh size ranging from a maximum of 25 × t to a minimum of t × t mm². The model constrains movement in all directions for the upper and lower connecting plates, except for allowing vertical movement at the bottom to facilitate vertical displacement simulations. The interaction between the shear keys and connecting plates uses a tie constraint to prevent relative motion, effectively merging them. The interface between the column, connecting plates, and shear keys is characterized by standard surface-to-surface contact with “hard contact” for normal interactions and a “penalty friction formulation” for tangential interactions, reflecting the practical assembly of these components. A friction coefficient of 0.3 was selected based on referenced interaction methods. Following tested models in Refs. [15,16,17,96], the lowest buckling mode shape from Figure 11a–c used an imperfection amplitude of L_c/600. It is the widely used method of applying imperfections to design the safe column, similarly used in Ref. [104]. In this study, material properties, including averaged yield and ultimate strength values of 380 and 434 MPa, respectively, and a modulus of elasticity and plastic strain of 206 GPa and 23%, were assumed to be consistent for both the tubes and shear keys, as well as the connecting plates joined to the shear keys. This assumption was made based on the fact that all components were constructed from the same mild steel type Q235, as per Ref. [97].

7. Numerical Analysis Results Analysis and Discussions on Modular Steel Columns Grouped with Shear-Keyed IMCs

7.1. Typical Deformed Modes

Figure 12a,b and Figure 13a,b illustrate the influence of shear keys; Figure 14a,b and Figure 15a,b examine the effects of tubes; Figure 16a,b and Figure 17a,b detail the combined impact of tubes and shear keys; and Figure 18a–c highlight the effect of the number of columns on typical deformation patterns. These figures collectively showcase the primary failure mechanisms, accompanied by stress contours at the ultimate deformation stage. Yellow areas, indicating stress values greater than the yield stress, demonstrate that the columns are uniformly stressed and remain entirely functional. Changes in shear-key thickness, as depicted in Figure 12b and Figure 13a,b, alter the failure modes observed in tubes and shear keys compared to those in Figure 12a. Figure 13a,b underscore the beneficial effects of increased shear-key height in shifting the failure location. Local buckling around shear keys is visible in Figure 12b, Figure 13a,b, Figure 14a,b, Figure 17a,b and Figure 18a–c, as well as at the mid-height of tubes, as shown in Figure 12a, Figure 15a,b, Figure 16a,b and Figure 18b,c. The primary failure mechanisms in adjacent tubes were typically symmetrical, though their positioning varied, being either uniform or disparate. This variation suggests that all tubes effectively withstand loads, as evidenced in Figure 14b and Figure 18a–c. Each side of the tubes exhibited local inward and outward sinusoidal (S-shaped) buckling. Failure modes on opposing sides of a tube were identical yet differed on adjacent faces. When adjacent tubes buckled symmetrically, visible contact occurred in the bulged regions on the inner sides. However, Figure 12a, Figure 13a, Figure 15c, Figure 16a, Figure 17a and Figure 18c show that buckling in one tube can alter the positioning of another, resulting in double S-shaped buckling without affecting the column’s capacity. Some parts of the tubes and shear keys, especially those above or below the failure region, remained elastic and are represented in blue or green. Shear keys in the regions experiencing tube buckling generally exhibited higher stress values, indicating the shear key’s role in stress distribution, as seen in Figure 16a,b, where the stresses on the bottom one and two shear keys were greater than in areas not experiencing buckling. Grey coloring represents the plastic region, typically observed in buckled areas, indicative of plastic buckling. Green-colored zones, which are less stressed than the yield strength, signify elastic buckling, as depicted in Figure 12a,b, Figure 13a,b, Figure 15a,b, Figure 16a,b and Figure 17a,b. This delineation emphasizes that modular steel shear-keyed grouped tubular columns display a combination of elastic and plastic local buckling. In cases of local elastic buckling, only an inward or outward pattern emerges. Conversely, plastic local buckling, characterized by simultaneous inward and outward buckling, produces an S- or double S-shaped waveform, appearing opposite on adjacent sides and identical on opposing sides.

7.2. Typical Column Capacity Behavior

Figure 19 illustrates the typical shortening behavior of modular steel tubular columns grouped with shear-keyed IMCs through generalized load-shortening curves, delineating three key phases: initial linear elasticity (I), a nonlinear phase (II), and a subsequent recession zone (III). The transition into the nonlinear phase and the extent of the nonlinear and recession stages distinctly impact the columns’ behavior, with the load represented by P and shortening by Δ. Initially, P increases linearly with Δ up to the yield point (P_y), indicating that the onset of yielding and the length of the elastic phase vary among different FEMs. This variation suggests that some columns, due to factors like reduced compressive or bending strength attributable to the shear keys, reach yielding earlier than others, affecting their yield and ultimate strength capacities. Beyond the yield point, the P-Δ curves adopt a parabolic trajectory towards P_u, marking the onset of local buckling as columns achieve their compression capacity. Following this, columns experience local buckling, which may manifest as either elastic or plastic depending on whether the material has yielded. The extent of the nonlinear phase differs across FEMs, with some displaying enhanced ductility—evident from the difference between ultimate and yield shortening (Δ_u − Δ_y). Factors such as increased rigidity or reduced slenderness—achieved through thicker or shorter tubes or longer shear keys—enhance compression capacity while potentially reducing ductility. The recession phase follows, characterized by a decrease in ultimate resistance alongside pronounced local buckling. The DI at this post-ultimate capacity stage offers additional insights. The observed behavior, including the tendency of tubes to buckle without offering further resistance, may be attributed to initial imperfections or out-of-straightness. This underscores the complex interplay of design factors that influence the structural response of shear-keyed grouped tubular columns.

7.3. Variations in Axial Compression Behavior Due to Structural Parameters

7.3.1. Impact of Shear-Key Dimensions

Figure 20a–c and Figure 21a–d present the load-shortening (P-Δ) curves, illustrating the impact of variations in the dimensions of shear keys, specifically their t_t and L_t. Figure 22a–d and Figure 23a–d represent the percentage rise or fall and maximum value (a;b) in ultimate compressive load capacity, axial shortening, initial stiffness, and ductility index as a result of varying shear-key thickness and length. The percentage changes in several structural parameters (including ultimate compressive load capacity, axial shortening, initial stiffness, and ductility index) concerning a specified baseline are computed in this study. Each parameter’s initial value at the beginning of the observation period or under specific initial conditions constitutes its baseline. The percentage increase or decrease is then calculated by comparing each subsequent measurement to this initial value, using the following formula: Percentage change (%) = [(Subsequent Value − Baseline Value)/Baseline Value] × 100. Figure 24a,b further highlights the impact of these variations on trends of P_u and K_e, showcasing improvements in the compressive behavior of modular steel tubular columns grouped with shear key connectors. As t_t increases from 15 mm to 20 and 25 mm, P_u has a noticeable increase from 3134 kN to 3377 kN and 3925 kN, respectively, alongside a 3% to 11% rise in K_e for L_t of 150 mm. Similarly, when L_t changes from 75 mm to 150, 250, and 300 mm, P_u shows a slight increase from 3777 kN to 3936 kN, 4069 kN, and 4188 kN for t_t increases of 25 mm, respectively, with K_e experiencing a moderate increase of 4%, 7%, and 10%. Figure 12a,b and Figure 13a,b suggest that while stress levels remain consistent, the location of failure shifts. This shift occurs as increasing the size of the shear key enhances the steel content, diminishes slenderness, and bolsters resistance against tube buckling. Consequently, these adjustments elevate the yield points and facilitate complete effectiveness or uniform stress distribution across the tubes. However, Figure 25a,b indicates a more nuanced relationship between pre- and post-ultimate ductility, hinting at the complex interplay between ductility and shear-key dimensions.

7.3.2. Impact of Tubes Dimensions

Figure 26a–c, Figure 27a–c, Figure 28a–c and Figure 29a–c illustrate the P-Δ curves, capturing the effects of changes in tube dimensions, specifically t_c and D × B. Figure 30a–d, Figure 31a–d, Figure 32a–d and Figure 33a–d represent the percentage rise or fall and maximum value (a;b) in ultimate compressive load capacity, axial shortening, initial stiffness, and ductility index as a result of varying tubes thickness and cross-sectional dimension. Figure 34a,b further explore their influence on P_u and K_e, revealing that variations in tube dimensions significantly enhance the compressive behavior of modular steel tubular columns grouped with shear-keyed IMCs. As t_c increases from 5 mm to 7, 8, and 9 mm, P_u experiences a substantial rise from 1403 kN to 2498 kN, 3599 kN, and 4044 kN, respectively, with K_e escalating by 41%, 69%, and 69% for an L_t of 150 mm. Additionally, an increase in D × B dimensions from 150 mm to 180, 200, and 220 mm results in a 44% to 61% and 65% growth in P_u, while K_e sees a significant boost from 858 kN/mm to 1197 kN/mm, 1289 kN/mm, and 1718 kN/mm. Figure 14a,b and Figure 15a,b underscore the significant role of tube dimensions not only in stress distribution but also in altering the location and patterns of failure. This outcome arises as enhancements in tube dimensions augment the steel content, diminish the slenderness of the tube, and bolster compressive resistance. With t_c increasing from 5 mm to 9 mm, the D/t_c ratio drops from 30 to 17, transitioning the cross-sectional classification from Class 3 to Class 1. These changes also promote a symmetrical failure pattern and ensure complete effectiveness or uniform stress distribution across the tubes. Figure 35a,b delve into the nuances of ductility, offering an in-depth analysis. They present a nuanced examination of the relationship between pre- and post-ultimate ductility, illustrating the intricate dynamics between ductility and tube dimensions.

7.3.3. Interplay between Tube and Shear-Key Dimensions

Figure 36a–c and Figure 37a–c present the P-Δ curves illustrating the impact of varying gaps between the tube and shear-key, as well as between adjacent tubes. Figure 38a–d and Figure 39a–d represent the percentage rise or fall and maximum value (a;b) in ultimate compressive load capacity, axial shortening, initial stiffness, and ductility index as a result of varying tube–key and tube–tube gap. Figure 40a,b explore how these gaps influence the P_u and K_e of modular steel tubular columns grouped with shear-key connectors. It is observed that an increasing gap between the tube and shear key significantly compromises the compressive behavior, while an initial increase followed by a decrease in the tube-to-tube gap has a mixed effect on performance. Specifically, as the gap between the tube and shear key expands from 0 mm to 4 mm and 6 mm, P_u experiences a decline from 3898 kN to 2964 kN and 2657 kN, marking reductions of 32% and 47%, respectively. Similarly, K_e sees a decrease from 1044 kN/mm to 625 kN/mm and 814 kN/mm, falling by 67% and 28%. Conversely, when the gap between tubes widens from 0 mm to 24 mm, 36 mm, and 50 mm, P_u exhibits a slight increase of 2% followed by a 1% reduction, while K_e initially increases from 1020 kN/mm to 1217 kN/mm and 1213 kN/mm, and then decreases to 1124 kN/mm, with increases of 16% and 9%. These trends suggest that while a small tube-to-tube gap can enhance the columns’ collective load resistance by promoting a unified action, a more significant gap diminishes this effect. Figure 16a,b and Figure 17a,b indicate that the gaps between the tube and shear key and between tubes do not markedly affect stress distribution or the location of failure. However, they do influence the failure patterns, with the sides of the tubes exhibiting inward buckling in some cases and outward buckling in others. This reversal of buckling direction indicates that an increase in the tube-to-shear-key gap lessens the integrative action between the shear keys and the composite tube, thereby reducing the shear keys’ protective influence on the tube ends. Meanwhile, a modest gap between tubes fosters a collective effect that enhances their axial load resistance as a cohesive unit, though an extensive gap reduces this capability. The relationship between pre- and post-ultimate ductility becomes less pronounced, as illustrated in Figure 41a,b, which delve into the details of ductility-related findings from the FEMs. This nuanced analysis underscores the complex dynamics between the spatial configurations within shear-keyed grouped tubular columns and their structural performance.

7.3.4. Impact of Varying Tube Quantities

Figure 42a–c displays the P-Δ curves for configurations with different numbers of tubes. Figure 43a–d represent the percentage rise or fall in and maximum value (a;b) of the ultimate compressive load capacity, axial shortening, initial stiffness, and ductility index as a result of varying tube quantities. Additionally, Figure 44 examines the effect of varying the number of tubes in a grouped column configuration with varying L_t. The analysis reveals a significant enhancement in the compressive behavior of shear-keyed grouped tubular columns as the number of tubes increases. Specifically, K_e sees an improvement from 1217 kN/mm to 2104 kN/mm and 2802 kN/mm, marking increases of 42% and 57%, respectively, as the tube count grows from 2 to 3 and then to 4. Concurrently, P_u escalates from 3915 kN to 6134 kN and 8102 kN, reflecting increases of 36% and 52%. This improvement is attributable to the augmented collective cross-sectional area provided by the increased quantity of tubes and shear keys, thereby bolstering the overall strength and stiffness. Moreover, as the tube count advances from 2 to 3 and 4 tubes, the P_u and K_e values are enhanced by factors of 1.6 and 2.1 and 1.7 and 2.3, respectively. This underscores the pronounced grouping effect that amplifies with the addition of more columns, consequently enhancing the compression behavior. Figure 18a–c indicate that while the failure location may vary, the number of tubes does not notably alter the stress distribution or the failure patterns. Further detailed examinations of ductility-related FEMs in Figure 45 reveal a diminished correlation between post-ultimate ductility. However, post-ultimate ductility experiences an uptick of 12% and 29%, highlighting that the structural integration provided by additional tubes notably improves the post-failure ductility, even if the pre-failure behavior shows a more complex relationship with the number of tubes involved. This nuanced analysis illustrates the critical role that tube quantity plays in influencing the structural integrity and resilience of shear-keyed grouped tubular columns.

8. Conventional Methods and New Approaches in the Design of Modular Steel Tubular Columns Grouped with Shear-Keyed IMCs

8.1. Conventional Methods

This section explores the conventional methods and methodology used to evaluate the structural integrity and resistance to buckling of tubular columns grouped with shear keys in MSSs. These practices are based on recognized codes and models. Conventional methods are used as a standard to assess the effectiveness and precision of newer approaches. By comprehending the fundamental principles established by these protocols, one can acknowledge the progress achieved by the newly proposed method in improving prediction precision and structural integrity.

8.1.1. The Li et al. Buckling Design Model

Drawing on experimental results, Li et al. [71] introduced a model for determining the buckling strength of modular grouped columns, encapsulated in Equations (2) and (3):

$$\frac{\sigma }{f_y}=\left\{\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}1, {\lambda }_P\le 0.62\\ \frac{0.95{\lambda }_P-0.30}{{{\lambda }_P}^{2.6}}, {\lambda }_P>0.62\end{array} ; {\lambda }_P=\frac{\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$t$}\right.}{56.8}\sqrt{\raisebox{1ex}{$f_y$}\!\left/ \!\raisebox{-1ex}{$235$}\right.} \right.$$

(2)

$$P=\sigma A_s$$

(3)

where $\frac{\sigma }{f_y}$ denotes the stress ratio, $\raisebox{1ex}{$h$}\!\left/ \!\raisebox{-1ex}{$t$}\right.$ defines the slenderness ratio, and ${\lambda }_P$ represents the normalized width-to-thickness ratio.

8.1.2. Indian Standard

IS800 [88,89] outlines the procedure for determining the buckling resistance of grouped columns, presenting Equations (4)–(9):

$$P_d=A_s{f}_{cd} \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1,2,3;P_d=A_e{f}_{cd} \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4$$

(4)

$$\left\{\begin{array}{c}D(or B)/t\epsilon \le 29.3, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1\\ D(or B)/t\epsilon \le 33.5, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 2\\ D(or B)/t\epsilon \le 42, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 3\\ D(or B)/t\epsilon >42, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4\end{array}\right.$$

(5)

$$f_{cd}=\frac{f_y/{\gamma }_{M0}}{\varphi +[{\varphi }^2-{\lambda }^2{]}^{0.5}}=\chi f_y/{\gamma }_{M0}\le f_y/{\gamma }_{M0}; \varphi =0.5\left[1+\alpha \left(\lambda -0.2\right)+{\lambda }^2\right]$$

(6)

$$\lambda =\sqrt{f_y/f_{cc}}=\sqrt{f_y(KL/r{)}^2/{\pi }^{2}E}$$

(7)

$$b_e\left(d_e\right)=2120t\sqrt{f_y}\left[1-\frac{465}{\frac{b}{t\left(\frac{d}{t}\right)\sqrt{f_y}}}\right], if\frac{b}{t\left(\frac{d}{t}\right)}>\frac{1435}{\sqrt{f_y}}; b_e(d_e)=b(d), if\le 1435/\sqrt{f_y}$$

(8)

$$A_e=A_s-2t\left(b-b_e\right)-2t(d-d_e)$$

(9)

where $P_d$, $f_{cd}$, $A_s$, d, b, $A_e$, d_e, b_e, $\lambda$, $\alpha$, $\chi$, $f_{cc}$, and ${\gamma }_{M0}$ represent the design compressive resistance, stress, cross-section area, length, width, effective area, length, width, slenderness, imperfection factor, stress reduction factor, Euler buckling stress, and partial safety factor. The code suggests ${\gamma }_{M0}$ = 1 for buckling members that revealed 71 (total 103) unsafe outcomes; thus, the modifications were suggested for ${\gamma }_{M0}$ = 2 that raised conservative findings to 100 with only 3 unsafe outcomes.

8.1.3. New Zealand Standard

NZS3404 [90] employs Equations (10)–(20) to calculate the compressive resistance of grouped members:

$$N_c={{\alpha }_{c}N}_s; N_s=\phi {k_{f}A}_n{f}_y; k_f=A_e/A_g$$

(10)

$$b_e=b\left(\raisebox{1ex}{${\lambda }_{ey}$}\!\left/ \!\raisebox{-1ex}{${\lambda }_e$}\right.\right)\le b; {\lambda }_e=b/t\sqrt{f_y/250}$$

(11)

$$A_e=A_s-2t\left(b-b_e\right)-2t(d-d_e); A_n=A_g$$

(12)

$${\lambda }_n=(KL/r)\sqrt{k_f}\sqrt{f_y/250}$$

(13)

$${\alpha }_a=2100({\lambda }_n-13.5)/{{\lambda }_n}^2-15.3{\lambda }_n+2050$$

(14)

$$\lambda ={\lambda }_n+{\alpha }_a{\alpha }_b$$

(15)

$$\eta =0.00326(\lambda -13.5)\ge 0$$

(16)

$${\alpha }_c=\xi \left\{1-\sqrt{1-{\left(\frac{90}{\xi \lambda }\right)}^2}\right\}; \xi =[\left(\lambda /90{)}^2+1+\eta \right]/\left[2(\lambda /90{)}^2\right]$$

(17)

where $N_c$, $N_s$, $k_f$, $A_g$, $A_e$, ${\lambda }_e$, $\phi$, ${\alpha }_a$, ${\alpha }_b$, and ${\alpha }_b$ represent the member, section compressive resistance, form factor, gross area, effective area, effective slenderness, strength, and slenderness reduction factor. The following specifications shown in Equations (18)–(20) are used for the member’s classification in NZS:

$$d/t\sqrt{f_y/250}\le 30 \mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{y} 1,2$$

(18)

$$d/t\sqrt{f_y/250}\le 40 \mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{y} 3$$

(19)

$$d/t\sqrt{f_y/250}\le 82 \mathrm{C}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}\mathrm{y} 4$$

(20)

The code suggests $\phi$ = 0.9 for buckling members, which revealed 68 (total 103) unsafe outcomes; thus, the modifications were proposed for $\phi$ = 0.5, which raised conservative findings to 97 with only 6 unsafe outcomes.

8.1.4. Canadian Standard

CSA-S16 [59] recommends specific standards for the classification and strength assessment of grouped members, outlined in Equations (21)–(23):

$$C_r=\phi A{F}_y(1+{\lambda }^{2n}{)}^{-\frac{1}{n}} \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1\text{–}3 ;C_r=\phi A_{eff}{F}_y(1+{\lambda }^{2n}{)}^{-\frac{1}{n}} \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4$$

(21)

$$\left\{\begin{array}{c}B(or D)/t\le 420/\sqrt{f_y}, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1\\ B(or D)/t\le 525/\sqrt{f_y}, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 2\\ B(or D)/t\le 670/\sqrt{f_y}, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 3\\ B(or D)/t>670/\sqrt{f_y}, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4\end{array}\right.$$

(22)

$$\lambda =\sqrt{F_y/F_e } ; F_e={\pi }^{2}E/{\left(\raisebox{1ex}{$KL$}\!\left/ \!\raisebox{-1ex}{$r$}\right.\right)}^2$$

(23)

where $C_r$, $\phi$, $\lambda$, $K$, $r$, and $A_{eff}$ represent ultimate strength, resistance factor of 0.90; strength ratio, effective length factor taken as 1.0, a radius of gyration, and effective area;$n$ is suggested as 1.34 for hot-rolled SHS. The code suggests $\phi$ = 0.9 for buckling members, which revealed 68 (total 103) unsafe outcomes; thus, the modifications were suggested for $\phi$ = 0.55, raising conservative findings to 100 with only 3 unsafe outcomes.

8.1.5. European Standard

Equations (24)–(30) are applied to determine the buckling resistance of grouped columns according to EN1993-1-1 [63]:

$$P_{u,b}={\chi f}_y{A}_s/{\gamma }_{M0} \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1,2,3$$

(24)

$$P_{u,b}={\chi f}_y{A}_{eff}/{\gamma }_{M0}; A_{eff}=A_s-2t{\rho }_{f}d-2t{\rho }_{w}b \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4$$

(25)

$$\left\{\begin{array}{c}D(or B)/t\epsilon \le 33, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 1\\ D(or B)/t\epsilon \le 38, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 2\\ D(or B)/t\epsilon \le 42, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 3\\ D(or B)/t\epsilon >42, \mathrm{C}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{s} 4\end{array}\right.; \epsilon =\sqrt{235/f_y}$$

(26)

$$\chi =1/{[\varphi +({\varphi }^2-{\overline{\lambda }}^2)}^{0.5}]\le 1; \varphi =0.5\left[1+\alpha \left(\overline{\lambda }-0.2\right)+{\overline{\lambda }}^2\right]$$

(27)

$$\overline{\lambda }=\sqrt{f_y{A}_s/P_{cr}};P_{cr}={\pi }^2{E}_{s}I/L^2$$

(28)

$${\rho }_f{(\rho }_w)=\left\{\begin{array}{cc}\hfill 1.0,& {\lambda }_f{(\lambda }_w)\le 0.673\\ \hfill \frac{{\lambda }_f{(\lambda }_w)-0.055(3+\psi )}{{{\lambda }_f}^2{{(\lambda }_w}^2)}\le 1.0,& {\lambda }_f{(\lambda }_w)>0.673\end{array}\right.; \psi =\frac{{\sigma }_{min}}{{\sigma }_{max}}$$

(29)

$${\lambda }_f{(\lambda }_w)=\sqrt{\frac{f_y}{{\sigma }_{cr}}}=\frac{d/t(b/t)}{28.4\epsilon \sqrt{k_{\sigma }}}$$

(30)

where $P_{u,c}$,$P_{u,b}$, $\chi$, $f_y$, $A_s$, $A_{eff}$, and ${\gamma }_{M0}$ represent the ultimate compressive resistance, buckling resistance, capacity reduction factor, yield strength, cross-section area, effective area, and safety factor. The standards [105], code [63,106], and research [15,16,17,107], recommend ${\gamma }_{M0}$ equal to 1.0. The code suggests ${\gamma }_{M0}$ = 1 for buckling members that revealed 80 (total 103) unsafe outcomes; thus, the modifications were suggested for ${\gamma }_{M0}$ = 1.85 that raised conservative findings to 100 with only 3 unsafe outcomes.

8.1.6. American Standard

AISC360-16 [60] calculates the strength of modular grouped columns using Equations (31)–(37) as follows:

$$P_u=f_c{A}_s\left(A_{eff}\right)$$

(31)

$$f_c=\left\{\begin{array}{c}Q\left[{0.658}^{Q{f}_y/f_e}\right]f_y , if Q{f}_y/f_e\le 2.25\\ 0.877f_e, if Q{f}_y/f_e>2.25\end{array}\right.; f_e={\pi }^2{E}_s/(\raisebox{1ex}{$KL$}\!\left/ \!\raisebox{-1ex}{$r$}\right.{)}^2$$

(32)

$$Q=\left\{1.0 \mathrm{i}\mathrm{f} \right.d/t\le 1.40\sqrt{E_s/f_y} \left(\mathrm{N}\mathrm{S}\right)$$

(33)

$$Q=\left\{Q_a{Q}_s \right. \mathrm{if} d/t>1.40\sqrt{E_s/f_y} \left(\mathrm{S}\right); A_{eff}=A_s-2t\left(b-b_e\right)-2t(d-d_e)$$

(34)

$${Q=Q}_a\left(Q_S=1\right); \mathrm{a}\mathrm{n}\mathrm{d} Q_a=A_{eff}/A_s (\mathrm{A}\mathrm{I}\mathrm{S}\mathrm{C} \mathrm{E}\mathrm{q}\mathrm{n}\text{-}7)$$

(35)

$$b_e=1.92t\sqrt{E_s/f_y}\left[1-\frac{0.34}{\left(b/t\right)}\sqrt{E_s/f_y}\right]\le b, if \raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$t$}\right.\ge 1.49\sqrt{\frac{E_s}{f_y}}, otheriwse=b$$

(36)

$$d_e=1.92t\sqrt{E_s/f_y}\left[1-\frac{0.38}{(d/t)}\sqrt{E_s/f_y}\right]\le d, if \raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$t$}\right.\ge 1.40\sqrt{\frac{E_s}{f_y}}, otherwise=d$$

(37)

where $f_y$ replaces $f_c$ per AISC7.2b, with $b$ or $d$ equivalent to B/D-2t_c, indicating the adjusted breadth or depth. The terms $Q_s$ and $Q_a$ denote reduction factors for unstiffened and stiffened elements, respectively [108]. Additionally, $b_e$ and $d_e$ are determined by Equations (36) and (37), representing the effective width and length. AISC360-16 applies a resistance factor of 0.90, as detailed in Section E1 of AISC 360-16 and Section 8.1 of SDCSS, for the design of axially compressed members [109]. The standard recommends φ = 0.9 for buckling members, which resulted in 68 (out of 103) unsafe outcomes. Consequently, adjustments to φ = 0.5 were proposed, enhancing conservative results to 100, with only 3 instances of unsafe outcomes.

8.1.7. Chinese Standard

The compressive strength criteria for modular grouped tubular columns in GB50017-2017 [66] are outlined using Equations (38)–(40):

$$P_u={\phi f}_y A_s; \phi =\left\{\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}1-{\alpha }_1{{\lambda }_n}^2,& \hfill if {\lambda }_n\le 0.215\\ \frac{1}{2{{\lambda }_n}^2}[K-\sqrt{K^2-4{{\lambda }_n}^2},& \hfill if {\lambda }_n>0.215\end{array}\right.$$

(38)

$$K=\left({\alpha }_2+{\alpha }_3{\lambda }_n+{{\lambda }_n}^2\right); {\lambda }_n=\frac{\lambda }{\pi }\sqrt{f_y/E_s}; \lambda =\frac{i}{L}; i=\sqrt{I/A_s}$$

(39)

$$\mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e} B=\left\{\begin{array}{c}{\alpha }_1=0.650\\ {\alpha }_2=0.965\\ {\alpha }_3=0.300\end{array}\right.; \mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e} C=\left\{\begin{array}{c}{\alpha }_1=0.730, {\lambda }_n\le 1.05\\ {\alpha }_2=0.906, {\lambda }_n\le 1.05\\ {\alpha }_3=0.595, {\lambda }_n\le 1.05\\ \mathrm{o}\mathrm{r}\\ {\alpha }_1=0.730, {\lambda }_n>1.05\\ {\alpha }_2=1.216, {\lambda }_n>1.05\\ {\alpha }_3=0.302, {\lambda }_n>1.05\end{array}\right.$$

(40)

where $\lambda$ denotes the slenderness ratio, L represents the unbraced length, and i defines the radius of gyration, which is $\sqrt{I/A_s}$.$A_s$ and ${\lambda }_n$ represents the cross-sectional area and normalized slenderness ratio, respectively. Furthermore, φ is the safety factor. Given that the original safety factor values (${\alpha }_1=0.650; {\alpha }_2=0.965; {\alpha }_3=0.300$) led to an overestimation in 83 instances (out of a total of 103). This study proposes revised values (${\alpha }_1=15.965; {\alpha }_2=1.80; {\alpha }_3=1.65$) for modular steel tubular columns grouped with shear-keyed IMCs. These adjustments yield 103 safe outcomes and enhance the accuracy of predictions, eliminating underestimations by 100%.

8.2. Newly Proposed Approach: Force Transmission Model

Shear-keyed tubular columns are members that are not homogeneous in composition and result in uneven distribution of stress at the ends and middle [110]. The presence of a non-welded shear-keyed IMC decreases the rotational stiffness of the tube ends, impacting the slenderness, initial stiffness, and buckling performance of the columns. This results in a weakening of the columns compared to homogeneous tubes without a shear-keyed IMC [86]. Consequently, an optimal design for the columns in these MSSs prevents flexural buckling and sway. The inadequate strength and rigidity of current IMCs render them incapable of ensuring efficient vertical and horizontal load transfer [31]. In order to ensure the safety of columns and prevent further deterioration of IMCs, modular steel structures are reinforced and braced against lateral deflection and instability caused by the P-Δ effect [110]. In MSS design, steel-braced moment-resisting frames are crucial for sufficient load resistance and preventing column flexural buckle and sideways instability ([4,39,40]). Therefore, in theoretical analysis, enforcing fixed–fixed boundary conditions at both ends of columns is preferred, as depicted in Figure 3, following Ref. [86]. This approach lays the theoretical groundwork for more detailed investigations into other boundary conditions, such as pinned–pinned and fixed–pinned cases, facilitating the development of a categorization system for modular steel shear-keyed grouped tubular columns and ensuring both conservative and economical design practices. The force transmission method and calculation model demonstrated under pure axial compression in Figure 46 and Figure 47 facilitated the simulation of buckling behaviors. This simulation was grounded in the premise, supported by buckling modes derived from the FEM of this study, as shown in Figure 11a–c, that all columns within a specific story would buckle simultaneously due to the interplay between shear keys and tubes. To formulate the load-deflection relationship, fourth-order differential Equations (44) and (45) were developed, utilizing the foundational Euler–Bernoulli beam Equations (41)–(43).

$$M=-\frac{EI}{\rho }=-EI\frac{d^{2}w}{d{x}^2}=EI{w}^{″} As w^{″}=\frac{1}{\rho }$$

(41)

$$M=-EI\frac{d^{2}w}{d{x}^2}=-EI{w}^{″}$$

(42)

$$Pw=-EI{w}^{″}$$

(43)

$${w_1}^4+{k_1}^2{w_1}^2=0 {k_1}^2=\frac{P}{E{I}_1} -L_1\le x\le 0$$

(44)

$${w_2}^4+{k_2}^2{w_2}^2=0 {k_2}^2=\frac{P}{E{I}_2} 0\le x\le L_2$$

(45)

In this context, M/$P$ is a bending moment/axial compressive load,$\rho /$x represents the radius of curvature/longitudinal coordinate,$w$ is the lateral displacement/deflection, ${EI}_1$ and $E{I}_2$ denote the flexural rigidity at the ends and mid-height, $I_1$ and $I_2$ indicate the moment of inertia at the ends and mid-height, while $k_1$ and $k_2$ symbolize the rigidity at the corresponding sections of modular grouped columns that are reinforced with shear-keyed IMCs, distinctly identified by L₁ and L₂. By deriving the general solution for Equations (44) and (45) and proceeding with their differentiation, Equations (46) and (47) are subsequently formulated.

$$\left\{\begin{array}{c}\begin{array}{c}{w}_1\left(x\right)=A_1\sin \left(k_{1}x\right)+B_1\cos \left(k_{1}x\right)+C_{1}x+D_1\\ {w_1}^{\prime }\left(x\right)=A_1{k}_1\cos \left(k_{1}x\right)-B_1{k}_1\sin \left(k_{1}x\right)+C_1\\ {w_1}^{″}\left(x\right)={-A}_1{k_1}^2\sin \left(k_{1}x\right)-B_1{k_1}^2\cos \left(k_{1}x\right)\\ {w_1}^{‴}\left(x\right)={-A}_1{k_1}^3\cos \left(k_{1}x\right)+B_1{k_1}^3\sin \left(k_{1}x\right)\end{array}\end{array}\right.$$

(46)

$$\left\{\begin{array}{c}{w}_2\left(x\right)=A_2\sin \left(k_{2}x\right)+B_2\cos \left(k_{2}x\right)+C_{2}x+D_2\\ {w_2}^{\prime }\left(x\right)=A_2{k}_2\cos \left(k_{2}x\right)-B_2{k}_2\sin \left(k_{2}x\right)+C_2\\ {w_2}^{″}\left(x\right)={-A}_2{k_2}^2\sin \left(k_{2}x\right)-B_2{k_2}^2\cos \left(k_{2}x\right)\\ {w_2}^{‴}\left(x\right)={-A}_2{k_2}^3\cos \left(k_{2}x\right)+B_2{k_2}^3\sin \left(k_{2}x\right)\end{array}\right.$$

(47)

$A_1$, $A_2$, $B_1$, $B_2$, $C_1$, $C_2$, $D_1$, and $D_2$ are unknown variables that can be determined by substituting the boundary conditions (BC) and continuity conditions (CC). Boundary and compatibility conditions outlined in Equation (48) can be incorporated into Equations (46) and (47), leading to the derivation of Equation (49).

$$\mathrm{B}\mathrm{C}=\left\{\begin{array}{c}{w}_1\left(-L_1\right)=0\\ {w_1}^{\prime }\left(-L_1\right)=0\\ {w_2}^{\prime }\left(L_2\right)=0\\ {w_2}^{‴}\left(L_2\right)=0\end{array}\right. ; \& \mathrm{C}\mathrm{C}=\left\{\begin{array}{c}{w}_1\left(0\right)=w_2\left(0\right)\\ {w_1}^{\prime }\left(0\right)={w_2}^{\prime }\left(0\right)\\ {w_1}^{″}\left(0\right)={w_2}^{″}\left(0\right)\\ {w_1}^{‴}\left(0\right)={w_2}^{‴}\left(0\right)\end{array}\right.$$

(48)

$$\left\{\begin{array}{c}-A_1\sin \left(k_1{L}_1\right)+B_1\cos \left(k_1{L}_1\right)-C_1\left(L_1\right)+D_1=0\\ A_1{k}_1\cos \left(k_1{L}_1\right)+B_1{k}_1\sin \left(k_1{L}_1\right)+C_1=0\\ A_2{k}_2\cos \left(k_2{L}_2\right)-B_2{k}_2\sin \left(k_2{L}_2\right)+C_2=0\\ {-A}_2{k_2}^3\cos \left(k_2{L}_2\right)+B_2{k_2}^3\sin \left(k_2{L}_2\right)=0\end{array}; \left\{\begin{array}{c}{A}_1=m\sqrt{m} A_2\\ B_1=m{B}_2\\ C_1=A_2{k}_2+C_2-m\sqrt{m} A_2{k}_1\\ D_1=B_2+D_2-m{B}_2\end{array}\right.\right.$$

(49)

Upon simplifying Equation (49), Equations (50)–(52) emerge, which can further be expressed through the relationships $m={EI}_1/E{I}_2={k^2}_2/{k^2}_1$, $n=L_1/L_2$, and $k_1{L}_1=n/\sqrt{m}{k}_2{L}_2$. These can be further condensed to form a determinant, as outlined in Equation (53). Through straightforward mathematical manipulation, this determinant can be simplified, leading to Equations (54)–(58). These equations are instrumental in calculating the buckling length and strength of modular steel tubular columns that are grouped with shear-keyed IMCs.

$$A_2[\left(m\sqrt{m} \right)k_1{L}_1\cos \left(k_1{L}_1\right)-m\sqrt{m}\sin \left(k_1{L}_1\right)] +B_2\left[\left(m\right)\cos \left(k_1{L}_1\right)+1+m{k}_1{L}_1\sin \left(k_1{L}_1\right)-m\right]+D_2=0$$

(50)

$$A_2\left[\left(m\sqrt{m} \right)k_1\cos \left(k_1{L}_1\right)+k_2-m\sqrt{m}{k}_1-k_2\cos \left(k_2{L}_2\right)\right]+B_2\left[\left(m\right)k_1\sin \left(k_1{L}_1\right)+k_2\sin \left(k_2{L}_2\right)\right]=0$$

(51)

$${-A}_2\left[{k_2}^3\cos \left(k_2{L}_2\right)\right]+B_2\left[{k_2}^3\sin \left(k_2{L}_2\right)\right]=0$$

(52)

$$\left|\begin{array}{ccc}\left(m\sqrt{m} \right)k_1{L}_1\cos \left(k_1{L}_1\right)-m\sqrt{m}\sin \left(k_1{L}_1\right) & m\cos \left(k_1{L}_1\right)+1+m{k}_1{L}_1\sin \left(k_1{L}_1\right)-m& 1\\ \left[\left(m\sqrt{m} \right)k_1\cos \left(k_1{L}_1\right)+k_2-m\sqrt{m}{k}_1-k_2\cos \left(k_2{L}_2\right)\right]& \left[\left(m\right)k_1\sin \left(k_1{L}_1\right)+k_2\sin \left(k_2{L}_2\right)\right]& 0\\ -\left[{k_2}^3\cos \left(k_2{L}_2\right)\right]& \left[{k_2}^3\sin \left(k_2{L}_2\right)\right]& 0\end{array}\right|=0$$

(53)

$$\sqrt{m} Tan\left(k_2{L}_2\right)\left[\cos \left(\frac{n}{\sqrt{m}}{k}_2{L}_2\right)+\frac{1}{m}-1\right]+\cos \left(\frac{n}{\sqrt{m}}{k}_2{L}_2\right)Tan\left(\frac{n}{\sqrt{m}}{k}_2{L}_2\right)=0$$

(54)

$${\mu }_1=\raisebox{1ex}{$L_{eff}$}\!\left/ \!\raisebox{-1ex}{$2L_2$}\right.=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2$}\right.k_2{L}_2;{\mu }_2=\raisebox{1ex}{$L_{eff}$}\!\left/ \!\raisebox{-1ex}{$L_t$}\right.=\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$2(n+1)$}\right.k_2{L}_2$$

(55)

$$P_m=\left[{\left(\raisebox{1ex}{${\mu }_2$}\!\left/ \!\raisebox{-1ex}{$L$}\right.\right)}^2{EI}_1\right]$$

(56)

$$I_1=I_c+I_t=\frac{1}{12}[\left(B\times D^3\right)-\left\{\left(B-2t_c\right)\left(D-2t_c{)}^3\right\}\right]+\frac{1}{12}[\left(b\times d^3\right)-\left\{\left(b-2t_t\right)\left(d-2t_t{)}^3\right\}\right]$$

(57)

$$I_2=I_c=\frac{1}{12}[\left(B\times D^3\right)-\left\{\left(B-2t_c\right)\left(D-2t_c{)}^3\right\}\right]$$

(58)

where $m$, $n$, ${\mu }_1$, ${\mu }_2$, $L_{eff}$, and $P_m$ represent the flexural rigidity ratio, length ratio of shear-keyed grouped columns with and without shear keys, buckling length coefficients in relation to column length without shear-key, and total column length, effective length, and critical load, respectively. Given that tubes and shear keys are assumed to be non-welded, there exists the potential for full or partial slip or rotation, unlike the non-slip condition observed in welded configurations. To accommodate this and ensure safety, the bending stiffness of modular steel tubular columns grouped with shear-key connectors is set to a minimum, accounting for a full-slip scenario. This results in a bending stiffness equivalent to double that of a single column’s stiffness in a double-column arrangement, expressed as 2(I_c). In contrast, in non-slip or welded conditions, the bending stiffness quadruples due to the combined action of the tubes and the number of shear keys represented as 4(I_c). Furthermore, for tri- and tetra-grouped tubes where the interior cross-section contributes to bending moment resistance, the full-slip scenario would lead to excessively conservative results. Consequently, a partial slip scenario is considered, which yields bending stiffness values of 4.5(I_c) for triple columns and 7(I_c) for tetra columns, taking into account the enhanced structural integrity provided by multiple shear keys and the integrated work of the tubes.

Conclusively, for IS800 [88,89], the initial assumption of a partial safety factor ${\gamma }_{M0}$ was set to 1 in the code, with a proposed modification to ${\gamma }_{M0}$ = 2. Similarly, NZS3404 [90] initially set the strength reduction factor φ to 0.9, with a proposed adjustment to φ = 0.5. In CSA-S16 [59], the resistance factor φ set by the code was revised from 0.9 to 0.55 after performing parametric studies on modular steel grouped tubular columns. In EN1993-1-1 [63], the capacity reduction factor ${\gamma }_{M0}$ was altered from 1 to 1.85. was altered from 1 to 1.85. In AISC360-16 [60], the resistance factor φ was adjusted from 0.9 to 0.5. For the GB50017-2017 [66], revisions to safety factor values from (${\alpha }_1=0.650; {\alpha }_2=0.965; {\alpha }_3=0.300$) to a new value (${\alpha }_1=15.965; {\alpha }_2=1.80; {\alpha }_3=1.65$) was proposed. The assumptions regarding equations accounted for the simultaneous buckling of columns within a specific story and considered the bending stiffness of grouped columns equivalent to double that of a single column’s stiffness in a double-column arrangement, expressed considering a full-slip scenario. Furthermore, for tri- and tetra-grouped tubes, a partial slip scenario is also taken into account, enhancing the accuracy of predictions.

8.3. Validation

Table 4. Compressive resistance details of parametrically studied shear-keyed grouped tubular columns based on code prediction equations.

D (mm)	B (mm)	tc (mm)	Lc (m)	IS800 Class	${\mathit{P}}_{\mathit{u},\mathit{I}\mathit{S}800}$ (kN)	NZS3404 Class	${\mathit{P}}_{\mathit{u},\mathit{N}\mathit{Z}\mathit{S}3404}$ (kN)	EC3 Class	${\mathit{P}}_{\mathit{u},\mathit{E}\mathit{C}3}$ (kN)	CSA Class	${\mathit{P}}_{\mathit{u},\mathit{C}\mathit{S}\mathit{A}}$ (kN)	AISC Class	${\mathit{P}}_{\mathit{u},\mathit{A}\mathit{I}\mathit{S}\mathit{C}}$ (kN)	GB Class	${\mathit{P}}_{\mathit{u},\mathit{G}\mathit{B}}$ (kN)
200	200	5	1.0	C4	2711	C3	2423	C4	1532	C4	1352	S	2204	B	2908
200	200	7	1.0	C3	3755	C2	3683	C2	4130	C2	3672	NS	3650	B	4026
200	200	8	1.0	C2	4269	C2	4187	C1	4696	C2	4174	NS	4148	B	4578
200	200	8	1.5	C2	4187	C2	4127	C1	4606	C2	4120	NS	4082	B	4462
200	200	9	1.0	C1	4776	C1	4686	C1	5254	C2	4670	NS	4644	B	5122
150	150	10	1.5	C1	3728	C1	3686	C1	4100	C1	3658	NS	3630	C	3752
180	180	10	1.5	C1	4598	C1	4539	C1	5058	C1	4524	NS	4486	C	4884
200	200	10	1.5	C1	5176	C1	5103	C1	5694	C1	5092	NS	5048	C	5410
250	250	10	1.5	C2	6619	C2	6507	C1	7282	C3	6494	NS	6448	B	7088
160	80	8	1.5	C1	2382	C1	3146	C1	2620	C1	2012	NS	2064	C	2206
200	120	8	1.5	C2	3301	C2	4083	C1	3502	C2	3102	NS	3088	B	3334
220	140	8	1.5	C3	3560	C2	4541	C2	4028	C3	3592	NS	3566	B	3864
250	180	8	1.5	C3	4560	C2	5227	C3	4944	C3	4422	NS	4384	B	4780

P_u,IS800, P_u,NZS3404, P_u,ec3, P_u,CSA, P_u,AISC, and P_u,GB denote the ultimate compressive resistance calculated according to the guidelines and formulas provided in each respective standard: with IS800 [88,89], NZS3404 [90], EC3:1-1 [63], CSA S16 [59], AISC360-16 [60], and GB50017 [66].

showcases the ultimate compressive strengths of modular steel columns grouped with shear-keyed IMCs derived from various international standards, including IS800 [88,89], NZS3404 [90], EC3:1-1 [63], CSA S16 [59], AISC360-16 [60], and GB50017 [66]. This comparison illustrates the variance in compressive resistance values according to different engineering standards. The effectiveness of a new approach, grounded in the force transmission principle of modular steel grouped tubular columns with shear key connectors, is validated through the analysis of P_u values from 103 FEMs of modular grouped tubular columns, detailed in Figure 48. A comparative analysis involving Figure 48 and Figure 49 juxtaposes FEM outcomes with predictions from the newly proposed model, IS800 [88,89], NZS3404 [90], EC3:1-1 [63], CSA S16 [59], AISC360-16 [60], GB50017 [66], and the Li et al. [71] model’s equations. The resulting mean analysis-to-prediction ratios stand at 1.66, 0.95, 0.96, 0.89, 1.00, 0.98, 0.90, and 0.86, respectively, suggesting the new model, on average, predicts conservative outcomes (mean > 1.00). The distribution of conservative (% safe) outcomes is 78 (76%), 31 (30%), 34 (33%), 22 (21%), 34 (33%), 34 (33%), 19 (18%), and 22 (21%). Existing models, including codes and the Li et al. model, generally overestimate results for modular grouped tubes with shear-keyed IMCs, with GB50017 (81% unsafe outcomes) and Li et al. (78% unsafe outcomes) showing a higher tendency towards unsafe predictions. However, the proposed model achieves 76% conservative estimates, with 25 predictions exceeding expected values by a tolerance within 20%. Enhancements to the model’s predictive accuracy and conservatism could be realized with an adjusted safety factor. Modifications to international standards such as IS800 [88,89], NZS3404 [90], EC3:1-1 [63], CSA S16 [59], AISC360-16 [60], and GB50017 [66], aimed at accounting for the nominal strength reduction in shear-keyed grouped tubular columns due to local buckling and the increased effective length factor from IMCs at both ends, are discussed. Figure 50 and Figure 51 further explore FEM outcomes in relation to the modified prediction equations of the standards mentioned above and the unmodified Li et al. [71] model and newly proposed model, demonstrating mean analysis-to-prediction ratios of 1.77, 1.57, 1.65, 1.63, 1.77, and 1.91, respectively. These modifications to international standards on average lead to safe outcomes (mean > 1.00), with IS800 [88,89], NZS3404 [90], EC3:1-1 [63], CSA S16 [59], AISC360-16 [60], and GB50017 [66] showing conservative results of 100 (97%), 97 (94%), 100 (97%), 100 (97%), 100 (97%), and 103 (100%), respectively, and only a minimal number of unsafe outcomes. By considering the actual behavior of shear-keyed grouped tubular columns, this enhancement in outcomes’ conservatism can be further optimized by proposing suitable safety factors for the revised equations in further study.

9. Conclusions

This study delved into the compressive behavior of modular steel tubular columns grouped with shear-keyed IMCs in MSSs. An FEM was developed and validated using axial compression tests conducted on both cold- and hot-rolled columns. Then, it was utilized to analyze 103 cases investigating the influence of shear keys, tube dimensions, and the number of tubes. The ultimate compression capacities were estimated using both models from the literature and the code prediction equations from IS800, NZS400, CSA-S16, EN1993-1-1, AISC360-16, and GB50017, along with the newly proposed buckling strength equations. The key findings include the following:

• The FEMs closely matched the experimental results, maintaining over- and underestimation errors within 20%, and accurately predicted both the failure modes and ultimate compression resistance of 28 cold-formed and hot-rolled tubes and multi-column walls, with a mean (Cov) for ultimate capacity at 1.00 (0.05).
• The generalized load-shortening behavior of modular steel grouped columns transitions through linear, nonlinear parabolic, and recession phases, ceasing load resistance upon reaching ultimate compressive strength, often marked by local buckling. A recession phase indicates a reduction in capacity alongside increased local buckling. Regions of the columns demonstrate either elastic or plastic buckling based on the stress being below or above the yield strength.
• Some sections of shear-keyed grouped columns do not yield; however, yielding occurred in most locations. Failures predominantly result from local buckling near shear keys or at the column’s mid-height in regions that are either yielding, not yielding, or a combination, showcasing elastic, plastic, or both types of local buckling. Elastic buckling results in either inward or outward deformation, whereas plastic buckling leads to an S-shaped waveform consistent across opposing sides but varying on adjacent sides. Adjacent tubes might buckle symmetrically or asymmetrically, with symmetrical buckling creating a double S-shaped pattern with bulged-out sections on the inner sides.
• Enhancements in the dimensions of shear keys (both thickness and height), tube dimensions (thickness and the ratio of length to width), and the number of tubes (increasing from 2 to 3 and then to 4) significantly enhance the compressive behavior of shear-keyed grouped columns by boosting both their strength and initial stiffness. The strength of columns grouped in threes and fours increases by factors of 1.6 and 2.1, respectively. On the contrary, the mutual effect between tubes and shear keys, such as an increasing gap between them, leads to a reduction in both strength and stiffness when the gap widens from 0 to 6 mm. Similarly, a gap between tubes initially contributes to an increase in strength and stiffness up to a gap of 36 mm but then leads to a decrease beyond a 50 mm gap. While shear-key dimensions notably affect failure modes, other factors exert less influence, and there is a weak correlation between pre- and post-ultimate ductility and these variables.
• Predictions based on the current Li et al. model and international standards such as IS800, NZS3404, EN1993-1-1, CSA-S16, AISC360-16, and GB50017 tended to produce non-conservative estimates, with approximately 80, 71, 68, 80, 68, 68, and 83 instances of overestimation, respectively. Conversely, the newly proposed theoretical equations yielded conservative results for 78 FEMs while overestimating 25 outcomes, primarily within a 20% range, achieving 76% conservative accuracy.
• Modified prediction equations for standards like IS800, NZS3404, EN1993-1-1, CSA-S16, AISC360-16, and GB50017 have yielded conservative predictions for the ultimate capacity of shear-keyed grouped columns, with outcomes of approximately 100, 97, 100, 100, 100, and 103 conservative predictions and only 3, 6, 3, 3, 3, and 0 instances of overestimation, respectively, enhancing conservatism to 97%, 94%, 97%, 97%, 97%, and 100%. These modifications underscore an improvement in the predictive accuracy and reliability of these standards for assessing the ultimate capacity of shear-keyed grouped tubular columns.

10. Future Work

This study primarily evaluated the axial compression performance of MSS tubular columns grouped with shear-keyed IMCs and proposed a theoretical equation for their design. The conclusions drawn mainly relate to the analyzed FEMs. Future studies will expand to cover tri- and tetra-neighboring modular grouped columns, incorporating experimental, numerical, and analytical assessments. These studies will involve tests, numerical analyses, and theoretical assessments on single, double, triple, and quadruple modular columns, considering variations in heights, cross-sectional lengths, and widths of columns, as well as different IMCs at the column ends (such as welded IMCs, shear-keyed IMCs, columns fitted with rotary-type IMCs, etc.). These investigations will provide insights into the effect of varying IMCs on buckling behavior and enable statistical comparison with the findings studied in this research, facilitating the proposal of design recommendations and advancements in MSSs. Moreover, further investigations will include columns with varying boundary conditions, such as pinned–pinned, fixed–pinned, and fixed–fixed boundary conditions, to understand their effect on failure and buckling behavior. Additionally, the scope will extend to include thin-walled cold-formed columns with varying steel types/strengths, innovative cross-sections, and composite columns, providing comprehensive findings for appropriate designs. Subsequently, further research and computation will be conducted to address the critical value of the slenderness ratio and stress–slenderness ratio in the context of the elastic-plastic buckling of shear-keyed grouped tubular columns. Finally, experimental, numerical, and theoretical studies will explore modular corner, middle, and interior frames with two, four, and eight columns fitted above and below varying IMCs, connecting them in modules under both sway and non-sway frame conditions, and propose the effective length factors, critical buckling load, and ultimate compressive strength equations for pinned, semi-rigid, and rigid situations of frames boundaries to propose a categorization system for rigid, pinned, and semi-rigid IMCs to improve interconnection behavior prediction and MSS dependability. This approach aims to enhance the practical applicability of MSSs and facilitate their progress in structural engineering.