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Abstract

This study investigates the structural performance of a newly proposed bolted connection for panelized modular structures, using high-strength bolts and extended plates, aimed at addressing Canada’s housing shortage, particularly in remote areas. A finite element model (FEM) was developed and validated against experimental data to evaluate the connection’s behaviour. A parametric study of 240 FEMs was conducted on unstiffened connections, examining the effects of extended plate thickness, bolt configuration, and bolt diameter. Results show these parameters significantly affect moment capacity, failure mode, and inelastic rotational behaviour. The bolt arrangement and quantity fostered a “strong column–weak beam” mechanism, enabling plastic hinging in the beam. Additionally, bolt diameter and plate thickness were found to govern ultimate rotation capacity and failure modes. To support practical application, design charts and a mathematical equation based on genetic regression were developed to predict ultimate capacity, thereby streamlining the design process and reducing trial iterations.

Keywords

beam-column bolted connection hollow structural section finite element model panelized modular structure design guidelines genetic regression

Introduction

Background

Canada is experiencing a severe housing shortage and an increasing rate of homelessness, especially in remote and northern regions. This crisis has been highlighted by several organizations, including the United Nations General Assembly on Adequate Housing (United Nations, 2019). These regions face unique engineering challenges related to material accessibility, extreme climate conditions, and logistical constraints (Elhadary et al., 2024b). In this context, modular housing has emerged as a practical and scalable solution, where transportation and lifting requirements pose significant challenges. This approach employs standardized building components or industrially fabricated modules, which are systematically assembled on-site through a mechanized installation process, enhancing efficiency and precision in construction. As an emerging technology, modular construction is transforming building practices across residential, commercial, institutional, and healthcare sectors. Countries such as Finland, Norway, and Sweden have adopted modular construction extensively approximately 45% of their residential properties are modular (Modular construction reports & industry analysis, 2023).

Modular construction is generally classified into two main categories: The two-dimensional (2D) panelized system and the three-dimensional (3D) volumetric system (i.e., a box-like structure). The 2D panelized system involves the off-site production of individual structural components or panels, transported and assembled on-site. In contrast, the 3D volumetric system entails creating fully finished three-dimensional modules off-site, which are transported and assembled into the final structure. The 2D panelized system has many advantages over the 3D volumetric system, including reduced transportation expenses, increased adaptability, and less material waste. The 2D system can be compactly packaged for transport, resulting in substantial cost savings, particularly in remote areas (Lawson et al., 1999). Furthermore, 2D is less constrained by transportation and lifting requirements, offering greater adaptability in various construction scenarios. In addition to its other advantages, the 2D panelized system significantly decreases the labour required for interconnection assembly. As a result, 2D panelized is generally favoured over the 3D volumetric system as the method of choice for modular construction. This preference is due to its efficiency and cost-effectiveness. The architectural framework for a two-dimensional panelized system consists of beam-floor modules and column modules, commonly known as a “panelized steel-modular” structure, where beam-column connections are used to join the columns and beams horizontally. These structures offer several technical advantages, making them an attractive alternative to traditional building materials. Unlike conventional timber structures. Cold-formed steel is a non-organic, non-combustible material that is resistant to mould, rot, and insect damage (American Institute of Steel Construction, 2021). Additionally, its lighter weight compared to concrete leads to significant savings in transportation and on-site handling (Steel Framing Industry Association (SFIA), 2024).

The connection of the 3D volumetric modular can be categorized into post-tensioned joints, locking devices, and bolted connections (Corfar and Tsavdaridis, 2022; Rajanayagam et al., 2021). Bolted connections are divided into three categories: fitting-to-fitting, beam-to-beam, and column-to-column joints. Various studies have proposed techniques to facilitate loading sharing and transmission between modular units, particularly for the column-to-column joint derived from the column splice joint. Numerous studies have proposed different methods for improving connection performance in modular structures. For example, (Choi et al., 2016; Deng et al., 2018; Gunawardena, 2016) suggested using plates and bolts around the perimeter of column sections to enhance the overall strength and stability. On the other hand, (Zhang et al., 2024) introduced a novel column-to-column connection using high-strength bolts, improving tensile behaviour. Other researchers, such as (Ma et al., 2021; Zhong et al., 2024), proposed incorporating steel boxes within column modules, with long bolts connecting the module columns. (Lian et al., 2021) examined the structural response to seismic loads of corner fitting connections in modular steel buildings, highlighting the significant impact of robust column-to-column connections in resisting lateral forces during an earthquake. Additionally, (Bazarchi et al., 2024) introduced an inbuilt component filled with grout, enhancing slip resistance between modules. (Chen et al., 2017) introduced an innovative modular connection utilizing plug-in devices and long bolts for vertical and horizontal connectivity through the module beams, offering improved ease of assembly and structural integrity. Finally, (Deng et al., 2017) analyzed various steel column connections in modular construction, including novel gusset plate configurations and bolted connections for open beam cross-sections such as channels, concluding that these solutions optimize the modular structure’s overall stiffness and performance. (Xu et al., 2020) proposed a method for connecting ceiling and floor beams using bolts along their length, which maximizes the benefits of combined double-beam action in modular steel buildings. In a similar context, (Lee et al., 2017) strengthened the rigidity of beam-to-beam connections by adding stiffener plates and ceiling brackets at the beam ends, significantly improving the modular system’s seismic performance. The third category is the fitting-to-fitting joint, where (Hwan et al., 2017) proposed a cubic steel bracket that facilitates the assembly of module units through bolts in various directions. Their experimental study demonstrated that steel brackets effectively resisted bending and shear forces. (Yu and Chen, 2018) proposed a connection detail for modular units resembling containers, which used an intermediate plate and a single bolt. Their study focused on the efficiency of the connection in terms of simplicity and cost-effectiveness, ensuring easy on-site assembly.

For 2D panelized steel modular structures, limited studies have been conducted on column-column-beam bolted connections, which are essential for rapid on-site assembly (Liu et al., 2017, 2019; Zhan et al., 2021). Liu et al. (2017) developed a bolted joint system that joins lower box-shaped columns via bolted flange connections and secures H-shaped beams using endplates and additional cover plates. Their experimental study included cyclic tests to assess the effect of bolt quantity and bolt slippage on joint stiffness and energy dissipation during seismic events. Their findings indicated that increasing bolt quantity significantly improved the energy dissipation capacity, enhancing the connection’s seismic resilience. Further, (Liu et al., 2020) improved the H-section beam through the addition of a trapezoidal stiffener. Their study demonstrated that moving the web bolts to the beam flange enhanced the structural resilience under seismic loading, which has implications for modular housing in seismic-prone areas. (Fadden et al., 2012, 2015; Fadden and McCormick, 2013) investigated the seismic response of welded HSS-to-HSS joints, considering both reinforced and unreinforced configurations. Their findings demonstrated that the thickness of the plates and reinforcement configuration affected the connection’s stiffness and inelastic rotation, which are crucial for ensuring stability in modular systems. More recently, (Elhadary et al., 2024b) proposed a bolted moment-resisting connection for HSS sections tailored to modular housing, as shown in Figure 1. Their experimental results demonstrated that increasing the plate thickness, bolt count, and bolt configuration significantly enhanced joint stiffness and capacity. This innovation offers a cost-effective solution for high-performance modular housing, especially suited to Indigenous communities requiring resilient, adaptable housing (Elhadary et al., 2024a; 2024b). However, existing experimental studies lack a systematic investigation into the influence of key geometric parameters, such as bolt diameter, number of bolts, bolt arrangement, and plate thickness, on overall connection behaviour. Specifically, the parameter ranges were not tested in a randomized or statistically robust manner, limiting the development of generalized design guidelines.

Figure 1.

Schematic diagram of the assembly method for four box modules with various column connection configurations (Elhadary et al., 2024a; 2024b).

Research significance

This study addresses that gap by evaluating the structural performance of a proposed HSS bolted connection for 2D panelized steel modular housing. A comprehensive parametric analysis using 240 finite element models (FEMs) investigates the influence of key geometric variables such as bolt number, bolt diameter, bolt arrangement, and extended plate thickness on connection capacity and failure mode. Based on the analysis, practical recommendations are proposed to mitigate premature failure and enhance connection reliability. Design charts were developed to aid in predicting the connection’s failure pattern according to selected parameters. These charts eliminate the need to design the connection by displaying the nominal moment of the connection as a ratio of the plastic moment of the beam, considering different bolt arrangements, extended plate thicknesses, and bolt diameters. A simplified symbolic regression equation was developed to predict the ultimate capacity of bolted connections by integrating key parameters such as bolt diameter, number of bolts, bolt arrangement, and extended plate thickness. This research holds significant implications for the future of housing in remote communities, potentially transforming the landscape of affordable and sustainable housing solutions.

Numerical study

Finite element model details

A 3D FEM of a steel HSS-to-HSS connection was developed using LS-DYNA software (Livermore software technology corporation (LSTC), 2007) and validated with the experimental results obtained by (Elhadary et al., 2024b). The geometric parameters and detailed dimensions of the six specimens tested experimentally are illustrated in Table 1 and Figure 2. The bolt arrangement (AR), defined in Table 1, represents the ratio between the perpendicular bolt pitch distance

(P_{n})

and the parallel bolt pitch distance

(P_{p})

. The FEMs were executed on the Digital Research Alliance of Canada (Digital Research Alliance of Canada, 2021) (i.e., SHARCNET). Each simulation required an average duration of 48 hours to achieve the designated termination point, encompassing 760 computational time steps. Figure 3 illustrates the boundary conditions applied on the FEM, in alignment with the experimental test conditions, where the column was fixed at the support, the beam was laterally restrained along the X-axis, and a prescribed motion in the Y-axis was applied to the bracket plate, which has dimensions of

340 \times 170 mm

and a moment lever arm of 1080 mm. All specimen components were defined as steel 350W and grade A325 high-strength long bolts of diameter

{5 / 8}^{″}

(15.875 mm) were used to assemble the joint. The steel’s elastic-plastic and hardening behaviours were modelled using a material model named “Mat_Piecewise_Linear_Plasticity”. The uniaxial tests were conducted following ASTM A370 (ASTM, 2014) using two Digital Image Correlation (DIC) cameras to accurately capture the material properties and true stress-strain behaviour of 350W steel and A325-grade high-strength bolts, as illustrated in Figure 4. The DIC system, a non-contact optical measurement technique, enabled precise monitoring of strain distribution and variation across the cross-sectional area. The key material properties obtained from these tests are summarized in Table 2. All parts of the model were created using eight-node 3D solid elements with lower integration and an hourglass model to prevent shear self-locking. A total of 390,000 solid elements were used in the FEM, with different mesh sizes applied to different sections of the connection areas, as illustrated in Figure 5. A 4 mm mesh was used for the HSS sections, and seven finer mesh layers were employed to model the profile thickness. Inflation layers were applied around the bolt holes to capture the stress gradient more effectively. This mesh strategy was developed based on a mesh sensitivity analysis, where various mesh sizes (i.e., 20 mm, 10 mm, 6 mm, 4 mm, and 3 mm) were evaluated by comparing the moment–rotation responses from FEM simulations and experimental results for different tested specimens, as illustrated in Figure 6. The interaction between bolts and steel sections was specifically defined to capture the load transfer and possible slip behaviour. The bolts were modelled as solid elements, and the contact between the bolt shank and the steel bolt holes was simulated using surface-to-surface mortar contact with a penalty friction formulation. A friction coefficient of 0.3 was applied to represent the sandblasted steel surface condition, consistent with the general frictional interactions in the connection. This frictional contact allowed for relative slip and bearing between the bolt and hole, enabling a realistic simulation of bolt hole deformation and bearing stresses. A 2 mm clearance between the bolt shank and the bolt hole was included in the model to accurately reflect the actual behaviour observed experimentally, allowing for initial slip and slight movement before bearing occurs. The same contact formulation was also applied to model the interfaces between the beam and the two extended steel plates, ensuring consistent simulation of frictional interactions throughout the connection. In contrast, the welded contacts were modelled using surface-to-surface-mortar-tied contact. Bolts were tightened to a snug-tight condition, indicating that no measurable pretension load was applied prior to testing. The model incorporated nonlinear material behaviour, large deformation, implicit non-linear time-step analysis, and the full Newton-Raphson method with a modified arc-length approach.

Table 1.

Summary of design parameters for test specimens (Elhadary et al., 2024b).

Spec. ID	Extended plate thickness (mm)	Number of bolts	Bolt Arrangement (AR)	Presence of stiffener
EP13-B6-AR1.5	13	6	1.5	Not included
EP10-B6-AR1.5	10	6	1.5	Not included
EP10-B6-AR1.0	10	6	1.0	Not included
EP13-B8-AR1.5	13	8	1.5	Not included
EP13-B6-AR1.5-ST	13	6	1.5	Included
EP10-B8-AR1.5	10	8	1.5	Not included

Figure 2.

Detailed dimensions of tested specimens: (a) EP13-B6-AR1.5, (b) EP10-B6-AR1.5, (c) EP10-B6-AR1.0, (d) EP13-B8-AR1.5, (e) EP13-B6-AR1.5-ST, and (f) EP10-B8-AR1.5.

Figure 3.

Boundary conditions of the proposed FEM and tested specimens (Elhadary et al., 2024b).

Figure 4.

The true stress-equivalent strain curves for tested specimens, the strain distribution utilizing the DIC camera, and a schematic diagram of the uniaxial test (Elhadary et al., 2024b).

Table 2.

Experimental material parameters (Elhadary et al., 2024b).

Profile	Material grade	Yielding stress $F_{y}$ $(MPa)$	Yielding elongation $ε_{y}$ $(%)$	Ultimate stress $F_{u}$ $(MPa)$	Ultimate elongation $ε_{u}$ $(%)$	Elasticity modulus $E$ $(GPa)$
Column	350W	366	0.18	465	15.3	204
Beam	350W	364	0.18	462	14.9	203
Extended plate	350W	367	0.18	470	15.5	204
High-strength bolts	A325	570	0.27	834	11.2	210

Figure 5.

Mesh configuration of the proposed FEM.

Figure 6.

Mesh sensitivity analysis of the moment–rotation response under various mesh sizes: (a) EP13-B6-AR1.5-ST, and (b) EP10-B8-AR1.5.

Model validation

Figure 7 illustrates the moment–rotation relationships derived from both the experimental tests and the developed FEM for six specimens. The applied moment was calculated by multiplying the load cell force by the 1080 mm distance from the loading point to the face of the column. Connection rotation was determined by subtracting the column rotation from the beam rotation. The lower initial stiffness and minor fluctuations observed in the FEM moment–rotation curves can be attributed to the modelled 2 mm bolt clearance, as shown in Figure 7. This clearance allowed an initial slip between the bolt shanks and the hole surfaces, with load transfer occurring primarily through plate–beam bearing. Once the bolts fully engaged with the hole surfaces, the connection began to resist loads through bolt bearing, resulting in an increase in stiffness. Table 3 summarizes the comparison between experimental and FEM results in terms of ultimate moment, ultimate rotation, initial stiffness, and failure modes for all tested specimens. The initial stiffness was calculated as the slope of the moment–rotation curve up to 10% of the peak moment to ensure it reflects the elastic behaviour. The FEM demonstrated good agreement with the experimental results, with FEM/EXP ratios ranging from 0.97 to 1.01 for ultimate moment, 0.91 to 1.06 for ultimate rotation, and 0.91 to 1.14 for initial stiffness. These findings indicate that the FEM effectively captures the overall nonlinear response of the connections. Figure 8 shows the failure mode derived from the proposed FEM and the corresponding specimens incorporated within the experimental program. The FEM effectively captured various failure modes that occurred during the loading of the specimens. Specimens such as EP13-B8-AR1.5, EP10-B6-AR1.0, EP10-B8-AR1.5, and EP13-B6-AR1.5-ST initially failed due to bolt bearing issues, leading to local buckling of the beam’s compression flange. In contrast, specimens like EP13-B6-AR1.5 and EP10-B6-AR1.5 exhibited block shear rupture after the onset of local buckling.

Figure 7.

Comparison between the moment-rotation curves of the test specimens and the FEM: (a) EP13-B6-AR1.5, EP13-B6-AR1.5-ST, (b) EP10-B6-AR1.5, EP10-B6-AR1.0, and (c) EP13-B8-AR1.5, EP10-B8-AR1.5 (Elhadary et al., 2024b; 2025b).

Table 3.

Comparison between FEM and experimental results.

Specimen ID	Ultimate moment (kN.m)		FEM/EXP	Ultimate rotation (rad)		FEM/EXP	Initial stiffness (kN.m/rad)		FEM/EXP	Failure mode
Specimen ID	Exp	FEM	FEM/EXP	Exp	FEM	FEM/EXP	Exp	FEM	FEM/EXP	Failure mode
EP13-B6-AR1.5	96.3	94.4	0.98	0.193	0.179	0.93	920	992	1.08	Block shear failure and local buckling of the beam
EP10-B6-AR1.5	90.2	88.3	0.98	0.161	0.152	0.94	759	856	1.14	Block shear failure and local buckling of the beam
EP10-B6-AR1.0	97.1	96.6	0.99	0.210	0.189	0.91	1020	1072	1.05	Bearing failure and local buckling of the beam
EP13-B8-AR1.5	93.8	94.03	1.01	0.154	0.163	1.06	1996	2103	1.06	Bearing failure and local buckling of the beam
EP13-B6-AR1.5-ST	87.1	86.2	0.99	0.170	0.171	1.01	4000	3623	0.91	Bearing failure and local buckling of the beam
EP10-B8-AR1.5	97.8	95.03	0.97	0.102	0.097	0.95	1448	1602	1.10	Bearing failure and local buckling of the beam

Figure 8.

Comparison of failure modes between the proposed FEM and experimental results for three specimens: (a) EP10-B6-AR1.0, (b) EP13-B6-AR1.5-ST, and (c) EP10-B8-AR1.5 (Elhadary et al., 2024b; 2025a).

Study parameters

A comprehensive parametric analysis involving 240 FEMs was conducted to investigate the influence of different geometric variables on the mechanical performance of the proposed unstiffened edge modular connection (i.e., L-shape). These models incorporate several parameters, including the thickness of the extended plate (T_pl = 8, 10, 13, 15 mm), the number of bolts (n = 4, 6, 8, 10), the bolt diameter (D = 12.7, 16, 19, 22, 25 mm), and the bolt arrangement ratio (AR = 1.5, 1.0, 0.6). The AR represents the ratio between perpendicular and parallel bolt pitch distances $P_{n} / P_{p}$ , where AR1.5 denotes a ratio of 1.5, AR1.0 signifies an equal ratio, and AR0.6 represents a ratio of 0.6. The interplay of these parameters is visually represented in Figures 9 and 10. The extended plate dimensions depended on the bolt arrangement and the number of bolts. While other parameters, such as the length of the specimen, cross sections of the beam and column, and the perpendicular pitch distance $(P_{n})$ , remain constant. The perpendicular pitch distance is calculated as half the section depth (H) $P_{n} = 0.5 H$ , while the parallel edge distance equals the normal edge distance. The specimens were coded as follows: the extended plate thickness (EP), the number of bolts (B), the bolt arrangement (AR) and the bolt diameter (D). For example, EP15-B8-AR1.0-D19 indicates a 15 mm plate thickness, eight bolts, an AR ratio of 1.0, and a bolt diameter of 19 mm. Utilizing the results extracted from the 240 FEMs, design charts and a simplified mathematical equation have been developed to predict failure modes and guide the selection of various parameters, aiding decision-making during the design stage.

Figure 9.

A visual representation of the interplay between various parameters to generate 240 FEMs.

Figure 10.

Geometric parameters incorporated in the study: (a) bolt arrangement; (b) extended plate thickness; and (c) number of bolts and bolt diameters (Elhadary et al., 2024b).

Results and discussion

A total of 240 FEMs exhibited a variety of failure modes depending on the geometric configuration of the connection. The predominant failure mechanisms included bolt bearing failure, block shear failure, bolt shear failure, rupture failure of the upper extended plate, and local buckling of the beam and local buckling of the column, as shown in Figure 11. The following sections provide a detailed analysis of how the studied parameters influence the failure mode, ultimate capacity, rotation behaviour, and ductility of the connection. In particular, the joint ductility index $(φ_{j})$ defined as the ratio between the ultimate rotation $(φ_{\max})$ , and the yield rotation $(φ_{y}),$ used to assess the deformation capacity of the joint. This index is a critical indicator for ensuring ductile behaviour and mitigating the risk of progressive collapse in steel moment-resisting frames.

Figure 11.

Failure modes of various specimens (a) EP10-B4-AR1.5-D12.7, (b) EP8-B4-AR1.5-D25, (c) EP15-B8-AR1.5-D25, (d) EP8-B8-AR1.5-D22, (e) EP8-B4-AR1.0-D22, and (f) EP15-B8-AR0.6-D16.

Effect of bolt arrangement

Three distinct bolt arrangements (i.e., AR1.5, AR1.0, and AR0.6) were examined in the study, each affecting the longitudinal length of the extended plate differently. The bolt arrangement was found to alter the connection failure mechanism (i.e., localizing the failure mode at the beam, column, or extended plate). The applied force on the beam is distributed via the bolt bearing and beam bearing on the extended plate. Figure 12 illustrates the impact of the AR on the distribution of forces through the bolts and lower extended plate for three specimens, namely EP15-B8-AR1.5-D22, EP15-B8-AR1.0-D22, and EP15-B8-AR0.6-D22. These specimens represent the dominant general behaviour of most of the studied parameters. This figure presents the resultant forces applied to the bolts and the lower extended plate. The bolts are initially subjected to bearing forces in the X-direction, which gradually increases sliding. Meanwhile, beam-bearing forces are applied in the Y-direction to the lower extended plate. The forces acting on the bolts and extended plate are determined by the contact algorithm between the beam and bolts and between the extended plate and the beam.

Figure 12.

Relationship between the force applied on the beam and the resultant forces distributed on the bolts and the lower extended plate for different bolt arrangements: (a) AR1.5, (b) AR1.0, and (c) AR0.6.

Upon examining the force distribution across different bolt arrangements, it was noted that the force is not equally distributed among the bolts. The outer row of bolts, specifically bolts numbers seven and eight, transferred almost all the bearing forces compared to the other bolts. The AR1.5 bolt arrangement triggered the bolt bearing failure at an earlier loading stage, reducing the beam bearing forces applied on the extended plate. This led to a reduction in the ultimate rotation of the connection, ultimately resulting in failure due to local instability of the beam and bearing-induced failure at the compression flange. For AR1.5 with an applied force of 40 kN, the outer bolts numbers 7 and 8 were able to reach the maximum bearing capacity of 100 kN and the lower extended plate was only affected by 65 kN, which explains the lower rotation of the extended plate. The AR0.6 bolt arrangement influenced the bearing stage at the outer bolts. Initially, it caused column rotation without triggering the bolt bearing mechanism until the applied force reached 35 kN, which led to a significant increase in the load on the lower extended plate, nearly doubling the force compared to AR1.5 (i.e., 65 kN to 110 kN). The outer bolts only carried 60 kN, which delayed the activation of the bearing mechanism, and the connection relied more on the rotational capacity of both the column and the extended plate. The altered force distribution caused extensive rotation of the lower extended plate, significantly affecting the upper extended plate. This resulted in rupture failure of the upper extended plate or, when the extended plate thickness exceeded 8 mm, caused local column buckling, an undesirable failure mode for the connection. The AR1.0 bolt arrangement exhibited a combined behaviour between the AR1.5 and AR0.6 configurations. It delayed the bolt-bearing mechanism’s activation by inducing rotation in the column, while also increasing the force applied to the extended plate by 35%. However, the AR1.0 arrangement reached the maximum bolt-bearing load at a later applied force of 55 kN, compared to 40 kN for AR1.5. This delay helped postpone the onset of beam instability caused by bearing failure at the lower flange. To maintain the principle of “strong column weak beam” and prevent any undesirable column or extended plate failure, bolt arrangements AR1.5 and AR1.0 are recommended, as these configurations enable the connection to reach its capacity via bearing-induced failure, subsequently resulting in beam instability, unlike bolt arrangement AR0.6, which showed the undesirable mode of failures, which happened regardless of other parameters studied.

Effect of the bolt quantity

The study examined the impact of the connection mechanism by utilizing four different bolt configurations: four, six, eight, and ten bolts, with various diameters. Figure 13, Figure 14, and Figure 15 illustrate failure modes experienced by the specimens depending on the number of bolts and bolt arrangements (i.e., AR1.0, AR1.5, and AR0.6) used in the connection, which played a crucial role in achieving moment capacity. It was observed that using four or six bolts with a 12 mm diameter was insufficient to resist shear forces for all bolt arrangements. The uneven distribution of forces across the bolt rows led to premature connection failure due to bolt shear, as shown in Figure 11(a). Increasing the bolt number to eight and ten, combined with AR0.6 or AR1.0 arrangements and a 12.7 mm diameter, caused the entire connection to rotate excessively before any bolt-bearing failure could occur. This led to undesirable failure by local buckling of the column, rather than the intended beam failure mode. However, with the bolt arrangement AR1.5 with eight and ten bolts, the connection failed by local buckling. This is because the increased number of bolts provided sufficient shear resistance to achieve the maximum connection capacity, and AR1.5 enhanced the bolt-bearing mechanism during the loading stage. Using four bolts with a 16 mm diameter, regardless of bolt arrangement, consistently resulted in early failure due to bolt shear, mainly in the outer bolt rows. For larger diameters (19 mm, 22 mm, and 25 mm), AR1.5 with four bolts caused block shear failure at the beam, as shown in Figure 11(b), even when using an 8 mm thick extended plate. However, AR1.0 with four bolts reduced force demand on the bolts, resulting in bolt bearing failure at the beam instead of block shear or local buckling. A similar trend was seen with AR0.6, especially when using 8 mm and 10 mm plates and a 19 mm bolt diameter. For AR1.5, increasing the bolt number from 6 to 8 and 10 (with diameters from 16 mm to 25 mm) enhanced capacity by an average of 11% and 18%, respectively. The added bolts improved force distribution and delayed beam local buckling. However, for the AR0.6 bolt arrangement, increasing the number of bolts from 6 to 8 and from 6 to 10 with different bolt diameters only increased the capacity by an average of 10% in both cases. In the latter case, AR0.6 depends on the ultimate strength of the column, which led to the undesirable local buckling failure mode of the column because it eliminated the bolt bearing force and caused the column to rotate by beam bearing on the extended plate until it reached its ultimate capacity and failed, as shown in Figure 11(f) and Figure 15. Based on these findings, it is recommended to use six bolts with AR0.6, regardless of diameter, to achieve a desirable beam local buckling failure mode. This configuration provided the highest connection capacity (i.e., 100 kN.m), outperforming even the AR1.5 arrangement with ten bolts.

Figure 13.

Relation between the number of bolts vs extended plate thickness vs ultimate moment of bolt arrangement AR1.5 for different bolt diameters: (a) D12,7; (b) D16; (c) D19; (d) D22; and (e) D25.

Figure 14.

Relation between the number of bolts vs extended plate thickness vs ultimate moment of bolt arrangement AR1.0 for different bolt diameters: (a) D12.7; (b) D16; (c) D19; (d) D22; and (e) D25.

Figure 15.

Relation between the number of bolts vs extended plate thickness vs ultimate moment of bolt arrangement AR0.6 for different bolt diameters: (a) D12.7; (b) D16; (c) D19; (d) D22; and (e) D25.

Influence of increasing the bolt diameter

The influence of different five-bolt diameters was investigated (i.e.,12.7 mm, 16 mm, 19 mm, 22 mm, and 25 mm). Utilizing a bolt diameter equal to 25 mm leads to an early capacity gain due to higher bearing resistance, and it reduces the ultimate rotation of the extended plate by an average of 60% compared to a bolt diameter of 16 mm. For bolt diameters of 19, 22, and 25 mm, the average ultimate moment capacity decreased by 10% for different bolt numbers and arrangements (i.e., AR1.0 and AR1.5) compared to the 16 mm bolt diameter. This can be attributed to the enlarged bolt-hole area at the beam flange, which caused a higher stress concentration earlier at the beam’s compression region due to the bearing forces, eventually leading to beam instability at a reduced capacity. Figure 16 illustrates the relationship between the nominal to plastic moment beam ratio, the bolt diameter to extended plate thickness ratio, and the number of bolts for bolt arrangements AR1.5 and AR1.0. Figure 16(b) shows how to determine a point on the 3D plot for a bolt diameter of 16 mm and a bolt number of 8 using arrows. The effect of different extended plate thicknesses on the different bolt diameters is shown for each group of bolt numbers. When the extended plate thickness is almost equivalent to the bolt diameter, the nominal capacity increases by 20% (relative to the plastic moment of the beam) for bolt AR 1.5. This occurs because it reduces the bearing forces on the extended plate, allowing it to transfer more force, as shown in the circle group of six bolts in Figure 16(a). Furthermore, increasing the number of bolts from 6 to 8 and from 8 to 10 boosts the capacity by an average of 10% in AR 1.5 and 20% in AR 1.0 due to the reduction in bearing forces. Based on the findings in Figure 16, it is recommended to maintain the bolt diameter to an extended plate thickness ratio below 2, as this strategy helps delay stress concentration occurring at the compression flange of the beam, thereby avoiding local buckling.

Figure 16.

Relation between the number of bolts and the ratios D/T_pl, and M_n/M_p _beam for two bolt arrangements: (a) AR1.5, and (b) AR1.0.

The bolt diameter significantly influenced the reduction of ultimate rotation from 0.2 rad to 0.1 rad, independent of the presence of stiffeners. This effect was particularly notable for bolt diameters of 19 mm and 22 mm in the AR1.5 bolt arrangement with extended plate thicknesses of 13 mm and 15 mm, as this configuration relies on bolt-bearing mechanics, as shown in Figure 17. However, using a 25 mm bolt diameter did not result in any further reduction in the ultimate rotation of the connection. For bolt arrangement AR1.0, increasing the bolt diameter from 16 mm to 19 mm or larger altered the loading mechanics, making it behave similarly to AR1.5, where load transfer depends on bolt bearing mechanics. This resulted in a significant reduction in ultimate rotation from 0.3 rad to 0.09 rad, representing a 70% decrease, as shown in Figure 18. This effect was particularly notable for extended plate thicknesses of 13 mm and 15 mm with 6, 8, or 10 bolts. For bolt configuration AR0.6, increasing the bolt diameter from 16 mm to 22 mm reduced the ultimate rotation by 50% (from 0.3 rad to 0.2 rad), as shown in Figure 19. This is because this configuration does not rely on bolt-bearing mechanics to gain capacity. However, despite the reduction, its ultimate rotation remains higher compared to other configurations. Additionally, increasing the bolt diameter to 25 mm had no further effect on reducing the ultimate rotation. Regarding ductility, the connection exhibited an average value of approximately 2.5, positioning it between limited and moderate ductility according to the National Building Code of Canada (NBCC, 2020). This trend was particularly observed with increasing bolt diameter, especially in bolt arrangements AR1.0 and AR1.5.

Figure 17.

Relationship between bolt count, extended plate thickness, ultimate rotation, and ductility for bolt arrangement AR1.5 with various bolt diameters: (a) D12.7, (b) D16, (c) D19, (d) D22, and (e) D25.

Figure 18.

Relationship between bolt count, extended plate thickness, ultimate rotation, and ductility for bolt arrangement AR1.0 with various bolt diameters: (a) D12.7, (b) D16, (c) D19, (d) D22, and (e) D25.

Figure 19.

Relationship between bolt count, extended plate thickness, ultimate rotation, and ductility for bolt arrangement AR0.6 with various bolt diameters: (a) D12.7, (b) D16, (c) D19, (d) D22, and (e) D25.

Influence of extended plate thickness

The extended plate thickness (i.e., 8 mm, 10 mm, 13 mm, and 15 mm) significantly influenced both the ductility and capacity of the proposed connection, as shown in Figures 17 –19. When an extended plate of 8 mm thickness was used regardless of the bolt quantity and bolt diameters, the connection experienced premature failure due to tensile rupture within the smaller/net cross-section of the plate at the level of the bolts, as shown in Figure 11(d). Such rupture failure of the extended plate was observed to be triggered by bolt arrangements AR1.0 and AR0.6 with six or more bolts, where the beam-bearing forces on the extended plates played a significant role, causing significant rotation of both plates and eventually resulting in the early failure of the upper plate, as illustrated in Figure 12. However, the AR1.5 bolt arrangement, particularly when combined with an 8 mm plate and 16 mm bolt diameter, effectively mitigated this rupture failure, as shown in Figure 20. This was due to the unequal force distribution, where the outer bolt row reached its maximum bearing resistance first, allowing the other bolt rows to start resisting forces without distributing them to the tensile extended plate. In contrast, when larger bolt diameters (i.e., 19 mm, 22 mm, 25 mm) were used with the same AR1.5 configuration and 8 mm plate, rupture again occurred. This was due to the higher bearing resistance provided by the larger bolts, which delayed the engagement of inner bolt rows. As a result, the tensile capacity of the plate was exceeded before full force redistribution could occur, as shown in Figure 20. Increasing the plate thickness from 10 mm to 13 mm and from 10 mm to 15 mm led to average ductility improvements of approximately 30% and 60%, respectively, across various bolt configurations. In terms of strength, the ultimate capacity increased by approximately 5% and 10%, respectively. These enhancements are attributed to the increased resistance to yielding and the plate’s ability to transfer more force to the column cap, even after local buckling at peak moment, as shown in Figure 11(c). Based on these findings, it is recommended that the nominal tensile force capacity of the extended plate’s net section be designed to resist at least 1.2 times the nominal force demand, where the force demand is calculated by dividing the ultimate nominal moment capacity of the connection by the beam depth. This ensures that the extended plate remains sufficiently strong to transfer the developed moment without premature failure. It is worth mentioning that the bolt arrangement remains the primary factor in controlling the failure mode, even when the thickness increased to 15 mm. The following sections will examine the combined effects of plate thickness, a bolt diameter of 12.7 mm, and varying numbers of bolts.

Figure 20.

Failure modes of different specimens with an 8 mm thick extended plate: (a) EP8-B8-AR1.0-D16; (b) EP8-B8-AR0.6-D16; (c) EP8-B6-AR1.5-D16; (d) EP8-B8-AR1.5-D22.

Failure mode prediction charts

A total of 240 FEMs and six test specimens (Elhadary et al., 2024b) were performed to cover the main design parameters, so it was crucial to summarize the failure modes to develop design charts. The failure mode charts are designed to help predict the failure mode of a connection by adjusting different design parameters. The connection’s failure modes are categorized into five types: (1) bolt shear failure, (2) beam block shear failure, (3) rupture of the tensile extended plate, (4) local buckling of the beam and (5) local buckling of the column. The first three types are considered premature failures, occurring suddenly at lower values before reaching the maximum capacity of the connection at a value lower than the plastic moment of the beam. On the other hand, the local buckling of the beam and column helped the connection reach a higher capacity; however, it can affect the connection’s integrity. To clarify, the failure mode should occur in the beam to adhere to the “strong column, weak beam” principle. These charts provide a streamlined way to achieve the desired connection strength and failure mode to achieve specified strength and failure modes by selecting the appropriate bolt diameter, number of bolts, and extended plate thickness without performing several iterations of full design analysis each time. Figure 21 shows three failure mode charts for different bolt arrangements (AR1.5, AR1.0, AR0.6), where each chart contains subplots: the first subplot illustrates the relationship between the nominal moment of the connection and the plastic moment of the beam, with the number of bolts ranging from 4 to 10. The subsequent subplots, categorized by extended plate thickness (8 to 15 mm), show the relationship between the number of bolts and failure modes for different extended plate thicknesses. The design charts were subject to specific constraints, including the requirement to maintain consistent column and beam cross-sections throughout the study. In particular, the beam’s cross-section was HSS 152 × 152 × 6.4, while the column’s cross-section was HSS 152 × 152 × 9.54, based on the experimental specimens by (Elhadary et al., 2024b). The charts can be utilized for two main purposes: (i) to select the desired failure mode for the connection by determining the required combination of extended plate thickness, number of bolts, and bolt diameter, and (ii) to select the required nominal connection moment and the available bolt diameter. Based on this selection, the required number of bolts can be determined. Then, by referring to the subplots, the failure mode can be determined based on the known number of bolts and the chosen extended plate thickness. Figure 21(a) illustrates the first method on the failure mode charts, where the preferred failure mode of the joint can be chosen, such as the local buckling of the beam. This choice can be made from the lower subplots (i.e., highlighted in gray colour), considering the thickness of the extended plate equal to 10 mm. Subsequently, the number of bolts and their diameter (indicated by the dashed line) can be determined, which will be equal to 6 and 16 mm, respectively, necessary to achieve the desired failure mode. Lastly, the upper graph can be used to accurately predict the ultimate moment of the joint based on that geometric combination.

Figure 21.

Failure modes prediction charts for different bolt arrangements; (a) AR1.5; (b) AR1.0; and (c) AR0.6.

Mathematical formula for capacity prediction

A machine learning technique, symbolic regression, was employed to develop a mathematical equation by searching for the best-fitting model without relying on predefined assumptions. This approach uses genetic programming to evolve and optimize complex model structures, expressing them in mathematical form. The process begins with a population of randomly generated mathematical expressions, typically represented as trees, which are evaluated based on their fitness, measuring how accurately the target data are predicted. The most effective models are then selected and modified through mutation (i.e., random changes) and crossover (i.e., the recombination of parts from different models) to generate new candidate solutions. This evolutionary process is repeated over multiple generations, during which the population is gradually refined to identify the most accurate mathematical representation, as illustrated in Figure 22. Utilizing 240 results obtained from the developed FEMs and processed through Eureqa software (DataRobot, 2024). The resulting dataset was split randomly into 70% for training, 30% for testing, and validation to predict the ultimate capacity of the connection. Equation (1) predicts the ultimate moment capacity of the bolted connection based on key parameters that govern its structural performance. The number of bolts (N) influences load distribution with diminishing returns as more bolts are added. Bolt diameter (D) affects shear and bearing strength, while bolt arrangement (AR) reflects the geometric layout, impacting stress distribution. Extended plate thickness (EP) relates to the plate’s ability to resist deformation and transfer loads. The nonlinear terms in the equation capture the complex interactions between these parameters. This equation is developed through regression analysis of experimental and numerical data, ensuring it reflects the physical response of the connection rather than being purely empirical. The configuration parameters used in the symbolic regression process are summarized in Table 4. The prediction performance of the developed model was evaluated using the Pearson correlation coefficient (R), Mean Absolute Error (MAE), and Standard Deviation of Absolute Error for both the training and testing datasets, as presented in equation (2), equation (3), and equation (4), respectively. Figure 23 illustrates a scatter plot of the actual versus predicted capacities, with the training R-value reaching 0.97 and the testing R-value at 0.921. During the training phase, the model achieved an MAE of 2.903 kN.m with a standard deviation of 2.388 kN.m, indicating accurate and consistent predictions. In the testing phase, the MAE slightly increased to 3.529 kN.m, with a standard deviation of 2.808 kN.m, demonstrating the model’s strong generalization to unseen data. As shown in Figure 24, over 70% of the training data exhibited prediction errors within the range of 1.839% to 2.892%, while more than 40% of the testing samples had errors between −3.225% and 2.374%. The largest observed error in the testing set was −14.42%, accounting for only 0.014% of the total samples. Table 5 provides a comprehensive comparison of ultimate capacities and failure modes for the tested specimens, including results from experimental tests, FEM, the developed predictive equation, and design charts. The table highlights the close agreement between predicted and observed capacities, with ratios of FEM/EXP and Equation/EXP ranging from approximately 97% to 106%. Moreover, the failure modes predicted by the different methods show strong consistency across all specimens, underscoring the robustness of the developed model and the prediction charts in accurately capturing critical failure mechanisms. The analysis highlighted each input parameter’s impact on the ultimate capacity, with the number of bolts contributing the most at 52.08%, followed by extended plate thickness at 22.96%, bolt arrangement at 13.29%, and bolt diameter at 11.67%. This ranking of feature importance is crucial for guiding the genetic regression algorithm, allowing for more efficient model optimization and ensuring the most influential parameters are prioritized in the final model.

M_{u l t} = a + \frac{b}{N} + \frac{c}{{(d + N)}^{D}} + e \times \frac{N}{(A R \times e^{E P})} - f \times A R \times D

(1)

Figure 22.

Workflow of symbolic regression via genetic programming.

Table 4.

Symbolic regression configuration parameters.

Parameter	Value	Description
Number of generations	10,000	Maximum number of evolutionary iterations
Population size	1000	Total number of candidate solutions per generation
Total search space	9.9 × 10⁶	Approximate number of candidate models explored
Mutation probability	15%	Probability of introducing random variations
Max tree depth	15	Limits the nesting of operations to control complexity
Max tree length	30	Limits the number of nodes in the expression tree to avoid overfitting

Where:

a = 131.01, b = - 136.39, c = - 1.4818 e^{19}, d = 20.783, e = - 4140.6, f = 0.64622

Pearson (R) = \frac{\sum_{i = 1}^{n} ({y_{i}}^{'} - \bar{y^{'}}) (y_{i} - \bar{y})}{\sqrt{\sum_{i = 1}^{n} {({y_{i}}^{'} - \bar{y^{'}})}^{2} \sum_{i = 1}^{n} {(y_{i} - \bar{y})}^{2}}}

(2)

M A E = \frac{1}{N} \sum_{i = 1}^{N} | y_{i} - {\bar{y}}_{i} |

(3)

{S t d}_{A b s} = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} (| y_{i} - {\bar{y}}_{i} | - M A E)}

(4)

Figure 23.

Predicted versus target moment capacity: (a) training dataset, (b) testing datasets.

Figure 24.

Error histogram for predicting ultimate moment: (a) training dataset, and (b) testing dataset.

Table 5.

Comparison of ultimate capacities and failure modes for specimens using experimental, FEM, developed equation, and design charts.

Specimen ID	Ultimate capacity					Failure mode (EXP/FEM/Design charts)	Comments
Specimen ID	EXP.	FEM	Developed equation	FEM/EXP	Equation/EXP	Failure mode (EXP/FEM/Design charts)	Comments
EP13-B6-AR1.5	96.3	94.4	93.6	0.98	0.973	Local buckling of the beam	All agree
EP10-B6-AR1.5	90.2	88.3	92.02	0.98	1.020	Local buckling of the beam	All agree
EP10-B6-AR1.0	97.1	96.6	96.81	0.99	0.997	Bearing failure and local buckling of the beam	All agree
EP13-B8-AR1.5	93.8	94.03	98.4	1.01	1.049	Bearing failure and local buckling of the beam	All agree
EP10-B8-AR1.5	97.8	95.03	97.45	0.99	0.996	Bearing failure and local buckling of the beam	All agree

Conclusions

This paper examines the mechanical behaviour of the corner HSS-to-HSS connection for panelized modular houses in remote Indigenous regions. Two hundred and forty finite element models were created for unstiffened extended plates to explore the effect of various geometric parameters on ultimate moment capacity, rotation, failure modes, and ductility. The studied parameters showed that the bolt arrangement, number of bolts, extended plate thickness, and bolt diameter significantly influenced the connection capacity gain and failure mode. The following conclusion is drawn from the above analysis.

• The FEM results showed a strong correlation with the experimental results, with an average error of 3%. It is recommended that the nominal force of the extended plate be at least 1.2 times the nominal force demand, where the force demand is calculated by dividing the ultimate nominal moment capacity of the connection by the beam depth, to avoid premature connection failure caused by rupture failure of the extended plate.

• The utilization of AR1.5 and AR1.0 bolt arrangements is recommended, as these configurations enable the connection to reach its capacity through bolt bearing failure, thereby adhering to the “weak beam strong column” principle. The AR0.6 bolt configuration had an unfavourable effect on the loading mechanism of the connection, especially when 8 and 10 bolts were used, leading to column failure due to local buckling, which is indicative of a “weak column strong beam” scenario. This study emphasizes the critical role of bolt arrangement in maintaining the structural integrity and performance of connections.

• The number of bolts used significantly influences the distribution of bearing force and maximum capacity. It was noted that using four bolts with larger diameters (i.e., 22 mm and 25 mm) caused early failures like block shear failure in the beam during loading. Similarly, using four bolts with a diameter of 12 mm also led to premature bolt shear failure. When the number of bolts increased to 8 or 10, particularly in AR0.6 and AR1.0 bolt arrangements, the failure was localized at the column, which is not a desirable failure. Six bolts are recommended for various bolt arrangements and bolt diameters of 16 mm or more. This strategy aligns with the “strong column weak beam” principle, which allows the plastic hinge to form at the beam.

• Increasing the bolt diameter from 16 mm to 22 mm was found to reduce the ultimate rotation of the connection by approximately 50%, without requiring any additional strengthening strategies. However, a further increase to 25 mm resulted in a reduction in connection capacity. This reduction was attributed to the enlarged bolt-hole area, which intensified stress concentrations at the beam’s compression flange, leading to block shear failure of the beam. It is recommended to maintain a bolt diameter to extended plate thickness ratio below 2, which minimizes stress concentration and postpones the onset of local buckling in the beam and bearing failure at the extended plate.

• Design charts are proposed for different bolt arrangements that can predict failure modes by altering design parameters. The accuracy of the failure mode chart is validated with the tested specimens for the proposed connection.

• A simplified symbolic regression equation was developed to predict the ultimate capacity of bolted connections, yielding an R-value of 0.97 for the training dataset and 0.921 for the testing dataset. These results validate the equation’s reliability and its effectiveness in accurately predicting the moment capacity of bolted connections.

Footnotes

ORCID iDs

Mostafa Elhadary

Ahmed Bediwy

Ahmed Elshaer

Author contributions

Mostafa Elhadary: Writing – original draft, Validation, Conceptualization, Formal analysis, Methodology, Software, Visualization. Ahmed Bediwy: Supervision, Funding acquisition, Review & editing paper. Ahmed Elshaer: Supervision, Conceptualization, Funding acquisition, Review & editing paper.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) under Grant No. RGPIN-2022-04755

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

All data, models, and code generated or used during the study appear in the submitted article.*

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Capacity and failure mode prediction of long-bolted connection in panelized modular houses

Abstract

Keywords

Introduction

Background

Research significance

Numerical study

Finite element model details

Model validation

Study parameters

Results and discussion

Effect of bolt arrangement

Effect of the bolt quantity

Influence of increasing the bolt diameter

Influence of extended plate thickness

Failure mode prediction charts

Mathematical formula for capacity prediction

Conclusions

Footnotes

ORCID iDs

Author contributions

Funding

Declaration of conflicting interests

Data Availability Statement

References