Flexural Strength of Cold-Formed Steel Unstiffened and Edge-Stiffened Hexagonal Perforated Channel Sections

Abstract

Cold-formed steel (CFS) channel beams are increasingly used as primary structural elements in modern construction due to their lightweight and high-strength characteristics. To accommodate building services, these members often feature perforations—typically circular and unstiffened—produced by punching. Recent studies indicate that adding edge stiffeners, particularly around circular web openings, can improve flexural strength. Extending this idea, attention has shifted to hexagonal web perforations; however, limited research exists on the bending performance of hexagonal cold-formed steel channel beams (HCFSBs). This study presents a detailed nonlinear finite element (FE) analysis to evaluate and compare the flexural behaviour of HCFSBs with unstiffened (HUH) and edge-stiffened (HEH) hexagonal openings. The FE models were validated against experimental results and expanded to include a comprehensive parametric study with 810 simulations. Results show that HEH beams achieve, on average, a 10% increase in moment capacity compared to HUH beams. However, when evaluated using current Direct Strength Method (DSM) provisions, moment capacities were underestimated by up to 47%, particularly in cases governed by lateral–torsional or distortional buckling. A reliability analysis confirmed that the proposed design equations yield accurate and dependable strength predictions.

1. Introduction

Perforations in the web of varying geometries are frequently introduced into cold-formed steel (CFS) channel members, such as floor joists and wall studs, to accommodate mechanical and electrical services [1,2,3]. Due to constraints on web depth, hexagonal perforations are often positioned along the beam span to improve accessibility for storage racks (see Figure 1). These openings can significantly affect the structural performance of hexagonal cold-formed steel channel beams (HCFSBs), notably by altering their moment capacity, bending behaviour, and susceptibility to buckling, owing to the reduced effective web area.

To enhance structural performance, a newer CFS channel beam having stiffened edge perforations (circular/elongated) is now commonly adopted in various countries [4,5,6,7,8,9,10,11]. These edge-stiffeners influence the stress flow and alter boundary conditions in the web, resulting in notable improvements in both compressive and flexural capacities (refer to Figure 2) [4,12]. However, there remains a gap in the literature concerning the bending strength and deformation behaviour of CFS channel sections incorporating hexagonal-shaped web holes—both unstiffened (HUH) and edge-stiffened (HEH (Figure 3)). Furthermore, current design standards, such as those by the American Iron and Steel Institute (AISI) [13] and the joint Australian/New Zealand standard (AS/NZS) [14], do not yet address design procedures for HEH configurations. This study aims to close this gap by investigating the flexural resistance of HCFSBs with HUH and HEH and proposes updated design guidelines. The specific HEH geometries examined are illustrated in Figure 3.

Significant efforts have been devoted to explore the bending behaviour and strength characteristics of CFS channel beams with different web perforation patterns [15]. Zhao et al. [16] investigated rectangular-shaped unstiffened web perforations, highlighting critical shortcomings in the DSM framework [13]. Moen et al. [17,18] studied beams containing unstiffened circular perforations and proposed refinements to the conventional design approach to better capture their moment capacity. Degtyareva et al. [19,20] focused on slotted unstiffened perforations and identified discrepancies in performance, leading to proposed modifications in the DSM equations to improve their predictive accuracy for flexural behaviour. Thirunavukarsu et al. [4] employed finite element analysis to simulate SupaCee CFS beams subjected to four-point bending with circular unstiffened holes. Additionally, Yu et al. [21,22] offered an analytical methodology specifically targeting the prediction of distortional buckling capacity.

Studies focusing on cold-formed steel (CFS) channel beams with edge-stiffened circular perforations remain relatively scarce. Yu et al. [3] observed that such stiffening led to a 13% improvement in flexural strength compared to beams without perforations. Investigations by Chen et al. [1,4] revealed that the current Direct Strength Method (DSM) provisions are generally conservative when applied to beams with edge-reinforced circular holes. Dai et al. [6] expanded upon this by conducting detailed finite element and parametric analyses, integrating machine learning techniques to enhance predictive accuracy. In addition, a wide array of prior studies [23,24,25,26,27,28,29,30] have assessed the bending strength and the buckling behaviour of CFS channel sections featuring solid, unperforated webs.

Previous research on CFS channel beams having elongated unstiffened perforations (EUH) has primarily focused on their behaviour under shear forces [31,32] and compressive loads [33,34,35,36,37]. Building on this foundation, Chandramohan et al. [38,39] and Wang et al. [40] examined elongated edge-stiffened web openings (EEH), revealing superior load-bearing performance relative to EUH sections under various loading conditions and proposing DSM-based design equations for predicting strength. Recently, Lawson et al. [41] introduced a method for assessing shear capacity in beams with both EUH and EEH by applying the Vierendeel bending mechanism.

While cold-formed steel beams are widely perforated for service integration, prevailing standards such as AISI [13] and AS/NZS [14] provide flexural capacity formulas only for circular unstiffened openings, leaving hexagonal and stiffened geometries undefined. As a result, practicing engineers often default to over-conservative assumptions or unsafe extrapolations. Prior investigations (e.g., Chen et al. [1]; Moen et al. [17,18]) have demonstrated capacity underestimations as high as 40–47% using unmodified DSM expressions. Furthermore, no design guidelines exist for combined buckling effects in beams with densely spaced or non-standard openings—despite their growing use in multi-service floor joists and composite slabs.

Various studies on innovative pre-engineered tubular cold-formed steel beam-column section with popular connectors were studied experimentally [42,43,44]. Experimental studies on V-stiffeners provided in various stainless steel lipped channel beams were studied, which showed a significant improvement in the flexural behaviour of [45].

Despite these advancements, there is a notable gap in research concerning the flexural behaviour of HCFSBs with either hexagonal unstiffened (HUH) or hexagonal stiffened edge (HEH) perforations under bending (four-point loading conditions).

This study addresses that gap by employing finite element (FE) analysis to assess the bending response and flexural strength of HCFSB’s with both HUH and HEH openings. The numerical models were calibrated using experimental benchmarks available in the literature to ensure modelling accuracy. Upon validation, an extensive parametric study was conducted, leading to the development of modified DSM equations. These revised formulations were further validated through a reliability-based statistical analysis.

2. Overview of the Experimental Study Conducted by Chen et al. [1]

To examine the flexural behaviour and bending strength of CFS channel beams with circular perforations—both stiffened and unstiffened edges—the developed numerical model was verified using four-point bending test data reported by Chen et al. [1]. A total of six experimental tests were conducted on specimens with varied web configurations, including differences in section height and the quantity of web openings, as summarized in

Table 1. Validation of FEA models using test results [1] for CFSCBs.

Specimen Details							Flexural Strength
Depth of wb d	Length L	Section Thickness t	Hole Depth dw	Hole Spacing	Hole Depth Ratio dw/d1	Length of Stiffener q	Test Results MEXP	FE Analysis MFEA	MEXP/MFEA	Failure Patterns *
(mm)		(mm)	(mm)	(mm)		(mm)	(kNm)	(kNm)
290	4000	2.11	140	-	0.49	-	16.7	17.9	0.93	D + L
290	4000	2.11	140	100	0.49	-	16.3	17.6	0.92	D + L
290	4000	2.11	140	50	0.49	-	15.7	16.5	0.95	D + L
290	4000	2.11	140	-	0.49	13	19.3	20.5	0.94	D + L
290	4000	2.11	140	100	0.49	13	19.8	21.2	0.93	D + L
290	4000	2.11	140	50	0.49	13	20.5	21.7	0.94	D + L
								Average value	0.93
								Variation coefficient	0.04

* D—distortional buckling; L—local buckling.

. The test arrangement from Chen et al. [1], which informed the numerical simulation methodology, is shown in Figure 4.

These six experimental datasets (Table 1) from Chen et al. [1] were employed to validate the finite element models constructed for beams with elongated perforations—specifically, edge-stiffened (EEH) and unstiffened (EUH) openings. After confirming the accuracy of the model through comparison with test results, the validated approach (described in Section 3) was extended for a broad parametric study.

3. FE Analysis

3.1. Overview

To investigate the flexural performance of HCFSBs with both unstiffened (HUH) and edge-stiffened (HEH) web openings under four-point loading, nonlinear elasto-plastic finite element (FE) analyses were performed using the ABAQUS platform [46]. The modelling approach accounted for centreline-based cross-sectional geometry, nonlinear material behaviour, and geometric imperfections. Further details of the simulation methodology are elaborated in the subsequent subsection.

3.2. Constitutive Material Behaviour

The mechanical properties used in the FE models were derived from tensile coupon tests conducted by Chen et al. [1]. For the parametric study described in Section 4, the material was modelled with an idealized elastic–perfectly plastic stress–strain curve following the Von Mises yield criterion. Engineering stress–strain data from the experimental tests were converted into true stress–strain representations for accurate implementation within the simulation environment.

$${\sigma }_{true}=\sigma (1+\epsilon )$$

(1)

$${\epsilon }_{true(pl)}=\ln (1+\epsilon )-{\displaystyle \frac{{\sigma }_{true}}{E}}$$

(2)

In this context, “σ”_true and “ε”_true refer to the true stress and true strain, respectively, whereas σ and ε indicate the engineering counterparts. The parameter E denotes the elastic modulus of the material.

3.3. Finite Element Meshing

The HCFSBs were modelled using S4R shell elements, which incorporate six degrees of freedom per node, while the loading and support blocks were constructed using C3D4 solid elements. To achieve an optimal balance between solution accuracy and computational cost, a mesh density of 5 mm × 5 mm was applied to the channel components, and a coarser 10 mm × 10 mm mesh was used for the loading fixtures. A refined mesh was employed in regions of high stress concentration—particularly around elongated openings and at the web–flange junctions. This meshing strategy was consistent with those reported in previous studies [1,23,38,39]. The mesh layout for specimen CH240-L4000-HUH1-D0.2-B2 is shown in Figure 5.

3.4. Application of Loading and Boundary Representation

In the FE simulations, “hard” contact definitions were used to simulate surface interaction between the beams and the rigid blocks, effectively eliminating penetration. Given their higher rigidity, the support and load blocks were assigned as master surfaces, while the channel web surfaces acted as slaves in the contact algorithm. The experimental boundary conditions outlined by Chen et al. [1] and Chandramohan et al. [23] were reproduced by modelling a simply supported system—axial translation and in-plane rotation were released in the roller support. Two reference points were defined based on the loading blocks to guide the application of vertical displacement (along the y-axis) while restricting translation and rotation about the z-axis. These constraints and the load application method are depicted in Figure 6a.

3.5. Geometrical Imperfections

To account for initial geometric imperfections in the FE simulations, a linear eigenvalue buckling analysis was conducted to derive representative imperfection shapes. This analysis was applied to all HCFSB models featuring either edge-stiffened (HEH) or unstiffened (HUH) perforations. The first mode shape, corresponding to the lowest buckling load, was selected to represent the imperfection pattern used in subsequent nonlinear analyses, consistent with approaches described in [1,4]. The amplitude of the applied imperfection was set at 0.15t, where t denotes the section thickness, as suggested in prior studies [4]. The IMPERFECTION feature available in ABAQUS [46] was employed to introduce these predefined shapes into the FE models. Figure 6b illustrates the initial imperfection configurations extracted from the first buckling mode for both HEH and HUH variants.

3.6. Validation

Table 1 exhibits a comparison between the FE analysis outcomes and the experimental data reported by Chen et al. [1]. For cold-formed steel channel beams (CFSCBs) with plain webs, as well as those featuring unstiffened and edge-stiffened circular web openings, the average ratio of experimental to numerical moment capacity (M_EXP/M_FEA) was found to be 0.93, with a coefficient of variation (COV) of 0.04 [1]. Figure 7 shows the failure patterns observed during both the physical tests and numerical simulations. As illustrated, beams with unstiffened circular web holes exhibited distortional–local buckling in the FEA, closely matching the experimental observations by Chen et al. [1]. Additionally, the load–displacement responses shown in Figure 8 highlight the strong consistency between the simulation and experimental results. Collectively, these findings confirm the high accuracy and reliability of the developed FE model.

4. Parametric Study

4.1. Overview and Specimen Coding

Using the validated modelling framework illustrated in Section 3, a detailed parametric investigation was performed, encompassing 810 finite element simulations of HCFSBs with both unstiffened (HUH) and edge-stiffened (HEH) web holes (see Figure 9). All specimens were subjected to four-point bending. Two cross-sectional depths (240 mm and 290 mm) were considered (illustrated in Figure 10), each having thicknesses of 1.83 mm and 2.11 mm, respectively. The geometric variation focused on the structural impact of hexagonal openings (see Figure 10), and the altered parameters were as follows:

Perforation depth-to-web depth ratio (d_w/d₁): 0.4, 0.5, and 0.6, respectively.

Perforation width-to-depth ratio (b_w/d_w): 2.0 to 3.0.

Stiffener length-to-web depth ratio (q/d₁): 0.04, 0.06, and 0.08, respectively.

Bent inner radius at the web and flange junction (r_q): 2 mm, 4 mm, and 6 mm, respectively.

Beam lengths of 4000 mm and 9000 mm were studied, with the corresponding constant moment regions (l_e) of 1200 mm and 2860 mm, respectively. The effect of 1, 3, or 5 hexagonal holes positioned symmetrically along the span was also examined.

Figure 11 presents the labelling convention used for HEH-type HCFSBs, where each identifier includes the web depth, beam length, web hole type, perforation depth-to-length ratio, stiffener length ratio, and corner radius. For instance, the designation “CH290-L4000-HEH1-D0.4-B2” specifies a section with a 290 mm web depth, 4000 mm beam length, HEH hole type, d_w/d₁ = 0.4, and b_w/d_w = 2.

4.2. Findings and Interpretations

Table 2. (a) FEA-derived flexural strength of CFSCBs with a beam length of 4000 mm. (b) FEA-derived flexural strength of CFSCBs with a beam length of 9000 mm.

(a)
Specimen	FEA-Based Flexural Strength (MFEA (kNm)) for CFSCB’s Featuring HUH and HEH
	HUH	HEH
		q/d1: 0.04			q/d1: 0.06			q/d1: 0.08
		rq: 2	rq: 4	rq: 6	rq: 2	rq: 4	rq: 6	rq: 2	rq: 4	rq: 6
CH240-L4000-HEH1-D0.4-B2	13.67	14.77	14.73	14.68	14.86	14.83	14.78	14.88	14.86	14.82
CH240-L4000-HEH3-D0.4-B2	13.17	14.24	14.19	14.11	14.44	14.41	14.43	14.70	14.66	14.60
CH240-L4000-HEH1-D0.4-B2.5	13.42	14.65	14.61	14.53	14.70	14.68	14.64	14.75	14.73	14.69
CH240-L4000-HEH3-D0.4-B2.5	12.33	13.39	13.34	13.24	13.65	13.48	13.40	14.38	14.31	14.30
CH240-L4000-HEH1-D0.4-B3	12.57	14.50	14.45	14.41	14.64	14.62	14.57	14.68	14.66	14.63
CH240-L4000-HEH3-D0.4-B3	12.10	13.20	13.13	13.01	13.47	13.37	13.27	14.28	14.25	14.14
CH240-L4000-HEH1-D0.5-B2	13.19	14.54	14.44	14.26	14.63	14.57	14.51	14.78	14.59	14.48
CH240-L4000-HEH3-D0.5-B2	12.87	14.18	14.07	13.93	14.28	14.21	14.08	14.35	14.27	14.16
CH240-L4000-HEH1-D0.5-B2.5	12.38	14.08	14.03	14.02	14.41	14.20	14.16	14.58	14.37	14.34
CH240-L4000-HEH3-D0.5-B2.5	12.07	13.90	13.76	13.53	14.09	13.99	13.88	14.18	14.09	13.98
CH240-L4000-HEH1-D0.5-B3	11.89	13.97	13.84	13.62	14.34	14.15	14.08	14.40	14.29	14.03
CH240-L4000-HEH1-D0.6-B2	12.79	14.10	13.90	13.70	14.48	14.09	13.89	14.71	14.43	13.98
CH240-L4000-HEH3-D0.6-B2	11.72	13.49	13.31	13.22	13.66	13.53	13.30	13.73	13.61	13.23
CH240-L4000-HEH1-D0.6-B2.5	12.56	13.52	13.41	13.32	13.88	13.65	13.49	13.98	13.87	13.74
CH240-L4000-HEH1-D0.6-B3	12.04	12.95	12.79	12.52	13.74	13.54	13.31	13.87	13.61	13.44
CH290-L4000-HEH1-D0.4-B2	16.95	19.63	19.57	19.54	19.93	19.83	19.72	20.03	19.97	19.86
CH290-L4000-HEH3-D0.4-B2	16.51	18.94	18.86	18.81	19.22	19.15	19.03	19.38	19.25	19.13
CH290-L4000-HEH1-D0.4-B2.5	16.30	19.23	19.20	19.09	19.37	19.33	19.31	19.45	19.40	19.35
CH290-L4000-HEH3-D0.4-B2.5	15.75	18.73	18.65	18.49	19.17	19.06	18.86	19.24	19.10	18.99
CH290-L4000-HEH1-D0.4-B3	16.12	18.97	18.92	18.77	19.29	19.25	19.22	19.44	19.32	19.26
CH290-L4000-HEH1-D0.5-B2	15.96	19.01	18.95	18.22	19.23	19.11	18.88	19.30	19.24	19.03
CH290-L4000-HEH3-D0.5-B2	15.46	17.34	17.17	16.93	17.60	17.47	17.25	17.80	17.59	17.40
CH290-L4000-HEH1-D0.5-B2.5	15.85	18.55	18.38	18.17	18.93	18.89	18.67	19.21	19.11	19.92
CH290-L4000-HEH1-D0.5-B3	15.80	18.25	18.07	17.92	18.48	18.33	18.24	18.68	18.42	18.26
CH290-L4000-HEH1-D0.6-B2	15.76	17.86	17.49	17.02	18.21	18.99	18.62	18.49	18.17	17.83
CH290-L4000-HEH1-D0.6-B2.5	15.65	17.77	16.92	16.49	17.81	17.64	17.41	18.24	18.94	18.65
CH290-L4000-HEH1-D0.6-B3	15.54	17.66	17.42	17.01	17.72	17.34	17.05	18.72	18.60	18.37
(b)
Specimen	FEA-Based Flexural Strength (MFEA (kNm)) for CFSCB’s Featuring HEH
	HUH	HEH
		q/d1: 0.04			q/d1: 0.06			q/d1: 0.08
		rq: 2	rq: 4	rq: 6	rq: 2	rq: 4	rq: 6	rq: 2	rq: 4	rq: 6
CH240-L9000-HEH1-D0.4-B2	7.20	7.34	7.28	7.27	7.37	7.33	7.32	7.39	7.36	7.35
CH240-L9000-HEH3-D0.4-B2	6.73	7.02	6.99	6.97	7.12	7.09	7.07	7.21	7.18	7.16
CH240-L9000-HEH5-D0.4-B2	6.36	6.80	6.75	6.71	6.95	6.91	6.87	7.09	7.05	7.01
CH240-L9000-HEH1-D0.4-B2.5	7.08	7.20	7.21	7.20	7.27	7.26	7.25	7.31	7.30	7.29
CH240-L9000-HEH3-D0.4-2.5	6.44	6.83	6.79	6.69	6.97	6.94	6.91	7.08	7.05	7.02
CH240-L9000-HEH5-D0.4-2.5	6.12	6.60	6.55	6.53	6.81	6.76	6.73	6.97	6.93	6.90
CH240-L9000-HEH1-D0.4-B3	6.95	7.13	7.11	7.0	7.20	7.19	7.18	7.25	7.24	7.23
CH240-L9000-HEH3-D0.4-B3	6.17	6.64	6.59	6.55	6.82	6.79	6.76	6.96	6.93	6.90
CH240-L9000-HEH5-D0.4-B3	6.02	6.56	6.54	6.50	6.79	6.76	6.76	6.97	6.94	6.94
CH240-L9000-HEH1- D0.5-B2	7.05	7.20	7.19	7.18	7.25	7.24	7.23	7.29	7.28	7.27
CH240-L9000-HEH3-D0.5-B2	6.34	6.74	6.72	6.71	6.88	6.86	6.82	7.00	6.96	6.94
CH240-L9000-HEH5-D0.5-B2	6.06	6.48	6.45	6.42	6.69	6.65	6.63	6.86	6.82	6.80
CH240-L9000-HEH1-D0.5-B2.5	6.86	7.06	7.05	7.03	7.14	7.12	7.11	7.19	7.18	7.17
CH240-L9000-HEH3-D0.5-B2.5	6.16	6.49	6.45	6.43	6.69	6.66	6.63	6.84	6.81	6.78
CH240-L9000-HEH5-D0.5-B2.5	5.98	6.43	6.42	6.40	6.67	6.65	6.60	6.81	6.80	6.78
CH240-L9000-HEH1-D0.5-B3	6.66	6.90	6.87	6.85	7.01	6.99	6.97	7.08	7.07	7.05
CH240-L9000-HEH3-D0.5-B3	5.87	6.32	6.31	6.30	6.57	6.56	6.55	6.74	6.73	6.72
CH240-L9000-HEH5-D0.5-B3	5.64	5.95	5.92	5.90	6.25	6.23	6.21	6.48	6.45	6.41
CH240-L9000-HEH1-D0.6-B2	6.88	7.07	7.06	7.06	7.14	7.13	7.12	7.19	7.18	7.18
CH240-L9000-HEH3-D0.6-B2	6.11	6.50	6.47	6.46	6.67	6.64	6.62	6.82	6.79	6.76
CH240-L9000-HEH5-D0.6-B2	5.98	6.24	6.21	6.19	6.35	6.31	6.29	6.48	6.45	6.41
CH240-L9000-HEH1-D0.6-B2.5	6.52	6.89	6.85	6.83	6.97	6.96	6.94	7.05	7.03	7.02
CH240-L9000-HEH3-D0.6-B2.5	6.16	6.36	6.34	6.32	6.54	6.52	6.51	6.71	6.69	6.67
CH240-L9000-HEH5-D0.6-B2.5	5.52	5.94	5.90	5.98	6.20	6.17	6.02	6.35	6.31	6.28
CH240-L9000-HEH1-D0.6-B3	6.34	6.61	6.59	6.57	6.75	6.74	6.72	6.86	6.85	6.84
CH240-L9000-HEH3-D0.6-B3	6.09	6.05	6.01	5.98	6.19	6.14	6.12	6.25	6.23	6.21
CH240-L9000-HEH5-D0.6-B3	5.31	5.65	5.63	5.61	5.85	5.83	5.81	5.94	5.92	5.90
CH290-L9000-HEH1-D0.4-B2	9.12	9.32	9.30	9.29	9.39	9.37	9.36	9.45	9.43	9.42
CH290-L9000-HEH3-D0.4-B2	8.28	8.84	8.79	8.76	9.03	8.99	8.96	9.20	9.16	9.13
CH290-L9000-HEH5-D0.4-B2	7.80	8.57	8.52	8.49	8.87	8.82	8.78	8.96	8.94	8.91
CH290-L9000-HEH1-D0.4-B2.5	8.92	9.17	9.14	9.12	9.26	9.24	9.22	9.34	9.32	9.31
CH290-L9000-HEH3-D0.4-B2.5	7.86	8.54	8.49	8.45	8.80	8.76	8.72	9.02	8.98	8.94
CH290-L9000-HEH5-D0.4-B2.5	7.77	8.36	8.35	8.34	8.51	8.48	8.47	8.78	8.73	8.69
CH290-L9000-HEH1-D0.4-B3	8.71	9.00	8.97	8.94	9.13	9.10	9.07	9.23	9.20	9.18
CH290-L9000-HEH3-D0.4-B3	7.60	8.32	8.26	8.24	8.65	8.61	8.58	8.91	8.88	8.85
CH290-L9000-HEH5-D0.4-B3	7.43	8.18	8.14	8.12	8.33	8.29	8.27	8.46	8.43	8.41
CH290-L9000-HEH1-D0.5-B2	8.84	9.12	9.11	9.09	9.22	9.21	9.19	9.30	9.29	9.27
CH290-L9000-HEH3-D0.5-B2	7.72	8.44	8.40	8.38	8.70	8.67	8.65	8.92	8.89	8.87
CH290-L9000-HEH5-D0.5-B2	7.65	8.35	8.33	8.31	8.56	8.53	8.52	8.78	8.75	8.73
CH290-L9000-HEH1-D0.5-B2.5	8.52	8.87	8.85	8.82	9.01	8.99	8.97	9.12	9.10	9.08
CH290-L9000-HEH3-D0.5-B2.5	7.56	8.23	8.21	8.20	8.45	8.43	8.41	8.78	8.76	8.75
CH290-L9000-HEH5-D0.5-B2.5	7.40	7.86	7.83	7.82	8.08	8.04	8.02	8.25	8.23	8.21
CH290-L9000-HEH1-D0.5-B3	8.20	8.57	8.54	8.50	8.76	8.73	8.70	8.91	8.88	8.86
CH290-L9000-HEH3-D0.5-B3	7.32	8.18	8.14	8.12	8.29	8.26	8.24	8.38	8.35	8.33
CH290-L9000-HEH5-D0.5-B3	7.14	7.75	7.72	7.70	7.84	7.81	7.79	8.06	8.02	7.99
CH290-L9000-HEH1-D0.6-B2	8.52	8.89	8.87	8.86	9.03	9.01	9.00	9.13	9.12	9.11
CH290-L9000-HEH3-D0.6-B2	7.61	8.26	8.24	8.23	8.54	8.52	8.50	8.76	8.74	8.72
CH290-L9000-HEH5-D0.6-B2	7.47	8.18	8.15	8.12	8.24	8.21	8.19	8.45	8.43	8.40
CH290-L9000-HEH1-D0.6-B2.5	8.07	8.50	8.47	8.44	8.69	8.67	8.65	8.84	8.82	8.80
CH290-L9000-HEH3-D0.6-B2.5	7.32	8.26	8.22	8.20	8.39	8.36	8.34	8.52	8.49	8.48
CH290-L9000-HEH5-D0.6-B2.5	7.11	7.75	7.72	7.70	7.98	7.95	7.92	8.09	8.05	8.03
CH290-L9000-HEH1-D0.6-B3	7.61	8.05	8.02	7.98	8.43	8.40	8.38	8.50	8.47	8.47
CH290-L9000-HEH3-D0.6-B3	7.05	7.87	7.85	7.83	8.19	8.17	8.16	8.28	8.26	8.24
CH290-L9000-HEH5-D0.6-B3	6.82	7.56	7.54	7.53	7.74	7.71	7.69	7.95	7.92	7.90

summarizes the flexural capacities of HCFSBs featuring HUH and HEH configurations obtained via finite element simulations, while Figure 12 illustrates the corresponding failure patterns for beams with lengths of 4 m and 9 m. The investigation assessed the impact of several geometric variables, including the perforation depth-to-web depth ratio (d_w/d₁), perforation width-to-depth ratio (b_w/d_w), edge stiffener length ratio (q/d₁), fillet radius at the stiffener junction (R_q), overall section depth (d), and beam span (L) on the bending resistance.

An increase in the d_w/d₁ ratios of 0.4, 0.5, and 0.6, respectively, resulted in a 7% reduction in flexural strength. Meanwhile, increasing b_w/d_w from 2.0 to 3.0 caused a more modest 5% drop, as depicted in Figure 13 and Figure 14. Variations in the q/d₁ ratio had a lesser influence, with a 10% enhancement in strength observed for HEH beams relative to HUH, while increasing q/d₁ from 0.04 to 0.08 produced only a 4% improvement (refer to Figure 15). The presence of edge stiffeners suppresses local buckling around the web holes, leading to distortional buckling in HCFSB members with the HEH.

Adjusting the fillet radius (r_q) from 2 mm to 6 mm had negligible effect, with a less than 1% change in moment capacity (Figure 16). In contrast, the section depth showed significant influence; expanding the web depth from 240 mm to 290 mm enhanced moment capacity by 28% (Figure 17). Beam length was also critical; as the length increased from 4000 mm to 9000 mm, moment capacity dropped by 51% (Figure 18). Shorter beams (4000 mm) predominantly exhibited distortional buckling, while longer, slender beams (9000 mm) displayed a combined failure mode of lateral–torsional and distortional buckling.

An increase in the number of web perforations from one to five led to a total flexural strength reduction of around 9%. These outcomes underscore the importance of carefully optimizing geometric design parameters to maximize structural performance while ensuring adequate moment resistance.

5. Comparison of FEA Results with the Existing Design Predictions

At present, the literature lacks dedicated design provisions or standards specifically addressing the flexural strength of HCFSBs with stiffened edges (HEH) or unstiffened (HUH) perforations. Nonetheless, the AISI [13] and AS/NZS [14] standards offer Direct Strength Method (DSM)-based approaches for evaluating the flexural strength of beams with unstiffened perforations in the web, considering local, distortional, and lateral–torsional modes of buckling. The parametric study methods employed in this research are based on the design formulations developed by Moen and Schafer [17,18]. Accordingly, the flexural strength results obtained were compared against predictions from these well-established methods, which are further discussed in the subsequent sections. Chen et al. (1) and Chandramohan et al. [23] presents the various formulas to find the local, distortional, and lateral–torsional modes of buckling.

A comparative evaluation of the moment strength of HCFSB with HUH derived from finite element analysis (FEA) and the values predicted using DSM equations from AISI [13] and AS/NZS [14] indicates that, on an average, the DSM estimates are approximately 23% lower when subjected to distortional or local buckling scenarios (refer to

Table 3. A statistical comparison between the finite element analysis (FEA) outcomes and the flexural strength estimated using the current DSM equations for HCFSBs with HUH sections.

Comparison	Mean	COV
Distortional buckling—MFEA/MAISI [13] and AS/NZS [14]	1.23	0.08
Lateral–torsional buckling—MFEA/MAISI [13] and AS/NZS [14]	0.54	0.11

and Figure 19). This demonstrates that the current DSM formulations [13,14] provide reasonably accurate predictions for an HCFSB with HUH subjected to these buckling modes. In contrast, for an HCFSB with HUH experiencing lateral–torsional buckling, the DSM-based design predictions fall short—by nearly 47%—relative to the FEA findings (see Table 3). The discrepancy between the FEA and DSM predictions for CFSCB with HUH under lateral–torsional buckling is highlighted in Figure 20.

6. Recommended Design Equation

As deliberated in Section 6, existing design codes do not offer dedicated equations for predicting the flexural strength of an HCFSB with HEH sections under distortional or lateral–torsional buckling. Additionally, the current DSM formulations tend to be overly conservative when estimating the moment resistance of an HCFSB with HUH profiles subjected to lateral–torsional buckling. To overcome these shortcomings, this study introduces modified DSM-based expressions tailored to an HCFSB with both HEH and HUH configurations by using the bivariate linear regression method. These equations are adapted from the foundational provisions in AISI [13] and AS/NZS [14] for cold-formed steel beams with unstiffened web perforations.

In line with the recommendations by Moen and Schafer [17,18], finite strip analysis was executed using the CUFSM’s hole module [44] to assess the gross sectional performance and the impact of web openings. This analysis provided key parameters including $M_{crdh}$, $M_{crlh}$, $M_{creh}$, $M_y,$ and $M_{ynet}$.

6.1. Distortional–Buckling Strength

For an HCFSB incorporating HEH, the moment resistance under distortional buckling is determined using the newly proposed equations (Equations (3) and (4)). In these expressions,

• $M_{bdp}$ denotes the predicted flexural strength against distortional failure;
• $M_{crdh}$ is the elastic distortional buckling moment for HUH sections;
• $M_y$ and $M_{ynet}$ represent the yield moments of the gross and net sections, respectively, where $M_y=Z_f\times f_y$;
• $f_y$ is the yield strength of the steel;

  $$For {\lambda }_d\le 0.873; M_{bdp}=M_y \le M_{ynet}$$

  (3)

  $$For {\lambda }_d>0.873; M_{bdp}=\left[1-0.82{\left({\displaystyle \frac{M_{ynet} }{M_{crdh} }}\right)}^{0.66}\right]M_{ynet}$$

  (4)

6.2. Lateral–Torsional Buckling Strength

The proposed design expressions for lateral–torsional buckling strength are given in Equations (5)–(7) for an HCFSB with HEH and Equations (8)–(10) for an HCFSB with HUH.

$$For M_{\mathit{creh}}<0.2M_{ynet}:{ M}_{bep}=10.61M_{\mathit{creh}}-1.64M_{\mathit{ynet}}$$

(5)

$$For 0.2M_{\mathit{ynet}}\le M_{\mathit{creh}}\le 0.4M_{\mathit{ynet}}:{ M}_{bep}=0.97M_{\mathit{creh}}+0.26M_{\mathit{ynet}}$$

(6)

$$For 0.4M_{\mathit{ynet}}<M_{\mathit{creh}}:{ M}_{bep}=1.3M_{\mathit{creh}}+0.1M_{\mathit{ynet}}$$

(7)

$$For M_{\mathit{creh}}<0.2M_{ynet}:{ M}_{bep}=9.06M_{\mathit{creh}}-1.39M_{\mathit{ynet}}$$

(8)

$$For 0.2M_{\mathit{ynet}}\le M_{\mathit{creh}}\le 0.4M_{\mathit{ynet}}:{ M}_{bep}=1.09M_{\mathit{creh}}+0.20M_{\mathit{ynet}}$$

(9)

$$For 0.4M_{\mathit{ynet}}<M_{\mathit{creh}}:{ M}_{bep}=1.3M_{\mathit{creh}}+0.1M_{\mathit{ynet}}$$

(10)

• $M_{bep}$ is the proposed bending capacity under lateral–torsional buckling;
• $M_{ynet}$ is the net section yield capacity;
• $M_{\mathit{creh}}$ denotes the corresponding elastic lateral–torsional buckling strength for an HCFSB with HUH.

Figure 21 presents the comparison between the FE analysis results and the flexural strengths obtained from the recommended DSM equations for an HCFSB with HEH (Figure 21a,b) and HUH (Figure 21c), respectively.

7. Reliability Assessment

To estimate the performance of the newly recommended DSM-based equations for an HCFSB with HEH sections, a reliability assessment was carried out. The procedure outlined in Equation (11) was adopted for this analysis. As per the AISI design specification [13], the required reliability index (β) for cold-formed carbon steel structural elements is a minimum of 2.5.

For an HCFSB with HEH sections, the results presented in

Table 4. Statistical comparison of FEA and recommended design strengths for HCFSBs with HEH and HUH.

Recommended Equations
	HEH		HUH
	Distortional: buckling $\left(M_{bdp}\right)$ Equations (3) and (4)	Lateral–torsional: buckling $\left(M_{bep}\right)$ Equations (5)–(7)	Lateral–torsional: buckling $\left(M_{bep}\right)$ Equations (8)–(10)
Sample size	243	486	54
Average value, Pm	1.00	1.00	1.00
Variation coefficient, Vp	0.06	0.08	0.10
Target reliability index, β	2.76	2.69	2.63
Design resistance factor, φ	0.85	0.85	0.85

confirm that the proposed DSM design equations achieved a reliability index (β) exceeding 2.5. This indicates that the developed equations can accurately and reliably predict the moment capacity of an HCFSB with HEH sections.

$$\phi =1.52{\mathrm{M}}_{\mathrm{m}}{\mathrm{F}}_{\mathrm{m}}{\mathrm{P}}_{\mathrm{m}}{\mathrm{e}}^{-\beta \sqrt{\{{{\mathrm{V}}_{\mathrm{m}}}^2+{{\mathrm{V}}_{\mathrm{f}}}^2+{\mathrm{C}}_{\mathrm{p}}{{\mathrm{V}}_{\mathrm{p}}}^2+{{\mathrm{V}}_{\mathrm{q}}}^2\}}}$$

(11)

where β = reliability index; $\phi$ = resistance factor (0.85); M_m and V_m = mean (1.1) and COV (0.1) of the material factor; F_m and V_f = mean (1.0) and COV (0.05) of the fabrication factor; P_m1 and V_P1 = mean and COV of FE results over proposed equation ($M_{FEA }/M_{Prop }$); V_q = COV (2.1) of load effect; C_p = correction factor $⌊{\displaystyle \frac{n+1}{n}}⌋⌊{\displaystyle \frac{n-1}{n-3}}⌋$; and $n$ = number of samples.

8. Conclusions

This study conducts a finite element (FE) analysis to explore the flexural behaviour of HCFSBs with HUH and HEH sections under four-point bending and validating the experimental data from the existing literature. Following validation, a parametric study was carried out using 810 FE analyses, considering material nonlinearity and geometric imperfections. This study examined how parameters such as (1) d_w/d₁, (2) d_w/b_w, (3) q/d₁, (4) R_q, (5) d, and (6) L impact on the flexural strength of HCFSBs with HUH and HEH sections. The key conclusions from this research are as follows:

(1)

Enhanced flexural capacity through edge stiffening: The finite element simulations showed that HCFSBs with HEH configurations exhibited, on average, a 10% higher bending capacity compared to those with HUH. Among the parameters analysed, factors such as d_w/d₁, section depth (d), and beam span (L) significantly influenced the flexural strength, whereas the web hole radius ratio (Rq) had minimal effect.

(2)

Identified shortcomings of current DSM design provisions: A comparative analysis revealed that the current DSM-based design provisions significantly underestimate the bending strength of HCFSBs with HUH sections. The predicted capacities were found to be 23% and 47% lower than FEA results for distortional and lateral–torsional buckling cases, respectively.

(3)

Development of modified DSM design equations: Utilizing FEA data, new DSM-oriented expressions were developed for estimating the moment resistance of HCFSBs with both HUH and HEH web hole configurations. These equations are valid for beams having the following ranges: (1) 240 mm≤ d ≤ 290 mm, (2) 0.4≤ d_w/d₁ ≤ 0.6, (3) 2.00 ≤ b_w/d_w ≤ 3.00, (4) 0.04 ≤ q/d₁ ≤ 0.08, and (5) 4000 mm≤ L ≤ 9000 mm.

(4)

Reliability verification of proposed formulations: The performance of the proposed DSM formulations for HCFSBs with HEH was verified through reliability analysis. The outcomes confirmed that the revised equations yield dependable estimates of flexural strength within acceptable safety margins.

• Future Research Directions:

• Experimental Validation: the findings should be corroborated through full-scale physical testing, especially for HEH configurations, to confirm model predictions and ensure applicability across diverse fabrication conditions.
• Sensitivity to Manufacturing Tolerances: future studies should assess how geometric imperfections from manufacturing (e.g., hole placement errors, edge quality) affect performance, particularly for thin-walled sections.
• Dynamic and Seismic Loading: investigation into the performance of these perforated beams under cyclic, fatigue, or seismic loads is essential for comprehensive design guidance.
• Integration into Codes: further collaboration with standards committees (e.g., AISI, AS/NZS) will be needed to validate these findings for formal inclusion in design codes.